key: cord-301829-6yrgkx96 authors: bhardwaj, rashmi; bangia, aashima title: data driven estimation of novel covid-19 transmission risks through hybrid soft-computing techniques date: 2020-07-25 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110152 sha: doc_id: 301829 cord_uid: 6yrgkx96 coronavirus genomic infection-2019 (covid-19) has been announced as a serious health emergency arising international awareness due to its spread to 201 countries at present. in the month of april of the year 2020, it has certainly taken the pandemic outbreak of approximately 11,16,643 infections confirmed leading to around 59,170 deaths have been recorded world-over. this article studies multiple countries-based pandemic spread for the development of the covid-19 originated in the china. this paper focuses on forecasting via real-time responses data to inherit an idea about the increase and maximum number of virus-infected cases for the various regions. in addition, it will help to understand the panic that surrounds this ncov-19 for some intensely affecting states possessing different important demographic characteristics that would be affecting the disease characteristics. this study aims at developing soft-computing hybrid models for calculating the transmissibility of this genome viral. the analysis aids the study of the outbreak of this virus towards the other parts of the continent and the world. a hybrid of wavelet decomposed data into approximations and details then trained & tested through neuronal-fuzzification approach. wavelet-based forecasting model predicts for shorter time span such as five to ten days advanced number of confirmed, death and recovered cases of china, india and usa. while data-based prediction through interpolation applied through moving average predicts for longer time spans such as 50-60 days ahead with lesser accuracy as compared to that of wavelet-based hybrids. based on the simulations, the significance level (alpha) ranges from 0.10 to 0.67, mase varying from 0.06 to 5.76, smape ranges from 0.15 to 1.97, mae varies from 22.59 to 6024.76, rmse shows a variation from 3.18 to 8360.29 & r(2) varying through 0.0018 to 0.7149. mase and smape are relatively lesser applied and novel measures that aimed to achieve increase in accuracy. they eliminated skewness and made the model outlier-free. estimates of the awaited outburst for regions in this study are india, china and the usa that will help in the improvement of apportionment of healthcare facilities as it can act as an early-warning system for government policy-makers. thus, data-driven analysis will provide deep insights into the study of transmission of this viral genome estimation towards immensely affected countries. also, the study with the help of transmission concern aims to eradicate the panic and stigma that has spread like wildfire and has become a significant part of this pandemic in these times. the world health organization (who) as on january 30, 2020 has announced 2019-2020 corona-genomic-virus a public health-emergency of international concern that can be abbreviated as pheic. situation further worsened worldwide which was declared pandemic on march 11, 2020 . till now, local transmission of this epidemic is being recording and increasing the count in countries including the six who regions. basically, their structure observed so far can be described as enveloped non-segmented positive-sense rnagenomic viruses having place in the clan of corona viridae majorly circulated in humans with other mammals. however, in most cases studied, individual related coronavirus infections are mild having identified two beta corona viruses: severe-acute-respiratory-syndrome-coronavirus (sars-cov) & middle-east-respiratory-syndrome-coronavirus (mers-cov). the outburst of ncovid-19 studied in detail through data-based modeling & forecast analysis [1] . detailed explanation of mathematical perspective to understand spread of infectious diseases is provided [2] . estimation of atmosphere pollutants through dynamic indicators, discussion of the meditating body complexity, statistical simulations towards dynamics of hiv, iot-based wireless transmissions having malware spread were modelled and studied in detail [3] [4] [5] [6] . coronavirus data analyzed for risk assessment and forecasts [7] . transmission data of the virus outbreak to atudy gov interventions [8] . towards tracking the rate of transmission of epidemic based on the data driven study of the situation was carried out [9] . study of a mathematical model towards dynamics of transmission and its control provided [10] . spatial spread relationships during coronavirus pandemic spread into the world via self-organizing maps analyzed [11] . who report on novel coronavirus in japan and mers-cov update has been surveyed [12, 13] . who report on coronavirus updated on january 19, 2020 [14] . the rate of spread of the epidemic in the scale-free networks [15] . as per the outcomes of this pandemic, efficiency of control strategies towards reduction of social mixing in china is modelled [16] . the complexity in the forecast accuracy of ncovid-19 pandemic is dealt with [17] . futuristic estimations computed via supervised learning of covid [18] . time series forecasting of the genomic virus spread in india applying genetic programming [19] . this pandemic outbreak is studied on the basis of training testing of multimodal data [20] . the molecules that may perhaps enter into host cell and cause acute respiratory syndrome targeting towards coronavirus studied [21] . study forecasted impending covid-19 spread cases for china plus some other regions using mathematical & traditional time-series prediction models [22] . mathematical model-based prediction at an early stage achieved for the outburst of this particular virus in china [23] . extensive exploration of pneumonia outbreak via corona-genome originating from bat species [24] . none of the authors have studied the wavelet based neuronal fuzzification hybrid model for the data of countrywise spread of covid-19 genome. in this article, forecasts of the country-based day to day basis data of confirmed, deaths and recovered cases. analysis has been carried out through the machine-learned wnf hybridization predicting for shorter time span and forecasts through interpolation alongwith moving averages method for longer time spans and performance measures through mase and smape which have not been applied in any of the studies yet. during exploration, daily data sets of china from december 31, 2019 to may 17, 2020 (a total of 139 days); for india from january 31, 2020 to may 17, 2020 (a total of 108 days) and for usa from is taken from january 20, 2020 to may 17, 2020 (a total of 119 days) trusted data sources provided by designated authorities. these three datatypes have been further divided into three data sets: confirmed cases, deaths cases and recovered cases respectively as mentioned in table conversion function is a function that converts a waveform into various rate of recurrence constituents. if conversion function is used in agreement with the scale then it is called wavelet transform, which converts the function alongwith the interval realm into the rate of recurrence realm. wavelet decomposition is carried out for records handling as with the help of wavelet demonstration, the non-stationarity of the economic and financial time series can be explained. possessing following characteristics: 6. finding mean. theorem: the root mean squared error is square root for mean squared errors calculated via actual outcomes and the expected quantities. remark: neuronal setup gets trained and tested through fuzzification having hybrid method for simulation of training and testing. for the country of china, the data has been analyzed under three main distinctions that are: confirmed cases, deaths' cases and recovery cases that are being recorded every day and provided through public bulletin from designated authorities as depicted through fig.7 to fig.21 . for the country of india, the data has been analyzed under three main distinctions that are: confirmed cases, deaths' cases and recovery cases that are being recorded every day and provided through public bulletin from designated authorities as depicted through fig.22 to fig.36 . it is the need of the hour to model the factors of covid-19 transmission to minimize its spread and the extent to which it can be harmful. since, china is the first country to record and report such cases so it is in a way the breeding place of this epidemic. thus, it is necessary to understand the scenario. prevention measures should be followed at its best so that the virus does not communicate to more people and to stop its breeding further. the wavelet decomposition depicts the data filtered through high and low pass filters filtering the noise in the sense normalizing for further computations. the trained responses are plotted with the actual data values to compare the scenario of confirmed, deaths and recovered cases respectively. simulations through time progression will aid in detailed study of virus structure dynamic evolution and perhaps indicate the emergence of randomness of the system. then the regression fit for the predicted data depicts the goodness of fit of predicted data upon the actual data. based on the simulations, the significance level (alpha) ranges from 0.10 to 0.67, mase varying from 0.06 to 5.76, smape ranges from 0.15% to 1.97%, mae varies from 22.59 to 6024.76, rmse shows a variation from 3.18 to 8360.29 & r 2 varying through 0.0018 to 0.7149. clearly, in this study smape and mase have lower performance errors and therefore effective in forecast. contribute towards better understanding of the scenario. thus, the daily datasets pertaining to those of usa have a great variability as compared to china and india. although, the spread has different timelines where india & america with the short time span have the greatest number of confirmed cases increasing uncontrollably at present. the forecast of 50-60 days ahead varying in every case helps to understand the clear picture of the pandemic spread and the manner in which the transmission rate may change in the following time periods in these three countries india, china and america. the outcomes of this study can provide an efficient learning and understanding of the future spread estimation and to eradicate the panic and stigmas of the people worldwide towards covid-19. also, it may aid to improve clinical strategies against this pandemic. the best alternative left for the mankind at this moment is to follow preventive measures such as no direct human interaction, self-quarantine, keeping the living area hygienic and maintaining social distance. data-based analysis, modelling and forecasting of the covid-19 outbreak the mathematical theory of lnfectious diseases. 2 nd edition statistical time series analysis of dynamics of hiv dynamic indicator for the prediction of atmospheric pollutants forensic investigations and risk management in mobile and wireless communications real-time forecasts and risk assessment of novel coronavirus (covid-19) cases: a data-driven analysis transmission dynamics of the covid-19 outbreak and effectiveness of government interventions: a data-driven analysis a data driven time-dependent transmission rate for tracking an epidemic: a case study of 2019-ncov early dynamics of transmission and control of covid-19: a mathematical modelling study. the lancet infectious diseases analysis of spatial spread relationships of coronavirus (covid-19) pandemic in the world using self organizing maps novel coronavirus -japan (ex-china). world health organization middle east respiratory syndrome coronavirus (mers-cov) -update:2 world health organization (who) epidemic spreading in scale-free networks the effect of control strategies to reduce social mixing on outcomes of the covid-19 epidemic in wuhan, china: a modelling study why is it difficult to accurately predict the covid-19 epidemic? covid-19 future forecasting using supervised machine learning models time series analysis and forecast of the covid-19 pandemic in india using genetic programming ai-driven tools for coronavirus outbreak: need of active learning and cross-population train/test models on multitudinal/multimodal data small molecules targeting severe acute respiratory syndrome human coronavirus nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study early prediction of the 2019 novel coronavirus outbreak in the mainland china based on simple mathematical model a pneumonia outbreak associated with a new coronavirus of probable bat origin authors thankful to ggsip university for providing research facilities. the author(s) declare that there is no conflict of interest. key: cord-288080-rr9e61ay authors: mohadab, mohamed el; bouikhalene, belaid; safi, said title: bibliometric method for mapping the state of the art of scientific production in covid-19 date: 2020-06-30 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110052 sha: doc_id: 288080 cord_uid: rr9e61ay global scientific production around the covid-19 pandemic, in the various disciplines on the various international scientific bibliographic databases, has grown exponentially. the latter builds a source of scientific enrichment and an important lever for most researchers around the world, each of its field and its position with an ultimate aim of overcoming this pandemic. in this direction, bibliometric data constitute a fundamental source in the process of evaluation of scientific production in the academic world; bibliometrics provides researchers and institutions with crucial strategic information for the enhancement of their research results with the local and international scientific community, especially in this international pandemic. the latest statistics indicate that there has been an exponential increase in the number of publications since the discovery of the covid-19 pandemic; the results provide a comprehensive view of interdisciplinary research in medicine, biology, finance and other fields. the number of publications in international databases aims to disseminate and share the contributions and advances of academic research from different groups of researchers from different universities and countries in the thematic of covid-19. bibliometrics [1] is a tool for mapping the state of the art in a field related to given scientific knowledge. so the use of bibliometric analysis [2] to identify and analyze the scientific performance of authors, articles, journals, institutions, countries through the analysis of keywords and the number of citations constitutes an essential element which provides researchers with the means to identify avenues and new directions in relation to a theme of scientific research. scientometrics [3] is considered as the science of measurement and the analysis of science which is based on an input set and an output set which uses bibliometrics in the field of study of publications. the latter is a meta-science which takes science as its object of study based on three elements of scientific activity: its inputs, its outputs and its impacts. thus, it makes it possible to map and broaden knowledge on a research field, by clarifying the links between the authors, the publications, the institutions, and other characteristics of the studied field. scientific publications [4] represent all publications in newspapers or conferences, either chapters in scientific books or scientific patents. all these types of publications represent the work of a researcher who publishes these works with the aim of circulating these results in databases which have broad international visibility and scientific credibility such as web of science, scopus… and renowned publishing houses such as elsevier, springer, wiley, etc.; but with all the efforts made, the benefits that can be drawn remain limited if we cannot manage this large mass of publication which is added every day to the thousands or millions of existing scientific papers. bibliometric data is used for: • measure and compare the scientific output of the researcher, research groups, institutions, regions or countries using indicators based on: -the number of publications. -the quotes received. -the collaborations. • identify the most important or influential journals in a given field. • monitor the evolution over time of a discipline or research subject. these data represent the main part of the data provided for each paper by the databases which allow bibliometrics to carry out statistical processing, and bibliometric analysis. according to statistics provided by johns hopkins university [5] until may 23, 2020, the death of more than 339,949 people worldwide, was the infection of 5,267,452, considerable efforts were made in the various disciplines relating to the treatment of this pandemic either from near or far. since the beginning of the year, covid-19 represents an increasing interest for researchers from all over the world, in response to this crisis, a lot of research was carried out in many fields of research (medical, biology, financial, ...) by several institutions and organizations, either public or private worldwide, each with their own means available. by reviewing most of the scientific databases, the search to identify the scientific output related to the subject of covid-19 [6] was carried out using a set of terms as search criteria, the language of the documents is the english because it is the universal language of research, all disciplines are authorized in order to provide a global view of covid-19 research in the various disciplines, research is limited to the period from early 2020 (beginning of the pandemic a been listed) so far. using the scopus search engine to search for the word "covid-19" and "coronavirus" from 01/01/2020 until 23/05/2020, we find 10,228 documents: -according to the authors: using the search engine of web of science to search for the word "covid-19" and "coronavirus" from 01/01/2020 until 23/05/2020 results in 5,161 documents: -according to the authors: -according to the country:  scopus:  africa: the exploitation of the bibliometric parameters available on the scientific data base on multiple field and discipline makes it possible to release relevant information which can meet the expectations of researchers, research teams and research institutes. the bibliometric analysis reveals to the researcher exact information for the construction of new research as in the case of our study on covid-19. this study was carried out on the basis of specific research using the three databases (scopus, web of science, pubmed) from the beginning of 2020 until 23/05/2020. the sample consists of 5,161 academic publications (web of science), 10,228 academic publications (scopus) and 7,991 academic publications (pubmed). the use of bibliometrics will contribute to the exploration and description of the existing scientific literature on the theme of covid-19. the steps taken to achieve the desired results are manifested as: the use of bibliometric tools plays an important role in guiding a particular field of study by collecting scientific data and synthesizing the results obtained. statistics from different bibliographic databases which differ either in terms of data volume or coverage constitutes a reliable source for bibliometric indicators [9] . choosing the right database, the right keywords and applying the filters that reflect the research objectives is a crucial step to have reliable results. among the credible scientific database which brings together most of the publishing houses known as elsevier, taylor & francis, springer…, we find scopus, web of science and for the medical field pubmed [10] equipped with different filters to refine the search and limit the results found. some researches try to analyze data coming from the various scientific databases, but there are structural differences between the platforms. thus the differences in the classification of information adopted by each of them builds an obstacle for an exploitation of the common data. for a good bibliometric analysis, we choose the following bibliometric data: -article title. -authors. -keywords. -number of citations. -year of publication. -journals. -type of documents. -institution. -country. -field of research. regarding the indicators used by scopus we find: -h-index [11] : is based on the highest number of articles with at least the same number of citations. -citescore: measures the average number of citations received per document published in the serial publication. -sjr: measures the weighted citations received by the periodical, the weighting of the citations depends on the domain and the prestige of the citing series. -snip: the standardized paper impact of the source which measures the actual citations received compared to the expected citations for the field of serial publication. regarding the indicators used by web of science we find: -h-index: the most used research indicator that measures both the productivity and the impact of an author's scientific production. -the impact factor: measures the importance of a review according to the number of citations received in a year. -journal citation reports: web of science product and an authoritative resource for impact factor data. in the present case study, the keywords employed are "covid-19" / "coronavirus" from the beginning of 2020 (date of the start of the pandemic). the search should focus mainly on the titles, keywords and abstracts of articles in each of the databases. then the results found for each of the three databases (scopus, web of science, pubmed) builds our separate database on which our bibliometric analysis will be applied. we export the data from scopus in format (.csv), web of science, pubmed in format (.txt). next, we use the vosviewer software [12] which represents a high-performance solution with numerous viewing options with co-quotation, co-word, co-author network analysis. through bibliometric analyzes we try to get the trends of scientific research in the theme of covid-19. in order to observe and evaluate the trends in publications in the thematic of covid-19, the vosviewer software was used to analyze the academic literature and examine the evolution of published articles, co-authorship, geographic area (country) of authors, co-citation, co-occurrence. the analysis of the authors belonging to the database allows to have a global view on the authors active in the thematic by offering the possibility to follow the work of these researchers by opening the door to achieve cooperation and partnerships. thus, the analyzes of research institutions and countries constitute an effective asset for finding the pillar institutions in each field, with the aim of seeking possible cooperation at the level of research institutions. the software used for viewing and mapping the structure of a research are including bibexcel, histcite, citespace, gephi, and vosviewer. for this work, we chose to work with vosviewer because it allows us to easily display and interpret the display of large bibliometric maps. in order to carry out the various analyzes previously cited and to examine the evolution of the articles published, we have for: we have 1 cluster which contains 12 items. we deduce that most institutions collaborate with each other on an international scale and not at the regional or continental level. -for countries: figure 18 : country organizations network in the "network visualization" display mode. we have 9 clusters distributed as follows: cluster 1-2-3: 5 items; cluster 4-5-6: 4 items; cluster 7-8-9: 3 items. as we see in figure 21 , the map indicates a large node representing china which means the great involvement of the chinese giant through these researchers in the various research fields related to covid-19. bibliometric studies are used to identify networks of researchers or to map the structure of researchers in a given research area. figure 19 : author co-authorship network in the "network visualization" display mode. we have 9 clusters distributed as follows: cluster 1:46 items; cluster2: 46 items; cluster3: 20 items; cluster 4:16 items; cluster 5:15 items; cluster 6:11 items; cluster 7: 11 items; cluster 8: 10 items; cluster 9: 10 items. the results clearly show that there are 9 groups of researchers who collaborate. two groups have a significant number of researchers despite an exponential increase in the number of publications since the start of the pandemic, international collaboration between the authors remains low. from the results found, it can be deduced that geographic proximity between institutions tends to strengthen the collaborative relationships of institutions. thus, it warns of the need to expand cooperation in other regions, countries or continents. -for countries: the analysis of the network of countries is an important form of analysis which makes it possible to visualize the most influential countries in a given field of research, thus it exposes the degree of scientific cooperation between the countries. we have 11 clusters distributed as follows: cluster 1: 7 items; cluster 2-3: 6 items; cluster 4: 5 items; cluster 5: 4 items; cluster 6-7-8: 3 items; cluster 9-10-11: 2 items. as we can see in figure 18 , the map shows a large node representing the countries and regions with the highest number of publications: china, united states, italy, england, france and spain.  pubmed: -for authors: figure 22 : author co-authorship network in the "network visualization" display mode. we have 6 clusters distributed as follows: cluster 1:27 items; cluster 2-3-4: 15 items; cluster 5: 7 items; cluster 6: 4 items. the results clearly show that there are 6 groups of researchers who collaborate with each other, a group has a large number of researchers, followed by a group that is distinguished by the number of researchers who compose them. -for institutions: figure 23 : author organizations network in the "network visualization" display mode. in 1 cluster with 13 items, we notice that there is a significant presence of italian medical institutions, the analysis of data from pubmed by vosviewer does not offer the possibility of analyzing the network of countries.  vosviewer: figure 24 : author keywords network in the "network visualization" display mode. we have 3 clusters distributed as follows: cluster 1: 6 items; cluster 2-3: 4 items. the results found build a map dividing the keywords into three groups with the minimum number of occurrences of a keyword fixed at 6 elements for the first group and 4 elements for the second and third group. the keyword "coronavirus" has the highest occurrence and total binding strength, other keywords with a high occurrence include "sars-cov-2", "covid-19". among the existing display means, there is the word cloud which is a practical tool allowing to have a dimensional visualization of the keywords most used in the database. for our case, we use wordle which is an analysis tool which makes it possible to display a word cloud which gives greater importance to the words which appear more frequently in the source text, for the three scientific databases already mentioned, we find: -scopus: from the figures (24-25-26-27) provided by vosviewer and wordle, a set of words related to the pandemic such as (covid-19, coronavirus, sars-cov-2, 2019-ncon) as synonyms used in scientific literature, so the appearance of terms (china, wuhan, usa) refers to the place of appearance of the pandemic and the countries that are conducting research to find the vaccine, too (medical, health, hospital, virology) refers to the most concerned research area, (zhang, wang) for the most productive researchers in the topic of covid-19 in scientific databases. since the onset of the pandemic, considerable effort has been invested by researchers worldwide depending on the fields and resources available, an exponential increase in scientific production has been recorded in the various databases around the covid-19. in this work, we opted for a statistical study for the data from the bibliographic databases scopus, web of science for the theme of covid-19. the scientific contribution of researchers from the usa and china shows a total involvement of institutions from these two countries, so for the african continent researchers from "south africa and egypt are the exception, while for the arab region saudi arabia and egypt are leading the efforts of the arab countries for this pandemic. afterwards, a bibliometric analysis method was adopted in order to map the state of the art on the theme of covid-19, so the three scientific databases (scopus, web of science, pubmed) were used. thus, the search must be precise and planned by combining the precision of the terms to be used and adequate filters to refine the results found, in order to conduct a relevant bibliometric analysis by analyzing the contributions of the authors, institutions, countries and the wordskeys. finally, it is well known that the method presented remains applicable for other scientific themes and not only for the covid-19 theme, it should be noted that the results obtained with the application of the proposed method may vary depending on the basis of scientific data chosen and the appropriate filters in order to present the evolution of published articles, co-authors, geographic area of the authors, co-citation, co-occurrence analysis and keywords. we wish to draw the attention of the editor to the following facts which may be considered as potential conflicts of interest and to significant financial contributions to this work. [or] we wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. we confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. we further confirm that the order of authors listed in the manuscript has been approved by all of us. we confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. in so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. bibliometrics, informetrics, scientometrics and librametrics: an overview an overview of qualitative comparative analysis: a bibliometric analysis a review of theory and practice in scientometrics ten guidelines for effective data visualization in scientific publications a comparison of bibliometric indicators for computer science scholars and journals on web of science and google scholar does the h-index for ranking of scientists really work? software survey: vosviewer, a computer program for bibliometric mapping participatory visualization with wordle key: cord-311544-7ihtyiox authors: sun, tingzhe; wang, yan title: modeling covid-19 epidemic in heilongjiang province, china date: 2020-05-29 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109949 sha: doc_id: 311544 cord_uid: 7ihtyiox the coronavirus disease 2019 (covid-19) surges worldwide. however, massive imported patients especially into heilongjiang province in china recently have been an alert for local covid-19 outbreak. we collected data from january 23 to march 25 from heilongjiang province and trained an ordinary differential equation model to fit the epidemic data. we extended the simulation using this trained model to characterize the effect of an imported ‘escaper’. we showed that an imported ‘escaper’ was responsible for the newly confirmed covid-19 infections from apr 9 to apr 19 in heilongjiang province. stochastic simulations further showed that significantly increased local contacts among imported ‘escaper’, its epidemiologically associated cases and susceptible populations greatly contributed to the local outbreak of covid-19. meanwhile, we further found that the reported number of asymptomatic patients was markedly lower than model predictions implying a large asymptomatic pool which was not identified. we further forecasted the effect of implementing strong interventions immediately to impede covid-19 outbreak for heilongjiang province. implementation of stronger interventions to lower mutual contacts could accelerate the complete recovery from coronavirus infections in heilongjiang province. collectively, our model has characterized the epidemic of covid-19 in heilongjiang province and implied that strongly controlled measured should be taken for infected and asymptomatic patients to minimize total infections. in december 2019, the coronavirus disease 2019 (covid-19) occurred in wuhan (hubei province) and then unfolded in china and all over the world [1] . the novel coronavirus has been a serious threat to public health [2] . china has taken strict measures such as quarantine, social-distancing, suspicious isolation, community surveillance, to restrain covid-19 outbreak by late february. however, heilongjiang has now become the province with most diagnosed patients in china (i.e. even worse than hubei province). massive imported cases emerge and continue to increase. specifically, a recent 'super spreader' or 'imported escaper' in heilongjiang province has led to tens of diagnosed or asymptomatic cases [3] . therefore, evaluating the epidemic of covid-19 in heilongjiang province may help provide effective strategies to restrict covid-19 transmission. recently, many mathematical models have been constructed to investigate the epidemic of covid-19. some models have made modifications based on the conventional 'seir' model [4] and concluded that strictly controlled interventions are critically important to impede covid-19 outbreak [5] [6] [7] [8] [9] [10] [11] . several other model instead established a stochastic transition model to evaluate the transmission of covid-19 and also emphasized the necessity of interventions such as social-distancing, isolation and quarantine [12, 13] . meanwhile, asymptomatic patients are covert cases which represent a serious threat to public health [14] . a few models have been developed to evaluate the role of coronavirus transmission based on asymptomatic cases [11, 15, 16] . however, all these models have ignored the covid-19 transmission from bats and unknown hosts. a recently proposed novel model has characterized these interactions among bats, unknown hosts and people in details [17] . they estimated the basic reproduction number (r0) to be ~2.4829 and indicated that model is locally asymptotically stable if r0 < 1 [17] . these important models have greatly improved our understanding about the epidemic of covid-19. in current work, we developed a mathematical model to characterize imported escaper and asymptomatic patients. we trained our model based on covid-19 epidemic in heilongjiang province from january 23 to march 25, by which the last confirmed patient was cured in heilongjiang province. using this model, we performed stochastic simulations and found that partial relief in strictly controlled interventions may contribute to the occurrence of diagnosed patients recently (from april 9 to april 19) provided that there is only one imported patient without surveillance [3] . meanwhile, we predicted that there is still an unidentified pool of asymptomatic patients. we suggested that strict or mild interventions should still be implemented to restrain a potential covid-19 outbreak. the number of experimentally confirmed patients and cured/recovered cases (january 23 to april 19) was obtained from health commission of heilongjiang province (http://wsjkw.hlj.gov.cn/, table s1 ). note that only domestic cases epidemiologically associated with the 'super spreader' between april 9 and april 19 were recorded and simulated. the total susceptible population was set to be the total λ describes the discharge rate from quarantine. βi (i=1, 2, 3) describes the averaged contact rate between s and c, a or i, respectively. specifically, β1 is the average contact rate between susceptible population and close contacts. β2 describes the average contact rate between susceptible population and asymptomatic patients. β3 denotes the averaged contact between susceptible population and infected patients. ε denotes self-recovery rate for asymptomatic cases since specific asymptotic patients can recover without treatments [18] [19] [20] . ν1 and ν2 represent the transition rate from close contacts to asymptomatic and diagnosed ones, respectively. ν3 is the transition from asymptomatic to diagnosed cases [18] [19] [20] . μ is the (maximum) average recovery rate. since the healthcare capacity in different provinces differ significantly owing to economic development, we further incorporated a threshold behavior for the recovery. we assumed that higher numbers of inpatients and quarantined cases will lower the average recovery rate. the health-care capacity will be improved if there are fewer diagnosed/quarantined patients. a hill function was used to characterize the threshold behavior. the threshold was represented as a parameter k and the cooperativity effect was described by the parameter n (see model 1). initial values for i and r were obtained from the official site (http://wsjkw.hlj.gov.cn/). the total susceptible population was set to be the total population in heilongjiang province as described above. other parameters and initial conditions (c and a) were estimated based on epidemic data from heilongjiang province. for parameter estimation, 400 stochastic runs using potterswheel toolbox [21] were implemented and the best fit parameter set was shown in table 1 . since the best fit for initial asymptotic population is <<1 (0.042736), we set initial value for a to be 0 since a should be an integer. we note that the parameter n in the best fit set was ~1.02545 suggesting that the threshold behavior in recovery rate has relatively low cooperativity. parameters were estimated by potterswheel using a trust-region method [21] . a χ 2 criteria was used for model identification (χ 2 /n<1, n is the number of data points) [21] . we did not incorporate death similar to recent assumptions [13, 22] . the τ-leap method was used for stochastic simulation [23] . ordinary differential equations (odes) were numerically solved using ode23s in matlab (r2018b). the 'ksdensity' function in matlab was used for kernel estimation. the model was first trained with epidemic data from heilongjiang province. using a reasonable initial guess, we performed unbiased parameter fitting by sampling parameters in logarithmic scale (totally, 400 sets). the top 200 fits were harvested. we noted that fitted initial values from 200 top fits for asymptomatic patient number 'a' approximated zero with mean value <0.1 and therefore the initial value 'a' for is set to be 0. the best fit was shown for infected and recovered patients by march 25 ( figure 1a ). notably, the fitted βi (i=1, 2, 3) values were to the order of 10 -10 to 10 -9 suggesting that strong interventions were implemented to restrain mutual contacts (table 1) . meanwhile, n is around 1 (1.02545) suggesting that the cooperativity was neglectable. the best fit parameter set was shown in table 1 . we further estimated a dynamic reproduction number based on ordinary differential equations [24] . the results showed that the reproduction number initially reached 5.0692 and then quickly dropped below 1 by feb 14 and finally reached ~0.8702 ( figure s1 ). therefore, our model could faithfully match the epidemic data in heilongjiang province and found that continuously strict measures indeed impeded the covid-19 transmission. we then performed stochastic simulations using τ-leap method [23] [3] . therefore, the number of infected patients was increased by 1 at march 19 in the stochastic model. however, the 'home quarantine' was failed as reported possibly owing to accidental egress by escalator or stairs [3] . we modeled this by increasing f (from 3 to 8) and then performed stochastic simulations. results suggested that even a moderate relief would not lead to the reported local outbreak (f=3,4 and 5, figure 2a ). increasing f to 6-8, however, will result in possible matches between stochastic simulations and reported epidemic data (black dots, epidemic data; colored curves: stochastic simulations, figure 2a ). the median values from 500 stochastic runs were shown for increasing f values (blue curves, figure 2b , the shaded areas denote 25% and 75% quartiles). therefore, we predicted that the mutual contacts for the 'super spreader' was significantly elevated to around 7. we further matched our simulations to the reported asymptomatic patients. setting f=7, stochastic simulation could match the epidemic data ( figure 2 ). however, we noted a large gap between the simulated number of asymptomatic patients to the reported cases (f=7, figure 3a ). since asymptomatic patients usually do not seek medical tests owing to lack of symptoms, the reported number of asymptomatic cases may be lower than the actual number ( figure 3a ) [18] [19] [20] . the underestimation was significant for f=6 or f=7 on april 19 (dashed line: the mean value, figure 3b ). the discrepancy seemed to grow if intervention strength was not altered ( figure 3b ). these simulations implied that the unidentified asymptomatic patients might be another serious concern since the asymptomatic cases can also be contagious [18] [19] [20] . since the confirmed patients have undergone an ongoing increase, we then evaluated the effect of establishment of strict interventions since april 20 (black dot, figure 4a ). strict interventions were implemented (f=1 or 2) since april 20. we measured the time to peak and the recovery time since april 9, at which the first reported domestic patient occurs in heilongjiang province [26] . we found that roughly ~33 days were used to reach the peak for f=1 whereas for f=2, the duration was increased to ~53 days ( figure 4b and 4c, left) . ~80 days were required for total recovery if f=1 whereas about 139 days were needed for f=2 ( figure 4b and 4c, right) . therefore, these simulations suggested implementation of strict interventions is strongly required to impede a potential outbreak. in this study, we constructed a modified model which incorporated threshold behavior in recovery rate and asymptomatic patients with significant difference to 'seir' model [4] . however, ongoing importation into china especially in heilongjiang province has been a serious threat recently. furthermore, the asymptomatic patients are not easily identified owing to lack of obvious symptoms [20] . we estimated the influence of asymptomatic patients based on a data-driven model from heilongjiang province. we anticipated that an area-or province-specific model may help unravel the underlying features of covid-19 epidemic. from simulation, we found that even one 'imported patient' or 'super spreader' reported recently can lead to up to ~ 40 infected patients and unpredictable pool of asymptomatic patients till april 19 [3] . we anticipated that the relaxed 'home quarantine' for this 'imported patient' [3] is unsuccessful and has initiated a local outbreak potentially. simulation suggested that the mutual contacts from the relaxed 'home quarantine' might possibly increase to f=7 compared with f=1 for strictly controlled interventions. the occurrence of the 'imported escaper' might be ascribed to the significantly longer latent period [2] , the weakened surveillance for covid-19 inpatient transfer in specific hospitals and the murky mismanagement for covid-19 tests [3] . all these deficiencies may possibly lead to a secondary blockage of some hospitals and communities for haerbin city in heilongjiang province. more seriously, the associated pool of asymptomatic patients was underestimated from official report. since asymptomatic patients are also contagious, the potential impact of these covert cases should attract extensive attention. as least about one month is still required if strict measures are immediately implemented. therefore, the 'imported escaper' argues for further epidemiological investigations as well as strongly controlled interventions. as more data are available, our model could be used for analyzing these novel epidemic data. liu et al. recently constructed a model with 'asymptomatic patients' [11] . however, asymptomatic populations in their model will eventually become diagnosed/confirmed patients with clear symptoms and are more likely to be 'pre-symptomatic' [14] , which is in stark contrast to recent definition (i.e. asymptomatic patients were those who never develop symptoms) [18] [19] [20] . we modeled the covid-19 epidemic induced by one imported patient without strict 'quarantine'. lessons from the emerging diagnosed patients recently in heilongjiang province since april 9 should catch attention as they are epidemiologically associated with this 'imported escaper'. furthermore, we recommended that the strict measures such as isolation, home or centralized quarantine should be re-established especially in haerbin city of heilongjiang province to lower the risk of a potentially secondary outbreak. finally, our model could also be applied elsewhere with adjusted parameters to monitor covid-19 epidemic. the authors declare no competing interests. this work is supported by national natural science foundation of china (31971185) and quality engineering project of anhui college education (2018jyssf086). ☐the authors declare the following financial interests/personal relationships which may be considered as potential competing interests: a novel coronavirus from patients with pneumonia in china china medical treatment expert group for, clinical characteristics of coronavirus disease 2019 in china van den broeck, p. vanhems, simulation of an seir infectious disease model on the dynamic contact network of conference attendees simulation of the clinical and pathological manifestations of coronavirus disease 2019 (covid-19) in golden syrian hamster model: implications for disease pathogenesis and transmissibility modeling analysis of covid-19 based on morbidity data in anhui evaluation of prevention and control interventions and its impact on the epidemic of coronavirus disease 2019 in chongqing and guizhou provinces the effect of control strategies to reduce social mixing on outcomes of the covid-19 epidemic in wuhan, china: a modelling study modeling the epidemic dynamics and control of covid-19 outbreak in china an updated estimation of the risk of transmission of the novel coronavirus (2019-ncov) predicting the cumulative number of cases for the covid-19 epidemic in china from early data a mathematical model for simulating the phase-based transmissibility of a novel coronavirus phase-adjusted estimation of the number of coronavirus disease asymptomatic and presymptomatic infectors: hidden sources of covid-19 disease estimating the effects of asymptomatic and imported patients on covid-19 epidemic using mathematical modeling beware of asymptomatic transmission: study on 2019-ncov prevention and control measures based on extended seir model modeling the dynamics of novel coronavirus (2019-ncov) with fractional derivative estimating the asymptomatic proportion of coronavirus disease 2019 (covid-19) cases on board the diamond princess cruise ship estimation of the asymptomatic ratio of novel coronavirus infections (covid-19) covert coronavirus infections could be seeding new outbreaks dynamical modeling and multi-experiment fitting with potterswheel interventions to mitigate early spread of sars-cov-2 in singapore: a modelling study stochastic models for regulatory networks of the genetic toggle switch estimation of the time-varying reproduction number outbreak in china, medrxiv novel coronavirus patients' clinical characteristics, discharge rate, and fatality rate of meta-analysis key: cord-268630-vu8yyisx authors: mohammad, mutaz; trounev, alexander title: implicit riesz wavelets based-method for solving singular fractional integro-differential equations with applications to hematopoietic stem cell modeling date: 2020-06-17 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109991 sha: doc_id: 268630 cord_uid: vu8yyisx riesz wavelets in [formula: see text] have been proven as a useful tool in the context of both pure and numerical analysis in many applications, due to their well prevailing and recognized theory and its natural properties such as sparsity and stability which lead to a well-conditioned scheme. in this paper, an effective and accurate technique based on riesz wavelets is presented for solving weakly singular type of fractional order integro-differential equations with applications to solve system of fractional order model that describe the dynamics of uninfected, infected and free virus carried out by cytotoxic t lymphocytes (ctl). the riesz wavelet in this work is constructed via the smoothed pseudo-splines refinable functions. the advantage of using such wavelets, in the context of fractional and integro-differential equations, lies on the simple structure of the reduced systems and in the powerfulness of obtaining approximated solutions for such equations that have weakly singular kernels. the proposed method shows a good performance and high accuracy orders. fractional differential modeling is widely used in many areas of applications in physics, engineering and many other major sciences. recently, atangana et al used fractional calculus in image processing [1] , modeling the dynamics of novel coronavirus (covid-19) [2] , finance [3] , dynamical systems [4] and many other applications and developments can also be found in [5] [6] [7] [8] . other interesting applications can be found in [9] [10] [11] [12] [13] [14] [15] [16] and references therein. it is important to mention that an extensive development of fractional calculus has been achieved and well established to describe many applications that using integro-differential equations (see for example [18] [19] [20] [21] 17] ). note that, the kernel of many of these integro-differential equations is singular with a fractional order and hence obtaining the exact solutions is challenging and sometimes even impossible. therefore, several methods involving integro-differential operators have been proposed to solve fractional differential equations. for a complete picture we refer to refs. (see [22] [23] [24] [25] ). * corresponding author. wavelets and their generalizations appear in a variety of advanced applications in filter banks analysis, in image processing and image recognition, transmission and storage [26, 28, 30, 27, 29] . this is largely due to the fact that wavelets have the right structure to capture the sparsity in "physical" images, perfect mathematical properties such as its multi-scale structure, sparsity, smoothness, compactly supported, and high vanishing moments. for example, the fbi center is using wavelets in their fingerprint database system for image reconstruction, see fig. 1 . note that, a function g ∈ l 2 (r ) is said to have vanishing moments of order m if it is orthogonal to the polynomials x s for all s = 0 , 1 , 2 , . . . , m − 1 . this property is connected to the multi-scale systems and its sparsity [26] . wavelet expansions have been successful in developing many numerical algorithms for investigating and solving various types of fractional, differential and integral equations (see for example [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [47] [48] [49] [50] [51] 46] ). motivated by the above contributions and properties, that are essential to develop efficient algorithms for the numerical solutions of a given fractional integro-differential equations (fides), the main goal of the proposed work is to develop an efficient algorithm based on riesz wavelets using the collocation method to solve fractional order of integro-differential equations with weakly singular kernels. before we proceed further, let us recall some definitions and notations needed for this paper. a function φ ∈ l 2 (r ) is called refinable if it satisfies the following equation is a finitely supported sequence and is called the refinement mask of φ. the corresponding wavelet function is defined by is finitely supported sequence and is called the high pass filter of ψ. in this paper, for f ∈ l 1 (r ) (which can be extended to l 2 (r ) ), we use the following fourier transform defined by the fourier series of the sequence a is defined by pseudo-splines have attracted many researcher due to their significant contribution to both numerical computations and analysis. the constructions of pseudo-splines tracked back to the wellknown work produced by daubechies et al. in [52, 53] . it is a family of refinable functions with compact support and have extensive flexibility in wavelets and applications. pseudo-splines known as a generalization of many well-known refinable functions such as the b-splines, interpolate and orthonormal refinable functions [28] .we refer the reader to [52] [53] [54] [55] [56] [57] [58] and references therein for more details. we use the smoothed pseudo-splines introduced in [58] to construct riesz wavelets and use it to apply our numerical scheme for solving different types of fides. pseudo-splines of order ( p, q ) of type i and ii, k φ (p,q ) , k = 1 , 2 , are defined in terms of their refinement masks, where note that, the refinement mask of the pseudo-splines of type i with order ( p, q ) is obtained using fejér-riesz theorem. using the fourier transform, the refinable pseudo-spline function generated using the above refinment masks is defined by (2.1) they are two types of smoothed pseudo-splines defined by its refinable masks. for r ≥ p , we have the smoothed refinable pseudosplines of type i (k = 1) , and ii (k = 2) with order ( r, p, q ), such where χ a is the indicator function on the set a . similarly, the refinement masks of both types of k φ ( r,p,q ) , for k = 1 , 2 , respectively are given by riesz wavelets have been studied extensively in the literature, for example, see [59] [60] [61] [62] and other references. riesz wavelet in l 2 (r ) if for any finitely supported sequence n j,k , = 1 , . . . , n; j, k ∈ z there exist positive numbers c and c such that if m defined in definition 2.1 defined in definition generates a riesz wavelet for l 2 (r ) , then we can conclude the following expansion for any function f ∈ l 2 (r ) such that 3) can be truncated by (2.4) in this section, we present the construction of riesz wavelet systems using smoothed pseudo-splines that will be used to solve some examples of fides. dong in [54] and bin in [62] for bsplines, and pseudo-splines proved that the system m defined in definition 2.1 forms a riesz wavelet for l 2 (r ) where the corresponding wavelet function ψ is defined by for smoothed pseudo-splines k φ (r,p,q ) , k = 1 , 2 , chuang in [58] proved the same result for the wavelet function in this case, it was also shown in [62] , for a given sequence of integer numbers m 1 , we have the following representation for the smoothed pseudo-splines refinable masks given by the corresponding refinable and wavelet functions are given by now we present some examples of riesz wavelets via some smoothed pseudo-splines of both types and different order. , we have the following refinable masks: then, m ( 1 ψ (6 , 2 , 1) ) and m ( 2 ψ (6 , 2 , 1) ) form riesz wavelet systems for l 2 (r ) . note that the vanishing moments for both systems is equal to 6. on how to plot these generators, we recommend han's book [27] for a complete analysis of graphing such wavelets. fig. 2 shows the graphs of the smoothed refinable functions and their corresponding riesz wavelets. , we have the following refinable masks: here we found 1 ˆ then, m ( 1 ψ (9 , 3 , 2) ) and m ( 2 ψ (9 , 3 , 2) ) form riesz wavelet systems for l 2 (r ) . fig. 4 shows the graphs of the smoothed refinable functions and its corresponding riesz wavelets. in this paper, we consider several types of fides integrodifferential equation in the sense of caputo fractional operator. let us first start with some basic definitions and notation preliminaries of the fractional calculus needed in this present work. definition 4.1. for a real function u ( t ) where t, α > 0, and n ∈ n we have the following fractional operators of order α, namely: • the caputo's fractional derivative, • the reimann-liouville fractional integral operator (fio), and for t > 0, we apply the collocation method based on riesz wavelets generated using some smoothed pseudo-splines refinable functions. first, we consider the following linear fide with weakly singular kernel type, such that u (m ) (0) = 0 for all m = 1 , 2 , . . . , n where n − 1 < α ≤ n, β 1 , β 2 are real numbers, 0 < θ < 1, f , and k are known functions, and u is the known function that needs to be numerically determined. the scheme works as per the following steps: 1. we create an ansatz for the function that needs to be approximated by choose a suitable collocation points based on the support of the riesz wavelets and the domain of the function. 5. solve the resulting system to get the coefficients v j,k . following the steps above, we have with a few algebra and after plugging the interpolating points, where i n v m u (μ) will generate the riesz wavelets operational matrix of fractional integration of order n . this generate a liner system of the form j e = f, where e and f are column vectors and j is a square matrix. solving this system leads to the coefficients v j,k needed for the approximated solution v m u (μ) . next, we study the hematopoietic stem cell (hsc) model based on fractional order derivatives. recent research results proved the usefulness of using the stem cells for medical and treatment purposes. for example, for it was used for treating type 1 diabetes patients, cancer cells, strengthen immune system to fight several infections and many more. most recently, the uae has used stem cells for the treatment of covid-19 to assess the effectiveness in fighting the virus. therefore, handling modeling that describe the dynamics of such cells is crucial. in this work, we consider the fractional model of hsc introduced in [63] and propose a generalization of the model based on atangana-baleanu fractional derivative as follows: (4.7) the system consists of 5 variables that describe the dynamics of the cells, namely: • eq. table 1 . accordingly, there will be 5 corresponding odes with nonnegative initial conditions related to specific needs. to solve the system of eqs. (4.3) -(4.7) , we consider the truncated expansion where α = b(α) 1 −α . then, the same procedure by the collocation methods will be applied to solve the ctl cells fractional order system. in this section, we use the riesz systems presented in section 3 to demonstrate and show the efficiency of the proposed method. we deal with the following examples. 0 . 2 , 0 . 01 , 0 . 1 , 0) . the graph of the variables x ( t ) and w ( t ) are shown in fig. 5 . table 2 and figs. 6 , 7 , 8 and 9 . the graphs of the numerical solutions using table 2 comparison results between the exact solution and its numerical approximation among different riesz wavelet systems. table 3 comparison results between the exact solution and its numerical approximation among different riesz wavelet systems. x exact table 3 and in this framework, the collocation method based on riesz wavelets has been applied to numerically solve fractional order type of integro-differential equations with singular kernel type. the riesz wavelets constructed via the smoothed pseudo-splines of type i and ii and using different orders. these examples of riesz wavelets have high vanishing moments orders and this effectively raised the approximation order of the numerical solution for the fide being handled. we showed some illustrations for the numerical solutions of two examples of fides with singular kernel, which is of a great interest in applications. the proposed algorithm showed high order of accuracy and excellent agreement with the exact solutions. it turns out that increasing the vanishing moments of the underlying wavelet function resulting in improving the accuracy orders of the numerical solutions. the utilized numerical algorithm can be expected to solve various types of fdes including the nonlinear case that will be considered in the future work. furthermore, we intended to study this important case on constructing appropriate higher dimensional smoothed pseudo-splines riesz wavelets in order to solve problems in higher dimensions of fractional integro-differential equations with singular kernels. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. mutaz mohammad: conceptualization, methodology, visualization, investigation, supervision, validation, writing -review & editing. alexander trounev: software, writing -original draft. a new application of fractional atangana-baleanu derivatives: designing abc-fractional masks in image processing modeling the dynamics of novel coronavirus (2019-ncov) with fractional derivative a fractional order optimal 4d chaotic financial model with mittag-leffler law fractional differential and integral operators with non-singular and non-local kernel with application to nonlinear dynamical systems new fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model chaos in a simple nonlinear system with atangana-baleanu derivatives with fractional order on the new fractional derivative and application to nonlinear fisher's reaction-diffusion equation decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena fractional calculus in the transient analysis of viscoelastically damped structures an analog simulation of noninteger order transfer functions for analysis of electrode process an efficient nonstandard finite difference scheme for a class of fractional chaotic systems the motion of a bead sliding on a wire in fractional sense new rheological problems involving general fractional derivatives within nonsingular power-law kernel on linear viscoelasticity within general fractional derivatives without singular kernel new general fractional-order rheological models with kernels of mittag-leffler functions fractional maxwell fluid with fractional derivative without singular kernel on a class of integro-differential equations modeling complex systems with nonlinear interactions qualitative analysis of an integro-differential equation model of periodic chemotherapy on fractional integro-differential equations with state-dependent delay application of fractional calculus to fluid mechanics analysis of a new partial integro-differential equation with mixed fractional operators fitted fractional reproducing kernel algorithm for the numerical solutions of abc-fractional volterra integro-differential equations legendre wavelets method for solving fractional integro-differential equations legendre wavelet collocation method combined with the gauss-jacobi quadrature for solving fractional delay-type integro-differential equations framelets and wavelets: algorithms, analysis, and applications. applied and numerical harmonic analysis construction of wavelets and framelets on a bounded interval compactly supported quasi-tight multiframelets with high balancing orders and compact framelet transforms derivative-orthogonal wavelets for discretizing constrained optimal control problems wavelet based methods for numerical solutions of two dimensional integral equations wavelet interpolation method for solving singular numerical solutions of fredholm-volterra integral. equations by using scaling function interpolation method numerical solutions of fredholm integral equations of the first kind by using coiflets generalized legendre expansion methods and functional differential equations legendre wavelet method for numerical solutions of partial differential equations gibbs phenomenon in tight framelet expansions gibbs effects using daubechies and coiflet tight framelet systems special b-spline tight framelet and it's applications on the gibbs effect based on the quasi-affine dual tight framelets system generated using the mixed oblique extension principle wavelets based simulation and visualization approach for unmixing of hyperspectral data biorthogonal-wavelet-based method for numerical solution of volterra integral equations m mohammad a numerical solution of fredholm integral equations of the second kind based on tight framelets generated by the oblique extension principle a collocation method via the quasi-affine biorthogonal systems for solving weakly singular type of volterra-fredholm integral equations applications of bi-framelet systems for solving fractional order differential equations bi-orthogonal wavelets for investigating gibbs effects via oblique extension principle fractional differential equations numerical solution of volterra integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions a predictor-corrector approach for the numerical solution of fractional differential equations haar wavelet approach to linear stiff systems framelets: mra-based constructions of wavelet frames smooth wavelet tight frames with zero moments pseudo-spline, wavelets and framelets pseudo box splines. applied and computational harmonic analysis properties of dual pseudo-splines wavelets and framelets from dual pseudo-splines a class of generalized pseudo-splines on compactly supported spline wavelets and a duality principle refinable functions and cascade algorithms in weighted spaces with holder continuous masks wavelets and pre-wavelets in low dimensions wavelets with short support single injection of cd8+ t lymphocytes derived from hematopoietic stem cells -mathematical and numerical insights we would like to thank the anonymous reviewers for carefully reading our manuscript and for giving such constructive comments which substantially helped improving the quality of the paper. key: cord-308115-bjyr6ehq authors: baba, isa abdullah; nasidi, bashir ahmad title: fractional order model for the role of mild cases in the transmission of covid-19 date: 2020-10-20 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110374 sha: doc_id: 308115 cord_uid: bjyr6ehq most of the nations with deplorable health conditions lack rapid covid-19diagnostic test due to limited testing kits and laboratories. the un-diagnosticmild cases (who show no critical sign and symptoms) play the role as a route that spread the infection unknowingly to healthy individuals. in this paper, we present a fractional order sir model incorporating individual with mild cases as a compartment to become smir model. the existence of the solutions of the model is investigated by solving the fractional gronwall's inequality using the laplace transform approach. the equilibrium solutions (dfe & endemic) are found to be locally asymptotically stable, and subsequently the basic reproduction number is obtained. also the global stability analysis is carried out by constructing lyapunov function. lastly, numerical simulations that support analytic solution follow. it was also shown that when the rate of infection of the mild cases increases, there is equivalent increase in the overall population of infected individuals. hence to curtail the spread of the disease there is need to take care of the mild cases as well. the outbreak of the novel strain of corona viruses (covid19) started late december in the wuhan province in china [1] . it became a global pandemic causing the devastating impact in terms of morbidity, infections and fatality in addition to socio-economic disaster. the virus source which is yet to be identified is said to have genetic linkages with severe acute respiratory syndrome (sars-cov) but less severe than middle east respiratory syndrome (mers-cov) [2] . the virus is transmitted to healthy persons via eyes, mouth and nose when an infected person produced respiratory droplets of cough and sneeze or as a result of contact with contaminated surfaces. the average incubation period from catching the virus to time onset of major symptoms (like fever, cough and sneeze) is between 2 -14days [3] . as the vaccine is not yet found, the control measures such as: social distancing, quarantine of suspected case, use of personal protective equipment (like face mask, hand globes, gown), regular hand sanitation using antibacterial agents (like soaps, sanitizer) and imposing the lockdown curfew(when necessary) are the effective intervention that mitigate the transmission of the infection. to execute these measures effectively, there is need to have an in depth study about the number of persons that each infected individual can infect, meanwhile a mathematical model describing the transmission dynamics of the disease should be established. in this regard,zhao and chen [4] developed a susceptible, un-quarantined infected, quarantined infected, confirmed infected (suqc) model to characterize the dynamics of covid-19 and explicitly parameterized the intervention effects of control measures. similarly, song et al. [5] developed a mathematical model based on the epidemiology of covid-19, incorporating the isolation of healthy people, confirmed cases and close contacts. tahir et al. [6] developed a mathematical model (for mers) inform of nonlinear system of differential equations, in which he considered a camel to be the source of infection that spread the virus to infective human population, then human to human transmission, then to clinic center then to care center. however, they constructed the lyapunov candidate function to investigate the local and global stability analysis of the equilibriums solution and subsequently obtained the basic reproduction number or roughly, a key parameter describing transmission of the infection. also,chen et al. [7] developed a bats-hosts-reservoir-people (bhrp) transmission network model for the potential transmission from the infection source (probably bats) to the human infection, which focuses on calculating . to suit korean outbreak, sunhwa and moran [8] established deterministic mathematical model (in form of seihr), in which they estimated the reproduction number and assessed the effect of preventive measures. similarly, lin et al. [9] modeled (based on seir) the outbreak in wuhan with individual reaction and governmental action (holiday extension, city lockdown, hospitalization and quarantine) in which they estimated the preliminary magnitude of different effect of individual reaction and governmental action. also yang and wang [10] proposed a mathematical model to investigate the current outbreak of the coronavirus disease (covid19) in wuhan, china. the model described the multiple transmission pathways in the infection dynamics, and emphasized the role of environmental reservoir in the transmission and spread of the disease. however, the model employed non-constant transmission rates which change with the epidemiological status and environmental conditions and which reflect the impact of the ongoing disease control measures. nonlocality is one of the main drivers of interest in fractional calculus applications. there are interesting phenomena that have what are called memory effects, meaning their state does not depend solely on time and position but also on previous state. such system can be very difficult to model and analyze with classical differential equations, but nonlocality gives fractional derivative built-in ability to incorporate memory effects [11] .fractional differential equations appear naturally in numerous fields of study including physics, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, and fitting of experimental data [12 -14, 23 -26] . recently there are many studies on epidemiological disease modeling using fractional order differential equations [27 -33] . the rieman-liouville fractional derivative is mostly used by mathematicians, but this approach is not suitable for real world physical problems since it requires the definition of fractional order initial conditions, which have no physically meaningful explanation yet. caputo introduced an alternative definition, which has the advantage of defining integer order initial condition for fractional differential equations [15] . in mostly poor and underdeveloped territories where there is no capacity of rapid diagnostic covid-19 test due to insufficiency of testing kits, the mild cases (who usually show no symptoms of the infection due to their strong and active immune system up to their recovery period) play a major role as a route that spread the disease to healthy individuals. here we build our model by incorporating the population of mild individuals into the compartmental sir model to become smir model in form of system of fractional order differential equations (fode) in the caputo sense. it should be noted that to our knowledge no model in literature considered the contribution of the mild cases of covid -19 in the proliferation of the pandemic. the paper is organized as follows: section 1is the introduction, section 2 is the preliminary definitions and theorems, the model formulation followed in section 3, section 4 is the stability analysis, section 5 gives numerical simulations and discussions and section 6 gives the conclusion. the notion of convergence of mittag-leffler function is fully discussed in [15] . theorem 1 [19] :the equilibrium solutions of the system is locally asymptotically stable if all the eigenvalues of the jacobian matrix evaluated at the equilibrium points satisfy despite the fact that almost 80% of the covid -19cases are mild who recover naturally (due to stronger and active immune system that fight against the virus), they still play a role as a route of transmission of the infection [2]. the model was formulated based on the assumption that new born of human are recruited into susceptible class ( ) at the rate . the susceptible individual who had contact with an infected at rate can developed mild symptoms and move to mild class ( ). based on [2], the mild patients with infectivity rate play the role as a routine that spread the infection, those with strong immunity recovered naturally at the rate , while some with critical illness became infected and moved to infectious class ( ) after the incubation period . the infectious individual may then recovered ( ) or died at the rates and respectively. figure 1 gives the schematic diagram describing the transmission dynamics of the disease. with the total population the linearity of the caputo operator yield we apply the laplace transform method to solve the gronwall's inequality (6) with initial condition ( ) linearity property of the laplace transform gives (7) to partial fraction gives taking the inverse laplace transform of (8) where ( ), ( ) are the series of mittag-leffler function (as in definition 4) which converges for any argument, hence we say that the solution to the model is bounded. thus, consider the system (1) through (4) written as ] proof reference to picard-lindelof theorem [20] we establish the following theorem. since and its closed set, then is complete metric space. the continuous system (10) can be transformed to equivalent integral equations as; (12) is equivalent to volterra integral equation that solves (10) . define an operator in now, we need to verify that (13) satisfies the hypothesis of contraction mapping principle. first to show hence the operator maps onto itself. secondly, to show that is a contraction, we have since by hypothesis , then is a contraction and has a unique fixed point.. thus, system (10) has unique solution. to obtain equilibrium solution, we set the system to zero and solve simultaneously as follows; considering ( ) ( ) , in (11)(14) we find the endemic equilibrium, consider system (1) then, we have the following jacobian matrix clearly, allthe eigenvalues have zero imaginary part ( ) therefore, hence by theorem 1 above, the disease free equilibrium is locally asymptotically stable. to obtain basic reproduction number which is a key parameter describing the number of secondary infections generated by a single infectious individual, we consider the eigen values above. for implies that this threshold quantity which if less than one disease free equilibrium will be stable and if greater than one it is unstable is what we termed as basic reproduction ratio . hence we define the endemic equilibrium points can be rewritten in the form of to derive the lyapunov candidate function for fractional order as in [24] , consider the family of quadratic lyapunov function and define the lyapunov candidate function as applying lemma 2 [23] above hence by theorem 3 above, the disease free equilibrium is globally asymptotically stable. case 2: at the endemic (positive) equilibrium, (28) becomes we had earlier established that the positive equilibrium is stable if . now consider the relations therefore back substitution the above relations into (30) yields where ( ( )) ( ) ( ) hence by theorem 3 above, the endemic equilibrium is globally asymptotically stable. in this chapter we carry out numerical examples to support the analytic results using parameter values in table 1 . for the variables we use ( ) ( ) ( ) ( ) when the rate of infection of the mild cases increases, there is equivalent increase in the overall population of infected individuals. hence to curtail the spread of the disease there is need to take care of the mild cases as well. in mostly poor and underdeveloped territories where there is no capacity of rapid diagnostic covid-19 test due to insufficiency of testing kits, the mild cases (who usually show no symptoms of the infection due to their strong and active immune system up to their recovery period) play a major role as a route that spread the disease to healthy individuals. here we build our model by incorporating the population of mild individuals into the compartmental sir model to become smir model in form of system of fractional order differential equations (fode) in the caputo sense. the existence of the solutions of the model was shown by solving the fractional gronwall's inequality using the laplace transform approach. two equilibrium solutions, disease free and endemic were obtained. both local and global stability of the equilibria were shown to depend on the magnitude of basic reproduction ratio. numerical simulations were carried out and dynamics of the populations were shown to vary for different values of it was also shown that when the rate of infection of the mild cases increases, there is equivalent increase in the overall population of infected individuals. hence to curtail the spread of the disease there is need to take care of the mild cases as well. world health organization (who). novel coronavirus-china modeling the epidemic dynamics and control of covid-19 in china the impact of isolation on the transmission of covid-19 and estimation of potential second epidemic in china. preprints (www.preprints.org) stability behavior of mathematical model of mers corona virus spread in population a mathematical model for simulating the phased-based transmissibility of a novel coronavirus estimating the reproductive number and the outbreak size of novel coronavirus (covid-19) using mathematical model republic of korea a conceptual model for the coronavirus disease 2019 (covid-19) outbreak in wuhan, china with individual reaction and governmental action a mathematical for the novel coronavirus epidemic in wuhan what is fractional calculus. cantor's paradise a survey on existence result for boundary value problem of nonlinear fractional differential equations the analysis of fractional differential equations: an application-oriented exposition using differential operators of caputo type stability and dynamics of a fractional order leslie-gower model fractional differential equations stability analysis of fractional differential system with rieman-liouville derivative differential equations: an introduction to differential equations, creative commons, 94105 differential equation of fractional order: methods, results and problems stability result for fractional differential equations with application to control processing stability analysis of caputo fractional-order nonlinear systems revisited lyapunov function for fractional; order systems volterra type lyapunov function for fractional order epidemic model analysis of caputo fractional-order model for covid-19 with lockdown analysis of meningitis model: a case study of northern nigeria mathematical modeling for adsorption process of dye removal nonlinear equation using power law and exponentially decaying kernels modeling chickenpox disease with fractional derivatives: from caputo to atangana-baleanu a new study on the mathematical modelling of human liver with caputo-fabrizio fractional derivative a mathematical model for covid-19 transmission by using the caputo fractional derivative epidemiological and clinical characteristics of the covid-19 epidemic in brazil a fractional differential equation model for the covid-19 transmission by using the caputo-fabrizio derivative on the mathematical model of rabies by using the fractional caputo-fabrizio derivative a mathematical theoretical study of a particular system of caputo-fabrizio fractional differential equations for the rubella disease model analysis of the model of hiv-1 infection of cd4^{+} t-cell with a new approach of fractional derivative we write to declare our interest in publishing our work titled "fractional order model for the role of mild cases in the transmission of covid-19 " with the journal " chaos, solitons and fractals". key: cord-299810-e57pwgnx authors: martelloni, gabriele; martelloni, gianluca title: modelling the downhill of the sars-cov-2 in italy and a universal forecast of the epidemic in the world date: 2020-07-01 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110064 sha: doc_id: 299810 cord_uid: e57pwgnx in a previous article [1] we have described the temporal evolution of the sars-cov-2 in italy in the time window february 24-april 1. as we can see in [1] a generalized logistic equation captures both the peaks of the total infected and the deaths. in this article our goal is to study the missing peak, i.e. the currently infected one (or total currently positive). after the april 7 the large increase in the number of swabs meant that the logistical behavior of the infected curve no longer worked. so we decided to generalize the model, introducing new parameters. moreover, we adopt a similar approach used in [1] (for the estimation of deaths) in order to evaluate the recoveries. in this way, introducing a simple conservation law, we define a model with 4 populations: total infected, currently positives, recoveries and deaths. therefore, we propose an alternative method to a classical sird model for the evaluation of the sars-cov-2 epidemic. however, the method is general and thus applicable to other diseases. finally we study the behavior of the ratio infected over swabs for italy, germany and usa, and we show as studying this parameter we recover the generalized logistic model used in [1] for these three countries. we think that this trend could be useful for a future epidemic of this coronavirus. we briefly review the historical evolution of the sars-cov-2 in the earth. in early december the sars-cov-2 appeared in wuhan, china. the disease caused by the new coronavirus has a name: "covid-19" (where "co" stands for corona, "vi" for virus, "d" for disease and "19" indicates the year in which it occurred). the oms director-general tedros adhanom ghebreyesus announced it on february 11, 2020, during the extraordinary press conference dedicated to the virus. the appearance of new pathogenic viruses for humans, previously circulating only in the animal world, is a widely known phenomenon (called spill over) and it is thought that it may also be at the basis of the origin of the new coronavirus (sars-cov-2). the scientific community is currently trying to identify the source of the infection. on december 31, 2019, the municipal health commission of wuhan (china) reported to the oms a cluster of cases of pneumonia of unknown etiology in the city of wuhan, in the chinese province of hubei. on january 9, 2020, the chinese center for disease prevention and control (cdc) reported that a new coronavirus (initially called 2019-ncov and now called sars-cov-2) has been identified as the causative agent and has been rendered publishes the genomic sequence. oms on march 11, 2020 declared that covid-19 can be defined as a pandemic. after notification of the epidemic by china, italy immediately recommended postponing unnecessary flights to wuhan and, subsequently, with the spread of the epidemic, to all of china. consequently, the latter has canceled all flights from wuhan. this disease does not save italy that has become a protected area with the dpcm signed on the evening of 9 march by the prime minister, giuseppe conte, who has extended the restrictive measures already applied for lombardy and the 14 northern provinces most affected by the coronavirus infection to the whole national territory. the new action comes into force on march 10 and will take effect until april 3. among the main innovations: it limits the movement of people, blocks sporting events, suspends teaching activities in schools and universities throughout the country until april 3. with the new ordinance of 22 march 2020 issued by the minister of health and the minister of the interior, from 22 march people are prohibited from moving with public or private trasportation in a municipality other than that in which they are located, except for proven work needs, absolute urgency or for health reasons. [2] many growth models have been very recently applied to study the evolution of the covid-19 infection [3, 4, 5, 6, 7, 8, 9, 10] . in [1] we tried to analyze the time evolution of the sars-cov-2 in italy, using a logistic model [11] at the beginning of the study and after with a generalization of that model. the logistic behaviour assumes that growth stops when maximum sustainable population density is reached through the carrying capacity k that depends on the environmental conditions. for example the ordinances of the prime minister g.conte, the people's hygiene habits are encoded in the carrying capacity k. we observe as the generalized model of [1] works very well until the april 7. after this date the large increase in the number of swabs meant that the logistical behavior of the infected curve no longer worked. at first in italy, pharyngeal swabs were initially made only on seriously ill people. this choice gave us the possibility to have a sample of the infected that we can describe with a single population model, after april 7 it becomes impossible. so we decided to use a different model to describe the new trend of the data and try to give different scenarios of the descent phase of the virus in italy, in the time window february 24-may 5. in [1] we described two different peaks, the peak of the infected and the deaths one. in this paper we analyze the peak of the currently infected and the downhill of the propagation of the sars-cov-2. to do this we define a new model similar to a sird (see for example [7] ), but without the population of supsceptibles, because there are no criteria on defining the susceptible ones. we consider three couple differential equations for infected i(t), deaths d(t) and recovery r(t) with the following conservation law where p (t) represents the currently infected (or positive). in the last part of this article we observe as the following ratio (infected i(t i ) over swabs s(t i )) is the most important parameter to describe the evolution of the sars-cov-2. indeed, we can describe the trend of this quantity only with a generalized logistic model with 4 parameters even with data after april 7. this behavior suggest us to use this model for a future epidemic of this virus. if we will able to perform a greater and constant number of swabs everyday, using this model, we may have better control over the contagion curve, and consequently over the number of deaths. our idea is to use a model that adapts to the data of the problem. we explain better. let's consider the following data: some comments about these data: the points 1) and 2) describe perfectly that the sample of infected is not clean; at the beginning of the contagion the swabs are performed only on the severe infected, after 1 month the number of swabs are increased of a factor 6 and consequently also the midly infected are detected. point 3) tells us that there is probably an incredible number of asymptomatics as a source of severe infected, we have no control about it. points 4), 5) indicate that while the death data is under control, the healed data are very oscillating in time. finally the points 6) tells us that contribution of asymptomatics, portrayed in [1] , changes in time, indeed from april 6-7 (14-15 days after the second ld, i.e. an incubation time ) the generalized logistic description fails. after these considerations we have decided to couple the following equations: with a conservation law where p (t) represents the currently infected (or positive). the parameters r 0 represents the rates of growth of epidemic, k is the carrying capacity for the classical logistic model, α is a constant in order to have a power low initial growth before ld, β is the exponent of the second term of equation 1 that represents the influence of asymptomatic; δ,a correction of the quadratic term of logistic, and γ are the constant parameters considering the influence of the government measures 1 , k f is a proportionality constant between deaths and total number of infected, while t d and t r are the delays of deaths and recoveries respect to infected respectively; the constant a represents the contribution of asymptomatic people as introduced in [1] and finally t 0 is the time of ld start. a brief consideration about the function f (t): the great variability of t r suggest us that only the parameter t r is not sufficient to describe correctly the function r(t), so we decided to introduce a coefficient time dependent. we present two different scenarios, in fig.1 we consider a linear approximation f (t) = a + bt, while in fig.2 we consider a quadratic approximation f (t) = a + bt + ct 2 . this choice is not random. indeed, considering the behaviour of the recovery time series in which a single recovery can heal with some delay in a window variable from few days to two months, the correct modeling could be a regressive linear function of type r(t) = n i=1 a i * i(t − t i ) (eventually introducing also no-linear term in the series), but in this way we introduce many degree of freedom how many are the coefficient a i of the regressive function. therefore, we consider an approximation using the two functions f (t) considered above. we desumed the following values for the principal parameters by means of 100 stochastic simulation using direct method gillespie algorithm adapted to nonautonomous differential equations: t r = 12 ± 1 for quadratic approximation. some comments about these values: with respect to [1] we observe that the total number of infected, positive, recovery and dead 10 5 figure 2: the scenario with a quadratic growth for the recoveries: the black curve represents the deaths, the red one for the infected, the green one for the recovery and the pink one for the currently infected. peak of the severe infected is correctly estimated, i.e. t 0 = 24 − 26 march and also the peak of the deaths, i.e. t 0 = 28 − 30 march; also the time delay t d remained the same; the same t r approaches the experimental lower limit in quadratic approximation. with respect to the logistic model r 0 is increased while the coefficient δ drops from the value 2 to the value 1.84, i.e. we are considering different models. in fig. 1-2 we observe as the peak of currently infected is close to april 20 and finally we give us our prevision for a linear growth for the recoveries close to july 10; for a quadratic growth we have close to june 20. the estimated numbers i(end) and d(end) are very close, but it is not surprising: the eqs. (3) and (4) for total infected head the model, while f (t) is present in eq. (6) that is only a proportionality equation. obviously a linear approximation for f(t) leads to a slower recovery curve and therefore a small increase of infected. now we consider the following parameter: that represents the number of infected normalized with the number of swabs s(t i ). we study this quantity with generalized logistic equation used in [1] : where α, r 0 , k and a have the same meaning used in the previous section. compared to the previous section we observe as studying the parameter i norm (t i ) we can describe the contagion with a simple logistic equation and without the phenomenological terms introduced in eqs.(3)(6) . in order to calibrate this model in the best way possible we use two algorithms, the first one based on simulated annealing [16] and the second one on optimized simplex [17] . we evaluate the function error defined as where x i is the real data at day i, y i (p) is the correspondent output of the model depending of vector parameter p and w i is a generic weight that we can use or can be equal to one. for our purpose we adopt as weight the derivative of data or the data at time (day) i: the use of derivative allows to calibrate better on average the curve, while the use of the data as weight permit to calibrate better the data of the last part of the curve. in fig. 3-4 cumulative rate real data model error +5% error -5% figure 6 : the scenario of italy minus lombardia with the derivative weight. for the parameters of fig. 4 r 0 = 0.178 ± 0.015, for the parameters of fig. 5 r 0 = 0.143 ± 0.015, and finally for the parameters of fig. 6 r 0 = 0.143 ± 0.015, we observe as the quantity i norm (t) is probably the most important quantity studying the evolution of the virus! we explain better: the contribution of asymptomatic people is essentially the same in lombardia and in the rest of italy, while the coefficient r 0 is larger if we consider italy compared to the scenario of italy minus lombardy; this consideration is extremely coherent with the data: the infected of lombardia region represent the 37% of all the italian infected. moreover the ratio infected over swabs is a very reliable parameter, we can describe correctly the italian situation only with 4 parameter and with a wellknown model. we stress that in the future if a nation is ready to carry out a large and constant number of swabs every day, using this model, we can have a reliable forecast of the epidemic! we consider also the scenario represented by eq. (19) for germany in fig. 7 and for usa in fig. 8 . for germany we study the time evolution of the sars-cov-2 in the time window march 8-may 11 and we obtain the following parameters for usa we study the contagion in the time window march 10-may 11 and we have these values for the parameters speed. let's try to justify this idea: a different speed may depend on population density, work habits and the number of swabs at the beginning of the epidemic. about the last consideration we imagine to immediately carry out a large number of swabs: knowing as soon as possible the largest possible number of infected means limiting the contagion and therefore the propagation speed of virus. we described the evolution of the sars-cov-2 in italy in the time window february 24-may 5. to do this we have built a phenomenological growth model adapted on the data of civil protection. with respect to a classical sir(d) model we did not consider the supsceptible population, because there are not medical evidences on which sample of the population can be ill. so we have considered three couple differential equations for infected i(t), deaths d(t) and recovery r(t) with the a conservation law including the currently positive population p(t). as the time delay between the onset of symptoms and healing t r days is a very oscillating parameter we introduced a sort of regressive function f (t) to modelling better this delay. so we described two scenarios of the end of epidemic: • i(end) = 247471, d(end) = 35235, close to july 10, for f (t) linearly approximated, • i(end) = 243766, d(end) = 34682, close to june 20, for f (t) in a quadratic approximation. obviously a linear approximation for f(t) leads to a slower recovery curve and therefore a small increase of infected. in the second part of this manuscript we described the time evolution of the normalized data that represents the number of infected normalized with the number of swabs s(t i ). we have studied this parameter on four different scenarios: • italy, • data of italy minus data of lombardia ( about 37% of the italian infected belong to the lombardia region ), • usa, • germany. so we have found that all the evolutions are governed by the same generalized logistic equation [1] , suggesting an universal feature of the propagation of sars-cov-2 virus. in particular the value of the parameter r 0 is in descending order compatible with the respective apparent cfr ( acfr ) • for italy r 0 = 0.175 and acf r = 14%, • for italy-lombardia r 0 = 0.143 and acf r = 11%, • for usa r 0 = 0.082 and acf r = 6%, • germany r 0 = 0.069 and acf r = 4, 5%. finally we suggest that the data i norm (t i ) is the most important parameter to control the propagation of the virus for a new inauspicious propagation of this virus in the world, because, knowing its universal feature, we can forward know the number of infected preparing a relevant number of swabs. analysis of the evolution of the sars-cov-2 in italy, the role of the asymptomatics and the success of logistic model early phylogenetic estimate of the effective reproduction number of sars-cov-2 emerging coronaviruses: genome structure, replication, and pathogenesis data analysis on coronavirus spreading by macroscopic growth laws covid19: an automatic, semiparametric estimation method for the population infected in italy analysis and forecast of covid-19 spreading in china a poisson autoregressive model to understand covid-19 contagion dynamics, ssrn -abstract-id=3551626 cdc covid-19 response team, severe outcomes among patients with coronavirus disease 2019 (covid-19) -united states how macroscopic laws describe complex dynamics: asymptomatic population and covid-19 spreading notice sur la loi que la population poursuit dans son accroissement on the nature of the function expressive of the law of human mortality and a new mode of determining life contingencies the simplex-simulated annealing approach to continuos non-linear optimization libelli parameter estimation of ecological models we thank many colleagues for interesting discussions, in particular andrea marzolla and domenico seminara. we also thank pierluigi blanc, s.o.c. infectious diseases 1 santa maria annunziata hospital, for stimulating discussions on technical subjects on which we had no knowledge.the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. key: cord-291227-dgjieg7t authors: mandal, manotosh; jana, soovoojeet; nandi, swapan kumar; khatua, anupam; adak, sayani; kar, t.k. title: a model based study on the dynamics of covid-19: prediction and control date: 2020-05-13 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109889 sha: doc_id: 291227 cord_uid: dgjieg7t as there is no vaccination and proper medicine for treatment, the recent pandemic caused by covid-19 has drawn attention to the strategies of quarantine and other governmental measures, like lockdown, media coverage on social isolation, and improvement of public hygiene, etc to control the disease. the mathematical model can help when these intervention measures are the best strategies for disease control as well as how they might affect the disease dynamics. motivated by this, in this article, we have formulated a mathematical model introducing a quarantine class and governmental intervention measures to mitigate disease transmission. we study a thorough dynamical behavior of the model in terms of the basic reproduction number. further, we perform the sensitivity analysis of the essential reproduction number and found that reducing the contact of exposed and susceptible humans is the most critical factor in achieving disease control. to lessen the infected individuals as well as to minimize the cost of implementing government control measures, we formulate an optimal control problem, and optimal control is determined. finally, we forecast a short-term trend of covid-19 for the three highly affected states, maharashtra, delhi, and tamil nadu, in india, and it suggests that the first two states need further monitoring of control measures to reduce the contact of exposed and susceptible humans. more significant compare to quantitative analysis. hence a suitable math-27 ematical model would not only able to represent the whole disease system 28 but also the study of the model would undoubtedly derive the precise nature 29 of the disease. it may forecast the behavioral aspect of the disease shortly. 30 although the primitive mathematical models on theoretical epidemiology 31 (see bernoulli [1] , hamer[13] , ross[40] . kermack according to the information received, it may take around one week to two 41 weeks for the exposure of symptoms of covid-19 of an infected person, 42 although during this period, that person able to infect other susceptible per-43 sons. however, there may be some infected persons whose infection is so mild 44 that the person would recover due to innate immunity even before the hos-45 pitalization. thus in this article, by the term 'infected' person, we will mean 46 those persons who are hospitalized. further, we assume that the medical 47 personals assisting covid-19 positive hospitalized individuals have taken 48 necessary protective items. thus to keep simplicity, we believe that only 49 exposed persons and asymptomatic infected persons can spread the disease. since here, we have assumed that the virus of covid-19 is spreading when 122 a vulnerable person comes into contact with an exposed person; therefore 123 we think that ρ 1 (0 < ρ 1 < 1) portion of susceptible human would maintain 124 proper precaution measure and ρ 2 (0 < ρ 2 < 1) portion of the exposed class 125 would take proper precaution measure for disease transmission (i.e., use of 126 face mask, social distancing and implementing hygiene). therefore the dis-127 ease can only be transmitted to the (1−ρ 1 )s portion of susceptible individuals 128 due to the contact of (1−ρ 2 )e portion of exposed individuals with a bi-linear 129 disease transmission rate β. we know that a person is whether infected by 130 the sars-cov-2 virus or not can be clinically detected using rt-pcr ex-131 amination and a person with negative results in the rt-pcr test may still 132 be covid-19 positive as it may take some days (from 7 to 21 days) to ex-133 press infection. therefore, the portion with positive covid-19 of the class 134 of population e is considered as infected, and they are hospitalized. let α 135 and b 2 be the portions of the exposed class goes to the infected class and 136 quarantine class, respectively. it should be noted that 0 < α + b 2 < 1 since 137 it would take quite a long time to get the output of the rt-pcr test, and 138 sometimes it requires more than one rt-pcr analysis for a single person for confirmation of covid-19. let among the quarantine classes of populations, 140 cq portion of communities move to infected level, and the b 1 q part would 141 become susceptible to the disease after the quarantine period. let η and σ 142 be respectively recovery rate of the hospitalized infected populations i and 143 exposed class e. let d be the natural death rate, which is common to all 144 classes of communities and δ be the covid-19 induced death rate. also, it 145 is statistically observed a person once recovered from the disease covid-19 146 has very little chance to become infected again for the same disease. hence, 147 we assume that no portion of the recovered population moves to the susconsisting of five first order differential equations shown as below: proof we assume that p = s + e + q + i + r. integrating the above inequality and by applying the theorem of differential equation due to birkhoff and rota [2], we get now for t → ∞, hence all the solutions of (1) that are initiating in {r 5 + } are confined in the region for any > 0 and for t → ∞. hence the theorem. interval" see (van den driessche and watmough [6] ). therefore the dimen-195 sionless quantity r 0 refers as the expectation of the spreading disease. there are several techniques are available for the evaluation of r 0 for an 197 epidemic spread. in our present research article we use the next generation 198 matrix approach [5, 9, 22] . now the classes which are directly involved for 199 spread of disease is only e, q, i. therefore from system (1) we have (2) the above system can be written as dy dt = φ(y) − ψ(y), equilibrium. now the jacobian matrix of φ and ψ at the disease free equilib-204 rium are respectively given by, the basic reproduction number (r 0 ) is the spectral radius of the of the matrix 207 (f v −1 ) and for the present model it is given by 3.3. equilibria the system has two possible equilibria. one is disease free equilibria 210 where infection vanishes from the system. it is given by where infection is always present in the system is called endemic equilibria, note it is observed from the expression of the above two equilibrium point 217 is that the disease free equilibrium e 0 is always feasible but the endemic theorem 3.2. the disease free equilibrium e 0 is locally asymptotic stable if proof. the jacobian matrix at the disease free equilibrium of the system (1) is given by now the characteristic equation of the system (1) at its disease free equilib-227 rium is given by (4) clearly all the eigen value of the jacobian matrix are negative if and only if 229 r 0 < 1. hence the system is locally asymptotically stable if r 0 < 1 and it is 230 unstable if r 0 > 1. hence the theorem. note here we see that the disease free equilibrium e 0 losses its stability 232 when the r 0 increases to its value greater than 1. so, we may conclude that 233 at r 0 the system (1) passes through a bifurcation around its disease free 234 equilibrium which are discussed in the next theorem.. theorem 3.3. the system (1) passes through a transcritical bifurcation 236 around its disease free equilibrium when r 0 = 1. proof. from the above analysis, it has been observed that when r 0 < 1 238 between the two equilibria, only the disease free equilibrium exists and lo-239 cally asymptotically stable where as r 0 > 1 is the threshold condition for proof. the jacobian matrix for the system (1) is given by it is clear from (5) that first two root are negative real and remaining roots 260 are the roots of the cubic polynomial. it is also observe that here c 1 , c 2 , 261 c 3 and c 1 c 2 − c 3 all are positive for any parametric value. hence following 262 the routh-hurwitz criterion we may conclude that the system (1) is locally 263 asymptotically stable around its endemic equilibrium e 1 . subject to the proposed model (1) . the parameters c 1 and c 2 corresponds as the weight constraints for the infected population and the control respectively. here the objective functional is linear in the control with bounded states. therefore it can be be showed by using standard results that an optimal control and corresponding optimal states exist [8] . now we need to find out the value of the optimal control m * (t) such that here we use the pontryagin's maximum principle [8, 28, 37 ] to derive the 275 necessary conditions for our optimal control and corresponding states. the lagrangian is given by the hamiltonian is defined as follows we minimize the hamiltonian with respect to the control variable m * (t). using the equations of the system (2) and (5), we obtain (13) we observe that the control parameter m does not explicitly occur in the 300 above expression, so next we calculate the second derivative with respect to 301 time. where using the state and co-state equations of systems (1) and (9), we simplify 304 the equation (14) and finally obtain the above equation can be written in the form and then we can solve the singular control as moreover in order to satisfy the generalized legendre-clebsch condition for the singular control to be optimal, we require d dm d 2 dt 2 ∂h ∂m = φ 1 (t) to be negative [25] . therefore we summarize the control profile on a nontrivial interval in the following way: hence the control is optimal provided φ 1 (t) < 0 and a ≤ − φ 2 (t) φ 1 (t) ≤ b. we study numerical results in two different cases, first for fixed control 308 and second when the control has been applied optimally. first, we consider 309 the values of parameters in table 1 , for numerical simulations. since δ is the 310 disease induced mortality rate and d is the natural death rate, hence δ > d. using these parameters and the initial conditions as s(0) = 500, e(0) = r 0 with respect to the relative change in its parameter ( table 1 ) . f (x 1 , x 2 , · · · , x n ) , for the parameter, to find the sensitivity of r 0 , we consider the parameters a, β, ρ 1 , ρ 2 , α, d, p, m, b 2 , σ 334 as r 0 is the functions of these parameters. the sensitivity index of r 0 with re-335 spect to the parameter β is given by similarly, we can find the sensitivity indices of r 0 with respect to the other 337 parameters. positive index indicates that r 0 is an increasing function of the corre-339 sponding parameter and negative index implies that r 0 is a decreasing func-340 tion of that parameter. for example, as γ r 0 β = 1 , it shows that if β is in-341 creased by 10% then the r 0 is also increased by 10%. again, as γ r 0 d = −0.409 342 implies that 10% increment in d will decrease r 0 by 4.09%. from table 2 , increases, which has been demonstrated in fig. 7 . 353 we know that the numerical value of basic reproduction number r 0 deter-354 mines the exact nature of the disease. from the table 1 and fig. 5, fig. 6 , we fit the proposed model (1) to the daily active infected, confirmed 413 (cumulative) infected, and recovered covid-19 cases in those three states 414 of india using the set of parameters as given in table 4 and the initial size of 415 the population from the table 5 . to fit these real data, we use the software 416 mathematica and then predict the behavior of covid-19 for those three 417 states on a short term basis. in fig. 11, fig. 12, and fig. 13 , we respectively 418 present the active covid-19 cases in maharashtra, delhi, and tamil nadu 419 for 91 days starting from 2nd march, 2020, till the 31st may 2020. also, in 420 fig. 14, fig. 15 and in fig. 16 , we present the cumulative confirmed (i.e., the 421 sum of active cases, recovered and death) covid-19 cases of maharashtra, 422 delhi, and tamil nadu, respectively, for the same period. 2 19 48 0 0 48 14 0 0 14 3 0 0 3 20 52 0 0 52 14 0 0 14 3 0 0 3 21 64 0 0 64 18 0 0 18 6 0 0 6 22 74 0 0 74 26 0 0 26 7 0 0 7 23 95 0 0 95 29 0 0 29 8 0 0 8 24 104 0 0 104 30 0 0 30 14 0 0 14 25 124 1 3 128 35 0 0 35 21 0 0 21 26 120 6 3 129 38 0 0 38 27 0 0 27 27 111 15 4 130 38 0 0 38 36 0 0 36 28 150 25 5 180 38 0 0 38 39 2 0 41 29 155 25 6 186 47 0 0 47 47 3 0 50 30 165 25 8 198 95 0 0 95 62 4 0 66 31 168 39 9 216 95 0 0 95 117 6 0 123 here, in both table 3a and 3b, the phrases 're' and 'conf' represents recovered and confirmed infected class respectively. parameter r 0 and found the most sensitive parameter, which has a positive 448 impact on r 0 is the disease transmission rate. the primary finding of this article is that we have derived a mathemati-484 cal model that can be used to study the qualitative dynamics of covid-19. the basic reproduction number and its sensitivity analysis would determine 486 the controlling procedure of the disease. also, we have incorporated the gov using the software mathematica, we try to fit our model (1) to references 528 essai dune nouvelle analyse de la mortalite causee par la 529 petite verole. mem. math. phy. acad. roy. sci. paris reproduction numbers and sub-threshold 543 endemic equilibria for compartmental models of disease transmission analysis and forecast of covid-19 spreading in 546 deterministic and stochastic optimal control the meta population dy-550 namics of an infectious disease: tuberculosis in possums 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singular controls for a general 593 sir-model with vaccination and treatment. discrete and continuous dy-594 namical systems optimal control applied to biological models. 596 crc press zhange 598 wy. hemographic fever with renal syndrome in china: mechanism on 599 two distinct annual peaks and control measures the reproductive number 602 of covid-19 is higher compared to sars coronavirus optimal control of epidemiological seir mod-605 els with l1-objectives and control-state constraints. hal-01101291 2014. 606 [32] ministry of health and welfare, government of india transmission potential of the novel coronavirus 609 (covid-19) onboard the diamond princess cruises ship. infectious dis-610 ease government 612 of india mathematical mod-614 eling of covid-19 transmission dynamics with a case 615 study of wuhan official updates coronavirus, covid-19 in india, government of india the 620 mathematical theory of optimal processes the 622 effect of control strategies to reduce social mixing on covid-19 epidemic in wuhan, china: a modelling study. the lancet 624 public health short-626 term forecasting covid-19 cumu-lative confirmed cases: 627 perspectives for brazil an application of the theory of probabilities to the study 630 of a priori pathometry: part i an epidemic model in a patchy environment the role of animal grazing in th spread of 635 chagas disease optimal vaccination policies for an 637 sir model with limited resources credit author statement manotosh mandal: conceptualization, methodology, software, formal analysis, investigation, validation, writing -original draft soovoojeet jana: conceptualization, validation, methodology, writing -original draft, writing -review & editing, visualization swapan kumar nandi: validation, formal analysis, investigation anupam khatua: investigation, resources, data curation sayani adak: resources, data curation t. k. kar: conceptualization, writing -review & editing, visualization 01 254 39 9 302 144 6 2 152 227 6 1 234 02 280 42 13 335 207 8 4 219 227 6 1 234 03 277 42 16 335 207 8 4 219 302 6 1 309 04 424 42 24 490 424 15 6 445 403 6 2 411 05 424 42 24 490 478 18 7 503 476 6 3 485 06 647 56 45 748 497 19 7 523 558 8 5 571 07 764 56 48 868 548 21 7 576 608 8 5 621 08 875 79 64 1018 546 21 9 576 664 19 7 690 09 946 117 72 1135 639 21 9 669 709 21 8 738 10 1142 125 97 1364 660 25 13 698 805 21 8 834 11 1276 188 110 1574 684 25 14 723 859 44 8 911 12 1426 208 127 1761 1025 25 19 1069 915 44 10 969 13 1619 217 149 1985 1103 27 24 1154 1014 50 11 1075 14 1948 229 160 2337 1452 30 28 1510 1104 58 11 1173 15 2250 259 178 2687 1501 30 30 1561 1111 81 12 1204 16 2437 295 187 2919 1504 42 32 1578 1110 118 14 1242 17 2711 300 194 3205 1551 51 38 1640 1072 180 15 1267 18 2791 331 201 3323 1593 72 42 1707 1025 283 15 1323 19 3075 365 211 3651 1779 72 42 1893 992 365 15 1372 20 3473 507 223 4203 1668 290 45 2003 1051 411 15 1477 21 3865 572 232 4669 1603 431 47 2081 1046 457 17 1520 22 4248 722 251 5221 1498 611 47 2156 943 635 18 1596 23 4594 789 269 5652 1476 724 48 2248 949 642 18 1629 24 5307 840 283 6430 1518 808 50 2376 911 752 20 1683 25 5559 957 301 6817 1604 857 53 2514 867 866 22 1755 26 6229 1076 323 8068 1702 869 54 2625 838 960 23 1821 27 6538 1188 342 8068 1987 877 54 2918 841 1020 24 1885 the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. key: cord-328069-a9fi9ssg authors: pathan, refat khan; biswas, munmun; khandaker, mayeen uddin title: time series prediction of covid-19 by mutation rate analysis using recurrent neural network-based lstm model date: 2020-06-13 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110018 sha: doc_id: 328069 cord_uid: a9fi9ssg sars-cov-2, a novel coronavirus mostly known as covid-19 has created a global pandemic. the world is now immobilized by this infectious rna virus. as of may 18, already more than 4.8 million people have been infected and 316k people died. this rna virus has the ability to do the mutation in the human body. accurate determination of mutation rates is essential to comprehend the evolution of this virus and to determine the risk of emergent infectious disease. this study explores the mutation rate of the whole genomic sequence gathered from the patient's dataset of different countries. the collected dataset is processed to determine the nucleotide mutation and codon mutation separately. furthermore, based on the size of the dataset, the determined mutation rate is categorized for four different regions: china, australia, the united states, and the rest of the world. it has been found that a huge amount of thymine (t) and adenine (a) are mutated to other nucleotides for all regions, but codons are not frequently mutating like nucleotides. a recurrent neural network-based long short term memory (lstm) model has been applied to predict the future mutation rate of this virus. the lstm model gives root mean square error (rmse) of 0.06 in testing and 0.04 in training, which is an optimized value. using this train and testing process, the nucleotide mutation rate of 400(th) patient in future time has been predicted. about 0.1% increment in mutation rate is found for mutating of nucleotides from t to c and g, c to g and g to t. while a decrement of 0.1% is seen for mutating of t to a, and a to c. it is found that this model can be used to predict day basis mutation rates if more patient data is available in updated time. the whole world is suffering by an ongoing pandemic due to coronavirus disease brought about by severe acute respiratory syndrome coronavirus 2 (sars-cov-2) [1] . it was an outbreak from wuhan, the capital of hubei province in china during december 2019 [2] . the virus was identified on 7th january and observed that it is spread by human-to-human transmission via droplets or direct contact [3, 4] . its infection has been estimated to be a mean incubation period of 6.4 days and a basic reproduction number of 2.24-3.58. since its identification, it has already been spread speedily over the whole globe, therefore who had declared covid-19 a global pandemic on 11th march 2020 [5] . the sars-cov-2 is a pathogenic human coronavirus under the betacoronavirus genus. in the recent decade, the other two pathogenic species sars-cov and mers-cov were outbreaks in 2002 and 2012 in china and the middle east, respectively [6] [7] [8] [9] . the complete genomic sequence (wuhan-hu1) of this large rna virus (sars-cov-2) was first discovered in the laboratory of china on 10th january [10] and placed in the ncbi genbank. the sars-cov-2 is a single positive-stranded rna virus having non-segmented in nucleic acid sequence [11] . although it is an rna virus but for simplicity of understanding the gene sequence has been given as dna type which means nucleobase uracil (u) has been replaced by thymine (t). the genomic sequence of sars-cov-2 virus shows about 79% and 50% similarity with the sar-cov and mars-cov, respectively [6] . sars-cov-2 performs mutation during replication of genomic information [12] . the mutation occurs due to some errors when copying rna to a new cell. mutations are mainly three kinds: base substitution, insertion, and deletion. further, in base substitutions, there are some more divisions: silent, nonsense, missense, and frameshift [13] . micro-level alteration of mutation rate is also detectable for virus infection in host immune systems and drastically change the virus characteristic and virulence. to understand viral evolution, the mutation rate is one of the crucial parameters [14] . furthermore, it is one of the most important factors for the assessment of the risk of emergent infectious disease, like due to sars-cov-2. therefore, an accurate estimation of this parameter finds a great significance [15, 16] . in connection to this and following the current pandemic, many researchers and scientists are working relentlessly to understand the evolution of sars-cov-2. asim et. al have performed phylogenetic analysis of sars-cov-2 virus based on the spike gene of the genomic sequence [17] . in this study, they described a detailed genomic sequence of the sars-cov-2 virus. they identified the factor of endemicity of sars-cov-2 and then focused to find out the next reservoir of the sars-cov-2 virus. based on the case study, the authors reported that all sequence of this virus is constituted in a single cluster without making any branching on this ongoing pandemic but not validated the findings with detailed statistical analysis. an analysis on gene signature of sars-cov-2 virus has been performed by ranajit and sudeep [18] . they estimate the ancestry rate of the european genome from the reference population by applying a statistical tool qpadm. then they applied pearson's correlation coefficient between various ancestry rate of european genome and performed statistical analysis on death/recovery ratio by using graphpad prism v8.4.0, graphpad software. in this study, they developed different linear regression models. finally, they performed genome-wide association analyses (gwas) among european and east asian genomes to examine single-nucleotide polymorphism (snp) which is correlated to the infection of the sars-cov-2 virus. from the snp association, they observed a huge difference in allele frequencies between european and eastern asian countries. debaleena et al. analyzed the statistical changes of signature from different variant of sars-cov-2 virus [19] . they calculated diversity, non-synonymous, synonymous, and substitution rates for each gene of the nucleotide sequence by using dnasp. they also employed time zone software for phylogenetic analysis and mutation detection for each gene. after that, they compared the sequence alignment of a protein of wuhan and india by using multiple sequence alignment. note that in their study, the mutation rate was not calculated based on the patient's genomic sequence. however, the contemporary literature shows adequate studies on the genomic sequence but very few studies on the mutation rate. therefore, the present study is designed to perform the mutation rate prediction for sars-cov-2 on the basis of the time series. unfortunately, the current data shows that the sars-cov-2 virus is highly infectious than the other harmful species of pathogenic human coronaviruses [20] . world populations are now suffering and are in great anxiety by observing the deadliest effect of this virus. but what can be done to stay healthy or avoid getting infected with the virus is still undiscovered. to stop sars-cov-2 virus, there is a critical need to invent proper vaccine and antibody based therapy against this virus [22] . scientists and researchers are trying their best to discover suitable drugs or vaccines to neutralize the effect of this virus on the human body, or at least in helping to create an effective resistance against the spreading out of this virus. for inventing proper drugs and vaccines against covid-19 rna viruses, genomic sequence and mutation analysis are crucially required. in fact, the viral mutation rate also plays a role in the assessment of possible vaccination strategies. in this regard, we performed a detailed study on the mutation rate of this virus using the available dataset in the ncbi genbank. from this dataset, we have analyzed the genomic sequence of 3408 patients from different countries for a period of 12 th january to 11th may 2020. we focus specifically on the mutations that have developed freely on different dates (homoplasies) as these are likely possibilities for progressing adjustment of sars-cov2 to its novel human host. specifically, we have calculated the base substitution mutation rates. due to the lack of necessary information for insertion and deletion, we have considered those as substitution mutations to ensure that no nucleotide goes out of count. it is expected that the present analysis would help to understand the changing behavior of this virus in the human body and set up strategies to combat the epidemiological and evolutionary levels. an adequate amount of gene dataset is currently available in the ncbi genbank which has the complete genome sequence of sars-cov-2. among the many entities, we have filtered the gene sequence, date of collection, and country of the sample. all genes are taken from the human body who are affected by covid-19. there are genes from almost 33 countries but china, australia and the united states has a considerable number of patients' data. though some countries like england, italy, france, spain, and brazil has a very high mortality rate but for the lack of data availability in the ncbi genbank till 15 th may, we were unable to calculate the mutation rates for these countries separately. therefore we have considered these countries along with others those have low gene data sequence available in the genbank as the rest of the world category to cover as much region as possible. table 1 shows the information about the gene dataset. in this dataset, there are also some partial genes. so we filtered them and take only with the level of the complete genome. there is a reference gene sequence of length 29903. finally, we have filtered the dataset by taking a maximum gene length of 29903 and a minimum of 29161 and avoided the copy sequences. with this filtering process, the total number of patients come down to 3068 from 3408, patients from china come down to 40 from 86, the united states come down to 1903 from 2103 and australia come down to 918 from 925. following the size of the available dataset, the mutation rate calculations have been set for four categories: china, united states (usa), australia and the rest of the world. furthermore, the dataset is arranged in a suitable way to separately calculate the nucleotide mutation and codon mutation. the first filtered dataset is to find the nucleotide mutation rate. then we have converted the four raw nucleotides (a=adenine, t=thymine, c=cytosine and g=guanine) into codon set. a codon consists of three nucleotides and forms a unit of genetic code in dna or rna. information given in table 2 is used to convert the gene sequence by their index number. for example, if three consecutive nucleotides are 'ttt' then it will be converted to 1, 'gct' is converted into 53, and so on. the conversion process has been shown in figure 1 . this process is important to understand the mutation in the codon sequence of covid-19. also, it helps to lower the computational complexity. gene mutates for many reasons. when rna tries to copy genetic codes form dna it may cause some error which causes mutation. also, errors in dna replication, recombination, and chemical damage in dna causes mutation. there are basically three types of mutations: base substitutions, deletions, and insertions. from this dataset, we can find out the three kinds of substitution mutation which are silent, missense, and nonsense. a silent mutation is the change of codon by which the resulting amino acid remains unchanged. if the resulting amino acid changes then it is called a missense mutation. on the other hand, when changing codon produces the stop signal for gene translation which causes a nonfunctional protein then it is called a nonsense mutation. these three types of substitution mutation of the observed dataset have been shown in figure 2 , where the missense rate is 34.3%, the nonsense mutation rate is 6.7% and the silent mutation rate is 0.8%. if the mutation type is missense then it can be said that the change of nucleotide has affected the protean generation, which may change the behavior of the virus. also, it is hard to identify the cure's gene sequence. the missense nucleotide mutation rate has been calculated by the given algorithm in figure 3 . after using this algorithm equation 1 has been used to convert the values in percentage. here, mutationrate is the final output array, mutation is the output array sized 4×4 containing raw values after applying the algorithm, lg is the length of a dataset which is 3068 for the full dataset, 40 for china, 918 for australia and 1903 for the usa, gs is the length of reference gene sequence which is 29903 in this dataset. in this process, we have calculated the nucleotide mutation rate for the prepared dataset. the mutation rate for china has been shown in figure 4 (a). it shows that a huge percent of thymine (t) are being mutated to other nucleotides but not producing the same amount of t again. also, a huge amount of adenine (a) is mutated to other nucleotides. comparing to t and a, cytosine (c) and guanine (g) were not changed much. after that, the mutation rate has been calculated for australia and the usa, and shown in figures 4(b) and 4(c). this is clear that all rates have a common factor of having the high mutation rate of t and a. but there is a significant increase in the mutation rate compared to china. this clearly indicates that this virus is very much active in changing its gene sequence. finally, the nucleotide mutation for the full dataset of 33 countries has been shown in figure 4 (d). it shows that c and g mutation rates are almost equal to the usa because there are more data of usa than any other countries. but some changes in t and a can be seen for the dataset for the rest of the world. these values vary on the availability of the data from different countries. the second processed and converted dataset that were prepared previously has been used here to calculate the codon mutation rate, and shown in figure 5 . changes in nucleotide cause changes in codon set, which later affects the protean directly. we have used the same algorithm shown in figure 3 for detecting the codon mutation rate. a small change has been made in the receiving array where array size was 4×4 for nucleotide but here the array size is 64×64 for codon mutation. after finding the codon mutations, equation 2 has been used to get the rates in percentage. here, codonmutation is the final output array, mutation is the output array sized 64×64 containing raw values after applying the algorithm, lg is the length of dataset which is 3068, gs is the length of the reference gene sequence which is 9967 in this converted dataset. the codon mutation rate for the full dataset has been shown in figure 5 . from the obtained value it is clear that codons are not frequently mutating like nucleotides. the diagonal values are 0 because that point codons are not changing comparing with reference gene sequence and heights codon mutation rate is 0.12%. in processed nucleotide mutation dataset, we have gene data from 12 th january 2020 to 11 th may 2020 discontinuously. these dates are in sorted ascending order which makes it easy to consider this as a time series dataset. in one particular date, this dataset has one or more patients. 3068 patients are in this dataset for 62 days. by taking all the patient we can find a time series dataset for patients shown in figure 6 . to obtain a day basis time-series dataset we have averaged the mutation rates for different patients in the same date. so the dataset becomes smaller and dates are in non-sequentially increasing order and the mutation rates for 62 days have shown in figure 7 . the low date availability makes it difficult to train a model in such a small amount of data. long short term memory network which is a type of recurrent neural network (rnn) has been used in deep learning. data has been organized as shown in table 3 where each set has a mutation rate of 12 patients. we have divided 80/20% data as training and testing. therefore, we get 2453 data for training and 614 for testing. an lstm model has been created with keras, a deep learning api of python and the structure has shown in figure 8 to train the dataset. first, the input layer got the prepared set of training data with 500 neurons. then it has been through a dense layer of 250 neurons with relu activation layer. after that 0.25 dropout has been used. another dense layer of 150 neurons has been used with relu activation. then again 0.25 dropout is used. finally, dense of 12 neurons has been used as an output layer with adam optimizer. this model gives root mean square error (rmse) of 0.06 in testing and 0.04 in training. after the train and testing process, the model seems to be working well. so we use the last 12 patients' mutation rate to predict one future patient's mutation rate and then take that patient and again make 12 patients' mutation rate by 11 old and 1 new patient and predicted. by this procedure, we have predicted 400 patients' mutation rate for future time, as shown in figure 9 . the nucleotide mutation rate of 400 th patient in future time has been shown in figure 10 . a little increment of mutation rate can be seen. if more continuous data can be found from different locations and date then this method can be applied to find the mutation rate for one particular date in the future. the covid-19 pandemic has almost stopped the world in this twenty-first century. the great spreading power mixing with mutation turns this virus greatly powerful and deadly. lockdown has limited the spreading power of this virus temporarily but the mutation power cannot be contained till now as no reliable vaccine has invented yet. in this research, we explain the nucleotide mutation rate and pattern in the codon mutation set. a rnn-based lstm model has been created to predict the future rate of mutation in person's body if effected with covid-19. with this model 400th patient in future time has been predicted. also, we have explained an lstm-rnn model for time series prediction based on patients' nucleotide mutation rate. by analyzing more patient data in updated time, this model can be used to predict day basis mutation rates. the situation may change if a reliable way of cure would be invented. also in this paper, the mutation rate is limited to base substitution only, insertion and deletion rate can be determined in further research. fig. 1 : nucleotide to codon indexing. coronaviridae study group of the international committee on taxonomy of viruses. the species severe acute respiratory syndrome-related coronavirus: classifying 2019-ncov and naming it sars-cov-2 laboratory testing of 2019 novel coronavirus (2019-ncov) in suspected human cases: interim guidance a novel coronavirus outbreak of global health concern who declares covid-19 a pandemic clinical features of patients infected with 2019 novel coronavirus in wuhan genomic characterisation and epidemiology of 2019 novel coronavirus: implications for virus origins and receptor binding genome composition and divergence of the novel coronavirus (2019-ncov) originating in china recombination, reservoirs, and the modular spike: mechanisms of coronavirus crossspecies transmission origin and evolution of pathogenic coronaviruses inhibition of sars-cov-2 replication by acidizing and rna lyase-modified carbon nanotubes combined with photodynamic thermal effect coronaviruses: an overview of their replication and pathogenesis emerging sars-cov-2 mutation hot spots include a novel rna-dependent-rna polymerase variant mutations: types and causes viral mutation rates increased fidelity reduces poliovirus fitness and virulence under selective pressure in mice quasispecies diversity determines pathogenesis through cooperative interactions in a viral population emergence of novel coronavirus and covid-19: whether to stay or die out? investigating the likely association between genetic ancestry and covid-19 manifestation emergence of multiple variants of sars-cov-2 with signature structural changes cryo-em structure of the 2019-ncov spike in the prefusion conformation models of rna virus evolution and their roles in vaccine design technical supports from the bgc university computer club has been acknowledged this research received no funding the authors declare no competing financial interest key: cord-316705-3wzurnfp authors: lalmuanawma, samuel; hussain, jamal; chhakchhuak, lalrinfela title: applications of machine learning and artificial intelligence for covid-19 (sars-cov-2) pandemic: a review date: 2020-06-25 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110059 sha: doc_id: 316705 cord_uid: 3wzurnfp background and objective: : during the recent global urgency, scientists, clinicians, and healthcare experts around the globe keep on searching for a new technology to support in tackling the covid-19 pandemic. the evidence of machine learning (ml) and artificial intelligence (ai) application on the previous epidemic encourage researchers by giving a new angle to fight against the novel coronavirus outbreak. this paper aims to comprehensively review the role of ai and ml as one significant method in the arena of screening, predicting, forecasting, contact tracing, and drug development for sars-cov-2 and its related epidemic. method: a selective assessment of information on the research article was executed on the databases related to the application of ml and ai technology on covid-19. rapid and critical analysis of the three crucial parameters, i.e., abstract, methodology, and the conclusion was done to relate to the model's possibilities for tackling the sars-cov-2 epidemic. result: this paper addresses on recent studies that apply ml and ai technology towards augmenting the researchers on multiple angles. it also addresses a few errors and challenges while using such algorithms in real-world problems. the paper also discusses suggestions conveying researchers on model design, medical experts, and policymakers in the current situation while tackling the covid-19 pandemic and ahead. conclusion: the ongoing development in ai and ml has significantly improved treatment, medication, screening, prediction, forecasting, contact tracing, and drug/vaccine development process for the covid-19 pandemic and reduce the human intervention in medical practice. however, most of the models are not deployed enough to show their real-world operation, but they are still up to the mark to tackle the sars-cov-2 epidemic. there are several disease outbreaks that invaded humanity in world history. world health organization (who), its co-operating clinicians and various national authorities around the globe fight against these pandemics to date. since the first covid-19 (coronavirus) disease case confirmed in china december 2019 wuhan district, the outbreak continues to spread all across the world, and on 30th january 2020 who declared the pandemic as an 1 cox-multivariant regression analysis revealed the model's significance towards and auxiliary tools for the healthcare expert. after evaluating 253 clinical blood samples from wuhan, researchers found eleven (bilirubin total, creatine kinase isoenzyme, glu, creatinine, kalium, lactate dehydrogenase, platelet distribution width, calcium, basophil, total protein, and magnesium) key relevant indices which can assist as a discrimination tool of covid-19 for healthcare expert toward rapid diagnosis [31] . the studies show that 11 relevant indices are extracted after employing the random forest algorithm with an overall accuracy of 95.95% and 96.97% specificity respectively. furthermore, the authors published that the tools were deployed and are available on web-server at http://lishuyan.lzu.edu.cn/covid2019_2/ to assist healthcare experts. the above studies give the evidence of an application of the expert system; designing rapid diagnosis was the main objective along with augmentation of accuracy. prompt and early detection reduce the spread of the disease and reserve more time to the healthcare expert to correspond to the next diagnosis to save more lives, resulting in low-cost medical expenditure. however, majority of the studied paper employed a single classification algorithm on individual data or more. therefore it is suggested to come up with a hybrid classification method applying more potential algorithm on multi-database or hybrid-database consisting of clinical, mammographic, and demographic data, as each type of data has a significant factor that could represent the true identity of the infected patients and deployment of the application in the real world. if a person diagnoses and is confirmed with covid-19, the next important step is contact tracing prevention of the wider spread of the disease. according to who, the infection spreads from person-to-person primarily through saliva, droplets, or discharges from the nose through contact transmission [32] . to take control on the spread of sars-cov-2, contact tracing is an essential public health tool used to break the chain of virus transmission [33] . the process of contact tracing is to identify and manage people who are recently exposed to an infected covid-19 patient to avoid further spread. generally, the process identifies the infected person with a follow-up for 14 days since the exposure. if employed thoroughly, this process can break the transmission chain of the current novel coronavirus and suppress the outbreak by giving a higher chance of adequate controls and helping reduce the magnitude of the recent pandemic. in this regard, various infected countries come up with a digital contact tracing process with the mobile application, utilizing different technologies like bluetooth, global positioning system (gps), social graph, contact details, network-based api, mobile tracking data, card transaction data, and system physical address. the digital contact tracing process can perform virtually real-time and much faster compared to the non-digital system. all these digital apps are designed to collect individual personal data, which will be analyzed by ml and ai tools to trace a person who is vulnerable to the novel virus due to their recent contacted chain. as shown in table 2, articles [34, 35] listed various countries competent with such ml and al-based contact tracing applications. studies show that over 36 countries successfully employed digital contact tracing use following centralized, decentralized, or hybrid of both techniques were proposed to lessen the effort and augment the effectiveness of the traditional healthcare diagnosis processes. concerning contact tracing, studies have proven the use of ml and ai in augmentation of contact tracing process against infectious chronic wasting disease [36] . after applying graph theory on infectious animal disease epidemics data, mainly shipment data between each farm, the resultant graph properties generated by the proposed model can be used to exploit to augment contact tracing more efficiently. moreover, the generated graphs have a potential prediction impact on the number of infections that can take place. however, there are still limitations in addressing the scenario, privacy, control over the data, and even data security breach. countries are working to overcome the challenges; some countries like israel "passed an emergency law to use mobile phone data" to tackle the current pandemic *37+. among the world contact tracing apps, some countries app violated privacy law and reported unsafe [35] so far they do the job acceptably by supplement the manual tracing process. however, virtually every country has their contact tracing application individually, as the outbreak continues to spread across the world, it becomes a global health emergency. to fight against the covid-19 as one, one should provide a standard de-facto centralized contact tracing application to trace every human being all around the world. also, it is reported that some specific query needs to address: "is it mandatory or voluntary?" "is the attempt clear or translucent?" "is information gathering lessened?" "will collected information be demolished as declared?", "is the data safe with the host" and "are there any restrictions or control on utilizing the information?". selective information shown in table 3 indicates the applications of ml and ai in forecasting and predicting the novel pandemic. a new novel model, that forecast and predicting 1-3 to 6 days ahead of total covid-19 patient of 10 brazilian states, using stacking-ensemble with support vector regression algorithm on the cumulative positive covid-19 cases of brazilian data was proposed, thus augmenting the short-term forecasting process to alert the healthcare expert and the government to tackle the pandemic [38] . recent studies suggested a novel model using a supervised multi-layered recursive classifier called xgboost on clinical and mammographic factor datasets. after applying the model, researchers found out those three significant key features (high-sensitivity c-reactive protein, lymphocyte and lactic dehydrogenase (ldh)) of the 75 features clinical and blood test samples result to be the highest rank of 90% accuracy in predicting and assessing covid-19 patient into general, severe and mortality rate [39] . furthermore, comparatively higher value in single lactic dehydrogenase appears to be a significant factor in classifying most patients in need of intensive medical care, as ldh degree related to various respiratory disorder diseases, namely asthma and bronchitis, and pneumonia. the forecast model employed decision rule to forecast rapidly and predict infected individuals at the highest risk, authorized patients to be manageable for intensive care, and possibly lessen the transience rate. a canadian based forecasting model using time-series was developed employing deep learning algorithm for the long-short-term-memory network, the studies found out a key factor intended for predicting the course with an ending point estimation of the current sars-cov-2 epidemic in canada and all over the globe [40] . the suggested model forecast ending point of this sars-cov-2 outbreak in canada will be around june 2020. based on the data collected from john hopkins university [3] , the prediction was likely to be accurate as newly infected cases have dropped rapidly and proven the applicability of the expert system in predicting and forecasting for the current pandemic outbreak by revealing key significant features. the real-time forecasting model was proposed combining the goodness of the wavelet-based forecasting model and autoregressive integrated moving average based time-series model [41] . the model solves the problem by generating short-term forecasts of the sars-cov-2 for various countries (india, united kingdom, canada, south korea, and france) to assist healthcare experts and policymakers as a preliminary cautioning module for each target country. real-time forecast and 10 days ahead, observed seven key features associated with dead rate. since the coronavirus epidemic fury, researchers and healthcare experts around the globe ubiquitously urged to develop a possible choice to tackle the development of drug and vaccine for the sars-cov-2 pandemic, and ml/ai technology constitutes to be an enthralling road. concerning the possibility of drug choice for infected patient's treatment, instant testing on the existing old marketable medicines for novel sars-cov-2 carrier in a human being is essential. researchers from taiwan are building a new model to augment the development of a novel drug [42] . after applying the ml and ai technology-based model on two datasets (one using the 3c-like protease constraint and other data-holding records of infected sars-cov, sars-cov-2, influenza, and human immunodeficiency virus (hiv)) using deep neural network on the eighty old drugs with potential for covid-19 treatment, the study suggested eight drugs, i.e., vismodegib, gemcitabine, clofazimine, celecoxib, brequinar, conivaptan, bedaquiline and tolcapone are found virtually effective against feline infectious peritonitis coronavirus. furthermore, other five drugs like homoharringtonine, salinomycin, boceprevir, tilorone and chloroquine are also found operational during ai experimental environment. a novel molecule transformer-drug target interaction model was proposed jointly by researchers from the us and korea to tackle the need for an antiviral drug that can treat the covid-19 virus [43] . [44] , starting with ml and ai-based pharmacophore computational analyzing on a limited size of in vitro infected carriers of the ebola virus. the study proposed an amodiaquine and chloroquine compound popularly used to treat the malaria virus. furthermore, after uncovering a decade of drug development based on ml and ai technology, a fusion of computational screening method with docking application and machine learning for choosing supplementary medication to investigate on sars-cov-2 was proposed [45] . researchers refer to the successful discovery of ebola [44] , and the zika virus [46] experience gain belief that the same model could be repeatedly utilized for drug discovery on covid-19 and future virus pandemic ahead. the selected review paper adopted various methodologies and technologies addressing the classical method of classification based on statistics to an advanced modern ai and ml algorithm. the use of computational tools, combined with docking application, was found to be more active in predicting the reusability of an existing old drug on covid-19 medication and dramatically minimize the level of a risk factor in the development of medicine more cost-effective process. during this urgency, the use of ml and ai can augment the drug development process by lessening the time slot on discovering a supplementary treatment and medication for the carrier by drawing a vast probability over security, manageability, and clinical information on the existing drug compound. issues and challenges found in this area were the limited resource of comprehensive hybrid data and real-life deployment of the application. since the outbreak of the novel sars-cov-2, scientists and medical industries around the globe ubiquitously urged to fight against the pandemic, searching alternative method of rapid screening and prediction process, contact tracing, forecasting, and development of vaccine or drugs with the more accurate and reliable operation. machine learning and artificial intelligence are such promising methods employed by various healthcare providers. this paper addresses on recent studies that apply such advance technology in augmenting the researchers in multiple angles, addressing the troubles and challenges while using such algorithm in assisting medical expert in real-world problems. this paper also discusses suggestions conveying researchers on ai/ml-based model design, medical experts, and policymakers on few errors encountered in the current situation while tackling the current pandemic. this review shows that the use of modern technology with ai and ml dramatically improves the screening, prediction, contact tracing, forecasting, and drug/vaccine development with extreme reliability. majority of the paper employed deep learning algorithms and is found to have more potential, robust, and advance among the other learning algorithms. however, the current urgency requires an improved model with high-end performance accuracy in screening and predicting the sars-cov-2 with a different kind of related disease by analyzing the clinical, mammographic, and demographic information of the suspects and infected patients. finally, it is evident that ai and ml can significantly improve treatment, medication, screening & prediction, forecasting, contact tracing, and drug/vaccine development for the covid-19 pandemic and reduce the human intervention in medical practice. however, most of the models are not deployed enough to show their real-world operation, but they are still up to the mark to tackle the pandemic. none. world health organization declares global emergency: a review of the 2019 novel coronavirus (covid-19) coronavirus disease (covid-2019) situation reports covid-19 dashboard by the the potential for artificial intelligence in healthcare intelligent decision making: an ai-based approach use of artificial intelligence in infectious diseases. artificial intelligence in precision health computer-based medical consultations: mycin machine learning for clinical decision support in infectious diseases: a narrative review of current applications an adaptive machine learning approach to improve automatic iceberg detection from sar images deep nonsmooth nonnegative matrix factorization network factorization network with semi-supervised learning for sar image change detection change detection in sar images by artificial immune multi-objective clustering a novel target detection method for sar images based on shadow proposal and saliency analysis machine-learning prognostic models from the 2014-16 ebola outbreak: data-harmonization challenges, validation strategies, and mhealth applications large-scale machine learning of media outlets for understanding public reactions to nation-wide viral infection outbreaks two-steps learning of fuzzy cognitive maps for prediction and knowledge discovery on the hiv-1 drug resistance automated diagnosis of hiv-associated neurocognitive disorders using large-scale granger causality analysis of resting-state functional mri covid-19 detection using deep learning models to exploit social mimic optimization and structured chest x-ray images using fuzzy color and stacking approaches breast cancer detection by leveraging machine learning. ict express machine learning methods for computer-aided breast cancer diagnosis using histopathology: a narrative review diabetic retinopathy detection through novel tetragonal local octa patterns and extreme learning machines machine learning and data mining methods in diabetes research artificial plant optimization algorithm to detect heart rate & presence of heart disease using machine learning classification models for heart disease prediction using feature selection and pca. informatics in medicine unlocked a hybrid machine learning approach to cerebral stroke prediction based on imbalanced medical dataset deep learning iot system for online stroke detection in skull computed tomography images artificial intelligence (ai) applications for covid-19 pandemic correlation of chest ct and rt-pcr testing in coronavirus disease 2019 (covid-19) in china: a report of 1014 cases application of deep learning technique to manage covid-19 in routine clinical practice using ct images: results of 10 convolutional neural networks automated detection of covid-19 cases using deep neural networks with x-ray images combination of four clinical indicators predicts the severe/critical symptom of patients infected covid-19 rapid and accurate identification of covid-19 infection through machine learning based on clinical available blood test results who: world health organization, 2020. health topic, coronavirus disease overview contact tracing in the context of covid-19 wikipedia: covid-19 apps mit: covid tracing tracker -a flood of coronavirus apps are tracking us. now it's time to keep track of them contact tracing for the control of infectious disease epidemics: chronic wasting disease in deer farms bbc: coronavirus: israel enables emergency spy powers short-term forecasting covid-19 cumulative confirmed cases: perspectives for brazil an interpretable mortality prediction model for covid-19 patients time series forecasting of covid-19 transmission in canada using lstm networks real-time forecasts and risk assessment of novel coronavirus (covid-19) cases: a data-driven analysis artificial intelligence approach fighting covid-19 with repurposing drugs predicting commercially available antiviral drugs that may act on the novel coronavirus (sars-cov-2) through a drug-target interaction deep learning model a common feature pharmacophore for fda-approved drugs inhibiting the ebola virus dã©jã  vu: stimulating open drug discovery for sars-cov-2. drug discovery today open drug discovery for the zika virus key: cord-280975-9hgtvm6d authors: sarkar, kankan; khajanchi, subhas; nieto, juan j. title: modeling and forecasting the covid-19 pandemic in india date: 2020-06-28 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110049 sha: doc_id: 280975 cord_uid: 9hgtvm6d in india, 1,00,340 confirmed cases and 3,155 confirmed deaths due to covid-19 were reported as of may 18, 2020. due to absence of specific vaccine or therapy, non-pharmacological interventions including social distancing, contact tracing are essential to end the worldwide covid-19. we propose a mathematical model that predicts the dynamics of covid-19 in 17 provinces of india and the overall india. a complete scenario is given to demonstrate the estimated pandemic life cycle along with the real data or history to date, which in turn divulges the predicted inflection point and ending phase of sars-cov-2. the proposed model monitors the dynamics of six compartments, namely susceptible (s), asymptomatic (a), recovered (r), infected (i), isolated infected (i(q)) and quarantined susceptible (s(q)), collectively expressed sarii(q)s(q). a sensitivity analysis is conducted to determine the robustness of model predictions to parameter values and the sensitive parameters are estimated from the real data on the covid-19 pandemic in india. our results reveal that achieving a reduction in the contact rate between uninfected and infected individuals by quarantined the susceptible individuals, can effectively reduce the basic reproduction number. our model simulations demonstrate that the elimination of ongoing sars-cov-2 pandemic is possible by combining the restrictive social distancing and contact tracing. our predictions are based on real data with reasonable assumptions, whereas the accurate course of epidemic heavily depends on how and when quarantine, isolation and precautionary measures are enforced. the ongoing coronavirus, sars-cov-2 epidemic has been announced a pandemic by the world health organization (who) on march 11, 2020 [1] , and in the first phase the govt. of india has announced 21 days nationwide lockdown from march 25, 2020 to april 14, 2020 , and in the second phase the lockdown has been extended to may 03, 2020 to prevent stage-iii spreading of the virus or human-to-human transmission [2] . to mitigate the unavoidable economic downturn. due to absence of any specific pharmaceutical inter35 ventions, government of various countries are imposing different strategies to prevent this outbreak and the lockdown is the most common one. as for examples, the measures adopted in this time incorporated social distancing, closing schools, universities, offices, churches, bars, avoid mass gatherings, other social places as well as contact of cases (quarantine, surveillance, contact tracing) [15] . on march 19, 2020 the govt. of india suspended all the international flights till march 22, 2020 [16] , and on march 23, 2020 40 the union govt. also suspended all the domestic flights till march 25, 2020 [17] to maintain the social distancing among the people. the prime minister of india has announced a 14 hours voluntary public curfew ('janata curfew') on march 22, 2020 as a precautionary measure to combat against covid-19. the govt. of india followed it up with lockdowns on march 23, 2020 to prevent the emanating threat in 75 districts across the country including major cities where the covid-19 infection was endemic [18] . 45 furthermore, on march 24, 2020 the govt. of india has ordered a nationwide lockdown for 21 days, overwhelming the entire 1.3 billion public in india [19] , and the lockdown has been extended to may 03, 2020 to prevent stage-iii spreading of the virus or human-to-human transmission [20] . predictive mathematical models play a key role to understand the course of the epidemic and for designing strategies to contain quickly spreading infectious diseases in lack of any specific antivirals or 50 effective vaccine [21, 22, 23, 24] . in the year 1927, kermack & mckendrick [25] developed a fundamental epidemic model for human-to-human transmission to describe the dynamics of populations through three mutually exclusive phages of infection, namely susceptible (s), infected (i) and removed (r) classes. mathematical modeling of infectious diseases are now ubiquitous and many of them can precisely depict the dynamic spread of particular epidemics. several mathematical models has been developed to study 55 the transmission dynamics of covid-19 pandemic. a bats-hosts-reservoir-people network model has been developed by chen et al. [26] to study the transmission dynamics of novel coronavirus. lin et al. [27] extended the seir (susceptible-exposed-infectious-removed) compartment model to study the dynamics of covid-19 incorporating public perception of risk and the number of cumulative cases. khajanchi et al. [28] studied an extended seir model to study the transmission dynamics of covid-19 and perform 60 analysis of viral dynamics using mathematical models have helped gain insights into the understanding of viral infections such as tuberculosis, dengue, and zika virus [32, 33, 34, 35, 36] . here, we developed 70 a new epidemiological mathematical model for novel coronavirus or sars-cov-2 epidemic in india that extends the standard seir compartment model, alike to that studied by tang et al. [37] for covid-19. the transmission dynamics of our proposed model for covid-19 is illustrated in the figure 1 . we develop here a classical seir (susceptible-exposed-infectious-recovered)-type epidemiological model 75 by introducing contact tracing and other interventions such as quarantine, lockdown, social distancing and isolation that can represent the overall dynamics of novel coronavirus or covid-19 (sars-cov-2). the model, named sarii q s q , monitors the dynamics of six compartments (classes), namely susceptible individuals (s) (uninfected), quarantined susceptible individuals (s q ) (quarantined at home), infectious but not yet symptomatic or asymptomatic infectious individuals (a), infected or infectious 80 with symptoms/clinically ill (i), isolated infected individuals (i q ) (infected or life-threatening or detected) and the recovered compartment (r) (no more infectious). the total size of the individuals is n = s + s q + a + i + i q + r. asymptomatic individuals have been exposed to the virus, but have not yet developed clinical symptoms of the covid-19 or sars-cov-2 [38]. in our model, quarantine describes the separation of coronavirus infected populations from the susceptible individuals before progression of 85 clinical symptoms, whereas the isolation refers to the dissociation of coronavirus infected populations with such clinical symptoms. the rate of change in each compartments at any time t is represented by the following system of nonlinear ordinary differential equations: the model is supplemented by the following non-negative initial values: herein, t ≥ t 0 represents time in days and t 0 indicates the starting date for the system of the coronavirus 90 epidemic. in our model construction, β s represents the probability of transmission per contact between an infective and a susceptible class, and ε s is denoted by the daily contact rate per unit of time. here the parameter β = β s ε s is explicitly associated with the measures like lock-down, social distancing, shaking hand, coughing and sneezing etc., which exactly decrease the number of social contacts. by enforcing con-95 tact tracing, a proportion ρ s , of individuals exposed to the coronavirus is quarantined. the quarantined classes can either move to the compartment s q or i q , depending on whether they are effectively infected individuals or not, whereas the another proportion 1 − ρ s , consists of populations exposed to the coronavirus who are missed from contact tracing and move to the infectious class i (once infected) or remaining in susceptible class s (if uninfected). then the quarantined classes, if uninfected (or infected), move to 100 the class s q (or i q ) at a rate of (1 − β s )ρ s ε s (or β s ρ s ε s ). those who are not quarantined individuals, but asymptomatic infectious individuals, will move to the asymptomatic compartment a at the rate of tined susceptible class due to fever and/or illness-like clinical symptoms. we symbolize ξ a , ξ i and ξ q are the rates of recovery individuals of asymptomatic class, symptomatic or clinically ill patients and isolated individuals, respectively. our model introduces some demographic effects by considering a proportional natural decay rate δ in each of the six individuals, and λ s represents the constant inflow of susceptible individuals. asymptomatic population develop to infected population at the rate γ a , so the average time 110 spent in the asymptotic class is 1 γa per unit time. in similar fashion, 1 γi represents the mean duration for infected individuals. we ignore the rate of probability of transforming susceptible again after having cured (recovered) from the disease infection. it is to be noted that our sarii q s q model did not take into account many important ingredients that take part a key role in the transmission dynamics of covid-19 such as the influence of the latency period, the inhomogeneous disease transmission network, the influence 115 of the measures already considered to fight the coronavirus diseases, the features of the individuals (for example, the influence of the stage-structure, individuals who are already medically unfit). some recent mathematical models incorporate asymptomatic such as in ndairou et al. [39] but others do not include them [40]. the basic reproduction number, symbolized by r 0 , is 'the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual' [41, 42] . the dimensionless basic reproduction number provides a threshold, which play a crucial role in determining the disease persists or dies out from the individual. in a more general way the basic reproduction number r 0 can be stated as the number of new infections created by a typical infective population at a disease free equilib-125 rium. r 0 < 1 determines on average an infected population creates less than one new infected population during the course of its infective period, and the infection can die out. in reverse way, r 0 > 1 determines each infected population creates, on average, more than one new infection, and the disease can spread over the population. the basic reproduction number r 0 can be computed by using the concept of next generation matrix [41, 42] . in order to do this, we consider the nonnegative matrix f and the 130 non-singular m −matrix v, expressing as the production of new-infection and transition part respectively, for the system (1), are described by the variational matrix of the model (1) computed at the infection free state ( the basic reproduction number r 0 = ρ(f v −1 ), where ρ(f v −1 ) represents the spectral radius for a 135 next generation matrix f v −1 . thus, the basic reproduction number of the system (1) is . (3) we calibrated our sarii q s q model for covid-19 to the daily new infected cases and cumulative table 7 . the description of the sarii q s q model are given in table 1 , list of key estimated parameter values are specified in table 2 and estimated initial population size are given in the table 3 . by calibrating the sarii q s q model parameters with real data up to 30 april 2020, we make an attempt to forecast the evolution of the epidemic in india and 17 145 provinces of india. in the model exploration, we did not consider the demographic effects because of the short epidemic time scale in compare to the demographic time scale, that is, λ s = δ = 0. to recognize the most influential parameters with respect to clinically ill infected population, we the prcc results has been shown in the figure 2 for six time points that represents the highest positively correlated parameters are the disease transmission rate β s , contact rate ε s of all the individuals, the probability rate γ a at which the asymptomatic individuals develops clinically symptoms and highly negatively correlated parameters are the quarantined rate ρ s of uninfected individuals, recovery rate ξ a 165 of asymptomatic infected individuals and the recovery rate ξ i of infected individuals, accounts the most uncertainty with respect to the infected individuals. thus, the prcc analysis yields these six parameters β s , ρ s , ε s , γ a , ξ a , and ξ i are the most influential parameters out of 9 parameters. therefore, we estimated these six parameters by using least square method. the most important challenge in any mathematical model based study is to estimate the model parameters and the initial population size. the solution of the sarii q s q model system (1) depends on both the parameter values and initial population size. the model parameters have been estimated assuming the initial population size and fitting the model simulation with the observed covid-19 cases. the assumed initial population sizes are presented in the table 3 . we have estimated six parameters, probability of 175 disease transmission (β s ), quarantined rate of susceptible individuals (ρ s ), contact rate of entire individuals ( s ), probability rate at which asymptomatic individuals develop clinical symptoms (γ a ), recovery rate of asymptomatic infected individuals (ξ a ) and rate of recovery for infected individuals (ξ i ) as these parameters are more sensitive in prcc analysis. the parameters are estimated from the observed daily new covid-19 or sars-cov-2 viruses. although, we have shown the plot validating model simulation optimize the error in parameter estimation [44] and errors are listed in the table 5 . first we have applied a five days moving average filter, which is a low pass filter, to smooth the random variation in the observed daily new covid-19 cases. the observed daily covid-19 cases are fitted with the model simulation by using least square method, which locally minimizes the sum of the square of errors. the square of sum of 185 the error computed as σ n i=1 (c(i) − s(i)) 2 , where c(i) represents the observed daily new covid-19 cases on i-th day, s(i) is the sarii q s q model simulation on i-th day and n is the sample size of the observed data. it has been observed that different set of parameter values can minimize the sum of the square of errors between the observed daily new covid-19 cases and the sarii q s q model simulation but we have considered the set of parameter values, which produce realistic r 0 . varying the random values of initially we have validated the model simulation with the observed covid-19 cases. the sources and duration of the observed data has been presented in table 7 . model simulated from the first date of coronavirus infection and up to 30 april, 2020 for whole india and for seventeen states of india. the model simulation fitted with the observed daily new covid-19 cases and cumulative covid-19 cases. the parameter values are taken from table 1 and the table 2 and the initial population size from table 205 3. to describe how best to minimize individuals impermanence and morbidity due to sars-cov-2, it is important to see the relative significance of various ingredients responsible for disease transmission. transmission of sars-cov-2 is directly related to the basic reproduction number r 0 . we compute the sensitivity indices for r 0 for the parameters of the sarii q s q model. this indices apprise us how 225 important each parameter is to disease transmission. sensitivity analysis is mainly used to describe the robustness of the model predictions to the parameters, as there are generally errors in collection of data and assumed parameter values. sensitivity indices quantify the relative change in a state variable when a parameter alters. the normalized forward sensitivity index for r 0 , with respect to the disease transmission coefficient β s can be defined as follows: which demonstrates that r 0 is a increasing function of β s . this implies that probability of disease transmission has a high influence on covid-19 control and management. the sensitivity indices of other parameters are given in the table 4 . in the table 4 , some of the indices are positive (and some are negative) which means if the parameter increases then increase the value of r 0 (and if the parameter increases then decrease the value of r 0 ). to control the outbreak of sars-cov-2, we must select the most 235 sensitive parameters who have most influence to reduce the diseases. as for example, the transmission rate β s has an impact in reducing the covid-19 diseases, which can easily be observed from the table 4 . therefore, we draw the contour plots for r 0 in the figure 8 and figure 9 dependence on the rate of disease transmission probability β s and the quarantine rate ρ s . contour plot shows that for the higher values of β s the reproduction number r 0 increases significantly, which means that the sars-cov-2 disease will 240 persist among the human and spread throughout the community if the public not take the preventive measures. thus, to control r 0 must reduce the disease transmission coefficient β s and increase the period of quarantine rate ρ s . thus, we may conclude that to end the covid-19 outbreak enhance the quarantine and reduce the probability of disease transmission following contact tracing, social distancing, limit or stop theaters and cultural programme etc. for set of parameter values in the table 1 and estimated parameter 245 values in the table 2 , we plot a bar diagram for the basic reproduction number r 0 in the figure 6 . from the bar-diagram in the figure 6 , it can be observed that the basic reproduction number r 0 for the state maharashtra is too high, which indicates that the substantial outbreak of the covid-19 in the state maharashtra. will be more accurate. however, this prediction gives us an overview of the pandemic, which will lead to decide future planning. in this study, we fitted sarii q s q model to forecast the pandemic trend over the period after 30 april, 2020 by using the observed data from the first day of infection to 30 we cov-2 and the evolution of epidemic become accessible at an unparalleled pace. howbeit, important questions still remain undetermined and precise answers for forecasting the transmission dynamics of the epidemic simply cannot be acquired at this stage. we emphasize the uncertainty of accessible authentic data, specially concerning to the accurate baseline number of infected individuals, which may guide to 345 the equivocal outcomes and inappropriate predictions by orders of size, as also identified by the other researches [45] . we hope that our predictions will be handy for govt. and different companies as well as the people towards making resolutions and considering the suitable actions that contain the spreading of the coronavirus to the possible stage. all the data used in this work has been obtained from official sources. all data supporting the findings of this study are in the paper and available from the corresponding author on request. observed data points are displayed in the red dot histogram and the blue curve represents the best fitting curve for the sariiqsq model. the first and third rows represents the daily new cases of coronavirus diseases, whereas the second and fourth rows represents the cumulative confirmed cases of covid-19. the estimated parameter values are listed in the table 2 . the initial values used for this parameter values are presented in the table 3 . observed data points are shown in the red dot histogram and the blue curve represents the best fitting curve for the sariiqsq model. the first and third rows represents the daily new cases of coronavirus diseases, whereas the second and fourth rows represents the cumulative confirmed cases of covid-19. the estimated parameter values are listed in the table 2 . the initial values used for this parameter values are 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france on the definition and the computation of the basic reproduction ratio r 0 in models for infectious diseases in heterogeneous populations reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission a methodology for performing global uncertainty and sensitivity analysis in systems biology avoidable errors in the modelling of out-460 breaks of emerging pathogens, with special reference to ebola 2019-novel coronavirus (2019-ncov): estimating the case fatality ratea word of caution campus quarantine (fengxiao) for curbing emer-465 gent infectious diseases: lessons from mitigating a/h1n1 in xi'an key: cord-317371-v7hmc9sj authors: zhang, xiaolei; ma, renjun; wang, lin title: predicting turning point, duration and attack rate of covid-19 outbreaks in major western countries date: 2020-04-20 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109829 sha: doc_id: 317371 cord_uid: v7hmc9sj in this paper, we employed a segmented poisson model to analyze the available daily new cases data of the covid-19 outbreaks in the six western countries of the group of seven, namely, canada, france, germany, italy, uk and usa. we incorporated the governments’ interventions (stay-at-home advises/orders, lockdowns, quarantines and social distancing) against covid-19 into consideration. our analysis allowed us to make a statistical prediction on the turning point (the time that the daily new cases peak), the duration (the period that the outbreak lasts) and the attack rate (the percentage of the total population that will be infected over the course of the outbreak) for these countries. the spread of coronavirus disease 2019 (covid-19) has become a global threat and the world health organization (who) declared covid-19 a global pandemic on march 11, 2020 [1] . as of april 9, 2020, there were 1,604,252 confirmed cases and 95,714 deaths from covid-19 worldwide [2] . the covid-19 pandemic has been greatly affecting peoples lives and the world's economy. among many infection related questions, governments and people are most concerned with (i) when the covid-19 outbreak will peak; (ii) how long the outbreak will last and (iii) how many people will eventually be infected. since the early spread of covid-19 in december of 2019 in wuhan, china, there have been tremendous efforts to understand the spread dynamics and to propose effective prevention and control strategies [3-7, 9, 10] . to stop the spread of covid-19 from the early epicenter-wuhan, china unprecedentedly locked down the entire city of wuhan on january 23. it was shown that the wuhan lockdown delayed the occurrence of covid-19 in other cities by 2.91 days and may have prevented more than 700,000 covid-19 cases outside of wuhan [8] . this massive lockdown was then later served as a model for several other countries battling covid-19 around the world. currently the development of vaccines is still in progress and there are no effective antiviral drugs for treating covid-19 infections. the only practical therapeutic option is hospitalization and intensive care unit management. thus predicting the peak time (or turning point), the duration and the final size of the outbreak for each country becomes crucial for policy makers and public health authorities to have informed decisions on appropriate interventions and resource allocations. however, since the covid-19 virus is a novel coronavirus, key infection parameters such as the mean incubation period and the mean infection period are not known. this, together with the complex contact patterns, makes predictions based on previously established compartmental models for other viruses very challenging. in this study, we simply regarded the daily new cases as a function of time t and coupled a power law with an exponential law. we also incorporated government's major interventions against the spread of covid-19 such as stay-at-home advises/orders, lockdowns, quarantines and social distancing into our modeling. by fitting the available daily new cases data to our model, we were able to identify the peak time of daily increased new cases, predict the duration, the final size and the attack rate of the outbreak for each country. more specifically, we analyzed the data (up to april 9) of canada, france, germany, italy, uk and usa (the six members of the group of seven (g7) countries). the data on daily new confirmed cases of covid-19 in these countries we used were taken from the wind database [11] and from the webpage on us and canada covid-19 live updates [12] . to identify the turning point and predict the further spread of covid-19 outbreaks while accounting for governments enforcement of stay-at-home advises/orders, social distancing, lockdowns, and quarantines against covid-19, we combine the power law with the exponential law for daily new cases based on a segmented poisson model. let y t be the daily new cases at day t since the first case was reported on day 1. our model takes the following form where µ t is the expectation of y t with segmented expressions given below. where α k , β k and γ k are regression parameters and k = 1, 2 correspond to periods before and after the day of major government actions (stay-at-home advises/orders, lockdowns, quarantines and social distancing) against covid-19 at day s. the advantage of our segmented poisson model is that the observed daily new cases before and after the day of major government actions are characterized integrally under a single model, but with separate mean curves. unlike the widely used log-transformed linear model, our poisson modeling approach enables us to deal with daily new cases as a count response with many zeros. in addition, our segmented poisson model allows us to account for governments' interventions at different stages dynamically by incorporating stage specific segments. as major government actions are taken when covid-19 outbreaks deteriorate seriously, the maximum number of daily new cases occurs during the period after the day of major government actions. it follows from (2.1) that the maximum value of y t is once the parameters α's, β's, and γ's have been estimated, we can then find the peak time t peak and make a prediction for the further spread of the outbreak. let n be the smallest integer of t such that y t ≤ 1 for t > t peak . then the outbreak would last for n days, that is, the duration of the outbreak is n days. in addition, the total cumulative number of infected individuals, i.e., the final size of the outbreak, can be estimated by then for a given country, the ratio of the total cumulative number of infected individuals and the total population (the 2020 population data was taken from worldometers [2]) would give the so-called attack rate of the covid-19 outbreak in that country. we applied our model to study the turning point and further spread of covid-19 outbreaks in the six western countries of g7, namely, canada, france, germany, italy, uk and usa. the parameter estimates together with their 95% confidence intervals for each of these six countries are displayed in table 1 and table 2 . using the 95% confidence intervals of β 2 and γ 2 , we can also find the range of the turning point computed by (min β 2 / max γ 2 , max β 2 / min γ 2 ). the estimated turning point, duration time, final size and the attack rate for each of the six major western countries are presented in table 3 . italy is the first country in this group (also in the world) whose cumulative confirmed cases overpass 100,000 (it occurred on march 30,2020). based on our estimate, italy's turning point is march 26 (in the range of (march 24∼march 28)), the outbreak is estimated to end around june 1, and the final size is 172, 451, which gives an infection attack rate of 0.285%. the observed data, our fitting and a 14-day prediction on the daily new cases and the cumulative cases are plotted in figure 1 . the usa now becomes the country with the most confirmed cases in this group (also in the world). our analysis found that usa's turning point is april 07 (in the range of april 03∼april 09)), the outbreak is expected to end in the early june (june 03), and the cumulative cases would be about 835, 158, i.e., the attack rate is 0.2523%. the fitting and prediction result is presented in figure 2 . the fitting and prediction for the other four countries are given in figure 3 . using the 95% confidence intervals of γ 2 , we could also give an upper bound for the final size. the upper bounds for the final size of usa, germany, uk, france, italy and canada were estimated to be 1.98 million, 1.27 million, 800 thousand, 750 thousand, 261 thousand, and 118 thousand, respectively. we have combined a power law with an exponential law with our segmented poisson model to analyze the covid-19 daily new cases data for six major western countries in the g-7 group. it is seen from figures 1-3 that the observed and estimated daily new cases are in good agreement. this together with the forecasted trend indicated that our model has well characterized the covid-19 outbreaks in these six major western countries. our analysis allowed us to identify/predict the turning point, to predict the further spread, the duration and the final size (the attack rate) of the outbreak of covid-19 in those six countries we studied. we found that among those six countries, france would ). if the current government actions remain unchanged, the outbreaks would likely to end at the beginning of june (ranging from may 21 to june 10) and the average duration of the outbreaks is 127 days (ranging from 115 to 138 days). it is seen from tables 1 and 2 that the estimated parameter γ 1 's are all negative (except for italy's, which is close to zero), and all γ 2 's are positive. this implies that if there were no major enforcement actions on control strategies such as lockdowns, social distancing, stayhome-advises/orders, then the covid-19 would have spread exponentially. for example, the total confirmed cases in the usa would have passed 1, 000, 000 on april 05, 2020. this indicates that the interentions/actions greatly reduced the outbreak sizes and flatted the epidemic curves. our prediction is based on the assumption that the current government interventions/actions would be imposed until the estimated end dates of the outbreaks. if those interventions were lifted or removed earlier cautiously based on scientific evidences, we would not expect any dramatic differences. on the other hand, if those interventions were lifted or removed earlier hastily without scientific support, our prediction would provide a reference to assess consequences of such irresponsible decisions. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. who director-general's opening remarks analysis and forecase of covid-19 spreading in china the attack rate of the covid-19 in a year real-time estimation of the risk of death from novel coronavirus (covid-19) infection: inference using exported cases trend and forecasting of the covid-19 outbreak in china analysis of covid-19 epidemic traced data and stochastic discrete transmission dynamic model an investigation of transmission control measures during the first 50 days of the covid-19 epidemic in china phase-adjusted estimation of the number of coronavirus disease nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study. the lancet declare the following financial interests/personal relationships which may be considered as potential competing interests the authors would like to thank the anonymous referee for his/her valuable comments and suggestions. xz was supported in part by the yunnan philosophy and social science planning project fund (hx2019082760); rm was supported by the natural sciences and engineering research council of canada (rgpin-2015-06124) and lw was supported by the natural sciences and engineering research council of canada (rgpin-2015-05686). key: cord-254195-k7e8g0ni authors: akinlar, m.a.; inc, mustafa; gómez-aguilar, j.f.; boutarfa, b. title: solutions of a disease model with fractional white noise date: 2020-04-30 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109840 sha: doc_id: 254195 cord_uid: k7e8g0ni we consider an epidemic disease system by an additive fractional white noise to show that epidemic diseases may be more competently modeled in the fractional-stochastic settings than the ones modeled by deterministic differential equations. we generate a new sirs model and perturb it to the fractional-stochastic systems. we study chaotic behavior at disease-free and endemic steady-state points on these systems. we also numerically solve the fractional-stochastic systems by an trapezoidal rule and an euler type numerical method. we also associate the sirs model with fractional brownian motion by wick product and determine numerical and explicit solutions of the resulting system. there is no sirs-type model which considers fractional epidemic disease models with fractional white noise or wick product settings which makes the paper totally a new contribution to the related science. fractional-stochastic calculus consist of fractional-order derivatives, integral operators or fractional brownian motion and a noise term representing uncertainty or randomness in modeling. these differential equations have found an outstanding role in efficiently modeling of many different phenomena in science, engineering and economics. epidemic diseases including coronavirus, brucellosis, chickenpox, dengue, ebola, mumps, influenza, measles, plague, sars, tetanus, tuberculosis, zika, west nile virus were studied in terms of some efficient mathematical models in the related scientific literature. these systems were generated by taking into account some facts such as duration of disease, availability and resistance against vaccination, immune systems of individuals in the population and so on. there are mathematical models employing deterministic [4] [5] [6] [7] , stochastic [1] [2] [3] , fractional-order [8] [9] [10] [11] [12] [13] [14] [15] system of differential equations. almost each of these models were generated by compartmental models considering each compartment as individuals of susceptible (denoted by s), infected (i), exposed (e), and recovered (r) ones. in the present research work, new sirs models both in deterministic and fractionalstochastic settings are proposed. we are concerned with endemic equilibrium and disease-free fixed points and numerical solutions of these models with different type of numerical techniques. we also consider additive fractional white noise representation of the sirs model and obtain numerical and exact solutions of these models. in the modeling of epidemic diseases via compartmental type mathematical models, there exists not any study considering fractional white noise, wick product and fractional-order operators all together. our driving motivation in this paper is to investigate the applicability of these significant and powerful techniques to the modeling of epidemic diseases. our contributions in this paper may be listed as: i) generating a new sirs model. ii.) considering this system in terms of stochastic equations and fractional white noise. iii) studying chaos at disease-free and endemic steady state points. iv) solving perturbed systems (perturbed from deterministic to fractional-stochastic) both numerically and explicitly (exactly). we provided figures illustrating the behavior of compartments in different environments. from this listed contributions, we can say that the present paper is totally a new contribution to mathematical biologists studying compartment models by fractional and stochastic differential equations. we believe that researchers in this area will employ the present paper and extend it to different models. we are concerned with a sirs system described as: in which s(t), i(t), r(t), stands for individuals of susceptible, infected and recovered, respectively, r(t) + i(t) + s(t) = m, represents population of all individuals, ρ and ξ are the birth and death rates, respectively, (we assumed that they are equal to each other,) ν is the probability of transmission rate between compartments s(t) and i(t), d is rate of recovery from the disease, k is the rate of being susceptible after recovered from disease. in our sirs model, we assume that the healed people are not infected and can not spread the disease any further among other individuals in the population. defining new variables a := ν m , e := k + ξ, c := d + ξ, n := ρm, re-expression of the system (1) is: fractional-order calculus [16-18, 48, 51] (or differential equations) generated by fractional-order derivative and integral operators found a significant place in applied and computational mathematics in recent years. they were employed in many different scientific work at mathematics, physics, economics and engineering. fractionalorder operators considers historical effects and have non-local computational ability which makes these operators more powerful and desirable in the modeling applications. there are many different fractional derivative or integral operators such as grünwald-letnikov, riemann-liouville, caputo, atangana-baleanu and so on. let g : (0, ∞) → r be a function. fractional-order integral operator of order β for g is defined by inhere γ(·) denotes gamma function. for (m ∈ n), caputo-type derivative operator of order β, m − 1 < β < m, of g(t) is given by . next, we express derivatives on left-hand-side of system (2) by caputo-type fractional-order time derivatives. therefore, we write the fractional-order sirs model as . disease-free equilibrium (or fixed-point or steady-state point) is determined by letting rightmost side of system (3) equal to 0, when i(t) ≡ 0. hence, disease-free equilibrium is given by theorem 2.1. consider d α t y(t) = g(t, y(t)), y(t 0 ) = y 0 , 0 < α ≤ 1, where d α t is caputo derivative of g(t, y(t)) : r + × r m → r m is a vector field. the disease free steady state pointê 0 is locally asymptotically stable if |arg(λ i )| > πα 2 , for i = 1, . . . , 4 where λ i , is the eigenvalue of jacobian g(t, y(t)) atê 0 . from equation det j n ξ , 0, 0 − λi 3×3 = 0, we obtain the eigenvalues as it is clear that λ 1 = −ξ < 0, and λ 3 = −e < 0, since ξ > 0 and e > 0. however, if λ 2 = an ξ − c > 0, the system becomes unstable, since |arg(λ 2 )| = 0 < πα 2 , for all 0 < α < 1. hence, system is stable locally asymptotically only if λ 2 < 0, that is possible when c > an ξ . now, we consider a quite useful and important number; namely, the basic reproduction number described in [40] among many others. it is a dimensionless number and typically denoted with r 0 . if r 0 < 0, then, the disease disappears and if r 0 > 0, the disease is disseminated amongst susceptible individuals (hosts.) finally, if r 0 = 0, this implies that an endemic disease exists in the population with a constant rate and infected hosts convey the disease to the susceptible individuals. hence, we conclude from the above computations that r 0 , for (3) is because r 0 > 0, the disease is disseminated amongst susceptible individuals. in particular, if any one of the coefficients ξ or c is equal to 0, then r 0 = 0, this implies that an endemic disease exists in the population with a constant rate and infected hosts convey the disease to the susceptible individuals. next, we consider the endemic steady-state point which is obtained when i(t) ≡ 0. hence, the endemic equilibrium point is which is determined by letting the rightmost part (3) equal to 0. therefore, where, it is possible to calculate the eigenvalues of this matrix for special parameter values and then study on stability, see e.g. [26] , of the system. notice that theorem 3.1. there exists a unique solution of (3) belonging to r 3 + . proof: after showing existence of non-negative solutions of the system, we present numerical solutions of (3). computational or numerical solutions of models in fractional calculus have been an active research area for scientists in recent years [21] [22] [23] [24] . we employ a numerical solution method; namely, trapezoidal rule, introduced and applied in, e.g. [19] , to obtain numerical solutions of the system (3). now, let us state the system (3) as where g ∈ c 2 [0, t ]. consider ∆ := c m . the discretization of the functions at the nodal points are used in the sense of approximate solution of u(t) depending on (α, g, ∆) is given by is approximate solution of (5) with the the mathematical models describing epidemic diseases are generated by deterministic, stochastic or fractional-order system of ordinary differential equations. each of these models explains the advantages of employed differential equation in their system. fractional-order models indicates that these types of models takes into account past effects of the system, stochastic models [27] [28] [29] 52 ] emphasis on probabilistic transmissions of the diseases among the individuals in the population and so on. to the best of our knowledge, there exists not any mathematical model for a epidemic disease which considers both fractional-order operators and white noise together. now, we perturb the fractional-order system (3) into a new system (mostly known as fractional-stochastic system) by an additive white noise to the-rightmost side of each equation in the system. our major aim in the section is to illustrate the use of both fractional-order operators and white noise together and test the effectiveness of resulting new model. it is possible to generate some other fractional-order stochastic sirs models simply by a new type of fractional order operator or the changing the way of adding the noise to the system. we consider the perturbed fractional-order stochastic sirs model: where dw i is a wiener process for each i = 1, 2, 3. disease-free point, obtained when i(t) ≡ 0, is the same point with the deterministic system (3). stability of the system (6) may be investigated by lyapunov stability method [25] which is applied to the system (6): hence, the system (6) is stable to 0, when we present an approximate solution of (6). by employing an euler type numerical method, discretized equations are expressed as s(n) = n (n) − ai(n − 1)s(n − 1) + ks(n − 1)r(n − 1) − ξs(n − 1) − c 1 s(n − 1) where ζ n , µ n , η n are gaussian random variables n (0, 1). the element, most of the sir, sis, seir, seirs type of epidemic disease models in the mathematical biology are generated by systems of ordinary differential equations. it is possible to assume that these models involve uncertainty, noise effects, undetermined outside forces and randomness in almost each time of the duration of the disease. these uncertainties may be strength of immune system of individuals, availability and resistance against vaccination, transmission rates of disease among individuals, and so on. the mathematical models in epidemiology is produced by means of either deterministic or stochastic systems of differential equations. we consider the deterministic fractional sirs system (2) by an additive fractional white noise [30] [31] [32] [33] which is stated via wick product. by using the equality: we present a numerical solution of the wick product added sirs model with an euler type numerical method. in the history of research so far, there exists not any mathematical model considering wick product interpretation of any epidemic disease. from this point of view, the present study is the first research work associating wick product with a sirs model. fractional brownian motion (fbm) is a process introduced by kolmogorov [41] , and employed by mandelbrot [42] . fbm is a gaussian process denoted typically by b h (t) depending on the hurst index, h, located in 0 < h < 1. fbm has zero mean and covariance function defined as when h = 1 2 ; b 1/2 (t) is a standard wiener process or brownian motion. some interesting properties of fbm including self-similarity, being centered gaussian process, having stationary increments, homogeneity in time, being symmetric and so on may be seen in, e.g. [43] [44] [45] [46] 50] . another quite useful and important tool in fbm and its applications is a product known as wick product. it is a paired (binary) type operation, typically denoted with a symbol , and it is employed in famous wiener-ito-chaos decomposition of function in l 2 −measurable space. wick product is a significant phenomena in stochastic and fractional stochastic calculus, mostly applied in the solutions of nonlinear differential equations in financial mathematics and fluid mechanics. a detailed treatment to wick product and its applications can be seen in ( [34] [35] [36] [37] [38] [39] 49] ). in this paper, we employ the wick product in the fractional brownian motion settings to represent fractional white noise. addressing an fractional wick-ito integral, following rule holds: in which the operator m is defined as where, the h n (x) defined in this equation is given as h n (x) = (−1) n e x 2 /2 d n dx n (e −x 2 /2 ), n = 0, 1, 2, . . . fractional white noise w h (t) also has a series representation given by where the operator m is defined by equation (10). now, by adding fractional white noise to the terms in the rightmost side of the system of equations (2), we get with some suitable supplementary conditions s(0) = s 0 , i(0) = i 0 , r(0) = r 0 . this system may be restated as writing in the integral form: it is further possible that we can express system (13) now, we solve the system (14) numerically employing an euler's type of numerical solution technique as follows: the in this section, we present two different exact (or sometimes known as explicit) solutions of the system given by (11) . in the first technique that we call it direct method, we solve the equations in the system (11) separately by considering each equation in the system alone. now, the first stochastic equation appearing in system of equations (11) . by using the identity given for total population of all indivials, we get that without loss of generality, let us ignore the term ρi(t)+ρr(t), since we deal with the population of susceptible individuals and by the fact that infected and recovered individuals are opposite to one another. then, ds(t) may be written as in a similar manner, the second equation in the system (11) may be expressed as finally, the third equation in the system (11) may be expressed as next, we present exact (or explicit) solutions of the system (2) having multiplicative fractional white noise by another method; namely, semi-martingale method [47] . let us write the equations of the system (2) with multiplicative fractional white noise. first equation in the system (2): from which s(t) is obtained as where, now, we express the second equation in the system (2) with multiplicative fractional white noise as follows: from which i(t) is obtained as finally, the third equation in the system (2) with multiplicative fractional white noise is: from which r(t) is obtained as fractional-stochastic differential equations and fractional brownian motion are very powerful modeling tools in science, engineering and economics. they found a very widespread applications areas in each of these areas. in the present research study, we pointed an original sirs model for a modeling of a possible epidemic disease under certain assumptions. we study on chaos (stability) analysis, numerical and explicit solutions of fractional-stochastic sirs models in fractional calculus and fractional brownian motion settings. as a future extension of the present research work, we plan to set up connections between sirs types models and control optimization problems. furthermore, because each of these epidemic disease models include parameters, we will study on problems of parameter estimation, identification and parameter sensitivity analyses of fractional-stochastic models in mathematical biology. use this form to specify the contribution of each author of your manuscript. a distinction is made between five types of contributions: conceived and designed the analysis; collected the data; contributed data or analysis tools; performed the analysis; wrote the paper. for each author of your manuscript, please indicate the types of contributions the author has made. an author may have made more than one type of contribution. optionally, for each contribution type, you may specify the contribution of an author in more detail by providing a one-sentence statement in which the contribution is summarized. in the case of an author who contributed to performing the analysis, the author's contribution for instance could be specified in more detail as 'performed the computer simulations', 'performed the statistical analysis', or 'performed the text mining analysis'. if an author has made a contribution that is not covered by the five pre-defined contribution types, then please choose 'other contribution' and provide a one-sentence statement summarizing the author's contribution. solutions of a disease model with fractional white noise author 1: prof. dr. mehmet ali akinlar ☒ conceived and designed the analysis specify contribution in more detail (optional; no more than one sentence) ☐ collected the data specify contribution in more detail (optional; no more than one sentence) ☒ contributed data or analysis tools specify contribution in more detail (optional; no more than one sentence) ☐ performed the analysis specify contribution in more detail (optional; no more than one sentence) ☐ wrote the paper specify contribution in more detail (optional; no more than one sentence) an introduction to stochastic processes with applications 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stochastic partial differential equations fractional white noise calculus and application to finance stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations new approach for the fornberg-whitham type equations traveling wave solutions of some important wick-type fractional stochastic nonlinear partial differential equations random fractional generalized airy differential equations: a probabilistic analysis using mean square calculus a novel method for a fractional derivative with non-local and nonsingular kernel modelling of transmission dynamics of nipah virus (niv): a fractional order approach the authors declare that there is no conflict of interests regarding the publication of this paper. we confirm that the manuscript has been read and approved by all named authors and that there are noother persons who satisfied the criteria for authorship but are not listed. we further confirm that there is nocompeting interest between the authors regarding the publication of this manuscript. key: cord-288894-2iaq3ayv authors: kumar, sachin; cao, jinde; abdel-aty, mahmoud title: a novel mathematical approach of covid-19 with non-singular fractional derivative date: 2020-07-01 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110048 sha: doc_id: 288894 cord_uid: 2iaq3ayv we analyze a proposition which considers new mathematical model of covid-19 based on fractional ordinary differential equation. a non-singular fractional derivative with mittag-leffler kernel has been used and the numerical approximation formula of fractional derivative of function [formula: see text] is obtained. a new operational matrix of fractional differentiation on domain [0, a], a ≥ 1, a ∈ n by using the extended legendre polynomial on larger domain has been developed. it is shown that the new mathematical model of covid-19 can be solved using legendre collocation method. also, the accuracy and validity of our developed operational matrix have been tested. finally, we provide numerical evidence and theoretical arguments that our new model can estimate the output of the exposed, infected and asymptotic carrier with higher fidelity than the previous models, thereby motivating the use of the presented model as a standard tool for examining the effect of contact rate and transmissibility multiple on number of infected cases are depicted with graphs. the fractional calculus is a classical branch has been developed recently to deal with a new discovered problems (see i.e. j. liouville and n. h. abel) [1] . also, more information and detail description have been discussed in kilbas et al. [2] , podlubny [3] , machado et al. [4] . fractional calculus generalize the differentiation and integration of integer order to real or fractional order. nowadays, this generalization is extended to the variable order and the differential equations and integral equation have real and variable orders have been discovered with a beautiful physical interpretation. control theory and stochastic process have many applications of fractional order differential equation [5] . the researchers have found many type of fractional derivative such as riemann-liouville, caputo, riesz, hadamard and grunwald-letnikov derivatives. the fractional differential equations (fdes) are obtained from ordinary ones by replacing the integer order to real order. fdes have many applications in science, engineering, biology, medical, finance, economics and groundwater flow [6, 7] . the applications of fdes are increasing day by day which leads to an urgent need to find the general solution of these fdes. to find out the analytical solution of these complicated * corresponding author. e-mail address: sachinraghav522@gmail.com (s. kumar) . fdes, one needs to invoke a numerical solution treatment. alarge number of methods have been discovered to deal with fdes and fpdes. some of them are eigen-vector expansion, homotopy perturbation method [8] , adomain decomposition method [9] , predictor-corrector method [10] , fractional differential transform method [11] and generalized block pulse operational matrix method [12] etc. the spectral method known as operational matrix method is very efficient method and easy to apply. it has a very desirable accuracy. some operational matrix based upon legendre wavelets [13] , chebyshev wavelets [14] , sine wavelets, haar wavelets [15] are available in literature. operational matrix based upon orthogonal and nonorthogonal polynomial are given in legendre polynomial [16] , laguerre polynomial [17] , chebyshev polynomial and semi-orthogonal polynomial as genocchi polynomial [18] . the novel corona virus was emerged first time in december, 2019, in wuhan city of china. the virus is a new type in its family. later world health organization (who) named it covid-19. due to this virus, any infected person faces many symptoms like as respiratory illness, cough, fever and difficulty in breathing [19, 20] . this spreads when a healthy person comes in a contact with the virus carried out by a infected person specially contact with the drops of cough and sneeze of infected person. some approximate solution of the time-fractional equations involving fractional integrals without singular kernel can be used to heed some light on the expected time development [21] [22] [23] . who has declared it as a pandemic due to widely spread of this virus. yet there is no medicine or vaccine to cure this virus infected people. only precautions can be adopted to keep ourselves safe. till the date 4, april, 2020, the number of confirmed covid-19 infected cases is 1,118,045 and 59,201 are dead due to this. the effect of this virus is more on the people of age greater than 40. the only cure is our precautions, we should have to quarantine ourselves in our homes to decreases contact rate, transmissible multiple. the human kind has the power to change the environment around us. there are some boundaries that should not be violated. in this present era, the intention of competition between humans, countries has developed so many powerful instruments to control on sea, air and ground. the human has created so many weapons like as guns, atom bomb, dangerous chemicals and nuclear bomb. so they violated the fundamental law of nature and led to so many natural disasters. we have forgotten that without nature we can not exist and we are just passenger. in this paper, we introduce a new and novel mathematical approach to study the behavior and dynamics of covid-19 with a new non-singular fractional derivative called mittag-leffler kernel's derivative. to solve the presented model, we use of a newly derived matrix with legendre collocation method. we will present some numerical treatments based on the number of infected people increases with increment in contact rate. the organization of thus article is as follows. in section 2 , some preliminary definition of fractional derivative and abc derivative are briefly discussed. the derivation of operational matrix of fractional differentiation based on orthogonal legendre polynomial on interval [0, a ] is derived in section 3 . the description of covid-19 model, its related data and procedure of numerical solution are given in section 4 . the results and discussion are presented in section 5 and the conclusion of all over article is given in section 6 . the definition of fractional integration and differentiation are available in literature [23] . there are mainly two types i.e., riemann-liouville and caputo [24, 25] . in starting, fractional derivatives with power law kernel are introduced. in recent years, many fractional derivative definitions with non-singular kernel are introduced as exponential kernel and mittag-leffler kernel. definition 1. we define the fractional integration of ( x ) of order the definition of riemann-liouville integration is given as follows definition 2. the definition of fractional differentiation with power law kernel is given in literature as follows with z ∈ [0, ∞ [ and n is an integer. the caputo definition has a similarity with integer derivative that is with m is a constant. where γ is floor function. all fractional operator are linear in nature as they follow the linearity property with m 1 and m 2 are constants. the caputo and riemann-liouville operator can be relate by the following expression [26] [27] [28] let a function ( x, t ) belongs to the sobolev space h 1 (0, 1). then this fractional derivative with mittag-leffler kernel which is also known as abc fractional derivative can be defined as . in this section we derive the legendre operational matrix of fractional differentiation on domain [0, a ], a ≥ 1, a ∈ n of mittag-leffler kernel derivative which is known as abc derivative. proof. from the definition (9) d n y l = 0 , l = 0 , 1 , . . . , n − 1 and for now, we use the series expansion formula of mittag-leffler function and evaluate the above integral as follows this is the desired approximation expression for abc derivative of here, we give the brief definition and property of extended legendre polynomial. we know that the legendre polynomial are orthogonal polynomial defined on interval [ −1 , 1] . by using the transformation z = 2 y −a a we transform the legendre polynomial from the interval [ −1 , 1] to the interval [0, a ]. the series form of this polynomial is given in the following expression the legendre polynomial are orthogonal on interval [ −1 , 1] with respect to the weight function 1. as we have extended the these polynomial to a larger interval [0, a ] so the orthogonality condition is changed according the transformation as follows with the help of these extended legendre polynomial a function χ( y ) belonging to l 2 [0, a ] can be written as a finite linear combination as the coefficient r j are determined as follows with the help of orthogonal condition where now in next theorem we will develop the legendre operational matrix of fractional differentiation on the domain [0, a ] with the help of eq. (13) . if we denote the column vector of extended legendre polynomial by n ( y ) then fractional differentiation of order n − 1 < γ < n is given by the formula, here r γ represents the operational matrix of fractional differentiation of order n × n. we can obtain this as the following where ξ i,j,l is defined by the following expression taking the help from the series expression of extended legendre polynomial and the definition of abc derivative to find out the ( i, j ) th element ϖ i,j of operational matrix r ρ , we perform the inner product as follows by using the orthogonal property of legendre polynomial we determined the above inner products value as follows where we use a numerical integration scheme known as simpson 1 3 rule and the interval of integration [0, b ] is divided into m equal sub parts with length of segment width h . putting the value of both of inner product in eq. (16) we obtained the following expression of assuming i, j = i l= ρ ξ i, j,l we get the final desired result we have derived the operational matrix of differentiation for fractional order on domain [0, b ]. but for the integer order elements of operational matrix is obtained as follows the function ζ j is defined as to study the measurement-induced by covid-19 transition due to the time development with different parameters and data, we performed the time-evolution using exact diagonalization. in this section we focus on cases dynamics interspersed with fractional order measurements in the two-dimension basis, and demonstrate the fact that there is a qualitative difference once initial values are introduced differently. we suppose that n p denotes the total population of people. this is divided into 5 categories as (i) s p -susceptible people. (ii) e p -exposed people (iii) i p -infected people (iv) a p -asymptotically infected people (v) r p -recovered people and n p = s p + e p + i p + a p + r p . the parameter π p denotes the birth rate of people and the parameter μ p represent the death rate of people in each case. the term η p s p i p represents the susceptible people will infected with a sufficient contact with infected people i p . here, η p is disease transmissibility coefficient. and the term ψη p a p s p denotes that susceptible people will be infected from asymptotically infected people with ψ transmissibility multiple of a p to i p and values of ψ belong to the closed interval [0,1]. the ψ = 0 states no transmissibility and ψ = 1 then this contact with asymptotically infected people will be treated as contact with infected people. the rate of susceptible people, from which they join the class of infected and asymptotatic class are denoted by ω p and ϱ p respectively. the removal and recovery rate from the class i p and a p to the class r p are depicts by the parameters τ p and τ ap respectively. the unknown function m is related to the seafood market or reservoir. the parameter η w is disease transmission coefficient from m to s p with term η w s p m . the contribution of virus from asymptomatically infected and symptomatic infected to the reservoir is denoted by ϱ p and ω p . the parameter λ denotes the removing rate of virus from reservoir. we present the model as follows [29] . the prescribed initial conditions for the above model are the value of used parameters are taken from the literature [29] ( table 1 ) we have derived the legendre operational matrix of fractional differentiation on domain [0, a ]. now we will use this newly operational matrix to find the solution of covid-19 model. so with the help of eq. (12) we will approximate the unknown functions present in our model as follows table 1 numerical value of used parameters for model (21 ψ 1 (x ) , . . . , ψ n−1 (x )) t is a column vector. now for approximating the left hand parts of all equations presented in model (19) we are operating the fractional operator of derivative on these sides and using eq. to find the solution we approximate the initial conditions by taking help of eq. (12) now using the eqs. (24) and (25) in our model we get the residual functions as now collocating eq. (25) and (24) at suitable points between the interval [0; a ], one gets a nonlinear system of algebraic equations. solving this system of equations and finding its dynamics, we discussed the numerical solution of our proposed model. we study the dynamics of susceptible, exposed, infected and asymptotically infected people using different fractional order. fig. 1 (a) shows the behavior between susceptible people versus time and fig. 1 (b) is indicates the relations between exposed people versus time. we see in fig. 1 (b) that number of exposed people increases with time. we observe that this growth increases as we increases the fractional order γ 2 from 0.7 to 1 while fig. 1 (a) shows that number of susceptible people decreases with time because they are getting into exposed or infected class. fig. 2 (a) and (b) are plotted between infected people i p ( t ) versus time and asymptotically infected people versus time respectfully. fig. 2 (a) predicts that number of infected people will increase with time. similar nature is also seen for the asymptotically infected people. the effect of fractional exponent on i p ( t ) and a p ( t ) is that it increases with increment in γ 3 and γ 4 from 0.7 to 1. as covid-19 spread with social contacting, touch with infected people and infected surfaces. we see that number of infected people increase exponentially with time. this behavior can be seen by taking data of italy and usa till today 4 april, 2020 as there infected people are increasing followed by this exponentially behavior. to study the behavior of this virus with contacting to infected or asymptotically infected people, we plotted the graph between the infected people i p ( t ) versus time and asymptotically infected people a p ( t ) versus time. we see in fig. 3 (a) that number of asymptotically infected people increases with time. and an important fact can be seen that it increases as contact rate η p increases. fig. 3 (b) shows that number of infected people increases rapidly like exponentially behavior and this number increases as contact rate increases. fig. 4 (a) and (b) show that the behavior of i p ( t ) and a p ( t ) with transmissibility condition ψ. we can observe that number of infected people and asymptotically infected people increases with increment in ψ. to study the behavior of model with different initial condition, we plotted two graphs fig. 5 (a) and (b). fig. 5 (a) is plotted between infected people versus time with different a p (0). we see as initial number of a p ( t ) increases the number of infected people also increases. similarly, from fig. 5 (b) , if at initially stage number of infected people is more than number of asymptotically infected people increases with this initially increment of infected people. measurement-induced covid-19 transitions represent an interesting new class of phase transition which shine light on the resilience of the present kind of viruses against a known one. they were initially explored for systems at ordinary differential equations dynamics and integrable models. in this work we have demonstrated that the nature of the measurement induced covid-19 transition can be well described by newly fractional calculus systems. the measurements have been made in a basis which is scrambled by different parameters and new controllers of the dynamics, then the transition from infected case to recovered case occurs at a nonzero measurement probability and can be controlled by changing the significant parameter, generalizing the previously studied chaotic systems. it is worth noting that one key difference between the model considered in this paper and previous models is that here there are more variables (spatial) disorder in the unitary part of the dynamics. this is noteworthy because we could derive an approximation formula for the fractional derivative of abc type of function ( t ≥ a ) on domain [0; a ]; a ≥ 1; a ∈ n : for the first time (as far as we know) and have developed the operational matrix of fractional differentiation with mittag-leffler kernel. the use of this newly derived matrix with legendre collocation method is applied to solve a system of fractional ordinary differential equation. we find out the dynamics of susceptible, exposed, infected and asymptotically infected people, that how is behave with different fractional fractional order. it is shown that the number of infected people increases with increment in contact rate. so if we want to stop this outbreak pandemic we should be quaran-tine to reduce the contact rate. the effect of transmissibility multiple is shown by graphical representation. effect on number on infected people and asymptotically infected people with different initial conditions. fractional calculus. in: fractals and fractional calculus in continuum mechanics theory and applications of the fractional differential equations fractional differential equations, to methods of their solution and some of their applications. fractional differential equations: an introduction to fractional derivatives recent history of fractional calculus application of fractional order calculus to control theory derivative with two fractional orders: a new avenue of investigation toward revolution in fractional calculus a numerical study of the nonlinear fractional mathematical model of tumor cells in presence of chemotherapeutic treatment homotopy analysis method for fractional ivps an eigenvector expansion method for the solution of motion containing fractional derivatives a predictor-corrector approach for the numerical solution of fractional differential equations a method for the numerical solution of the integro-differential equations numerical solution of fractional differential equations using the generalized block pulse operational matrix application of legendre wavelets for solving fractional differential equations solving a nonlinear fractional differential equation using chebyshev wavelets haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations on legendre polynomial approximation with the vim or ham for numerical treatment of nonlinear fractional differential equations laguerre polynomial solutions of a class of initial and boundary value problems arising in science and engineering fields novel identities for genocchi numbers and polynomials mathematical prospective of coronavirus infections in bahrain, saudi arabia and egypt modelling the spread of covid-19 with new fractal-fractional operators: can the lockdown save mankind before vaccination? some new hypergeometric transformations via fractional calculus technique new approximate solution of the time-fractional nagumo equation involving fractional integrals without singular kerne oscillatory properties of a certain class of mixed fractional differential equations continuous family of solutions for fractional integro-differential inclusions of caputo-katugampola type fractional modelling and the leibniz (l-fractional) derivative as viscoelastic respondents in polymer biomaterials chaos in a simple nonlinear system with atangana-baleanu derivatives with fractional order numerical approximation of riemann-liouville definition of fractional derivative: from riemann-liouville to atangana-baleanu validity of fractal derivative to capturing chaotic attractors modeling the dynamics of novel coronavirus (2019-n-cov) with fractional derivative the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. key: cord-299312-asc120pn authors: khoshnaw, sarbaz h.a.; shahzad, muhammad; ali, mehboob; sultan, faisal title: a quantitative and qualitative analysis of the covid–19 pandemic model date: 2020-05-25 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109932 sha: doc_id: 299312 cord_uid: asc120pn global efforts around the world are focused on to discuss several health care strategies for minimizing the impact of the new coronavirus (covid-19) on the community. as it is clear that this virus becomes a public health threat and spreading easily among individuals. mathematical models with computational simulations are effective tools that help global efforts to estimate key transmission parameters and further improvements for controlling this disease. this is an infectious disease and can be modeled as a system of non-linear differential equations with reaction rates. this work reviews and develops some suggested models for the covid-19 that can address important questions about global health care and suggest important notes. then, we suggest an updated model that includes a system of differential equations with transmission parameters. some key computational simulations and sensitivity analysis are investigated. also, the local sensitivities for each model state concerning the model parameters are computed using three different techniques: non-normalizations, half normalizations, and full normalizations. results based on the computational simulations show that the model dynamics are significantly changed for different key model parameters. interestingly, we identify that transition rates between asymptomatic infected with both reported and unreported symptomatic infected individuals are very sensitive parameters concerning model variables in spreading this disease. this helps international efforts to reduce the number of infected individuals from the disease and to prevent the propagation of new coronavirus more widely on the community. another novelty of this paper is the identification of the critical model parameters, which makes it easy to be used by biologists with less knowledge of mathematical modeling and also facilitates the improvement of the model for future development theoretically and practically. this work reviews and develops some suggested models for the covid-19 that can 19 address important questions about global health care and suggest important notes. then, we 20 suggest an updated model that includes a system of differential equations with transmission 21 parameters. some key computational simulations and sensitivity analysis are investigated. 22 also, the local sensitivities for each model state concerning the model parameters are 23 computed using three different techniques: non-normalizations, half normalizations, and 24 full normalizations. 25 results based on the computational simulations show that the model dynamics are 26 significantly changed for different key model parameters. interestingly, we identify that 27 transition rates between asymptomatic infected with both reported and unreported 28 symptomatic infected individuals are very sensitive parameters concerning model variables 29 in spreading this disease. this helps international efforts to reduce the number of infected 30 individuals from the disease and to prevent the propagation of new coronavirus more 31 widely on the community. another novelty of this paper is the identification of the critical 32 model parameters, which makes it easy to be used by biologists with less knowledge of 33 mathematical modeling and also facilitates the improvement of the model for future 34 development theoretically and practically. the idea of chemical kinetic theory is an important approach for understanding and 126 representing the biological process in terms of model equations. the important assumptions 127 to build such models are model states, parameters, and equations. this is because it helps 128 the investigation of mathematical modeling effectively and easily [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] infected, shows symptoms of covid-19, and detected by the government, either 185 from a rapid test or from voluntary action to report to the hospital. we assume that 186 all individuals in this class will get a specific treatment and supervision, whether it's 187 through monitored isolation or treatment in the hospital. 188 5. recovered group (r). this group present individual who get recovered from 189 covid-19, and had a temporal immunity. 190 the transmission diagram which illustrates the interaction between each group described in 191 figure 1 . where 208 209 furthermore, the model's constant parameters and initial states with their definitions are 210 described in table 1 . using equations (2)-(4), the model dynamics are described by the 211 following system of non-linear ordinary differential equations 212 the model initial populations are expressed in the following equation 213 co 263 mp 264 utat 265 ion 266 al 267 sim 268 ulat 269 ions 270 for 271 the 272 mo 273 del 274 stat 275 es 276 giv 277 en 278 in system (9) parameter has an effective role in the dynamics of . 419 3. the transition rate has also affected asymptomatic infected people, reported 420 symptomatic infected people, and unreported symptomatic infected people. it 421 can be seen that the model dynamics for such states become more flat when the 422 value of is increased. this is an important key element for controlling this 423 disease. practically discussed and investigated in this area. have improvements in interventions and healthcare programs. 510 the used model in this paper has further improved based on the computational results using 511 matlab for different initial populations and parameters. some main results can help in 512 understanding the suggested model more widely and effectively. by using computational 513 simulation, we identify some key critical parameters that have a great role in spreading this 514 virus among the model classes. one of the identified key parameters is the transmission rate 515 between asymptomatic infected and reported symptomatic individuals. this is an important 516 finding in the understanding of the covid-19 and how this virus spreads more quickly. 517 some other critical model parameters have investigated in this paper. for example, the 518 transmission parameter between asymptomatic infected and unreported symptomatic 519 individuals has a great impact on the dynamics of the model states. besides these findings 520 provide additional information about estimations and predictions for the number of infected 521 individuals. accordingly, our results in identifying key parameters are broadly consistent 522 with clinical and biological findings. 523 remaining issues are subject to sensitivity analysis. this is also an important issue that can 524 be further studied. we have applied the idea of local sensitivity to calculate the sensitivity 525 of each model state concerning model parameters for the updated model of the covid-19. 526 three different techniques are investigated which are non-normalizations, half 527 normalizations, and full normalizations. these provide us an important step forward to 528 identify critical model elements. by using local sensitivity approaches we concluded 529 that almost all model states are sensitive to the critical model parameters { } . this 530 becomes a great step forward and helps international attempts regarding the covid-19 531 pandemic outbreak. this may help to reduce the number of infected individuals from the 532 disease and to prevent the coronavirus more widely in the community. it can be concluded 533 that the identified factors can be controlled to reduce the number of infected individuals. 534 overall, our results demonstrate a strong effect of the key critical parameters on the 535 spreading covid-19. 536 therefore, based on the effect of each involved parameters over the model states, more 537 suggestions and interventions can be proposed for controlling the covid-19 disease. that 538 will be useful for any interventions and vaccination programs. accordingly, the healthcare 539 communities should pay more attention to the quarantine places for controlling this disease 540 more effectively. it can be strongly suggested that anyone in the quarantine places should 541 be separated from the others and should use only their separate equipment, bedroom, and 542 toilet to prevent the transmission of the virus through the touching of shared surfaces. 543 another suggestion is that reducing the contact between asymptomatic-symptomatic groups 544 and susceptible groups, this is effectively minimizing the number of infected people. it 545 seems necessary to plan a certain strategy to put the asymptomatic infected individuals on 546 quarantine places sooner rather than later. future research on identifying key critical 547 elements might extend the explanations of the new covid-19 more widely. it will be 548 important that future research investigates more suggested transmissions between the 549 model groups. for example, the model will further improve by adding two transmission 550 paths, one of them is between unreported symptomatic infected and reported symptomatic 551 infected, the other one is between asymptomatic infected and recovered individuals. 552 553 history and recent advances in coronavirus 555 discovery human coronaviruses: insights 558 into environmental resistance and its influence on the development of new antiseptic 559 strategies a 563 novel coronavirus from patients with pneumonia in china early transmission dynamics in 567 of novel coronavirus-infected pneumonia clinical features of patients 570 infected with 2019 novel coronavirus in wuhan pneumonia 573 of unknown etiology in wuhan, china: potential for international spread via commercial 574 air travel nowcasting and forecasting the potential domestic and 576 international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling 577 study estimationofthereproductivenumberofnovelcoronavirus (covid-19) and 581 the probable outbreak size on the diamond princess cruise ship: a data-driven analysis a discrete stochastic model of the covid-19 outbreak: 584 forecast 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authors: rafiq, danish; suhail, suhail ahmad; bazaz, mohammad abid title: evaluation and prediction of covid-19 in india: a case study of worst hit states date: 2020-06-19 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110014 sha: doc_id: 311054 cord_uid: dwns5l64 in this manuscript, system modeling and identification techniques are applied in developing a prognostic yet deterministic model to forecast the spread of covid-19 in india. the model is verified with the historical data and a forecast of 30-days ahead is presented for the 10 most affected states of india. the major results suggest that our model can very well capture the disease variations with high accuracy. results also show a steep rise in the total cumulative cases and deaths in the coming weeks. the advent and spread of 2019 novel coronavirus (sars-cov-2) has posed a global health crisis with a sharp rise in cases and deaths since its first detection in wuhan, china, in december 2019. the infection causes illness ranging from common cold to extreme respiratory disease and death [1] . currently, the prime epidemiological risk factor for 2019 novel corona-virus disease includes close contact with infected individuals with an incubation period of 2 − 14 days [2] . the case mortality rate is projected to range from 2 to 3% [3] . various drugs are being assessed in line with previous researches into therapeutic treatments for sars and mers, however, there is no robust evidence for any significant improved clinical outcome [4] . apparent risk of acquiring the disease has led many governments to institute a variety of control procedures like quarantine, isolation and lock-down measures. despite rigorous global containment measures, the frequency of the novel corona-virus disease continues to rise, with over 4.5 million confirmed cases and over 300, 000 deaths worldwide as on 17 th may, 2020 [5] . although countries around the world have enhanced capacity building of the laboratory systems and response procedures, yet, there is a need for proper disease surveillance systems. comprehending the initial transmission of the virus and analyzing the effectiveness of control measures are crucial in assessing the prospects for continued transmission in newer locations. this necessitates tracking the course of the pandemic to be able to foresee its emergence for a better response. prospective studies on modeling and forecasting of the epidemic have been carried out to provide analytical predictions on the size and end phase of the spread. wu, et al. [6] have used a susceptible exposed infectious recov-email address: danish_pha2007@nitsri.net (corresponding author) (danish rafiq) ered (seir) meta-population model to simulate the epidemic across all major cities in china. early dynamics of transmission and control of covid-19 within and outside wuhan has also been studied using a stochastic transmission dynamic model [7] . another study used the seir compartmental model to predict the feasibility for conducting the summer olympics of 2020 in japan [8] . similarly, abdullah, et al. [9] presented a stochastic sir model to predict the spread of covid-19 in kuwait. a classical seir type mathematical model is also presented in [10] to study the qualitative dynamics of covid-19 in india. further work has been carried out in [11] , with special focus on the transmissibility of super-spreader individuals in wuhan, china. besides the above mentioned compartmental models, some other methods have been used to model and forecast the covid-19 spread. for example, in [12] , a data-driven estimation method like long short-term memory (lstm) is used for the prediction of total number of covid-19 cases in india for a 30-days ahead prediction window. in [13] , daily forecasts of covid-19 activity from global epidemic and mobility model (gleam), an agent-based mechanistic model is used as an one of the inputs to produce stable and accurate forecasts two days ahead of current time. harun, et al. [14] have used box-jenkins (arima) and brown/holt linear exponential smoothing methods to estimate and forecast the number of covid-19 cases in the g8 countries. al-qaness et al. [15] have incorporated a modified version of flower pollination algorithm (fpa) coupled with the salp swarm algorithm (ssa) to forecast the number of confirmed cases of covid-19 for ten days in china. as on 17 th may 2020, india observed a total cases of 90, 927 with 2, 872 deaths [16, 17] . the very first case was reported on 30 th january 2020, in a coastal state of kerela (southern india) when a student returned from wuhan, china. subsequently, the number of positive cases in in-dia rose rapidly due to the arrival of many passengers via airways [18] . an overview of the spread of covid-19 in india is shown in figure (1) . it can be easily seen that the virus has spread to entire country with the worst hit states being maharashtra (30,706 cases), gujarat (10,988), tamil nadu (10,588), delhi (9,333), rajasthan (4,960), and madhya pradesh (4,789). figures (2) and (3) show the trend of rising new cases and deaths in india. this manuscript demonstrates a control-theoretic, datadriven estimation technique to derive a time-series model from the historical data collected from [5, 16] up-to 17 th may 2020. the model is then used for the prediction of the total number of cases and deaths in most affected states of india for the next 30 days. the paper is sectioned as follows: section (2) describes the system identification method employed. section (3) presents the predicted cases and deaths along-with some discussions. finally, conclusions are presented in section (4). to estimate the spread of covid-19 in india, we used a predictive error minimization (pem) based system identification technique to identify a discrete-time, single-input, single-output (siso) model [19] [20] [21] . different models very identified for different states based on the data collected. the models were then verified on the testing data and upon validation, the models were used to predict the total number of cases and deaths for the next 30-days in the 10 worst hit states in india. the discrete-time, identified model can be realized in the state-space from given as: where the y(t) represents total number of cases or deaths of a particular area which is proportional to system state vector x(t) ∈ r n , u(t) is the time series input and t s is the sampling interval. here, the unknowns to be identified are a ∈ r n×n , k ∈ r n×1 and c ∈ r 1×n which are in canonical form. here, n is the dimension of the state-space model. the identification problem can thus be posed as to selecting a model set m (θ) (indexed by a finite dimensional parameter vector θ) and evaluating a member from the set which best describes the recorded input-output relation according to a given criterion. one such criteria as given in [22] is defined as : where (t, θ) = (y 0 −ŷ 0 , ..., y n −ŷ n ) is referred as the prediction error, l(.) is a scalar measure of fit and z(t) = [y t (t), u t (t)] and n is length of data-set. typical choices of l(t, θ, ) can be seen in [22] . the identified model thus minimizes the 1-step ahead prediction and the error (t, θ) between the measured y(t) and predicted valuesŷ(t) is used to make the future prediction about the system. the prediction error identification estimate is thus given as: here, we have taken: figures (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) show the dynamics of the forecasted response for most infected states of india along-with a 10step predicted response comparison with the validation data. further results are presented in table (1). as seen from table (1), maharashtra has recorded the highest number of covid-19 cases accounting for 36% of the total countrys caseload. it has also witnessed the sharpest rise in covid-19 deaths with mumbai being the epicenter of the pandemic in india. the constant influx of tourists, reliance on public transportation and population destiny have cumulatively made the metropolitan city hospitable for corona virus. even though the state is conducting more tests, the violation of physical distancing rules by individuals particularly in containment zones results in the mixing of infected with healthy population. moreover, unlike other red zones of maharashtra, mumbai faces shortage of icu beds and dedicated covid-19 hospitals. according to the prediction made herein, it would be inevitable that mumbai and its suburbs would continue to see an upsurge in the number of cases and deaths for at least up to 17 th june 2020. gujarat has recorded the second highest covid-19 mortality rate in the country in spite of reporting its first case as late as march 20. the covid-19 mortality rate of ahmedabad city is 6.8%, which is double the national average. officials acknowledge that while gujarat had its guard up sufficiently fast, there was a delay in testing. even by mid of march, the daily average was as less as 15 tests per day, going up to 200/day by the end of march. according to the data driven identification scheme employed herein, the mortality rate in gujarat may increase as high as 15.2% up to 17 th june 2020. tamil nadu, although being the third worst hit indian state in terms of covid-19 cases has witnessed the least number of mortalities with 1 among 143 positive cases succumbing to the disease (see fig 6) . this is attributed to its credibility as a trusted medical center of the country. chennai has the highest medical tourism in india with the states average being above the national average in the health sector. this may be the reason that the predictable mortality rate of tamil nadu projected in this study is least among the rest of the states in consideration (see table (1)). as per our prediction based on data up to 17 th may 2020, delhi along with other states would continue to see marginal surge in the number of covid-19 cases owing to the relaxations in lock-down measures. the impact of removing the curbs will be more evident by the mid of june 2020. the under-funding of the healthcare system, paucity of testing labs, violations of the lock-down protocols and inadequate quarantine facilities arranged by states and union territories are the biggest hurdles in combating the spread. the study concerns the spread of covid-19 in india. a control-theoretic approach is used to develop an epidemic model to simulate and predict the disease variations of 10 most affected states of india. results depict a rapid increase in the number of cases in the coming days. however, it is pertinent to mention that the future estimation provided, is subject to certain system parameters and can vary based on the external inputs like lock-down measures, social-distancing, vaccine/drug development, rapid testing, etc. information provided by our model could help establish a realistic assessment of the situation for the time-being and in the near future in order to apply the appropriate public health measures. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. carlos,mild or moderate covid-19 three months of covid-19: a systematic review and meta-analysis coronavirus: covid-19 has killed more people than sars and mers combined, despite lower case fatality rate clinical features of patients infected with 2019 novel coronavirus in wuhan, china, the lancet nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study early dynamics of transmission and control of covid-19: a mathematical modelling study prediction of the epidemic peak of coronavirus disease in japan forecasting the spread of covid-19 in kuwait using compartmental and logistic regression models a model based study on the dynamics of covid-19: prediction and control mathematical modeling of covid-19 transmission dynamics with a case study of wuhan prediction for the spread of covid-19 in india and effectiveness of preventive measures a machine learning methodology for real-time forecasting of the 2019-2020 covid-19 outbreak using internet searches, news alerts, and estimates from mechanistic models modeling and forecasting for the number of cases of the covid-19 pandemic with the curve estimation models, the box-jenkins and exponential smoothing methods optimization method for forecasting confirmed cases of covid-19 in china on the consistency of prediction error identification methods dynamical effects of overparametrization in nonlinear models improved structure selection for nonlinear models based on term clustering system identification -theory for the user, appendix 4a method for the solution of certain problems in least-squares algorithm for least-squares estimation of nonlinear parameters the levenberg-marquardt algorithm: implementation and theory, numerical analysis on the decay rate of hankel singular values and related issues ministry of human resource development (mhrd), new delhi, india, is duly acknowledged. 2. author 1 would like to thank asiya batool for fruitful discussions. the authors declare no potential conflicts of interest regarding the publication of this paper. key: cord-269363-drjj705k authors: nenchev, vladislav title: optimal quarantine control of an infectious outbreak date: 2020-07-28 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110139 sha: doc_id: 269363 cord_uid: drjj705k this paper studies the optimal control of an infectious spread based on common epidemic models with permanent immunity and no vaccine availability. assuming limited isolation control and capacity constraints on the number of infections, an optimal quarantine control strategy that balances between the total number of infections and the overall isolation effort is derived from necessary optimality conditions. the specific optimal policy is then obtained by optimizing the switching times of this generalized strategy. in the case of a newly emerged disease, these results can be used in a data-driven receding horizon manner to improve actions as more data becomes available. the proposed approach is applied to publicly available data from the outbreak of sars-cov-2 in germany. in particular, for minimizing the total number of infections or the number of isolated individuals, the simulations indicate that a sufficiently delayed and controlled release of the lock-down are optimal for overcoming the outbreak. the approach can support public health authorities to plan quarantine control policies. since the first recorded covid-19 infections in wuhan, china, the disease has spread worldwide. there has been a plurality of attempts to interpret and predict the development of the disease based on public data using first-principles epidemiological [1] [2] [3] , data-driven [4] , or mixed models [5, 6] . based on these predictions, a wide variety of temporal isolation policies have been established in most affected countries. however, evaluating the effectiveness of these responses is notoriously difficult due to the reliability of available data, and since their impact on the outbreak evolution can only be measured with a substantial delay. mathematical analysis of first-principle models of biological systems can provide valuable insight, e.g., on their general properties [25] , or how to control infectious disease outbreaks [7] . in the simplest model, the population is divided into the susceptible s , infected i and recovered r groups (therefore called the sir model), and their evolution over time is represented as a set of coupled ordinary differential equations. augmenting the state by exposed e individuals, leads to the so-called seir model. applications of outbreak controls minimizing various different cost functions have been proposed for the sir model, e.g., [8] [9] [10] [11] [12] to name a few, as well as heuristically tailored policies [13, 14] . although s(e)ir models can be too simplistic for many real outbreaks, they often allow deriving complete analytical solutions, thus providing a foundation e-mail address: vladislav.nenchev@gmail.com for rigorous mathematical results upon which more complex models can be built. an issue of practical concern for many disease outbreaks without an available vaccine, such as for sars-cov-2 as of june 2020, is minimizing the overall quarantine effort or the final outbreak size, while respecting control and capacity constraints on the current number of infections. to account for quarantine measure (similar to [5] and [15] ), the sir model is extended by a quarantined subpopulation q . infected individuals are moved depending on the applied control rate from i to q , where they cannot cause additional infections. then, using the direct adjoining approach [16] and applying pontryagin's minimum principle (pmp) to the corresponding state-and input-constrained control problem, in this paper it is shown that bang-bang or bang-boundary controls are admitted in a particular sequence to solve the corresponding optimal control problem. to obtain the specific optimal policy, the switching times between the bang-boundary subarcs are determined in an induced optimization problem. analogous results can be obtained for the seir model. upon an outbreak of a previously unknown disease, better model parameter estimates can be obtained as more data becomes available, and the induced optimization problem can be recomputed in a data-driven receding horizon manner to improve actions. the proposed methods are applied to publicly available data until june 2020 from the outbreak of sars-cov-2 in germany. in particular, minimizing the total outbreak size or the total number of isolated individuals are studied. the findings of this work can be used to evaluate if the policies that have been established in practice resemble the optimal policy, and support public health authorities in implementing effective future quarantine strategies. the remainder of this paper is structured as follows. in section 2 , a mathematical statement of the problem is provided. in section 3 , the optimal quarantine policy under isolation control and capacity constraints, and, in section 4 , the data-driven receding horizon control scheme are described. then, in section 5 , the approach is applied based on covid-19 data from germany until june 2020. finally, the main results and future work are discussed in section 6 . we address epidemics with no vaccine availability, in which the only available control is based on isolation. let the entire population p be divided into three sub-populations: susceptible s ; infected i ; and recovered r . births' and deaths' effects are neglected. the recovered population is to be interpreted as those who can no longer infect others, and, who cannot become reinfected by the disease. this is known as the classic epidemiological sir model [7] . to include quarantine or isolation control, the model is augmented by introducing a time varying quarantine strength control u ( t ) and a corresponding quarantined (and also recovered from the disease) population q . then, with x = [ s, i, r, q] t ⊆ r 4 + the sub-populations' evolution is governed by the following system of coupled nonlinear ordinary differential equations: where the population p = 4 i =1 x i (t) , and the infection b and recovery m rates are assumed to be constant in time. note that the model assumes homogeneous mixing among the sub-populations, as well as uniform susceptibility and disease progress for every individual in the population. the sir model assumes a direct transition from susceptible to infected. it can be easily augmented by including exposed individuals e to account for a significant incubation period during which individuals have been infected but are not yet infectious themselves. this so-called seir epidemiological model [7] has been employed in several prior studies, such as the sars outbreak, e.g., [17, 18] , as well as the covid-19 outbreak [1] [2] [3] . since the seir model did not provide any additional qualitative insights in the studied problem, for simplicity, the sir model is analyzed, and details for the seir model can be found in the appendix. an important characteristic of epidemics is the reproduction number thus, applying the control u ( t ) results in reducing the effective reproduction number r t . determining the maximal quarantine control u max is highly dependent on local quarantine policies and population specifics, directly corresponding to the minimally achievable reproduction number (2) . another important parameter of epidemics is the fraction α s of severe conditions among infected individuals requiring hospitalization and admission to an intensive care unit. highly infectious diseases (with a high r t ) causing many severe conditions in the infected population (corresponding to a high α s ), may quickly lead to an overload of available treatment resources and, thus, cause a large number of additional potentially avoidable deaths. assuming that the overall available capacity is described by α c , this capacity limitation can be described as the in this work, the goal is to obtain an optimal quarantine control policy u ( t ), t ∈ [0, t f ] for a fixed final time t f , that minimizes a weighted combination of the total number of infections and the overall number of quarantined individuals at time t f . note that the overall number of quarantined individuals corresponds to the applied level of containment and can, thus, be linked to the economical impact of the outbreak. the total number of infections is denoted by the sum of the final recovered and final quarantined populations, and is directly correlated to the overall number of deaths. this can be captured by the cost function where w ∈ [0, 1]. the control is limited to u ( t ) ∈ [0, u max ] at all times, such that u = 0 corresponds to an uncontrolled epidemics evolution, and u = u max to the case, when maximal possible isolat f ] exists, this leads to the following (mayer type) optimal control problem. (4) is minimized subject to the dynamics (1) , while respecting the constraint (3) at all times. in the following, using the direct adjoining approach and applying pontryagin's minimum principle, necessary optimality conditions are derived for the optimal control problem above. further, it is shown that the obtained locally optimal solutions constitute an optimal generalized strategy under certain assumptions. as (3) is a pure state constraint, (higher order) time derivatives are used to obtain the control u ( t ) that yields a system evolution along the active state constraint. since u ( t ) appears linearly in the state equation of x 2 (1) , the first total time derivative of the function h ( x ( t )) (3) contains the control explicitly: . if τ 1 and τ 2 are maximal, let τ 1 be the entry-time and τ 2 the exit-time of the boundary arc. on the boundary arc, thus, the control on a boundary arc is obtained from the boundary control (6) depends on x 1 and is, therefore, not guaranteed to be in the feasible control set [0 , u max ] . thus, assume that to derive first order necessary optimality conditions by applying pmp, a lagrange multiplier η(t) ∈ r associated with the state constraint (3) is introduced. for that, based on the aforementioned regularity and boundedness conditions on a boundary arc, and supposing that the state space constraint is not active at the initial and final time, i.e., h ( x (0)) < 0, h ( x ( t f )) < 0, the direct adjoining approach [16] is applied. the corresponding augmented hamiltonian is given by with the adjoint variable λ ∈ r 4 . as shown in maurer et al. [19] , there exist an absolutely continuous λ( t ) and a piecewise absolutely continuous multiplier function η( t ), such that the following for the sir model, the following holds: as the hamiltonian (7) is linear in u ( t ), the optimal control will depend on the sign of the switching function, given by as suggested in ledzewicz and schättler [20] , with the total time derivative of (11) and the costates (8) , holds on a boundary arc with t ∈ [ τ 1 , τ 2 ]. this yields the explicit expression for the multiplier assuming that a solution of problem 1 exists and the control u ( t ) has finitely many switching times, the following proposition holds. the generalized optimal control policy for problem 1 is if t 3 > t 2 , the tangency constraints (16) must also hold. proof. on interior arcs, where the state constraint is inactive h ( x ( t )) < 0, from the minimum condition (10) , the optimal control is the switching function (11) is zero ⇐⇒ λ 2 (t) = 1 or x 2 (t) = 0 . the latter case is trivial and will be neglected in the following. to fulfill λ 2 (t f ) = 0 , λ 2 (0) > 1 and ˙ λ 2 (0+) ≤ 0 , such that the optimal control policy may start with a zero segment for [ t 0 , t 1 ). if ) > 0 and σ (t 1 −) < 0 assuming u (t 1 −) = 0 , and as −˙ σ (t 1 )(0 − u max ) > 0 , a second segment for t ∈ [ t 1 , t 2 ) may have the control u max . on a boundary arc, given the regularity and boundedness conditions for the boundary control, the minimum condition ) ≤ u max (from the boundedness condition of the boundary control) and ˙ x 2 (t 2 +)( ū b ) = 0 , and thus, the continuity conditions on the boundary σ ( substituting relevant equations into (13) yields , resulting from the boundedness condition of the boundary control, completes the additional tangency constraints (16) . if h ( x ( t )) < 0 and σ (t) = 0 for t ∈ [ ˜ τ 1 , ˜ τ 2 ) , a singular arc may occur. since ∂ σ /∂ u = 0 holds trivially, the singular control u s can be obtained from d σ /d t = 0 with η(t) = 0 and λ 2 (t) = 1 , which yields u s = b p x 1 − m = u b . the optimality of the corresponding singular arc can be checked with the generalized legendre-clebsch condition of first order, which yields − ∂ ∂u transitioning from an interior arc to a singular arc and vice versa at time τ ∈ [0, t f ] violates the continuity conditions if the condition above holds for any junction time τ , then μ( t 2 , τ ) > 0 for the boundary arc and μ( τ , τ ) ≤ 0 for the singular arc would have to hold simultaneously. thus, a singular arc will not be part of the optimal solution. finally, since h (x (t 3 −)) = 0 and h 1 (x (t 3 −)) = 0 if t 2 = t 3 , ˙ x 2 (t 3 )(u = 0) < 0 and the optimal control (14) ends with a zero segment for t ∈ [ t 3 , t f ], which also satisfies the transversality condition (9) . if u b (x 1 (t 2 )) ≤ u max does not hold, by analogous derivations to the proof above, it can be shown that adding a finite number n of sequences of a zero arc followed by an u max -arc, until a switch to the boundary arc at time t 2 n +2 becomes possible with u b (x 1 (t 2 n +2 )) = u max , fulfills the necessary optimality conditions. for simplicity, this case is excluded in the following elaborations. as a consequence of the above proposition, solutions of problem 1 will exhibit the structure of the generalized bangboundary control policy (14) . note that the specific optimal policy strongly depends on the choice of w as well as the model parameters. what remains to be determined are the optimal switching times. instead of directly optimizing the switching times t 1 , t 2 , t 3 , t 4 , introduce the arc durations ξ i = t i − t i −1 , assuming t 4 = t f . thus, the induced optimization problem reads (1) , (3) , (14) , (15) , (16) this optimization problem can be simplified further. if w = 1 , i.e., the goal is to minimize only the total number of infections, the op(3) is violated. for other values of w , the optimal control is given by (14) , which denotes two possible policies -one containing a boundary arc, and one without a boundary arc. note that (1) is explicitly solved on the boundary arc, as already used in the proof of proposition 1 . thus, solving (17) can be replaced by solving the three aforementioned boundary value problems and choosing the policy that yields the minimal cost. corresponding derivations for the seir model can be found in the appendix. the induced optimization problem (17) can be recomputed in a data-driven manner to obtain an improved quarantine action, as shown in the following. while a fast response upon an epidemic outbreak is essential, for a newly emerged disease, like in the case of covid-19 at the beginning of 2020, disease characteristics are highly uncertain. as more data and insights become available over time, it can be expected that the parameter estimates of the models can be improved continuously. this can be achieved, e.g., by minimizing the mean square error loss function between measured state data ˜ (1) . analogously, this can be done for the seir model with p = { b, m, c} . at any time τ , using the parameter estimates p τ obtained by (18) and the measured state ˜ x (τ ) , the induced optimization problem (17) can be re-solved with x 0 = ˜ x (τ ) . assume that the parameter estimates are updated whenever new data becomes available. based on that, a data-driven receding horizon approach is adopted, where the control is obtained by re-computing (17) , only when the current parameter estimates deviate from the parameter estimates used in the previous computation step evaluated by a function ζ by a pre-defined threshold δ. the similarity function ζ was assumed to be a simple absolute deviation in this work, but it can contain more complex considerations like, e.g., the covariance of the estimates. this leads to algorithm 1 . note that, in general, receding horizon control approaches with uncertain parameters cannot guarantee that hard constraints will not be violated. thus, to increase satisfaction probability, the capacity constraint (3) can be tightened conservatively based on the confidence of the current parameter estimate p t . the proposed method is applied to data from the sars-cov-2 outbreak in germany until june 2020. two cases are considered -w = 0 and w = 0 . 5 , which correspond to minimizing only the overall number of isolated individuals and a trade-off between the overall number of isolated individuals and the total size of the outbreak, respectively. the optimization problems are solved with scipy's bvp and minimize routines. most parameters are taken from a report of the german robert koch institute [21] -a susceptible population of x 1 (0) = 80 . 10 6 and an initial number of infections x 2 (0) = 10 0 0 , corresponding to the number of registered infections in germany on march 8, 2020. the initial numbers of recovered and quarantined individuals are both 0. for the seir model, an initial exposed individuals number of 100 is assumed. further, a basic reproduction number r t = 2 . 0 is assumed in the uncontrolled models, as well as an inverse mean infectious period in days of m = 0 . 091 . the percentage of infected individuals admitted to an intensive care unit is 1.2% and the mortality is 0.5%. the overall number of intensive care units in germany was α c = 40 . 10 3 by the end of march 2020 [22] . the assumed maximum control value u max is defined such that it leads to an effective reproduction number ˜ r t = 1 . 0 , i.e., with eq. (2) given by u max = 0 . 091 . for the receding horizon control approach, publicly available data from rki from march 1 until june 1, 2020 to re-estimate the parameters p 1 = { b, c, m } for this time period, and assume the actual parameters r t = 2 . 5 , c = 0 . 25 and u max = 0 . 068 thereafter. the control is re-computed according to algorithm 1 . figs. 1 and 2 show the model state trajectories, the capacity constraint and the control over time for the uncontrolled (left) and the optimally controlled (right) with the aforementioned parameters for the sir and seir model, respectively. it is notable that for both cases, i.e., for w = 0 and w = 0 . 5 , respectively, the optimal solution is given by a bang-boundary control, when using the sir or the seir model. applying the optimal control to the sir model under the given assumptions, the pandemic is predicted to lead to an expected total number of cases of 54 084 981, and a total number of deaths of 270 424. the simulations also indicate that the turnaround (final peak of active cases) should occur 26 weeks after the start of the outbreak. the overall outbreak should be completely overcome after 44 weeks, without a second wave of infections. this closely resembles predictions made in an der heiden and buchholz [21] and [23] . fig. 3 shows the result with the data-driven receding horizon control. due to the high estimate of the reproductive number in the initial phase, the maximal isolation control starts earlier, and, overall, applies more isolation due to the parameter uncertainty. note that the control maintains the capacity constraint. in practice, a toggling between bang-bang controls could be prevented by introducing a hysteresis, when a new control policy is computed in algorithm 1 . applying the optimal control under the given assumptions, the pandemic is predicted to lead to an expected total number of cases of 49 004 201, and a total number of deaths of 245 021. note that the lower number of infections is a consequence of the longer period of applied quarantine control compared to the preceding simulation example. another consequence of the longer quarantine is a slightly delayed turnaround that should occur 30 weeks after the start of the outbreak. the overall outbreak should be completely overcome after 50 weeks, without a second wave of infections, and while control and capacity constraints are maintained at all times. the optimal quarantine control of an infectious spread was studied assuming an sir epidemic model with permanent immunity and no vaccine availability. the control strategy, which minparameters : initial parameters p 0 , initial times t 0 , 1 , t 0 , 2 , t 0 , 3 , threshold δ, final time t f output : optimal switching times t 1 , t 2 , t 3 1: if p t−1 = ∅ then 2: (18) with data x | [0 ,τ ] and initial parameter value p t−1 . 5: if ζ (p t−1 , p t ) > δ then 6: t 1 , t 2 , t 3 ← solve (17) initialized with previous t 1 , t 2 , t 3 7: p t−1 ← p t . 8: end if 9: return t 1 , t 2 , t 3 imizes the final outbreak state assuming that only limited quarantine control is available and capacity constraints have to be respected, was derived from necessary optimality conditions. the solution is to possibly delay the control action first, and then, if needed, apply maximal isolation. if the capacity boundary is reached, a boundary control maintaining the infection numbers may be applied afterwards, until the infection numbers begin to drop. this result was used in a data-driven receding horizon control approach to improve actions as more data becomes available for a previously unknown disease. the application of the proposed schemes to publicly available data from the outbreak of sars-cov-2 in germany indicate that a sufficiently delayed and controlled release of the lockdown are optimal, and would result in overcoming the outbreak within one year. note that a constant overall population was assumed throughout this work. effects resulting from population exchange could be addressed, e.g., as suggested in xue et al. [24] , leading to a stochas-tic induced optimization problem. an interesting remaining question is how to determine the maximal isolation control value, as well as how to map the boundary control values to practical isolation actions. a promising approach for addressing both topics could be using machine learning techniques, e.g., building upon ideas from [5] . another research direction could be to embed the parameter estimation uncertainty into the data-driven optimization to achieve an improved exploration-exploitation trade-off. the author declares that he has no competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. and c denote the exposure and infection rates, respectively. the reproduction number remains as defined in (2) . analogously to the elaborations provided for the sir model, the most important equations are derived as: following an analogous approach, proposition 1 can be proven for the seir model, yielding an identically structured optimal policy. novel corona virus 2019-ncov: early estimation of epidemiological parameters and epidemic predictions. medrxiv estimation of the transmission risk of the 2019-ncov and its implication for public health interventions nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study prediction of criticality in patients with severe covid-19 infection using three clinical features: a machine learning-based prognostic model with clinical data in wuhan. medrxiv quantifying the effect of quarantine control in covid-19 infectious spread using machine learning. medrxiv early dynamics of transmission and control of covid-19: a mathematical modelling study infectious diseases of humans: dynamics and control an optimal isolation policy for an epidemic optimal isolation policies for deterministic and stochastic epidemics an explicit optimal isolation policy for a deterministic epidemic model optimal control of epidemics with limited resources time-optimal control strategies in sir epidemic models strategic release of lockdowns in a covid infection model. medrxiv optimal policies for control of the novel coronavirus disease (covid-19) outbreak effective containment explains subexponential growth in recent confirmed covid-19 cases in china a survey of the maximum principles for optimal control problems with state constraints modelling the sars epidemic by a lattice-based montecarlo simulation extension and verification of the seir model on the 2009 influenza a(h1n1) pandemic in japan optimization methods for solving bang-bang control problems with state constraints and the verification of sufficient conditions a local field of extremals for single input systems with state space constraints modellierung von beispielszenarien der sars-cov-2-epidemie 2020 in deutschland projecting the spread of covid19 for germany. medrxiv2020 a data-driven network model for the emerging covid-19 epidemics in wuhan, toronto and italy robustification as a tool in modeling biochemical reaction networks the author thanks m.c. lüffe for enriching discussions. including quarantine control analogously to (1) , the subpopulationsâ evolution in the seir model is governed by the following system of coupled nonlinear ordinary differential equations: key: cord-332618-8al98ya2 authors: barraza, néstor ruben; pena, gabriel; moreno, verónica title: a non-homogeneous markov early epidemic growth dynamics model. application to the sars-cov-2 pandemic date: 2020-09-18 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110297 sha: doc_id: 332618 cord_uid: 8al98ya2 this work introduces a new markovian stochastic model that can be described as a non-homogeneous pure birth process. we propose a functional form of birth rate that depends on the number of individuals in the population and on the elapsed time, allowing us to model a contagion effect. thus, we model the early stages of an epidemic. the number of individuals then becomes the infectious cases and the birth rate becomes the incidence rate. we obtain this way a process that depends on two competitive phenomena, infection and immunization. variations in those rates allow us to monitor how effective the actions taken by government and health organizations are. from our model, three useful indicators for the epidemic evolution over time are obtained: the immunization rate, the infection/immunization ratio and the mean time between infections (mtbi). the proposed model allows either positive or negative concavities for the mean value curve, provided the infection/immunization ratio is either greater or less than one. we apply this model to the present sars-cov-2 pandemic still in its early growth stage in latin american countries. as it is shown, the model accomplishes a good fit for the real number of both positive cases and deaths. we analyze the evolution of the three indicators for several countries and perform a comparative study between them. important conclusions are obtained from this analysis. -we present a mathematical model based on a new stochastic process described by a pure birth process. -the proposed model matches the subexponential growth on the early stage of an epidemic. -the mathematical expression of the cumulative case incidence and cumulative death curves is obtained, with a quite accurate fit in both cases. -the model contains two parameters, the immunization and infection rates. the behavior in time of those parameters allows to assess the evolution of the outbreak. -we obtain a new indicator, the mean time between infections. this indicator allows not only to monitor the epidemic growth but also to predict the peak of cases. the coronavirus disease transmitted by the sars-cov-2 appeared in china by the end of 2019. more than 17000000 of cases and 675000 deaths were reported since then. the lack of a vaccine and an effective treatment for this disease forced the governments to adopt lockdown actions in order 5 to protect the population and to slow down the spread of the outbreak. despite those actions, health systems in several countries resulted overwhelmed. many mathematical models were developed in order to predict the behavior of the outbreak in several ways, most of them based on the well known sir (susceptible-infectious-recovered) models. we present in this work a different 10 approach having the advantage of a rapid and clear interpretation of its parameters. we obtain this way a quite useful new indicator, the mean time between infections (mtbi). modeling contagion has attracted the interest of statisticians for decades. these models are useful to model the spread of contagious diseases in a popula15 tion or plantation. a well-known example is the polya urn model, where balls of different colors are extracted from an urn in such a way that when a given ball is extracted, not only is it put back but also a certain number of balls of the same color is added. the probability of getting a ball of that color in the next drawing is thus increased, modeling a contagious process. contagion modeling 20 has been already applied in many areas of engineering, see for example [1, 2, 3]. the polya stochastic process is obtained as the limit from the polya urn model in a similar way as the poisson process is obtained from a bernoulli process. as it happens with the homogeneous and non-homogeneous poisson processes, the polya stochastic process can be described by the pure birth equation 25 2 (see chapter 17 of [4] ). the main difference between these two types of pure birth processes is that, in the polya process, the event rate is not just a function of time but also of the number of individuals in the population. there are also cases where the event rate depends only on the number of previous events but not on time, like the yule process. however, since the probability of births rises 30 due to an increase in the population while the probability of an individual birth remains constant, it does not describe a true contagion process. the stochastic mean of the polya process is a linear function of time, as will be shown. this characteristic makes the polya process unsuitable for application in many real cases of disease growth dynamics and engineering. hence, 35 we propose a different model based on a pure birth process with an event rate that, like polya's, depends on both the elapsed time and the number of previous events, but with a different functional form. we obtain this way a mean number of events that is a nonlinear function of time, with either an increasing or decreasing first derivative provided a given parameter is greater or less than 40 one. an important application of our proposed model is the study of the spreading of diseases. we obtain this way a process controlled by two competitive phenomena, infection and immunization, i.e. two opposing forces. hence, we get a model with two parameters: the transmission and immunization rates. the mean value function shows different behavior according to whether the ratio between infection and immunization rates is greater or lower than one. in the former case, its second derivative is positive and the mean value curve is convex, whereas on the latter case its second derivative is negative and the resulting curve is concave. the limit case of infection/immunization ratio γ/ρ = 1 is also 50 possible, resulting on a mean value function that grows linearly over time. the evolution of the parameters shows how well the pandemic is being controlled, either by lowering the infection rate via isolation and quarantine measures or by increasing the immunization rate with a vaccination campaign. we obtain this way three useful indicators: the infection/immunization ratio, the immunization 55 rate and the mean time between infections (mtbi). our model is quite suitable to be applied to early epidemic growth dynamics, where exponential models fail (this matter is discussed in [5] ). since ours falls in the category of subexponential models, the results presented in this article support the idea developed in the cited reference. another advantage of our 60 approach is that, as a subset of the subexponential cases, we are able to model cases with linear and sublinear (concave) cumulative case incidence curves, as it will be shown. as an interesting and quite important application, we apply our model to the recent sars-cov-2 pandemic as an alternative to the sir models. our 65 theoretical values match real data, which allows us to assess the evolution of the oubtreak and to predict its trend a few days ahead. the parameters can be easily estimated through simple methods, such as least squares, applied either to positive cases or deaths. we show applications to data from several countries with good agreement. this paper is organized as follows: motivation and related work are exposed in section 2, the proposed model is presented in section 3, applications to the sars-cov-2 pandemic are developed in section 4 and some discussions on the results we obtained are presented in section 5. the final conclusions are exposed in section 6. our main motivation is to obtain a model that describes an epidemic outbreak at its first stage, before it reaches the inflection point in the case incidence curve, which is useful to monitor how contagion is spreading out. this 80 first stage corresponds to an r naught greater than one. our model is inspired in the polya-lundberg process, which comes from the contagious urn model as explained in the previous section. since the mean value function of the polya-lundberg process is a linear function of time (see appendix b), we introduce a modification in the event rate in order to get a mean value function that grows 85 subexponentially with either positive or negative concavity as we observe in the early epidemic growth curves usually reported. there exists a wide variety of models that can be used to describe the evolution of an epidemic. main standard is held by the so called compartmental 90 models, i.e., the family of sir based models (sir, seir, sirs, etc. see for instance [6, 7, 8, 9] ). these models consist of a set of differential equations that take into account transitions between different compartments. the first classical sir model was porposed by kermack and mckendrick [10] . due to the recent covid-19 coronavirus pandemic, most of the scientific 95 community is dedicated to study its behavior, both by using sir based models and introducing new ones. in [11] , a stochastic term is introduced in the system of differential equations to simulate noise in the detection process. in [12] the authors use an arima based method to correct errors arising from the delay between the day the sample is taken and the day a diagnosis is made, which compartmental models can be found in [15] . in [16, 17] some well-known ma-110 chine learning and time series algorithms (like arima and svr) are used to learn from a present dataset and forecast the case incidence curve up to the following six days. combinations of sir models embedded with some random components was proposed in very recent articles, see for instance [18, 19, 20] . a theoretical framework to model epidemics by a non homogeneous birth-and-115 death process was proposed in [21] . an extensive analysis of epidemics at their early stage was performed in [5] . in this study, the authors propose a phenomenological model (see also [22, 23] ) and analyze the application of sir based models with either exponential and subexponential behavior. our approach is quite different. on one hand, our 120 model is stochastic whereas theirs is deterministic. on the other hand, we obtain our contagion model as a special case of a very general and well-known probabilistic model, the pure birth process. it is interesting to compare the differential equations the cited authors arrive with ours, which will be done in a future publication. furthermore, and unlike them, we are also able to model 125 cumulative case incidence curves that grow with negative concavity. the model we propose is based on a pure birth process. these processes describe the evolution of a population where individuals can only be born. we are then modeling the epidemic spread as births in a population, where every 130 birth corresponds to a new infection case. our novel approach consists of finding a proper incidence rate that describes the contagion phenomenon. we develop next the fundamentals of pure birth processes and our proposal. we propose to get the probability of having r infections in a given time t 135 from a pure birth stochastic differential equation. pure birth processes describe the behavior of a population where individuals can only be born and are not allowed to die. all the individuals are assumed to be identical. the probability of having r individuals in the population at a given time t, p r (t), is given by the solution of the following differential equation, see [4, 24] : (1) assuming we have 0 individuals at time 0, we impose the following initial conditions on eq. (1): p 0 (0) = 1 and p r (0) = 0 for r ≥ 1. to see that p r (t) is the process given by eq. (1) is also markovian, where the number of individuals corresponds to the state of the system, s r (t). the only transitions 145 allowed 2 are s r (t) → s r (t + dt) and s r (t) → s r+1 (t + dt) with probabilities: the dependence of λ r (t) on t defines the type of process. if λ r (t) is a function of t only or a constant, then the process is non-homogeneous or homogeneous respectively, with independent increments in both cases. if λ r (t) is also a function of r, the process has dependent increments. from eq. (1), the probability of having no births in a certain time interval greater than t − s given the population has r individuals by the time s is given by the well-known exponential waiting time: the mean number of infections in a given time is under certain conditions (see appendix b to verify those are satisfied by 2 actually, other transitions have probabilities that are higher order infinitesimals. our model) we have the following differential equation for the mean value: depending on the proposed function for λ r (t), the mean number of failures may or may not be easily obtained. our formulation allows us to calculate the mean time between infections (mtbi), this is, the mean time between births in a pure birth process. from eq. (3), we can predict the mean time between infections (mtbi) after r infected individuals were detected by the time s using a model with incidence rate λ r (t) as: here, z is a normalizing constant to consider cases where the probability of 155 having no infections in an infinite time interval is greater than zero and is given by eq. (7) (see appendix c for a full deduction): eq. (6) gives the expected time to the next infection given that by the time s the infected population consists on r individuals, this is, the mean time between two consecutive detections. this indicator is, in most cases, u-shaped (see fig. 1 ). an exceptional case is also shown in fig. 4b , and will be discussed later). it is expected to decrease at first, showing the acceleration of the spread (infections occur more often). when the cumulative case incidence curve reaches the inflection point and the epidemic starts to mitigate, mtbi shows a flattening portion (infections occur 165 at a constant rate). then, due to the deceleration of the epidemic (infections occur less often) mtbi tends to get larger again. this curve resembles the bathtub curve, well known in reliability engineering, which is also obtained when this class of models is applied to reliability studies, see [25] . 3.2. proposed incidence rate 170 we propose a two-parameter incidence rate given by the following expression: the parameters involved in eq. (8), indicate how fast the outbreak is spreading. we can see that the γ parameter is related to the strength of the contagion effect, since it is a factor of the number of infections, and the ρ parameter is related to the outbreak mitigation, either by natural immunization or due to external measures. that is the way how these two parameters, once estimated, 175 are useful to monitor the disease progress and the impact of actions taken by health institutions. these actions can affect either the γ parameter via the population solation and quarantine, or the ρ parameter by, for instance, vaccination campaigns. as we will show, the ratio between both rates is the exponent of 9 time t on the mean value function (eq. (11)), and it determines the curve's 180 concavity, provided it is greater or less than one. we can compare our proposed incidence rate with other formulations by rewriting eq. (8) as follows: the numerator of the second term in eq. (9) can be compared with the numerator of the incidence rate analyzed in [26] (eq. 1: is the number of infections, β the transmission probability per contact and c the 185 contact rate) by replacing i(t) with r and β c with γ; our γ parameter can be interpreted as the transmission rate in the same way. in that proposal, the n (t) denominator corresponds to the total population, since the authors consider the per capita incidence rate. as an interesting remark, it can be seen that it also coincides with that of the polya-lundberg process. the first term in eq. (9) is 190 equal to λ 0 (t), hence it can be considered as the initial incidence rate. on the other hand, following the usual behavior of ρ, this term rapidly vanishes with time. being the ratio γ ρ the coefficient of r in the contagion rate (eq. (8)), it takes into account both the infection and immunization rates, so it can be directly rethe evolution of these two parameters, the ratio γ ρ and ρ as a function of time, is indicative of how well the epidemic is being controlled by actions taken from health institutions. we expect a strong decrease in γ ρ , and in consequence, a strong increase in ρ. our definition of the outbreak parameters deserves a detailed analysis. the well-known definition of the basic reproduction number r 0 assumes all the population is susceptible and no immunization is present. this would imply ρ to be zero; in that case, eq. (8) reduces to λ r (t) = λ r. which is the event rate of another classic stochastic model, called the yule 205 process. in that case, the event rate given by eq. (10) grows linearly with the population size, which results in an exponential expression for the mean value function (see [27] for more details about the yule process). this means that, if we allow our ρ to eventually be zero, the yule process becomes a special case of ours, hence exponential growth can also be modeled. however, as it can be 210 seen in table 1 , even the worst case scenarios (very sharp incidence curves) differ from exponential growth, being this another empyrical confirmation of chowell's thesis presented in [5] . it should be remarked that our proposed rate (eq. (8) (5), as demonstrated in appendix b. from the incidence rate (eq. (9)), we are able to get the exact solution of p r (t) (see appendix a) and therefore we can show that m (t) is finite and eq. as seen from eq. (11), the mean value obtained from our model is a nonlinear function of time with a positive or negative concavity whether γ ρ is greater or lower than one, as shown in fig. 2 . this expression is the functional form that for our proposed model, it is straightforward to obtain an expression for the mtbi by inserting eq. (8) into eq. (6). this leads to eq. (12): replacing r by the mean number of infections given by eq. (11) we get eq. (13) which is a useful conditional expression, as shown in appendix c. in this section, we show the indicators obtained from several countries' re-235 ports. data were taken up to early july 2020 from https://ourworldindata. org/coronavirus-source-data. for asian and european countries, and also the usa, only data from the early epidemic stages were considered for analysis, this is, up to the inflection point. real data and fitted curves are shown in figs. 3 and 5. we fit eq. (11) to the incidence and death curves of the recent cov-2 pandemic in order to obtain the three mentioned indicators. the case incidence curves are shown in fig. 3 , while in order to show that the proposed model fits well not just positive concavities as shown before, we consider the data from uruguay, which presents cumulative number of deaths curves are shown in fig. 5 . the parameters and the coefficient of determination are shown in table 2 . in this section we show the evolution of the γ ρ parameter obtained from our model as a function of time, for the analyzed countries. in order to obtain the evolution of this parameter over time, we perform several estimation runs with successive subsets of the dataset and record each estimate. this picture is indicative of how effective the measures taken by different 265 countries have been, or how fast the outbreak is reaching the mitigation stage. as the transmission rate approaches the immunization rate, this parameter reaches the value of 1 as a limiting case. curves are depicted in fig. 6a . as previously discussed, the strength force that contains the outbreak is 270 determined by the immunization rate ρ parameter, and we expect it to increase due to actions taken by governments. then, we analyze the evolution of this parameter in the same way as it was done for γ ρ in the previous section. values of ρ as a function of time are depicted in fig. 6b. it is interesting to analyze the current behavior of the parameters for latin american countries, since they are still at the first stage of the pandemic and where, with the exception of brazil, the lockdown was relaxed and turned strict again more than once. curves are depicted in fig. 7 . spreading out and the inflection point has not been reached yet. this indicator is useful in order to estimate the moment when the curve will start to flatten. in fig. 9 we depict the mtbi for china in order to see the flattening section 285 of this indicator as the cumulative cases of incidence reaches its inflection point. by the time we are writing this report (early july 2020), contrarily to europe and the usa, latin american countries have not yet reached the inflection point in the cumulative incidence curve, which can also be seen through the mtbi characteristics. this inflection point in the cumulative incidence curve 290 corresponds to a minimum in the mtbi indicator, as seen in fig. 8 for italy. it table 3 . cumulative incidence figure 9 : mean time between infections and cumulative cases incidence for china. the well goodness of fit obtained for several countries shows that the spreading is governed by the same law. although with different parameter values, the cumulative case incidence curve follows the same mathematical expression. those parameters indicate the level of contagion and the effectiveness of the actions taken by governments. the model fits rather well the number of in-300 fected and deaths in the population previous to the inflection point, at the early pandemic growth stage, though its prediction beyond four or five days ahead is generally too pessimistic and overestimates the actual data. the mtbi minimum values indicated in table 3 show an interesting result. in r that implements our proposed model, and which was used to obtain the shown curves, is available at [28] . it can be easily proved that eq. (1) has a unique solution and is given by (see also [29] ): is a probability mass function provided that ∞ r=0 p r (t) = 1; this is shown by following the same steps as in section 4 of chapter xvii from [4] (see also [30] , section 4.3.2), where it is proved for the case λ r (t) depends on r but not on t. the authors are working on a future publication where this property will be properly generalized to functions that depend on t as well. recall our proposed functional form of λ r (t): let the function µ r (t) be: it can be seen from eq. (a.3) that µ r (t) = λ r (t) and µ r (0) = 0 ∀r ≥ 0. we now define the auxiliary function which does not depend on r. note that µ ∆ (0) = 0. it is a fact that the following equality holds: the proof can be done by differentiating the right side of eq. (a.5) to show it's indeed a primitive of the left side's integrand. since the functions are also equal on t = 0, fundamental theorem of calculus yields that the equalty is valid for every t. considering the initial condition p 0 (0) = 1 (the beginning population is 0), the probability mass functions are given by: this is demonstrated by induction over r, using eq. (a.6) as inductive hypothesis and p 0 (t) from eq. (a.1). in fact, we can rewrite eq. (a.6) as: using the fact hat µ r (t) = µ 0 (t) + rµ ∆ (t) and replacing µ 0 (t), µ 0 (0) and µ ∆ (t) by their expression yields: (a.8) it should be noted that, since the expression eq. (a.1) is also valid for the case λ r (t) = ρ γ ρ +r 1+ρ t , we can define µ r (t) and µ ∆ (t) so that the same properties are achieved. therefore, following the same steps we get the expression of the 365 pmfs for the polya-lundberg process. a similar procedure provides the pmfs for the yule process. multipliying eq. (1) by r and summing we obtain since the condition lim r→∞ r λ r (t) p r (t) = 0. (b.1) is attained for our model, we get ∞ r=1 r p r (t) = ρ 1 + γ ρ r 1 + ρ t p r (t) = ρ 1 + ρ t the same procedure that leads to eq. (b.3) is also valid for the polya-lundberg process. replacing λ r (t) by its expression and solving the differential 370 equation results in a linear mean value function. in the same way, with λ r (t) = λ r, we get an exponential mean value function, which corresponds to the yule process. appendix c. mean time between infections calculation for the contagion model 375 we begin from the exponential waiting time (eq. 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bieliūnas, martynas; skarbalius, gediminas; misiulis, edagaras; navakas, robertas title: simplified model of covid-19 epidemic prognosis under quarantine and estimation of quarantine effectiveness date: 2020-07-29 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110162 sha: doc_id: 298626 cord_uid: duvzwxv0 a simplified model of covid-19 epidemic dynamics under quarantine conditions and method to estimate quarantine effectiveness are developed. the model is based on the daily growth rate of new infections when total number of infections is significantly smaller than population size of infected country or region. the model is developed on the basis of collected epidemiological data of covid19 pandemic, which shows that the daily growth rate of new infections has tendency to decrease linearly when the quarantine is imposed in a country (or a region) until it reaches a constant value, which corresponds to the effectiveness of quarantine measures taken in the country. the daily growth rate of new infections can be used as criteria to estimate quarantine effectiveness. the 2019-2020 coronavirus outbreak has started since december 29th, 2019 in wuhan, hubei province, people's republic of china, and has progressively expanded through almost all countries. this ongoing pandemic of coronavirus disease 2019 (covid-19) caused by severe acute respiratory syndrome coronavirus 2 (sars-cov-2). in order to prevent the spread of the disease, many of the countries affected by the disease has been put under quarantine, which has led that 4,5 billion people (58 % of whole world population) around the globe faced some form of lockdown [1] . however, the remaining problem of the current covid-19 disease is that second wave of the epidemic is expected after imposed quarantine is discontinued in each country. on the basis of epidemiological data provided by european centre for disease prevention and control [2] and estimation that the number of infections can be four times higher than the number of confirmed cases [3] , it can be evaluated that for most countries only less than 10 % of the population of the infected country/region will acquire the immunity after first wave of epidemic controlled by the effective quarantine. that would require next 5-6 similarly controlled waves to achieve herd (population) immunity of 50-70 % in order to stop further spread of disease. such scenario is improbable as it would require a significant amount of time, during which the individual immunity could be lost because even today, when more than 4.8 million people have been infected in more than 188 countries and territories (20 may 2020, [4] ), it is not clear for how long time the recovered patients have the immunity. however, hope for sars-cov-2 antiviral vaccine or sufficiently effective antiviral drugs, or sars-cov-2 virus mutation to less aggressive strain [5] gives us chance to survive through the course of the controlled epidemic. therefore, forecasting the spread of the pandemic on the basis of mathematical models are extremely important for decisions how to prepare countries in order to avoid overloading of health system and manage other related problems. valuable information that could be obtained from modelling is forecast of the expected time and number of most active infected cases and the effectiveness of applied infection control measures. it is current global trend, that the experience and available data from already affected countries are used to model the pandemic dynamics in other countries before the epidemic has reached the peak or to estimate effectiveness of various scenario of the next wave management [6] . most popular epidemic dynamics models of covid-19 are based on transmission model for a directly transmitted infectious disease, such as standard compartment models of disease sir [7, 8] , or more advances derivates, such as seir and similar models [9] [10] [11] [12] . many of the models, which are used to forecast the covid-19 epidemic, do not accurately capture the transient dynamics of epidemics; therefore, they give poor predictions of both the epidemic's peak and its duration [13] , because calibration of parameters are based on dynamics of such non-reliable epidemiological data as number of active infectious cases. we propose to build epidemic analysis and model on the dynamics of rate of new infection cases as more reliable epidemiological data together with an assumption of effectiveness to isolate registered infectious during imposed quarantine. other epidemiological parameters, such as numbers of active infectious, deaths, recovered, severe patients and others, can be forecasted from the forecasted dynamics of the rate of new infectious cases. the proposed approach is based on sir model. the simplest sir model consists of three compartments: s for the number of susceptible, i for the number of infectious, and r for the number of removed (recovered, deceased or immune) individuals. these variables (s, i, and r) represent the number of people in each compartment at a particular time. we denote the total population size by . the dynamics of the simplest sir system (excluding birth and death) can be described by the following set of ordinary differential equations [11] : where γ is the rate of recovery or mortality, β is the infectious rate controlling the rate of spread that represents the probability of transmitting disease from infectious individual to susceptible individual. the disease transition rate β/ni is defined as a product of β and a probability of disease transmission during a contact between an infectious individual and a susceptible individual. in that case, is the average number of contacts per person per time unit and can be defined by the typical time between contacts = 1⁄ . the transition rate between i and r defined by γ and is estimated from typical time until recovery = 1⁄ . in case of isolation or self-isolation, γ can be defined by the average number of days that a person is infectious (before they are isolated or self-isolate), = 1⁄ . the dynamics of the infectious class depends on the following ratio: the so-called basic reproduction number (or basic reproduction ratio) of an infection and represents the average number of infections generated by one individual over the course of the infectious period. the estimation model parameters and , as well as 0 , may vary depending on country due to methodological issues, including different assumptions and choice of parameters, utilized models, used datasets and estimation period. in addition, during the spread of the sars-cov-2 virus infection, it was found that the model parameters are varying according to the dynamics of transmission of the novel coronavirus outbreak as well as the case reporting rate, which requires to build up more sophisticated and complex models [14] .the model parameters are constantly calibrated according to the last epidemiological data because forecast is constantly refreshed. the wide and rapid spread of the disease forced many countries to impose quarantines, entry bans and other restrictions during the pandemic to reduce the movement of population and recent travellers in most affected regions [15] . global restrictions that apply to all foreign countries and regions have also been imposed in other countries preventing their own citizens to travel overseas. these measures, especially quarantine, helped to suppress the spread of the disease within the population of various countries with different effectiveness. other important property of the pandemic is that total number of infected cases is much smaller than population size of infected country or region due to quarantine measures: this assumption allows to simplify epidemiological models used to simulate codiv-19 disease spread under quarantine. consequently, on the basis of eq. ( 3 ), the parameter of sir model can be estimated as the number of new registered cases of infection to number of active cases ratio: where is the rate of new infected cases the number of daily new infected cases is defined as so, expected number of new cases in next day could be predicted by the today number of active infected individuals multiplied by the infectious rate in general, the infectious rate is time dependent and usually can be described by a complex function with additional parameters that must be daily calibrated according to last epidemiological data. behaviour of the infectious rate during quarantine may differ in different countries, which reduces possibilities to build correct model of the infectious rate on the basis of epidemiological data from other countries. furthermore, there is no clear criteria to relate influence of quarantine actions with the infectious rate . as illustration, the dynamics of the parameter estimated from active infected individuals and new cases by eq. ( 4 ) is demonstrated in figure 1 [17] . in order to damp daily and weekly fluctuations, , and estimated the parameter were smoothed by moving average, where each average is calculated over a sliding window of length = 7 days: where is any parameter to be smoothed. as can be seen, the dynamics of the parameter demonstrates similar non-linearly decreasing behaviour after the start of the quarantine for almost all countries. in addition, an estimation of the number of active infected individuals depends on the number of recovered active cases of infections , which in turn depends on testing protocols, tests number, delays in testing and other circumstances, which are different in each country. consequently, this makes = − unreliable parameter for model calibration. for example, in lithuania the first pool consisting of 54 recovered cases of 999 total cases was reported only after 30 days from the first infection registration [18] , which can be explained only by troubles in recording of recovered cases and can be also expected in other countries. in contrary, the number of daily new infected cases , despite of countries specificity, is still the most reliable parameter allowing to estimate disease spreading. in addition, other epidemiological parameters, such as numbers of active infectious, deaths, recovered, severe patients and others, can be forecasted from the forecasted dynamics of the number of daily new infectious cases. in order to build model of covid-19 disease spread during quarantine, we make simplifying assumptions, that:  registered infected individuals do not spread virus to health individuals, because of an effective isolation of the registered infected individuals;  no imported infections. new cases of infectious individuals are generated by previously infectious individuals until they are registered; therefore, the number of new cases of the day is dependent on the number of new cases generated during previous day and the effectiveness of imposed quarantine actions: 1) restriction of social contacts and mobility, 2) identification of infectious individual and his contact tracing as soon as possible, 3) isolation of infectious individuals. let us analyse hypothetical simplified case. the averaged time period, during which an infected individual is registered and isolated after infection, is denoted . during this time period, the infected individual is infectious and infects individuals. in the end of this period, the infected individual is isolated and does not take more participation in the process of the epidemic spread. during the time period [ − , ], new infected cases are generated, from which ( − , ) are registered and isolated. these new registered infected cases will then generate more cases and the number of new registered infection cases for next time period [ , + ]: where is the growth rate of new cases and can be associated with the effective reproduction number [19] . the parameter is time dependent and > 1 means that number of new cases is increasing, while < 1 -decreasing. according to eq. ( 6), the parameter can expressed as follows: it is evident that in limits of → 0, we have which, together with eq. ( 5 ), gives differential equation for disease dynamics during quarantine: keeping in mind eqs. ( 4 ) and ( 5 ), the parameter can be related to the infectious rate as follows let us define 0 as the growth rate of new infection cases when epidemic starts and spreads without control. because of 0 > 1, the number of new cases is exponentially increasing. it is expected that the population in any country would start to behave more safely even though no official quarantine actions were taken; consequently, the growth rate of new cases is expected to slowly decrease before strict quarantine rules are imposed. the growth rate of new infected cases can be defined as 0 = ( = ) > 1 at the quarantine start time = . because of 0 > 1, the number of new cases is still increasing. let us assume that the imposed quarantine is ideally effective, which means that all infected individuals are isolated until the end of the time span and do not have contact with other individuals during time span [ , + ]. furthermore, no new cases will be generated for the next time period [ + , + 2 ]. in such case, the growth rate of new cases becomes constant and equal to 0 after the quarantine starts: ( > ) = = 0 (figure 2 (a) ), which means that the supposed effectiveness of the quarantine is equal to 1 and epidemic is stopped immediately. if the quarantine is less effective, then the growth rate of new cases is non zero constant ( > ) = > 0 , which leads to slower spread of the disease in case of 0 > > 1 and suppression of the epidemic in case of 1 > > 0 (figure 2 (b) ). consequently, quarantine effectiveness can be measured as the zero effectiveness = 0 means that the growth rate of new cases during quarantine remains the same as before: = 0 . therefore, in order to suppress the disease, the growth rate of new cases must be below 1, which means that quarantine effectiveness must be greater than 1 − 1 0 ⁄ . in realistic scenario, the growth rate of new cases does not change sharply at the time because of time lag due to incubation period, infections generated by non-registered infected individuals and so on. in addition, it takes a time for people to adjust to the quarantine requirements after the beginning of the quarantine; therefore, there is a time lag before people start to strictly follow the rules. consequently, the growth rate of new cases decreases from initial value 0 until reaches constant value satisfying the effectiveness of applied quarantine at the time ( figure 3 ). − depends on the properties of covid-19 disease, such as incubation period and time span of individual being infectious, and how quickly quarantine actions are implemented. as a result, the angle = − (( 0 − ) ( − ) ⁄ ) depends on − and the effectiveness of applied quarantine actions, because the case of = 0 corresponds to situation when applied measures does not improve quarantine effectiveness and remains constant. the case of approaching to 90 demonstrates exceptional effectiveness of the applied measures. negative corresponds to increasing the growth rate of new cases , what signalizes about critical state of broken quarantine. in addition, for a big country with heterogeneous population distribution across the country, it is expected to have several infection clusters and, therefore, wider time period − will be for decreasing stage (and consequently sharper φ) due to different start time of disease in each infection cluster. so, angle φ depends on the quarantine effectiveness, durations specific for the disease and homogeneity of the infected country or region. the proposed parameters and together with analysis of the population mobility and social contacts [20] can be used to estimate effectiveness of country or region lockdown measures. in this paper we will analyse only . in order to predict covid-19 disease spread in infected country or region with imposed quarantine, a model of the growth rate of new cases needs to be developed. it is possible to build up such model speculatively in general; however, it is reasonable to analyse dynamics of in various countries. we analysed covid-19 pandemic data from various countries [2, 17] . the daily growth rate of new infection cases was estimated on the basis of the registered daily new infection cases , ( ) (defined by eq. ( 7 )) and smoothed by moving average according to eq. ( 9 ) with sliding window of length = 7 days in order to damp daily and weekly fluctuations: in this case, the parameter ( ) can be interpreted as the average daily reproduction number [19] . as illustration, the dynamics of ⟨ ⟩, obtained as the parameter smoothed by moving average according to eq. ( 9 ) with sliding window of length = 7 days, is demonstrated in figure 4 as can be seen, that despite of fluctuations, the scenario described in figure 3 can be identified in the dynamics of the parameter : chaotic behaviour of before quarantine demonstrates linear decreasing until some certain value after quarantine. let us analyse some examples in greater detail. 16 ) is presented in figure 5 (b) by solid blue line. if to average fluctuations over supposed straights, the daily growth rate of new cases was almost constant or slowly decreasing before quarantine, started to decrease sufficiently after quarantine had been imposed and had somewhat linear dependency on time during quarantine time, which allowed to approximate by descending straight line ( figure 5 consequently, the model of the daily growth rate of new cases can be described as follows: where 0 = 1.058 is value in the beginning of the quarantine occasional new infections occurring during last 4 weeks demonstrate that efficiency of applied quarantine actions was sufficient to suppress substantially the epidemic in the country, but the effectiveness is still not enough to decimate the infection. let us note, that small numbers of daily new infection cases ( < 20) are registered after ′ = 30 days, and therefore, the proposed method cannot not be used for such small numbers, because of too casual nature of new infections. consequently, the epidemic end cannot be forecasted accurately. we will shortly overview some other countries demonstrating applicability and possible shortcomings of the proposed approach. the next example of the similar scenario is switzerland. the government of switzerland announced that no lockdown would be implemented; however, some restrictions were implemented [21] . on 13 march 2020, the federal council decided to cancel classes in all educational establishments until 4 april 2020 and banned all events (public or private) involving more than 100 people. furthermore, the borders were closed, and border control was enacted. on 16 march 2020, the federal council announced further measures and a revised ordinance. measures included the closure of bars, shops and other gathering places until 19 april 2020, but leaved open certain essentials, such as grocery shops, pharmacies, (a reduced) public transport and the postal service. since 20 march 2020, all events or meetings over 5 people were prohibited, and economic activities would continue including construction. the daily growth rate of new cases began decreased before first official restrictions of social contacts (figure 6 (b) ) while daily numbers of new cases were relatively small. during three following weeks after the restrictions initiation, continued to decrease from 1.207 on 13 march 2020 until 0.927 and it is expected for to remain constant until the end of the epidemic ( figure 6 ). [2] and the calculated growth rate of new cases are presented in figure 7 . the modelled was overestimated (figure 7 (b)), because was approximated by one straight in the time interval from (corresponding to 21 march 2020) to = + 14 days (figure 7 (a) ). to improve the model of , the change of alert level from 2 to 4 on fifth day after the beginning of the quarantine must be considered. accordingly, must be approximated by 2 straights in the time interval [ , ] . the similar scenario is demonstrated in iceland, where universities and secondary schools were closed on 16 march 2020. furthermore, public gatherings of over 100 were banned on the same day [23]. the same quarantine conditions were applied for the whole quarantine period. nevertheless, there were three stages during the time interval [ , ], = + 25 days (figure 8 (b) ). the first stage of decreasing lasted for 6 days since the beginning of the quarantine. then the next stage followed up for 7 days, in which the fluctuates around ≈ 1. the third and the final stage of decreasing took place for 12 days, during which reached value of = 0.863. therefore, model for the whole period of 25 days, which describes the behaviour of by one decreasing line the similar scenario to australia and switzerland cases developed in austria (figure 9) , where the epidemic started on 2020 february 25, 2020 and the quarantine was imposed on 2020 march 16 [24] . however, slight increase of the daily growth rate after one month of quarantine may be related to the easter celebration, which started after good friday, 2020 april 10. after the easter, the number of daily new cases fluctuates around 65 cases per day. the case of poland illustrates the scenario when the proposed method to predict epidemic dynamics under quarantine does not work straightforward, because of country specificity. strict rules of the country lockdown were implemented on 12 march 2020 [21], and it was expected, that quarantine effectiveness would be the same as in the cases described above. however, during four following weeks after the beginning of the quarantine, decreased from 1.256 only to 1.0 and then stayed fluctuating around ≈ 1 till now (20 may 2020) (figure 10) , what shows the low effectiveness of the taken quarantine actions. as in austria case, the most reliable reason is the easter celebration. during the expected next wave of the covid-19, such specificity of the country can be taken in to account. italy has been under quarantine since march 9, 2020 [16] . it seems that italians are too tired of the country lockdown, that they have strong intentions to finish the lock-down as soon as possible. therefore, italy celebrated the easter more quietly without visible breaks of the quarantine restrictions ( figure 11 ). however, a high value of the daily growth rate of new infection cases during the imposed quarantine = 0.960 suggests that quarantine actions are not sufficient and, there is little hope to reach the end of epidemic as fast as australia having = 0.883. denmark is an example in which the proposed model cannot be applied to predict the epidemic dynamics, and clarification of the reasons for such discrepancy requires more detailed analysis of epidemic situation in the country ( figure 14) . however, the last 35 days is below 1, what demonstrates that social distancing helps to reduce infection rate. situation in russia serves as an example of ineffective quarantine actions for at least first two weeks of the country lockdown. this can be explained by huge size of the country and heterogeneous distribution of the population across the country, which is reason for sequence of arising infection clusters in different locations/regions at different times, despite that national quarantine was imposed on 30 march, 2020. consequently, the daily growth rate of new infection cases = 1.146 remained above 1 for 18 days with almost zero angle of inclination = − ( ) ≈ 0 (figure 15 (b) ), what demonstrates that effectiveness of the quarantine actions was insufficient during this period. for example, only from 12 may, moscow's residents will be required to wear face masks and gloves in public transport and public places [26] , while face masks in public places are one of the most important obstacles for virus spread [27] . consequently, the daily growth rate crossed critical value = 1 only after 40 days from the national lockdown start, but it gives hope that eventually the epidemic reached peak of the daily new infection cases . because of heterogeneous distribution of the population, analysis and forecasting of epidemic situation must be done at region level to generate results with practical value. like russia, united states of america is another huge country by the population number and size. the dynamics of the new cases and the daily growth rate of new infection cases fluctuating around ≈ 1 till now in usa show that quarantine is not effective enough ( figure 16 ) and suggested emergence of new clusters of infection. the forecast, made on the basis of the current dynamics of the epidemic, shows that epidemic will not be defeated in the usa until the end of this year. it should be taken into account, that differences of covid-19 statistics across various states are huge: cases per 100 000 people are varying more than 10 times [28] . therefore, overall us data can suffer from too high level of generalisation. again as in russia, analysis and forecasting of epidemic situation should be done at state level to generate results with practical value. scenario realised in sweden, where the quarantine started on 10 march, 2020 [29], is specific. due to soft conditions of the quarantine (technically most of eu countries would not attribute swedish regime a quarantine, just a gradual restriction of some social activities), it seems that the new cases is achieved maximum only in 6 weeks days after the lockdown start and then the stabilised at = 1 (figure 17 ). further dynamics of the epidemic is still unclear because no country has experience of such situation. the experience gained in sweden is very important and will be used by other countries for the second wave management in the future. the experience gained during the first wave of covid-19 pandemic could help countries to be better prepared for the next wave, which is expected to take place in the autumn of 2020. we proposed the simplified approach, which allows to quantifiable estimate effectiveness of the imposed quarantine conditions/restrictions and compare between countries, forecast the epidemic spread and take appropriate decisions. the proposed approach is based on the time series data of registered daily infection rate as most reliable epidemiological data together with the following main assumptions:  total number of infections is much smaller than a population size of the infected country or region;  registered infected individuals are effectively isolated during the imposed quarantine. the observed dynamics of the pandemic in examined countries shows that the daily growth rate of new infection cases has a tendency to decrease linearly during the imposed quarantine period until reaching a constant value, which corresponds to the effectiveness of taken quarantine measures in the country. the proposed parameters , and together with the analysis of the population mobility and social contacts can be used to estimate the effectiveness of the lockdown measures. on the basis of these parameters, the countries experiencing ongoing epidemic can use the proposed approach to study effectiveness of taken quarantine measures in other countries yet affected by the covid-19 disease. the proposed approach has a limitation because it cannot be directly applied for a country with large population size, which might have several epidemic clusters due to the heterogeneous population distribution across the country. in this case, each cluster must be analysed separately. also, the approach cannot be applied when the daily infection cases is too few because of casual nature of emerging new infections. consequently, the epidemic finish cannot be forecasted accurately. on the basis of the proposed approach, more complex models of epidemic forecasting can be developed. a demographic scaling model for estimating the total number of covid-19 infections on the origin and continuing evolution of sars-cov-2 quantifying the effect of quarantine control in covid-19 infectious spread using machine learning the sir model and the foundations of public health modelling the covid-19 epidemic and implementation of population-wide interventions in italy an seir infectious disease model with testing and conditional quarantine nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study the mathematics of infectious diseases stability of epidemic models with waning immunity dynamics of covid-19 epidemics: seir models underestimate peak infection rates and overestimate epidemic duration an updated estimation of the risk of transmission of the novel coronavirus (2019-ncov) joint european roadmap towards lifting covid-19 containment measures 2020 novel coronavirus (covid-19) cases data n svarbiausia informacija apie koronavirusą (covid-19) n.d different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures see how your community is moving around differently due to covid-19 n частичное возобновление работы, масочный и перчаточный режим wearing face masks in the community during the covid-19 pandemic: altruism and solidarity n tracking the pandemic: how quickly is the coronavirus spreading state by state? n data availability the data used to support the findings of this study are available from the corresponding author upon request. data for the infected cases count from various countries is obtained from the data we also would like to thank tadas markūnas and aušra džiugytė for collecting of covid-19 epidemiological data. ad developed the method, analysed data and post-process data. mb proposed the conception of the model and analysed data. gs, em and rn performed data processing. all authors wrote and approved the manuscript. we declare no competing interests. correspondence and requests for materials should be addressed to ad.declaration of interests + the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.☐the authors declare the following financial interests/personal relationships which may be considered as potential competing interests: key: cord-283291-lj3k53px authors: brugnago, eduardo l.; silva, rafael m. da; manchein, cesar; beims, marcus w. title: how relevant is the decision of containment measures against covid-19 applied ahead of time? date: 2020-08-12 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110164 sha: doc_id: 283291 cord_uid: lj3k53px the cumulative number of confirmed infected individuals by the new coronavirus outbreak until april 30(th), 2020, is presented for the countries: belgium, brazil, united kingdom (uk), and the united states of america (usa). after an initial period with a low incidence of newly infected people, a power-law growth of the number of confirmed cases is observed. for each country, a distinct growth exponent is obtained. for belgium, uk, and usa, countries with a large number of infected people, after the power-law growth, a distinct behavior is obtained when approaching saturation. brazil is still in the power-law regime. such updates of the data and projections corroborate recent results regarding the power-law growth of the virus and their strong distance correlation between some countries around the world. furthermore, we show that act in time is one of the most relevant non-pharmacological weapons that the health organizations have in the battle against the covid-19, infectious disease caused by the most recently discovered coronavirus. we study how changing the social distance and the number of daily tests to identify infected asymptomatic individuals can interfere in the number of confirmed cases of covid-19 when applied in three distinct days, namely april 16(th) (early), april 30(th) (current), and may 14(th) (late). results show that containment actions are necessary to flatten the curves and should be applied as soon as possible. how relevant is the decision of containment measures against covid19 applied ahead of time? since the first infection of the coronavirus in december 2019, observed in wuhan (china), the virus has spread around the world very quickly and nowadays 215 countries, areas, or territories 1 report confirmed cases of the infection. innumerable scientists in distinct areas are using their knowledge in the battle against the still evolving covid-19 outbreak around the globe. the daily analysis of data about the spreading of the virus and possible interpretations that allow us to track and control the virus are of most relevance. it is a timely appeal to find explanations and models which may allow email addresses: elb@fisica.ufpr.br (eduardo l. brugnago 1 ), rmarques@fisica.ufpr.br (rafael m. da silva 1 ), cesar.manchein@udesc.br (cesar manchein 2 ), mbeims@fisica.ufpr.br (marcus w. beims 1 ) 1 "territories" include territories, areas, overseas dependencies and other jurisdictions of similar status [1]. us to understand the evolution of the viruses better, saving lives and avoiding economic and social catastrophes [2] . in the battle against the covid-19 spreading, some models focus on the geographical spread of the virus [3, 4] , while others remain restricted to a given area, or country, but analyze the local temporal development of the epidemic. in the context of diseases, in 1760, daniel bernoulli proposed a mathematical model of disease propagation and showed the efficiency of the preventive inoculation technique against smallpox [5] . this model included susceptible and immune individuals [6] . later on, kermack and mckendrick [7] came up with the susceptible-infected-recovered (sir) model. during the last years, other more sophisticated models have been proposed like the delayed sir epidemic model [8] , the susceptible-exposed-infected-recovered (seir) model [9, 10, 11] and its modified versions [12, 13, 14, 15, 16] . both approaches, brazil, (c) uk, and (d) usa, excluding days with less than 100 infected. the black-continuous curves represent the function ∝ t µ that fit the time-series, with exponent µ for each country. the insets display the same curves but in the log-log plot. the geographical spread, and the local temporal evolution, are of most relevance. more recently, the use of heterogeneous effects in the sir model [17] and the viral infections in the presence of latently infected cells [18] have been analyzed. while the present work focuses on non-pharmacological containment measures, in the battle against the covid-19, studies about the use of ivermectin are in progress [19] , a drug used in malaria spreading scenarios which allows the transition from the prevalence to the eradication of the disease [20] . it is well-known that the decisive quantity used to regulate the dynamical evolution of epidemics, in general, is the average reproductive number r 0 , which gives the number of secondarily infected individuals generated by a primary infected individual. while for values r 0 < 1 the number of newly infected individuals decreases exponentially, for 1 < r 0 < ∞ it increases exponentially [21, 22] . starting from the primordial exponential solution put up by verhulst in 1838, the well known logistic model for the law of population growth [23] , models were improved more and more in the last decades to better describe the nonlinear and complex comportments which occur in our environment. in fact, in many realistic systems, power-law functions are the law of growth (or decrease), as in the branch-ing processes with a diverging reproductive number [21] , in scale-free networks and small worlds [24] , and in foraging in biological systems [25] . indeed, recent investigations showed a power-law growth of the cumulative number of infected individuals by the new coronavirus [12, 13, 26, 27] , which might be typical of small world networks [28] and possibly related to fractal kinetics and graph theory [29] . recently, we have shown that power-law growth is observed in countries from four distinct continents [12] until march 27 th , 2020. the considered countries were: brazil, china, germany, italy, france, japan, spain, the republic of korea, and the united states of america (usa). one leading observation was that after an initial time with a low incidence of newly infected people, the growth of the cumulative number of confirmed cases for all studied countries followed a power-law. the distance correlation [30, 31, 32, 33] between these countries was found to be very strong and suggest a universal characteristic of the virus spreading. one of the goals of the present work is to update to april 30 th , 2020 the time-series analysis for the covid-19 growth for the countries brazil and usa. we included belgium and united kingdom (uk) on this list and leave out the other countries which are reaching the saturation regime. meaning initial condition n country population. depends on the country s individuals susceptible to infection. exposed individuals, latent cases. adjusted from data i s symptomatic infectious cases. i s (t 0 ) = c(t 0 ) i a asymptomatic and mild infectious cases. . we call to attention that the values in the vertical axis in fig. 1 change for different countries. initial data, regarding the days with less than 100 infected individuals, were discarded. the black-continuous curves represent the function ∝ t µ that fits the time series, and the exponent µ for each country is indicated in each panel. the insets display the same curves but in the log-log plot. straight lines in the log-log plot indicate power-law growth. the only country for which the power-law growth still takes place is brazil, as shown in fig. 1(b) . the reason is that it is still away from the saturation point. this is different for belgium, uk, and usa, as can be seen in figs. 1(a), 1(c), and 1(d), respectively. dashed-black lines in these three panels are projections in case the powerlaw would have guided the growth. besides the above updates, in this paper, we describe in detail the modified seir model which was used recently [12] to propose strategies to flatten the power-law curves. it is shown how to adjust the parameters of the model to real data. furthermore, using the same model we discuss what would be the effect of early, current, and late non-pharmacological actions to flatten the curves of the four countries shown in fig. 1 . this clearly shows that each day lost by delaying non-pharmacological actions can cost many lives. the paper is divided as follows. in sec. 2, we present in detail the model used in this work. section 3 discusses the effect of containment actions on the total number of confirmed infected cases applied in three distinct days and sec. 4 summarizes our results. in this section we describe in detail the model used to reproduce the realistic data of the who and to predict the effect of strategies used to flatten the power-law curves. the model that we used is the modified seir model described by the following six ordinary differential equations (odes) [12] in addition to these equations, we compute the cumulative number of confirmed cases c of covid-19 from the following ode: through this variable, the parameters (θ, κ s ) can be adjusted, as described later on. table 1 brings together all variables and parameters of the model and their meaning. in the case of the variables, the initial conditions are also presented and in the case of the parameters, the predefined values obtained from preceding studies are also listed. the highlight lines in table 1 call to attention to the variable c, which is the main quantity analyzed in this work, to the adjustable parameters θ and κ s , and to the strategic parameter κ a . after the adjustment, the parameters θ and κ a will be varied to give rise to specific strategies. worth to mention that θ = γr 0 , where r 0 is the basic reproductive number without social distance actions, and γ is the interaction factor between individuals. this factor comprises the parameters of isolation and social interaction. larger social distance implies smaller values of θ, which is equivalent to reduce r 0 . the distinction between θ and r 0 allows us to identify the direct effects of the actions in the battle against the pandemic. thus, the ideal situation would be to find θ < 1. condensing the explanation of the schema, starting from the left, susceptible individuals s develop into exposed individuals e by a rate θ(i s + αi a )/(n t inf ) which, after a latent time t lat , become symptomatic i s or asymptomatic i a with the rate (1 − β)/t lat and β/t lat , respectively. applying daily tests in a rate κ s (κ a ) to identify symptomatic (asymptomatic) infected individuals, they are immediately sent to quarantine q, staying there for a time t ser before recovering (r). on the other hand, infected individuals who have not been tested are sent to the class r after the infection time t inf . since no vaccine has been developed until today, the model does not contain an immunization term. no rigid quarantine is taken into account. furthermore, in eq. (5), the factor t ser dividing q represents a rate of exit from the quarantine (to the group r). from the dynamical point of view, the model is non-chaotic. this allows us to discuss the asymptotic behavior of the relevant quantities. in fact, multigroup epidemiological models of seir type have been shown, in general, to be asymptotically stable [35, 36] . the fixed point is found by assuming zero for all time derivatives of the variables in the epidemiological model, furnishing (s * , e * , i * s , i * a , q * , r * ) = (s * , 0, 0, 0, 0, n − s * ), as well as the total number of confirmed cases goes to c * , where the stars denote the fixed point and s * and c * depend on initial conditions and parameters. for s * = 0 we have r * = n and for s * = 0 we obtain r * = n −s * . since all variables and parameters from the model are positive, from eq. (1) we realize that s(t) always decreases and that asymptotically lim t→∞ s(t) = s * ≥ 0. furthermore, it is possible to rewrite eq. (2) as thus, for sufficently small values of ds/dt close to s * , e(t) decreases exponentially and lim t→∞ e(t) = e * = 0. with a similar analysis we conclude that i * s = i * a = q * = 0, dr/dt = 0 in eq. (6), and dc/dt = 0 in eq. (7) . thus, the dynamics always reaches the fixed point s * which is stable for all considered parameters. it is known that for systems composed of differential equations with r unknown parameters, 2r + 1 experiments with real data are needed to obtain all the information that is potentially available about the parameters [37] . since in our case we have only two adjustable parameters (r = 2), we need at least 5 real data to adjust parameters correctly. this minimum value is automatically taken into account in all numerical simulations when adjusting the parameters. empty circles in fig. 3 are the real data for the cumulative number of confirmed cases of covid-19 for the four countries analyzed. to find the best values for the pairs (θ, κ s ) = (θ ef f , κ ef f s ) that fit the real data and the best time-series split in periods p i , we performed simulations varying θ ∈ [0.5, 5.0] using a step equal to 0.1 and κ s ∈ [0.0, 1.0] using a step equal to 0.05 and testing different combinations of periods p i , always obeying the minimum amount of real data requested in each period. the goal of these simulations is to minimize the mean square error between the numerical results and real data. thereon, we need five pairs of parameters in fig. 3 (a), namely p 1 , p 2 , p 3 , p 4 , and p 5 for belgium, and six pairs of parameters for the other countries, seen in fig. 3 (b)-3(d). details of the adjustable parameters are given in table 2 . the initial condition e(t 0 ) for the variable e(t) is determined inside the first period p 1 of the data considering the interval e(t 0 ) ∈ [c(t 0 )/5, 10c(t 0 )] using a step equal c(t 0 )/5, where c(t 0 ) is the cumulative number of confirmed cases obtained from the who data for the first day in p 1 . we do not start the parameter adjustment from the first day of reported infections, but later on. the model produces better results in such cases. after adjusting the parameters to the real data, the black-continuous curves in fig. 3 display the results of integrating equations of the model. we observe that these curves nicely reproduce the data in all cases. when real data are not available anymore, the black-continuous curves represent projections of the cumulative number of infected individuals until the day 150, considering that the pair (θ ef f , κ ef f s ) found in the last period will not be changed. in this section, we discuss the effects of distinct strategies applied in different days on the total number of infected individuals. essentially we discuss two strategies: (i) vary the degree of the social distance; (ii) for a constant value of the social distance, vary the number of daily tests that allow identifying and isolating the infected asymptomatic individuals. as mentioned before, fig. 3 displays the real data (empty circles) and black-continuous curves, which were adjusted to fit the data. the results are shown for belgium in fig. 3(a) , brazil in fig. 3(b) , uk in fig. 3(c) , and usa in fig. 3(d) . during the integration of the odes of the model, it is possible to change the parameter θ, which represents the amount of social distance. therefore, we changed this parameter from 0.5 to 2.5 using a step of 0.002 in three distinct dates, namely april 16 th (greendashed curves), april 30 th (blue-dashed curves), and may 14 th (red-dashed curves). curves with dark colors are related to θ = 2.5, and light colors to θ = 0.5. worth mention that high values of θ mean low degrees of social distance, what can potentialize the epidemic spread. we see, in the case of belgium, for example, that strong social distance strategies (θ = 0.5) can flatten the curves for the three distinct days. however, their efficiency in flattening the curves becomes less for later days (see blue and red light dashed curves). on the other hand, if social distance strategies are relaxed to θ = 2.5 in belgium when compared to θ ef f = 0.7 obtained for the last period p 5 (see table 2 ), the number of infected people increases very much. furthermore, if you wait longer to relax the social distance, days april 30 th (blue-dashed curves) or may 14 th (reddashed curves), for example, the total number of infected cases diminishes. essentially the same behavior is observed for all the other countries analyzed. please see figs. 3 in all these simulations the values κ s = 1.0 and κ a = 0 were kept fixed. at next, we keep the social distance parameter constant at θ = 2.0, set κ s = 1.0, and change the daily rate of identification of asymptomatic infected individuals. the choice for this strategy is that without tests it is impossible to recognize that asymptomatic individuals are infected. in the simulations, we varied κ a from 0.1 to 1.0 using a step of 0.009. results are presented in fig. 4 for the same countries from fig. 3 . for better visualization, we start the plot at later times when compared to fig. 3. figures 4(a) , 4(c), 4(e), and 4(g) display the total cumulative number of confirmed infected cases and figs. 4(b), 4(d), 4(f), and 4(h) show the cumulative number of only symptomatic infected individuals. strategies are again applied in days april 16 th (green-dashed curves), april 30 th (bluedashed curves), and may 14 th (red-dashed curves). curves with dark colors are related to κ a = 1.0 and curves with light colors to κ a = 0.1. this constant can be interpreted as follows: κ a = 0.1, for example, represents a daily rate of identification and isolation of 10% of all asymptomatic infected individuals. this represents a huge number of daily tests for countries like brazil and usa, which have a large population. to make it possible to compare the projection tendencies, for which κ a = 0, and the scenarios shown in fig. 4 , we compute the cumulative number of symptomatic infectious cases (b). in this quantity, asymptomatic cases or those with mild symptoms are not considered. similar to the c variable, b is an auxiliary variable of the model, obtained by integrating the ode let us discuss results from fig. 4 taking just one country: uk. in fig. 4 (e) we observe that the strategy of realizing tests on asymptomatic individuals increases the total number of confirmed cases in-stantly. for the largest value κ a = 1.0 for example, the dark green curve increases very much on the day april 16 th . for κ a = 0.1, the light green curve barely changes in this day. however, after around 15 days, both curves cross each other, and the dark green curve asymptotically converges to a much smaller value than the light-green curve, which shows that the realization of a huge amount of daily tests to identify and isolate asymptomatic individuals is also an efficient strategy that could be applied to relax the social distance (increase the value of θ). nevertheless, it is important to mention that, for countries with large populations, values κ a ≈ 1.0 are not practical parameters. the same behavior can be observed when the tests are applied in days april 30 th (blue-dashed curves) and may 14 th (red-dashed curves). essentially an analogous interpretation is valid for the other countries. one difference is observed in brazil. in fig. 4(c) we can see that all the dashed curves cross the black-continuous curve of the tendency, meaning that even the late actions were able to diminish the number of infected individuals. it occurs because of the constant value θ = 2.0 is lower than the θ ef f = 2.1 obtained in the last period p 6 for brazil (see table 2 ). this is not the case for the other countries since black-continuous curves had a smaller value of θ ef f at p 6 , or p 5 for belgium (see table 2 ), when compared to θ = 2.0 used in fig. 4 . at next, we discuss some projections for the cumulative number of symptomatic infected individuals, shown in figs. 4(b) for belgium, 4(d) for brazil, 4(f) for uk, and 4(h) for usa. now, we take the example of the usa. when increasing the value of θ from θ ef f = 1.10 (see table 2 ) to θ = 2.0 and setting κ a = 1.0, on april 16 th , the dark green curve tends to flatten the growth of the cumulative number of symptomatic infected cases. on the other hand, for κ a = 0.1, the tendency is to increase such quantity when compared to the blackcontinuous curve. this projects bad news for the usa in case they relax the social distance (increase the value of θ) and apply a small number of tests to the identification of asymptomatic infected individuals. similar behavior is observed for the other countries. the only difference is for brazil, shown in fig. 4(d) , where the black-continuous curve is lying above the dashed curves once the value of θ used in these strategies is lower than the θ ef f obtained in p 6 for brazil. the projections tend to get worst as the application day of the strategy is delayed. the cumulative number of confirmed cases of covid-19 until april 30 th , 2020, is demonstrated for four exemplary countries: belgium, brazil, uk, and usa, representing three distinct continents. after an initial period with a low incidence of newly infected people, a power-law growth of the number of confirmed cases is observed. for each country, we found a distinct growth exponent. usa leads the increasing rate, followed by uk, brazil, and belgium. for belgium, uk, and usa, countries with a large number of infected individuals, the power-law growth gave place to a distinct behavior when approaching saturation. brazil is still in the power-law regime. such updates of the data and projections corroborate recent results regarding the power-law growth of the cumulative number of infected individuals by the new coronavirus and its strong correlation between different countries around the world [12] . furthermore, we study a variation of the well known seir epidemic model [16, 38] for predictions using (or not) distinct government strategies applied on three distinct dates, namely april 16 th (early action), april 30 th (current action), and may 14 th (late action). the main goal is to show that time is one of the most important weapons we have in the battle against the covid-19. recently, it has been shown that there is a short time window for which it is possible to avoid the spread of the epidemic [39] . in this work, the authors applied the richards growth model to study the fatality curves of some countries. their findings show that, in general, the efficiency of an intervention strategy decays quickly as the adoption time is delayed, corroborating that time is essential in containing an outbreak. in our case, in the three days mentioned above, we applied two strategies: (i) distinct degrees of social distance (vary θ), and (ii) distinct degrees of identification of asymptomatic individuals (vary κ a ). in the first strategy, we change the values of θ from 0.5 to 2.5, meaning strong and essentially no social distance containments, respectively. in the second strategy, we change κ a from 0.1 to 1.0. this can be interpreted as identifying daily 10% of all asymptomatic individuals when κ a = 0.1. the ideal case is represented using κ a = 1.0, when all asymptomatic infected people are identified each day. results for all countries convince us that nonpharmacological strategies must be applied as soon as possible. these include social distance and a large number of testing and immediate isolation of asymptomatic infected individuals. furthermore, time delays in applying such strategies lead to an irreversible catastrophic number of infected people. and they also acknowledge computational support from prof. c. m. de carvalho at lftc-dfis-ufpr (brazil). c. m. also thanks fapesc (brazilian agency) for financial support coronavirus: why you must act now power-law distribution in the number of confirmed covid-19 cases a mathematical model for the spatiotemporal epidemic spreading of covid19, medrxiv essai d'une nouvelle analyse de la mortalite causee par la petite verole et des avantages de l'inoculation pour la prevenir daniel bernoulli's epidemiological 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distance correlation ttest of independence in high dimension distance correlation detecting lyapunov instabilities, noise-induced escape times and mixing decay of the distance autocorrelation and lyapunov exponents early transmission dynamics in wuhan, china, of novel coronavirus-infected pneumonia modeling and dynamics of infectious diseases (series in contemporary applied mathematics) stability of the endemic equilibrium in epidemic models with subpopulations for differential equations with r parameters, 2r + 1 experiments are enough for identification global dynamics of a seir model with varying total population size modelling fatality curves of covid-19 and the effectiveness of intervention strategies the authors thank cnpq (brazil) for financial support (grant numbers 432029/2016-8, the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.author contributions e.l.b. contributed to the model implementation. r.m.s. and c.m. collected and analyzed the data, and m.w.b. mainly wrote the paper. all authors contributed to the discussion and analysis of the results and the final compilation of the work. key: cord-019114-934xczf3 authors: zhan, xiu-xiu; liu, chuang; sun, gui-quan; zhang, zi-ke title: epidemic dynamics on information-driven adaptive networks date: 2018-02-16 journal: chaos solitons fractals doi: 10.1016/j.chaos.2018.02.010 sha: doc_id: 19114 cord_uid: 934xczf3 research on the interplay between the dynamics on the network and the dynamics of the network has attracted much attention in recent years. in this work, we propose an information-driven adaptive model, where disease and disease information can evolve simultaneously. for the information-driven adaptive process, susceptible (infected) individuals who have abilities to recognize the disease would break the links of their infected (susceptible) neighbors to prevent the epidemic from further spreading. simulation results and numerical analyses based on the pairwise approach indicate that the information-driven adaptive process can not only slow down the speed of epidemic spreading, but can also diminish the epidemic prevalence at the final state significantly. in addition, the disease spreading and information diffusion pattern on the lattice as well as on a real-world network give visual representations about how the disease is trapped into an isolated field with the information-driven adaptive process. furthermore, we perform the local bifurcation analysis on four types of dynamical regions, including healthy, a continuous dynamic behavior, bistable and endemic, to understand the evolution of the observed dynamical behaviors. this work may shed some lights on understanding how information affects human activities on responding to epidemic spreading. the spreading dynamic is one of the core issues in network science [1] [2] [3] , where most of the related researches focus on epidemic spreading and information diffusion in recent years. much of the work to date focuses on the analysis of these two processes independently, such as the spread of single contagion [4] [5] [6] or concurrent diseases [7, 8] , and the diffusion of various kinds of information (e.g., news [9] , rumor [10] , innovation [11] .). however, the epidemic spreading process is closely coupled with the corresponding disease information diffusion (or saying individuals' awareness of the disease) in the real world. for instance, during the severe acute respiratory syndrome (sars) outbreak in china in 2003, overwhelming number of disease reports have been posted. these kind of information about sars may affect the individuals' behavior in keeping away from sars and thus help to make the disease under control [12, 13] . therefore, disease information diffusion may play an important role in the control of the epidemic outbreak, but it is not easy to quantitatively measure the strength of its impact [14] . nowadays, some models have been proposed to model the interaction between epidemic spreading and information diffusion on complex networks [14] [15] [16] [17] . the fundamental assumption is that, when a disease starts to spread in the population, people may get the disease information from their friends or media before the advent of the epidemic and take some preventive measures to keep away from being infected [15, 18, 19] . by depicting preventive measures as the reduction of transmitting probability [20, 21] or particular states of individuals (immune or vaccination) [22] , previous models showed that the disease information diffusion indeed inhibits the epidemic spreading significantly (reduce the epidemic prevalence as well as enhance the epidemic threshold) [15, 23] . therefore, the emergence of mutual feedback between information diffusion and epidemic spreading [14] exhibits the intricate interplay between these two types of spreading dynamics. the interplay between these two types of spreading dynamics is similar to the competing epidemics [24, 25] to some extent, that is to say, there is a competitive mechanism between epidemic spreading and the ini.e., the network structure stays fixed when the two processes are spreading on the network. however, individuals would sometimes cut off the connections with the infected ones when they become aware of the disease, leading to the change of network structure. consequently, how to characterize the mutual spreading process on the adaptive networks is a crucial issue we want to address in this work. generally, the network dynamic researches could be classified into two lines: (i) one is the dynamics of the network , which focuses on the time evolution of network structure [26] [27] [28] ; (ii) the other is considered as the dynamics on the network , which concerns the state change of the nodes (or interactions) on networks, such as the epidemic spreading and information diffusion process [29, 30] , the evolutionary game [31] and so forth. currently, researchers became to study how the epidemic would spread on adaptive networks, i.e., considering one epidemic spreading process on dynamical changing networks [32] . in [32] , the author proposed a model by considering that the susceptible individuals are allowed to protect themselves by rewiring their links from the infected neighbors to some other susceptible ones [33] [34] [35] . many researches indicate that segregating infected (or susceptible) individuals with the adaptive behavior is an efficient strategy to reduce the fraction of susceptible-infected ( si ) interactions, as well as hinder the outbreak of the whole epidemic spreading [36] [37] [38] . in addition, abundant temporal behaviors are presented to illustrate the spreading dynamics on the adaptive network, such as the coexistence of multiple stable equilibrium and the appearance of an oscillatory region, which are absent in the spreading dynamics on static networks [32, 39] . besides the edge rewiring strategy, the link cutting or temporarily deactivating is also a commonly used rule in the adaptive models [40, 41] . in this work, we consider a more complicated case that two dynamical processes (i.e., epidemic spreading and disease information diffusion) are spreading on adaptive networks. therefore, three dynamical processes are coupled in this case, we aim to illustrate how the adaptive behavior can affect the interplay between epidemic spreading and information diffusion. the adaptive behavior is aroused by the individuals awareness of the disease. in this model, those who have been informed of the emergence of disease can break their neighbouring connections to prevent further infection. additionally, epidemic spreading and disease information diffusion are described by the si and sis model, respectively. the disease information generation of the infected individuals is considered to form a mutual feedback loop between these two types of spreading dynamics [20] . therefore, the effect of information diffusion on epidemic spreading could be interpreted by two aspects: (i) reduce the epidemic spreading probability with protective measures; and (ii) cut off si links with the information-driven adaptive process. both numerical analyses based on the pairwise approach and simulation results indicate that the information diffusion and the adaptive behavior of the nodes can inhibit the epidemic outbreak significantly. in addition, we present a full local bifurcation diagram to show the abundant dynamical behaviors in the proposed model. the paper is organized as follows. in section 2 , we give a detailed description of the model as well as mathematical expressions based on the mean-field model and the pairwise model. in section 3 , we first analyze the case of epidemic and disease information spreading on static network, i.e., the case of no adaptive behavior is taken into account. we further give the results of how the epidemic and disease information spreading processes interact with each other on adaptive network. the sensitivity analysis of the parameters and dynamical characterization of the model is given in the end of section 3 . we conclude the paper with some future directions of the work in section 4 . we give a detailed illustration of our model in fig. 1 . the vertical transformation describes the diffusion of disease information by an sis model, where individuals can be at one of the two states: (i) + : indicates that the individuals have known the existence of the disease, denoted as the informed ones; (ii) −: indicates that the individuals have not known the existence of the disease. at each time step, the informed nodes will transmit the information to their unknown ( −) neighbours with probability α, and each informed individual may forget the information of the disease with a probability λ. besides, the one who has been infected by the disease will become to know the information of the disease with a corresponding rate ω [14, 16] . in the horizontal transformation of fig. 1 , the epidemic spreading is described by an si model. each node is at one of two states, susceptible (s) or infected (i). the disease can be transmitted through the si links, where the s-state individuals could be infected with the probabilities β, σ i β, σ s β and σ si β respectively through s − i − , s − i + , s + i − and s + i + links, where σ i , σ s and σ si are the impact factors of the information on epidemic spreading. generally, when people know the occurrence of the disease (informed individuals), they would like to take some measures to protect themselves, leading to the reduction in infectivity (0 < σ s , σ i < 1). in particular, the influence coefficient of the epidemic spreading probability through s + i + links could be calculated as σ si = σ s σ i , with the assumption of the independent effect of the infection probability. additionally, we consider an information-driven adaptive process which the informed individuals would reduce physical contacts to protect themselves or their friends. that is to say, the informed susceptible individuals ( s + ) will keep away from their infected neighbors to protect themselves from being infected, and informed infected individuals ( i + ) will also avoid contacting their susceptible neighbors to prevent the epidemic from further spreading. consequently, the edge-breaking rule of adaptive behavior is adopted [40] . thus, at each time step, the s + ( i + ) state individuals will break the links connected to their i ( s )-state neighbors with rate r s ( r i ) respectively. specially, the breaking rate of the s + i + pairs could be interpreted as 1 − (1 − r s )(1 − r i ) with the independent assumption. it is worth noting that the deactivation of si links only represents the avoidance of physical contacts between the sand i -state individuals. that is to say, the edge-breaking process will not affect the diffusion of disease information for it can be transmitted through other types of connections such as phone, internet and so forth. the dynamic of the epidemic spreading degen erates to a classical si model when we set r s = r i = 0 , i.e., there is no edge-breaking in this case. according to the model described above, the spreading process can be summarized as follows. at the beginning, an individual is randomly selected as the i + node, which is considered as the seed of both the epidemic spreading and information diffusion, and all other individuals are set as s − ones. at each time step, (i) the infected individuals would transmit the disease to their susceptible neighbors with the corresponding probabilities; (ii) the informed individuals would transmit the disease information to their uninformed neighbors; (iii) the informed individuals can forget the information; (iv) the informed individuals would also break the links with their relevant neighbors by considering the adaptive mechanism. finally, the spreading process would be terminated when the size of the infected individuals becomes stable. firstly, we develop theoretical analysis to depict the dynamic processes of both information diffusion and epidemic spreading. in particular, mean-field analysis and the pairwise analysis are adopted. let χ be the state variable, thus [ χ] denotes the expected values of individuals of different types on the population (e.g. [ s + ] and [ s + i + ] represent the expected number of informed susceptible nodes and expected number of links connecting an informed susceptible node to an informed infected node respectively). therefore, with the classical mean-field approach, we can obtain: comparatively, with the pairwise approach, we can obtain: where, the first terms of eqs. (1) and (2) describe the infection of the s + -state individuals, the second terms describe the information acceptance of the i − -state individuals, the third terms describe the information generation of the i − -state individuals and the last terms represent the information loss of the i + -state individuals. simultaneously, the full set of differential equations based on those two approaches can be illustrated in appendix a . by the way, the adaptive process could be described by the last terms of in the pairwise approach of eq. (4) . it should be noted that the pairwise analysis is based on a wellknown closure approximation given by with the assumption that the degree of each individual obeys poisson distribution [42, 43] . in general, it might be very hard to get exact solutions of such complex differential equations, thus we give numerical solutions of the equations instead of the theoretical analysis in the following analysis. in this work, we perform our model on the er network with a total population of n = 10 , 0 0 0 and average degree k = 6 unless otherwise stated. moreover, all the simulation results are given by 10,0 0 0 realizations. we first consider a simple case of no adaptive behavior when the epidemic and disease information are spreading in the network, i.e., the case of spreading on static network. fig. 2 gives the simulation result of the fraction of infected nodes evolving with time for various information diffusion probabilities α, with the epidemic spreading probability β = 0 . 3 . for the si process, the whole population would be infected when β > 0 for the connected social networks, resulting in that the final infected density equals to 1 for all the values of α in fig. 2 . that is to say, the disease information diffusion cannot avoid the epidemic spreading to the whole population when we perform our model on static network. however, we find that the disease information diffusion can slow down the epidemic spreading when we increase the value of α. furthermore, the time cost for the whole population becomes infected when α = 1 is about three times longer than that of α = 0 . in this sense, the diffusion of the disease information can slow down the epidemic spreading significantly. in addition, the inset of fig. 2 indicates that the epidemic spreading can enhance the disease information diffusion. actually, according to model illustrated in fig. 1 , on the one hand, we realize that the epidemic spreading could be influenced by information diffusion where the epidemic spreading probability of the informed individuals would change; and on the other hand, the information diffusion could be influenced by the epidemic spreading where the social disease information level (namely info in the inset of fig. 2 ) would be higher if more people are infected for the information generation, denoted by the parameter ω. in this way, a mutual feedback between disease spreading and information diffusion emerges: higher prevalence of the infected individuals makes more disease information generated in the population, which in turn gives rise to more informed individuals, thereby weakening the spread of epidemic. fig. 3 shows a comparison of the evolution of infected density from the numerical analysis according to eqs. (3) and (4) and the simulation results on er network. infected density curve based on the classical mean-field approach is much quicker than that of the simulation result, which would be caused by the mean-field assumption on the si model. in the mean-field assumption, the iand s -state individuals are well-distributed in the system. however, in the si process, the i -state individuals are all well clustered, resulting in that many i -state individuals have no chance to contact the s -state individuals. in this way, the classical mean-field approach can not exactly describe the si model. however, such problem is not so significant in the pairwise approach, which consider the time evolution of the links as well. fig. 3 shows that the infected density curve of the pairwise approach finds good agreement with the simulation results. in this part, we shall present the spreading dynamics with the information-driven adaptive process, the results are shown in fig. 4 . different from the results of fig. 2 , the saturation value of the infected density at the final state is much smaller than 1 in fig. 4 (a) . that is to say, with the adaptive process based on the information diffusion, many individuals could avoid being infected via reducing some contacts. in addition, we also plot the numerical solution based on the pairwise approach in fig. 4 (a) . it can be seen that the pairwise solution is not well consistent with simulation for the spreading dynamic on the adaptive network. the difference might be caused by the network structure variation in the adap-tive process, where the assumption of the pairwise approach is the poisson degree distribution. this conjecture is proved in fig. 4 (b) , where the degree distribution of the original network is approximate to the poisson-distribution with mean degree around 6 (pink circle markers), while the distribution of the network at the final state (gray diamond markers) deviates from the original distribution. in addition, fig. 4 (a) shows that the difference becomes larger with the increase of time, where the degree distribution deviates more away from the original distribution when the process goes on. the information-driven adaptive process can not only slow down the speed of epidemic spreading, but also can diminish the epidemic prevalence at the final state significantly according to figs. 2 and 4 . for simplicity, we assume r s = r i = r in the following analysis. in order to exhibit the influence of information diffusion in detail, we show the full phase diagram α − β with r = 0 . 1 in fig. 5 , the color gives the infected density in the final state for each combination of α and β. the fig. 5 (a) and (b) are the numerical solution of the pairwise approach and the simulation result, respectively. as stated previously, the numerical solution is not very statistics of haggle network, where n, e, c represent the number of nodes, the number of links, clustering coefficient of each system respectively. in order to intuitively demonstrate the epidemic spreading and the information diffusion process on adaptive network, we show the simulation results of those two types of spreading processes for various α on two different networks, i.e., a 100 × 100 lattice with degree k = 4 as well as a real-world network, e.g., haggle network [44] . the contacts in haggle network represent connection between people measured by carrying wireless devices. the statistics of the network is given in table 1 . the visualization of how epidemic and disease information interact with each other for these two networks are given in fig. 6 and fig. 7 , respectively. taking lattice as an example, we present four kinds of different levels of information spreading processes (corresponding to different α), and observe how the information diffusion affects the spreading of epidemic. in addition, as the adaptive edge-breaking process is merely executed on the epidemic spreading process, while these edges can still transmit information, thus the density of informed people can still maintain at a high level in the network. for each α in fig. 6 , firstly we give the fraction of the infected individuals at each time step (the red curve in each subfigure). for some particular time steps, we show the states of each individual with the gridding patterns, where the red dots and the gray dots represent the infected and informed individuals respectively (the contact networks and the un-informed susceptible individuals are not shown in the figures). we can intuitively see the distribution of the infected and informed individuals and conclude that when the diffusion of information is slower than the epidemic, we cannot stop the epidemic from spreading ( fig. 6 (a) and (b) ), however, when the information is diffusing faster, the epidemic will be trapped into an isolated area and cannot spread anymore ( fig. 6 (c) and (d)). furthermore, the visualization of these two processes on haggle network displays similar results as the results on lattice. the sensitivity of the edge-breaking probability on epidemic spreading dynamics. the phase diagram in fig. 5 shows the impact of information diffusion rate α on the epidemic spreading dynamics. in general, the adaptive edge-breaking probability r s and r i are also important parameters in affecting the epidemic spreading process. fig. 8 illustrates the epidemic prevalence in the final state versus the adaptive edge-breaking rate ( r ) for various information diffusion rate α. it can be found that the epidemic prevalence diminishes with the increase of r , i.e., the epidemic could be controlled if people are very sensitive with the disease information and subsequently keep away from the infected. it should be noted that there is no disease information diffusion when α = 0 , but with considering the information generation, the infected individuals could stop contacting with the susceptible neighbors to impede the further spreading of epidemic. with the increase of α, the epidemic prevalence reduces sharply versus r and the continuous transition could be observed. by the way, it will change to a total isolation of infected individuals for r = 1 , which seems to be the most effective way in controlling the contagion [45, 46] . dynamical characterization of the information-driven rewiring. in order to deeply characterize the complex dynamical features of the proposed process, we concentrate on the distribution of the infected density in the final state ( i * ) rather than the simple average value [32, 39] . fig. 9 shows four different types of dynamical behavior by calculating the distribution of the final fraction of infected for various β and r . for the distribution of fig. 9 (a) , we have carried out 10,0 0 0 realizations of the infected density, and above 94% of the infected density is 0.0 0 01, and the maximal is 0.0 0 07, i.e., the infected density i * → 0, thus we consider this distribution indicates a healthy state (the disease can't spread out) under the parameters setting here. similarly, as to the distribution of fig. 9 (d) , above 90% of the infected density is higher than 0.8, indicates a case of endemic state (epidemic outbreaks). whereas the case illustrated in fig. 9 (c) is very different, where the infected density i * is around either zero or a nonzero value. this indicates that a bistable state [32] is located in this model, where healthy state and endemic state are both stable in this case. in addition, a continuous dynamic behavior can also be observed in particular parameter settings ( fig. 9 (b) ). according to the dynamical behavior illustrated in fig. 9 under different parameter sets, bifurcation diagram of the density of the infected as a function of infected probability β for different values of the edge-breaking rate r is given in fig. 10 (a) . without the adaptive edge-breaking mechanism ( r = 0 ), the disease can spread out only if β > 0 for the si process. when r > 0, the dynamical behaviors become more complicated, where the discontinuous phase transitions, bistable, oscillatory are observed. a fast edge-breaking (large r ) leads to a broad healthy and bistable state range (shows by the range in the arrows) in fig. 10 (a) . in fig. 10 (b) , we give a full r − β bifurcation diagram according to our simulation results, and we can clearly identify the areas of healthy, a continous dynamic behavior, bistability and endemic state in this model. at last, we present the dependence of the average value of infected density over 10,0 0 0 independent realizations on r and β in fig. 10 (c) , where the changing of the density is consistent with the area classification in fig. 10 (b) . in order to understand the interplay between the dynamics on the network (the spread of epidemic spreading and disease information) and the dynamics of the network (the time varying of network links), we present two types of spreading dynamics with si and sis process respectively on an information-driven adaptive network, where the individuals who have known the disease information would probably cut off their links with others. firstly, we illustrate the mutual feedback between epidemic spreading and information diffusion without considering the edge-breaking process ( r s = r i = 0 ), where the high epidemic prevalence preserves high disease information level, which in turn slows down the epidemic spreading. in this case, the numerical analysis based on the pairwise approach is consistent with the simulation result very well. secondly, the results are very different when the informationdriven edge-breaking process is considered ( r s , r i > 0). the epidemic cannot spread out if the spreading probability is smaller than the threshold (shown in fig. 5 ). in addition, the disease spreading and information diffusion pattern on the lattice as well as on a real-world network give visual representations that the disease might be trapped into an isolated field with informationdriven adaptive process. therefore, the information-driven adaptive process can inhibit the epidemic spreading significantly that it can not only slow down the epidemic spreading speed, but also reduce the epidemic prevalence. finally, we give the local bifurcation analysis on four types of dynamical phenomena, including healthy, a continuous dynamic behavior, bistable and endemic, indicating that the state changes from healthy to a continuous dynamic behavior, bistable, endemic state as β increases. in summary, we study the dependence of the epidemic spreading on the disease information diffusion and the informationdriven adaptive process, with considering the simplest spreading model (si) and adaptive process (edge-breaking). recent researches show the different features between the epidemic and the information diffusion [47, 48] , and this difference would also impact the interplay between epidemic spreading and disease information diffusion significantly. another area 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networks: coevolution of disease and topology emergence of scaling in random networks collective dynamics in 'small-world' networks temporal networks epidemic spreading in scale-free networks information spreading on dynamic social networks heterogeneous networks do not promote cooperation when humans play a prisoner's dilemma epidemic dynamics on an adaptive network adaptive coevolutionary networks: a review adaptive networks effects of community structure on epidemic spread in an adaptive network epidemic spreading on weighted adaptive networks susceptible-infected-recovered epidemics in dynamic contact networks the structure of coevolving infection networks epidemic threshold and topological structure of susceptible-infectious-susceptible epidemics in adaptive networks epidemics in adaptive social networks with temporary link deactivation intermittent social distancing strategy for epidemic control representing spatial interactions in simple ecological models the effects of local spatial structure on epidemiological invasions impact of human mobility on opportunistic forwarding algorithms probing into the effectiveness of self-isolation policies in epidemic control quarantine-generated phase transition in epidemic spreading how events determine spreading patterns: information transmission via internal and external influences on social networks avalanche outbreaks emerging in cooperative contagions tiirec: a tensor approach for tagdriven item recommendation with sparse user generated content denote [ χ] as the expected values of individuals of different types described in section 2.2 , the epidemic spreading is depicted by the parameters β, σ i β, σ s β and σ si β, while the diffusion of disease information is controlled by the parameters: α, λ, ω. all these parameters have been explained in section 2.1 . according to the model described above, the differential equations of the meanfield approach ( eq. (3) ) and pairwise approach ( eq. (4) ) are given as follows. key: cord-295116-eo887olu authors: chimmula, vinay kumar reddy; zhang, lei title: time series forecasting of covid-19 transmission in canada using lstm networks() date: 2020-05-08 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109864 sha: doc_id: 295116 cord_uid: eo887olu on march 11(th) 2020, world health organization (who) declared the 2019 novel corona virus as global pandemic. corona virus, also known as covid-19 was first originated in wuhan, hubei province in china around december 2019 and spread out all over the world within few weeks. based on the public datasets provided by john hopkins university and canadian health authority, we have developed a forecasting model of covid-19 outbreak in canada using state-of-the-art deep learning (dl) models. in this novel research, we evaluated the key features to predict the trends and possible stopping time of the current covid-19 outbreak in canada and around the world. in this paper we presented the long short-term memory (lstm) networks, a deep learning approach to forecast the future covid-19 cases. based on the results of our long short-term memory (lstm) network, we predicted the possible ending point of this outbreak will be around june 2020. in addition to that, we compared transmission rates of canada with italy and usa. here we also presented the 2, 4, 6, 8, 10, 12 and 14(th) day predictions for 2 successive days. our forecasts in this paper is based on the available data until march 31, 2020. to the best of our knowledge, this of the few studies to use lstm networks to forecast the infectious diseases. every infectious disease outbreak exhibits certain patterns and such patterns needed to be identified based on transmission dynamics of such outbreaks. intervening measures to eradicate such infectious diseases rely on the methods used to evaluate the outbreak when it occurs. any outbreak in a country or province usually occurs at different levels of magnitude with respect to time i.e. seasonal changes, adaptation of virus over time. usually patterns exhibited in such scenarios are non-linear in nature and this motivates us to design the system that can capture such non-linear dynamic changes. with the help of these non-linear systems, we can describe the transmission of such infectious diseases. in [1] [2] a transmission model for malaria and in [3] a mathematical model for analysing dynamics of tuberculosis has been developed to study the transmission using mathematical models. in [4] a laplacian based decomposition is used to solve the non-linear parameters in a pine witt disease. a modified sirs model in [5] successfully helped to control the syncytial virus in infants. similarly mathematical models presented in [6] , [7] helped clinicians to better understand the characteristics of human liver and transmission of dengue outbreak. most of the data driven approaches used in previous studies [8] are linear methods and often neglects the temporal components in the data. they depend upon regression without non-linear functions and failed to capture the regressive (ar) methods overwhelmingly depends on assumptions and such models are difficult for forecasting realtime transmission rates. wide range of statistical and mathematical models [9] [10] have been proposed to model the transmission dynamics of current covid-19 epidemic. in many cases, these models are not able to fit the given data perfectly and accuracy is also low while predicting the growth of covid-19 transmission. r0 is a popular statistical method specifically used to model an infectious disease. often referred as âăÿreproduction numberâăź because, the infections reproduce itself with respect to time. r0 forecasts the number of people can get the infection from the infected person. in this model, an extra weight is applied to the person who never infected the current disease nor vaccinated. if the value of r0 of a disease is 10, then the infected person will spread the disease to 10 other people surrounding him. in [11] authors used r0 method to find the infection rate of novel virus on diamond princes cruise ship [11] . however, in such method it is difficult to find the starting point of the infectious disease by identifying patient zero and the people he interacted with during his incubation period. it is worth noting that mathematical models presented in [12] , [13] , [14] can be used to solve the complex non-linear patterns of infectious diseases. even though these epidemiological models are good at capturing vital components of an infectious disease, parameters of these models required several assumptions. such hypothesized parameters would not fit the data perfectly and precision of such models will be low. meanwhile, in engineering applications [15] , model parameters are calculated with the help of real-time data. similar approach was used in this research to find the model parameters instead of assumptions. in order to overcome the barriers of statistical approaches, we developed the deep learning based network to predict the real-time transmission. our model could help public health care providers, policy makers to make necessary arrangements to tackle the rush of potential covid-19 patients. this experiment is based on the data sets of confirmed covid-19 cases available until march 31, 2020. artificial intelligence and mobile computing are one of the key factors for the success of technology in health care systems [16] . in the world of smart devices, data is being generated in the unprecedented way than ever before and promoted the role of machine learning in healthcare [16] . the world today is more connected than ever before this helped to share the real time infectious data between the countries. the distinctive feature of artificial intelligence is its flexibility, domain adaptation and economical to integrate with existing systems. over the last few weeks, many researchers came up with several mathematical models to predict the transmission of novel corona virus [17] [18] . the major drawbacks of the existing models are linear, non-temporal and several assumptions while modelling the network. first of all, the covid-19 is a time series data set and it is highly recommended to use the sequential networks to extract the patterns from it. second of all, the data we are dealing with is dynamic in nature so by using statistical and epidemiological models, results are often vague [19] [20] . in [21] , [22] , [23] , [24] researchers used deep learning based lstm networks to forecast covid-19 infections. the lstm models used in the above networks could not able to represent the spatio-temporal components simultaneously. in this paper we addressed the above problem by modifying the internal connections. in our modified lstm cells, we have established the alternative connections between the input and output cells. this type of connections not only helps the networks to preserve spatio-temporal components, but also to transfer the historical information to the next units. in this paper, we made an effort to predict the outbreak of covid-19 based on past transmission data. first of all, coherence of input data needs to be analyzed in order to find the key feature i.e. number of new cases reported with respect to the previous day infections. after selecting the key parameters of the network, several experiments was conducted to find the optimal model that can predict future infections with minimum error. previous studies on covid-19 predictions, did not considered the recovery rate while developing the model. in this research, we considered the recovery rate as one of the features while building our model. from the design point of view, when a crisis occurs, algorithms tend to assign high probability and completely neglects the previous information which leads to biased predictions. we addressed this issue in our literature and solved this by using lstm networks. our results are expected to alert the public health care providers of canada to prepare themselves for the crisis against covid-19. with the help of this real-time forecasting tool, front-line clinical staff will be alerted before the crisis. the rest of this paper is structured as follows: section ii describes methods, datasets and lstm models used in this paper. in section iii, we have discussed our findings and in section iv, concussion and future work was discussed the covid-19 data used in this research is collected from johns hopkins university and canadian health authority, provided with number of confirmed cases until march 31, 2020. the data set also includes number of fatalities and recovered patients by the end of each day. the dataset is available in the time series format with date, month and year so that the temporal components are not neglected. a wavelet transformation [25] is applied to preserve the timefrequency components and it also mitigates the random noise in the dataset. the fundamental point to represent and forecast the trends of current is to select conventional functions to fit the data. the covid-19 dataset is divided into training set (80%) on which our models are trained and testing set (20%) to test the performance of the model. a large part of real-world datasets are temporal in nature. due to its distinctive properties, there are numerous unsolved problems with wide range of applications. data collected over regular intervals of time is called time-series (ts) data and each data point is equally spaced over time. ts prediction is the method of forecasting upcoming trends/patterns of the given historical dataset with temporal features. in order to forecast covid-19 transmission, it would be effective if input data has temporal components and it is different from traditional regression approaches. a time series (ts) data can be break downed into trend, seasonality and error. a trend in ts can be observed when a certain pattern repeats on regular intervals of time due to external factors like lockdown of country, mandatory social distancing, quarantines etc. in many real-world scenarios, either of trend or seasonality are absent. after finding the nature of ts, various forecasting methods have to be applied on given ts given the ts, it is broadly classified into 2 categories i.e. stationary and non-stationary. a series is said to be stationary, if it does not depend on the time components like trend, seasonality effects. mean and variances of such series are constant with respect to time. stationary ts is easier to analyze and results skilful forecasting. a ts data is said to nonstationary if it has trend, seasonality effects in it and changes with respect to time. statistical properties like mean, variance, sand standard deviation also changes with respect to time. in order to check the nature (stationarity and non-stationarity) of the given covid-19 dataset, we have performed augmented dickey fuller (adf) test [26] on the input data. adf is the standard unit root test to find the impact of trends on the data and its results are interpreted by observing p-values of the test. if p is between 5-1%, it rejects the null hypothesis i.e. it does not have a unit root and it is called stationary series. if p is greater than 5% or 0.05 the input data has unit root so it is regarded as non-stationary series. before diving into the model architecture, it is crucial to explain the internal mechanisms of lstm networks and reasons behind using it instead of traditional recurrent neural networks. recurrent lstm networks has capability to address the limitations of traditional time series forecasting techniques by adapting nonlinearities of given covid-19 dataset and can result state of the art results on temporal data. each block of lstm operates at different time step and passes its output to next block until the final lstm block generates the sequential output. as of this writing, rnns with blocks (lstm) are the efficient algorithms to build a time series sequential model. the fundamental component of lstm networks is memory blocks, which was invented to tackle vanishing gradients by memorizing network parameters for long durations. memory block in lstm architecture are similar to the differential storage systems of a digital systems. gates in lstm helps in processing the information with the help of activation function (sigmoid) and output is in between 0 or 1. reason behind using sigmoid activation function is because, we need to pass only positive values to the next gates for getting a clear output. the 3 gates of lstm network are represented with the following equations below: where: = function of input gate = function of forget gate = function of output gate = coefficients of neurons at gate (x) −1 = result from previous time step = input to the current function at time-step t = bias of neurons at gate (x) input gate in the first equation gives the information that needs to be stored in the cell state. second equation throws the information based on the forget gate activation output. the third equation for output gate combines the information from the cell state and the output of forget gate at time step âăÿtâăź for generating the output. the internal block diagram of lstm block used in this study is shown in 1 the motivation behind initiating self-loops is to create a path so that gradients or weights can be shared for long durations. especially, this is useful while modelling deep networks where vanishing gradient is a frequent issue to deal with. by adjusting weights as self-looped gates, we can adjust the time scale to detect the dynamically changing parameters. using the above techniques, lstms are able to produce the state-of-the-art results in [27] . the network architecture used in this study is shown in 2 the methods used in this study are based on data guided approaches and are completely different from previous studies. our approaches and predictive outcomes will provide assistance for restricting the infections and possible elimination of current covid-19 pandemic. we trained our network with data until march 31, 2020 reported by canadian health authority. in this study we found that policies or decisions taken by government will greatly affect the current outbreak.several studies on forecasting of coid-19 transmission are based on the r0 method however, they didn't include the sensitivity analysis to find the important features. we examined our model predictions using mean square error (mse). in figure 4 we plotted the total number of confirmed cases and forecasted covid-19 cases in canada as a function of time. from the figure we can observe that, canada didnâăźt witness its peak yet and it is expected number of cases will soon increase exponentially despite the social distancing. although our model achieved better performance when compared with other forecasting models, it is unfortunate that transmissions are following increasing trend. the rate of infections in usa, italy and spain are growing exponentially meanwhile, the number of infections in canada are increasing linearly in figure 3. if canadians follow the regulations strictly, the number of confirmed cases will soon decline. in our lstm model-1 we trained and tested our network on canadian dataset; the rmse error is 34.83 with an accuracy of 93.4% for short term predictions in canada. meanwhile, based on our testing/validation dataset the rmse error is about 45.70 with an accuracy of 92.67% for long term predictions. the predictions of lstm model are shown in 4 with solid red line. it shows that our model was able to capture the dynamics of the transmission with minimum loss. from the figure 4 we can say that canada witnessed linear growth in cases until march 16 2020 after its first confirmed case. the current epidemic in canada is predicted to continue until june 2020. our second lstm model-2 is trained on italian dataset to predict short-term and long-term infections in canada. for short term predictions, the rmse error is about 51.46 which is higher than previous model. accord-covid-19 forecasting using lstm networks ing to this second model within 10 days, canada is expected to see exponential growth of confirmed cases. it was a challenging task to forecast the dynamics of transmission based on small dataset. even though covid-19 outbreak started in canada around early january, the consistent epidemiological data wasn't released until early february. because of small dataset several statistical models struggled to select the optimal parameters and several unknown variables led to uncertainty in their predictions. lstm model is different from statistical methods in many ways for instance, the proposed lstm network fits the real-time data and without any assumptions while selecting hyperparameters. it was able to overcome the parameter assumptions using cross validation and achieved better performance by reducing the uncertainty. after reaching the inflection point, the recovery rate will start decrease rapidly and death rate may increase at the same time as shown in figure 5 . in order to find the trend of the infections we decomposed the given series and the trend of infections is increasing with respect to time. further, number of infections followed increasing trend from sunday to tuesday and followed decreasing trend until saturday as shown in figure 6 . as we are still under the stage of dilemma about the current situation of covid-19 because, the accuracy of our estimates is bounded with a lot of external factors. so, it is recommended to conduct the follow-up study after this experiment to be more precise about the dynamics of this novel infectious disease. the actual number of cases might be higher than the cases reported by the government because, of the backlog of test results and some people will be immune before even testing. all the above factors may lead to discrepancy of our model estimations. even though we addressed data imbalance by using statistical methods like interpolation and re-sampling yet we couldnâăźt represent patients who are on incubation period or not tested. other problem while modelling current pandemic is that, people covid-19 forecasting using lstm networks figure 4 : predictions of the lstm model on current exposed and infectious cases (red solid line). the red dotted lines represents the sudden changes from where number of infections started following exponential trend. the black dotted lines in the figure represents the training data or available confirmed cases travelling between the provinces. based on our sensitivity analysis our projections may go down if current trials on potential vaccines achieves fruitful results. finally, in order to minimize the bias on our training algorithm we introduced regularization. further, by training our network inversely, we found that outbreak in canada started around early january but, it was not reported until january last week. even without the knowledge of 1st case, our inverse training will help governments to better understand the outbreak of covid-19 and helps then to prevent such outbreaks in future. the patterns from the data reveals that prompt and effective approaches taken by canadian public health authorities to minimize the human exposure is showing a positive impact when compared with other countries like usa and italy 3. rate of transmission in canada is following linear trend while in usa is witnessing an exponential growth of transmissions. however, it is too early to draw the conclusions about the current epidemic. after simulations and data fitting, our model predicted canada would reach peak within 2 weeks from now. however, the current outbreak resembles early 20th century spanish flu [28] , which killed millions of people and lasted for 2 covid-19 forecasting using lstm networks years. based on our model simulations, the current covid-19 pandemic is expected to end within 3 months from now. due to some unreported cases, a small number infection clusters may appear until december 2020. however, recent technological improvements and international cooperation between countries may even reduce the duration current pandemic. to sum up, this is the first study to model the infections disease transmission model to predict the gravity of covid-19 in canada using deep learning approaches. based on our current findings, provinces that have implemented social distancing guidelines before the pandemic has less confirmed cases than other provinces 3. for instance, saskatchewan issued social distancing guidelines 2 weeks ahead than quebec which has half of the confirmed cases in canada. our results could help canadian government to monitor the current situation and use our forecasts to prevent further transmissions. we confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. in so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. we further confirm that any aspect of the work covered in this manuscript that has involved human patients has been conducted with the ethical approval of all relevant bodies and that such approvals are acknowledged within the manuscript. irb approval was obtained (required for studies and series of 3 or more cases) written consent to publish potentially identifying information, such as details or the case and photographs, was obtained from the patient(s) or their legal guardian(s). the international committee of medical journal editors (icmje) recommends that authorship be based on the following four criteria: 1. substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data for the work; and 2. 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manuscript has been read and approved by all named authors. we confirm that the order of authors listed in the manuscript has been approved by all named authors. the corresponding author declared on the title page of the manuscript is: this author submitted this manuscript using his/her account in editorial submission system. we understand that this corresponding author is the sole contact for the editorial process (including the editorial submission system and direct communications with the office). he/she is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. we confirm that the email address shown below is accessible by the corresponding author, is the address to which corresponding author's editorial submission system account is linked, and has been configured to accept email from the editorial office of international journal of women's dermatology: someone other than the corresponding author declared above submitted this manuscript from his/her account in editorial submission system: we understand that this author is the sole contact for the editorial process malaria transmission dynamics of the anopheles mosquito in kumasi, ghana bifurcation analysis of a mathematical model for malaria transmission mathematical analysis of the transmission dynamics of hiv/tb coinfection in the presence of treatment semianalytical study of pine wilt disease model with convex rate under caputo-febrizio fractional order derivative a new fractional hrsv model and its optimal control: a non-singular operator approach a new study on the mathematical modelling of human liver with caputofabrizio fractional derivative a new fractional modelling and control strategy for the outbreak of dengue fever bridging the gap between evidence and policy for infectious diseases: how models can aid public health decision-making application of the arima model on the covid-2019 epidemic dataset forecasting of covid-19 confirmed cases in different countries with arima models estimation of the reproductive number of novel coronavirus (covid-19) and the probable outbreak size on the diamond princess cruise ship: a datadriven analysis the fractional features of a harmonic oscillator with position-dependent mass new aspects of time fractional optimal control problems within operators with nonsingular kernel a new feature of the fractional euler-lagrange equations for a coupled oscillator using a nonsingular operator approach deep learning for real-time gravitational wave detection and parameter estimation: results with advanced ligo data covid-19 forecasting using lstm networks the âăĳinconvenient truthâăi̇ about ai in healthcare preliminary estimation of the basic reproduction number of novel coronavirus (2019-ncov) in china, from 2019 to 2020: a data-driven analysis in the early phase of the outbreak transmission potential and severity of covid-19 in south korea updating of covariates and choice of time origin in survival analysis: problems with vaguely defined disease states strong consistency of least-squares estimation in linear regression models with vague concepts machine learning approach for confirmation of covid-19 cases: positive, negative, death and release multiple-input deep convolutional neural network model for covid-19 forecasting in china prediction for the spread of covid-19 in india and effectiveness of preventive measures neural network based country wise risk prediction of covid-19 wavelet transform domain filters: a spatially selective noise filtration technique lag order and critical values of the augmented dickey-fuller test insights into lstm fully convolutional networks for time series classification a pandemic warning? no funding was received for this work. vinay kumar reddy chimmula: conceptualization of this study, methodology, software, writing -original draft preparation,critical revision of the manuscript for important intellectual content. lei zhang: data curation,critical revision of the manuscript for important intellectual content,supervision and material support, regular feedback after each update. manuscript title: the authors whose names are listed immediately below certify that they have no affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers' bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript. the authors whose names are listed immediately below report the following details of affiliation or involvement in an organization or entity with a financial or non-financial interest in the subject matter or materials discussed in this manuscript. please specify the 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above.author's name (fist, last) signature date all persons who meet authorship criteria are listed as authors, and all authors certify that they have participated sufficiently in the work to take public responsibility for the content, including participation in the concept, design, analysis, writing, or revision of the manuscript. furthermore, each author certifies that this material or similar material has not been and will not be submitted to or published in any other publication before its appearance in the hong kong journal of occupational therapy. please indicate the specific contributions made by each author (list the authors' initials followed by their surnames, e.g., y.l. cheung). the name of each author must appear at least once in each of the three categories below. conception and design of study: vkr chimmula acquisition of data: vkr chimmula analysis and/or interpretation of data: vkr chimmula; l zhang drafting the manuscript: vkr chimmula, revising the manuscript critically for important intellectual content: l zhang; vkr chimmula approval of the version of the manuscript to be published (the names of all authors must be listed):vkr chimmula, l zhang. key: cord-309758-2rnhrbeq authors: batistela, cristiane m.; correa, diego p.f.; bueno, átila m; piqueira, josé roberto c. title: sirsi compartmental model for covid-19 pandemic with immunity loss date: 2020-10-29 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110388 sha: doc_id: 309758 cord_uid: 2rnhrbeq the coronavirus disease 2019 (covid-19) outbreak led the world to an unprecedented health and economic crisis. in an attempt to respond to this emergency, researchers worldwide are intensively studying the dynamics of the covid-19 pandemic. in this study, a susceptible infected removed sick (sirsi) compartmental model is proposed, which is a modification of the classical susceptible infected removed (sir) model. the proposed model considers the possibility of unreported or asymptomatic cases, and differences in the immunity within a population, i.e., the possibility that the acquired immunity may be temporary, which occurs when adopting one of the parameters ([formula: see text]) other than zero. local asymptotic stability and endemic equilibrium conditions are proved for the proposed model. the model is adjusted to the data from three major cities of the state of são paulo in brazil, namely, são paulo, santos, and campinas, providing estimations of duration and peaks related to the disease propagation. this study reveals that temporary immunity favors a second wave of infection and it depends on the time interval for a recovered person to be susceptible again. it also indicates the possibility that a greater number of patients would get infected with decreased time for reinfection. in rio de janeiro, brazil, social distancing measures were introduced on march 17, 2020 [8] , and the government of the state of são paulo, brazil, decreed quarantine on march 23, 2020. by june 25, 2020, the state of são paulo had 248,587 confirmed cases with 13, 759 deaths. an important question related to patients' immunity after recovery was raised. differences in immunity after recovery have been reported in humans [12] , and in experiments with rhesus macaques [13] . the experiment in [12] collected plasma from 175 recovered covid-19 patients, and sars-cov-2-specific neutralizing antibodies (nabs) were detected from day 10 to 15, after the onset of the disease and remained thereafter. nonetheless, the nabs levels were variable in the cohort. among them, 52 ( ≈30%) patients developed low levels of these antibodies, in 10 ( ≈6%) the nabs levels were undetectable, and 25 ( ≈14%) developed high levels. in [14] , data suggests an imbalance between the control of virus replication and activation of the adaptive immune responses. in some patients, lung cells remain vulnerable to infection due to immune system deficiencies. the virus continues replicating while the immune system attacks infected cells, also killing healthy cells in the vicinity, thereby severely inflaming the lung tissues. this seems to be the mechanism, which makes some patients severely ill weeks after their initial infection. additionally, sars-cov-2 probably induces immunity like other coronaviruses; however, this mechanism is not fully understood [15] . the economic crisis due to the pandemic is another important issue. according to [16] the mortality rate of covid-19 is not necessarily correlated with the global economic crisis, since governments, companies, consumers, and media have reacted to the economic shock. however, a global recession seems to be inevitable, and its duration and intensity will depend on the success of the measures taken to prevent the spread of the disease. taking the whole scenario into account, researchers worldwide are intensively studying and developing mathematical models of the covid-19 outbreak. the knowledge on the dynamics of this pandemic is important to estimate the duration and peaks of the outbreak. macro-modeling of propagation of infectious diseases has been a field of research since the kermack and mckendrick proposition [17] [18] [19] [20] [21] . it is a dynamic model that classifies individuals in a population as susceptible -infected -removed (sir) [22] . since then, several modifications of the sir model [23, 24] such as, the susceptible -exposed -infected -removed (seir) model, [25, 26] have been proposed for epidemic modeling. in addition, studies considering time delay [27] and fractionalorder [28] [29] [30] dynamical systems have been applied to disease propagation and could be useful to model the covid-19 outbreak, and other biological systems [31, 32] . in [33] a sird (sir with delay) model had been adjusted to covid-19 spread in china, italy, and france. the results have shown that the recovery rates were similar for the three countries. in [34] , a mixed analytical-statistical inverse problem is used to predict the covid-19 progression in brazil. a siru model, susceptible -infected -removed -unreported, was used to estimate the direct problem and a bayesian parameter was used for the inverse problem. compartmental models considering immigration and home isolation, or quarantine, were proposed by [10] . all the situations presented an infection-endemic equilibrium with results showing that home isolation or lock down mitigates the infection probability. in this work the proposed model is a modification of the original compartmental sir model of , which includes a sick ( s ick ) population compartment representing nodes of the networks that manifest the symptoms of the disease. although, several relevant questions are raised about the spread of the disease, we will focus on two of them in this study. the first one is associated with the group of asymptomatic individuals, who constitute the majority of the cases of the disease, and the second is related to the possibility that the immunity acquired by an infected person may be temporary. as the mechanism of the spread of this disease is associated with the number of infected people and contact of this group to the susceptible group, it is important that the group of asymptomatic or unreported individuals be studied. asymptomatic individuals are the group of infected people who do not manifest the symptoms, but can spread the disease by establishing effective contact with susceptible individuals. to infer the behavior of asymptomatic individuals, the infected compartment will be divided into infected people who manifest the symptoms (sick) and those who do not, based on public data. in addition, the influence of acquired immunity among individuals who have recovered from the infection is also being studied. if the acquired immunity is permanent, then the pattern of spread is described as temporal. on the other hand, if the immunity is temporary, it is important to understand the time interval after which the individual becomes susceptible again, changes in the dynamics of disease spread, and, the influence of the time interval for this reinfection on the temporal response. the proposed susceptible -infected -removed -sick (sirsi) model also considers birth and death of individuals in the given population and introduces a feedback from those in the recovered group who did not gain immunity or lost their immunity after a period of time. the proposed sirsi model presents both a disease free and an endemic equilibrium. the influence of the re-susceptibility feedback is investigated either analytically or numerically. the parameters of the proposed sirsi model are numerically fit to the pandemic situation of three cities in são paulo state, in brazil, namely, são paulo, the capital of the state, santos, on the coast and approximately 80 km away from são paulo, and campinas, in the interior of the state and approximately 90 km distant from the capital são paulo. the paper is organized as follows, section 2 presents the sirsi compartmental mathematical model. section 3 presents the equilibrium points and stability conditions, showing the possible existence. the proposed sirsi model can be seen in fig. 1 . in this model, the susceptible population s is infected at a rate α when contacted by an infected individuals from i. the susceptible compartment also receives a population, at a rate 1 /γ , who did not gain complete immunity after recovering or who lost their immunity after a period of time. the compartment irepresents the infectious population in the incubation stage prior the onset of symptoms. infection transmission during this period has been reported in [35] [36] [37] [38] jid: chaos [m5g; october 29, 2020; 10:20 ] population can be asymptomatic or symptomatic. the period between infection and onset of symptoms 1 /β 2 , ranges from 3 to 38 days with a median of 5.2 days. once the infected individual is tested positive and the case is documented, it is moved to the s ick compartment and those who do not develop severe symptoms become asymptomatic. [39] reported that the estimate of the number of infections originating from undocumented cases is as high as 86%, including infected individuals with mild, limited and lack of symptoms. in other recent studies [40, 41] , it was found that, 20% -40% of positive tested patients were asymptomatic. in this work, and [42] , under-reporting of asymptomatic cases is considered. the size of this population could be 7 times bigger than that of the documented cases and this group recovers within the period 1 /β 1 . some of the individuals within this population could develop symptoms. in [43] , it has been reported that the average time between infection and the onset of symptoms can be 4.6 days. once the case is documented it should be moved to the s ick compartment, which consists of those patients with severe symptoms seeking medical attention. in [44] it was reported that this population could represent up to 19.9% of the total documented cases, of which 13.8% are severe cases and 6.1% require intensive care. the sick population recovers within the period 1 /β 3 or removed at a rate σ (see fig. 1 ). the effect of social distancing measures in the infected patients and corresponding deaths toll are shown in fig. 1 . the parameter θ is introduced (1) , and subject to the constraint 0 < θ < 1 . consequently, the sirsi model is given by (1) , where λ and δ are the birth and death rates, respectively. the number of documented cases is a key information that should be noted from (1) because the number of confirmed cases is publicly available, and will be used to fine-tune the model. although data on the number of new cases are added daily, it tends to be less representative of the dynamics. here, the model given by (1) is considered, such that ˙ to investigate the influence of the introduction of feedback from the recovered individuals with no immunity, the equilibrium points from (1) must be determined and their stability must be discussed. assuming the realistic hypothesis that α = 0 , i.e., susceptible can be converted into infected, the equilibrium points are f (x * ) = 0 . using the hartman-grobman theorem [45] the local stability of the equilibrium points can be determined by the eigenvalues of the jacobian matrix computed on each equilibrium point. the jacobian of (1) is given by (2) . the disease-free ( section 3.1 ) and the endemic ( section 3.2 ) equilibrium points are determined in the following section. the disease-free equilibrium is a state corresponding to the absence of infected individuals, i.e. , i * = 0 . applying this condition to belonging to the first octant of r 4 and given by: considering p 1 , the corresponding jacobian [45] by the laplace determinant development [46] , the eigenvalues of (4) are the elements in the diagonal, that is, the eigenvalues ξ 1 ,ξ 3 and ξ 4 are real and negatives, indicating asymptotically stable directions. consequently, ξ 2 determines the equilibrium stability. if α(1 − θ ) λ/δ < (β 1 + β 2 ) the eigenvector associated indicates an asymptotically stable direction. for α(1 − θ ) λ/δ > (β 1 + β 2 ), the equilibrium point p 1 is unstable, with the onset of a bifurcation in the parameter space. the endemic equilibrium points are characterized by the existence of infected people in the population, that is, (i * = 0 ). therefore, there is an endemic equilibrium point p 2 = x * = (s * , i * , r * , s * ick ) t in the first octant of r 4 given by consequently, the existence condition for the endemic equilibconsidering p 2 , the corresponding jacobian [45] with eigenvalues ξ given by with any further effort to analytically obtain ξ eigenvalues becomes difficult due to the complexity of the coefficients. a possible alternative approach is to perform numerical calculations. however, some insight for the model with feedback γ can be obtained, in terms of bifurcations and stability, analyzing the eigenvalues when γ = 0 and γ = 0 . endemic equilibrium exists even with γ = 0 . in this case, the eigenvalues are consequently, ξ 1 and ξ 2 are real and negative, indicating asymptotically stable directions. the conclusion is the same for ξ 4 , because it is real and negative or complex with negative real part. the eigenvalue ξ 3 must be analyzed more accurately because it can be the conjugated to ξ 4 in a complex case, i.e., in situations with < 0 , indicating asymptotic stability. however, in a real case, i.e. when , then the endemic equilibrium point is unstable. however, in the case of instability due to ξ 3 , the existence condition (6) for the endemic equilibrium is not satisfied. consequently, if the endemic equilibrium point p 2 exists, it is asymptotically stable. analyzing how γ changes the endemic situation, in a case with γ → ∞ is described by following the same existence condition given by (6) . under the assumption γ → ∞ , the coefficients from eq. (8) can be approximated by and, consequently, (8) is rewritten as assuming that at least one root ξ goes to infinity as γ → ∞ : then, ξ 1 = −γ , and γ → ∞ , so, one eigenvalue goes to −∞ as γ → ∞ . to find an approximation to the other three roots, the characteristic equation is rewritten in eq. (13) . it can be also assumed that the other three roots are finite following in this section the parameters of the proposed sirsi model (1) (see fig. 1 ) are numerically adjusted to fit the curve of confirmed symptomatic cases of three major cities in the state of são paulo -brazil, using publicly available data from the state data analysis system -seade ( sistema estadual de análise de dados 2 ) [47] . the total populations in each of the cities were obtained from the same source and it is shown in table 1 . birth and death rates were calculated using λ and δ respectively. linear interpolation was necessary as the data from the public repository were out of date. results are shown in fig. 2 . one of the first measures to contain the spread of the virus was the imposition of social distancing. this intervention, represented by θ in the model, focuses on reducing the possible contact between infected and susceptible individuals, and hence it is introduced as a factor of the transmission rate, i.e. α(1 − θ ) . the time series corresponding to the daily measures of this index along with their mean for each one city considered are shown in fig. 3 . although this time variation resembles a 7-day periodic function, especially on the second half of the register, the mean of this measure is used as a representative value. information about the duration of antibody response to sars-cov-2 is limited; however, currently it is known that the immunity acquired is temporary. recent research highlights that immune response and antibody protection after recovery may depend on the severity of the infection, in some cases this protection can last for as little as 12 weeks while in others, no antibody protection is observed [49] [50] [51] [52] [53] [54] . to assess the influence of this behavior on disease spread, we set the feedback parameter γ at constant values γ ∈ [ 0 . 00 0 . 01 0 . 02 0 . 03 0 . 04 ] in order to map possible scenarios especially those related to second waves of infection. the disease transmission rate of symptomatic individuals prior to hospitalization is estimated to be between 1.12 and 1.19, while that of asymptomatic cases is between 0.1 and 0.6. in the model (1) this parameter is represented by the product α(1 − θ ) , thus, making α ∈ [ 0 . 1 1 . 0 ] . the mean time between infection and onset-of-symptoms for confirmed cases, 1 /β 2 , is 2 days, making the time from onset-of-infection to full recovery, 1 /β 1 , is considered to be between a few days to two weeks (5 to 15 days). the time period for a symptomatic patient to overcome the disease, 1 /β 3 , and the time between hospitalization to death, 1 /σ, are both considered to be between 5 to 20 days [43, 44, 55, 56] . the trust-region reflective least-squares algorithm [57] [58] [59] [60] along with a 4th-order runge-kutta integrator were used to fit parameters in model (1) to actual data collected from public repositories for each one of the three cities into consideration. all parameters and initial conditions computed are normalized with respect to the total population in each case. for fitting, it was set s 0 ∈ [0 , 1] ,i 0 ∈ [0 , 1] with initial guess s i 0 = 99 . 9% and i i 0 = 0 . 1% , the initial condition for s ick and r was set to zero. results are shown in tables 2-7 in order to assess the influence of the parameter γ in the endemic equilibrium, the eigenvalues were plotted for the set of fit parameters found, for γ ∈ [0 , 2) . in fig. 4 the eigenvalues are shown for the endemic equilibrium of each city, and for each one of the fit sets. for γ = 0 the eigenvalues are stable for são paulo and santos, and as γ increases, the eigenvalues move towards the left-hand side of the complex plane, whereas for campinas the eigenvalues are unstable for γ = 0 . in fig. 5 a closer in this section numerical experiments conducted using the matlab-simulink [61] for two different conditions are described. first, the sirsi model is fit with the real data for the s icks population and different values of the parameter γ . consequently, the simulation for the infected population i can be inferred from the proposed model. the numerical experiments, as in section 4 , were conducted for three major cities in the state of são paulo, namely, são paulo, campinas, and santos. the initial condition is x 0 = (s 0 , i 0 , s ick 0 , r 0 ) t , where s 0 and i 0 are the normalized susceptible and infected populations, which are considered as free parameters as they can be modified by the fitting algorithm. 0.000e + 00 0.000e + 00 0.000e + 00 0.000e + 00 0.000e + 00 r 0 0.000e + 00 0.000e + 00 0.000e + 00 0.000e + 00 0.000e + 00 s 0 1.008e + 00 1.008e + 00 1.007e + 00 1.007e + 00 1.005e + 00 i 0 8.749e-06 9.148e-06 9.364e-06 9.614e-06 1.776e-05 s ick 0 0.000e + 00 0.000e + 00 0.000e + 00 0.000e + 00 0.000e + 00 r 0 0.000e + 00 0.000e + 00 0.000e + 00 0.000e + 00 0.000e + 00 the fig. 6 shows that the sirsi model gives a good adjustment for the data of confirmed cases in the infected population. considering that the acquired immunity is permanent, i.e. , γ = 0 , and that the isolation rate is constant, the peak of the infection occurs soon after july 2020, and the disease will not persist, until the end of the year. on the other hand, assuming that immunity is not permanent and adopting a reinfection rate γ = 0 . 01 such that every 100 days a recovered individual becomes susceptible again, the model predicts a decrease in the confirmed cases and a new wave of infection in the first half of 2022. decreasing the time interval in which a recovered person becomes susceptible ( γ = 0 . 02 ) to 50 days, the model indicates a second wave of infection in the first half of 2021. decreasing the time interval in which a recovered person becomes susceptible ( γ = 0 . 04 ) to 25 days, the model indicates that number of confirmed infected will be reduced by almost two thirds by the end of this year and the number of infected cases with continue to reduce. in this case, the number of confirmed cases will remain higher than the others. in fig. 7 , the infected compartment i inferred from the sirsi model is presented, showing that the peak of infection is close to july 2020. increasing γ reduces the time for a recovered person to become susceptible again, causing the peaks in fig. 7 to increase, when compared to the curves for lower values of γ . this behavior, however, cannot be observed in fig. 6 , indicating that the increase in re-susceptibility feedback gain γ possibly contributes to the increase of asymptomatic or unreported infected cases. in addition, it seems that if recovered individuals acquire permanent immunity γ = 0 , the number of infected people tends to zero by the end of 2020. for γ = 0 . 01 , there is a small increase in january 2022. for γ = 0 . 02 , a new wave of confirmed cases can be seen in fig. 6 , and the increase in the infected population is also observed in fig. 7 . for são paulo, the numerical experiments show that considering any reinfection rate, there will be confirmed infected cases and unreported infected cases until 2023, indicating the need for a control strategy and study on preventive inoculation. fig. 8 shows the sirsi model adjusted to the data of confirmed cases in santos. assuming that the immunity acquired is permanent, γ = 0 , and isolation rate is constant, the peak of the confirmed cases in santos will occur very close to july 2020, similar to that in são paulo (see fig. 6 ). cities, since the second wave of infection in the city of santos occurs prior to são paulo. when γ = 0 . 04 , after the peak of the confirmed cases, a second wave can be observed in the numerical results before the end of 2020, delaying the reduction of the confirmed cases. for santos, the numerical experiments show that when γ = 0 and γ = 0 . 03 , the number of confirmed cases tends to zero in the beginning of 2023. the infected compartment i inferred from the sirsi model is presented in fig. 9 , showing that the peak of infection is close to july 2020. the increase in re-susceptibility feedback gain γ , reduces the time for an infected person to be susceptible again, causing higher peaks when compared to the curves for lower values of γ . this behavior does not occur in fig. 8 indicating that the increase in feedback possibly contributes to the increase in asymptomatic or unreported infected cases. this situation is similar to that observed for são paulo. in addition, when γ = 0 , the number of infected people i tends to zero before the end of the 2020 (see the curve for γ = 0 in detailed study of the dynamics, because together with the situation in which the infected individual acquires permanent immunity, these rates suggest that the number of confirmed cases (see fig. 8 ) and asymptomatic infected individuals tends to zero more quickly. in the situation in which a recovered person becomes susceptible in 25 days, it is observed that the infection persists in the population for a longer time, as shown by the curve with γ = 0 . 04 in fig. 9 justifying an accurate set of public strategies. for permanent acquired immunity, γ = 0 , and constant isolation rate, the peak of confirmed infection cases occur in the beginning of the second half of 2020. considering the re-susceptibility feedback gain γ > 0 , in fig. 10 , it seems that with the increase in γ the time necessary for the number of confirmed cases tending to zero is slightly higher, which is different from that of santos and são paulo. however, campinas does not present a second wave of infection, even with the variation of γ . the general behavior of campinas, concerning the sensitivity analysis for γ , presents different results from that of santos and são paulo. it can be noticed in fig. 10 that the observed data are far from the peak of infection predicted by the sirsi model. at this point, more data is needed for any further qualitative analysis. observing the eigenvalues for the city of campinas (see figs. 4 and 5 ) it can be noticed that they are all real, indicating that there is no oscillatory behavior in the dynamics of the model. depending on the new data this situation might change. the infected compartment of campinas presents the peak of infection close to the beginning of the second half of 2020, fig. 11 . increase in the reinfection parameter causes an increase in the infection peak as shown in fig. 10 indicating that the increase in feedback possibly contributes to the increase in either asymptomatic or unreported infected cases. the proposed sirsi model was fit to publicly available data of the covid-19 outbreak, providing estimates on the duration and peaks of the outbreak. in addition, the model allows us to infer information related to unreported and asymptomatic cases. the proposed model with re-susceptibility feedback adjusted to the confirmed infection data, suggests the possibility that recovered patients may have temporary immunity γ > 0 or even permanent γ = 0 . considering the situation in which immunity is temporary, there is a second wave of infection which, depending on the time interval for a recovered person to be susceptible again, indicates a second wave with a greater or lesser number of reinfected patients. if the time interval is larger (larger γ ), the second wave of infection will have a greater number of infected people when compared to that of a shorter time interval. the qualitative behavior for são paulo and santos is similar in terms of the sensitivity analysis of the re-susceptibility feedback gain γ . the bigger the value of γ , the shorter the time for a recovered person to become susceptible to reinfection, increasing the number of unreported or asymptomatic cases. it is suggested to collect more data for the city of campinas because the data of the confirmed infected cases presents a certain distance from the peak of the infection. the dynamics of the model may undergo some significant changes, given the sensitivity of the model to disturbances. it is inferred from the proposed model that if the acquired immunity is temporary, a second wave of infection is a serious possibility. in addition, the number of asymptomatic patients increases if the acquired immunity lasts for a short period of time. data are publicly available with [47, 48] . there is no conflict of interest between the authors. world health organization who timeline -covid-192020 world health organization declares global emergency: a review of the 2019 novel coronavirus (covid-19) ministério da saúde do brasil 2020 world health organization who coronavirus disease (covid-19) dash-board2020[online development of new hybrid model of discrete wavelet decomposition and autoregressive integrated moving average (arima) models in application to one month forecast the casualties cases of covid-19 the effect of control strategies to reduce social mixing on outcomes of the covid-19 epidemic in wuhan, china: a 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wuhan on the implementation of a primal-dual interior point method recursive quadratic programming methods based on the augmented lagrangian trust region methods on the parameters estimation of hiv dynamic models numerical computing with matlab, other titles in applied mathematics key: cord-337760-joti9nwg authors: buldú, javier m.; antequera, daniel r.; aguirre, jacobo title: the resumption of sports competitions after covid-19 lockdown: the case of the spanish football league date: 2020-06-04 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109964 sha: doc_id: 337760 cord_uid: joti9nwg in this work, we present a stochastic discrete-time seir susceptible-exposed-infectious-recoveredmodel adapted to describe the propagation of covid-19 during a football tournament. specifically, we are concerned about the re-start of the spanish national football league, la liga, which is currently –may 2020– stopped with 11 fixtures remaining. our model includes two additional states of an individual, confined and quarantined, which are reached when an individual presents covid-19 symptoms or has undergone a virus test with a positive result. the model also accounts for the interaction dynamics of players, considering three different sources of infection: the player social circle, the contact with his/her team colleagues during training sessions, and the interaction with rivals during a match. our results highlight the influence of the days between matches, the frequency of virus tests and their sensitivity on the number of players infected at the end of the season. following our findings, we finally propose a variety of strategies to minimize the probability that covid-19 propagates in case the season of la liga was re-started after the current lockdown. the propagation of the virus sars-cov-2 officially started at the beginning of december 2019 in wuhan (china), where the first covid-19 victim was diagnosed with a new type of coronavirus. the virus first spread over different states in china before reaching other countries. on march 11th 2020, the world health organization (who) declared covid-19 a pandemic, pointing to more than 118000 cases of the coronavirus illness in over 110 countries around the world [1] . the evolution of the pandemic, which is (in may 2020) still affecting many countries worldwide, has been a matter of debate, since different strategies can be adopted to mitigate the spreading of covid-19, some of them with unclear or unpredictable consequences. due to the novelty of this unforeseen pandemic, the use of mathematical models is being extremely useful to predict the dynamics of the coronavirus spreading and the effects of different policies on the eventual reduction of the number of affected individuals. despite there are different approaches for modelling the pandemics, both continuous-time and discrete-time sir-based models are probably the most extended approaches. the susceptible-infected-recovered (sir) model was first proposed by kermack and mckendrick in 1927 [2] , and consists of a compartmental model where individuals are split into three different states: (i) susceptible (s), when they are sane, (ii) infected (i), when they have the virus and (iii) recovered (r). more sophisticated models include more possible states, such as deceased (d) in the sird model [3] or exposed (e) in the seir model [4] . the latter model has been extensively applied to describe the exponential growth of the number of individuals infected by sars-cov-2, the effects of quarantine and confinement measures and, ultimately, to evaluate an adequate way of leaving confinement measures without increasing the risk of a second outbreak [5, 6, 7, 8, 9, 10, 11] . for example, peng et al. collected the epidemic data from five different chinese regions and estimated the effects of the quarantine over all of them, forecasting the decrease of the number of infected individuals region by region [5] . on the other hand, radulescu et al. introduced a compartmental model consisting of dividing the population into age groups and analysing how the number of infected individuals was related to each age group. with this model, the effects of several social measures were simulated (closing campuses, schools or restaurants), showing different impacts at each age group [6] . regarding the vast (and recent) scientific literature about the seir model applied to covid-19, we can remark one of its significant merits: it can be easily adapted to describe a diversity of scenarios. in this manuscript, we present a discrete-time seir-type mathematical model that describes the spreading of the coronavirus during a sports competition. the motivation behind our study is that there has been a lively debate about whether sports competitions that were not completed before the coronavirus crisis should be re-started or, ultimately, cancelled [12, 13, 15] . on the one hand, it is not the first time that epidemic diseases have threatened sports competition. for example, as pointed in [13] , the 2014 fifa world cup in brazil overlapped with a period in which dengue risk was close to its maximum at three cities where matches were carried out [14] . furthermore, attendants and players had to take special precautions due to zika, a mosquito-transmitted disease. despite the risks, the competition continued without significant problems regarding the number of individuals infected by dengue or zika. on the other hand, many voices have claimed that sports competitions should be cancelled, not only for the high risk of athletes being infected during a competition but also due to the inability to be adequately treated in case of injury due to the saturation of hospitals [15] . however, to the best of our knowledge, this debate has not been confronted with mathematical models that describe the propagation of sars-cov-2 between athletes. here, we are concerned about the eventual re-start of the spanish national league, which is currently suspended with 11 pending fixtures, and focus on the optimum strategies to minimize the propagation of covid-19 among the players in case the competition was re-started after the current lockdown. we designed a mathematical model that incorporates the interaction of players during training sessions, leading to intra-club spreading, and during matches, responsible for inter-club contagions. furthermore, we incorporated the use of tests to evaluate its consequences in identifying and confining those players that already have been infected. the model, whose main parameters were based on the scientific literature concerning the infection and recovery periods of covid-19, could be easily adapted to describe other kinds of sports competitions just by modifying the number of players and teams participating in the tournament. in seir models [16] , a disease propagates through a network of individuals whose dynamical state can be either susceptible (s, healthy and susceptible to be infected), exposed (e, infected but in the latent period -period from infection to infectiousness-and therefore unable to infect other individuals), infectious (i, infected and able to infect other individuals), and removed (r, which includes (i) recovered individuals after having suffered the infection and therefore immune and (ii) deceased people). players can be in different states: susceptible (s), exposed (e), infectious (i) and quarantined (q). in case players are detected to be infected by the virus, they remain confined (indicated by the c suffix). after confinement, players undergo a quarantine before being eligible to play again. parameters β train , β match and β ext account for the probability of becoming exposed (e) during training, matches or externally (player social circle) respectively. probability σ describes the transition from exposed to infectious (i). probability γ controls the transition from infectious to recovered (r) or quarantined (q). finally, γ q is related to the quarantine period a player must follow after recovery. figure 1 represents a sketch of our discrete-time model. the time is discretised in days, and every random event is calculated once a day. the individuals (players from now on) can be infected at any time (that is, any day of the season) from people different to the players (technical staff of the team, family, etc.) with a probability β ext . the second source of infection occurs during the training period, where they can be infected from other players of their own team with probability β train . finally, during the matches, players are exposed to infection from the players of their own team and the adversary team with probability β match . 1 once a player has been infected and becomes exposed, he/she has a probability σ of finishing the latent period and become infectious. exposed and infectious players have, respectively, probabilities µ e and µ i of being detected as infected by covid-19 via a virus test or because they show disease symptoms. if this is the case, players will be confined at their homes remaining in two possible states: exposed e c or infectious i c . asymptomatic infectious players (belonging to class i, but not detected by virus tests), and confined infectious players, overcome the disease with probability γ. note that confined players that have been recovered will remain quarantined (class q) during a convalescence until they are prepared for playing again and become recovered (r) with probability γ q . the days between virus tests n test and the days between matches n match are two critical variables for controlling the number of infected players during the championship, and therefore their influence in the model should be studied carefully. note that the virus tests should be done in this context via polymerase chain reaction (pcr) controls. the reason is that fast antibody or antigen detectors are only reliable more than a week after the infection, and in many cases even after the patient has already shown symptoms. this fact would allow the infectious (but not identified as infected) players to spread the virus for several days, making the control of the disease a hard task. our model can be applied to a diversity of competitions related to team contact sports, but we have focused on the re-start of the spanish male national league. therefore, we considered a competition with m = 20 teams composed of l = 25 players, the latter being the upper limit of players that can be registered by a team in the competition. the generalization to liga iberdrola (spanish feminine first division football league, with 16 teams), to the masculine or feminine football leagues of other countries, or even to another team sports (such as basketball, handball, rugby, etc.) is direct. every team plays a match every n match days (in particular in days that are multiples of n match ), and during the n match − 1 days in between fection rate β typically used in sir models, which is the average number of contacts per person per time multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject. while β is a rate (and can therefore be larger than 1), our parameters are probabilities (and ≤ 1 in consequence). the players train at their own stadiums. we supposed no resting days, as there is a clear interest for finishing the leagues as soon as possible, but including them in the model is trivial. we represented the training dynamics of the players, and contacts between them, using social networks instead of mean-field contacts. in this way, players' social networks during the team training followed a random structure of connections (different for each team but maintained during all simulations) and were generated using an erdös-rényi model [17] with a probability p = 0.2 of connecting two players. this was done to describe the internal professional and friendship dynamics that every player has during training times and also during lunch time, etc. during training time, the infectious players (class i) might infect their neighbours in the social network with probability β train . during the match day, every infected player on the pitch can infect any other player of its own team or the adversary with probability β match (here we used a mean-field approach due to the inevitable contact dynamics that players follow during a match). note that players cannot avoid voluntarily the contact with other players in the contest (with the exception, perhaps, of celebrating a goal, that could be forbidden if necessary), and therefore the contagion probability during a match might be more significant than expected at first glance. also, as a third infection source, players can be infected any day from their social circle with probability β ext . finally, in order to minimise the spreading of the disease, a virus test is done to all 500 players every n test days (in particular in days that are multiples of n test , and before the match if it coincides with a match day). players that yield a positive result are immediately confined. there is a wide range of values in the recent literature regarding each of the parameters that define the different steps of the disease (see table 1 for a summary of the parameters of the model). the latent period σ −1 is the average time from infection to infectiousness, the incubation period is the average time from infection to the appearance of the first symptoms, and the infectious period γ is the average time that the patient is infectious. depending on the virus, the latent period can be shorter or larger than the incubation period. in the case of covid-19 the latent period is 1 or 2 days shorter on average than the incubation period, which makes it especially easy for the disease to spread among the population during the time in which people are infectious but asymptomatic. regarding the mean incubation period, in [20] it was shown to be around 5 days, similar to that of sars, table 1: summary of the main parameters used in the model: probability of being infected during the training period β train , during a match β match and from the player's social circle β ext ; latent period σ −1 , infectious period γ −1 and quarantine period γ −1 q ; probability of being detected as exposed (by virus test) µ e and as infectious (by virus test or by symptoms) µ i ; number of days between virus tests n test and matches n match . and in [21] it was affirmed that it could be as short as four days. note, however, that this quantity was not used in our model. in [18] , it was used a mean latent period σ −1 of 3 days and a mean infection period of γ −1 = 5 days, based on the wuhan data. we selected these values because they were also used in other more recent studies [19] . note, however, that these are mean values: in [22] it was shown that the probability that patients with mild symptoms infected other people was very low after a week from the appearance of symptoms, but these means that in mild cases of covid-19 patients can be infectious for as much as 10 days. furthermore, we have fixed the quarantine period γ −1 q to be five days, but varying slightly this quantity would not affect the results substantially. the probability β ext of being infected during a day from the player's social circle will slowly decrease as more and more individuals in the country recover from the disease, but for simplicity we have supposed it constant during the whole league, and one order of magnitude lower than the expected value β ext based on available spanish statistics. the reason is that players will be for sure either quarantined during the rest of the league (and in that case β ext = 0) or at least their social life will be very restricted during that time. in order to obtain a plausible value for β ext , we have used the data resulted from a mass virus testing campaign developed during the first days of may 2020 to 62400 spanish citizens belonging to 30000 different homes. the main result of that study is that around 5% of the spanish population (2.3 × 10 6 out of 46.9 × 10 6 citizens) has been infected during the pandemic, and this represents 10 times the detected cases so far (229540 on 14th may 2020). as at this date around 500 new infections were detected per day, we can extrapolate that n inf ≈ 500 × 9 were not detected infected cases, and therefore n inf /γ people would be infective during the latent period. in summary, where the latent period is γ −1 = 5 days and the basic reproductive number (i.e. the expected number of infections generated by one case in a population where all individuals are susceptible to infection) is, according to the spanish health authorities, r 0 ≈ 1 at this time. on the other hand, there is not available experimental data to obtain precise values for the infecting probabilities β train and β match , so we have fixed them at moderate values and checked that slight variations did not qualitatively change the results. in particular, as on average each player is in contact, during the training time, with a fraction p = 0.2 of the total number of players in the team (l = 25), a first-order basic calculation yields that he/she will infect around plβ train = 0.5 other players per training day, as far as all other players are susceptible. during a match, nonetheless, an infectious player can infect any of the other 21 players in the field, and will infect on average 21β match = 0.21 players per match (supposing again that the rest are susceptible). in summary, at the beginning of la liga, and in the improbable situation that an infected player skipped all virus tests, he/she would infect around two other players during the latent period, and this quantity would decrease with time as more and more players become infected and then recovered. we believe this is a plausible result taking into account that r 0 > 4 at the beginning of the pandemic and r 0 ≈ 1 after two months of absolute quarantine of the whole population of the country, and players would have an intermediate situation with a controlled but not quarantined behaviour. the value of the probability µ i of being detected as infectious, either because a player shows disease symptoms or because the virus test yields a positive result, has been considered to be within the window [0,1], being 0 in case of not doing any test and being asymptomatic, and 1 when tests have 100% sensitivity. however, when the sensitivity of the test is not analysed, we considered a value of 0.9 which is close to the typical one attributed to pcr tests. concerning the probability of detecting an exposed individual, we set it as µ e = µ i /3, i.e., three times less than detecting an infected individual through the same test. the reason is that the viral load of an exposed individual is lower than that of an infectious one, therefore reducing the probability of a positive test result. we simulated between 10 4 and 10 5 seasons using our discrete-time model and obtained the main statistics of the accumulated number of infected players at time t, n(t). importantly, the seed of all simulations contained one player of the league who is already infected at the first day of the tournament (i.e., n(0) = 1). by doing so, the epidemic spreading begins at day one instead of any random day of the season, and therefore time t should be understood as days after the first infection. figure 2 analyses the influence that the number of days between tests and matches, n test and n match , have on the accumulated number of infected players n(t) along the rest of the season (i.e., 11 matches and the training days in between). 10 4 independent simulations were performed, and the mean values of n(t),n(t), are plotted in the figure. in fig. 2a we see how the mean accumulated number of infected players n(t) changes when the number of days between matches n match is modified within the interval {3, 4, 5, 6, 7}, i.e., we set the minimum and the maximum number of days between matches to 3 and 7, respectively. interestingly, we observe that it is convenient to reduce the time between matches to the minimum. the reason is twofold. on the one hand, with n match being the lowest, the competition would last fewer weeks, and therefore the players would be exposed for less time. on the other hand, the probability of being infected is higher during a training day than during a match day, since players are more exposed to physical contact with other players during training. for these reasons, the higher the number of days between matches, the higher the slope of the curves of fig.2a . in fig. 2b we show the different evolution of the mean value of the accumulated number of infected playersn(t) when pcr tests are or are not table 1 , with µ i = 0.9, unless specified otherwise. the seed of all simulations contained one player infected at the first day of the tournament. (a) influence of the number of days between two consecutive matches, n match , onn(t). in this simulation, pcr tests with 90% sensitivity were carried out every n test = 7 days. in (b) we compare the outcome of not doing any tests during the rest of the season and doing them every n test = 7 days (matches played every 7 days), while in (c) we focus on the number of days n test between each pcr control (matches played every 4 days, closer to the optimum frequency of 1 every 3 days). (d) influence of test accuracy µ i onn(t) (pcr tests and matches carried out every 4 days). performed. matches are played with a separation of 7 days, in this case. we can observe how skipping the tests increases substantially the number of infected players. these results show that conducting a coronavirus detection test is essential to prevent its spread among la liga teams. however, it is necessary to take into account the frequency and reliability of such tests. to investigate this issue, we assume that it is decided to play, for example, every 4 days, a measure close to the most favourable scenario of 3 days, although not so extreme. in fig. 2c we see how important it is to perform tests as often as possible, ideally every day. as the tests are more separated over time, the risk of infecting more players inevitably increases. finally, it is possible to simulate how important the accuracy of the tests is and the consequences of making use of low sensitive methods. figure 2d shows how the mean value of infected playersn(t) increases as the reliability of the tests µ e and µ i decreases. these results support the convenience of performing pcr testing, whose accuracy is estimated to be substantially larger than any other method. as mentioned above, the curves shown in fig. 2 are the mean values n(t) obtained after m = 10 4 simulations of the model. while the standard deviation of the meann(t), σn = σ n / √ m, is so small that would be hardly distinguishable from the curves in any of the plots, the standard deviation of n(t), σ n , is on the contrary very large -in some cases of the order of the mean n-and shows that the evolution of a single process is highly unpredictable. to cast light on this question, in fig. 3 we have plotted the probability function of the accumulated number of infected players n(t) (i.e., probability of obtaining n(t) = 1, 2, 3... accumulated infected players after t days, calculated as the normalised histogram of 10 5 simulations of the process) when matches and pcr controls are carried out every 7 days (green curve in fig. 2a ) after t = 4 days, 20 days, and at the end of the league (t = 78 days). in the first days of the competition (fig. 3a) , the disease starts to spread in the team of the so-far unique infected player. as expected, a poisson distribution approximates accurately its function probability (note that in this and further calculations of approximations to the data, we subtract the initial infected individual from the series and shift the obtained curve 1 position in the x-axis). however, the disease soon spreads towards other teams and the function distribution becomes more complex: at moderate values of the time (t = 20, fig. 3b ) the probability function presents a hump that certifies that the curve is in fact the consequence of several spreading processes, that is, the addition of intra-and interteam spreading, plus the potential infections coming from outside the league. furthermore, when the season reaches its end (t = 78, fig. 3c ) the curve presents an exponential-like tail, and at that time the standard deviation σ n is almost as large as the meann (as it is verified in exponential probability density functions). note that, while a normal approximation is not accurate at the end of the league, when t grows substantially (many weeks after the end of the season, and therefore not shown) the probability function becomes a gaussian, as expected from the central limit theorem. finally, let us show that the statistical behaviour of the system at the end of the season is compatible with that of a compound poisson process (cp p ), that is, a stochastic process with jumps, where the jumps arrive randomly according to a poisson process. first, the function probability of the number of infected teams m inf (t) at t = 78 is indeed accurately approximated by a poisson distribution (fig. 3d) . second, the mean and standard deviation of the number of infected players at the end of the season, obtained numerically, (n = 5.6 and σ n = 4.8) agree (with an error of 6% and 8% respectively) with those obtained by the theoretical expressions typical of compound poisson processes [23] for t = 78: wheren t eam ± σ n t eam = 4.6 ± 4.1 is the accumulated number of infected players at the team where the infection started, andm inf ± σ m inf = 1.4 ± 0.6 is the accumulated number of infected teams. in summary, in spite of the complex particular details of the model here presented and sketched in fig. 1 , its statistical behaviour can be described as the addition of several processes, each of them happening in a different team, and where the infection dynamics from one team to another follows a random poisson process. "all models are wrong, but some are useful". this famous statement, attributed to the statistician george box, sums up the usefulness of our model. although it is not possible to predict the exact number of infected individuals, the model allows describing, in a qualitative way, the influence that different measures can have to mitigate the spreading of the coronavirus during a sports competition. in that sense, we first investigated the effects of reducing the number of days between matches. interestingly, we observed that this fact resulted in a substantial reduction in the number of infected players at the end of the season (see fig. 2a ). this result could seem counterintuitive, since reducing the frequency of matches would promote the spreading speed of the coronavirus between teams. however, the fact that the season would be more compressed also results in a reduction of the exposure time to external infections, being the latter a factor with a major influence on the final number of infected players. next, we investigated the consequences of modifying the number of days between pcr tests. let us remark that antibody and antigen tests should be ruled out in this context because they are not effective until the disease is well advanced. as expected, we observed that the tests should be as accurate as possible and should be carried out continuously along with the competition, with the optimum scenario being one test per day (see figs. 2b-d). the analysis of the statistical behaviour of the numerical results yields that, in spite of the complexity of the model, the system at the end of the season can be described as a compound poisson process (see fig. 3 ). regarding the large unpredictability associated to it we conclude that, while qualitative results are clear and lead to behaviour strategies easy to follow by the sports organizations, obtaining precise predictions for a single realisation -the real case-is not possible. this is in agreement with recent work that warns about the strong sensitivity to parameter values in epidemics modelling [24] . we must also note that applying all the measures suggested by the model involves a cost. on the one hand, reducing the time between matches can be very physically demanding. the recovery time after a match would be reduced and the risk of injury would increase. to reduce this risk, teams should increase player rotations. regarding the tests, football clubs should provide the necessary support and means to carry out such a high number of tests in such a short time. without an adequate policy in this regard, the risk of reinfection in a competition would skyrocket. players will also pay a personal cost to control the eventual spreading of coronavirus. minimising their contacts with other individuals would mean limiting their travels, public events and, in general, reducing interactions with people outside the family environment. in fact, maintaining them in confinement during the rest of the season would be obviously the optimum situation. finally, although the results shown here are focused on the resumption of the men's spanish national league, the conclusions are equally valid for the women's competition. furthermore, the model could be adapted to any competition in which matches involve some physical contact between players, such as basketball, handball or rugby. a contribution to the mathematical theory of epidemics the sird epidemial model global stability for the seir model in epidemiology epidemic analysis of covid-19 in china by dynamical modeling management strategies in a seir model of covid 19 community spread the effectiveness of the quarantine of wuhan city against the corona virus disease 2019 (covid-19): wellmixed seir model analysis modelling the epidemiological trend and behavior of covid-19 in italy modeling the covid-19 outbreaks and the effectiveness of the containment measures adopted across countries mathematical assessment of the impact of 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assessment of hospitalized patients with covid-2019 introduction to probability models predictability: can the turning point and end of an expanding epidemic be precisely forecast? conceptualization, methodology. jacobo aguirre: data curation. jacobo aguirre & javier m. buldú: numerical simulations. jacobo aguirre & javier m. buldú: writing-original draft preparation. jacobo aguirre, daniel r. antequera & javier m. buldú: writing-reviewing and editing the authors acknowledge e. lázaro and j.a. sánchez-brunete for fruitful conversations on virological and pharmaceutical aspects of covid-19, respectively, p. catalán for advice on the statistical analysis of the results, and j. iranzo for assistance on the calculation of the model parameters. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. key: cord-258018-29vtxz89 authors: cooper, ian; mondal, argha; antonopoulos, chris g. title: a sir model assumption for the spread of covid-19 in different communities date: 2020-06-28 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110057 sha: doc_id: 258018 cord_uid: 29vtxz89 in this paper, we study the effectiveness of the modelling approach on the pandemic due to the spreading of the novel covid-19 disease and develop a susceptible-infected-removed (sir) model that provides a theoretical framework to investigate its spread within a community. here, the model is based upon the well-known susceptible-infected-removed (sir) model with the difference that a total population is not defined or kept constant per se and the number of susceptible individuals does not decline monotonically. to the contrary, as we show herein, it can be increased in surge periods! in particular, we investigate the time evolution of different populations and monitor diverse significant parameters for the spread of the disease in various communities, represented by countries and the state of texas in the usa. the sir model can provide us with insights and predictions of the spread of the virus in communities that the recorded data alone cannot. our work shows the importance of modelling the spread of covid-19 by the sir model that we propose here, as it can help to assess the impact of the disease by offering valuable predictions. our analysis takes into account data from january to june, 2020, the period that contains the data before and during the implementation of strict and control measures. we propose predictions on various parameters related to the spread of covid-19 and on the number of susceptible, infected and removed populations until september 2020. by comparing the recorded data with the data from our modelling approaches, we deduce that the spread of covid-19 can be under control in all communities considered, if proper restrictions and strong policies are implemented to control the infection rates early from the spread of the disease. in december 2019, a novel strand of coronavirus (sars-cov-2) was identified in wuhan, hubei province, china causing a severe and potentially fatal respiratory syndrome, i.e., covid-19. since then, it has become a pandemic declared by world health organization (who) on march 11, which has spread around the globe [1, 2, 3, 4, 5] . who published in its website preliminary guidelines with public health care for the countries to deal with the 5 pandemic [6] . since then, the infectious disease has become a public health threat. italy and usa are severely affected by covid-19 [7, 8, 9] . millions of people are forced by national governments to stay in self-isolation and in difficult conditions. the disease is growing fast in many countries around the world. in the absence of availability of a proper medicine or vaccine, currently social distancing, self-quarantine and wearing a face mask have been emerged as the most widely-used strategy for the mitigation and control of the pandemic. 10 in this context, mathematical models are required to estimate disease transmission, recovery, deaths and other significant parameters separately for various countries, that is for different, specific regions of high to low reported cases of covid-19. different countries have already taken precise and differentiated measures that are important to control the spread of the disease. however, still now, important factors such as population density, insufficient evidence for different symptoms, transmission mechanism and unavailability of a proper vaccine, makes it difficult 15 to deal with such a highly infectious and deadly disease, especially in high population density countries such as india [10, 11, 12] . recently, many research articles have adopted the modelling approach, using real incidence datasets from affected countries and, have investigated different characteristics as a function of various parameters of the outbreak and the effects of intervention strategies in different countries, respective to their current situations. it is imperative that mathematical models are developed to provide insights and make predictions about the 20 pandemic, to plan effective control strategies and policies [13, 14, 15] . modelling approaches [8, 16, 17, 18, 19, 20, 21] are helpful to understand and predict the possibility and severity of the disease outbreak and, provide key information to determine the intensity of covid-19 disease intervention. the susceptible-infected-removed (sir) model and its extended modifications [22, 23, 24, 25] , such as the extended-susceptible-infected-removed (esir) mathematical model in various forms have been used in previous studies [26, 27, 28] to model the spread of 25 covid-19 within communities. here, we propose the use of a novel sir model with different characteristics. one of the major assumptions of the classic sir model is that there is a homogeneous mixing of the infected and susceptible populations and that the total population is constant in time. in the classic sir model, the susceptible population decreases monotonically towards zero. however, these assumptions are not valid in the case of the spread of the covid-19 virus, since new 30 epicentres spring up around the globe at different times. to account for this, the sir model that we propose here does not consider the total population and takes the susceptible population as a variable that can be adjusted at various times to account for new infected individuals spreading throughout a community, resulting in an increase in the susceptible population, i.e., to the so-called surges. the sir model we introduce here is given by the same simple system of three ordinary differential equations (odes) with the classic sir model and can be used to gain 35 a better understanding of how the virus spreads within a community of variable populations in time, when surges occur. importantly, it can be used to make predictions of the number of infections and deaths that may occur in the future and provide an estimate of the time scale for the duration of the virus within a community. it also provides us with insights on how we might lessen the impact of the virus, what is nearly impossible to discern from the recorded data alone! consequently, our sir model can provide a theoretical framework and predictions that 40 can be used by government authorities to control the spread of covid-19. in our study, we used covid-19 datasets from [29] in the form of time-series, spanning january to june, 2020. in particular, the time series are composed of three columns which represent the total cases i d tot , active cases i d and deaths d d in time (rows). these datasets were used to update parameters of the sir model to understand the effects and estimate the trend of the disease in various communities, represented by china, south korea, india, australia, usa, italy and the state of texas in the usa. this allowed us to estimate the development of covid-19 spread in these communities by obtaining estimates for the number of deaths d, susceptible s, infected i and removed r m populations in time. consequently, we have been able to estimate its characteristics for these communities and assess the effectiveness of modelling the disease. the paper is organised as following: in sec. 2, we introduce the sir model and discuss its various aspects. in 50 sec. 3, we explain the approach we used to study the data in [29] and in sec. 4, we present the results of our analysis for china, south korea, india, australia, usa, italy and the state of texas in the usa. section 5 discusses the implications of our study to the "flattening the curve" approach. finally, in sec. 6, we conclude our work and discuss the outcomes of our analysis and its connection to the evidence that has been already collected on the spread of covid-19 worldwide. 55 2. the sir model that can accommodate surges in the susceptible population the world around us is highly complicated. for example, how a virus spreads, including the novel strand of coronavirus (sars-cov-2) that was identified in wuhan, hubei province, china, depends upon many factors, among which some of them are considered by the classic sir model, which is rather simplistic and cannot take into consideration surges in the number of susceptible individuals. here, we propose the use of a modified sir model with characteristics, based upon the classic sir model. in particular, one of the major assumptions of the classic sir model is that there is a homogeneous mixing of the infected i and susceptible s populations and that the total population n is constant in time. also, in the sir model, the susceptible population s decreases monotonically towards zero. these assumptions however are not valid in the case of the spread of the covid-19 virus, since new epicentres spring up around the globe at different times. to account for this, we introduce here a sir model that does not consider the total population n , but rather, takes the susceptible population s as a variable that can be adjusted at various times to account for new infected individuals spreading throughout a community, resulting in its increase. thus, our model is able to accommodate surges in the number of susceptible individuals in time, whenever these occur and as evidenced by published data, such as those in [29] that we consider here. our sir model is given by the same, simple system of three ordinary differential equations (odes) with the 70 classic sir model that can be easily implemented and used to gain a better understanding of how the covid-19 virus spreads within communities of variable populations in time, including the possibility of surges in the susceptible populations. thus, the sir model here is designed to remove many of the complexities associated with the real-time evolution of the spread of the virus, in a way that is useful both quantitatively and qualitatively. it is a dynamical system that is given by three coupled odes that describe the time evolution of the following three 75 populations: 1. susceptible individuals, s(t): these are those individuals who are not infected, however, could become infected. a susceptible individual may become infected or remain susceptible. as the virus spreads from its source or new sources occur, more individuals will become infected, thus the susceptible population will increase for a period of time (surge period). furthermore, it is assumed that the time scale of the sir model is short enough so that births and deaths (other than deaths caused by the virus) can be neglected and that the number of deaths from the virus is small compared with the living population. based on these assumptions and concepts, the rates of change of the three populations are governed by the following system of odes, what constitutes our sir model where a and b are real, positive, parameters of the initial exponential growth and final exponential decay of the infected population i. it has been observed that in many communities, a spike in the number of infected individuals, i, may occur, which results in a surge in the susceptible population, s, recorded in the covid-19 datasets [29] , what amounts to a secondary wave of infections. to account for such a possibility, s in the sir model (1), can be reset to s surge at any time t s that a surge occurs, and thus it can accommodate multiple such surges if recorded in the published data in [29] , what distinguishes it from the classic sir model. the evolution of the infected population i is governed by the second ode in system 1, where a is the transmission rate constant and b the removal rate constant. we can define the basic effective reproductive rate r e = as(t)/b, as the fate of the evolution of the disease depends upon it. if r e is smaller than one, the infected population i will decrease monotonically to zero and if greater than one, it will increase, i.e., if di(t) thus, the effective reproductive rate r e acts as a threshold that determines whether an 100 infectious disease will die out quickly or will lead to an epidemic. at the start of an epidemic, when r e > 1 and s ≈ 1, the rate of infected population is described by the approximation di(t) dt ≈ (a − b) i(t) and thus, the infected population i will initially increase exponentially according to i(t) = i(0) e (a−b)t . the infected population will reach a peak when the rate of change of the infected population is zero, di(t)/dt = 0, and this occurs when r e = 1. after the peak, the infected population will start to decrease exponentially, following i(t) ∝ e −bt . thus, eventually (for t → ∞), the system will approach s → 0 and i → 0. interestingly, the existence of a threshold for infection is not obvious from the recorded data, however can be discerned from the model. this is crucial in identifying a possible second wave where a sudden increase in the susceptible population s will result in r e > 1, and to another exponential growth of the number of infections i. the data in [29] for china, south korea, india, australia, usa, italy and the state of texas (communities) are organised in the form of time-series where the rows are recordings in time (from january to june, 2020), and the three columns are, the total cases i d tot (first column), number of infected individuals i d (second column) and deaths d d (third column). consequently, the number of removals can be estimated from the data by since we want to adjust the numerical solutions to our proposed sir model (1) to the recorded data from [29] , for each dataset (community), we consider initial conditions in the interval [0, 1] and scale them by a scaling factor f to fit the recorded data by visual inspection. in particular, the initial conditions for the three populations are set such that s(0) = 1 (i.e., all individuals are considered susceptible initially), is the maximum number of infected individuals i d . consequently, the parameters a, b, f and i d max are adjusted 115 manually to fit the recorded data as best as possible, based on a trial-and-error approach and visual inspections. a preliminary analysis using non-linear fittings to fit the model to the published data [29] provided at best inferior results to those obtained in this paper using our trial-and-error approach with visual inspections, in the sense that the model solutions did not follow as close the published data, what justifies our approach in the paper. a prime reason for this is that the published data (including those in [29] we are using here) are data from different countries 120 that follow different methodologies to record them, with not all infected individuals or deaths accounted for. in this context, s, i and r m ≥ 0 at any t ≥ 0. system (1) can be solved numerically to find how the scaled (by f ) susceptible s, infected i and removed r m populations (what we call model solutions) evolve with time, in good agreement with the recorded data. in particular, since this system is simple with well-behaved solutions, we used the first-order euler integration method to solve it numerically, and a time step h = 200/5000 = 0.04 that 125 corresponds to a final integration time t f of 200 days since january, 2020. this amounts to double the time interval in the recorded data in [29] and allows for predictions for up to 100 days after january, 2020. obviously, what is important when studying the spread of a virus is the number of deaths d and recoveries r in time. as these numbers are not provided directly by the sir model (1), we estimated them by first, plotting the data for deaths d d vs the removals r d m , where r d m = d d + r d = i d tot − i d and then fitting the plotted data with the nonlinear function where d 0 and k are constants estimated by the non-linear fitting. the function is expressed in terms of only model values and is fitted to the curve of the data. thus, having obtained d from the non-linear fitting, the number of recoveries r can be described in time by the simple observation that it is given by the scaled removals, r m from the sir model (1), less the number of deaths, d from eq. (3), the rate of increase in the number of infections depends on the product of the number of infected and susceptible individuals. an understanding of the system of eqs. (1) explains the staggering increase in the infection rate around the world. infected people traveling around the world has led to the increase in infected numbers and this results in a further increase in the susceptible population [14] . this gives rise to a positive feedback loop leading to a very rapid rise in the number of active infected cases. thus, during a surge period, the number of susceptible individuals increases and as a result, the number of infected individuals increases as well. for example, as of 1 march, 2020, there were 88 590 infected individuals and by 3 april, 2020, this number had grown to a staggering 1 015 877 [29] . 135 understanding the implications of what the system of eqs. (1) tells us, the only conclusion to be drawn using scientific principles is that drastic action needs to be taken as early as possible, while the numbers are still low, before the exponential increase in infections starts kicking in. here, we have applied the sir model (1) considering data from various countries and the state of texas in the usa provided in [29] . assuming the published data are reliable, the sir model (1) can be applied to assess the spread of the covid-19 disease and predict the number of infected, removed and recovered populations and deaths in the communities, accommodating at the same time possible surges in the number of susceptible individuals. figures 1-17 show the time evolution of the cumulative total infections i tot , current infected individuals, i, recovered 155 individuals, r, dead individuals, d, and normalized susceptible populations, s for china, south korea, india, australia, usa, italy and texas in the usa, respectively. the crosses show the published data [29] and the smooth lines, solutions and predictions from the sir model. the cumulative total infections plots also show a curve for the initial exponential increase in the number of infections, where the number of infections doubles every five days. the figures also show predictions, and a summary of the sir model parameters in (1) and published data in [29] 160 for easy comparisons. we start by analysing the data from china and then move on to the study of the data from south korea, india, australia, usa, italy and texas. the number of infections peaked in china about 16 february, 2020 and since then, it has slowly decreased. the from the plots shown in figs. 3 and 4 , it is obvious that the south korean government has done a wonderful job in controlling the spread of the virus. the country has implemented an extensive virus testing program. there has also been a heavy use of surveillance technology: closed-circuit television (cctv) and tracking of bank cards and mobile phone usage, to identify who to test in the first place. south korea has achieved a low fatality rate (currently one percent) without resorting to such authoritarian measures as in china. the most conspicuous part of the south korean strategy is simple enough: implementation of repeated cycles of test and contact trace measures. to match the recorded data from india with predictions from the sir model (1), it is necessary to include a 180 number of surge periods, as shown in fig. 5 . this is because the sir model cannot predict accurately the peak number of infections, if the actual numbers in the infected population have not peaked in time. it is most likely the spread of the virus as of early june, 2020 is not contained and there will be an increasing number of total infections. however, by adding new surge periods, a higher and delayed peak can be predicted and compared with future data. in fig. 5 , a consequence of the surge periods is that the peak is delayed and higher than if no surge periods were 185 applied. the model predictions for the 30 september, 2020 including the surges are: 330 000 total infections, 700 active infections and 7 500 deaths, whereas if there were no surge periods, there would be 130 000 total infections, 700 active infections and 6 300 deaths, with the peak of 60 000, which is about 40% of the current number of active cases occuring around 20 may 2020. thus, the model can still give a rough estimate of future infections and deaths, as well as the time it may take for the number of infections to drop to safer levels, at which time restrictions can 190 be eased, even without an accurate prediction in the peak in active infections (see figs. 5 and 6). a surge in the susceptible population was applied in early march, 2020 in the country. the surge was caused by 2 700 passengers disembarking from the ruby princes cruise ship in sydney and then, returning to their homes around australia. more than 750 passengers and crew have become infected and 26 died. two government enquires 195 have been established to investigate what went wrong. also, at this time many infected overseas passengers arrived by air from europe and the usa. the australian government was too slow in quarantining arrivals from overseas. from mid-march, 2020 until mid-may, 2020, the australian governments introduced measures of testing, contact tracing, social distancing, staying at home policy, closure of many businesses and encouraging people to work from home. from figs. 7 and 8, it can be observed that actions taken were successful as the actual number of infections as of early june, 2020, the peak number of infections has not been reached. when a peak in the data is not reached, it is more difficult to fit the model predictions to the data. in the model, it is necessary to add a few surge periods. this is because new epicentres of the virus arose at different times. the virus started spreading in washington state, followed by california, new york, chicago and the southern states of the usa. the need to add surge periods shows clearly that the spread of the virus is not under control. in the usa, by the end of may, 2020, the number of active infected cases has not yet peaked and the cumulative total number of infections keeps getting bigger. this can be accounted for in the sir model by considering how the susceptible population changes with time in may. during that time, to match the data to the model predictions, surge periods were used where the normalized susceptible population s was reset to 0.2 every four days. what is currently happening in the usa is that as susceptible individuals become infected, their population decreases, with 215 these infected individuals mixing with the general population, leading to an increase in the susceptible population. this is shown in the model by the variable for the susceptible population, s, varying from about 0.06 to 0.20, repeatedly during may. until this vicious cycle is broken, the cumulative total infected population will keep growing at a steady rate and not reach an almost steady-state. the fluctuating normalized susceptible variable provides clear evidence that government authorities do not have the spread of the virus under control (see figs. 9 220 and 10). the plots in figs. 11 and 12 show that the peak in the total cumulative number of infections has not been reached as early as june, however, the peak is probably not far away. if there are no surges in the susceptible population, then one could expect that by late september, 2020, the number of infections will have fallen to very 225 small numbers and the virus will have been well under control with the total number of deaths in the order of 2 000. in mid-may, 2020, some restrictions have been lifted in the state of texas. the sir model can be used to model some of the possible scenarios if the early relaxation of restrictions leads to increasing number of susceptible populations. if there is a relatively small increase in the future number of susceptible individuals, no series impacts occur. however, if there is a large outbreak of the virus, then the impacts can be dramatic. for example, at the end 230 of june, 2020, if s was reset to 0.8 (s = 0.8), a second wave of infections occurs with the peak number of infections occurring near the end of july, with the second wave peak being higher than the initial peak number of infections. subsequently, the number of deaths will rise from about 2 000 to nearly 5 000, as shown in figs. 13 and 14. if governments start lifting their containment strategies too quickly, then it is probable there will be a second wave of infections with a larger peak in active cases, resulting to many more deaths. figure 15 shows clearly that the peak of the pandemic has been reached in italy and without further surge periods, the spread of the virus is contained and number of active cases is declining rapidly. the plots in panels (a), (b) in fig. 16 are a check on how well the model can predict the time evolution of the virus. these plots also assist in selecting the model's input parameters. 240 the term flattening the curve has rapidly become a rallying cry in the fight against covid-19, popularised by the media and government officials. claims have been made that flattening the curve results in: (i) reduction in the peak number of cases, thereby helping to prevent the health system from being overwhelmed and (ii) in an increase in the duration of the pandemic with the total burden of cases remaining the same. this implies that social 245 distancing measures and management of cases, with their devastating economic and social impacts, may need to continue for much longer. the picture which has been widely shown in the media is shown in fig. 17(a) . the idea presented in the media as shown in fig. 17 (a) is that by flattening the curve, the peak number of infections will decrease, however, the total number of infections will be the same and the duration of the pandemic will be longer. hence, they concluded that by flattening the curve, it will have a lesser impact upon the demands in 250 hospitals. figure 17(b) gives the scientific meaning of flattening the curve. by governments imposing appropriate measures, the number of susceptible individuals can be reduced and combined with isolating infected individuals, will reduce the peak number of infections. when this is done, it actually shortens the time the virus impacts the society. thus, the second claim has no scientific basis and is incorrect. what is important is reducing the peak in the number of infections and when this is done, it shortens the duration in which drastic measures need to be taken and not lengthen the period as stated in the media and by government officials. figure 17 shows that the peak number of infections actually reduces the duration of the impact of the virus on a community. mathematical modelling theories are effective tools to deal with the time evolution and patterns of disease outbreaks. they provide us with useful predictions in the context of the impact of intervention in decreasing the 260 number of infected-susceptible incidence rates [30, 31, 32] . in this work, we have augmented the classic sir model with the ability to accommodate surges in the number of susceptible individuals, supplemented by recorded data from china, south korea, india, australia, usa and the state of texas to provide insights into the spread of covid-19 in communities. in all cases, the model predictions could be fitted to the published data reasonably well, with some fits better than others. for china, the actual 265 number of infections fell more rapidly than the model prediction, which is an indication of the success of the measures implemented by the chinese government. there was a jump in the number of deaths reported in mid-april in china, which results in a less robust estimate of the number of deaths predicted by the sir model. the susceptible population dropped to zero very quickly in south korea showing that the government was quick to act in controlling the spread of the virus. as of the beginning of june, 2020, the peak number of infections in india 270 has not yet been reached. therefore, the model predictions give only minimum estimates of the duration of the pandemic in the country, the total cumulative number of infections and deaths. the case study of the virus in australia shows the importance of including a surge where the number of susceptible individuals can be increased. this surge can be linked to the arrival of infected individuals from overseas and infected people from the ruby princess cruise ship. the data from usa is an interesting example, since there are multiple epicentres of the virus 275 that arise at different times. this makes it more difficult to select appropriate model parameters and surges where the susceptible population is adjusted. the results for texas show that the model can be applied to communities other than countries. italy provides an example where there is excellent agreement between the published data and model predictions. thus, our sir model provides a theoretical framework to investigate the spread of the covid-19 virus within 280 communities. the model can give insights into the time evolution of the spread of the virus that the data alone does not. in this context, it can be applied to communities, given reliable data are available. its power also lies to the fact that, as new data are added to the model, it is easy to adjust its parameters and provide with best-fit curves between the data and the predictions from the model. it is in this context then, it can provide with estimates of the number of likely deaths in the future and time scales for decline in the number of infections in communities. our 285 results show that the sir model is more suitable to predict the epidemic trend due to the spread of the disease as it can accommodate surges and be adjusted to the recorded data. by comparing the published data with predictions, it is possible to predict the success of government interventions. the considered data are taken in between january and june, 2020 that contains the datasets before and during the implementation of strict and control measures. our analysis also confirms the success and failures in some countries in the control measures taken. strict, adequate measures have to be implemented to further prevent and control the spread of covid-19. countries around the world have taken steps to decrease the number of infected citizens, such as lock-down measures, awareness programs promoted via media, hand sanitization campaigns, etc. to slow down the transmission of the disease. additional measures, including early detection approaches and isolation of susceptible individuals to avoid mixing them with no-symptoms and self-quarantine individuals, traffic restrictions, and medical treatment have 295 shown they can help to prevent the increase in the number of infected individuals. strong lockdown policies can be implemented, in different areas, if possible. in line with this, necessary public health policies have to be implemented in countries with high rates of covid-19 cases as early as possible to control its spread. the sir model used here is only a simple one and thus, the predictions that come out might not be accurate enough, something that also depends on the published data and their trustworthiness. however, as the model data show, one thing that is 300 certain is that covid-19 is not going to go way quickly or easily. health organization, coronavirus disease (covid-19) outbreak nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study novel coronavirus (covid-19) cases, provided by jhu csse estimation of the transmission risk of the 2019-ncov and its implication for public health interventions the effect of human mobility and control measures on the covid-19 epidemic in china naming the coronavirus disease (covid-19) and the virus that causes it an epidemiological forecast model and software assessing interventions on covid-19 epidemic in china extended sir prediction of the epidemics trend of covid-19 in italy and compared with hunan, china quantifying the effect of quarantine control in covid-19 infectious spread using machine learning predictions for covid-19 outbreak in india using epidemiological models covid-19: india imposes lockdown for 21 days and cases rise mohfw, coronavirus disease 2019 (covid-19). available online on the predictability of infectious disease outbreaks the effect of travel restrictions on the spread of the 2019 novel coronavirus (covid-19) outbreak. science epidemics with mutating infectivity on small-world networks early 340 dynamics of transmission and control of covid-19: a mathematical modelling study. the lancet infectious diseases modified seir and ai prediction of the epidemics trend of covid-19 in china under public health interventions analysis and forecast of covid-19 spreading in china, italy and france a data-driven network model for the emerging covid-19 epidemics in wuhan, toronto and italy estimation of covid-19 dynamics on a back-of-envelope: does the simplest sir model provide quantitative parameters and predictions modeling the impact of mass influenza vaccination and public health interventions on covid-19 epidemics with limited detection capability three basic epidemiological models the mathematics of infectious diseases the basic epidemiology models: models, expressions for r0, parameter estimation, and applications the sir model and the foundations of public health global analysis of the covid-19 pandemic using simple epidemiological models a modified sir model for the covid-19 contagion in italy mathematical modeling of covid-19 transmission dynamics with a case study of wuhan mod-370 elling the covid-19 epidemic and implementation of population-wide interventions in italy the effectiveness of quarantine of wuhan city against the corona virus disease 2019 (covid19): a wellmixed seir model analysis ), e0230405. declaration of competing interest i am attaching herewith a copy of our manuscript entitled "a sir model assumption for the spread of covid-19 in different communities" co-authored by ian cooper and chris g. antonopoulos in favor of publication in your esteemed journal chaos, solitons & fractals. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work am is thankful for the support provided by the department of mathematical sciences, university of essex, uk to complete this work. key: cord-325862-rohhvq4h authors: zhang, yong; yu, xiangnan; sun, hongguang; tick, geoffrey r.; wei, wei; jin, bin title: applicability of time fractional derivative models for simulating the dynamics and mitigation scenarios of covid-19 date: 2020-06-04 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109959 sha: doc_id: 325862 cord_uid: rohhvq4h fractional calculus provides a promising tool for modeling fractional dynamics in computational biology, and this study tests the applicability of fractional-derivative equations (fdes) for modeling the dynamics and mitigation scenarios of the novel coronavirus for the first time. the coronavirus disease 2019 (covid-19) pandemic radically impacts our lives, while the evolution dynamics of covid-19 remain obscure. a time-dependent susceptible, exposed, infectious, and recovered (seir) model was proposed and applied to fit and then predict the time series of covid-19 evolution observed over the last three months (up to 3/22/2020) in china. the model results revealed that 1) the transmission, infection and recovery dynamics follow the integral-order seir model with significant spatiotemporal variations in the recovery rate, likely due to the continuous improvement of screening techniques and public hospital systems, as well as full city lockdowns in china, and 2) the evolution of number of deaths follows the time fde, likely due to the time memory in the death toll. the validated seir model was then applied to predict covid-19 evolution in the united states, italy, japan, and south korea. in addition, a time fde model based on the random walk particle tracking scheme, analogous to a mixing-limited bimolecular reaction model, was developed to evaluate non-pharmaceutical strategies to mitigate covid-19 spread. preliminary tests using the fde model showed that self-quarantine may not be as efficient as strict social distancing in slowing covid-19 spread. therefore, caution is needed when applying fdes to model the coronavirus outbreak, since specific covid-19 kinetics may not exhibit nonlocal behavior. particularly, the spread of covid-19 may be affected by the rapid improvement of health care systems which may remove the memory impact in covid-19 dynamics (resulting in a short-tailed recovery curve), while the death toll and mitigation of covid-19 can be captured by the time fdes due to the nonlocal, memory impact in fatality and human activities. fractional calculus can provide a promising tool in modeling biological phenomena, as reviewed recently by lonescu et al. [1] . for example, fractional-derivative equation (fde) models have been applied to capture complex dynamics in biological tissues [2] , tumor growth [3] , dna sequencing [4] , drug uptake [5] , and salmonella bacterial infection in animal herds [6] . most recently, the fde models have been applied to model the pine wilt disease [7] , the human respiratory syncytial virus (hrsv) disease [8] , the harmonic oscillator with a position-dependent mass [9] , human liver using caputo-fabrizio fractional derivative [10, 11] , and tumor-immune surveillance [12] . motivated by these successful applications, this study tests whether the fdes can be applied to model the dynamics and mitigation scenarios of coronavirus, an emerging and critical research area that has not been focused by the fractional calculus community before. the novel coronavirus disease 2019 (covid-19) outbreak, a respiratory illness that started (first detected) in late december 2019, is a pandemic infecting >336,000 people in more than 140 countries with the average fatality rate of 4.4% globally (data up to 3/22/2020) [13] . the covid-19 pandemic is infiltrating almost every aspect of life, damaging global economy, and altering both man-made and natural environments. urgent actions have been taken but further effective and efficient strategies are promptly needed to confront this global challenge. to address this challenge and promptly guide the next efforts, it is critical to model the covid-19 outbreak. mathematical models are among the necessary tools to quantify the covid-19 and, therefore, testing the applicability of such fde models under this new global pandemic is the primary objective motivating this study. this study aims to model the covid-19 evolution dynamics (i.e., transmission, infection, recovery, and death evolution) for representative countries with apparent coronavirus cases, including china, the united states (u.s.), italy, japan, and south korea using mathematical models, most specifically the fde models described previously. it should be noted that the covid-19 spread in these countries experienced different starting (initiation and detection) times, an important fact to consider when applying these models. for example, china had passed the peak of the coronavirus outbreak, finally reaching a milestone with no new local infections on 3/19/2020 (79 days from the onset, 12/31/2019, in wuhan, china), while the u.s. coronavirus cases soared past 10,000 on the same day. this study will apply the core characteristics of the covid-19 outbreak obtained in china to estimate the covd-19 spread in the u.s., as well as other countries where the number of affected people has not yet reached its number of peak cases. in addition, this new pandemic may last for a relatively longer time than expected [14] . no vaccine against sars-cov-2 (severe acute respiratory syndrome coronavirus 2) is currently available [15] . indeed, a vaccine for prevention and infection control may not be ready before march 2021 for the covid-19, considering a minimum of 4~5 weeks for trials and at least 1 year for safety evaluation and final deployment. efficient strategies are therefore needed to mitigate the covid-19 outbreak. possible non-pharmaceutical scenarios, such as isolation of cases and contact tracing, can be evaluated using mathematical models [16] in order to identify the most efficient mitigation strategies going forward. this is another major motivation and the secondary task of this study using the fde models. to address the questions mentioned above, this study is organized as follows. section 2 proposes an updated seir model with an fde-based component for covid-19, where "s", "e", "i", and "r" stand for susceptible, exposed, infectious, and recovered people, respectively [17] . this model is then applied to fit and predict the covid-19 spread in various provinces and major cities in china, resulting in abundant datasets to derive the core characteristics of the covid-19 6 dynamics in evolution. section 3 predicts the spread of covid-19 in the other countries using the knowledge that is gained from the china case study. section 4 proposes a fully lagrangian approach with the time fde to model the spatiotemporal evolution of covid-19, and then applies it to evaluate non-pharmaceutical scenarios to mitigate the virus spread. section 5 reports the main conclusions, specifically the feasibility of fdes for capturing covid-19 evolution. in the appendix, the impacts of the non-singular kernel and fractional derivative type on model results are further discussed for readers particularly interested in these fractional calculus techniques. the main contributions of this work, therefore, include 1) the first application of fdes in modeling the evolution of the covid-19 death toll, 2) an updated seir model with a transient recovery rate to better capture the dynamics of covid-19 pandemic within china and for other countries, and 3) a particle-tracking approach based on stochastic bimolecular reaction theory to evaluate the mitigation of the spread of the covid-19 outbreak. several mathematical models have been applied for epidemic analysis of covid-19. the most widely used one, to date, is the well-known seir model. for example, peng et al. [18] (using data up to 2/16/2020) proposed a generalized seir model to successfully estimate the key epidemic parameters of covid-19 in china, and predicted the inflection point and ending time of confirmed covid-19 cases. the seir model was also applied by li et al. [19] (using the observed data up to 2/6/2020) to compare the effect of city lockdowns on the transmission dynamics for different cities in china. the sir model with time-dependent transmission and recovering rates was used by chen et al. [20] (using data up to 2/20/2020) to analyze and predict the number of confirmed cases of covid-19 in china. the sir model was extended by wang et al. [21] (using data up to 2/12/2020) to incorporate various time-varying quarantine protocols for assessing interventions on the covid-19 epidemic in china. the seir model and its modifications were also successfully applied by others [22] [23] [24] [25] , mostly for assessing the early spreading of covid-19 in china. previous applications of the popular seir model, however, may contain high uncertainty since they had limited data access for only a short period of the covid-19 outbreak. as will be shown below, the covid-19 dynamics have changed dramatically in the last three months, likely due to the improvement/adjustment of screening/testing techniques, public hospital system capabilities, and the government's control policies for contagious diseases. an updated, much better version of the seir model is therefore needed and can be reliably built now for china as more detailed and complete datasets are available that includes the coronavirus outbreak data (i.e., # of people infected) well beyond the peak in china. other models have also been used for specific purposes related to covid-19, including the global metapopulation disease transmission model to predict the impact of travel limitations on the epidemic spread [26] , the transmission model for risk assessment [27] , and the synthetic contact matrix model for quantifying reproductive ratios for covid-19 [28] . to the best of our knowledge, the classical stochastic epidemic models, such as the discrete or continuous time markov chain model and the stochastic differential equation model, have not yet been applied for covid-19 spread scenarios. a preliminary stochastic model built upon the fde for evaluating covid-19 spread in a city will be developed and applied in section 4. this section focuses on the deterministic model. the classical seir model considers constant parameters, which may not adequately capture time-dependent dynamics of epidemic (the covid-19) spread. hence, the seir model is updated in section 2.1, and then tested in section 2.2. the classical seir model containing four populations (s, e, i, and r) takes the form [29] : where is the stock of susceptible population; is the number of persons exposed to or in the latent period of the disease; is the stock of infected population; is the stock of recovered population; r denotes the number of susceptible people whom the infected people contact daily; n is the sum of all the four groups of people ( ( ) ( ) ( ) ( ) (which is a constant representing the constancy of population n); p is the constant rate of infection (i.e., representing the probability for the infected people to transform the susceptible people into infected ones); is the constant rate for the exposed person to be transformed into an infected one; and is the constant recovery rate (defining the speed for the infected person to be cured or expired). to allow for possible time-sensitive rates and time nonlocal-dependency for covid-19 evolution, we revise model (1) as: where represents the number of deaths (which is one component of i); is the rate for the healthy, susceptible person to be transferred to an infected one from exposed persons (note that covid-19 patients in the incubation period might be contagious too); and is the number of healthy, susceptible people that are contacted by exposed people daily. now, the infection rates can change with time, and the infected persons are removed from the risk of infection via a time-dependent rate term of ( ). if the recovered individuals can return to a susceptible status due to, for example, a loss of immunity, then the partial differential equation (pde) (2d) for the time rate of change of requires one more (sink) term: , where is the rate of the recovered individuals returning to a susceptible status. we replace the integer-order derivatives in the classical seir (1) by the fractional-order derivatives in (2) , to capture the possible nonlocal impact, such as the memory impact or any apparent delay, in the covid-19 outbreak. herein we use the death evolution (2e) as an example. hence, the fde (2e) containing the death probability of (while the other patients are cured) is modified as: which is the caputo fractional derivative [30, 31] with order ( ) (note that all the other indexes , , , and in model (2) are in the same range of (0, 1]). the caputo fractional derivate was selected here because the initial condition for caputo derivative takes the same form as that for the integer-order pde. particularly, the caputo derivative allows the utilization of initial values (of the integer order) with known physical interpretations. when the order =1, model (2e) reduces to the classical integer-order pde for the death evolution. the fractional pde (2e) is used here for two reasons. first, the evolution of deaths and cures may be characterized by a random process, due to the fact that the exact time for the (recovered) person to be initially infected is unknown (i.e., the patient that died or was cured today may have been diagnosed yesterday or last week). second, some patients may not be treated in time after being infected, making the death toll evolve with a time memory. therefore, we extend the classical mass-balance equation of death cases using the fde to characterize the random property and memory impact embedded in the temporal evolution of mortality. the fractional pde (2e) and its classical version will be compared below using real (observed) data. we apply the seir model (2) does not exhibit any late-time tailing behavior, which supports the application of the integer-order, time-local model (2c). the fast decline of the late time "current infected population" is most likely due to the improvement of health care facilities, which tends to accelerate the recovery of infected people and remove the possible memory impact on the disease recovery evolution. other complex seir models were also applied to model covid-19 spread, such as the one proposed by tang et al. [32] which has 12 rates/probabilities and 8 groups of people in seir. numerical results show that, compared with the seir model (2), the complex model proposed by tang et al. [32] (with solutions shown by the dotted lines in fig. 1a ) accurately fits the observed data at the early stage, but then overpredicts the spread of covid-19 observed after 2/12/2020. the dynamics of transmission for covid-19, especially the recovery rate, therefore, changed over time in china, likely due to the time-dependent conditions (i.e. improvements/adjustment)s in medical care as mentioned above. a dynamic time-local seir model, therefore, may be preferred for modeling covid-19 spread in china. in addition, compared to the fde (2e), the best-fit solution using the classical pde for death evolution (see the black, dotted line in fig. 1a ) slightly overestimates the late-time growth of mortality. the actual death toll in hubei province grew slower than that estimated by a constant rate model, indicating that the memory impact may affect the late-time dynamics of death and can be better captured by the fde (2e). the best-fit solutions using model (2) fit the evolution of the infected and recovered populations well for the data recorded from hubei province and three large cities closely related (such as in transportation or economic cooperation) to wuhan, china ( fig. 1) . the model was also shown to predict well the observed time series of covid-19 spread from 3/8/2020 to 3/22/2020 for most places, except for shanghai city (fig. 1d) . this is likely due to the number of overseas covid-19 cases imported into shanghai, whereby the number of cases were observed to increase rapidly after 3/3/2020, causing inconsistency in the affected population and failure of the model. shanghai pudong international airport, one of the two airports located in shanghai city, is the eighth-busiest airport in the world and the busiest international gateway of mainland china. when excluding the coronavirus cases imported from overseas, model (2) predicts the covid-19 spread evolution data in shanghai (fig. 2) . therefore, model (2) works well for various places in china that may not be experience a significant influx of imported cases, as external sources can easily break the internal evolution, especially the asymptotic status, of covid-19 in china. the resultant time-dependent recovery rate ( ) is depicted in fig. 3 , where the rate fitted by the latest observation data point within the fitting period (i.e., 3/7/2020) remains stable in the following prediction period. the best-fit recovery rate is the highest for shanghai (except for the impulse of ( ) for wenzhou as discussed below), which is expected since shanghai has the best public health system of all of these cities. contrarily, hubei shows the lowest recovery rate, likely due to its delayed response and the relatively limited public health capability at the beginning of the outbreak compared with shanghai. wenzhou exhibits an impulse in the infection and recovery dynamics of coronavirus (fig. 1b) , different from that observed for the other places studied herein. on february 27, 2020, the number of wenzhou's infected people suddenly declined, combined with the sudden increase of the number of people cured. this abrupt change can be effectively captured by the seir model (2) with an impulse in the recovery rate ( ). this recovery impulse is most likely due to a new hospital, the no. 2 affiliated hospital of wenzhou medical university, recently built in this city in early february, which significantly improved the public health system. the first discharged cases of coronavirus from this hospital appeared in late february 2020, resulting in the sudden increase in the total recovery rate. in addition, a relatively large number of people working in wuhan returned to wenzhou in late january, and it appears that the improved efficient screening process successfully identified the number of infected cases. the new cases were ~29,000 from 1/24/2020 to 1/31/2020 in wenzhou (with an average of 3,600 new patients per day), who were then immediately centralized for treatment. it appears that this fast response helped to alleviate the spread of coronavirus in wenzhou. the best-fit parameters of model (2) are listed in table 1 , and the initial values for each group of people are listed in table 2 . we reveal three behaviors in model parameters. first, the best-fit "s"-shaped recovery rate ( ) (fig. 3) can be described by the sigmoid function ( ) ( ) (where a, b, and c are factors), showing that the recovery rate increases exponentially before reaching a stable condition. this increase is likely due to the fact that healthcare facility capabilities improved with time (with an accelerating rate) before reaching their asymptote or maximum capacity. second, the rates and probabilities (r, , p, and ) affecting the covid-19 transmission/infection evolution slightly change in space and remain stable for a given site ( table 1 ). the small spatial fluctuation of these rates may be due to the similar strategies implemented by we also introduce an index c to quantify the infection severity of covid-19 at different places: where is the maximum number of cumulative infected people at the given site. a smaller c represents a greater infection severity of coronavirus. there is a power-law relationship between the regional population n and the maximum cumulated number of infected people (fig. 4) . this empirical formula may be used to approximate the largest cumulative number of infections, which will be applied in the subsequent section for predicting the covid-19 evolution outside of china, where the coronavirus infection has not yet reached its peak number of cases. different countries are applying different modes to slow the the spread of the covid-19. in the next sub-sections, we discuss several representative countries and then fit/predict the virus spread there. to to decrease the acceleration of covid-19 spread, italy's mode is now similar to china: lockdown of the full population. the predicted covid-19 spread in italy is plotted in fig. 6a . although italy has followed china's mode of national isolation, the number of infected people increased rapidly from 3/14/2020 to 3/19/2020 (~495 new cases per day). to account for the delayed national quarantine compared with china, we decrease the c index (while also increasing the upper limit of the cumulative infection). the covid-19 evolution prediction results show that there may be a turning point in the next two weeks when the current infected cases begin to decline. we also separate the death toll from the number of recovered cases. south korea's mode of combating the spread of covid-19 is through fast detection and tracking of the disease. south korea is using efficient mobile diagnoses tests and accurate tracing of infected cases to maintain a low death rate even with a large infected population. the mobile method can test 20,000 people per day (the maximum capability on 3/12/2020), and apps for cell phones and/or credit cards can accurately track the routes of infected people with the help of local government (without an invasion of privacy), so that warnings can be immediately delivered to the general population to obviate the places with high risk. the current infected population may have passed its peak number of cases around 3/20/2020, and the prediction shows that the covid-19 outbreak may be well controlled in ~35 days from 3/22/2020 (fig. 6b) . japan's mode of combating the spread of covid-19 is compatible with that of the u.s., in addition to other changes such as enhanced education/outreach and rapid treatment of the infected cases. specific policies include social distancing (which might be a key barrier to the spread of the novel coronavirus), personal hygiene, and quarantine of the infected cases. the current data and modeling results (fig. 6c) show that japan has an efficient way so far to limit the maximum population infected and slow the spread of covid-19, while this outbreak may last for a while. the model-predicted covid-19 spread in the u.s., using the fitted infection and recovery rates from china (fig. 5) , reveals the impact of one possible mitigation scenario for covid-19: coronavirus lockdowns, which have now been implemented by some states in the u.s. such as california and new york. other non-pharmaceutical options can and should also be evaluated using mathematical models, considering the recent surge of infected cases in the u.s. when the number of infected persons is initially small compared to that of susceptible people, the infected and susceptible people are not well mixed and hence the system is not homogeneous. under such conditions, a stochastic model is needed, as the deterministic, continuum models (such as the seir model) assume well-mixing of components for a homogeneous system [33, 34] . hence, this section develops and applies a stochastic model to evaluate non-pharmaceutical scenarios for mitigating the covid-19 with a small number of initial infections. the random walk based stochastic model for covid-19 spread is analogous to a mixing-limited bimolecular reaction-based mechanism/condition [35] . when a reactant a particle (representing a susceptible individual a) meets a reactant b particle (representing an infectious person b), a chemical reaction may occur if the collision energy is large enough to break the chemical bond (meaning that the susceptible person a may be infected if satisfying additional criteria such as a and b are close enough, and b touches his/her face after receiving coronavirus from a). therefore, the condition of a being infected is not deterministic but rather a random, probability-controlled process. this probability is related to various factors, such as the duration that a and b are in contact, the infectivity rate, and the distance between the two people, which may be characterized parsimoniously by the interaction radius r that controls the number of reactant pairs (susceptible + infectious) in a potential reaction (infection) [35] . hence, the core of the random walk stochastic model to simulate covid-19 spread is to define the interaction radius the analogous development and similarities between bimolecular reactions and the sir model can also be seen from their governing equations. the time-dependent sir model takes the form [36] : where ( ) and ( ) denote the transmission rate and recovery rate at time t, respectively. the rate equations for irreversible bimolecular reaction take the form [35] : where , , and denote the concentrations of a, b, and c, respectively; and ( ) is the forward kinetic coefficient of reaction. equations (5a) and (5b) are functionally similar to equations (4a) and (4b), respectively, if the recovery rate ( ) . therefore, following the argument in zhang et al. [35] and lu et al. [37] , we derive analytically the interaction radius r for the sir model (4): where v denotes the volume of the domain, is the time step in random walk particle tracking, is the mass (or weight) carried by each a particle, is the initial concentration of a (which can be assumed to be the normalized value 1 here), and denotes the initial number of susceptible people. the movement of a, b, and c can be described by the following time fde [35] : where denotes a, b, and c, respectively; denotes the fractional capacity coefficient (which controls the ratio between the immobile and mobile population); denotes the mean moving speed; and denotes the macrodispersion coefficient. after defining the interaction radius r, the particle tracking scheme proposed by zhang et al. [35] and lu et al. [37] with particle trajectories defined by the time fde (7) can be applied to model the transmission of coronavirus between the susceptible and infectious people. in addition to pharmaceutical strategies including vaccine and therapeutic drug development, and herd immunity that may either take a while or have a high risk, non-pharmaceutical scenarios can be tested. several particle-tracking based stochastic models were proposed recently [38] to evaluate non-pharmaceutical scenarios to mitigate coronavirus spreading in a city. here we evaluate three related scenarios (described below) using the stochastic model proposed above. in the stochastic model, we assume that 10 days after being infected, the person will be removed because of being cured or expired (dead). this is because the median disease incubation period has been estimated to be 5.1 days [39] . for simplicity purposes, the interaction radius r (6) remains constant, since the constant interaction radius was found to be able to efficiently capture the temporal variation of effective reaction rates in mixing controlled reaction experiments or simulations [35, 37] . the initial number of a and b particles is 10,000 and 4, respectively. the lagrangian solutions of the covid-19 outbreak for the three scenarios are depicted in fig. 7 . modeling results for scenarios 1, 2, and 3 show a peak in the curve of newly infected people at time t=28, 65, and 32, respectively, demonstrating that the virus spreads the fastest for the scenario without mitigation constraints (i.e., scenario 1, where the number of the total infected people or cases increases by one order of magnitude every 10 days in the rising limb), as expected. however, the value for this peak (scenario 1, =198 people) is lower than that for scenario 3 (=267 people), although the total number of the infected people for scenario 1 (9,336) is slightly larger than scenario 3 (9,322). this may be due to a greater separation of infection cases for the higher number of initial coronavirus carriers in scenario 1, which causes a lower and relatively flatter covid-19 evolution peak compared to that of scenario 3. scenario 2 has the lowest peak value (=121 people) and the most-delayed peak in the curve for new cases, and the total infection time is almost doubled compared to the other two scenarios, indicating that people living with strict social distancing may also suffer from a much longer period of covid-19 threat. it is also noteworthy that the overall trend of the solution (simulation) of scenario 1 (initial surge without special constraints, fig. 7a ) is similar to that for italy which had a delayed response to the covid-19 outbreak initially (fig. 6a) , and scenario 3 solution (a lower peak value and a longer duration due to social distancing, fig. 7b ) is similar to that observed in japan where social distancing actions have been implemented (fig. 6c) . the simulated particle plumes plotted in fig. 8 reveal the subtle discrepancy between the three mitigation scenarios. scenario 1 assumes that four initial cases were initially located on the right side of the city, while the whole population (10,000 susceptible persons) was distributed randomly in the 1ï�´1 domain (fig. 8a) . the trajectory of each person is assumed to follow (two-dimensional) brownian motion with retention (described by eq. (7)), to capture the random vector for each displacement and the random waiting time between two consecutive motions (described by the time fractional derivative term in eq. (7)). the virus moved quickly from east to west (figs. 8b and 8c) , spreading over the entire city before all the infected people were cured or expired (dead) at time t=69 (fig. 8d) . a total of 664 susceptible people (6.6% of the total population) distributed randomly around the city were never infected. scenario 2 assumes that social distancing can reduce the infection probability, which can be characterized by a smaller reaction rate or a smaller interaction radius in our lagrangian approach. the virus was spreading at a much lower rate from west to east than that in scenario 1 (figs. 8e~8g), reaching a stable condition (i.e., # of cases neither increasing or decreasing) at a later time (t=125) and leaving more susceptible people unaffected (2,734 total, which is 27.3% of the population). therefore, social distancing is effective in limiting the spread of coronavirus among people. note, however, this scenario assumes that every person in this city strictly maintains social distancing; otherwise a surge of infections may occur the same way as that shown in scenario 1. scenario 3 assumes self-quarantine. notably, not all of the infected people can be effectively quarantined due to the following: 1) people can be infected without coronavirus symptoms; 2) people in the incubation period can transmit the infection; and 3) limited health care facilities and capabilities for the large influx of patients. for example, according to imai et al. [40] , many infected people could not be appropriately screened initially in wuhan city, china. under this condition, we assume that 50% of people infected and diagnosed are immediately quarantined, while the remaining infected people (fig. 8i) can still cause the spread of coronavirus (figs. 8j~8l). self-quarantine, therefore, may not be as effective as maintained social distancing. fractional calculus provides a useful tool to modeling complex dynamics in biology, and this study extended the fde for modeling the coronavirus outbreak. the covid-19 is a pandemic third, a stochastic model based on the lagrangian scheme for the time fdes, analogous to a mixing-limited reaction mechanism model, showed that self-quarantine may not be as effective as strict social distancing, since not all the infected people can be diagnosed and immediately quarantined. while strict social distancing can apparently slow covid-19 spread, the pandemic may last longer. this is another case that fractional calculus may be used to explore covid-19 outbreak. therefore, one of the main contributions of this study is to extend the application of fdes to model dynamics and mitigation scenarios of the coronavirus spread. this appendix quantifies the potential impacts of the non-singular kernel and the type of the fractional derivative on the covid-19 death toll simulation. first, the nonsingular time-fractional definitions also provde promising modeling tools for real-world fractional dynamics. for example, the atagana-baleanu fractional derivative is defined as [41] : where ( ) â�� ( ) represents the single-parameter mittag-leffler function. we employ this definition to extend the seir model and then use the finite difference method to simulate the evolution of the covid-19 death toll, which leads to the following solver: (fig. 9) , which is consistent with the conclusion in the previous study (see [42] ). the resultant death roll peak is lower than that simulated by the conventional kernel function, but it does not mean that the kernel used in the caputo fractional derivative is no longer valid. indeed, the solution of the caputo fractional derivative can capture well the obsvered death peak (fig. 1a) . applications of the nonsingular kernel functions deserve further study in the future. second, the riemann-liouville fractional derivative is defined as [43] : the caputo fractional derivative listed in section 2.1 and the riemann-liouville fractional derivative (9) are related by [44] : where , and the operators and represent the caputo and riemann-liouville fractional derivatives, respectively. â�� the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. credit author statement: the role of fractional calculus in modeling biological phenomena: a review fractional calculus models of complex dynamics in biological tissues modeling with fractional difference equations fractional dynamics in dna simulation of drug uptake in a two compartmental fractional model for a biological system on fractional sirc model with salmonella bacterial infection semi-analytical study of pine wilt disease model with convex rate under caputo-febrizio fractional order derivative a new fractional hrsv model and its optimal control: a non-singular operator approach the fractional features of a harmonic oscillator with position-dependent mass a new study on the mathematical modelling of human liver with caputo-fabrizio fractional derivative new aspects of time fractional optimal control problems within operators with nonsingular kernel a new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator johns hopkins coronavirus resource center a mathematical model for the novel coronavirus epidemic in wuhan, china covid-19 -new insights on a rapidly changing epidemic feasibility of controlling covid-19 outbreaks by 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risk assessment of novel coronavirus covid-19 outbreaks outside china estimation of country-level basic reproductive ratios for novel coronavirus (covid-19) using synthetic contact matrices linear model of dissipation whose q is almost frequency independent-ii impact of absorbing and reflective boundaries on fractional derivative modes: quantification, evaluation and application estimation of the transmission risk of the 2019-ncov and its implication for public health interventions an introduction to stochastic processes with applications to biology discrete and continuous sis epidemic models: a unifying approach evaluation and linking of effective parameters in particle-based models and continuum models for mixing-limited bimolecular reactions networks: an introduction lagrangian simulation of multi-step and rate-limited chemical reactions in multi-dimensional porous media why outbreaks like coronavirus spread exponentially and how to "flatten the curve the incubation period of coronavirus disease 2019 (covid-19) from publicly reported confirmed cases: estimation and application report 2: estimating the potential total number of novel coronavirus cases in wuhan city new fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model time fractional derivative model with mittag-leffler function kernel for describing anomalous diffusion: analytical solution in bounded-domain and model comparison fractional integrals and derivatives stochastic models for fractional calculus resources. bin jin: supervision, funding acquisition key: cord-258235-khdyxiwe authors: chakraborty, tanujit; ghosh, indrajit title: real-time forecasts and risk assessment of novel coronavirus (covid-19) cases: a data-driven analysis date: 2020-04-30 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109850 sha: doc_id: 258235 cord_uid: khdyxiwe the coronavirus disease 2019 (covid-19) has become a public health emergency of international concern affecting 201 countries and territories around the globe. as of april 4, 2020, it has caused a pandemic outbreak with more than 11,16,643 confirmed infections and more than 59,170 reported deaths worldwide. the main focus of this paper is two-fold: (a) generating short term (real-time) forecasts of the future covid-19 cases for multiple countries; (b) risk assessment (in terms of case fatality rate) of the novel covid-19 for some profoundly affected countries by finding various important demographic characteristics of the countries along with some disease characteristics. to solve the first problem, we presented a hybrid approach based on autoregressive integrated moving average model and wavelet-based forecasting model that can generate short-term (ten days ahead) forecasts of the number of daily confirmed cases for canada, france, india, south korea, and the uk. the predictions of the future outbreak for different countries will be useful for the effective allocation of health care resources and will act as an early-warning system for government policymakers. in the second problem, we applied an optimal regression tree algorithm to find essential causal variables that significantly affect the case fatality rates for different countries. this data-driven analysis will necessarily provide deep insights into the study of early risk assessments for 50 immensely affected countries. in december 2019, wuhan city of china became the centre of an outbreak of pneumonia of unknown cause, latter named as coronavirus disease 2019 (covid19) , which raised intense attention not only within china but internationally [13; 32] . the covid-19 pandemic is the most significant global crisis since the world war-ii that affected almost all the countries of our planet [4] . as of april 4, 2020, an outbreak of covid-19 has resulted in 11, 16, 643 confirmed cases with reported deaths of 59,170 worldwide [22] . on march 11, who publicly characterized covid-19 as a "global pandemic", and shortly after that, the united states declared covid-19 outbreaks a national emergency. the covid-19 has caused a great threat to the health and safety of people all over the world due to its widespread and potential harm. thus, the studies of the novel covid-19 epidemics and its future development trend has become a cutting-edge research topic at this moment. we are therefore motivated to ask: (a) can we generate real-time forecasts of daily new covid-19 cases for countries like canada, france, india, south korea, and the uk? (b) what are the probable causal variables that have significant impacts on the case fatality rates for the profoundly affected countries? to answer the first question, we study classical and modern forecasting techniques for which the prediction accuracy largely depend on the availability of data [28] . in outbreaks of covid-19 epidemics, there are limited data available, making predictions widely uncertain. from previous studies, it was evident that the timing and location of the outbreak facilitated the rapid transmission of the virus within a highly mobile population [29] . in most of the affected countries, the governments implemented a strict lockdown in subsequent days of initial transmission of the virus and within hospitals, patients who fulfill clinical and epidemiological characteristics of covid-19 are immediately isolated. the constant increase in the global number of covid-19 cases is putting a substantial burden on the health care system for canada, france, india, south korea, and the uk. to anticipate additional resources to combat the epidemic, various mathematical and statistical forecasting tools [21; 34] and outside china [20; 36; 10] were applied to generate short-term and long-term forecasts of reported cases. these model predictions have shown a wide range of variations. since the time series datasets of covid-19 contain both nonlinear and nonstationary patterns, therefore, making decisions based on an individual model would be critical. in this study, we propose a hybrid modeling approach to generate short-term forecasts for multiple countries. in traditional time series forecasting, the autoregressive integrated moving average (arima) model is used predominantly for forecasting linear time series [6] . but in recent literature, the wavelet transformation based forecasting model has shown excellent performance in nonstationary time series data modeling [27] . thus, combining both models may accurately model such complex autocorrelation structures in the covid-19 time-series datasets and reduce the bias and variances of the prediction error of the component models. in the absence of vaccines or antiviral drugs for covid-19, these estimates will provide an insight into the resource allocations for the exceedingly affected countries to keep this epidemic under control. besides shedding light on the dynamics of covid-19 spreading, the practical intent of this data-driven analysis is to provide government officials with realistic estimates for the magnitude of the epidemic for policy-making. the second problem is connected with the global concern of health and mortality due to the significant covid-19 outbreaks. mortality is crudely estimated using a statistic, the case fatality rate (cfr), which divides the number of known deaths by the total number of identified cases [18; 5; 30] . during the current phase of this global pandemic, it is criti-cally important to obtain reliable estimates of the overall cfr. the estimates of cfr are highly dependent on several country-specific demographic parameters and various disease characteristics. a key differentiation among the cfr of different countries can be found by determining an exhaustive list of causal variables that significantly affect cfr. in this work, we put an effort to identify critical parameters that may help to assess the risk (in terms of cfr) using an optimal regression tree model [7] . the regression tree has a built-in variable selection mechanism from high dimensional variable space and can model arbitrary decision boundaries. the regression tree combines case estimates, epidemiological characteristics of the disease, and heath-care facilities to assess the risks of major outbreaks for profoundly affected countries. such assessments will help to anticipate the expected morbidity and mortality due to covid-19 and provide some critical information for the planning of health care systems in various countries facing this epidemic. the rest of the paper is organized as follows. in section 2, we discuss the data, development of the hybrid model, and experimental results for short-term forecasts of covid-19 for canada, france, india, south korea, and the uk. in section 3, country-wise cfr datasets, method, and results for finding critical parameters are presented. we discuss the assumptions and limitations of our findings in section 4. finally, the discussions about the results and policy recommendations are given in section 5. we focus on the daily figures of confirmed cases for five different countries, namely canada, france, india, south korea, and the uk. the datasets are retrieved by the global change data lab 1 ). all these datasets are collected from the starting date of the disease for the respective countries to april 4, 2020. in this section, we first briefly discuss these datasets, followed by the development of the proposed hybrid model, and finally, the application of the proposed model to generate short-term forecasts of the future covid-19 cases for five different countries. all these datasets and codes to be used in this section are made publicly available at https://github.com/indrajitg-r/covid for the reproducibility of this work. five univariate time-series datasets are collected for the real-time prediction purpose of covid-19 cases for india, canada, france, south korea, and the uk. several previous studies have forecasted future covid cases for china and a few other countries using mathematical and traditional time series forecasting models, for details see [29; 21; 34; 20; 36] . we try to nowcast the covid-19 cases of five different countries based on their past cases. for india and uk, we consider the daily laboratory-confirmed cases from january 30, 2020, through april 4, 2020 and from january 31, 2020 through april 4, 2020, respectively, for model building. daily covid-19 cases data for canada, france, and south korea are taken for the time period january 20, 2020 through april 4, 2020, january 25, 2020 through april 4 2020, and january 26 through april 4 2020, respectively. the dataset for india contains a total of 64 observations, 65 observations for the uk, 70 observations for canada, 71 observations for france, and 76 for south korea. for these five countries the outbreaks of covid-19 started almost from the same timeline and the epidemic curves still not showing the sharp diminishing nature, just like china. we limit our attention to trended and non-seasonal models, given the patterns, observed in table 1 . note that we follow a pragmatic approach in that we assume that the trend will continue indefinitely in the future in contradiction with other s-curve or deterministic sir modeling approaches which assume convergence. training data acf plot pacf plot to forecast confirmed cases of covid-19, we adopt hybrid time series forecasting approaches combining arima and wavelet-based forecasting techniques. the proposed hybrid model overcome the deficiencies of the single time series models. before describing the proposed methodology, we give a brief description of the individual models to be used in the hybridization. arima is a classical time series model, used for tracking linear tendencies in stationary time series data. arima model is denoted by arima(p, d, q). the parameters p and q are the order of the ar model and the ma model respectively, and d is the level of differencing [9] . arima model can be mathematically expressed as follows: where y t denotes the actual value of the variable under consideration at time t, ε t is the random error at time t. the φ i and θ j are the coefficients of the arima model. the basic assumption made by the arima model is that the error series follows zero mean with constant variance, and satisfies the i.i.d condition. building an arima model for any given time series dataset can be described in three iterative steps: model identification (achieving stationarity), parameter estimation (the autocorrelation function (acf) and the partial autocorrelation function (pacf) plots are used to select the values of parameters p and q), and model diagnostics checking (finding the 'best' fitted forecasting model using akaike information criterion (aic) and the bayesian information criterion (bic)) [15] . wavelet analysis is a mathematical tool that can reveal information within the signals in both the time and scale (frequency) domains [27] . this property overcomes the basic drawback of fourier analysis and wavelet transforms the original signal data (especially in the time domain) into a different domain for data analysis and processing. wavelet-based models are most suitable for nonstationary data, unlike arima [23] . most epidemic and climatic time-series datasets are nonstationary; therefore, wavelet transforms are used as a forecasting model for these datasets [11; 2] . when conducting wavelet analysis in the context of time series analysis, the selection of the optimal number of decomposition levels is vital to determine the performance of the model in the wavelet domain. the following formula for the number of decomposition levels, w l = int[log(n)] is used to select the number of decomposition levels, where n is the time-series length. the wavelet-based forecasting (wbf) model transforms the time series data by using a hybrid maximal overlap discrete wavelet transform (modwt) algorithm with a 'haar' filter. daubechies wavelets can produce identical events across the observed time series in so many fashions that most other time series prediction models cannot recognize [3] . the necessary steps of a wavelet-based forecasting model, defined by [2] , are as follows. firstly, the daubechies wavelet transformation and a decomposition level are applied to the nonstationary time series data. secondly, the series is reconstructed by removing the high-frequency component, using the wavelet denoising method. and, lastly, an appropriate arima model is applied to the reconstructed series to generate out-of-sample forecasts of the given time series data. for the covid-19 datasets, we propose a hybridization of stationary arima and nonstationary wbf model to reduce the individual biases of the component models [24] . the covid-19 cases datasets for five different countries are complex in nature. thus, the arima model fails to produce random errors or even stationary residual series, evident from figure 1 . the behavior of the residual series generated by arima is mostly oscillatory and periodic; thus, we choose the wavelet function to model the remaining series. several hybrid models based on arima and neural networks are available in the field of time series forecasting; see for example [35; 1; 12; 19; 8; 25] . algorithm 1 proposed hybrid arima-wbf model 1 given a time series of length n, input the in-sample (training) covid-19 daily cases data. 2 determine the best arima(p, d, q) model using the in-sample (training) data. • arima parameters p, d, and q values are selected using the procedures described in section 2.2.1. • obtain the predictions using the selected arima(p, d, q) model for the in-sample data and generate required number of out-of-sample forecasts. • obtain the residual series (ε t ) by subtracting arima predicted values from the original training series. 3 train the residual series (ε t ) generated by arima by the wbf model, as described in section 2.2.2. • select the number of decomposition level using the formulae w l = int[log(n)] and boundary is chosen to be 'periodic'. • obtain in-sample predictions (ε t ) using the wbf model and generate required number of out-of-sample forecasts.. motivated by the above discussion, we propose a novel hybrid arima-wbf model which is a two-step pipeline approach. in the first step of the proposed hybrid approach, an arima model is built to model the linear components of the epidemic time series, and a set of outof-sample forecasts are generated. in the second phase, the arima residuals (oscillatory residual series) are remodeled using a mathematically-grounded wbf model. here, wbf models the left-over autocorrelations (in this case, the oscillatory series in figure 1 ) in the residuals which arima could not model. the algorithmic presentation of the proposed hybrid model is given in algorithm 1. the proposed model can be looked upon as an error remodeling approach in which we use arima as the base model and remodel its error series by wavelet-based time series forecasting technique to generate more accurate forecasts. this is in relevance to model misspecification in which disturbances in the nonlinear time series of covid-19 cases cannot be correctly modeled with the arima model. therefore, if the error series generated by arima is adequately modeled and incorporated with the forecasts, the performance of the out-of-sample estimates can be improved, even though marginally at times. remark. the proposed hybrid approach contradicts other mathematical and traditional forecasting modeling approaches applied to covid-19 data. we choose two completely diverse models for hybridization, one from classical forecasting literature and another from modern forecasting approaches. five time series covid-19 datasets for canada, france, india, south korea, and the uk are considered for training the proposed model and the component models. the datasets are nonlinear, nonstationary, and non-gaussian in nature. we have used root mean square error (rmse), mean absolute error (mae), to evaluate the predictive performance of the models used in this study [17] . since the number of data points in both the datasets is limited thus going for advanced deep learning techniques will simply over-fit the datasets [14] . we start the experimental evaluation for all the five datasets with the classical arima(p,d,q) using 'forecast' [16] statistical package in r software. to fit an arima model, we first specify the parameters of the model. using acf plot and pacf plot (see table 1 ), we can decide the value of the parameters of the model. we have also performed unit root tests for stationarity check and all the datasets were found nonstationary. the 'best' fitted arima model is chosen using aic and bic values for each training dataset. the fitted arima models for five datasets are as follows: arima(1,2,1) for india, arima(1,1,2) for canada, arima(0,1,1) for france, arima(2,1,0) for south korea, and arima(2,2,2) for the uk. we employ a pre-defined box-cox transformation set to λ = 0 to ensure the forecast values stay positive. as the arima model is fitted, forecasts are generated for 10-time steps (5 april 2020 to 14 april 2020) for all the five datasets. we also compute training data predicted values and calculate the residual errors. plots for the residual series are given in figure 1 . it is interesting to see that the error series (residuals) generated by arima are oscillating and nonstationary for all the datasets. these seasonal oscillations can be captured through the wavelet transform, which can decompose a time series into a linear combination of different frequencies. these residual series as in figure 1 ) satisfy the admissibility condition (zero mean) that forces wavelet functions to wiggle (oscillate between positive and negative), a typical property of wavelets. thus, we remodel the residuals obtained using the arima model with that of the wbf model. the value of wavelet levels is obtained by using the formula, as mentioned in algorithm 1. wbf model was implemented using 'waveletarima' [26] package in r software with 'periodic' boundary and all the other parameters were kept as default. as the wbf model is fitted on the residual time series, predictions are generated for the next ten time steps (5 april 2020 to 14 april 2020). further, both the arima forecasts and wbf residual forecasts are added together to get the final out-of-sample forecasts for the next ten days (5 april 2020 to 14 april 2020). the hybrid model fittings (training data) for five countries, namely canada, france, india, south korea and the uk are displayed in figures 2(a) , 3(a), 4(a), 5(a) and 6(a) respectively. the real-time (short-term) forecasts using arima, wbf, and hybrid arima-wbf model for canada, france, india, south korea, and the uk are displayed in figures 2(b) , 3(b), 4(b), 5(b) and 6(b) respectively. the predicted values for the training covid-19 cases datasets of the proposed hybrid model for five countries are further used for model adequacy checking and based on actual and predicted test outputs, we computed rmse and mae for all the datasets and reported them in table 2 . the performances of the proposed hybrid arima-wbf model are superior as compared to the individual models for canada, france, and the uk, whereas, for india and south korea, our results are competitive with arima. it is often true that no model can be universally employed in all circumstances, and this is in relevance with "no free lunch theorem" [33] . even if in a very few cases hybrid arima-wbf model gave lower information criteria values (in terms of rmse and mae for training data), we still can opt for the hybrid model given the asymmetric risks involved as we believe that it is better to take decisions based on a hybrid model rather than depending on a single one at least for this pandemic. we produced ten days ahead point forecasts based on all the three models discussed in this chapter and reported then in figures 2-6 . our model can easily be updated on a daily or periodic basis once the actual values are received for the country-wise covid-19 cases. remark. please note that this is not an ex-post analysis, but a real, live forecasting exercise. thus, these real-time short-term forecasts based on the proposed hybrid arima-wbf model for canada, france, india, south korea, and the uk will be helpful for government officials and policymakers to allocate adequate health care resources for the coming days. at the outset of the covid-19 outbreak, data on country-wise case fatality rates due to covid-19 were obtained for 50 affected countries. the case fatality rate can be crudely defined as the number of deaths in persons who tested positive for covid-19 divided by the confirmed number of covid-19 cases. in this section, we are going to find out a list of essential causal variables that have strong influences on the cfr. the datasets and codes of this section are made publicly available at https://github.com/indrajitg-r/covid for the reproducibility of this work. in the face of rapidly changing data for covid-19, we calculated the case fatality ratio estimates for 50 countries from the day of starting the outbreak to 4 april 2020 from the following website 2 . a lot of preliminary analysis is done to determine a set of possible variables, some of which are expected to be critical causal variables for risk assessments of covid-19 in these affected countries. previous studies [22; 30; 18; 5] have suggested that the total number of cases, age distributions, and shutdown period have high impacts on the cfr values for some of the countries. along with these three variables, we also considered seven more demographic structures and disease characteristics for these countries as input variables that are likely to have a potential impact on the cfr estimates. therefore, the cfr modeling dataset consists of 50 observations having ten possible causal variables and one numerical output variable (viz. cfr), as reported in table 3 . the possible causal variables considered in this study are the followings: the total number of covid-19 cases (in thousands) in the country till 4 april, 2020, population density per km 2 for the country, total population (in millions) of the country (approx.), percentage of people in the age group of greater than 65 years, lockdown days count (from the starting day of lockdown till april 4, 2020), time-period (in days) of covid-19 cases for the country (starting date to april 4, 2020), doctors per 1000 people in the country, hospital beds per 1000 people in the country, income standard (e.g., high or lower) of the country and climate zones (e.g., tropical, subtropical or moderate) of the country. the dataset contains a total of 8 numerical input variables and two categorical input variables. for the risk assessment with the cfr dataset for 50 countries, we apply the regression tree (rt) [7] that has built-in feature selection mechanism, easy interpretability, and provides better visualization. rt, as a widely used simple machine learning algorithm, can model arbitrary decision boundaries. the methodology outlined in [7] can be summarized into three stages. the first stage involves growing the tree using a recursive partitioning technique to select essential variables from a set of possible causal variables and split points using a splitting criterion. the standard splitting criteria for rt is the mean squared error (mse). after a large tree is identified, the second stage of rt methodology uses a pruning procedure that gives a nested subset of trees starting from the largest tree grown and continuing the process until only one node of the tree remains. the cross-validation technique is popularly used to provide estimates of future prediction errors for each subtree. the last stage of the rt methodology selects the optimal tree that corresponds to a tree yielding the lowest cross-validated or testing set error rate. to avoid instability of trees in this stage, trees with smaller sizes, but comparable in terms of accuracy, are chosen as an alternative. this process can be tuned to obtain trees of varying sizes and complexity. a measure of variable importance can be achieved by observing the drop in the error rate when another variable is used instead of the primary split. in general, the more frequent a variable appears as a primary split, the higher the importance score assigned. a detailed description of the tree building process is available at [17] . the rationale behind the choice of rt as a potential model to find the important casual variables out of 10 input variables for the cfr estimates is the simplicity, easy interpretability, and high accuracy of the rt algorithm. we apply an optimal rt model to the dataset consisting of 50 different country samples and try to find out potential casual variables from the set of available variables that are related to the case-fatality rates. rt is implemented using 'rpart' [31] package in r with "minsplit" equals to 10% of the data as a control parameter. we have used rmse, co-efficient of multiple determination (r 2 ), and adjusted r 2 (adjr 2 ) to evaluate the predictive performance of the tree model used in this study [17] . an optimal regression tree is built with 7 variables with 'minsplit' = 5 with equal costs for each variable. the estimates of the performance metrics for the fitted tree are as follows: rmse = 0.013, r 2 = 0.896, and adjr 2 = 0.769. a variable importance list from the rt is given in figure 7 and the fitted tree is provided in figure 8 . from the variable importance plot based on the complexity parameter of the rt model (also see figure 7 ), seven causal variables are obtained out of 10 potential input variables having higher importance. these seven causal variables that significantly affect the cfr for 50 most affected countries are the followings: total number of covid-19 cases in the country (in thousands), percentage of people in the age group of greater than 65 years, total population (in millions) of the country, doctors per 1000 people in the country, lockdown period (in days) for the country, time-period (in days) of covid-19 cases for the country, and hospital beds per 1000 people in the country. our results are consistent with previous results obtained by [30; 18; 5] , where the authors suggested that the total number of cases, age distributions, and shutdown period have high impacts on the cfr estimates. but interestingly, we obtained four more essential causal variables that will provide some new insights into the study of risk assessments for covid-19 affected countries. out of these 7 numerical input variables, there are four control variables (number of cases, people of age group > 65 years, lockdown period, and hospital beds per 1000 people) present that can be managed to fight against this deadly disease. once these variables are taken care of, the respective country may reduce their case fatality rate at a significant rate. x.x3 x.x9 x.x8 x.x5 x.x6 x.x7 x.x2 x.x4 x.x1 figure 7 : variable importance percentages affecting the cfr based on a complexity parameter in rt figure 8 shows the relationship between the important causal variables and cfr. in figure 8 , the tree starts with the total number of covid-19 cases as the most crucial causal variable in the parent node. in each box, the top most numerical values suggest the average cfr estimates based on the tree. one of the key findings of the tree is the following rule: when the number of cases of a country is greater than 14,000 having a population between 14 to 75 million are having second highest case fatality rate, viz., 10%. similarly, one can see all the rules generated by rt to get additional information about the relationships between control parameters and the response cfr variable. x.x1 >= 0.99 x.x1 < 14 x.x4 >= 7.4 x.x4 < 19 x.x7 >= 3.4 x.x4 >= 21 x.x5 < 21 x.x4 < 6.6 x.x2 >= 75 x.x6 < 76 x.x2 < 14 x.x8 < 1.5 yes no x.x1 >= 0.99 x.x1 < 14 x.x4 >= 7.4 x.x4 < 19 x.x7 >= 3.4 x.x4 >= 21 x.x5 < 21 x.x4 < 6.6 x.x2 >= 75 x.x6 < 76 x.x2 < 14 x we made some simplifying assumptions to carry out the analysis of covid-19 datasets. the assumptions are listed as follows: (a) the virus mutation rates are comparable for different countries; (b) the recovered persons will achieve permanent immunity against covid-19; (c) we ignore the effect of climate change (also spatial data structures) during the shortterm predictions. along in this line, we presented two different approaches to deal with two inter-connected problems on covid-19. in the first problem of short-term predictions for covid-19 outbreak in five countries, we proposed a hybrid methodology combining arima and wbf models. in the second problem of risk assessment, we found some important factors affecting case fatality rates of covid for 50 highly affected nations. however, there may exist a few more controllable factor(s), and some disease-based characteristics that can also have an impact on the value of cfr for different countries, can be regarded as future scope of the study. the covid-19 outbreaks globally present a significant challenge for modelers, as there are limited data available on the early growth trajectory, and epidemiological characteristics of the novel coronavirus have not been fully elucidated. in this study, we considered two alarmingly important problems relevant to ongoing covid-19 pandemic. the first problem deals with the real-time forecasts of the daily covid-19 cases in five different countries. we proposed a hybrid arima-wbf model that can explain the nonlinear and nonstationary behavior present in the univariate time series datasets of covid-19 cases. ten days ahead forecasts are provided for canada, france, india, south korea, and the uk. the proposed model can be used as an early warning system to fight against the covid-19 pandemic. below we present a list of suggestions based on the results of the real-time forecasts. 1. since we presented a real-time forecast system unlike an ex-post analysis, thus one can regularly update the actual confirmed cases and update the predictions, just like it happens in weather forecasting. 2. the forecasts mostly show oscillating behavior for the next 10 days and reflect the impact of the broad spectrum of social distancing measures implemented by the governments, which likely helped stabilize the epidemic. 3. the short-term forecasts don't necessarily show any stiff decay sooner; also, these five countries are not going to face any unlike uplifts in the number of cases too. 4. guided by the short-term forecasts reported in this paper, the lockdown period can be adjusted accordingly. secondly, we assessed the risk of covid-19 by finding seven key parameters that are expected to have powerful associations with that of case fatality rates. this is done by designing an optimal regression tree model, a simplified machine learning approach. the model is very flexible, easily interpretable, and the more data will come, one can just incorporate the new data sets and rebuild the trees to get the updated estimates. rt provides a better visual representation and is easily interpretable to be understood by a broader audience. quantification of the outbreak risks and their dependencies on the key parameters will support the governments and policymakers for the planning of health care systems in different countries that faced this epidemic. experimental results suggest four control variables out of seven highly influential variables that will have a significant impact on controlling cfr. below we present a point by point discussion of the control variables affecting cfr and preventive actions to be taken by the governments. 1. the number of covid cases of the country can be reduced by enforcing social distancing strategies. 2. number of people of age group > 65 years should be specially taken care of and isolated. 3. lockdown time period can be extended if the country faces a sharp increase in the number of cases and or deaths. 4. the number of hospital beds should be increased by making special health care arrangements in other places to deal with this emergency due to covid-19. forecasting nonlinear time series with a hybrid methodology forecasting time series using wavelets wavelet-based nonlinear multiscale decomposition model for electricity load forecasting modeling and 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study. the lancet infectious diseases early transmission dynamics in wuhan, china, of novel coronavirus-infected pneumonia serial interval of novel coronavirus (covid-19) infections. international journal of infectious diseases comparative study of wavelet-arima and wavelet-ann models for temperature time series data in northeastern bangladesh ensembles for time series forecasting a hybrid arima-svm model for the study of the remaining useful life of aircraft engines wavelet methods for time series analysis forecasting the novel coronavirus covid-19 real-time forecasts of the covid-19 epidemic in china from february 5th to february 24th cm-mid ncov working group, et al. estimating the infection and case fatality ratio for covid-19 using age-adjusted data from the outbreak on the diamond princess cruise ship. medrxiv package 'rpart'. available online: cran a novel coronavirus outbreak of global health concern no free lunch theorems for optimization nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study time series forecasting using a hybrid arima and neural network model preliminary estimation of the novel coronavirus disease (covid-19) cases in iran: a modelling analysis based on overseas cases and air travel data key: cord-290952-tbsccwgx authors: ullah, saif; khan, muhammad altaf title: modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study date: 2020-07-03 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110075 sha: doc_id: 290952 cord_uid: tbsccwgx coronavirus disease (covid-19) is the biggest public health challenge the world is facing in recent days. since there is no effective vaccine and treatment for this virus, therefore, the only way to mitigate this infection is the implementation of non-pharmaceutical interventions such as social-distancing, community lockdown, quarantine, hospitalization or self-isolation and contact-tracing. in this paper, we develop a mathematical model to explore the transmission dynamics and possible control of the covid-19 pandemic in pakistan, one of the asian countries with a high burden of disease with more than 100,000 confirmed infected cases so far. initially, a mathematical model without optimal control is formulated and some of the basic necessary analysis of the model, including stability results of the disease-free equilibrium is presented. it is found that the model is stable around the disease-free equilibrium both locally and globally when the basic reproduction number is less than unity. despite the basic analysis of the model, we further consider the confirmed infected covid-19 cases documented in pakistan from march 1 till may 28, 2020 and estimate the model parameters using the least square fitting tools from statistics and probability theory. the results show that the model output is in good agreement with the reported covid-19 infected cases. the approximate value of the basic reproductive number based on the estimated parameters is [formula: see text]. the effect of low (or mild), moderate, and comparatively strict control interventions like social-distancing, quarantine rate, (or contact-tracing of suspected people) and hospitalization (or self-isolation) of testing positive covid-19 cases are shown graphically. it is observed that the most effective strategy to minimize the disease burden is the implementation of maintaining a strict social-distancing and contact-tracing to quarantine the exposed people. furthermore, we carried out the global sensitivity analysis of the most crucial parameter known as the basic reproduction number using the latin hypercube sampling (lhs) and the partial rank correlation coefficient (prcc) techniques. the proposed model is then reformulated by adding the time-dependent control variables u(1)(t) for quarantine and u(2)(t) for the hospitalization interventions and present the necessary optimality conditions using the optimal control theory and pontryagin’s maximum principle. finally, the impact of constant and optimal control interventions on infected individuals is compared graphically. modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study coronavirus disease is the biggest public health challenge the world is facing in recent days. since there is no effective vaccine and treatment for this virus, therefore, the only way to mitigate this infection is the implementation of non-pharmaceutical interventions such as social-distancing, community lockdown, quarantine, hospitalization or self-isolation and contact-tracing. in this paper, we develop a mathematical model to explore the transmission dynamics and possible control of the covid-19 pandemic in pakistan, one of the asian countries with a high burden of disease with more than 100,000 confirmed infected cases so far. initially, a mathematical model without optimal control is formulated and some of the basic necessary analysis of the model, including stability results of the disease-free equilibrium is presented. it is found that the model is stable around the disease-free equilibrium both locally and globally when the basic reproduction number is less than unity. despite the basic analysis of the model, we further consider the confirmed infected covid-19 cases documented in pakistan from march 1 till may 28, 2020 and estimate the model parameters using the least square fitting tools from statistics and probability theory. the results show that the model output is in good agreement with the reported covid-19 infected cases. the approximate value of the basic reproductive number based on the estimated parameters is r 0 ≈ 1.87. the effect of low (or mild), moderate, and comparatively strict control interventions like social-distancing, quarantine rate, (or contact-tracing of suspected people) and hospitalization (or self-isolation) of testing positive covid-19 cases are shown graphically. it is observed that the most effective strategy to minimize the disease burden is the implementation of maintaining a strict social-distancing and contact-tracing to quarantine the exposed people. furthermore, we carried out the global sensitivity analysis of the most crucial parameter known as the basic reproduction number using the latin hypercube sampling (lhs) and the partial rank correlation coefficient (prcc) techniques. the proposed model is then reformulated by the novel coronavirus infectious disease, caused by the coronavirus is commonly known as covid19 . it has become the greatest challenge the history has ever seen. started from wuhan city of china earlier this year, it spread to the rest of the world in a few months and was declared a pandemic by the un. it has paralyzed life across the globe. the main cause of the virus is yet to be discovered, but it is presumed that it has emerged in one the biggest animal market in the chinese city of wuhan [1, 2, 3] . so far, it has engulfed more than 210 countries of the world and according to who statistics, the virus has affected around 6 million people across the world and more than 300 thousand people have died so far [1, 2, 3] . the recovery rate is higher than the mortality rate. however, the ratio varies from country to country and region to region. usa is the most affected country which is the epicenter of the virus followed by europe [2] . scientists are struggling to discover or invent a vaccine for the treatment, but it is yet to be discovered. the question, how long it will last? is on everyone's lips. although research is in a very early stage and with the passage of time things will unfold. the scientists are trying to dig out the main symptoms and causes of transmission. however, according to the information available the main symptoms are high fever, severe chest pain, continuous dry coughing, body aches, headache and difficulty in the respiratory system. the spread, according to available information, is droplets, produced by an infected person during coughing and sneezing and physical contacts, etc. [4] . covid-19 pandemic has caused great damage not only to human lives and health it has multidimensional effects. it not only exposed the weak health infrastructure even in the most advanced countries of the world, but also badly affected the world economy. almost the entire world is on lockdown and all the economic and business activities are halted. it has shocked the largest economies of the world, i.e. china and usa where the economic slowdown is observed since the outbreak of coronavirus. the third world countries, particularly pakistan, are the prime targets of the economic devastation. millions of people have lost their jobs in the past few months. poor countries are unable to repay their debts. they are even unable to support poor citizens who cannot earn their livelihood. the social and political lives of the nations are affected badly. people across the world have severed social relations. they avoid meeting each other even in the gravest times. governments of all the countries have diverted their attention and resources to cope with the challenge of this mysterious disease and thus no political activity is visible. like other countries in the world, covid-19 pandemic poses a huge threat to both humans health and economy in pakistan. this infection is even more devastating in pakistan because the implementation of non-pharmaceutical interventions i.e., make social-distancing and community lockdowns are certainly very tough for a society like pakistan. the government is unable to afford a strict nationwide lockdown. the covid-19 first case was reported on 26 february 2020 in karachi which was a student come back from iran. later, only within three weeks, the infection spread in all four provinces, gilgit-baltistan, azad jammu and kashmir, and the federal territory of islamabad. the total number of confirmed infected cases raised to 2291 across the nation and 252 new cases were reported on 1 april. due to the rapid growth of infected people across the country the government of pakistan decided to put the whole nation under strict lockdown and later extended twice until 9 may, due to a worse situation. currently, the situation in pakistan is worse and the number of confirmed cases crossed the cases reported in mainland china. pakistan is placed 17th in the list of highly reported infected cases and deaths by countries, territories, or areas [5, 6] with a total of more than 100,000 confirmed covid-19 cases. about 31,000 patients are fully recovered and 1900 people lost their lives due to this deadly infection [7, 8] . mathematical models are very useful in helping us to understand the transmission dynamics and control of emerging and re-emerging communicable diseases. one of the main challenges that mankind is facing nowadays is predicting the severity and suggest suitable public health intervention strategies to curtail the covid-19 pandemic. recently, a number of mathematical models have been proposed to explore the transmission patterns of covid-19 pandemic. in [9] , the authors formulated a deterministic model to explore the impact of various public interventions on the dynamic and mitigation covid-19 in ontario, canada. in [10] , a mathematical model based on nonlinear differential equations is presented to study the dynamics of covid-19 infection in highly affected countries that are china, italy, and france. a fractional-order covid-19 model with the atangana-bleanu-caputo operator is proposed by khan and atangana [11] and implemented the model to analyze the infection in wuhan. the role of lockdown in the absence of effective vaccines and treatment in order to mitigate the covid-19 pandemic is analyzed in [12] . the author has used the novel fractional-fractal operators to formulate the proposed mathematical model [12] . in [13] , a transmission model is formulated to predict the cumulative covid-19 cases in italy, uk, and usa. the transmission dynamics of covid-19 in mexico are studied using mathematical and computational models in [14] . the influence of non-pharmaceutical controls, including quarantine, hospitalization or self-isolation, contact-tracing, and the use of a face mask on the dynamics of covid-19 pandemic in the population of new york state and the entire usa is studied in detail in [15] . the present study proposes a new transmission model to analyze the dynamics and impact of non-pharmaceutical interventions on the covid-19 in pakistan. initially, we develop the model without optimal control variables and provide a good fit to the reported cases and then estimate the model parameters using a nonlinear least square curve fitting approach. further, the model is reformulated by adding two times dependent control variables. the rest of the manuscript is organized as: the mathematical formulation of the covid-19 model is presented in section 2. basic mathematical analysis, including stability results of disease-free equilibrium is explored in section 3. the model fitting to reported cases and estimation of parameters is done in section 4. the simulation results of without optimal control model are shown in section 5. the global sensitivity analysis and its graphical interpretation are depicted in section 6. the optimal control problem and its analysis of the covid-19 infection is presented in 7. finally, a brief concluding remarks are given section in 8. this section presents a brief description of the proposed model to study the dynamics and possible control of the covid-19 pandemic. the model is developed by dividing the whole human population at any time t, denoted by n (t) into eight mutually exclusive sub-groups depending on the disease status. these sub-groups are susceptible s(t), exposed or latent e(t), infected with the disease symptoms (or symptomatically infectious) i(t), asymptotically infectious having no clinical symptoms i a (t), quarantined q(t), hospitalized i h (t), critically infected (or intensive care) patients i c (t) and the recovered/removed individuals r(t), so that the infected individuals showing mild symptoms of the disease are also placed in the epidemiological class i a (t). the quarantine and isolation should be either at home or the specific centers or hospitals designated by the government. further, the group i h stand for the patients admitted into hospital also contains those with clinical symptoms of the disease who are self-isolating at home. we further assume that hospitalized people may also transmit the infection after interacting with susceptible people. the transmission dynamics of the covid-19 disease is expressed through the following system of the nonlinear differential equation: (1) the corresponding initial conditions are the birth rate is denoted by λ and the natural mortality rate in all groups is denoted by µ. the susceptible people acquired covid-19 infection when they interact with the infected people in i, i a and i h compartments. the force of infection is where the parameter β shows the effective contact rate, i.e., contacts capable of leading to infection transmission and the parameter 0 ≤ ψ ≤ 1 accounts for the assumed reduction in disease transmissibility of asymptomatic infected individuals in comparison to symptomatic one. similarly, ν is used for the infectiousness rate due to hospitalized covid-19 patients. it is noticed from the transmission patron of covid-19 that the asymptomatic individuals are comparatively dangerous relative to the individuals in the i h class because they are not aware of the infection and are capable to transmit the infection. the latent individuals develop an infection after completion of incubation period and become infected at the rate ω and a fraction denoted by ρ enters to the symptomatic class after showing disease symptoms and the remanding with no (or mild) symptoms join the asymptomatic compartment i a (t). the exposed individuals who have interaction with covid-19 infected patients are detected (via contact-tracing) and placed in quarantine at the rate κ which further moves to hospitalized class if they are tested positive with covid-19 infection. the symptomatically-infectious people are hospitalized at the rate η which further moves to critically-infected class i c at the rate φ if they are serious and need critical care. the parameters ζ 1 and ζ 2 represent the recovery rates in i and i a groups respectively. further, the recovery rates of quarantined, hospitalized and critically infected classes are shown by φ 1 , φ 2 and φ 3 respectively. finally, the covid-19 induced mortality for individuals in the i, i h and i c classes are respectively shown by ξ 1 , ξ 2 and ξ 3 . for simplicity, let us denote then, the above model (1) can be written as we present some basic and necessary analytical results of the covid-19 model (3) lemma 1. let p(0) ≥ 0 denotes the initial data and p(t) = (s, e, i, i a , q, i h , i c , r) are the model variables, then all solutions of the model (3) will be non-negative for all t > 0. (3): it can be further written as hence, the solution of (5) can found as below following a similar procedure, it can be shown that p(t) > 0, ∀ t > 0. in order to prove the second part we have 0 < p(0) ≤ n (t), and then the addition of all equations of the covid-19 model (3) we have the dynamics of the covid-19 mathematical model (3) will be studied in the following closed and biologically feasible region. the region defined in the closed set ∆ ⊂ r 8 + , is positively invariant for the model (3) with non-negative initial conditions in r 8 + . proof. as in lemma 1, it follows from the summation of all equation of the covid-19 model (3), it is clear that the solution of and (6) is given in the following inequality, in particular, n (t) ≤ λ/µ, if n (0) ≤ λ/µ. therefore, the region ∆ is positively invariant as will as attracts all the possible solution trajectories in r 8 + . the propose model (3) has a disease free equilibrium (df e), given by next, we investigate the most important and crucial threshold quantity known as the basic reproduction number and generally denoted by r 0 . this parameter measures the average number of new covid-19 infected cases generated by a typical infected individual when introduced into a completely susceptible population. the most common approach used to obtain r 0 is the next-generation method presented in [16] . the next generation matrices obtained from the model (3) are given as follows: the corresponding jacobian matrices f and v evaluated at df e is given as below: hence, utilizing the definition r 0 = ρ(f v −1 ) (where ρ(.) represents the spectral radius), we derived the following expression for the basic reproduction number: interpretation of r 0 in order to interpret the basic reproductive quantity we split the expression for r 0 as follows: where, the first term r 0i in (7) shows the average number of new covid-19 infections generated by symptomatically-infectious individuals in the i class. this term contains the product of the infection rate in the i class (the disease transmission rate), β, the fraction of exposed people that completed the incubation period and move to the symptomatic stage ( ρω k 1 ) and the average period spend in the i compartment ( 1 k 2 ). the second constituent reproduction number r 0a represents the number of new covid-19 infection cases generated by asymptomatically-infectious individuals in class i a . it is the product of infectious rate due to asymptomatic coivd-19 individuals (βψ), the fraction of latent people that completed the incubation period and move to the asymptomatic stage ( ρ(1−ω) k 1 ) and the average period spent in the asymptomatic class ( 1 k 3 ). similarly, the third constituent reproduction number r 0h expresses the new covid-19 infection cases generated by hospitalized/isolated individuals. in particular, the first term in r 0h represents the contribution into the hospitalized class by symptomatic infectious individuals (in class i). it is the product of infectious rate due to hospitalized individuals (βν), the fraction of latent individuals that completed the incubation period and move to the asymptomatic stage ( ρ(1−ω) k 1 ), the portion of individuals that left the symptomatic class i and move to the hospitalized class i h ( η k 2 ), and the average duration in the hospitalized class ( 1 k 5 ). finally, the second term in r 0h expresses the contribution of quarantined individuals into the hospitalized class. in this part, we will prove the local and global stability of the model around the df e. the epidemiological implication of the stability result of df e case is that a small influx of covid-19 infections cases will not generate a covid-19 outbreak if r 0 < 1. (3) is locally asymptotically stable if r 0 < 1 and unstable otherwise. proof. the jacobian matrix j w 0 obtained at the dfe w 0 is as follows: clearly, from the above jacobian matrix j w 0 , the eigenvalues −µ, −µ and −k 6 have negative real part. there remanning eigenvalues can be obtained through the equations given below: the coefficients involved in (8) are as follow: where, clearly, c j for j = 1..., 5 are all positive if r 0 < 1. further, it is easy to show the remaining routh-hurtwiz condition for the fifth order polynomial (8) . thus, the df e is locally asymptotically stable if r 0 < 1. the global dynamics of df e, w 0 of the covid-19 transmission model is studied in the following result. theorem 2. the system (3) at w 0 is globally asymptotically stable if r 0 <1, and unstable for r 0 >1. proof. let we consider the following lyapunov function, in order to prove the required result: where b i , for i = 1, 2, · · · , 5, used for some unknown positive constants. differentiating the function z(t) with respect to t and using the solutions of system (3), we obtain: we obtained after some simplification hence, it is obvious that if r 0 < 1 then dz(t) dt < 0. therefore, the largest compact invariant set in ∆ is the singleton set w 0 and using the lasalle's invariant principle [17] , w 0 is globally asymptotically stable in ∆. the present section investigates the data fitting using model (3) to the confirmed reported covid-19 infected cases in pakistan. the disease situation in pakistan is becoming worse day by day and currently, the cumulative reported cases are higher than china. in this study we consider the covid-19 confirmed cases from march 01, 2020, till may 28, 2020, reported in pakistan. the data is obtained from [7, 8] . in order to parameterize the model, we utilized two approaches: some of the demographic parameters are estimated from the literature. we assume the time unit is days and the estimation procedure of the parameters is as follows: • the birth rate λ: the total population of pakistan estimated by un for the year 2020 is about n (0)=220,309,834 [18], therefore, the parameter λ is obtained from λ/µ = n (0), and it is assumed that this is the limiting population in the disease absence, so that λ = 8939 per day. • mortality rate due to coronavirus ξ 1 : the death rate due to this novel infection in pakistan is 2.2% so far [8] , therefore the covid-19 induced mortality rate is estimated as ξ 1 = 0.022. the remaining biological parameters are fitted from the reported infected cases plotted in figure 1 . to do this we used the non-linear least-square curve fitting technique followed in [19, 20] . we briefly present the main steps in this statistical technique. in order to present the main theme of the algorithm, firstly, the model (3) can be comprehensively expressed as the function f depends on time t, the vectors of dependent or state variables y and unknown parameters θ to be estimated. the purpose of using the least square technique is to estimate the best values of model parameter which is obtain by minimizing the error between the reported data pointsỹ t l and the solution of the model y t l associated with the model parameters θ. the objective function used in the minimization procedure is given asθ where n denotes the available actual data points. to obtain the model parameters, we aimed to minimize the following objective function minθ subject to eq.(10). for more detail about this technique please see [21, 19] and reference therein. we investigate the proposed model fit to the reported covid-19 infected cases in pakistan using the above approach. the reported cases are shown in figure 1 . the model is solved using ode45 (rk4 technique) package which is a solver for the initial value problem in matlab. then, we implemented the lsqcurvefit package to fit the model to real data and to estimate the parameters. the best fit to the reported data via our model is depicted in figure 2 . it can be seen that the model simulation is in good agreement with the real data. the estimated and fitted parameters are given in this section is devoted to perform the simulation results of the covid-19 transmission model (3) . the model is solved numerically using fd45 package in matlab which is base on the range-kutta fourth-order method. the estimated parameter values given in table 1 are utilized in the simulation process in order to study the impact of various possible non-pharmaceutical interventions against the spread of covid-19 in pakistan. in the graphical results, the various non-pharmaceutical control parameters are taken at their baseline values given in table 1 (unless otherwise stated in captions). the effect of parameter β that represents the impact of effective contacts is shown in figure 3 . we have analyzed the impact of baseline social-distancing, mild social-distancing (10% reduction in β), moderate social-distancing (30% reduction in β) and comparatively strict socialdistancing (35% reduction in β) on the disease transmission. it is observed from figure 3 that the enhancement in social-distancing significantly reduced the burden of cumulative new infected cases. it is further observed that the implementation of a highly-effective social-distancing strategy (i.e., at least 35% reduction in β) dramatically reduces the cumulative new infected cases. thus, this graphical interpretation suggests that strict socialdistancing measures that reduces the contacts between people including staying 2 meters apart or more preferably staying at home should be implemented by the government. the impact of parameter ψ representing the infectiousness rate due to asymptomaticallyinfected individuals is depicted in figure 4 . it can be seen that the reduction in ψ also significantly reduces the cumulative number of newly confirmed covid-19 infected cases. this interpretation shows that the people who do not even know that they are infected (i.e., those with mild or no symptoms), are significantly contributing the disease burden. further, the influence of parameter ν the infectiousness rate due to covid-19 patient admitted in the hospital are depicted in figure 5 . it is observed that the reduction in this parameter has no reasonable impact on disease transmission. obviously, the hospitalized covid-19 patients are isolated and no one is allowed to meet him. the health care facilities are also supposed to follow strict standard operating procedure (sop) during the treatment and look after of hospitalized patients. therefore, these infected individuals do not contribute greatly to the disease burden. we further simulate the covid-19 model (3) by using the baseline parameter tabulated in table 1 and various of κ by increasing with different level i.e, mild, moderate, and strict rates. the resulting behavior is depicted in figure 6 showing that a strict quarantine or contact-tracing policy (up to 70% enhancement to its baseline) is needed to reduce the disease burden in pakistan. finally, the impact of hospitalization or selfisolation of tested positive cases (η) is plotted in figure 7 . it is observed that this strategy is comparatively less effective than social-distancing and quarantine interventions. these graphical interpretations emphasize that once a covid-19 infected case is diagnosed via testing, that case must be rapidly isolated and his/her contacts quickly traced (via effective contact-tracing) and placed in quarantine. global sensitivity analysis is one of the important aspects of mathematical modeling not only for epidemic models but in all sciences. the global sensitivity analysis of the threshold quantity r 0 is used to measure the effect of changes in the dominant factors of the model and to point out the most influential parameters of the model that greatly influence the prevalence of infection. furthermore, the sensitivity results provide a pathway to set effective and suitable control strategies to curtail the disease in a community. more specifically, this analysis is helpful to explore how the initial inputs to the model contribute to the system outputs. a latin-hypercube sampling approach (lhs) coupled with the partial rank correlation coefficient (prcc) is commonly used for this purpose [22] . this technique provides prcc and the corresponding p-values for each parameter the use of which can help estimate the level of uncertainty in an epidemic model. the higher prcc and smaller p-value of a parameter indicating that it has a substantial effect on simulation behavior. the graphical prcc results of the covid-19 model associated parameters taken in this analysis are shown in figure 8 , while the numerical values of prcc and corresponding p-values are placed in table 2 . it is observed from table 2 and figure 8 that β is considered to be the most sensitive parameter with high prcc value with a positive sign followed by ψ, ν, and ω. moreover, µ and κ, ξ 1 and η have competitively high prcc with negative sign and zero p-values. previously, we analyzed the impact of non-pharmaceutical interventions with constant rates. in this section we formulate an optimal control problem for covid-19 with the inclusion of two time dependent controls in the model (3). the resulting control problem is presented in (13) . these controls are chosen on the basis of global sensitivity results. the control variable u 1 (t) is used for the enhancement of effective contact-tracing policy to quarantine the exposed individuals which was previously taken as a constant parameter. the time dependent control variable u 2 (t) is used to enhance the hospitalization or selfisolation of diagnosed covid-19 infected cases (following testing). thus, the resulting control model after incorporating the aforementioned control variables is formulated via the following system: subject to the non-negative initial conditions. in order to minimize this covid-19 infection, we are aimed to minimize the cost function given as: where the expressions a i for i = 1, 2, · · · , 5, are the constants and representing the balancing cost factors while t f represents the final time. we consider the quadratic objective functional because the intervention is nonlinear, for more details see the work and references therein [23, 24, 20, 25] . onward our main objective is to investigate an optimal controls u * 1 , u * 2 for quarantine and hospitalization respectively such that the associated control set is given by the lagrangian and hamiltonian for the above optimal control system is defined by and where λ j , for j = 1, 2 · · · , 8, are the adjoint variables. we use the pontryagin's maximum principle [26] in order to solve the covid-19 optimal control problem (13) . to do this, let u * 1 , u * 2 are the desired optimal solution then, the corresponding conditions of pontryagin's maximum principle used in solution process are as follows: utilizing the conditions mentioned above (17), we present the solution of optimality system in the following theorem. theorem 3. the optimal controls u * 1 , u * 2 and the solutions s * , e * , i * , i * a , q * , i * h , i * c and r * of the corresponding control system (13) that minimize the objective functional j(u 1 , u 2 ) over ω. there exists adjoint variables λ i , where i = 1, 2, · · · , 8, along with transversality conditions λ i (t f ) = 0 such that furthermore, the associated optimal controls u * 1 and u * 2 are given by proof. the desired results (18) and the transversality conditions are obtained by utilizing the conditions specified in (17) for the hamiltonian function given (16) and with the settings s = s * , e = e * , i = i * , i a = i * a , q = q * , i h = i * h , i c = i * c and r = r * . further, to obtain the equations (19) for the control characterization, we use the condition ∂h(t,u j * ,λ j ) ∂u j = 0 given in (17) for j = 1, 2, and the equations (19) are presented. we present and discuss the graphical results of the covid-19 model (3) with constant quarantine and hospitalization/isolation control measures and the model (13) with time dependent control interventions and compared both results. to perform the simulations, both models are solved numerically using the rk4 technique. the estimated and fitted parameters given in table 2 are used in the simulation results. the time level is considered up to 150 units (days). the weight and balancing constants are chosen as a 1 = 0.001, a 2 = 0.01, a 3 = 0.001, a 4 = 300, and a 5 = 200. it should be noticed that the weights constant values taken in the simulations are theoretical as they were chosen only to carried out the control strategies developed in this study. in figure 9 (a-d), we have depicted the impact of hospitalization control only and keeping the quarantine or case tracing control inactive (i.e., u 1 = 0 and u 2 = 0). the control profile for this strategy is shown in 9 (e). it is observed that although the control u 2 is kept 100% for the first 70 days, but still it has no significant impact on the different infected individuals as seen 9 (a-d). thus, only the hospitalization or self-isolation intervention is not enough for the control of covid-19 pandemic in pakistan. the impact of only quarantine optimal control by keeping the hospitalization control zero (i.e., u 1 = 0 and u 2 = 0) is shown in 10 (a-d). the corresponding control profile for this case is depicted in 10 (e). it can be seen that by maintaining a strict quarantine control is very effective in minimizing the spread of the covid-19 infection. finally, the impact of both optimal controls on the dynamics of covid-19 burden is analyzed. the graphical results are depicted in figure 10 (a-d) , while the control profile is shown in 10 (e). it can be observed from 10 (a-d) that the number of exposed, symptomatically-infected, asymptomatically-infected, and critically-infected individuals are decreasing very significant when the optimal quarantined and hospitalization controls are applied rather than the constant case. the effectiveness can be viewed from the difference between the peaks of the two graphs. from the control profile depicted in figure 10 (e), it can be seen that the control u 1 is kept initially at 100% for 140 days and gradually reduces towards the end of the intervention, while the control u 2 is set to 55% and then immediately increases to 90% in the initial days and then gradually reduced during the rest of the intervention. the covid-19 pandemic has rapidly spread out to most of the regions of the world and has severe public health and socio-economic burden in developed and devolving countries including pakistan. the number of reported cases in pakistan is increasing and more than 100,000 confirmed cases have been reported till 05 june 2020. in the absence of a safe and effective vaccine or antiviral, the whole human's community is being focused on the use of non-pharmaceutical interventions against the covid-19 pandemic. in this study, we formulated a mathematical model in order to study the dynamics of covid-19 pandemic in pakistan, and used it to assess the community-wide impact of the various control and mitigation strategies. initially, we developed the model and presented some mathematical analysis, including positivity and stability results of the disease-free equilibrium. it is proven that the disease-free equilibrium is stable both locally and globally when r 0 < 1. the model is parameterized from the covid-19 confirmed cases reported till may 28, 2020 in pakistan while some parameters are estimated from literature. the findings show that the model predicted infected curve is in good agreement to the real infected cases. the estimated numerical value of the basic reproduction is obtained as r 0 ≈ 1.87 showing the alarming situation of the pandemic in pakistan. the control and mitigation strategies should be implemented to bring the threshold quantity r 0 to a value less than unity. after the estimation of model parameters, we simulated the model to explore the effectiveness of various control strategies implemented in pakistan. firstly, we presented the impact of three effectiveness levels (i.e., low or mild, moderate and strict) of social-distancing in curtailing the burden of covid-19. the simulation results revealed that although the implementation of mild social-distancing decreased the covid-19 burden significantly (as measured in terms of shifting and lowering the peak of daily infected cases), still a strict social-distancing measures should be implemented and maintained for an extended period of time to avoid a significant outbreak in pakistan. further, we simulated the model to assess the effect of various levels of quarantine and hospitalization or self-isolation interventions. with a highly-effective quarantine intervention (enhanced by 70% to its baseline value) a dramatic reduction in the pandemic peak was observed. on the other hand increase of hospitalization intervention of confirmed cases had no significant influence on the pandemic burden. finally, the proposed covid-19 model is reformulated by the inclusion of two time-dependent control variables in order to assess the impact of optimal control measures on disease dynamics. we simulated the control model and compared the effect of constant and optimal time-dependent control measures on disease burden. it is observed from the simulation results of the control covid-19 model that the infected individuals significantly decrease with the implementation of both time-dependent control measures. although it is impressive that pakistan ramps up daily diagnostic covid-19 testing and contact tracing to have a realistic measure of the burden of the nationwide pandemic and emphasize personal hygiene and hand washing, physical-distancing, wearing face masks in public. still, this study suggests that the implementation of basic non-pharmaceutical interventions, particularly social-distancing, quarantine (or self-isolation or stay at home) should be strictly observed in the future to avoid the worse scenario in pakistan. it is also believed that the present study will be beneficial to the decision-making in combating the disease. in the near future, we will extend the present model by introducing the fractional operators with local and nonlocal kernel to gain more insights about the dynamics of covid-19 pandemic. an interactive web-based dashboard to track covid-19 in real time. the lancet infectious diseases world health organization coronavirus disease (covid-19) situation reports presumed asymptomatic carrier transmission of covid-19 covid-19 coronavirus pandemic world health organization coronavirus disease (covid-19) situation reports covid-19 coronavirus pandemic covid-19 coronavirus pandemic in pakistan quantifying the role of social distancing, personal protection and case detection in mitigating covid-19 outbreak in ontario analysis and forecast of covid-19 spreading in china, italy and france modeling the dynamics of novel coronavirus (2019-ncov) with fractional derivative modelling the spread of covid-19 with new fractal-fractional operators: can the lockdown save mankind before vaccination? forecasting the cumulative number of confirmed cases of covid-19 in italy, uk and usa using fractional nonlinear grey bernoulli model modeling and prediction of covid-19 in mexico applying mathematical and computational models mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel coronavirus reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission the stability of dynamical systems modeling the transmission dynamics of tuberculosis in khyber pakhtunkhwa pakistan optimal control strategies for dengue transmission in pakistan parameter estimation in nonlinear dynamical systems a methodology for performing global uncertainty and sensitivity analysis in systems biology optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model media coverage campaign in hepatitis b transmission model mathematical formulation of hepatitis b virus with optimal control analysis the maximum principle. the mathematical theory of optimal processes no conflict of interest exists regarding the publications of this paper. the authors declare that there is no conflict of interests regarding the publications of this work. key: cord-342955-vf3c6ksm authors: lu, jingjing; teng, zhidong; li, yingke title: an age-structured model for coupling within-host and between-host dynamics in environmentally-driven infectious diseases date: 2020-06-21 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110024 sha: doc_id: 342955 cord_uid: vf3c6ksm in this paper, an age-structured epidemic model for coupling within-host and between-host dynamics in environmentally-driven infectious diseases is investigated. the model is described by a mixed system of ordinary and partial differential equations which is constituted by the within-host virus infectious fast time ordinary system and the between-host disease transmission slow time age-structured system. the isolated fast system has been investigated in previous literatures, and the main results are introduced. for the isolated slow system, the basic reproduction number r(b0), the positivity and ultimate boundedness of solutions are obtained, the existence of equilibria, the local stability of equilibria, and the global stability of disease-free equilibrium are established. we see that when r(b0) ≤ 1 the system only has the disease-free equilibrium which is globally asymptotically stable, and when r(b0) > 1 the system has a unique endemic equilibrium which is local asymptotically stable. with regard to the coupled slow system, the basic reproduction number r(b), the positivity and boundedness of solutions and the existence of equilibria are firstly obtained. particularly, the coupled slow system can exist two positive equilibria when r(b) < 1 and a unique endemic equilibrium when r(b) > 1. when r(b) < 1 the disease-free equilibrium is local asymptotically stable, and when r(b) > 1 and an additional condition is satisfied the unique endemic equilibrium is local asymptotically stable. when there exist two positive equilibria, under an additional condition the local asymptotic stability of a positive equilibrium and the instability of other positive equilibrium also are established. the numerical examples show that the additional condition may be removed. the research shows that the coupled slow age-structured system has more complex dynamical behavior than the corresponding isolated slow system. there are many viral infectious diseases in the world, such as viral influenza, japanese encephalitis, measles, infant paralysis, viral hepatitis, rabies, aids, ebola, sars, novel coronavirus pneumonia caused by 2019-ncov, and so on. these diseases not only damage the health and happiness of individuals and families, but also have a great impact on society and the country. hence, further research in virus area appears to be very critical and necessary, which present leading treatments and control measures for diseases. the previous researches have done a lot on the virus model, and obtained some good results. see, for example, feng et al. [1, 2] , cen et al. [3] , feng et al. [4] , wen et al. [5] , mideo et al. [6] , coombs et al. [7] , gilchrist and coombs [8] , korobeinikov [9] , larly, based on the assumptions about the dependence of betweenhost variables, they illustrated the possible occurrence of a conflict between natural selection at the individual and population levels. similarly, the model in [1] still have multiple attractors when reproduction number r 0 < 1 due to a backward bifurcation. it's not hard to find models [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] are based on the assumption that individuals in each class are homogeneous. for example, infected individuals have the same infectivity and by-product mortality during their period of infection. this assumption is reasonable when modeling infectious diseases such as influenza and sexually transmitted diseases. however, infectivity experiments on hiv/aids epidemic recognized the importance of variable infectivity in the transmission dynamics of infectious diseases in [16] . furthermore, researchers proposed stage-structured models described by ordinary differential equations (odes). to be more realistic and reasonable, we introduce continuous infection age, which also increased the difficulty of research. in recent years, many papers have studied age-structured model, we just list [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] . so far, works on coupled within-host and between-host dynamics with infection age are very rare. in [29] , the authors proposed an infection-age structured hiv-1 model linking within-host and between-host by introducing viral load-dependent transmission rate and different transmission rates at different ages of infection are discussed. this makes us want to extend the model in [1] by introducing age-dependent mortality due to infection at the population level. there are two time scales in our model, one is the dynamic evolution time in the host s , the other is the dynamic evolution time between the hosts t . usually, the dynamic time at the cellular level is much faster than that at the population level. that is to say, s is a faster time variable, t is a slower time variable (see [1] [2] [3] [4] [5] ). the concepts of fast time and slow time were introduced in [31] early. our main purpose of this paper is to explore the potential effect of within-host dynamics on the between-host transmission dynamics in environmentally-driven infectious diseases. we construct the following nested model with two time scales and infection age: (1) (t,a ) ∂t + ∂i (t,a ) ∂a = −(μ + α(a )) i (t, a ) , where s = s(t ) represents the number of susceptible at time t ; i ( t, a )represents the density of infectious individuals with age of infection a at time t ; e = e(t ) represents the level of environment contamination at time t (0 ≤ e ( t ) ≤ 1); t = t (s ) , t * = t * (s ) and v = v (s ) denote the densities of healthy cells and infected cells, and the parasite load related to the degree of environment contamination at time s , respectively. the parameters in systems (1) and (2) have the following meanings: c is the recruitment rate of cells which free from infection, k is the infection rate of cells, m represents the natural mortality of infected cells, d denotes the infection-induced mortality rate of infected cells, p is the parasite production rate by an infected cell, c is the clearance rate of parasites within-host, h is the recruitment rate of hosts, β is the infection rate of hosts in a contaminative environment, μ denotes the natural death rate of hosts, α( a ) is the disease-induced mortality of hosts at age a, γ is the clearance rate of virus in environment. as we know, the level of environmental pollution can be related to the number of infected individuals and the withinhost virus concentrations with the form θ iv . meanwhile, the viruscontaminated environment generates an increase on within-host virus concentrations is given by g ( e ). the article is arranged as follows. in section 2 , we introduce some main results of the isolated fast system (3) . in section 3 , the isolated slow system (6) is discussed. the basic reproduction number r b 0 and some criteria on the positivity, boundedness, existence and local asymptotic stability of equilibria are stated and proved. furthermore, we obtain global asymptotic stability of disease-free equilibrium. in section 4 , the coupled slow system (25) is discussed. we obtain basic reproduction number r b , positivity and boundedness of solutions, and the existence of equilibria. particularly, it is show that the number of equilibria depends on the magnitude of basic reproduction number and the sign of h m , and the system can have two positive equilibria when r b < 1. the local asymptotic stability of equilibria stated and proved under the certain conditions and the instability of other positive equilibrium also is established. in section 5 , the numerical examples are presented to illustrate the main conclusions obtained in section 4 . finally, in section 6 , we give a conclusion. in fast time system (1) , we assume that the environmental virus concentration e ( t ) is a constant e and 0 ≤ e ≤ 1, where e = 0 means there is no virus in the environment, e > 0 represents there has virus in the environment, and e = 1 indicates that the virus in the environment reaches its maximum. thus, fast time system (1) becomes into an isolated within-host virus infection system we introduce the following assumption for function g ( e ). from the biological background of system (3) , we assume that any solution ( t ( s ), t * ( s ), v ( s )) of system (3) satisfies the following initial conditions the fast time system (3) has been investigated in [1] [2] [3] [4] , and in [5] for a discrete analog of system (3). the complete dynamical properties of system (3) have been established. here, we summarize these results in the following. firstly, with regard to the positivity and boundedness of the solutions and the existence of equilibria for system (3) we have the following results. (4) is positive for all s ≥ 0 and ultithe within-host reproduction number is defined as follows here we see that k indicates the probability that the virus will come into contact with healthy cells in the host and infect them, 1 c is the survival time of the virus, p m + d represents the amount of virus released by within-host infected cells during their survival period, t 0 is the number of healthy cells at the beginning of the infection. therefore, r w represents the number of cases in which an infected cell infects healthy cells in their survival time at the early stages of infection. lemma 2. let e = 0 , then system (3) always has infection-free equilibrium u 0 = (t 0 , 0 , 0) , and when r w > 1, system (3) has a unique infectious equilibrium lemma 3. let e > 0, then system (3) always has a unique positive furthermore, on the global asymptotic stability of the infection-free equilibrium and infectious equilibrium of system (3) , the following theorems are established. ( a ) if r w ≤ 1, then infection-free equilibrium u 0 is globally asymptotically stable. ( b ) if r w > 1, then infectious equilibrium u 1 is globally asymptotically stable. theorem 1 shows that when there is no toxin in the environment, viral infection in the host depends on the basic reproductive number r w . when r w ≤ 1, the within-host virus will eventually be eliminated, thus infected cells will eventually be cleared. when r w > 1, viral infection in the host will continue, the virus will eventually stabilize at a positive balance v 1 , and meanwhile the healthy cells and infected cells eventually stabilize at the positive equilibrium level t 1 and t * 1 . let e > 0 in system (3) , then infectious equilibrium u 2 is globally asymptotically stable. theorem 2 indicates that when there is always virus in the environment and the environmental virus directly affects the virus infection in the host, it will directly increase the amount of virus in the host, then the virus infection always exists in the host. meanwhile, healthy cells, infected cells and the virus eventually stabilize at a positive equilibrium ( ˜ t (e) , ˜ t * (e) , ˜ v (e)) in the host which related to the environment contamination level e . moreover, lemma 3 shows that as the virus in the environment is gradually cleared, that is, e → 0, the amount of virus and infected cells in the host will be gradually removed when the basic reproduction number r w ≤ 1; when r w > 1, even if the virus in the environment is gradually removed, the virus and the infected cells in the host will eventually still stabilize at a positive balance v 1 and t * 1 . so as time goes on, the virus in the host will be excreted by the host and enter the environment after a period of time, thereby increasing the amount of virus in the environment. when the amount of virus in the environment reaches a certain level, it will infect other healthy hosts in the environment, so that the virus begins to spread among the population (that is, between the hosts). then, infectious disease is prevail among hosts caused by the virus and it's epidemic regular pattern expressed by the age-structured sie model (2). we consider that the virus in the environment does not affect the virus infection in the host, that is, we have g ( e ) ≡ 0 in the fast time system (1) . when the basic reproduction number r w ≤ 1, the amount of virus in the infected person will eventually be cleared and the disease caused by the virus will eventually be eliminated without an epidemic. therefore, in the following discussion, we always assume the basic reproduction number r w > 1. it is obvious that the dynamics of the fast system of virus infection in the host is much faster than that of the slow system of disease transmission between the hosts, so that we can assume that the state of the fast system has reached its limit equilibrium state without further changes in the state of the slow system. namely, when r w > 1, we . we emphasize that this equilibrium state v 1 doesn't depend on the amount of virus in the environment e . furthermore, we assume that the amount of virus excreted into the environment by the infected person is θ iv 1 (1 − e) . thus, system (3) becomes into an isolated between-host disease transmission system we introduce the following assumptions. ( h 2 ) function α( a ) is nonnegative and lipschitz continuous for all a ≥ 0 with lipschitz constants m α , and belongs to l 1 we take the phase space of system (6) by x = r 1 + × l 1 from the biological background of system (6) , we assume that the initial value for any solution of system (6) is defined by x 0 := (s(0) , i 0 (·) , e(0)) ∈ x satisfying the following conditions it follows from the standard theory of differential equation, we can obtain that system (6) has a unique solution (t, x 0 ) = (s(t) , i (t, ·) , e(t )) satisfying the initial condition (0 , x 0 ) = x 0 . on the positivity and ultimate boundedness of solutions for system (6) , we have the following results. the solution ( t, x 0 ) of system (6) with initial value (7) is nonnegative for all t > 0 and ultimately bounded. particularly, we proof. the nonnegativity of solutions can be proved by using the similar method given [23] , and we hence omit the proof. now, we prove the ultimate boundthus, e ( t 1 ) ≥ 0. from the third equation of system (6) , we have this leads to a contradiction. hence, 0 ≤ e ( t ) ≤ 1 for all t ≥ 0. this shows that ( t, x 0 ) is ultimately bounded. this completes the proof. ( t, x 0 ) of system (6) is defined for all t > 0. therefore, the continuous semi-flow : r + × x → x defined by system (6) takes the following form clearly, system (6) always has a disease-free equilibrium w 0 = we define the between-host reproduction number as follows r b0 = 0 (μ+ α(s ))d s is the probability for an individual becomes an infected person after a period of time a. we rewrite here we see that β represents the probability for the virus in the environment contacts the healthy person and makes them infected, 1 γ is the survival time of the virus in the environment, θv 1 ∞ 0 π (a ) da is the amount of an infected person discharging virus into the environment and s 0 is the number of healthy people in the environment at the beginning of the epidemic. therefore, r b 0 indicates the number of infected cases in which an infected person infects susceptible in their survival time at the early stages of infection. if system (6) has an endemic equilibrium w 1 ( s , ī (a ) , ē ) , then it must satisfy the following equations: it follows from the first equation of (9) that integrating the second equation of (9) from 0 to a one can get ī (a ) = ī (0) π (a ) . we obtain from the third equation of (9) that on substituting (10) and (11) into the fourth equation of (9) , we have therefore, hence, if r b 0 > 1, then system (6) has a endemic equilibwe first consider the local stability of disease free equilibrium w 0 ( s 0 , 0, 0). linearizing system (6) at equilibrium w 0 we have 11 e λt , we obtain the following eigenvalue problem: (14) integrating the second equation of (14) from 0 to a we can obtain it follows from the fourth equation of (14) that substituting (15) and (16) into the third equation of (14) yields let us denote the left hand of (17) by f ( λ). obviously, therefore, f ( λ) is a decreasing function. if r b 0 > 1, due to the continuity and differentiability of f ( λ), there has a unique positive root. accordingly, equilibrium w 0 is unstable when we now claim that if r b 0 < 1, equilibrium w 0 is locally asymptotically stable. if not, eq. (17) has at least one root λ 1 = a 1 + ib 1 satisfying a 1 ≥ 0 thus f (λ 1 ) = 1 . in this case, one has this produces a contradiction. hence, if r b 0 < 1, all roots of characteristic equation has negative real parts. accordingly, equilibrium w 0 is locally asymptotically stable if r b 0 < 1. therefore, we get the following conclusion. now, we give a result on the global asymptotic stability of disease-free equilibrium w 0 of system (6) . theorem 4. for system (6) , if r b 0 ≤ 1, then disease-free equilibrium w 0 is globally asymptotically stable. the proof of theorem 4 is given in appendix a. we are now in a position to study the local stability of endemic equilibrium w 1 ( s , ī (a ) , ē ) of system (6) . linearizing system (6) at we obtain the following eigenvalue problem: integrating the second equation of (19) from 0 to a we can obtain from the third equation of (19) one can get substituting (20) and (22) into the fourth of (19) and combined with (21) , we obtain the characteristic equation of system (6) at equilibrium w 1 of the form where we claim that all roots of eq. (23) have negative real parts. otherwise, eq. (23) has at least one root λ 2 = a 2 + ib 2 satisfying a 2 ≥ 0. in this case, this produces a contradiction. hence, all roots of the characteristic eq. (23) have nonnegative real parts when in conclusion, we have the following result. theorem 5. for system (6) , then endemic equilibrium w 1 is locally asymptotically stable when r b 0 > 1 . when r b 0 > 1, whether endemic equilibrium w 1 of system (6) also is globally asymptotically stable still is an important open problem. from the discussions in this section, we see that the isolated slow system (6) has the well dynamical properties. here, we establish the almost complete conclusions. that is, when r b 0 ≤ 1 the disease-free equilibrium is globally asymptotically stable, and when r b 0 > 1 the endemic equilibrium w 1 is locally asymptotically stable. but, unfortunately, we do not get the global asymptotic stability of w 1 when r b 0 > 1. from theorems 3 -6 we can obtain that the transmission and prevalence of diseases caused by virus infection of isolated between-host slow system is completely determined by the basic reproduction number r b 0 . when r b 0 ≤ 1, even if viral infection continues in the host, but the epidemic eventually dies out among the hosts; when r b 0 > 1, infectious diseases will spread and prevail between the hosts, and eventually become an endemic disease. when the virus in the environment has an effect on the virus infection in the host, we assume that this effect causes the amount of virus in the host to increase, and use the function g ( e ) to express this increase. in this way, the dynamic change of the virus in the host will satisfy the equation dv dt = g(e) + pt * − cv . when e > 0, due to the dynamics of the fast system of virus infection in the host is much faster than that of the slow system of disease transmission between the hosts, we can assume that the state of the fast system has reached its equilibrium state without further changes in the state of the slow system. namely, we have (2) , and further have the following coupled slow system: in section 2 , we obtain that for any particularly, we also have that ˜ similarly to isolated slow system (6) , the phase space of system (25) is taken by x = r 1 + × l 1 , and the initial condition of any solution of system (25) is defined by x 0 := it is clear that system (25) has a unique solution firstly, on the positivity and boundedness of solutions for system (25) , we have the following results which is similar to isolated slow system (6) . the solution ( t, x 0 ) of system (25) with initial value x 0 is positive for all t > 0 and ultimately bounded. particularly, we also we give the between-host reproduction number for system (25) when r w > 1 as follows: rewriting firstly, system (25) always has a disease-free equilibrium w 0 ( s 0 , it follows from the first equation of (27) integrating the second equation of (27) from 0 to a one can get it follows from the third equation of (27) that obviously, 0 < ˆ e < 1 . substituting (28) and (30) into the fourth equation of (27) and combining with (29) , we can obtain we denote the functions as follows: and is clear that we can confirm the existence of positive equilibrium if one obtains the existence of ˆ e which satisfy the corresponding conditions. furthermore, it is easy to observe that ˆ e is the possible zeros of h ( e ) in (0,1). based on the reproduction number r b , we have the following results. (25) always has a disease-free equilibrium w 0 ( s 0 , 0, 0) . ( ii ) system (25) has two positive equilibria ˆ proof. it is obvious that system (25) always has a diseasefree equilibrium w 0 ( s 0 , 0, 0). by calculating, we have because we can not judge the sign of h (e) , we continue to compute hence, h (e) < 0 for all 0 < e ≤ 1. this shows that h ( e ) is an upper convex function. assume that condition ( a ) holds, then owing to h (0) < 0, h (1) < 0 and h m > 0, h(e) = 0 has only two positive roots. hence, system (25) has only two positive equilibria ˆ if condition ( b ) holds, then from h(0) = 0 and h m > 0, we easily obtain that h(e) = 0 has a unique positive root ˆ e 3 . hence, endemic equilibrium ˆ exists and is unique. if condition ( c ) holds, then from h (0) > 0, h (1) < 0, we easily obtain that h(e) = 0 has a unique positive root ˆ e 3 . hence, endemic equilibrium ˆ exists and is unique. at last, we prove ( iv ). if r b = 1 then h(0) = 0 , this combined with h m = 0 show that h ( e ) no zero in (0,1). if h m < 0 then we have that h(e) = 0 has no root. therefore, there is only diseasefree equilibrium w 0 ( s 0 , 0, 0). this completes the proof. comparing lemma 7 with foregoing lemma 5 , we see that the dynamical behavior of coupled slow system (25) is more complex than the isolated slow system. lemma 7 shows that due to the feedback effect of virus in the environment on the virus infection in the host, the coupled between-host disease transmission slow time system produces two positive equilibrium when the basic reproduction number r b < 1. this will likely lead to backward bifurcation. as we all know, the occurrence of backward bifurcation will bring great uncertainty and complexity to the treatment and control of infectious diseases. the proof of theorem 6 is given in appendix b. theorem 6 shows that when the basic reproduction number r b < 1, because the coupled slow system may have two positive equilibrium at this time, the infectious disease will be extinct only when the number of infected individuals is relatively small. when this number is relatively larger, the transmission dynamics of disease will be complicated in the between-host, and there may even be new disease outbreaks. the proof of theorem 7 is given in appendix c. the proof of theorem 8 is given in appendix d. the epidemiological implication of theorem 7 and theorem 8 is that when basic reproduction number r b is less than unity, a small influx of infected individuals into the community would generate large disease outbreaks. this situation has made it difficult to control the disease, prompting researchers to look for smaller between-host basic reproduction number to control the transmission of disease. it is completely similar to the proof method of theorem 8 (see appendix d), we can easily get that equilibrium ˆ w 3 is locally asymptotically stable when γ μ+ βˆ e 3 ≤ 1 . this completes the proof. theorem 9 indicates that when the coupled slow system has only one endemic equilibrium, as long as the condition is satisfied, the infectious disease will be long-term popular in the coupled slow system, and it will become into an endemic disease. the condition γ μ+ βˆ e 3 ≤ 1 is a purely mathematical condition, it is only used in the mathematical argument of theorem 9 . comparing with theorem 5 for the isolated slow system, we here have an interesting open problem. that is, whether endemic equilibrium ˆ w 3 of system (25) also is locally asymptotically stable only when r b > 1. from discussions in this section, we see that the coupled slow system (25) has the complex dynamical behaviors. particularly, there exists the backward bifurcation. that is, when r b < 1 and is close to 1, then system (25) has two positive equilibria ˆ w 1 and ˆ w 2 , and from theorems 7 and 8 we get that ˆ w 1 is unstable, and ˆ w 2 is locally asymptotically stable. in this section, we give some numerical examples to verify the theoretical results obtained in theorems 6 -9 . for the convenience of numerical simulations, we take the function g(e) = we in the following examples, where w is a positive constant. example 1. in systems (1) and (2) (0 . 05(a − 1) )) 2 + 20 , 0 . 2) , respectively. thus, the conclusion obtained in theorem 6 is verified by the numerical example. example 2. here we verify theorems 7 and 8 . we will give the following two cases to show that the condition γ μ+ βˆ e 2 ≤ 1 may be a purely mathematical condition, which is only used in the mathematical argument of theorem 8 . case ( i ). we take the parameters in systems (1) fig. 2 (a ) − (c) show that the positive equilibrium ˆ is locally asymptotically stable, and the positive equilibrium ˆ the initial value ( s (0), i 0 ( a ), e (0)) in fig. 2 tively. in addition, the diagram of function h ( e ) with e ∈ [0, 1] is also given in fig. 2 ( d ) . case ( ii ). we take the parameters in systems (1) and (2) therefore, theorem 8 can not be applied. the numerical simulations given in fig. 3 (a ) − (c) show that the posiis still locally asymptotically stable, and the positive equilibrium ˆ spectively. in addition, the diagram of function h ( e ) with e ∈ [0, 1] is also given in fig. 3 ( d ) . example 3. now, we verify theorem 9 . we will give the following two cases to show that the condition γ μ+ βˆ e 3 ≤ 1 is a purely mathematical condition, which is only used in the mathematical argument of theorem 9 . case ( i ). the parameters are chosen in systems (1) tively. in addition, the diagram of function h ( e ) with e ∈ [0, 1] is also given in fig. 4 ( d ) . case ( ii ). we choose the parameters in systems (1) and (2) (−0 . 05 a ) . by calculation, we obtain r b = 25 . 868 > 1 , and system (25) has a unique endemic equilibrium ˆ . 51 and ˆ e 3 = 0 . 9467 . we also have γ μ+ βˆ e 3 = 1 . 926 > 1 . therefore, theorem 9 can not be applied. the numerical simulations given in fig. 5 (a ) − (c) show that the endemic equilibrium ˆ (0 . 05(a − 1) )) 2 + 10 , 0 . 33) , respectively. in addition, the diagram of function h ( e ) with e ∈ [0, 1] is also given in fig. 5 ( d ) . in this paper we propose an age-structured epidemic model for coupling within-host and between-host dynamics in environmentally-driven infectious diseases. the model is described by a mixed system of ordinary and partial differential equations which is divided into a fast time system of ordinary differential equations and a slow time age-structured system by using the idea of perturbation theory. for isolated fast system (3) , the dynamical behavior has been investigated in [1] [2] [3] [4] [5] . we only list the main results in this paper. for isolated slow system (6) , we firstly obtain the positivity and ultimately boundedness of solutions, the basic reproduction number r b 0 and the existence of equilibrium. next, as main results, the local and global stability of equilibrium by using the linearization method and lyapunov functions, respectively. we obtain that the disease-free equilibrium is globally asymptotically stable if r b 0 ≤ 1, and the endemic equilibrium is locally asymptotically stable if r b 0 > 1. however, an open problem is to establish the concrete criterion for the global stability of the endemic equilibrium. for coupled slow system (25) , we firstly obtain the existence of positive equilibrium with the help of basic reproduction number r b and h m . particularly, we get that system (6) can have two positive equilibria ˆ and h m > 0, and a unique endemic equilibrium ˆ w 3 when r b > 1. this shows that system (6) can undergo a backward bifurcation at r b = 1 . we further establish a series of criteria for the local stability of disease-free equilibrium and positive equilibrium. that is, the disease-free equilibrium is locally asymptotically stable if r b < 1, the positive equilibrium ˆ w 1 is unstable when r b < 1 and h m > 0, the endemic equilibrium ˆ w 2 is locally asymptotically stable if r b < 1, h m > 0 and γ μ+ βˆ e 2 ≤ 1 , and the unique endemic equilibrium ˆ w 3 is locally asymptotically stable if r b > 1 and γ μ+ βˆ e 3 ≤ 1 . in general, we just want the local stability of the equilibrium of system (25) to be only related to the basic reproduction number there are some interesting and important problems which can be further investigated. in this paper we do not investigate the persistence of disease. for isolated slow system (6) and coupled slow system (25) , when basic reproduction numbers r b 0 > 1 and r b > 1, whether we also can obtain the uniform persistence of disease. in addition, in view of the important role of nonlinear incidence rate in epidemic models, whether the main conclusions established in this paper also can been extended to the model with nonlinear incidence. these problems are very challenging and will be solved in the future. the main idea of this paper was proposed by zhidong teng. jingjing lu prepared the manuscript initially and performed all proof. we denote (s 0 , 0 , 0) . therefore, the global asymptotic stability of w 1 follows from lasalle's invariance principle. this completes the proof. proof. linearizing system (25) at equilibrium w 0 we have let x 3 (t) = x 31 e λt , y 3 (t, a ) = y 31 (a ) e λt , z 3 (t) = z 31 e λt , we obtain the following eigenvalue problem: if λ = −γ or λ = −μ, then λ is a root which has negative real part. if λ = −γ and λ = −μ, then y 31 (0) = 0. integrating the second equation of (33) from 0 to a we can obtain it follows from the third equation of (33) that substituting (34) and (35) into the fourth equation of (33) yields the characteristic equation of the slow system at w 0 : obviously, . therefore, f 2 ( λ) is a decreasing function. if r b > 1, due to the continuity and differentiability of f 2 ( λ), there has a unique positive root. accordingly, equilibrium w 0 is unstable when r b > 1. assume that equilibrium w 0 is unstable when r b < 1. then eq. (36) has at least one root λ 3 = a 3 + ib 3 satisfying a 3 ≥ 0 thus f 2 (λ 3 ) = 1 . in this case, one has this leads to a contradiction. therefore, equilibrium w 0 is locally asymptotically stable when r b < 1. this completes the proof. proof. let ˆ be any positive equilibrium of system (25) . linearizing the system (25) at equilibrium ˆ w we have let x 4 (t) = x 41 e λt , y 4 (t, a ) = y 41 (a ) e λt , z 4 (t) = z 41 e λt , we obtain the following eigenvalue problem: . hence, we can rewrite the form of (38) as follows: , then λ is a root which has neg, then y 41 (0) = 0. we can get following equation from the first equation of (39) x 41 = − βˆ s z 41 λ+ μ+ βˆ e . integrating the second equation of (39) from 0 to a we can obtain y 41 (a ) = y 41 (0) e − a 0 (λ+ μ+ α(s ))d s . it follows from the third equation of (39) that z 41 = . hence, the characteristic equation of system (25) at equilibrium ˆ w can be established as follows: now, we consider equilibrium ˆ . based on the definition and properties of f ( e ), g ( e ) and q ( e ), we have the following results: at present, we obtain the inequality f 3 (0) > 1, according to the expression of f 3 ( λ), we also acquire lim λ→∞ f 3 (λ) = 0 . consequently, the characteristic equation of system (25) at equilibrium ˆ w 1 at least has a unique positive part thanks to the continuity of f 3 ( λ). in other words, equilibrium ˆ w 1 is unstable. this completes the proof. proof. from (40) , we can obtain the characteristic equation of system (25) at equilibrium ˆ w 2 can be established as follows: rewriting the characteristic equation with the following form: assume that equilibrium ˆ w 2 is unstable when r b < 1, h m > 0 and γ μ+ βˆ e 2 ≤ 1 , then the characteristic equation has at least one root λ 4 = a 4 + ib 4 satisfying a 4 ≥ 0. in this case, we introduce some notations as follows: the characteristic root λ 4 is substituted into the lhs and rhs , we can get following result: thus, from (31) , we have re ( lhs ) ≤ γ . from the definition of h ( e ) and ˆ e 2 is the zero of h ( e ), we from h ( e ) < 0 for all 0 ≤ e ≤ 1, we get h ( e ) is a strictly monotone decreasing function in [0,1]. we assume h m = h(e * ) , 0 < e * < 1, then 0 < e * < ˆ e 2 < 1 and h (e * ) = 0 . hence, . hence, we have we further have ∂ f (x,y ) ∂x = 0 ⇔ (x + μ) 2 − y 2 = 0 , ∂ f (x,y ) ∂y = 0 ⇔ (x + μ) y = 0 . this shows that f ( x, y ) has only a stationary point (−μ, 0) in r 2 , and this stationary point is not in the first quadrant. thus, the minimum of f ( x, y ) in r 2 + is reached at the boundary coupled within-host and between-host dynamics and evolution of virulence a mathematical model for coupling within-host and between-host dynamics in an environmentally infectious disease emerging disease dynamics in a model coupling within-host and between-host systems a model for coupling within-host and between-host dynamics in an infectious a discrete time analogue for coupled within-host and between-host dynamics in 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giardini editori e stampatori dynamical systems we are very grateful to the reviewers for their careful reading and constructive comments of our manuscript, which grately improved our paper. this research was supported by the national natural science foundation of china (grant nos. 11771373 ) and the college scientific research project of xinjiang uygur autonomous region, china (grant nos. xjedu2018y021). the authors declare that they have no conflict of interest. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. key: cord-301035-dz8642qx authors: rasheed, jawad; jamil, akhtar; hameed, alaa ali; aftab, usman; aftab, javaria; shah, syed attique; draheim, dirk title: a survey on artificial intelligence approaches in supporting frontline workers and decision makers for covid-19 pandemic date: 2020-10-10 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110337 sha: doc_id: 301035 cord_uid: dz8642qx while the world has experience with many different types of infectious diseases, the current crisis related to the spread of covid-19 has challenged epidemiologists and public health experts alike, leading to a rapid search for, and development of, new and innovative solutions to combat its spread. the transmission of this virus has infected more than 18.92 million people as of august 6, 2020, with over half a million deaths across the globe; the world health organization (who) has declared this a global pandemic. a multidisciplinary approach needs to be followed for diagnosis, treatment and tracking, especially between medical and computer sciences, so, a common ground is available to facilitate the research work at a faster pace. with this in mind, this survey paper aimed to explore and understand how and which different technological tools and techniques have been used within the context of covid-19. the primary contribution of this paper is in its collation of the current state-of-the-art technological approaches applied to the context of covid-19, and doing this in a holistic way, covering multiple disciplines and different perspectives. the analysis is widened by investigating artificial intelligence (ai) approaches for the diagnosis, anticipate infection and mortality rate by tracing contacts and targeted drug designing. moreover, the impact of different kinds of medical data used in diagnosis, prognosis and pandemic analysis is also provided. this review paper covers both medical and technological perspectives to facilitate the virologists, ai researchers and policymakers while in combating the covid-19 outbreak. first china, then worldwide [16] to tackle the pandemic of covid-19, scientists and medical experts are trying to come up with state-of-the-art solution to control the spread, identify infected patients, monitor virus growth and develop vaccines. many researchers are working with computer-based diagnosis techniques using clinical blood samples, radiography images, and respiratory-related datasets. ai and machine learning techniques are helping medical personnel in monitoring, analysis and prediction of various critical applications such as survival mortality assessment, forecasting models, virions sequence formation and drug discovery models. in order to help stop the pandemic, the early detection of positive cases is crucial as it facilitates the containment and isolation of the virus, thus reducing its spread. currently, the standard procedure followed for identification of covid-19 positive cases is performed by reverse-transcription polymerase chain reaction (rt-pcr) [22] . to further improve and hasten the early detection process, there is room for the development of new and innovative tools that may improve the early detection and tracking of the virus [23, 24] . as there is a clear need for these innovative technical solutions, there has, logically, been a large amount of rapid-paced development of these tools. however, due to the speed and number of these developments, it is easy to become overwhelmed, thus limiting the ability to critically examine, explore, and understand what a given solution does, how it performs, and whether or not it may or may not be beneficial. thus, with this in mind, the main objective of this comprehensive review is to provide an in-depth account of covid-19 and various advancements in ai techniques that have been currently developed to help manage or combat covid-19. the review was conducted between july and august 2020, using the keywords -covid-19‖ and (combined with one of the following: -ai‖, -machine learning‖, -deep learning‖) as well as -covid-19‖ and -ai‖ and (combined with one of the following: -clinical blood samples‖, -forecasting models‖, -drug discovery models‖). this approach is, admittedly, broad, but allowed for a large number of relevant papers to be identified. additionally, due to the nature of this study being a relatively new research subject, pre-print papers (such as those submitted to arxiv) were not precluded. as recommended by [25] , we selected google scholar as our electronic bibliographic databases to remove any kind of biasness for any specific scientific publisher. we then excluded papers not written in english, duplicates, guest editorials, poster sessions, and blogs. once the papers were identified, they were analyzed and grouped based on their aims and approaches. based on this conducted study, it is hoped that this paper will serve as a common ground for both medical experts and computer scientists to better understand the current state-of-the-art associated with covid19 with this in mind, the main contributions of this study are as follows:  detailed history, characteristics, taxonomy, symptoms, behavior, and patterns of covid-19 are outlined.  presentation and discussion of ai (both traditional ml and advanced dl) approaches applied to covid-19 in a variety of aspects and domains.  description of major covid-19 diagnosis techniques using clinical blood samples, radiography images, respiratory-related data for various critical applications such as survival mortality assessment, forecasting models, virions sequence formation and drug discovery models.  a detailed discussion on issues, challenges, and recommendations to tackle the pandemic through ai and timely facilitate effective decision making is presented. in order to meet these objectives, the rest of the paper is organized as follows. section 2 lays out the historical overview, origin and means of transmission, and global dispersion of the virus. section 3 highlights the novel ai techniques that have been applied to covid-19, and furthermore, it explores non-applied, but potentially useful, ai-based applications for vaccine discovery, mortality and survival rate prediction, and outbreak forecasting. section 4 discusses the challenges in covid-19 research. finally, the review is concluded in section 5 by presenting a summarized overview of the conducted studies. the type of coronaviruses that cause infections in humans belongs to a subfamily called coronavirinae which is part of a larger [26]. the name corona comes from the fact that the viral particle has spike-like projections or structures on its outer layer that can be seen under a scanning electron microscope. the rna of coronaviruses is single-stranded, positive sense, has a diameter of around 80 to 120 nm and a nucleic acid length from 26 to 32 kbs [27] . they are classified under four variants or genera entitled as alpha (α), beta (β), gamma (γ) and delta (δ) [27] . it has been observed that mammals typically get infected by the α-and β-cov while the other two (γ-and δ-cov) mostly infect birds. from the α-cov group, hcov-nl63 and hcov-229e have been found to affect humans, normally causing a mild respiratory illness, similar to the common cold. from the β-cov group, sars-cov and mers-cov both tend to cause moderate to severe respiratory illnesses with a higher morbidity and mortality rate [28, 29] . the most recent addition to the coronaviruses is that of sars-cov-2, which became known in late 2019 after a number of atypically presenting pneumonia cases were identified in wuhan china; it was initially speculated that these cases might originate from the huanan seafood market [30] . the hallmark sign of many of the cases was the development of certain symptoms including dry cough with fever, breathing difficulty and a-typical presentation of pneumonia. on january 7 th 2020, bangladesh have also registered a high number of cases; figure 1 shows the top 15 countries for coronavirus infections and deathsbased on the latest who situation reports [34] . the disease itself is so pervasive due to the ease with which it is able to transmit itself, via small respiratory droplets (such as those created when talking, sneezing, coughing, and breathing) and has a higher level of danger to a person's health due to the current absence of targeted pharmaceutical solutions such as vaccines or antiviral drugs. pandemics are an ever-present threat to human society, health, and wellbeing and, therefore, attention must be dedicated to understanding how they occur, how they can be prevented, and how we can strategically mitigate the negative effects they cause. the current pandemic associated with covid-19 is unique in that it represents the first and largest pandemic in modern times where new technological solutions, e.g. ai, are not only applicable to the pandemic, but able to be created by a large group of stakeholders from scientists to doctors to ai-hobbyists alike. as the pandemic has caused great disruption to normal day-to-day operations and created a sense of unknown amongst the public, many motivated scientists and citizens have tried to assist in the covid-19 response by developing their own unique ai-based tools to solve a large number of problems, in a variety of applied domains, such as: coivd-19 disease detection and classification, mortality rate prediction and severity assessment, outbreak forecasting and tracking, biological insight of sars-cov-2 strain, and drug discovery. in order to explore the different approaches that are currently being developed and trialed, this section presents a selection of the different ways in which ai-based solutions have been applied to the covid-19 pandemic. in order to provide a visual representation of how an ai-based covid-19 system may work, a generic model has been created in figure 2 . the system can process various types of raw data including images, blood samples, crowed sourced data, even gps information for tracking positive cases, etc., which can be stored in a repository. data mining techniques are usually applied to clean the raw data that is and then stored in a database for further processing and retrieval. ml and dl methods can be applied to this data for better visualization, diagnosis, and forecasting, which can help the end-users making better decisions and taking actions to mitigate the disease. early diagnosis of covid-19 is important to ensure better outcomes for those who have been infected. at the beginning of the pandemic, due to the sense of the unknown that surrounded the new disease, one way that was being used to diagnose patients was via radiographic imagery (such as ct scans or x-rays) of the lungs. table ii . there has also been a large number of dl-based approaches proposed to identify covid-19 from radiographic imagery in the literature. for example, brunese et al. [47] exploited the transfer learning technique and fine-tuned dl-based modified visual geometry group (vgg-16) model. the proposed framework was divided into 2 models that considered 6523 chest x-rays (2753 belongs to patients with pulmonary diseases, 250 of covid-19 infected patients, while 3520 related to healthy patients). the first model discriminates between healthy and pulmonary diseases patients, while the second model classifies the chest xray (cxr) of pulmonary disease patient as covid-19 or other pneumonia. in the end, if a cxr is classified as covid-19, the framework provides a visualization of the cxr highlighting the potentially sars-cov-2 infected area. from the experimental data, it was observed that the discriminatory model (model 1) achieved accuracy, sensitivity, specificity and f-measure of 96%, 96%, 98% and 94% respectively, whereas the disease classification model (model 2) yielded an accuracy of 98%, sensitivity of 87%, specificity of 94%, and an f-measure of 89%. similarly, to identify sars-cov-2 infected cases from cxr images, various cnn frameworks such as vgg19, mobilenet v2, inception, xception, and inception resnet v2 have been evaluated using transfer learning by apostolopoulos and mpesiana [48] . from their conclusion, it was observed that mobilenet v2 outperformed the other tested state-of-the-art architectures by securing 99.10% sensitivity, 97.09% specificity, 97.40% accuracy on the two-class problem and 92.85% accuracy on the threeclass problem. in that paper, the models were trained and evaluated on a dataset of 1427 cxr images that constitutes 700 images of bacterial pneumonia infected patients, 224 images of covid-19 infected cases, and 504 belonging to normal patients. mobilenet v2 also performed well yielding a sensitivity, specificity, two-class problem accuracy, and three-class other identified dl-based methods and approaches are highlighted in table iii . another type of covid-19 diagnostic tool that has been developed is related to respiratory wave data, such as developed in wang et al. [114] . in this paper, the authors proposed a respiratory pattern classification model that was based on a novel gru neural network [115] , which extends gru networks with bidirectional and attentional mechanisms (bi-at-gru) by using time-series real-world and stimulated respiratory data. the model detects and distinguishes the tachypnea respiratory pattern, a more rapid respiration symptom that occurs in covid-19 infected patients, from 6 other viral infection patterns. a novel respiratory stimulated model (rsm) was used to generate stimulated breathing patterns to fill the scarce real-world data. the trained model was evaluated on real-world data captured by a depth camera. the proposed bi-at-gru model resulted in precision, recall, f1-score, and accuracy of 94.4%, 95.1%, 94.8% and 94.5% respectively. table v lists the other identified covid-19 ai-based applications that utilize respiratory or coughing data. outside of just diagnosis of covid-19, ai-based tools have also been applied to identifying the severity of the disease in a patient, as well as to develop and optimize treatment strategies. for example, in order to attempt to help to reduce the mortality of covid-19, and to optimize potential treatment strategies, a novel dl-based framework, combined with a multivariate logistic regression model, was proposed by bai et al. [119] . multi-layer perceptron (mlp) was used to convert each statistical sample containing 75 clinical data characteristics to a 40-dimensional feature vector. the proposed model was then evaluated using a dataset of 133 patients and achieved an overall accuracy of 89.1% and 95.4% auc under 5-fold cross-validation repeated 5 times. the researchers found six key risk factors (age, lymphocyte, comorbid with hypertension, albumin, hypersensitive c-reactive protein level, and progressive consolidation in ct images) plus fibrosis in ct as a predictive factor for malignant progression. albahri et al. [120] also proposed an ml-based rescue framework that was combined with a novel multi-criteria decisionchen et al. [121] developed a fatality rate predictive model for severe sars-cov-2 patients based on five distinct ml approaches, namely: elastic net, bagged flexible discriminant analysis (bagged-fda), logistic regression, random forest and partial least squares regression. the model used data of 183 severely infected covid-19 patients, out of which 115 survived and found four key features and clinical pointers (age, d-dimer level, lymphocyte count, and high-sensitivity c-reactive protein level) for survival/mortality assessment. moreover, the developed models were evaluated on an external validation set that consisted of 64 severe covid-19 confirmed cases, out of which 33 were survivals. the experimental results concluded that the logistic regression model was to be preferred over other ml models due to its high interpretability and simplicity; it obtained sensitivity, specificity and auc of 89.2%, 68.7% and 89.5% respectively. table vi shows other identified ai-based approaches related to severity and fatality assessment models. one of the first areas where ai was applied to the covid-19 pandemic was related to the development of outbreak forecasting models, which had the potential to help decision makers understand the potential progression of covid-19 in their area. one such approach was developed by kavadi et al. [132] who suggested a partial derivative regression and non-linear machine learning (pdr-nml) model to predict the covid-19 transmission dynamics in india. the presented model achieved better performance in terms of accuracy and prediction time over linear regression and state-of-the-art ai-based models. it secured an accuracy of 99.7% with a prediction time of 5750 ms for 5000 samples. another approach was offered by carrillo-larco and castillo-cara [133] who presented a model based on k-means clustering that categorized countries sharing similar numbers of confirmed sars-cov-2 cases. in this study, researchers not only used covid-19 prevalence data (deaths, confirmed cases etc.), but also considered open access predictors like the prevalence of diseases such as hiv/aids, diabetes, and tuberculosis in 156 countries along with their standard health system metrics, air quality, and social-economic parameters such as the gross domestic production. a third approach was developed by hu et al. [134] , who attempted to develop an ai-based modified auto-encoder (mae) that modeled multiple public health interventions by using real data to forecast a potential covid-19 pandemic outbreak in a large geographic region and subsequently calculating the potential impact of interventions to curb the pandemic. the proposed architecture consisted of two single auto-encoders, each having three feedforward neural network layers. the used data pertained to confirmed positive covid-19 cases and covid-19 attributed deaths across 152 countries over the period of january 20 th to march 16 th , 2020. the estimate includes pandemic peak time, end time, and future cases. it concludes that mae performed better in forecasting daily new cases in china (with an error of 0.00134) when compared with susceptible-exposedinfected-recovered (seir) model as used by sameni [135] . beyond conventional ml-based approaches, advanced dl techniques have also been used to estimate future cases. paul et al. [136] suggested a convolutional lstm-based multivariate spatiotemporal model to forecast the pandemic outbreak at the world-level. the proposed framework converted the spatial features into groups of temporal/non-temporal component-based 2d images and used data from usa and italy to train the network. the model performed well for predicting the number of cases over a 5-days period, with a mean absolute percent error (mape) of 0.3% and 5.57% for italy and usa respectively. other identified approaches for the development of potential covid-19 medications and vaccinations are highlighted in table viii . the investigation of this paper reveals several ai-based approaches that have been proposed as potential ways to help, with the covid-19 pandemic, covering everything from initial diagnoses via image diagnostics up to the presentation of models that help to understand the spread of covid-19 and identify potential new outbreak areas. by providing this comprehensive overview and identification of the current state-of-the-art, this structured review may help assist medical and research stakeholders currently involved in the covid-19 response. when it comes to potential barriers to the use of ai for covid-19 responses, one of the foremost inaccuracies is related to the failure of forming diagnostic programs that fail differentiate by symptomatic basis. therefore, in order to attain more accurate and precise prediction models, the patient's history of other existing ailments such as diabetes, heart, liver or other chronic conditions along with his age and gender should also be taken into account. the significance of these factors greatly influences the characteristics of covid-19 for a specific patient compared with other pneumonia diseases known so far. as many countries around the world are opening up their data related to covid-19, it becomes possible for anyone to create new ai-based services, and, additionally, if these statistics are provided in an effective way, tools for the prediction of covid-19 diagnoses and spread could, in theory, be developed. another issue regarding the covid-19 pandemic is urgency. as the government needs high-quality statistical data in almost real-time to understand the current epidemiological situation in their area and to regulate quarantine in that same area, effort should be focused on improving the availability and provision of data first. as at the beginning of the pandemic, data was often not of high quality (though this depends on a large amount on which country's data is being examined) the initially used scientific datasets were often constructed in a quick manner and may not be as precise or adequate as one would ideally like. while considering the statistical analysis of the covid-19 disease, the crucial challenge is to secure real-time high-quality data. as this disease is still in a rapid growth phase, the fast nature of data makes it extremely difficult to produce reliable models. collection of regularly updated region-specific values of covid-19 is crucial in order to overcome this issue. a final challenge, and one of the most important, is the lack of understanding of epidemiological theory, beliefs, and knowhow of many ai researchers. this leads to a situation where many of the ai-based systems are created without input from medical professionals, thus limiting their usefulness and reliability. thus, to this end, it is important that any implemented aibased system aims to involve medical experts in the design and implementation. whilst it does appear to be the case that many ai-based approaches have been proposed, there is a limited real-world implementation of these tools. what this paper has shown, however, is that there is clearly a wide variety of potential methods that could be applied. moving forward, it is imperative for ai-designers and researchers to work together with medical professionals to create and develop these systems that are applicable to real-world datasets and relevant to the work of the given stakeholder group (e.g. policymakers or health care workers). ai is not a cure-all that can fix the covid-19 crisis, but it does have the potential to augment a given person's ability to handle and manage the crisis, thus, potentially, limiting the spread of the virus and leading to potentially better outcomes for infected patients. as covid-19 has permeated throughout the world, there is no one who has not felt its impact in some way. thus, scientific and medical researchers are working rapidly to find a viable cure against the disease. one tool that may well help aid this effort is artificial intelligence (ai) by assisting front-line workers in numerous ways. this paper has initially presented the details of the covid-19 viruses including taxonomy, signs and symptoms, cause of spread, affected body organ, and hazardous impact on human society over the last century. secondly, we mentioned the structural genome and origin of covid-19 infectious virus, compared it with other coroviridae family viruses, its transmission behavior and impact on global health. we presented an overview of potential use cases and ai-based implementations currently being developed and trialed. for instance, technological experts have developed ai-based programs for the rapid analysis, scanning and diagnosis of the coronavirus through pneumonia radiographic images or clinical blood sample data. these approaches offer an additional way to detect covid-19, in addition to the more commonly used pcr test. as testing capabilities continue to improve, ai-based tools become more useful to assist in clinical settings rather than to replace medical professionals. however, where ai is likely to see an increased use and benefit moving forward in the future, especially as more and higher quality data becomes available, is its use in predicting potential new areas of outbreak and forecasting the spread of covid-19. these models are based on parameters such as the number of positive patients in different regions of a city and tracking their movements to anticipate the potential spread through contact. for example, it could be possible to apply an ai-based model to data gathered via a contact tracing application or sewer covid-19 prevalence data, though other data could also be used, to better predict the scale and size of a new outbreak. the results thus obtained could help to manage and implement precautionary actions by defining guidelines in pandemic affected areas. it is important that ai-based tools are relevant, accurate, and impactful. in order to do this, cooperation is needed between ai-based researchers, medical professionals, and governmental agencies. thus, further research should identify real-world empirical examples of ai being used on real-world data. this paper has made initial steps in gathering and highlighting the current state-of-the-art, but it does not differentiate between examples working -in the wild‖ and those that are done under laboratory conditions. so, while in theory, these approaches may well be useful, until they are evaluated within an applied context it is hard to quantify their level of usefulness, in fact, it may well be the case that ai-based tools actually inhibit effectiveness if they discreetly contain some level of bias, or replace the decision making capability of a more competent human actor. ☒ the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐the authors declare the following financial interests/personal relationships which may be considered as potential origin and evolution of pathogenic coronaviruses covid-19: animals, veterinary and zoonotic links zoonoses. in: encyclopedia of food and health defining epidemics in computer simulation models: how do definitions influence conclusions? defining epidemics in computer simulation models: 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novel coronavirus (sars-cov-2) through a drug-target interaction deep learning model crispr-based surveillance for covid-19 using genomically-comprehensive machine learning design potential 2019-ncov 3c-like protease inhibitors designed using generative deep learning approaches dna double-strand break repair in mammalian cells. view project matrix-isolated systems modeling view project competing interests: key: cord-261599-ddgoxape authors: nabi, khondoker nazmoon; abboubakar, hamadjam; kumar, pushpendra title: forecasting of covid-19 pandemic: from integer derivatives to fractional derivatives date: 2020-09-21 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110283 sha: doc_id: 261599 cord_uid: ddgoxape in this work, a new compartmental mathematical model of covid-19 pandemic has been proposed incorporating imperfect quarantine and disrespectful behavior of the citizens towards lockdown policies, which are evident in most of the developing countries. an integer derivative model has been proposed initially and then the formula for calculating basic reproductive number [formula: see text] of the model has been presented. cameroon has been considered as a representative for the developing countries and the epidemic threshold [formula: see text] has been estimated to be ∼ 3.41 [formula: see text] as of july 9, 2020. using real data compiled by the cameroonian government, model calibration has been performed through an optimization algorithm based on renowned trust-region-reflective (trr) algorithm. based on our projection results, the probable peak date is estimated to be on august 1, 2020 with approximately 1073 [formula: see text] daily confirmed cases. the tally of cumulative infected cases could reach ∼ 20, 100 [formula: see text] cases by the end of august 2020. later, global sensitivity analysis has been applied to quantify the most dominating model mechanisms that significantly affect the progression dynamics of covid-19. importantly, caputo derivative concept has been performed to formulate a fractional model to gain a deeper insight into the probable peak dates and sizes in cameroon. by showing the existence and uniqueness of solutions, a numerical scheme has been constructed using the adams-bashforth-moulton method. numerical simulations enlightened the fact that if the fractional order α is close to unity, then the solutions will converge to the integer model solutions, and the decrease of the fractional-order parameter (0 < α < 1) leads to the delaying of the epidemic peaks. in the late december 2019, the world health organization (who) was enlightened about sudden new cases of a unique pneumonia and the specific cause of this infection was an enigma for the health authorities of the city of wuhan in china. a few days later, it has been unearthed that this pneumonia is caused by a virulent virus and it has been named officially as the novel coronavirus (2019-ncov). the modes of transmission are similar to the viruses responsible for the previous epidemics of sars and mers [31] . however, this virus is more contagious and is the cause of millions of deaths worldwide. to date, there is no clinically established treatment or specific vaccine for covid-19. nevertheless, according to the suggestions of different prudent infectious disease specialists, several countries have started using chloroquine combined with azithromycin as alternative drugs [15] . several non-pharmaceutical interventions such as physical distancing, wearing face masks in public places, home quarantine, isolation and countrywide lockdown policies have been promoted to curb the spread of the virus [27] . cameroon, a country in sub-saharan africa, had its first confirmed case on march 05, 2020. it was a 58-year-old french citizen, arrived by yaoundé airport on february 24, 2020 [22] . with an aim to quell the spread of the virus, the government undertook several policies such as closing the borders, confinement measures, prohibiting large gatherings of people, closures of all kinds of educational institutions (from kindergarten to university), wearing face masks in public places and so on. in addition, cameroonian government implemented contact-tracing strategy together with mass-media sanitisation campaigns. contact-tracing consists of search, or identify all individuals who have had a contact with a confirmed case and force them to be quarantined if they are covid-19 positive [24] . the scarcity of medical resources in maximum sub-saharan african countries compelled authorities to request asymptomatic cases of covid-19 to be self-quarantined in their home while taking medication (conventional or not) following proper health guidelines [16] . the main problem encountered by health workers is the stigmatization of covid-19 patients. indeed, like the cases of hiv/aids in africa, stigma pushes confirmed cases or those who have had direct contact with a confirmed case of covid-19 to run away from hospitals, and in this way they are contributing to the rapid spread of the virus in the community [23, 37] . different types of mathematical models have been playing notable role in predicting the transmission dynamics of infectious diseases and different effective control measures can be designed to limit the community transmission. several models have already been proposed to predict the evolution of covid-19 and to study the impact of control measures imposed by different governments [1, 26] . although these models have similarities, each of them presents specifications related to the evolution of covid-19 and various control measures deployed by several respective governments. a model has been considered in [1] , describing the interactions among the bats and unknown hosts, human population and population of virus in reservoir (seafood market). atangana-baleanu derivative concept has been implemented successfully. however, confined or quarantined class and isolated class have not been considered in this study. in a recent study, nabi [26] has proposed a new susceptible-exposed-symptomatic infectious-asymptomatic infectious-quarantined-hospitalized-recovered-dead (sei d i u qhrd) compartmental mathematical model and calibrated model parameters to project the future dynamics of covid-19 for various covid-19 hotspots. although he considered a quarantine compartment, he did not consider the possibility of those quarantined people to contract the virus. although this consideration goes well with certain developed countries (france, italy, russia and spain), certain realities [27] of covid-19 pandemic in several developing countries and maximum african countries have not been taken into account in any model. in case of cameroon, several crucial factors such as violation of containment measures and negligence towards confinement measures must be taken into account to understand the transmission dynamics of covid-19. importantly, numerous daily wage-earners are compelled to go out to fetch food for their family members contravening all confinement orders. in this way, they are getting infected and spreading the viruses to their family members as well as neighbors [27] . it is therefore prudent to envisage future outbreak dynamics of covid-19 in developing countries with imperfect confinement. moreover, some people flee quarantine in hospitals to return to their families due to the stigmatization of covid-19 patients. henceforth, considering perfect quarantine would be highly debatable if we want to model and forecast the transmission dynamics of covid-19 in developing countries. fractional calculus or non-integer order calculus, a branch of mathematical analysis, contains the theory of fractional-order derivatives operators. the history of non-integer order calculus is more than 300 years old and modern fractional calculus is a rapidly evolving field both in theory, analysis and applications with a view to handling complicated real-world problems. a plethora of concepts regarding fractional derivatives have already been introduced and applied by researchers in various branches of science and engineering [13, 17, 32] . atangana-baleanu (ab), caputo-fabrizio (cf), and caputo derivatives are the most commonly used derivatives in solving real-world problems. grunwald-letnikov fractional operators have been applied comprehensively in the field of image processing. ab, cf, and caputo fractional derivatives have different kernel properties. caputo derivative is defined with power-law type kernel (non-local but singular), caputo-fabrizio with exponentially decaying type kernel (non-singular) and ab derivative in the caputo sense has mittag-leffler type kernel [36] . a new generalised caputo type fractional derivative has been introduced in [30] . asymptotic stability of generalised caputo fdes has been introduced by baleanu in [5] . integer-order derivative operators have lucid geometric and physical interpretations, which clearly simplify their applications in tackling real-world challenges. recently, various techniques have been proposed by researchers to describe the solution of non-linear fractional differential equations. a new technique for solving non-linear volterra integro-differential equations in atangana-baleanu derivative sense has been introduced in [12] . another technique with the uses of genocchi polynomials has been proposed in [35] . a new technique to solve multi-variable order differential equations in ab sense has been introduced by ganji et al. [11] . in this paper, a new compartmental covid-19 model has been studied rigorously with the help of caputo fractional derivatives. the advantage of applying caputo fractional derivatives to solve the proposed covid-19 model is the dynamics of the model can be observed more deeply using the real-time cameroon data. in this framework, real-time data can be compared with the model outputs in a precise manner. fractional derivatives can be a suitable alternative to integer derivatives in studying complex dynamics as there are always some limitations of integer derivatives described in [2] . podlubny has given the geometric and physical significance of riemann-liouville fractional integral in [33] . as compared to the riemann-liouville fractional derivative, caputo fractional derivative has the identical physical interpretation [33] . in a physical model, fractional derivatives manage systems with memory that ensure that the evolving system state is reliant on its past states also. the aim of this work is to forecast the probable time and size of the epidemic peaks of the novel coronavirus outbreak in cameroon by studying a realistic compartmental model using the robust concept of caputo fractional derivative. a compartmental mathematical model of covid-19 progression dynamics has been proposed incorporating the efficacy of confinement measures and imperfect quarantine. estimation of parameters has been performed by using real-time data, followed by a projection of the evolution of the disease. global sensitivity analysis is applied to determine the influential mechanisms in the model that drive the transmission dynamics of the disease. afterwards, the integer model is transformed into a fractional model in caputo sense and the existence and uniqueness of solutions have been presented. using the adams-bashforth-moulton scheme, a numerical scheme for the fractional model has been constructed. several numerical simulations are performed to compare the result obtained by the integer-order model with the fractional model outputs. the entire paper is organized as follows. model formulation and basic properties are presented in section 2. section 3 is devoted to model calibration using real data of reported cases of covid-19 in cameroon, global sensitivity analysis of the proposed model, and forecasting of the disease future dynamics. in section 4, the compartmental model is transformed into a fractional model in the caputo sense, following by the existence and uniqueness of solutions, construction of a numerical scheme and numerical simulations in which the coefficient of fractional order vary. the paper ends with some insightful observations and fundamental findings. a compartmental differential equation has been proposed to describe the transmission dynamics of the covid-19. the spread starts with an introduction of at least one infected human into a susceptible population. the human population has been categorized into seven compartments according to their epidemiological status, i.e {s(t), c(t), e(t), a(t), q(t), h(t), r(t)} which represent the number of susceptible individuals, confined individuals, infected individuals in incubation period, asymptomatic infectious individuals (undetected but infectious) including those who fled quarantine, quarantined individuals or confirmed infected individuals, hospitalised and recovered cases. so, the total human population, denoted by n , at any time can be represented by n (t) = s(t) + c(t) + e(t) + a(t) + q(t) + h(t) + r(t). the formulation of the model are subject to some assumptions listed bellow. • migration is not considered. indeed, concerning migration, the initial time of our model is taken when the disease is already inside the region or country and that international migrations are prohibited [26] . • vital dynamic (birth and natural deaths) is ignored. the idea is to observe what may happen in the short-term. indeed, since for example to have a birth we need around 9 months on an average, to have a natural death rate which is the inverse of the average life expectancy i.e. 1/55 per year in some developing country like cameroon and no one would like the disease to stay so long. • confinement is not perfect. indeed, in some developing countries, because of insufficient financial income in households, people do not respect the confinement measures imposed by the government, and are forced to go out to work to provide for their families. • the immunity is perfect. indeed we assumed, without confirmed proof of lost of immunity, that recovered individuals develop certain immunity against the disease [9, 29] . • hospitalized patients (in respiratory support) can not spread the disease as they remain under closed supervision. the flow diagram of the proposed model is depicted on figure 1 , where susceptible individuals decrease either via confinement at a rate c, or via infection by a direct contact with an infectious individual (a or q). the transmission rate from unquarantined asymptomatic carriers (a) (respectively quarantined symptomatic carriers (q)) is β (respectively ηβ). the modification and dimensionless parameter 0 ≤ η < 1 accounts for the assumed reduction in transmissibility of quarantined symptomatic carriers relative to unquarantined asymptomatic carriers. since the confinement is not perfect, confined individuals move to the latent class at a rate (1 − )λ where 0 ≤ ≤ 1 is the efficacy of confinement. this is in contrast with a variety of seir-style models recently employed in, e.g. [9, 29] , where the authors consider that confined individuals are not exposed to an infection. indeed, in developing country such as cameroon, since the confinement is not perfect, it is practical that an infected individual can infect a confined person by returning back home. this phenomenon was also observed at the beginning of the epidemic in a developed country such as italy, where old confined individuals had been infected by their grandchildren who are asymptomatic carrier [16] . after the incubation period of 1/γ, a proportion q of infected individuals move to the asymptomatic class a and the other (1−q) to the quarantine class. it important to note that asymptomatic class a include all infectious individual who are not in quarantine and those who has fled the quarantine. the rate of transition from unquarantined asymptomatic infectious to quarantine infectious, unquarantined asymptomatic infectious to hospitalised class, unquarantined asymptomatic infectious to recovered class and hospitalised to recovered class are r 1 σ 1 , r 2 σ 1 , (1 − r 1 − r 2 )σ 1 and σ 3 respectively. r 3 σ 2 , r 4 σ 2 and (1 − r 3 − r 4 )σ 2 are transition rates of quarantined asymptomatic infectious to unquarantined asymptomatic infectious, quarantined asymptomatic parameter description β infectious contact rate η infectiousness factor for quarantined infected carrier p transition rate from confined class to unconfined class c transition rate from unconfined class to confinement class confinement efficacy γ transition rate from exposed class to infectious class q fraction of exposed that become quarantined carriers σ 1 transition rate from unquaratined class to quarantined class and recovered class σ 2 transition rate from quaratined class to unquarantined class, hospitalised and recovered σ 3 transition rate from hospitalised class to recovered class r 1 fraction of unquaratined infectious that become quarantined infectious r 2 fraction of unquaratined infectious that become hospitalised r 3 fraction of quarantined infectious that become unquarantined infectious r 4 fraction of quarantined infectious that become hospitalised δ a disease induced death rate, unquarantined infectious δ q disease induced death rate, quarantined infectious δ h disease induced death rate, hospitalised infectious to hospitalised class, and quarantined asymptomatic infectious to recovered class respectively. δ a , δ q and δ h are the disease-induced death rate of unquarantined asymptomatic infectious, quarantined asymptomatic infectious and hospitalised patients respectively. the above assumptions led to the following nonlinear system of ordinary differential equations, the parameters are described in table 1 . we set x = (s, c, e, a, q, h, r) the vector of state variable, let f : r 7 → r 7 the the right hand side of system (2), which is a continuously differentiable function on r 7 . according to [39, theorem iii.10 .vi], for any initial condition in ω, a unique solution of (2) exists, at least locally, and remains in ω for its maximal interval of existence [39, theorem iii. 10 .xvi]. hence model (2) is biologically well-defined. here, we prove that all state variables of model (2) are non-negative for all time, i.e, solutions of the model (2) with positive initial data remain positive for all time t > 0. the following result can be obtained. is a solution of (2) with positive initial conditions. let us consider e(t) for t ≥ 0. it follows from the third equation of system (2) that since e(0) ≥ 0, it follows that e(t) ≥ 0 for t ≥ 0. we proceed with the same for a(t), q(t), h(t) and r(t). now, it remains to prove that s(t) and c(t) are also positive. assume the contrary, and lett such that s(t) = 0 and c(t) ≥ 0. from the first equation of (2), it follows that ds(t) dt | t=t = pc(t), which means that s(t) < 0 for t ∈ (t − ζ,t) with ζ a small positive constant. this leads to a contradiction. thus s(t) ≥ 0 for t ≥ 0. we proceed with the same to prove that c(t) ≥ 0. (2) with positive initial conditions are bounded by the total population n 0 . where n 0 is equal to the total population. it follows that for all t ≥ 0, we have s(t) ≤ n 0 , in what follows, we study the model (2) in the following set which is positively-invariant and attracting region for the model (2). now, we define the following manifold, in which any point is a disease-free equilibrium (dfe) of the model (2) note that w ⊂ d. in the following, we will work with the largest disease-free equilibrium point denoted by x 0 (see [14] ). assuming s + c = n 0 , it follows that using notations in [38] , matrices f and v for the new infection terms and the remaining transfer terms are, respectively, given by then, the control reproduction ratio is defined, following [38] , as the spectral radius of the next generation matrix, f v −1 : where ρ(·) represents the spectral radius operator. the formula for control reproduction number has been formulated. indeed, the insightful epidemic threshold, r c calculates the average number of new secondary covid-19 cases generated by a covid-19 positive individual in a population where a certain fraction of susceptible people are confined. the control of covid-19 pandemic is governed by the application of some control measures which contribute to bring down r c under unity [38] . hence, we claim the following result which is a direct consequence of the next generation operator method [38, theorem 2] . lemma 3. if r c < 1, the disease-free equilibrium x 0 is locally asymptotically stable and unstable if r c > 1. proof. note the two last equations of the system (2) are uncoupled to the remaining equations of the system. since the total population, n 0 , is constant, we have s + c = n 0 − (e + a + q + h + r). so, the local stability of the covid-19 model (2) can be studied through the remaining system of state variables (e, a, q). thus, we obtain that the jacobian matrix associated to these variables is given by the eigenvalues of j are the roots of the following characteristic polynomial: note that a 2 is always positive. a 0 and a 1 are positive provide that r c < 1. thus, it follows that j has all its eigenvalues with negative real parts. hence, the disease-free equilibrium, x 0 , is locally asymptotically stable whenever r c < 1. remark 1. lemma 3 implies that if r c < 1, then a sufficiently small flow of infected individuals will not generate an outbreak of covid-19, whereas for r c > 1, epidemic curve reaches a peak by growing exponentially and then decreases to zero as t → ∞. the better control of the covid-19 can be established by the fact that the dfe x 0 is globally asymptotically stable (gas). in this context, we claim the following result. theorem 1. if r c < 1, then the manifold, w, of disease-free equilibrium points of the model (2) is gas in d. proof. assume that r c < 1. let x = (s, c, r) and y = (e, a, q) . since we deduce that thus from [38, theorem 2] , it follows that s(f − v ) < 0 if and only if r c < 1 where s(m ) is the stability modulus of the matrix m . therefore, the trajectories of the auxiliary system whose right-hand side (6) converges to zero whenever r c < 1. since all the state variables are non-negative and f − v is a metzler matrix, by the comparison theorem [18] , it follows that lim shows that w is an attractive manifold. moreover, w is locally asymptotically stable when r 0 < 1. we conclude that in system (2), the manifold w is globally asymptotically stable when r c < 1. in the absence of confinement measures, i.e. = 0, r c converges to the basic reproduction number, r 0 , given by using (5), it follows that in the next section, numerical simulations have been performed rigorously. parameters considered in our simulations are either estimated from real data or related to the covid-19 outbreak data in cameroon. as in many countries in sub-saharan africa, the actual situation of the covid-19 in cameroon and its economic impact are mixed. indeed, after the appearance of the first case confirmed on march 07, 2020 until the introduction of control measures by the government of cameroon (closure of public places, in particular schools, universities, drinking places and other places of entertainment from 6 p.m. and so on), the statistics advanced by the cameroonian health authorities do not reflect, according to certain non-governmental organizations, the actual situation of the covid-19 epidemic in cameroon. since no nongovernmental organization is empowered to carry out tests, the only statistics we use here to calibrate our model are those communicated by the cameroonian ministry of public health during her daily press briefings [25] . also, these control measures decreed by the government have really impacted certain economics activity sectors. indeed, the fact of the closing of the drinking places at other places of distractions, without forgetting the prohibition of the gatherings of more than 50 persons, a lot of people found themselves in technical unemployment, and of this fact was forced to leave for go to affect activities that may allow them to support themselves and their families. the number of covid-19 positive cases (npc) per day, the number of recovered cases (nrc) per day and the number of death cases (ndc) per day in cameroon, have been taken in the summarized table online on the evolution site of covid-19 [34] , from early march to early july 2020. time=0 corresponds to march 7, 2020 and time=124 stands for the day july 08, 2020. as of july 10, 2020, the total death toll is 359, the total number of recovered cases is 11,525 and the total number of positive cases is 14,916 in cameroon [34] . the model (2) calibration has been performed using a newly developed optimization algorithm based on trust-region-reflective (trr) algorithm, which can be regarded as an evolution of levenberg-marquardt algorithm [26] . this robust optimization procedure can be used effectively for solving nonlinear least-squares problems. this algorithm has been implemented using the lsqcurvefit function, which is available in the optimization toolbox in matlab. necessary model parameters have been estimated using this optimization technique. daily infected cases data have been collected from a trusted data repository, which is available online [34] . a 7-day moving average of the daily reported cases has been used for our model calibration due to moderate volatile nature of real data fig. 2 . it has been observed that the number of daily testing in cameroon has been really inconsistent. with an aim to capture the real outbreak scenario, the 7-day moving average has been used in this regard. according to recent statistics, the total population of cameroon is found to be 26,548,238 [34] . for initial conditions, we take s(0) = 26533750, c(0) = e(0) = a(0) = h(0) = r(0) = 0 and q(0) = 1. the solutions have been obtained after resolving the following optimization problem numerically. where ψ = {β, σ 1 , σ 2 , σ 3 , r 1 , r 2 , r 3 , r 4 , δ a , δ q , δ h , η, p, c, , γ, q}, is displayed in table 2 parameter probable range base value trr output references β 0. table 2 : calibration of the model parameters using a newly developed optimization algorithm based on well-known trust-region-reflective algorithm. partial rank correlation coefficient (prcc) method has been carried out to quantify the most dominant mechanisms, which are significantly responsible for the transmission dynamics of covid-19 disease. prcc is a global sensitivity analysis method, which is used to quantify the relationship between model response function (outputs) and model parameters (sampled by latin hypercube sampling method) in an outbreak setting [28] . prcc values are ranged between -1 and 1. a negative prcc value indicates a negative correlation between the model output and respective input parameter, whereas a positive prcc value depicts a positive correlation between response function and corresponding model parameter. again, similar result has been found, when the confined infectious individual class (q) has been used as the model response function fig. 7 . in this case, the prcc indexes are estimated to be 0.885 and −0.69 accordingly. importantly, the public health implication of this scenario is that, the transmission dynamics of covid-19 can be controlled effectively in the community by curtailing the direct infectious contact rate and strengthening the confinement measures. direct transmission rate can be limited by promoting several non-pharmaceutical measures such as maintaining physical distance carefully, wearing face-masks in public places and following health guidelines properly. however, it is often really challenging for the government of a developing country to implement strict confinement policies [27] . thanks to the memory effect which represents an advantage of the fractional derivative compared to the ordinary derivative, the theory and the application of fractional calculus have been widely used to model dynamic processes in the fields of science, engineering and many others [3, 4] . before presenting the fractional model with caputo derivative, we give the definition of fractional derivative in caputo sense, and some useful results (see [20] ). here, we remind a definition and some results about caputo (c) derivative. definition 1. the caputo non-integer order derivative of g ∈ c k −1 is presented as lemma 4. if 0 < β < 1 and m is an integer (non-negative), then ∃ the +ve constants c β,1 and c β,2 only dependent on β, s.t and (m + 2) β+1 − 2(m + 1) β+1 + m β+1 ≤ c β,2 (m + 1) β−1 . where c is a +ve constant independent of m & h. by replacing the integer derivative order of the ode covid-19 model (2) by the fractional-order model in caputo sense, we obtain the following fractional model in this portion of the study, we will provide the existence of the unique solution for the proposed fractional-order model under the caputo fractional operator by applying fixedpoint results. in this setting, b is the banach space of real-valued continuous functions defined on an interval i with the associated norm , with e(i) denotes the banach space of real-valued continuous functions on i and the associated sup norm. for convenience the proposed system (11) can be rewritten in the equivalent form given below. by applying the caputo fractional integral operator, the above system (12) , reduces to the following integral equation of volterra type with the caputo fractional integral of order 0 < α < 1. now we prove that the kernels g 1 ; g 2 ; g 3 ; g 4 ; g 5 ; g 6 and g 7 fulfil the lipschitz condition and contraction under some assumptions. in the following theorem, we have proved for g 1 and can proceeds for the rest in a similar patron. theorem 2. let us consider the following inequality 0 ≤ π i ηp h αk 4 + ξ < 1. the kernel g 1 satisfies the lipschitz condition as well as contraction if the above inequality is satisfied. proof. for s and s 1 we proceed as below: since a(t) and q(t) are bounded functions, i.e, a(t) ≤ k 1 and q(t) ≤ k 2 , by the property of norm function, above inequality can be written as . here v 1 = {β(k 1 + ηk 2 ) + c} which implies that hence, for g 1 the lipschitz condition is obtained and if an additionally 0 ≤ β(k 1 + ηk 2 ) + c < 1, we obtain a contraction. the lipschitz condition can be easily verified for the rest of the cases and given as follows: recursively, the expressions in (13) can be written as the difference between successive terms of system (12) in recursive form is given below: with below initial conditions s 0 (t) = s(0), c 0 (t) = c(0), e 0 (t) = e(0), a 0 (t) = a(0), q 0 (t) = q(0), h 0 (t) = h(0), and r 0 (t) = r(0). taking norm of the first equation of (17), we obtain using lipschitz condition (14) we obtained thus, we have similarly, for the rest of equations in system (12) we obtained from above we can write that now, we claim the following result which guaranteed the uniqueness of solution of model (12) . theorem 3. the proposed fractional epidemic model (12) has a unique solution for t ∈ [0, t ] if the the following inequality holds proof. as we have shown that the kernels condition given in (14) holds. so, by considering the eqs. (20) and (29) , and by applying the recursive technique we obtained the succeeding results as below: therefore, the above mentioned sequences exist and satisfy φ 1n (24) and employing the triangle inequality for any k , we obtain where 1 γ(α) b α v i < 1 by assertion and t i = 1 γ(α) v i b α n , i = 1, 2, ..., 7. therefore, s n , c n , e n , a n , q n , h n , r n are regarded as the cauchy sequence in g(j). hence, they are uniformly convergent as described in [20] . applying the limit theory on eq. (16) when n → ∞ indicates that the limit of these sequences is the unique solution of the model (12) . ultimately, the existence of a unique solution for (12) has been achieved. in view of the fact that most of the fractional differential equations (fdes) do not have exact analytic solutions, so approximation and numerical techniques must be employed. numerous analytical and numerical methods have been put up to solve the fdes. for numerical solutions of the system (11) one can apply the generalized adams-bashforth-moulton method. to provide the exact solution by means of this algorithm, we take into account the following nonlinear fractional differential equation [19] : this equation is equivalent to volterra integral equation: diethelm and neville [7] employed the predictor-correctors scheme based on the adams-bashforth-moulton algorithm [6] . so we use again this same technique to find the solution of the projected model (11) . for α ∈ [0, 1], 0 ≤ t ≤ t and setting h = t /n and t n = nh, for n = 0, 1, 2, ..., n ∈ z + , the solution of the projected model is where and theorem 4. the numerical method (28)-(29) is conditionally stable. proof. lets 0 ,s j (j = 0, ..., n + 1) ands p n+1 (n = 0, ..., n − 1) be perturbations of s 0 , s j and s p n+1 , respectively. then, the following perturbation equations are obtained by using eqs. (20) and (29) using the lipschitz condition, we obtain where ζ 0 = max 0≤n≤n {|s 0 | + h α m a n,0 γ(α+2) |s 0 |}. also, from eq. (3.18) in [20] we derive where γ 0 = max{ζ 0 + h α m a n+1,n+1 γ(α+2) η 0 }. c α,2 is a positive constant only depends on α (lemma 4) and h is assumed to be small enough. applying lemma 2 concludes |s n+1 | ≤ cγ 0 . which completes the proof. we replace the integer order model to the fractional order model to study the true cameroonian data. the advantage to use caputo fractional derivative to study the current true data is to predict the epidemic peaks more clearly at different fractional order values α. parameter values use here are found in table 2 , and for which the basic reproduction number, r 0 is equal to 3.41. for initials conditions, we take s(0) = 26548238, c(0) = e(0) = a(0) = h(0) = r(0) = 0, q(0) = 1, and a fixed time step size of h = 2 −8 . the value of the fractional-order derivatives are α = 1, 0.9, 0.8, 0.7, 0.6, 0.5. firstly, figures have been analysed at integer order (denoted by red line) for the given model and then we analyse the nature of the compartmental model at different fractional values to give more peak predictions for the real-time data. it has been observed that onset of the epidemic peaks is delayed when the value of order α decreases. in figures 8 and 9 , one can see that when α −→ 1 the solutions of our fractional model (11) converge to the solutions of the integer model (2) . from above simulations, it can be conveyed that there are so much uncertainties of epidemic peaks for the given real-time data range. caputo derivative works well to study these biological phenomenons. robust forecasting results of covid-19 outbreak can be achieved by applying our model structure as imperfect quarantine and unsound confinement which are harsh realities in developing countries have been taken into account meticulously. a compartmental mathematical model has been formulated and studied to predict the evolution of the novel coronavirus disease in cameroon. after presenting the model with integer derivatives, the basic reproductive number r 0 of the model has been computed, which measures the number of secondary covid-19 positive cases caused by a single infectious person in a completely susceptible cameroonian population. the dynamics of the disease is impacted by the value of this epidemic threshold. indeed, if r 0 < 1, then a sufficiently small influx of infected individuals will not generate an outbreak of the covid-19, and if r 0 > 1, then a sufficiently small influx of infected individuals will generate an outbreak of the covid-19. using real data communicated by the cameroonian government, our model has been calibrated using a newly developed optimization algorithm based on wellknown trust-region-reflective (trr) algorithm. with the result of this model calibration, the probable peak date is estimated to be on august 01, 2020 with approximately 1073 (95% ci : 714 − 1654) daily new confirmed cases, and which corresponds to approximately ∼ 20, 100 (95% ci : 17, 343 − 24, 584) cumulative infected cases. as of july 11, according to the recent statistics communicated by the health authorities of cameroon, nearly 14,916 confirmed cases have already been identified. this difference between the "real data" communicated by the government and the predictions of our model can be explained by the fact that the fear that this disease creates within populations pushes the wealthy individuals towards private hospitals which not only give false test results but overbidding the treatment of this disease [8] . global sensitivity analysis has been performed implementing partial rank correlation coefficient (prcc) analysis to quantify the most crucial parameters, which significantly impact the progression dynamics of covid-19. it has been unearthed in our analysis that the direct infectious contact rate (β) and the efficacy of confinement measures ( ) are the most influential parameters in controlling the outbreak of covid-19. this ignites the public health implication that by reducing physical contact between 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differential equations the authors thank the editor and the anonymous reviewer for their comments which permitted us to improve the manuscript. this work does not have any conflict of interest. no funding is received for this study. table 2 key: cord-333162-gwmvsoru authors: malki, zohair; atlam, el-sayed; hassanien, aboul ella; dagnew, guesh; elhosseini, mostafa a.; gad, ibrahim title: association between weather data and covid-19 pandemic predicting mortality rate: machine learning approaches date: 2020-07-17 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110137 sha: doc_id: 333162 cord_uid: gwmvsoru nowadays, a significant number of infectious diseases such as human coronavirus disease (covid-19) are threatening the world by spreading at an alarming rate. some of the literatures pointed out that the pandemic is exhibiting seasonal patterns in its spread, incidence and nature of the distribution. in connection to the spread and distribution of the infection, scientific analysis that answers the questions whether the next summer can save people from covid-19 is required. many researchers have been exclusively asked whether high temperature during summer can slow down the spread of the covid-19 as it has with other seasonal flues. since there are a lot of questions that are unanswered right now, and many mysteries aspects about the covid-19 that is still unknown to us, in-depth study and analysis of associated weather features are required. moreover, understanding the nature of covid-19 and forecasting the spread of covid-19 request more investigation of the real effect of weather variables on the transmission of the covid-19 among people. in this work, various regressor machine learning models are proposed to extract the relationship between different factors and the spreading rate of covid-19. the machine learning algorithms employed in this work estimate the impact of weather variables such as temperature and humidity on the transmission of covid-19 by extracting the relationship between the number of confirmed cases and the weather variables on certain regions. to validate the proposed method, we have collected the required datasets related to weather and census features and necessary prepossessing is carried out. from the experimental results, it is shown that the weather variables are more relevant in predicting the mortality rate when compared to the other census variables such as population, age, and urbanization. thus, from this result, we can conclude that temperature and humidity are important features for predicting covid-19 mortality rate. moreover, it is indicated that the higher the value of temperature the lower number of infection cases. cently, researchers found that a notable association be22 tween the weather variables (temperature and humidity) 23 and the regions that have major covid-19 outbreaks. 24 moreover, these regions are located at the same temper-25 ature zone in the northern hemisphere [4] . 26 even though the pandemic is becoming a global is-27 sue, the most infected areas include outbreak epicentres 28 such as parts of northeastern united states, chinas cen-29 tral province of hubei, south korea, japan, iran, italy, 30 spain, germany, and england, all of which share an av-31 erage temperature of 5c to 11c and 47% to 79% hu-32 midity in january and february 2020. for italy, regions 33 with a temperature higher than 15 degrees celsius and 34 75% humidity have less spread of covid-19 cases. 35 therefore, we hypothesise that the spread of the virus 36 will decrease in the area with higher temperature and 37 humidity than areas with average records. from our exmelin et al. [6] conducted a study to analyze the 73 spatial evolution of coronavirus pandemic around the 74 world using unsupervised neural network namely self-75 organizing maps. the researchers concluded that the 76 clustering abilities of self-organizing maps enable to 77 group countries based on covid-19 confirmed cases. the main contribution of our work includes: • find the best predictive model for daily confirmed to regions that are below this threshold (see figure 6 ). based on the experimental results as shown in table 379 2, we have proved that climatic conditions such as temtable 3 . these models help to unin table 1 . clinical 537 features of patients infected with 2019 novel coronavirus 538 in wuhan, china potential for global spread of a novel coron-544 avirus from china geographic information systems and covid-548 19: the johns hopkins university dashboard temperature and latitude 554 analysis to predict potential spread and seasonality for covid-555 19 covid-19 spreading in rio de janeiro, brazil: 559 do the policies of social isolation really work? anal-564 ysis of spatial spread relationships of coronavirus 565 (covid-19) pandemic in the world using self organiz-566 ing maps temperature, population and longitu-571 dinal analysis to predict potential spread for covid-19 temperature de-575 creases spread parameters of the new covid-19 case dynamics demystifying a hidden trend: do tempera-579 ture variations affect covid-19 virus spread? seira 607 model for the spread of covid-19 among heterogeneous 608 populations on a 613 comprehensive model of the novel coronavirus (covid-19) 614 under mittag-leffler derivative real-time forecasts and risk assess-619 ment of novel coronavirus (covid-19) cases: a data-driven 620 analysis a 624 model based study on the dynamics of covid-19: prediction 625 and control multiple ensem-630 ble neural network models with fuzzy response aggregation for 631 predicting covid-19 time series: the case of mexico, health-632 care covid-19 jhu data web scrap and cleaning funda-650 mentals of machine learning introduction to scikit-learn ehsanes saleh, mohammad arashi, in-660 troduction to ridge regression introduction to ensemble learn-665 ing predicting hard rock pil-669 lar stability using gbdt, xgboost, and lightgbm algorithms short-term weather 673 forecast based on wavelet denoising and catboost chinese control conference (ccc) a comparative study of prediction 679 and classification models on ncdc weather data, inter-680 national getting started with scikit-learn for machine learn-685 ing decision tree classifier: a detailed sur-689 review of ordinary least 696 squares and generalized least squares credit author statement zohair malki: supervision and project administration guesh dagnew: data curation, writing-original draft preparation key: cord-346185-qmu1mrmx authors: velásquez, ricardo manuel arias; lara, jennifer vanessa mejia title: forecast and evaluation of covid-19 spreading in usa with reduced-space gaussian process regression date: 2020-05-22 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109924 sha: doc_id: 346185 cord_uid: qmu1mrmx in this report, we analyze historical and forecast infections for covid-19 death based on reduced-space gaussian process regression associated to chaotic dynamical systems with information obtained in 82 days with continuous learning, day by day, from january 21(th), 2020 to april 12(th). according last results, covid-19 could be predicted with gaussian models mean-field models can be meaningfully used to gather a quantitative picture of the epidemic spreading, with infections, fatality and recovery rate. the forecast places the peak in usa around july 14(th) 2020, with a peak number of 132,074 death with infected individuals of about 1,157,796 and a number of deaths at the end of the epidemics of about 132,800. late on january, usa confirmed the first patient with covid-19, who had recently traveled to china, however, an evaluation of states in usa have demonstrated a fatality rate in china (4%) is lower than new york (4.56%), but lower than michigan (5.69%). mean estimates and uncertainty bounds for both usa and his cities and other provinces have increased in the last three months, with focus on new york, new jersey, michigan, california, massachusetts,... (january e april 12(th)). besides, we propose a reduced-space gaussian process regression model predicts that the epidemic will reach saturation in usa on july 2020. our findings suggest, new quarantine actions with more restrictions for containment strategies implemented in usa could be successfully, but in a late period, it could generate critical rate infections and death for the next 2 month. able on the center for systems science and engineering at johns hopkins university [6] , the available data analyzed is considered between january 21 th 2020 and april 39 12 th 2020, included, with a feedback process in a neural network applied; it allows 40 to examined the information in real time in each state, at fig. 1 • . with covid-19, the spatial representation of the disease by using gis plat-73 form allows to verified the "material, population and social psychology at three 74 scales: individual,group and regional" [9] ; in this case, with gis technology is 75 necessary to implement big data techniques, for cross validations and analysis. • associated to gpr the idea is to "utilize nonlinear diffusion map coordinates 79 and formulate a deterministic dynamical system on the system manifold" [12] . a interesting approach has a reduced-space data-driven dynamical system with 81 an evaluation, with efficiency with low intrinsic dimensionality. • an "advantage of employing gpr to reconstruct the reduced-order dynamics 83 is the simultaneous estimation of the dynamics and the associated uncertaint" with an input xn ∈ r and "noisy scalar output yn" [11] . for infection population, where: where: 119 θ 1 : is a hyper-parameter with maximum covariance in chaotic systems. 132 in eq. (9), bayes rule is written with a normalized process to find (f, f * ) in eq. (10). in eq. (11), associates to "conditioning the joint gaussian prior distribution on 134 the observations, resulting in the closed-form gaussian distribution" [12] . with eq. (12), f * , the mean and covariance should be directly added to obtain eq. (13). 136 finally, in eq.(14) makes feasible to use up to more than twelve thousands of training x ∈ r in the eq. (16), individually node in the hidden layer has "linear combination" [18] 152 with a chain events. the hidden layers are composed as following: in eq. (18) describes a matrix of weights, it is analogous to regression coefficient in where: 172 h: it is a description of the hidden layer, due to restriction of the "linearly trans-173 formed and passed to the output layer". however, china has implemented robust strategies for distancing control and quar-206 antine around this country, therefore, the recovery rate is 90%. far away from new 207 york, with recovery rate of 9.42% and on the other hand, cities as texas is 13.09%. besides, hawaii has different rate of 56.79% for the restrictions initiated with more 209 influence in their population. combining a mathematical model with multiple datasets, fig. 6 and fig. 7 , we 211 found that the median daily rt of covid-19 in usa probably varied between 2.6 212 and 5.6 in april, 2020, before 3 weeks of travel restrictions were introduced. we in traditional methods, we detect mistakes in the forecast associated to old pan-269 demic researches, and as we can see, the covid-19 has behaved in unpredictable 270 and changing ways, according to the factors used in its treatment. therefore, we 271 think that the pandemic behaves as a dynamic-chaotic system, which sets new start-272 ing points that do not intersect and changes its behavior according to the way it 273 is treated. consequently, this study could be a real contribution for understanding 274 this problem with a good tool of processing a systematic review. as we can see 275 in the fig. 9 , the example started with three countries, with three initial states. 276 hence they have been evolving together until some days, but after a month, they grants/financial support. none. credit authorship contribution statement. conflict of interest. there is no conflict of interest in this work. isolation, quarantine, social distancing 325 and community containment: pivotal role for old-style public health measures in 326 the novel coronavirus (2019-ncov) outbreak the covid-19 pandemic in the usa: what might we 328 expect? potential impact of 330 seasonal forcing on a sars-cov-2 pandemic real-time forecasts of the covid-19 epidemic in china from february analysis and forecast of covid-19 spreading in china an interactive web-based dashboard to track 338 covid-19 in real time real-time estimation and prediction of mortality caused by covid-19 with 342 patient information based algorithm propagation analysis and prediction of the covid-19, infectious disease mod-347 elling covid-19: challenges to gis with big data why is it 354 difficult to accurately predict the covid-19 epidemic? gaussian processes in machine learning. the mit reduced-space gaussian process re-359 gression for data-driven probabilistic forecast of chaotic dynamical systems, 360 mit industrial liaison program nonparametric forecasting of low-362 dimensional dynamical systems converting data into knowledge for preventing failures in power transformers corrosive sul-367 phur effect in power and distribution transformers failures and treatments, engi-368 neering failure analysis pattern recognition and image preprocessing removal of impulse noise in color images 373 based on convolutional neural network gradient-based learning applied to 376 document recognition an overview of deep learning in medical imag-378 harmonic failure 381 in the filter of static var compensator automatic diagnosis of skin diseases 1 acknowledgments. key: cord-259846-oxbmtend authors: naik, parvaiz ahmad; zu, jian; owolabi, kolade m. title: global dynamics of a fractional order model for the transmission of hiv epidemic with optimal control date: 2020-06-18 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109826 sha: doc_id: 259846 cord_uid: oxbmtend in this paper, a nonlinear fractional order epidemic model for hiv transmission is proposed and analyzed by including extra compartment namely exposed class to the basic sir epidemic model. also, the infected class of female sex workers is divided into unaware infectives and the aware infectives. the focus is on the spread of hiv by female sex workers through prostitution, because in the present world sexual transmission is the major cause of the hiv transmission. the exposed class contains those susceptible males in the population who have sexual contact with the female sex workers and are exposed to the infection directly or indirectly. the caputo type fractional derivative is involved and generalized adams-bashforth-moulton method is employed to numerically solve the proposed model. model equilibria are determined and their stability analysis is considered by using fractional routh-hurwitz stability criterion and fractional la-salle invariant principle. analysis of the model demonstrates that the population is free from the disease if [formula: see text] and disease spreads in the population if [formula: see text]. meanwhile, by using lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. furthermore, for the fractional optimal control problem associated with the control strategies such as condom use for exposed class, treatment for aware infectives, awareness about disease among unaware infectives and behavioral change for susceptibles, we formulated a fractional optimality condition for the proposed model. the existence of fractional optimal control is analyzed and the euler-lagrange necessary conditions for the optimality of fractional optimal control are obtained. the effectiveness of control strategies is shown through numerical simulations and it can be seen through simulation, that the control measures effectively increase the quality of life and age limit of the hiv patients. it significantly reduces the number of hiv/aids patients during the whole epidemic. epidemiology mainly deals with the infectious diseases and predicts their occurrence, transmission as well as control in a population. it identifies the factors responsible for disease spread, facilitates treatment quality and health services, provides necessary measures for prevention, treatment, planning in order to improve the efficiency and effectiveness of health services [1] . hiv is a retrovirus which is discovered in 1981 in usa among the gay community causes an aids a severe life intimidating ailment. at present, there is no vaccine or cure for aids, that makes it an incurable disease with high mortality rate (there are almost 25 million deaths by aids per year worldwide), also it spread quickly affecting about 14,0 0 0 new case/day. the time duration for hiv to as an attractor. wang et al. [4] studied a delayed fractional order sir model with saturated incidence and treatment functions. they have provided the sufficient conditions that guarantee the existence of equilibria and discussed the global stability results for both disease-free equilibrium as well as endemic equilibrium by constructing a suitable lyapunov functions. almeida [5] in his paper studied a fractional seir epidemic model in presence of treatment. he analysed the model and his main focus was on the fractional differential equations in order to describe the dynamics of certain epidemics. further, he proved the local stability for both equilibria. carvalho et al. [6] provided a hiv/hcv coinfection fractional order model to understand the impact of hiv viral load on the coinfection. their main motive in the model was to provide good fits to real data from patients suffering from several diseases such as hiv, hcv, dengue fever and many more. they have numerically suggested that the hiv viral load impacts impressively the severity of the hcv infection. also, by their results they showed that the treatment efficacy is also influential over the natural progression of hcv on the hiv/hcv coinfection. recently, kheiri and jafari [7] analysed a multi-patch hiv/aids epidemic model with fractional order derivatives and investigated the effect of human movement on the spread of hiv/aids epidemic among patches. they derived the basic reproduction number r 0 of the model and studied the local as well as global stability of the equilibria on the basis of r 0 . they have shown that the system is stable if r 0 < 1 and it becomes unstable if r 0 > 1 . they also obtained the sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. besides this, they formulated a fractional optimal control problem in which the state and costate equations are given in term of the left fractional derivatives. they incorporated in the model time dependent controls in order to control the spread of hiv/aids epidemics. they also derived the necessary conditions for the fractional optimal control in their proposed model. the effect of varying the fractional order on the disease spread is also studied in their model. researchers have continuously studied the fractional order models of hiv disease dynamics and provided many well-known mathematical techniques for the solution of these models for the dynamics of hiv epidemics [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] . besides this, a number of studies on fractional order modeling of other infectious diseases can be found in the literature [20] [21] [22] [23] . the fractional order derivative not only find its application on modeling infectious diseases but in other fields as well like vibration equation [24] and so on. the optimal control theory is developing fast and its various applications are extensively used in many fields of science and engineering [25] . this theory for linear systems has been highly improved [26] , however, the nonlinear optimal control problem (ocp) has become a strong topic and should be deeper investigated [27] [28] . jajarmi and baleanu [29] proposed a new approach based on the modal series method and eigenvalue decomposition technique to solve a class of nonlinear optimal control problems. they have also investigated the convergence analysis of their suggested technique. jajarmi et al . [30] proposed a new approach for the optimal control of time-varying delay systems with external persistent matched disturbances. in their internal model principle, they converted original time-delay model with disturbance into an augmented system without any disturbance. then, they selected a quadratic performance index for the augmented system to form an undisturbed time-delay optimal control problem. the necessary optimality conditions are then derived in terms of a two-point boundary value problem involving advance and delay arguments. at the end they finally provided a fast iterative algorithm for the latter advance-delay boundary value problem. they also investigated the convergence of the new iterative technique. the purpose of dealing with fractional order systems is the memory and hereditary properties which are the complex behav-ioral patterns of biological systems gives us more realistic way to model hiv/aids systems. in the fractional order models, the memory property allows the integration of more information from the past which predicts and translates the models more accurately. also, the hereditary property describes the genetic profile along with age and status of the immune system. because of such properties fractional order calculus have found wide applications to model dynamics processes in many well-known fields of science, engineering, biology, medicine and many other [31] [32] . saeedian et al. [33] formulated sir epidemic model with the inclusion of memory effect and studied its behavior along the memory effect on the disease spread with the help of fractional derivatives. rihan [34] provided a class of fractional order differential models of biological systems with memory, such as dynamics of tumor-immune system and dynamics of hiv infection of cd4 + t cells. communicable diseases have been a cause of global concern throughout the history of mankind. its outbreak severely affects the morbidity and the mortality rates across the globe. it is therefore important to implement the control measures to prevent and control the disease spread among the populations. kheiri and jafari [35] formulated a fractional optimal control epidemic model of hiv/aids with random testing and contact tracing. in their model, they have incorporated the control measures of condom use and antiretroviral therapy for the control of spread of hiv/aids in the susceptible population. they have presented a forward-backword sweep numerical method based on adams-bashforth-moulton method for the solution of their model. agrawal [36] formulated a fractional optimal control problem by using the reimann-liouville fractional derivatives and presented a numerical method for its solution. bashir et al. [37] presented a fractional optimal control for a kinetic model and provided a numerical scheme for its solution. going by the antecedents, we have seen clearly that modeling of physical and real-life scenarios with the fractional order derivatives is much more accurate when compared with the integer order cases. this assertion has been demonstrated a number of research papers, monographs and books, see for example [38] [39] [40] [41] . in view of these achievements, we are motivated in this research work by modeling the control and analysis of sei 1 i 2 r dynamics of hiv disease transmission using the caputo fractional order operator which is most suited for modeling the biological and physical facts [42] [43] [44] [45] [46] [47] . the choice of using the caputo derivative is due to the fact that, if the given function is a constant, then the caputo derivative of that function gives zero. primarily, the caputo operator computes an ordinary differential equation, followed by a fractional integral to obtain the desired order of fractional derivative. more importantly, the caputo fractional differential equation (fdo) permits the use of local initial conditions to be included in the derivation of the model. in the present paper, we propose and analyze a fractional optimal control problem, in which the state and co-state equations are given in terms of the caputo fractional derivatives. this approach simplifies the use of fractional numerical methods to solve the state and co-state equations. fractional optimal control problems can be regarded as a generalization of classic optimal control problems for which the dynamics of the control system are described by fractional differential equations. we incorporate into the model time dependent controls such as condom use for exposed individuals, treatment for infected female sex workers, awareness about the disease among unaware infectives and behavioral change for susceptibles in order to reduce the risk of the spread of hiv/aids disease. conditions for fractional optimal control of the disease are derived and the state and co-state equations are characterized by caputo fractional derivatives. the numerical solution of the proposed fractional optimal control problem is obtained by using generalized adams-bashforth-moulton method. furthermore, the efficacy of order of fractional derivative, the control strategies and the value of objective functional is investigated. the structure of the paper is designed as: in the next section 2 , some preliminary results required for the formulation of mathematical model is provided. development of the proposed mathematical model and its well-posedness is discussed in section 3 . in section 4 , we discuss the mathematical analysis of the proposed fractional order sei 1 i 2 r epidemic model along with equilibrium points and the stability of equilibrium points. in section 5 , the fractional optimal control problem is formulated and discussed. also, in this section, the necessary conditions for the optimality of proposed fractional optimal control problem is provided. furthermore, in section 6 , application of the generalized adams-bashforth-moulton method is performed on the proposed model and the numerical simulations are done to validate the analytical studies. in section 7 , numerical results are given to illustrate the capability of generalized adams-bashforth-moulton method and the behavior of the obtained solutions is also discussed in this section. finally, section 8 concludes all the major findings of the present research study. researchers have continuously extended the definitions of fractional order derivatives like the riemann-liouville, the caputo, caputo-fabrizio, atangana-baleanu, the grunwald-letnikov, the weyl, the marchaud, the riesz, and the miller and ross [48] [49] [50] [51] [52] . recently, many new definitions of fractional derivative [53] have hugely evolved, going from the derivatives with nonsingular kernel and new riemann-liouville fractional derivative without singular kernel to the two-parameter derivatives with non-singular and non-local kernel [54] [55] [56] . definition 2.1. a real function ψ( t ), t > 0 is said to be in the space c η , η ∈ r , if there exists a real number l > η, such that ψ (t) = t l ψ 1 (t) , where ψ 1 ( t ) ∈ c [0, ∞ ) and it is said to be in the space where κ > 0 and (.) is a well-known gamma function. (3) definition 2.4. the caputo fractional derivative of ψ( t ) order κ > 0 is defined as where the operator c 0 d κ t satisfies the following two basic properties: the definition 2.3 and definition 2.4 are not equivalent to each other, and their difference is expressed by the caputo operator c 0 d κ t , has advantages for differential equations with initial values. in the case of riemann-liouville and caputo derivatives, respectively, the initial values are usually given as [57] rl assume that the function d κ t ψ (t) , satisfies some smoothness conditions in every finite interval (0, t ), t ≤ t . choosing the grid 0 = τ 0 < τ 1 < ... < τ n +1 = t = ( n + 1 ) u, τ n +1 − τ n = u, and using the classical notation of finite differences, the laplace transform of the caputo fractional derivative of ψ( t ) order κ > 0 is defined as definition 2.6. the laplace transform of the function where e κ, κ 1 is the two-parameter mittage-leffler function with κ, κ 1 > 0. further, the mittage-leffler function satisfies the following equation [58] e κ, κ 1 to describe the transmission dynamics of hiv epidemics, we have generalized the basic sir epidemic model by including more compartments, to one in which population is divided into five sub-classes, the susceptible population s ( t ), the exposed population e ( t ), the infective population that don't know they are infected i 1 ( t ), the infective population that know they are infected i 2 ( t ), by means of medical screening or otherwise and recovered population r ( t ). the proposed model is considered as the generalization of the original kermack-mckendrick model [8] , where only three compartments were considered, but here the exposed compartment is included contains those susceptible males in the population who have sexual intercourse with the female sex workers as a result by having sexual contact they are exposed to the infection. furthermore, the infected class is divided into two sub-classes namely infected female sex workers who are unaware about their disease status and the infected female sex workers who knows their disease status. thus, the model takes the following form [ 3 , 15 , 18 ] . for the understanding of hiv disease dynamics, the total population n ( t ) is divided into five sub-population compartments namely susceptible, exposed, infected but unaware, infected but aware and recovered such that the following description is associated to the above classical model: the susceptibles are recruited at a rate , β 1 is the per capita rate for susceptibles individuals with unaware infectives, β 2 is the per capita rate for susceptibles individuals with aware infectives, λ is the natural death rate unrelated to aids, σ is the break through into infected class, θ is the rate of unaware infectives to become aware infectives by screening or testing, ρ is the rate by which types of infectives develop aids and d is the aids related death rate. it may further be noted that we further extend the above ordinary differential model to the following fractional order system of order κ, with σ , ρ > 0 being the rate that exposed individuals become infectious and recovery rate, respectively and λ ≥ 0 being the infection related death rate. the purpose of considering the fractional order case is the significant uniqueness of these varieties of fractional order systems with non-local characteristics (memory) and hereditary properties that have not been seen with the integer-order differential operators which widely exists in biology. also, using fractional order differential equations can help us to reduce the errors arising from the neglected parameters in modelling real life phenomena. in each case, we replace the ordinary derivative by a fractional derivative. thus, our proposed fractional order model for hiv disease transmission has the form subject to the initial conditions and if κ = 1 , then system (9) reduces to an integer order system (8) . it is clear that the variable r ( t ) does not appear in the first four equations, thus it is meaningful to consider the reduced system (9) as: (11) subject to the positive initial conditions here, it is assumed that the functions and their caputo fractional derivatives are continuous at t ≥ 0. the existence, uniqueness, and non-negativity of the solution of system (11) are analyzed. the schematic diagram of the proposed fractional order sei 1 i 2 r epidemic model (9) is shown in fig. 1 . in this section, we first prove the existence and uniqueness of positive solution, then the basic reproduction number and the existence conditions for both equilibria (disease-free equilibrium and endemic equilibrium) are obtained, finally, the conditions for the stability of both the equilibria are obtained. let us denote r 4 for the proof of the main theorem about the non-negativity of the solutions, we recall the following lemma [ 3 , 8 , 15 ] . this completes the proof. t for the initial value problem given by (11) along initial conditions (12) on t ≥ 0 in (0, κ) and the solution will remain in r 4 + . furthermore, the solutions are all bounded. proof. according to lin [58] from the theorem 3.2 [58] and remark 3.2 [58] , we can determine the solution on (0, ∞ ) by solving the model (11) along initial conditions (12) which is not only existent but also unique. subsequently, we have to explain the nonnegative domain r 4 + , is positively invariant region. from model (11) , we find on each hyperplane bounding the non-negative orthant, the vector field points into r 4 + . furthermore, from system (11) thus, by lemma 4.1 , in the case of hiv infection, the total population n ( t ), i.e., the subpopulations s ( t ), e ( t ), i 1 ( t ) and i 2 ( t ) are bounded. by positivity means the population survives and boundedness refers as a natural restriction to growth as a consequence of limited resources. this completes the proof of the theorem 4.1 . therefore, the biologically feasible region for the system (9) is for the equilibrium points, setting the right-hand side of the system (11) equal to zero, we obtain equilibrium points as after simplification, the system (13) gives the disease-free equithus, the proposed nonlinear fractional order sei 1 i 2 r epidemic model has at most two equilibria namely disease-free equilibrium in order to study the local stability of the disease-free equilibrium, we first compute the basic reproduction number by using next generation matrix method [61] [62] [63] [64] (11) can be written as by the next generation matrix method, the matrices ғ and ѵ at the disease-free equilibrium point − d 0 are obtained by where ғ is non-negative and v is a non-singular m-matrix. therefore, the basic reproduction number denoted by r 0 which is considered as the spectral radius of the next generation matrix v −1 at the disease-free equilibrium − d 0 is thus given by it shows that if r 0 < 1 , then the disease does not spread in the population and the infection dies. on the other hand, if r 0 > 1 , then the disease persists in the whole population. now, we will discuss the local stability analysis of equilibrium points. for this, we state the results in the form of theorems and prove them. proof. to prove the above theorem 4.2 , the general jacobian matrix and the matrices corresponding to each equilibrium point will be obtained. therefore, the jacobian matrix is given by therefore, by the routh-hurwitz stability conditions for fractional order systems [65] , the necessary and sufficient condition for various fractional order models. therefore, the disease-free equilibrium of system (11) is asymptotically stable if all of the (14) . hence, a sufficient condition for the local asymptotic stability of the equilibrium points is that the eigenvalues γ this confirms that fractional order differential equations are, at least, as stable as their integer order counterpart. by solving the characteristic equation, the eigenvalues can be obtained as the simplification allows us to get the following algebraic equation this implies, therefore, the roots of the characteristic equation are , satisfy the condition given by (14) . therefore, all the eigenvalues have negative real parts if r 0 < 1 . this completes the proof of the theorem 4.2 . in the next theorem 4.3 , we discuss the local asymptotic stability of the endemic equilibrium of the system given by (11) . proof. the jacobian matrix of the system (11) evaluated at endemic equilibrium − d * is given as the characteristic equation of the linearized system is in the form now, the discriminant of the polynomial − p (γ ) = γ 3 + ϑ 1 γ 2 + ϑ 2 γ + ϑ 3 is described by [ 3 , 12 , 15 ] and using the construction of results by ahmed et al. [ 19 , 66 ] , following fractional routh-hurwitz conditions associated with are observed. we have the following result. i if d (− p ) > 0, then the necessary and sufficient condition for the equilibrium point to be locally asymptotically stable is ϑ 1 > 0, the global existence of the solution of the fractional differential equation always becomes a most important concern, which is carry out in the following section. theorem 4.4. [ 12 , 58 ] , assume that the function : r + × r 4 → r 4 satisfies the following conditions in the global space: 1) the function ( t, ψ( t )) is lebesgue measurable with respect to t on r . 2) the function ( t, ψ( t )) is continuous with respect to ψ( t ) on ( t, ψ ( t ) ) ≤ α 1 + α 2 ψ (t) , for all most every t ∈ r and all ψ (t) ∈ r 4 . here α 1 , α 2 are two positive constants and has a unique solution. proof. from the theorem 4.4 , we obtain the unique solution on (0, ∞ ) by solving the system (11) . firstly, lin [58] discussed the proof of theorem and shows that the solution is not only exist but also unique. in theorem 4.1 , we already proved that the solution of model (11) will remain in r 4 + . ( [ 52 , 67 ] ) let ψ (t) ∈ r + be a continuous and derivable function. then, for any time instant t ≥ 0, and where κ ∈ (0, 1). note that for κ = 1 , the inequalities in (20) and (21) becomes equalities. now, we provide the global stability results of the equilibria in the following theorems by considering the lyapunov direct method. (11) is globally asymptotically stable in , if r 0 ≤ 1 and unstable when r 0 > 1 . to prove this, we define a lyapunov function φ 1 ( t ) given using the disease-free steady state condition of model (11) , s 0 = λ , we have from the equation (22) as in addition, we know that c 0 d κ t φ 1 (t) | ( 11 ) = 0 , if and only if s(t ) = s 0 and i 1 (t) = 0 .substituting i 1 (t) = 0 into (11) , one can directly obtain e(t ) = 0 . using i 1 (t) = e(t ) = 0 again in (11) , then i 2 (t) = 0 . therefore, the maximum invariant set for { ( s, e, i 1 , i 2 ) ∈ 0 : c 0 d κ t φ 1 (t) | ( 11 ) = 0 } is the singleton set -d 0 . according to the lasalle's invariance principle [61] [62] [63] [64] , we know that all solutions in 0 converge to -d 0 .therefore, the disease-free steady state of model (11) is globally asymptotically stable when r 0 ≤ 1 . this completes the proof of the theorem 4.6 . (11) is globally asymptotically stable in , when r 0 > 1 . to prove this, we define a lyapunov function φ 2 ( t ) given using the endemic conditions, therefore, φ 2 ( t ) is bounded and non-increasing. further, the limit of φ 2 ( t ) exits as t → ∞ . in addition, we know that c lasalle's invariance principle [61] [62] [63] [64] , we know that all solutions in * converge to − d * .therefore, the endemic equilibrium of proposed model (11) is globally asymptotically stable when r 0 > 1 . this completes the proof of the theorem 4.7 . in this section, we extend the basic model (8) by including some particular control measures aimed at controlling the spread of the hiv infection and formulate the fractional optimal control problem by proposing the control objectives. the aim of the control measures is to reduce the infection in the population and thus there is the need to formulate the optimal control problem to achieve this goal. the first control function v 1 ( t ) represents the behavioral change for susceptibles which reduced the number of exposed by a factor ( 1 − v 1 (t) ) . the control v 1 ( t ) is proposition of the susceptible individuals who change their sexual habits per unit of time. the second control function v 2 ( t ) is the use of condoms to the exposed individuals who are going to have sexual interaction with the female sex workers. the third control function v 3 ( t ) represent the enhancement of the strength of treatment for the infected individuals. the fourth control function v 4 ( t ) is the awareness source among the unaware infectives about their disease status. under these control measures the proposed model (8) all formulas and models should be left aligned . (24) with the non-negative initial conditions the control is completely effective when v i (t) = 1 and the control is not effective when v i (t) = 0 , for i = 1 , 2 , 3 , 4 i.e., 0 ≤ v i ( t ) < 1. our focus is to minimize the number of exposed individuals under the cost of applying control measures which can be done by consider the following fractional optimal control problem to minimize the objective functional given by subjected to the state system given in (24) along non-negative initial conditions (25) . in eq. (26) , q represent the positive weight constant of the exposed population, while -p 1 , -p 2 , -p 3 , and -p 4 are positive weight constants for behavioral change, personal protection, treatment strategy and awareness source respectively. the subjected to the state system given in (24) , where the control set is defined as the lagrangian ɫ and hamiltonian h for the fractional optimal problem (24) (28) are respectively given by [ 35 , 68-69 ] (29) and this further implies, where λ s , λ e , λ i 1 , λ i 2 and λ r are the adjoint variables. we have to prove the necessary conditions for the optimality of the fractional system (24) . for the optimal control v ( t ), that minimizes the performance index (31) subjected to the dynamical constraints (32) with initial conditions π ( 0 ) = π 0 (33) where π ( t ) and v ( t ) are the state and control variables, respectively, l and ω are differentiable functions, and 0 < κ ≤ 1. we have the following theorem. if ( π , v ) is a minimizer of (31) under the dynamic constraint (32) and the boundary condition (33) , then there exists a function λ such that the triplet ( π , v, λ) satisfies proof. for the proof of theorem 5.1 , readers are suggested to see [ 7 , 35-36 ] , where the authors have given the proof in detail. this completes the proof of the theorem 5.1 . and r * be optimal state solutions with associated optimal control variables v * 1 , v * 2 , v * 3 , v * 4 for the optimal control problems (24) and (26) . then there exist adjoint variables λ s , λ e , λ i 1 , λ i 2 and λ r satisfying with transversality conditions or boundary conditions furthermore, the control functions v * 1 , v * 2 , v * 3 and v * 4 are given by all form ulas and mode ls shou ld be left alig ned . proof. the adjoint system (35) i.e., λ s , λ e , λ i 1 , λ i 2 and λ r are obtained from the hamiltonian h as in this section numerical solution for the proposed fractional order sei 1 i 2 r epidemic model (9) is presented. because no analytical solution to the nonlinear fractional system (9) is available, we use the technique so-called generalized adams-bashforth-moulton method [ 3 , 35 , 37 ] to obtain the numerical solution of the system (9) . in this algorithm, we derive the predictor-corrector scheme for obtaining the numerical solution of the nonlinear fdes. to provide the estimated solution by means of this algorithm, consider the subsequent nonlinear fractional differential equation [ 3 , 35 , 37 ] with the following initial conditions now, with operating by the fractional integral operator on the equation (37) , we can obtain on the solution ψ( t ) by solving the following equation: this equation (39) is equivalent to the volterra integral equation. diethelm et al. [70] [71] [72] used the predictor-corrector scheme based on the adams-bashfort-moulton algorithm to integrate (39) . setting h = t n , t n = nh and n = 0 , 1 , 2 , ..., n ∈ z + , the equation (39) can be discretized as follows: where a q,n +1 = all form ulas and mode ls shou ld be left alig ned . (41) and the predicted value ψ p h ( t n +1 ) is determined by the error estimate is in which p = min (2 , 1 + κ ) . in this subsection, we solve numerically the nonlinear fractional sei 1 i 2 r epidemic model using the proposed method. in view of the generalized adams-bashforth-moulton method, the numerical scheme for the proposed model (9) is given in the following form [73] [74] [75] [76] [77] further, the quantities , r h ( t q )), are computed from the following functions, in addition, the quantities are computed from equations (43) (47) respectively, at the points t n +1 , n = 1 , 2 , 3 , ..., m. for the fractional optimal control problem (24) discussed in section 5 , similar procedure is followed for the numerical results. therefore with π = s, e, i 1 , i 2 , r and the control v . similarly, for the adjoint system we have the coefficients a q,n +1 , b q,n +1 are given by equations (41) and (42) respectively. in order to justify our theoretical findings, we introduced in this section some numerical experiments obtained for different instances of fractional power κ for the hiv epidemic model without control (9) and with control (24) along with adjoint variable systems and the control strategies. we present the numerical results for the model (9) when all control measures are absent and also to examine the role of fractional order κ on the hiv disease spread. then, we simulate the fractional optimal control of the model and investigate the effect of the controls introduced in the model on the spread of epidemics. we use the generalized adams-bashforth-moulton method for the simulation of both the systems and use the values of parameters described in table 1 . in this subsection, we present numerical results for fractional system (9) and allow the values of κ to varies from k = 0 . 75 to k = 1 as seen in the fig. 2 . it is clear from the fig. 2 that fractional order has significant effect on dynamic behavior of all the components. we observe that when the derivative order κ is reduced from 1, the memory effect of the system increases, and therefore the infection grows slowly and the number of hiv-infected population and aids people increases for a long time. also, undiagnosed hiv-infected population in some societies refuse to per-form hiv test for reasons such as stigma and fear of identification due to lack of knowledge about the disease. this results in a delay in identification of hiv-infected individuals, an increase in the undiagnosed hiv-infected population, fast progress of aids and an increase in people diagnosed with aids. on the other hand, the experience or knowledge of individuals about the disease causes susceptible and exposed individuals to take different precautions, such as behavioral change, vaccination, treatment and condom use, against infection transmission. this leads to a slow growth of infection among the population. therefore, from the numerical results in fig. 2 , we conclude that the derivative order κ (0.75 ≤ κ ≤ 1) can play the role of precautionary measures against infection transmission, treatment of infection and delay in accepting hiv test. existence of attractors for some fractional order κ for different population groups are given in fig. 3 . thus, the results from fig. 3 shows that there is tendency of each population class to exist and enter into permanence with time. numerical results for the difference of integer-order and fractional order are given in figs. 4-5 . it is clearly visible from figs. 4-5 that the differential equations with fractional order derivative have rich dynamics and describe biological systems better than traditional integer-order models. from the above discussion and numerical results in figs. 2-5 , we conclude that the derivative order κ can play the role of experience or knowledge of individuals about the past of the disease. therefore, the numerical results confirm that differential equations with fractional order derivative have rich dynamics and describe biological systems better than traditional integer order models. as a result, our numerical results are more logical than the results of other articles on the modeling of the hiv/aids epidemic and other models with integer-order derivative due to the presence of the fractional derivative order κ (0.75 < κ ≤ 1). now, we investigate the effect of the control measures introduced in the model on the spread of the epidemic. we consider the following strategies and examine the corresponding numerical results. in this subsection, we present the numerical results for the model (24) when all control measures are present. the results are obtained in different ways by applying control strategies in the following five ways. in the first control strategy, we set the control measures v 3 = 0 , v 4 = 0 and active the control measures v 1 = 0 . 5 , v 2 = 0 . 5 namely the behavioral change for susceptible individuals and condom use for the exposed individuals which is shown in fig. 6 , for different values of fractional order κ. analysis of control strategy-1 predicts that susceptible and exposed individuals greatly decrease after implementing control measure v 1 , v 2 . in fig. 6 , we observe that this control strategy results in a significant decrease in the number of undiagnosed hiv-infected population and aids people for a long time compared with the case without control. fig. 6 shows that, by applying the strategy-1, the value of r 0 will be less than 1 for more time when κ is reduced from 1. this means that by decreasing κ from 1, we can control the spread of disease over a longer period of time. therefore, the presence of the fractional derivative order κ in the model increases the use of condom control and behavioural control in the population. the control v 1 is proportion of the susceptible individuals who change their sexual habits per unit time. the class r , the removed class, represents the number of people who have greatly changed their sexual habits such that they cannot easily be infected through sexual contact. people in the class r take on safe habits and keep these habits in the rest of their lives. the importance of class r is that it emphasizes the need for prevention for a disease like hiv that has no treatment. therefore, increasing the members of this class plays an important role in controlling the spread of disease. strategy-2. using only condom use control in the second control strategy, we set the control measures v 1 = 0 , v 3 = 0 , v 4 = 0 and active the control measure v 2 = 0 . 5 namely the condom use for the exposed individuals which is shown in fig. 7 , for different values of fractional order κ. analysis of control strategy-2 predicts that exposed individuals greatly decrease after implementing control measure v 2 . the results in fig. 7 , further show that condom use is the main control measure which can be helpful in controlling the disease more properly. this is because the control is applied to exposed class which is the main source from which the virus can transmit and spread due to the fact that this class is easily available for virus during their sexual contact with infected female sex workers. strategy-3. using only treatment control in the third control strategy, we set the control measures v 1 = 0 , v 2 = 0 , v 4 = 0 and active the control measure v 3 = 0 . 7 namely the efficiency of treatment given to the aware infected individuals which is shown in fig. 8 , for different values of fractional order κ. analysis of control strategy-3 predicts that infected individuals greatly decreases after implementing control measure v 3 . this is due to the fact that treatment of diagnosed hiv-infected population results in an increase in the level of cd4 + t-cells of this class. therefore, this strategy prolongs the lifespan of hiv-infected patients and delays the onset of aids. fig. 8 shows that differential equations with fractional order derivative have rich dynamics and describe biological systems better than traditional integerorder models. strategy-4. using treatment control and awareness control in the fourth control strategy, we set the control measures v 1 = 0 , v 2 = 0 and active the control measures v 3 = 0 . 5 , v 4 = 0 . 5 namely the efficiency of treatment given to the aware infected individuals and the awareness source for unaware infected individuals which is shown in fig. 9 , for different values of fractional order κ. analysis of control strategy-4 predicts that infected individuals greatly decreases after implementing control measures v 3 and v 4 . in fig. 9 , we observe that this control strategy results in a signif-icant decrease in the number of aware infected hiv people and unware infected people compared with the case without control. in last control strategy, we activate all the control measures v 1 = 0 . 5 , v 2 = 0 . 5 , v 4 = 0 . 5 and v 4 = 0 . 8 namely the behavioral change for susceptible individuals, condom use for exposed individuals, efficiency of treatment given to the aware infected individuals and the awareness source for unaware infected individuals which is shown in fig. 10 , for different values of fractional order κ. analysis of control strategy-5 predicts that susceptible individuals and exposed individuals decreases with the control measures v 1 and v 2 while infected individuals greatly decrease after implementing control measures v 3 and v 4 . by adding the behavioral change control v 1 or condom use control v 2 to the art treatment control v 3 , we see from figs. 6-9 , that the strategies 1-4 result in a decrease in the hiv infected population and aids people compared with the case without control. with implementing all the control effort s, we observe that the strategy-5 results in a significant decrease in the hiv-infected population and aids people compared with the case without control. with comparison of the strategies, we see that the strategy-5 is better than the other strategies in control and reduction of the spread of hiv/aids epidemic. therefore, by applying the strategy-5, we can increase the life time and the quality of life for those living with hiv and decrease significantly the number of hiv-infected population and aids people. on the other hand, in human societies, the process of evolution and control of the epidemic is associated with memory. when a disease spreads in a society, the experience or knowledge of individuals about the past of the disease helps susceptible individuals to take different precautions, such as behavioural change, treatment, awareness and condom use against infection transmission. also, the experience or knowledge can lead to the screening measures of entry and exit between different groups. it is noticeable from fig. 10 that due to the memory property of fractional derivatives, the derivative order κ affects the values of the controls. we see that the maximum levels of the controls are reduced when κ limits to 1. on the other hand, the memory effect characterized by fractional derivative is reduced when κ limits to 1. therefore, by reducing the memory effect, the maximum levels of the controls are reduced. in the current study, we have introduced a nonlinear sei 1 i 2 r fractional order epidemic model for the transmission dynamics of hiv epidemics. the non-negative solution of the model is provided by using the generalized mean value theorem. we obtained the basic reproductive number r 0 , which perform as a threshold parameter in the disease status. the existence of equilibria and their asymptotical stability results using fractional routh-hurwitz stability criterion is discussed. we established and investigated the stability analysis of the fractional order model with respect to the values of r 0 . the disease-free equilibrium is locally asymptotically stable if r 0 ≤ 1 . for r 0 > 1 , using theorem 4.3 and corollary 4.1 , we investigated the local stability of the positive endemic equilibrium state − d * . meanwhile, global asymptotic stability of the disease-free and endemic equilibrium point is investigated by constructing a suitable lyapunov functions. additionally, we investigated the optimal control problem by the application of the optimal control theory. we used the pontryagin's minimum principle to provide the necessary conditions needed for the existence of the optimal solution to the optimal control problem. furthermore, generalized adams-bashforth-moulton method is applied to obtain a numerical solution of the proposed fractional order sei 1 i 2 r epidemic model (9) and the fractional optimal problem (24) . the re-sults obtained shown that the adams-bashforth-moulton method is an accurate and effective technique for obtaining the numerical solution of the proposed nonlinear fractional order sei 1 i 2 r epidemic model. lastly, the theoretical results are verified by numerical simulations to measure the efficacy and impact of controls on the transmission of the hiv/aids disease. from the numerical simulation, the size of the exposed population is significantly reduced under the controlled conditions. this proposes that if all four control measures v 1 (behavioral change for susceptibles), v 2 (condom use by the exposed individuals), v 3 (strength of treatment for the infected individuals), v 4 (awareness source among the unaware infectives) are employed for the same period of time and continue for a considerable period of time, the spread of hiv disease through prostitution could be restricted. in this manner, the fractional order optimal control method can progress the value of the necessary control measures. this recommends that personal precautional measures, periodic monitoring by medical professionals and researchers should be done to control the transmission of the hiv disease dynamics. the recently emerged virus namely novel coronavirus (covid-19) which has originated from wuhan the capital city of hubei providence of mainland china in december 2019 is a major threat to mankind at present in the whole world. application of fractional order derivatives to model the new outbreak of coronavirus and other trending diseases are left for future research. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. modeling the role of acquired immune response and antiretroviral therapy in the dynamics of hiv infection dynamics of an sir model with nonlinear incidence and treatment rate hiv/aids epidemic fractional-order model dynamic analysis of a delayed fractional-order sir model with saturated incidence and treatment functions analysis of a fractional seir model with treatment hcv coinfection model: a fractional-order perspective for the effect of the hiv viral load stability analysis of a fractional order model for the hiv/aids epidemic in a patchy environment contributions to the mathematical theory of epidemics. iii-further 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adams-bashforth scheme with atangana-baleanu-caputo fractional derivative to model chaotic problems numerical solution of some fractional dynamical systems in medicine involving non-singular kernel with vector order the first author is very grateful to xi'an jiaotong university for the assistant professor position provided to him. also, the authors would like to thank the reviewers and editors of this paper for their careful attention to detail and constructive feedback that improved the presentation of the paper greatly. the study was supported by grants from the china postdoctoral science foundation (grant nos. 2019m663653 and 2014m560755), the national natural science foundation of china (grant nos. 11971375, 11571272, 11201368 and 11631012), the national science and technology major project of china (grant no. 2018zx10721202) and grant from the natural science foundation of shaanxi province (grant no. 2019jm-273). the funding body did not play any roles in the design of the study and in writing the manuscript. key: cord-337256-b3j3kg73 authors: wang, peipei; zheng, xinqi; li, jiayang; zhu, bangren title: prediction of epidemic trends in covid-19 with logistic model and machine learning technics date: 2020-07-01 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110058 sha: doc_id: 337256 cord_uid: b3j3kg73 covid-19 has now had a huge impact in the world, and more than 8 million people in more than 100 countries are infected. to contain its spread, a number of countries published control measures. however, itâs not known when the epidemic will end in global and various countries. predicting the trend of covid-19 is an extremely important challenge. we integrate the most updated covid-19 epidemiological data before june 16, 2020 into the logistic model to fit the cap of epidemic trend, and then feed the cap value into fbprophet model, a machine learning based time series prediction model to derive the epidemic curve and predict the trend of the epidemic. three significant points are summarized from our modeling results for global, brazil, russia, india, peru and indonesia. under mathematical estimation, the global outbreak will peak in late october, with an estimated 14.12 million people infected cumulatively. an outbreak of atypical pneumonia [coronavirus disease 2019 (covid19)] caused by severe acute respiratory syndrome coronavirus 2 (sars-cov-2) since late december 2019 has made huge impact on people life and work. the virus may spread from bats to humans through another intermediate host and cause 5 severe respiratory syndrome [1] , characterized by strong human-to-human transmission through the air [2, 3] . the world health organization (who) declared an international emergency on 31 january, 2020. since initial identification, despite of strict control, now it becomes a pandemic in global, which is a big threat and challenge to world health and economy [4] , the disease has spread 10 to over 100 countries across the world ( figure 1 ). as of 3:33 pm on 16 june, 2020, a total of 8,044,683 covid-19 cases have been reported worldwide, with 437,131 deaths and 3,883,243 survivors, with an overall case fatality rate of 5.43% [5] . john hopkins university is offering the current data [6] . the infectivity of covid-19 is greater than that of influenza, with an estimated r0 15 value (the basic reproduction number, representing viral infectivity) of 2.28 [7] . therefore, it is of striking significance to predict the pandemic trends of infection worldwide. many scholars have developed a number of predicting methods for the trend forecasting of covid-19, in some severe countries and global [8, 9] , debating 30 about mathematical model, infectious disease model, and artificial intelligence model. the models based on mathematical statistics, machine learning and deep learning have been applied to the prediction of time series of epidemic development [10, 11] . logistic is often used in regression fitting of time series data due to its simple principle and efficient calculation. for example, in the 35 coronavirus case, logistic growth is characterized by a slow increase in growth at the beginning, fast growth phase approaching the peak of the incidence curve, and a slow growth phase approaching the end of the outbreak, i.e., the maximum of infections. wu et al. [12] the rest of this paper is structured as follows. in section 2, data sources and our proposed method are introduced detailed. experiments and results analysis are given in section 3. section 4 presents the discussion of our research. finally, in section 5, the main contribution of this paper is summarized. across the whole of china retrieved from an archived news-site (sohu) [14] was used for logistic training. the logistic model originated from the modeling of population growth in 85 ecology [15] . as an improvement on the malthus population model [16] , in 1838, pierre franois verhulst published the logistic equation: where q, r and k indicate the population size, intrinsic growth rate and maximum population size that the environment could carry, respectively. dq/dt represents the growth of the population. r and k are constants number and is a trend function used to analyze the non-periodic changes of time series. s(t) a periodic term, reflecting the periodic change, such as the periodicity of a week or a year. h(t) is the influence of an occasional day or days, such as a holiday. t we create an instance of the prophet class and then call its fit and predict methods. the input to prophet is always a time series with two features: date 135 ds and value y. here in our study, ds is the date of day, and y is the accumulated cases in a particular country. in this paper, we integrate the most updated covid-19 epidemiological data before jun 16, 2020 into the logistic model to fit the cap of epidemic to modeling the logistic growth of covid-19, q, r and k in equation logistic growth is characterized as follows [23] : where q(t) is the number of cases at time t, a is constant, b could be considered 150 as incubation rate and k is the cap value, the maximal number of cases for q(t). therefore, the number of cases at the very beginning is k/(1 + a), and the key point is ln a/b at which the cumulative situation curve turns, when rapid increases in the number of cases are replaced by slow increases. we initialize a,b and k randomly and update it by using nonlinear least squares [24] . the if the current day in time series is less than t f ast , it means the key point is still ahead and the growth is increasing maybe exponential. otherwise, it means that virus spread has been controlled and growth is going towards the end. we set the time t max with maximum cases is three times of t f ast for the first kind of growth while it is more 20 days than current day for the second kind of growth. then the estimated top number of infections q top can be calculated as follows: the aforementioned top number of infections q top will be feed into prophet model with actual time series data. we perform around five to six months ahead forecasting by using prophet, with 95% prediction intervals and logistic growth 180 type. no tweaking of seasonality-related parameters and additional regressors are performed. in three time series are constructed from our collected data, namely confirmed 195 cases, recovered cases and death cases series. in our experiment, we assume that each of these three sequences has a peak, in other words, the epidemic will end eventually. obviously, the number of active confirmed cases is equal to the number of accumulated confirmed cases minus the number of recovered and deaths. we first apply logistic model to fit the curve and calculate the 200 time with fastest growing rate, then use prophet to make a prediction. it is worth noting that there are three significant points in our forecasting results, as is described in table 1 . the first is the maximum number of existing infections (the time when the blue line reach peak in figure 5 ), i.e., epidemic epidemic peak point this peak means that the active infections has reach the top value and since then, the number of active cases will decrease. the fastest growth point after this point, the epidemic gradually slows down and finally becomes stabled. turn point this point occurred when the number of cumulative cured exceeds the number of active confirmed cases, marks an early victory in the control of the epidemic. we chose brazil, russia, india, peru and indonesia as the forecast countries, worldwide. we could summarize our basic predictions as follows: as is shown in table 2 , the fastest growth point of those five countries has already passed. with current intervention, the total epidemic size in brazil is values were compared separately ( figure 6 ). the results found that there was overall a good fit between our projected and reported data. according current management capacity, especially in icu care [26] . once the outbreak exceeds the combination of national health resources, it will take a long time to recover. in our predicted results, by the late july or mid of august 2020, the healing rate will increase and brazil will have about 1.7 million confirmed cases at the end. our study highlighted another key point, the strict control measures adopted 1.35%, 4.90% and 2.98%, respectively and there are still some increasing cases every day (figure 7) . because of the rapid outbreak in south american and european countries, the medical system nearly collapsed in a short period of time [27] . after strict control, the growth rate of the epidemic gradually slowed down, and the days when the cumulative cure was greater than the existing 285 diagnosis came in mid-june, which means that south american and european countries still have a long way to fight in strictly controlling the outbreak. while more data is needed to make more detailed predictions, these models could help predict future confirmed cases if the spread of the virus does not change in a way beyond expectation. as we all know, this virus is new and has 290 the ability to spread seriously. this characteristic may affect all our predictions, but to our best knowledge at the time we spent writing this paper, the proposed model is effective. in this article, a forecasting method with logistic and prophet model is however, as is shown in figure 6 , all of our predictions are based on the assumption of there will be a maximum of outbreak, and the epidemic curve is modeled based on a full logistic curve. in real world, there maybe some small 305 peak during the pandemic due to different intervention of the government and different public cooperation. besides, when we forecasting the epidemic in some countries, the effects of input cases and spatial influence between countries are not taken into account. to address the aforementioned limitations, the following aspects are worthy evolutionary history, potential intermediate animal host, and cross-species analyses of 330 sars-cov-2 clinical characteristics of coronavirus disease 2019 in china early transmission dynamics in wuhan, china, of novel coronavirusinfected pneumonia will covid-19 generate global preparedness? coronavirus disease (covid-19) situation report 147 an interactive web-based dashboard to track covid-19 in real time estimation of the reproductive number of novel coronavirus (covid-19) and the prob-350 able outbreak size on the diamond princess cruise ship: a data-driven analysis modified seir and ai prediction of the epidemics trend 355 of covid-19 in china under public health interventions modelling and predicting the spatio-temporal spread of coronavirus disease 2019 (covid-19) in italydoi: dx analysis and forecast of covid-19 spreading in china, italy and france statistical analysis of forecasting covid-19 for upcoming month in pakistan generalized logistic growth modeling of the covid-19 outbreak in 29 provinces in china and in the rest of the world forecasting at scale combatting sars (in chinese) simulation of rice biomass accumulation by an extended logistic model including influence of meteorological factors an essay on the principle of population sars epidemiology modeling lstm network: a deep 385 learning approach for short-term traffic forecast an improved neural network-based approach for short-term wind speed and power forecast multiple-instance learning approach via bayesian extreme learning machine automatic forecasting procedure logistic model-based forecast of sales and generation of obsolete computers in the u.s an adaptive nonlinear least-squares algorithm covid-19 projections arboviral diseases and covid19 in 405 brazil: concerns regarding climatic, sanitation and endemic scenario, arboviral diseases and covid19 in brazil: concerns regarding climatic, sanitation and endemic scenariodoi the who regional office for europe, the european observatory on health systems, policies, covid-19 health system response monitor 410 (hsrm) there is no conflict of interest in this work. key: cord-355419-8txtk0b3 authors: feng, liang; zhao, qianchuan; zhou, cangqi title: epidemic in networked population with recurrent mobility pattern date: 2020-06-25 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110016 sha: doc_id: 355419 cord_uid: 8txtk0b3 the novel coronavirus (covid-19) has caused a global crisis and many governments have taken social measures, such as home quarantine and maintaining social distance. many recent studies show that network structure and human mobility greatly influence the dynamics of epidemic spreading. in this paper, we utilize a discrete-time markov chain approach and propose an epidemic model to describe virus propagation in the heterogeneous graph, which is used to represent individuals with intra social connections and mobility between individuals and common locations. there are two types of nodes, individuals and public places, and disease can spread by social contacts among individuals and people gathering in common areas. we give theoretical results about epidemic threshold and influence of isolation factor. several numerical simulations are performed and experimental results further demonstrate the correctness of proposed model. non-monotonic relationship between mobility possibility and epidemic threshold and differences between erdös-rényi and power-law social connections are revealed. in summary, our proposed approach and findings are helpful to analyse and prevent the epidemic spreading in networked population with recurrent mobility pattern. recently, the novel coronavirus (covid-19) has caused a global crisis and more than 4 million people of more than 200 countries have been infected up to now (14th may, 2020) [1] . the primary measures taken by many governments are home quarantine and maintaining social distance, with the aim to break transmission of this infective virus and halt the spread of pandemic. although most public places are locked down, supermarkets and drugstores which are essential for daily life remain open. different from commonly used homogeneous mixing approaches [2, 3] , we give an analysis of epidemic spreading in population following a structured network with recurrent mobility pattern in this work. the influences of network structure and human mobility on epidemic spreading have received lots of attention in recent years. on one hand, homogeneous mixing assumption among individuals is often invalid and variations of social interaction network bring large differences in the propagation process of virus [4] [5] [6] [7] [8] [9] [10] . on the other hand, human mobility also greatly affects the peak and duration of epidemic outbreak [11] [12] [13] [14] [15] , and recurrent patterns between people and their familiar locations (e.g. workplace, supermarket) often dominate the behaviour of human mobility [16, 17] . however, the interactions of individuals in public places provide another route of virus transmission, which makes the analyse of epidemic spreading much more difficult. one widely used approach to analyse epidemic spreading in complex networks is metapopulation model, which divides the whole population into several geographical structured parts [13, 18] , and contacts among individuals in the same subpopulation are assumed to be well-mixed. though some challenges remain [19] , a useful strategy of analysing the impact of human mobility behaviour on epidemic spreading is to integrate metapopulation model with reaction-diffusion process [20, 21] . each subpopulation can be formulated as a node and edges between different metapopulations indicate the mobility probabilities of individuals. reaction-diffusion process has been widely studied in physics, for epidemic spreading, diffusion often refers to individual movements between different places and reaction indicates the contagion process within each place after human behaviour. in addition to the way of using a unipartite network to model metapopulations for analysis of epidemic spreading with recurrent mobility pattern, heterogeneous networks are adopted recently [22, 23] . all these methods take the assumption individuals in the same subhttps://doi.org/10.1016/j.chaos.2020.110016 0960-0779/© 2020 elsevier ltd. all rights reserved. population will contact with each other in a well-mixed fashion, however, each individual only interacts with his neighbors in real social networks. in this paper, we utilize a heterogeneous network to represent the social connections among individuals and recurrent human mobility pattern between population and locations. there are two types of connections: one is the edge between any two individuals, and the other is connection between individual and common areas. the social network may follow different network structures, such as erdös-rényi or power-law networks. in order to model the dynamics of virus spreading, we apply the discrete-time markov chain method in the context of susceptible-infected-susceptible (sis) infection process like [8, 20, 23, 24] . with a mobility possibility, each agent will choose to get into public places which he connects to or stay in social network. since there are no fixed individuals and connections among them, we assume people gathering in a public area will have contacts with all other individuals. the agent who chooses to stay in social network will have contacts with his remaining neighbors, and he may get infected if and only if he has some contagious neighbors who do not go outside at that time step. in this paper, detailed formulation of epidemic model in this kind of networked population with recurrent mobility pattern is introduced, and we also give theoretical results of the epidemic threshold. besides, the decay of epidemic infection below threshold and impact of isolation factor are presented. as far as we know, this is the first attempt to analyse epidemic spreading in networked population with human recurrent mobility by using social contacts network among individuals. the remaining of this paper is organized as follows. in section 2 , we give the formulation of epidemic model for virus spreading in networked population with recurrent mobility pattern, along with theoretical results of epidemic threshold. the analyse of epidemic threshold from the view of non-linear dynamical system (nlds) and its decay property are introduced in section 3 . experimental results on two types of social networks, erdös-rényi and power-law networks are showed in section 4 , and we conclude this work with a summary in section 5 . the heterogeneous graph of networked population in our paper consists of two different parts. one is composed of m individuals with specific social network structure, and the other is n public places. different from previous works [20, 21, 23] which only consider connections between metapopulations and common areas, we discard the well-mixed assumption in metapopulation and suppose there is a contact network among individuals. we formulate an epidemic model of virus propagating in networked population with recurrent mobility pattern between individuals and public areas. the social connections are defined by a m × m matrix a , where a ij is 1 if individual i has a contact with agent j . the edges between individuals and locations are dependent on a m × n matrix b , where b ij is 1 if agent i will visit place j if he goes out, and 0 otherwise. here, we assume edges are unweighted, and undirected, hence a is a symmetric matrix. in order to simulate the recurrent mobility pattern between individuals and public places, at each time step, individuals will go to all public places which they connect to with a possibility of p . after movements of individuals, virus propagates in both remaining networked population and common areas independently. we also force individuals in common places to return back to social network at the end of step, for the purpose of ensuring recurrent mobility patterns of individuals. fig. 1 gives an example of epidemic spreading at time t when some individuals get into public places with possibility p . it should be noted that, the edges of both a and b for a specific heterogeneous network remain unchanged during the whole simulation. however, at each time step, since some individuals will go outside, the remaining a and resulting infection process based on b in common areas might be different. for dynamics of epidemic spreading among individuals, we adopt the popular sis model. there are two states, susceptible and infective, for each individual. we assume the homogeneous spreading of virus where the infection rate is β and recovery rate is μ. a susceptible individual will become infected by a possibility of β when contacting with a contagious agent, and individuals who got infected at previous time steps will recover with possibility μ and become susceptible again at each time step. the states of population at t + 1 are only dependent on those at t , thus a discretetime markov chain can be taken to model the dynamics of epidemic spreading. the possibility individual i is infective at the beginning of time t is denoted as p i,i,t , and susceptible p i,s,t where p i,s,t = 1 − p i,i,t for sis model. the evolution of p i,i,t can be formulated as where the left part is possibility when i has already got infected and remains infective, and right is possibility that susceptible individual i becomes infective at time t where i ( t ) is the infection possibility. different from formulation in traditional networked populations [5, 6, 22, 24, 25] , i ( t ) consists of two different components where p is the mobility possibility, d i ( t ), c i ( t ) are infection possibilities when agent i stays in social contact network or gets into public places. in this paper, we define the possibility of a susceptible individual getting infected when interacting with k contagious people as where we use a i j = 1 for neighbors of i only in the contact network a and assume probabilities p j,i,t are independent of each other. for individuals in common areas, we take the assumption of well-mixed fashion used in [3, 13, 20, 21, 23] , which means every one will have a contact with each other in the same place. therefore, we can get where j refers to a public place and b ij takes the value of 1 for places j which individual i connects to. combining eq. 2 and eq. 3 into i ( t ), eq. 1 can be reformulated as when whole population is near the critical onset of epidemic outbreak, the infective possibility for each individual and corresponding infection probability are negligible which means p i,i,t+1 = p i,i,t = p i,i 1 and β 1. by using approximations (1 − β ) n ≈ 1 − nβ, and neglecting high-order terms o ( β 2 ), we can reduce eq. 4 into if we use p i = [ p 1 ,i , p 2 ,i , · · · , p m,i ] t to indicate infected possibilities of all m individuals, eq. 5 can be easily rewritten as hence, the infection threshold of epidemic spreading can be obtained [5, 6] where λ max ( q ) is the largest eigenvalue of matrix q . similar to [23] , we consider the impact of isolation factor γ where 0 ≤ γ ≤ 1 which constrains infected individuals from going into common places and they can only have contacts with social neighbors who do not go outside. therefore, we can get i,t and the epidemic threshold with isolation factor is in the last section, we derive the epidemic threshold for networked population with recurrent mobility pattern of individuals. on one hand, when p equals 0, no individual goes out and virus spreads only through contact network a , eq. 7 turns into situations in [5, 6, 24, 25] . on the other hand, when all individuals get into corresponding common areas which means p = 1 , eq. 7 becomes dynamics of epidemic spreading in bipartite networks as discussed in [3, 22] . here we give some theoretical analysis of epidemic threshold from the view of non-linear dynamical system ( nlds ) and introduce the exponential decay property of infective individuals when β is below epidemic threshold. for sis model, the transitions of individual i can be described by (9) when no one gets infected in system, the equilibrium point is p i = [0 , 0 , · · · , 0] t and p s = [1 , 1 , · · · , 1] t where the numbers of 0 and 1 are both m . according to [26] where we can easily get j | p where i is the identity matrix and q = β(1 − p) 2 a + β p 2 bb t . therefore, the system is asymptotically stable at equilibrium point p if all the eigenvalues are less than 1 in absolute value. assume the eigenvector is [ v 1 , v 2 ] t , and corresponding eigenvalue is λ j , we can get following equations although when v 2 = 0 , λ j = 1 , it is related to ∂p s,t+1 ∂p s,t which does not cause instability when system is at equilibrium point p . then we just need λ max ((1 − μ) i + q ) < 1 which means (11) hence the obtained epidemic threshold is consistent with result of eq. 7 . by using the same strategy, we can also easily prove that epidemic threshold of sir model is the same as sis model. recall that we use approximations (1 − β ) n ≈ 1 − nβ, if we take the high-order terms o ( β 2 ) into account, the following results can be obtained by combining above results with eq. 4 , dynamic of infection possibility for individual i becomes and transitions for all people satisfy which indicates λ i,w < 1 for every i , therefore where c is a constant. hence, the values of p i will exponentially decrease over time if β is less than the epidemic threshold. in order to validate the correctness of proposed model, we evaluate results from monte carlo simulations with theoretical predictions of eq. 4 . two different types of network structures are used in the following experiments: erdös-rényi network and power-law network. we keep the edges between individuals and locations fixed, which means b is unchanged while changing structure of social connections a among individuals. in order to better analyse the impact of mobility possibility p on different networks, total number of edges among individuals remains almost the same for erdös-rényi and power-law network. a comparison between simulation and theoretical results is showed in fig. 2 . the average infected ratio for the whole population is defined as η = m j p j,i /m. we run 20 times for each mobility possibility p . for each simulation, we run 400 time steps and use average value of η in the last 100 steps as the steady infection ratio. as we can see, numerical solutions of proposed model have good correspondences with results of monte carlo simulations. for small mobility possibility p = . 2 , the epidemic threshold of powerlaw network is significantly smaller than that of erdös-rényi net-work. as explicated in [27] , the heterogeneity of degree distribution in power-law network makes the largest eigenvalue larger than erdös-rényi network, and this makes virus more easily break out among individuals. when p increases, the differences between erdös-rényi and power-law network both in epidemic threshold and steady infection ratio become less obvious. this can be explained by the fact that, when more individuals go outside, more and more spreads of epidemic take place in common areas while the connections b between individuals and common locations are the same for both social contact networks. in the next, we analyse the impact of recurrent mobility possibility p on epidemic threshold β min with different numbers of contact edges. from fig. 3 , all curves show non-monotonic behaviours where epidemic threshold achieves its largest value β min for a specific value of p . similar non-monotonic phenomena are also founded in [20, 23] . also, the largest epidemic thresholds in power-law network are smaller than those in erdös-rényi network. besides, when total edges of social contacts ( e ) increase, β min decreases and p increases for the synthetic networks. after that, we investigate how isolation factor γ influence epidemic spreading and the results are plotted in fig. 4 . when γ is 1, there are no isolations for infected individuals, and the curve behaves consistently with fig. 3 . for small isolation factors ( γ < 0.04 in fig. 4 ) , the restriction of infected individuals makes β min monotonically increase with mobility possibility. interestingly, in erdös-rényi network, all curves nearly intersect at one point and similar phenomenon is also founded in [23] which can be explained by the effective contact number of infected neighbors as discussed in [2, 4] , while curves in power-law network show more obvious dispersion due to the heterogeneity of degree distribution. at last, we demonstrate the total number of infected individuals for different infection rates β at different time steps in fig. 5 . as we can see, when β < β min , the spread dies out exponentially, and becomes an epidemic otherwise. the experimental results have good agreements with outcomes of eq. 13 . we can also find that, for the same scale of threshold, such as 2 β min and 5 β min , the number of steady infected individuals in power-law network is relatively smaller than erdös-rényi network. this indicates that although epidemic threshold in power-law network is usually smaller, there will be more infected population in erdös-rényi network due to the homogeneity of individual's social contacts with their neighbors when infection rate increases by the same scale. in this paper, we propose an epidemic model for networked population with recurrent mobility pattern. a heterogeneous network is used to represent the structure containing different connections. there are two types of edges, social connections among individuals and mobility connections between individuals and common areas. a detailed formulation by discrete-time markov chain method to describe the dynamics of epidemic spreading is given, and we derive theoretical results about epidemic threshold. several simulations on both erdös-rényi and power-law social networks are conducted and experimental results verify the correctness of our model and analysis. the non-monotonic relationship between epidemic threshold and mobility possibility indicates there is an optimal value which will make virus hard to spread. the influences of different values of isolation factor which restricts infected individuals from getting into common areas are analysed, and more obvious dispersion appears in power-law network because of heterogeneity of degree distribution. in addition, we demonstrate and prove the exponential decay of infection when epidemic rate is less than threshold. in summary, this pa-per not only provides an approach to model epidemic spreading in networked population with recurrent mobility pattern, but offers a tool to analyse different network structures and social measures, such as restricting infected individuals. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. mathematical tools for understanding infectious disease dynamics epidemic processes in complex networks mean field theory of epidemic spreading with effective contacts on networks epidemic thresholds in real networks epidemic spreading in real networks: an eigenvalue viewpoint mean-field modeling approach for understanding epidemic dynamics in interconnected networks discrete-time markov chain approach to contact-based disease spreading in complex networks dynamics and control of diseases in networks with community structure sir dynamics in random networks with communities human mobility patterns predict divergent epidemic dynamics among cities multiscale mobility networks and the spatial spreading of infectious diseases reaction-diffusion processes and metapopulation models in heterogeneous networks interaction between epidemic spread and collective behavior in scale-free networks with community structure optimal control and stability analysis of an epidemic model with population dispersal memory in network flows and its effects on spreading dynamics and community detection recurrent host mobility in spatial epidemics: beyond reactiondiffusion intervention threshold for epidemic control in susceptible-infected-recovered metapopulation models seven challenges for metapopulation models of epidemics, including households models critical regimes driven by recurrent mobility patterns of reaction-diffusion processes in networks the spreading of infectious diseases with recurrent mobility of community population vaccination and epidemics in networked populations-an introduction epidemic spreading in localized environments with recurrent mobility patterns analysis, estimation, and validation of discrete-time epidemic processes got the flu (or mumps)? check the eigenvalue! differential equations, dynamical systems, and linear algebra spectra of random graphs with given expected degrees this work is supported in part by national natural science key: cord-349841-eigcqb1b authors: boukanjime, brahim; caraballo, tomas; fatini, mohamed el; khalifi, mohamed el title: dynamics of a stochastic coronavirus (covid-19) epidemic model with markovian switching date: 2020-10-16 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110361 sha: doc_id: 349841 cord_uid: eigcqb1b in this paper, we analyze a stochastic coronavirus (covid-19) epidemic model which is perturbed by both white noise and telegraph noise incorporating general incidence rate. firstly, we investigate the existence and uniqueness of a global positive solution. then, we establish the stochastic threshold for the extinction and the persistence of the disease. the data from indian states, are used to confirm the results established along this paper. today, the world is facing the ongoing covid-19 pandemic, caused by the sars-cov2 coronavirus. the novel coronavirus has been a serious threat to public health [4] . in late december 2019, the disease covid-19 was first discovered in wuhan (hubei province) and caused the first pandemic of this century. the virus appears to be transferred mostly through narrow respiratory 5 droplets by coughing, sneezing, or peoples interaction in close proximity (usually less than one meter) with each other for a certain time frame. however, it might be possible that other unobserved environmental exposures may have facilitated the rate the disease spreads through human-to-human transmission. in [13] , it is reported that covid-19 infected individuals generally develop symptoms, including mild respiratory symptoms and fever, on an average of 5-6 days after infection (mean 5-6 days, range 1-14 days). at the present, there is no effective treatment for covid-19 in the world. therefore the only way to stop the spread of this disease is to quarantine or isolate the initially infected population as showed by guide line of world health organization. by the end of june 2020, the covid-19 virus has infected more than 10 927 025 people and died at least 521 512 in all over the world [27] . however, since the randomness of population mobility and uncertainty of 15 control measures, methods of predicting covid-19 and then preventing and controlling the disease for public health departments still remain unclear. recently, the novel coronavirus covid-19 has attracted much attention from many researchers and various comprehensions have been made to deepen understanding and grasping the valuable inferences through mathematical modeling [14, 15, 16] . therefore, it is of great significance to 20 establish and study the model of infectious diseases. mathematical modelling is an important decision tool that can be useful to analyze the spread and understand the level of manageability and the effect of prevention and control mechanisms applied to the pandemic. a numerous number of models are being used to project the current covid-19 pandemic. wang et al. [17] developed an seir model to estimate the epidemic trends in wuhan, assuming the prevention and control 25 measures were either sufficient or insufficient to control the epidemic. hellewell et al. [18] developed a transmission model and found that highly effective contact tracing and case isolation are enough to control a new outbreak of covid-19 within three months in most scenarios. in another recent work, chakraborty and ghosh [28] have considered a hybrid arima-wbf model to forecast various covid-19 affected countries throughout the globe. several other models established 30 a stochastic transition model to evaluate the transmission of covid-19 and also emphasized the necessity of interventions such as social-distancing, isolation and quarantine [19, 21] . in [24] , mandal et al. consider a mathematical model of covid-19 where human populations are subdivided into five time-dependant classes, namely, susceptible s(t), exposed e(t), quarantined q(t), hospitalized infected i(t) and recovered or removed r(t). they have assumed that the virus covid-19 is spreading when a vulnerable person comes into contact with an exposed person. the model is a system of five first order ordinary differential equations shown as below: where all parameters are positive numbers. a is the constant recruitment rate to the susceptible population; β stands for the disease transmission rate; ρ 1 (0 < ρ 1 < 1) is portion of susceptible human would maintain proper precaution measure and ρ 2 (0 < ρ 2 < 1) represents portion of the exposed class would take proper precaution measure for disease transmission (i.e, use of face mask, social distancing and implementing hygiene). therefore (1 − ρ 1 )s denotes portion of susceptible individuals due to the contact of (1 − ρ 2 )e portion of exposed individuals; α and b 2 are the portions of the exposed class going to the infected class and quarantine class, respectively. however, b 1 and c represent the portions of the quarantine class moving to susceptible class and infected class, respectively. η and ν stand for the recovery rate of hospitalized infected population i and exposed class e; µ denotes the natural death rate and δ is the covid-19 induced death rate. according to the theory in [24] , the system (1) always has a disease-free equilibrium r 0 = aβ(1 − ρ 1 )(1 − ρ 2 ) µ(b 2 + α + ν + µ) > 1, there exists a unique endemic 35 equilibrium e * = (s * d , e * d , q * d , i * d , r * d ), where s * d = b2+α+ν+µ β(1−ρ1)(1−ρ2) , e * d = (b 1 + c + µ) aβ(1−ρ1)(1−ρ2)−µ(b2+α+ν+µ) β(1−ρ1)(1−ρ2){b2(c+µ)+(α+ν+µ)(b1+c+µ)} , q * d = b 2 aβ(1−ρ1)(1−ρ2)−µ(b2+α+ν+µ) β(1−ρ1)(1−ρ2){b2(c+µ)+(α+ν+µ)(b1+c+µ)} , 40 i * d = {α(b1+c+µ)+b2c}{aβ(1−ρ1)(1−ρ2)−µ(b2+α+ν+µ)} β(1−ρ1)(1−ρ2){b2(c+µ)+(α+ν+µ)(b1+c+µ)}(η+µ+δ) , r * d = ηi * d +νe * d µ . [24] established the following theoretical results about the stability of the equilibriums: (i) if r 0 < 1, then the disease-free equilibrium e 0 of system (1) is locally asymptotically stable. (ii) if r 0 = 1, the system (1) passes through a transcritical bifurcation around its disease-free equilibrium. (iii) if r 0 > 1, then the endemic equilibrium e * of system (1) is locally asymptotically stable. in fact, the covid-19 epidemic model is unavoidably subjected to the environmental noise, which made the parameters involved in the system often fluctuate randomly around some average values as the surrounding environment fluctuation. see [1, 2, 3, 6, 9, 10, 20, 29] and references therein for epidemic models with environmental noise. therefore, it is necessary to include random fluctuations in the process of covid-19 modelling. in this paper, we propose a stochastic covid-19 model adopting a generalized incidence function [25, 26] as follows: where b(t) is a real-valued brownian motion and σ 2 > 0 represents the intensity of the white noisė . the function f (·) is generally assumed to be a non-negative twice continuously differentiable x < f (0) for any x > 0). note that the covid-19 epidemic models may be perturbed by telegraph noise which can causes the system to switch from one environmental regime to another [22] . mostly the switching between environmental regimes is often memoryless and the waiting time for the next switching follows the exponential distribution [23] . hence the regime switching can be modelled by a continuous time markov chain (r(t)) t 0 with values in a finite state space s = {1, 2, ..., n }. then model (2) disturbed by white noise and telegraph noise develops to where markov chain r(.) is f t -adapted but independent of brownian motion b(.) and defined on of markovian chain is defined by in this paper, we assume that γ ij > 0 for i, j = 1, ..., n with j = i. this assumption assures that the markov chain r(t) is irreducible, which implies that it has a unique stationary distribution π = (π 1 , π 2 , ..., π n ), which can be determined by solving the following equation subject to n k=1 π k = 1, π k > 0, for any k ∈ s. for convenience, we denote for any fixed vector to begin the analysis of the model, we define the subsets throughout this paper, we carry out the case of small noises: the structure of the rest of the paper is as follows: in section 3, we show the existence and uniqueness of a global positive solution to the system (3). in the sections 4 and 5, we study the 55 existence of a stochastic threshold for the extinction and the persistence in mean of the disease. in last section, we present some numerical simulations to demonstrate our main theoretical results. to study the dynamical behaviour of an epidemic model, we firstly need to consider whether the solution is global and positive. in this section, we will prove there is a unique global positive proof. obviously, the coefficient of model (3) are locally lipschitz continuous, so there is a unique lowhere τ e is the explosion time [31] . if τ e = ∞ a.s., then this local solution is global. to this end, where throughout this paper we set inf ∅ = ∞ (∅ denotes the empty set). clearly, τ n is increasing for any n ≥ n 1 and t ∈ [0, τ n ), we have it then follows that the non-negativity of this function can be obtained from u − 1 − log u 0 for any u > 0. let n n 1 and t > 0 be arbitrary. for any 0 t min{τ n , t }, applying the generalized itô's formula [31] to v yields where lv : r 5 + → r is defined by )ǎ µ f (0) + 5μ +b 1 +b 2 +α +ν +č +η +δ where the inequality f (e) e ≤ f (0) is used and k is a positive constant which is independent of 80 s, e, q, i, r and k. the remaining part of the proof is similar to [30] and hence is omitted here. this completes the proof. it is most crucial to deal with the conditions for the extinction of diseases when their dynamics is under investigation. this section is devoted to establish sufficient conditions so that the covid-19 85 goes out of the population. first, we need to define the following number at this stage, we have the following theorem: proof. applying the generalized itô's formula to log e, we have moreover, for any k ∈ s, the function defined by using the 90 boundedness of the solution and the fact that f (e) ≤ f (0)e, we obtain integrating (5) from 0 to t and then dividing by t on both sides lead to it follows from the ergodic property of r(t) that lim sup taking the superior limit on the both sides of (6), (7) and making use of the large number theorem for local martingales, we get which implies that substituting this into the third equation of system (3), we obtain for all ω ∈ λ, t > t, thus by the comparison theorem we get recalling that p(λ) = 1, hence we obtain similarly, when lim t→∞ e(t) = 0 a.s., and lim t→∞ q(t) = 0 a.s., then by using the same approach as above, we can conclude that remark 1. obviously, the quantity r 0 s is smaller than r 0 . hence, the extinction of the disease in system (3) could be ensured even if the condition r 0 < 1 is not verified. to investigate epidemic dynamical system, we are also interested in when the disease persists 110 in host population. in this section we need to assume that the function f (·) · is c−lipschitz and we have the following result on the prevailing behaviour of the covid-19 disease. for some positive constants m i , i = 1, · · · , 4. proof. applying the itô's formula on the function e → log e, we get where using the fact that s ≤ǎ µ and f (e) e ≤ f (0), one can easily show that from the first equation of (3), we can easily claim that now, let u and v in (0,ǎ/μ). without loss of generality, assume that u < v. the monotonocity and consequently, substituting this inequality into (11) and making use of (9), we obtain where then, applying the generalized itô's formula on the function v (e, k) = log e +ω(k) and using (12), since the generator γ is irreducible, then for p 0 = (p (1), · · · , p (n )) with p (k) = f ǎ µ f (0), k , there is a solutionω = (ω(1), · · · ,ω(n )) to the system γω = −p 0 + n k=1 π k p (k)1, where 1 is the unit vector of r n . that is, substituting the above equality into the inequality (13), integrating from 0 to t and dividing by t, we obtain the large number theorem for local martingales and the boundedness of s implies that lim from the third equation of (3), we can establish that taking the inferior limit and making use of both the boundedness of q and the inequality (14), we get lim inf where m 2 =b 2 m 1 / b 1 +č +μ . following the same way, we can easily claim that lim inf where m 3 = (αm 1 +ĉm 2 ) / η +μ +δ and m 4 = (νm 1 +ηm 3 ) /μ. this makes finish of the proof. are assumed to be more likely than the first one to show the applicability of the analytical results established along this paper. the parameter values chosen are given in [24] and reported in the table 1 . to demonstrate the effect of telegraph noise on the dynamics of covid-19 disease, in addition to data of table 1 , we set the following settings: then, by direct computation, we obtain r 0 s = 0.6303 < 1. in other words, the conditions of the theorem 4.1 hold and the fig. 2 shows the empirical means and the standard deviations of the solution to (3) in a 5 × 10 3 samples, as well as the trajectories of the deterministic system (1) without switching for different values of the three considered regimes. so the stochastic process (3) for covid-19 disease switches over the states 1, 2 and 3 before going to extinction. on the other hand, when the following parameter values are considered, this paper investigates a stochastic epidemic model describing covid-19 dynamics affected 125 by mixture of environmental perturbations modeled by white and telegraph noises. by means of lyapunov approach, the existence and positivity of a global solution is well proved. in terms of a stochastic threshold r 0 s , the extinction and the persistence in mean of the covid-19 epidemic are investigated. particularly, under small noises, the condition r 0 s < 1 is sufficient to reduce the daily number of confirmed infectives and make the coronavirus disease 2019 extinct. reciprocally, the 130 persistence of this novel epidemic is inevitable once r 0 s stays away from unity. based on the data from different states of india, we performed numerical simulations in order to support and illustrate the main results of this paper. although many important contributions are made in literature to draw the dynamical properties of the covid-19, some of them still unidentified and much more efforts are recommended to 135 make it more comprehensible and help humanity to overcome the current pandemic. as a further suggestion, other improvements such us time varying parameters can be considered to make the studied covid-19 model more realistic. a task which we leave for next works. ☒ the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐ the authors declare the following financial interests/personal relationships which may be considered as potential competing interests: a stochastic hepatitis b epidemic model driven by lévy noise analysis of a deterministic and a stochastic epidemic model with two distinct epidemics hypothesis a stochastic analysis for a triple delayed siqr epidemic model with vaccination and elimination strategies china medical treatment expert group for, clinical characteristics of coronavirus disease 2019 in china early transmission dynamics in wuhan, china, of novel coronavirus-infected pneumonia analysis of a stochastic distributed delay epidemic model with relapse and gamma distribution kernel phase-adjusted estimation of the number of coronavirus disease 2019 cases in wuhan a mathematical model for simulating the phase-based transmissibility of a novel coronavirus a stochastic threshold of a delayed epidemic 170 model incorporating levy processes with harmonic mean and vaccination the effect of a generalized nonlinear incidence rate on the stochastic sis epidemic model van 175 den broeck w, vanhems p. simulation of an seir infectious disease model on the dynamic contact network of conference attendees an algorithmic introduction to numerical simulation of stochastic differential equations a time delay dynamic system with external source for the local outbreak of 2019-ncov modelling the spread of covid-19 with new fractal-fractional operators: can the 185 lockdown save mankind before vaccination? modeling the dynamics of novel coronavirus (2019-ncov) with fractional derivative phase-adjusted estimation of the number of coronavirus disease feasibility of controlling covid-19 outbreaks by isolation of cases and contacts a mathematical model for simulating the phase-based transmissibility of a novel coronavirus a control treatment for a stochastic epidemic model 195 with relapse and crowly-martin incidence phase-adjusted estimation of the number of coronavirus disease stochastic population dynamics under regime switching stationary distribution of stochastic population systems under regime switching a model based study on the dynamics of covid-19: prediction and control stochastic analysis of a two delayed epidemic model incorporating lévy processes with a general non-linear transmission stationary distribution and threshold dynamics of a stochastics sirs model with a general incidence real-time forecasts and risk assessment of novel coronavirus (covid-19) cases: a data-driven analysis the effect of stochasticity with respect to reinfection and nonlinear transition states for some diseases with relapse environmental noise suppresses explosion in population dynamics stochastic differential equations with markovian switching brahim boukanjime: conceptualization, writing -original draft, writing -review & editing tomás caraballo: validation, investigation, conceptualization, writing -original draft, writing -review & editing methodology, validation, formal analysis, writingoriginal draft, writing -review & editing writing -original draft, writing -review & editing the authors would like to thank the editor and reviewers for their careful reading of the key: cord-319804-i5oprni9 authors: mahajan, ashutosh; sivadas, namitha a; solanki, ravi title: an epidemic model sipherd and its application for prediction of the spread of covid-19 infection in india date: 2020-07-28 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110156 sha: doc_id: 319804 cord_uid: i5oprni9 originating from wuhan, china, in late 2019, and with a gradual spread in the last few months, covid-19 has become a pandemic crossing 9 million confirmed positive cases and 450 thousand deaths. india is not only an overpopulated country but has a high population density as well, and at present, a high-risk nation where covid-19 infection can go out of control. in this paper, we employ a compartmental epidemic model sipherd for covid-19 and predict the total number of confirmed, active and death cases, and daily new cases. we analyze the impact of lockdown and the number of tests conducted per day on the prediction and bring out the scenarios in which the infection can be controlled faster. our findings indicate that increasing the tests per day at a rapid pace (10k per day increase), stringent measures on social-distancing for the coming months and strict lockdown in the month of july all have a significant impact on the disease spread. india where it has reached at alarming level.  impact of lockdown and the number of tests conducted per day on predictions of containment is studied.  purely asymptomatic cases and spread from them as well as exposed in incubation period considered.  increasing the tests per day by 10k every day, stringent measures on social-distancing and strict lockdown in july have significant impact on the disease spread. the outbreak of novel corona virus disease (covid-19) caused by severe acute respiratory syndrome coronavirus 2 (sars-cov-2), originated from a wet market in wuhan, china, is now widespread in the world and has severely affected many counties including india. the first case of covid-19 in india was reported on january, 30 2020 at thrissur, kerala, in a student who had returned from china. india has reached the fourth position in the world in the number of confirmed covid-19 cases and presently has 449,613 confirmed cases and 14,162 deaths as of june 23, 2020, which is really an alarming situation. social distancing is the best method for mitigating this pandemic until an effective medicine or vaccine is invented [1] . the first nationwide lockdown is ordered by the prime minister on march 24 for 21 ladays and further extended the lockdown till may 3 by relaxing certain substantial fields [2] . later, though the lockdown period is extended to june 30, the freedom is given to the states to impose restrictions assessing the situations in the respective states. mathematical modeling and simulation are helpful for predicting the transmission of the epidemic and to implement necessary actions for its control. mathematical models for the epidemic have a major role to make predictions of the transmission dynamics of the disease and thus assist the authority to take necessary movements for the containment. several epidemic models are already reported in the literature, however, the covid-19 is different type of infection showing certain special characteristics. the transmission of the disease from the persons who are infected without showing any symptoms (asymptomatic cases) is one of the special characteristics of this disease [3] and to be considered in the modeling. also, the disease can be spread from the infected who is in the incubation period [4] . in this study, we present a new mathematical model named sipherd, incorporating the aforementioned characteristics of the covid-19. many mathematical models for the infectious disease spread are reported in the literature, and the classical and widely used method is sir model described in [5] . an approximate spatial epidemiological model of the covid-19, is initially proposed in [6] in which the spread of the disease within between and the countries is analysed. the infected and undetected cases and the spread of covid-19 from those persons are incorporated in [7] but this study is associated with only china. a modified compartmental sir model is discussed in [8] . in this study, the total population of the country is divided into eight compartments, and this work is carried out only for italy, and also, the model does not undertake the purely asymptomatic cases of infected. a different compartmental model seir [9] predicts the dynamics of the transmission of the covid-19 for certain countries, and the impact of quarantine of the infected persons are also studied in it. another improved sir model is depicted in [10] , and the time dependency of the parameters of the sir model is also examined in this. prediction of transmission of covid-19 using curve fitting algorithms are reported in [11] , [12] and [13] . a stochastic mathematical model is proposed in [14] to analyse the impact of covid-19 on the healthcare system in india. assessment of the preventive measures of covid -19 such as lockdown and prediction of its spread in india is studied in [15] . the seir epidemic and regression model is extended for predicting and evaluating the transmission of covid-19 in india [16] . progression of the epidemic in india is also determined using the mathematical modelling in [17] and [18] . in india, complete lockdown is limited to the containment zones and hotspots from june 1 on-wards. the forbidden activities in the places outside the containment zones are re-opened in a phased manner with the conditions to follow the standard operating guidelines given by the health ministry of india. inter-state and intra-state travel is allowed in the present situation without any pass/permission, religious places, restaurants, shopping malls and hospitality facilities are allowed to open whereas the educational institutions, entertainment zones, international air travel and railway services remain prohibited as of june 23. despite all these restrictions, the infection is growing exponentially, and policymakers need to consider different ways for the containment of the disease. the most hit cities in india by covid-19 pandemic are mumbai, delhi and chennai with a collective population of around 40 million. in this paper, we bring out different possible ways for better control of the infection spread. we employ an improved mathematical model sipherd [19] for the covid-19 pandemic embedding the purely asymptomatic infected cases and the transmission of the disease from them. the model simulations bring out the efficacy of different ways for the containment, by predicting the total number of active and confirmed cases, total deaths, and daily new positive cases considering various social distancing/lockdown conditions and the number of tests done per day. we model the evolution of the covid-19 disease by dividing the population into different categories as listed below which is described in detail in [19] [20] . the sipherd model equations are for the defined entities (s,i,p,h,e,r,d) are a set of coupled ordinary differential equations (1 to 7). the population is divided into different categories, as susceptible (s), exposed (e), symptomatic (i), purely asymptomatic (p), hospitalized or quarantined (h), recovered (r) and deceased (d). where, tr and td are the delay associated with the recovery and death respectively with respect to active cases h. the various parameters seen in fig 1 and their optimized values for india covid-19 data are listed in table i.all fractions add up to unity that can also be seen from summing the above equations. the detection of the asymptomatic and symptomatic cases can be taken dependent on the number of tests done per day (tp d ). where, µ0, µ1, ν0, and ν1 are positive constants. the effectiveness of the tests increases if contact tracing is performed. so far in india contact tracing is performed well, we assume that the increased tests are also performed on the suspects more carefully and the detection probability increases with increasing tests linearly. since the severe cases are going to approach for the tests, one component of the detection probability is not taken dependent on the number of tests. recovery of asymptomatic cases is taken faster than the symptomatic cases. the total confirmed cases are the addition of the active cases, extinct cases, and a part of the recovered that were detected. this can be written as the set of coupled ordinary differential equations for the model can be solved numerically for a given set of parameters and initial values. it is however important that the parameters are determined accurately so that the model demonstrates the real situation of the infection spread. we take into account the data sets of the total number of confirmed cases, active cases, cumulative deaths and tests done per day, and find the model parameters that generates the best possible match between the actual data and model. for this purpose, a cost function is written in terms of errors between the actual and solver data sets. the minimizer of the cost gives the optimized set of parameters. the model and the optimization codes are implemented in matlab. the number of total positive or confirmed cases, present active cases and deaths are collected from [21], [22] , and the number of tests per day from [23], which is plotted in supplementary material fig.s1 a. the parameters determined by our model for covid-19 spread in india are listed in table i and the simulation data from the model is compared with the actual data in supplementary material fig.s6 . the parameter values related to the characteristics of the disease are discussed in more detail in [19] . for the available data till june 21, 2020, we run the model for first 109 days i.e. till june 20 to extract parameters listed in table i, and then with the extracted parameters, the model is run for 500 days starting from march 4. the current increase in tests per day is around 2.5k, we assume the same trend for the test per day and take the current value of transmission rate to generate the simulated data for the prediction. two scenarios are considered for lockdown and social distancing conditions. one possible scenario is that the conditions are kept the same, and the second one is that they are made stricter by taking into account some measures after june 21 such that the transmission rates α and β decrease by 5% . these measures can include restrictions on travel, large gathering of people for social events, distribution of low-cost masks and hand sanitizers in hot spots. test per day assumed to be increased by 2.5k, which is close to the current trend and taken saturated at 1 million for both the scenarios. the mortality rate is calculated from the data and is improved in steps from initial value 3.17e-3, 3.14e-3, 2.29e-3, 2.79e-3 on march 4 to june 21 as seen in supplementary material fig.s1 .b. for the future, mortality rate is taken improved to a fixed value 2e-3. a comparison of the predictions for the two scenarios i.e. with and without the stringer measures is plotted in fig. 2 . reproduction number variation with time and evolution of the undetected infected cases can be seen in supplementary material fig.s2 . the total number of reported cases is predicted to be around 12 million. this number appears very high. however, compared to the usa reported cases 2.4 million, the number is reasonable, given the fact that india's population is roughly four times higher. we also study one more possible scenario in which a total lockdown is imposed for july 2020. the rate of transmission of infection is going to decrease in the imposed lockdown, and we take the α and β values to decrease by 10% from the current value. the prediction with 2.5k increase in tests per day and saturation at 1 million tests, is compared in fig. 5a,b and in fig. 5d , we plot the prediction for the daily new cases. detailed plots and evolution of infected can be seen in supplementary material fig.s5 for this condition of stricter lockdown for july and relaxed after that. we compare the effect of testing on the prediction of total, active and death cases in fig. 3 a,b. total, active and extinct cases are plotted for the coming months if tests per day are increased by 2.5k, 5k and 10k per day after june 21 and saturated at 1 million as seen in the initial reproduction number 3.2 is seen go down to 1.5 after the imposition of lockdown on 20th day i.e. march 24. after the second lockdown on 56th day i.e. april 27 it has reached 1.32. it shows a downward trend further and goes below 1 on october 5,2020. if tests per day are increased 10k per day, then the reproduction number comes below one on july, 16, which can also be seen in fig. 5c . the reproduction number is seen to go below one on july 30 if the transmission rate decreases by 10% due to lockdown in july. the initial basic reproduction number 3.2 is in the range of the mean reported value [24] [25] . a sensitivity study is carried out for the different parameters, as seen in supplementary material fig.s4 and s5. the parameters are increased and decreased by 5% from the optimized values to see the changes in the outcomes. it can be seen in supplementary material fig.s5 that the model prediction are most sensitive to the transmission rates α3,β3 and γ3. just a 5% change in these estimated parameters gives a huge change in total, active and extinct cases. supplementary material fig.s4 v. conclusion sipherd model is employed for covid-19 spread that considers purely asymptomatic category of infected cases in addition to the symptomatic, and the disease spread by the exposed. the effect of lockdown on the rates of transmission of infection and the influence of tests per day on detection rates has been incorporated in the model. with the current trend, total infections would be 12 million when disease ends and can lead to 362k total deaths. our findings suggest that increasing the number of tests at 10k per day with highly efficient contact tracing, rather than the current 2.5k per day rise leads to a reduction of 9 million reported cases and reduction of 270k in total extinct cases. in the absence of a vaccine, the infection can last long till the end of this year and number of deaths could be around 350k if social distancing conditions and increase in tests remain at the current trend. age-structured impact of social distancing on the covid-19 epidemic in india assessment of 21 days lockdown effect in some states and overall india: a predictive mathematical study on covid-19 outbreak estimating the asymptomatic proportion of coronavirus disease 2019 (covid-19) cases on board the diamond princess cruise ship, yokohama, japan, 2020 transmission of 2019-ncov infection from an asymptomatic contact in germany containing papers of a mathematical and physical character application of the be-codis mathematical model to forecast the international spread of the 2019-20 wuhan coronavirus outbreak mathematical modeling of the spread of the coronavirus disease 2019 (covid-19) considering its particular characteristics. the case of china a sidarthe model of covid-19 epidemic in italy on an interval prediction of covid-19 development based on a seir epidemic model a time-dependent sir model for covid-19 with undetectable infected persons preliminary estimation of the basic reproduction number of novel coronavirus (2019-ncov) in china, from 2019 to 2020: a data-driven analysis in the early phase of the outbreak predictions of 2019-ncov transmission ending via comprehensive methods artificial intelligence forecasting of covid-19 in china healthcare impact of covid-19 epidemic in india: a stochastic mathematical model prediction for the spread of covid-19 in india and effectiveness of preventive measures seir and regression model based covid-19 outbreak predictions in india predictions for covid-19 outbreak in india using epidemiological models prediction of covid-19 disease progression in india: under the effect of national lockdown estimation of undetected symptomatic and asymptomatic cases of covid-19 infection and prediction of its spread in usa an epidemic model sipherd and its application for prediction of the spread of covid-19 infection for india and usa the reproductive number of covid-19 is higher compared to sars coronavirus novel coronavirus 2019-ncov: early estimation of epidemiological parameters and epidemic predictions ☒ the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.☐the authors declare the following financial interests/personal relationships which may be considered as potential competing interests: eclaration of interest statement key: cord-308296-43gmzqa6 authors: alkahtani, badr saad t.; alzaid, sara salem title: a novel mathematics model of covid-19 with fractional derivative. stability and numerical analysis date: 2020-06-17 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110006 sha: doc_id: 308296 cord_uid: 43gmzqa6 a mathematical model depicting the spread of covid-19 epidemic and implementation of population covid-19 intervention in italy. the model has 8 components leading to system of 8 ordinary differential equations. in this paper, we investigate the model using the concept of fractional differential operator. a numerical method based on the lagrange polynomial was used to solve the system equations depicting the spread of covid-19. a detailed investigation of stability including reproductive number using the next generation matrix, and the lyapunov were presented in detail. numerical simulations are depicted for various fractional orders. uncertainties around the spread of covid-19 have lead many researchers to understand investigation in many field of technology, science and engineering in the last five months since its appearance in wuhan-china last december-2019 many mathematical models were suggested in the last five months with the aim to understand the dynamics spread of the novel deathly disease [10] . many journal have launch special issues on covid-19, in field of science, technology and engineering, with the aim together all novel results changing from theoretical to practical point of view. of course so far many new results have been collected many many data are ready although they are still being collected. mathematician while they do not provide a cure nor a vaccine for any infectious disease however the mathematical models can help in many ways [11] . for example, their result are very useful to predict the future behavior of the spread and even control it. technique like markov chan, fuzzy, stochastic, monte-carlo approach and many others are very useful in this process [5] [6] [7] [8] . on the other hand fractional differential operators are used to include into mathematical models the effect of non locality often divide by power process, fading memory process and cross-over [12, 15] . in this paper we consider, the model suggested in 9. in this section, we recall some basic definitions and properties of fractional calculus theory which are useful in the next sections. definition 3.1. let u be a function not necessarily differentiable, and ϑ be a real number such that ϑ > 0, then the caputo derivative with ϑ order with power law is given as [13] then the new caputo derivative of fractional order is given by: where m (ϑ) is a normalization function such that m (0) = m (1) = 1 [4] . but, if the function u = h 1 (a, b) then, new derivative called the caputo-fabrizio fractional derivative can be defined as in addition, now after the introduction of a new derivative, the associate anti-derivative becomes important, the associated integral of the new caputo derivative with fractional order was proposed by losada and nieto [14] . with the function f differentiable then, the definition of the new fractional derivative (atangana-baleanu derivative in caputo sense) is given as where m (ϑ) has the same properties as in the case of the caputo-fabrizio fractional derivative. it should be noted that we do not recover the original function when ϑ = 0 except when at the origin the function vanishes. to avoid this kind of problem, the following definition is proposed. definition 3.4. let u ∈ h 1 (x, y), y > x, ϑ ∈ [0, 1] and not necessary differentiable then, the definition of the new fractional derivative (atangana-baleanu fractional derivative in riemann-liouville sense) is given as [1] . definition 3.5. the fractional integral associate to the new fractional derivative with nonlocal kernel (atangana-baleanu fractional integral) is given as [1] : when alpha is zero we recover the initial function and if also alpha is 1, we obtain the ordinary integral. in this section we consider the model suggested in [9] . here s(t) is the class of susceptible, i(t) is the class of infected asymptomatic infected undetected, d(t) is the class of asymptomatic infected, detected, h(t) is the healed class, parameters therein and their physical meaning and interpretation can be found in [9] . since all parameters used in the model are positive. if the initial assumptions are positive then all the classes are positive, for the models with classical and non-local sperators. we start with classical caseṡ however the product βd(t) + γa(t) + δr(t) is positive. since all classes should have same sign, thusi̇ with d(t) and a(t) being positive ∀t ≥ 0, we have that since e(0) is positive or zero τ and t (τ ) are positive, then e(t) ≥ 0 ∀t ≥ 0. to prove for classes δ(t) and a(t), we define the following norm ∀f ∈ c[a, b] since all the other classes are positive then ∀ t > 0 with the non local operator, we only show the positiveness for caputo derivativė thus, following the procedure suggested before s * (αi * + βd * + γa * + δr * ) = 0, i * = 0 thus h * = 0 the disease equilibrium is ζ + + λ b , 0, 0, 0, 0, 0, 0 although the reproductive number was given in [9] , we only present the next generation matrix associated to the model. we choose the 5 classes of infected. from the above, the matrix the model suggested will lead to endemic situations iḟ i(t),ȧ(t),ḋ(t),ṙ(t),ṫ (t) > 0 ∀t ≥ 0 that is to say simplifying i, then we get theorem 5.1. the disease free equilibrium are asymptotically globally stable within the acceptable interval if r 0 < 1 and unstable if r 0 > 1. proof :the proof will be achieved the use of the lyapunov function defined by we present the lyapunov associate to the model using the fundamental theorem of calculus, we convert the above to so that x(t))dt (6.5) and by subtracting (6.6) from (6.5), we get so the solution is, f (t n−1 , x n−1 ) (6.8) so for the equation (4.1) the solution is {−s n−1 (t n−1 )[αi n−1 (t n−1 ) + βd n−1 (t n−1 ) + γa n−1 (t n−1 ) + δr n−1 (t n−1 )]} (6.9) {s n−1 (t n−1 )[αi n−1 (t n−1 ) + βd n−1 (t n−1 ) + γa n−1 (t n−1 ) + δr n−1 (t n−1 )] −[ + ζ + λ]i n−1 (t n−1 )} (6.10) ηd n−1 (t n−1 ) + θa n−1 (t n−1 ) − (v + ξ)r n−1 (t n−1 ) (6.13) µa n−1 (t n−1 ) + vr n−1 (t n−1 ) − (σ + τ )t n−1 (t n−1 ) (6.14) λi n (t n ) + ρd n (t n ) + κa n (t n ) + ξr n (t n ) + σt n (t n ) λi n−1 (t n−1 ) + ρd n−1 (t n−1 ) + κa n−1 (t n−1 ) + ξr n−1 (t n−1 ) + σt n−1 (t n−1 ) τ t n−1 (t n−1 ) (6.16) in this section, we present numerical simulation for different values of fractional α. the numerical simulation are presented in figure 1 , 2, 3, 4, 5, 6, 7 and 8. we observed that with this model all classes are increasing exponentially. in this paper, we considered a set of 8 nonlinear ordinary differential equations to model the spread of covid-19 in a given population. the model is comprised of susceptible class, 5 sub-classes of infected, recovered and death. we presented the positivity of each class as function of time, for classical and fractional case. we used the concept of next generation matrix to derive the reproductive number, we presented a detailed study of stability of equilibrium points. numerical simulations are presented for different values of fractional orders. new fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model the role of power decay, exponential decay and mittag-leffler functions waiting time distributions: application of cancer spread new numerical approach for fractional differential equations a new definition of fractional derivative without singular kernel investigation on fractional and fractal derivative relaxation-oscillation models a mathematical model for simulating the transmission of wuhan novel coronavirus. biorxiv novel coronavirus: where we are and what we know short term outcome and risk factors for adverse clinical outcomes in adults with severe acute respiratory syndrome (sars) modelling the covid-19 epidemic and implementation of population-wide interventions in italy modelling the covid-19 epidemic and implementation of population-wide interventions in italy a mathematical model of treatment and vaccination interventions of pneumococcal pneumonia infection dynamics modeling the dynamics of novel coronavirus (2019-ncov) with fractional derivative translated from the 1987 russian original, revised by the authors properties of the new fractional derivative without singular kernel numerical analysis for the fractional diffusion and fractional buckmaster's equation by two step adam-bashforth method the authors would like to extend their sincere appreciation to the deanship of scientific research at king saud university for funding this group no. rg-1437-017. 6.2. covid-19 model with atangana-baleanu derivative. considering equation (4.1) , the modified cancer model with atangana-baleanu derivative derivative is given as abc 0 d α t s(t) = −s(t)(αi(t) + βd(t) + γa(t) + δr(t)) abc 0 d α t i(t) = s(t)(αi(t) + βd(t) + γa(t) + δr(t)) − ( + ζ + λ)i(t)we consider the following fractional differential equation let us consider the following fractional differential equationthe above equation can be converted to a fractional integral equation by applying the fundamental theorem of fractional calculus:at a given point = t n+1 , n = 1, 2, 3, . . . , the above equation is reformulated asand at the point t = t n , n = 1, 2, 3, . . . , we havewhich on subtraction yieldsso for the equation (6.17) the solution ish n+1 − h n = λi n (t n ) + ρd n (t n ) + κa n (t n ) + ξr n (t n ) + σt n (t n )n ϑ + 1 + λi n−1 (t n−1 ) + ρd n−1 (t n−1 ) + κa n−1 (t n−1 ) + ξr n−1 (t n−1 ) + σt n−1 (t n−1 ) dear editor, this is to confirm that both authors have done the work. we have done equal work in this paper. we confirm that we are both aware of the submission in this special issue in chaos solitons and fractal key: cord-352990-0uglwvid authors: nadim, sk shahid; chattopadhyay, joydev title: occurrence of backward bifurcation and prediction of disease transmission with imperfect lockdown: a case study on covid-19 date: 2020-08-17 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110163 sha: doc_id: 352990 cord_uid: 0uglwvid the outbreak of covid-19 caused by sars-cov-2 is spreading rapidly around the world, which is causing a major public health concerns. the outbreaks started in india on march 2, 2020. as of april 30, 2020, 34,864 confirmed cases and 1,154 deaths are reported in india and more than 30,90,445 confirmed cases and 2,17,769 deaths are reported worldwide. mathematical models may help to explore the transmission dynamics, prediction and control of covid-19 in the absence of an appropriate medication or vaccine. in this study, we consider a mathematical model on covid-19 transmission with the imperfect lockdown effect. the basic reproduction number, r(0), is calculated using the next generation matrix method. the system has a disease-free equilibrium (dfe) which is locally asymptotically stable whenever r(0) < 1. moreover, the model exhibits the backward bifurcation phenomenon, where the stable dfe coexists with a stable endemic equilibrium when r(0) < 1. the epidemiological implications of this phenomenon is that the classical epidemiological requirement of making r(0) less than unity is only a necessary, but not sufficient for effectively controlling the spread of covid-19 outbreak. it is observed that the system undergoes backward bifurcation which is a new observation for covid-19 disease transmission model. we also noticed that under the perfect lockdown scenario, there is no possibility of having backward bifurcation. using lyapunov function theory and lasalle invariance principle, the dfe is shown globally asymptotically stable for perfect lockdown model. we have calibrated our proposed model parameters to fit daily data from india, mexico, south africa and argentina. we have provided a short-term prediction for india, mexico, south africa and argentina of future cases of covid-19. we calculate the basic reproduction number from the estimated parameters. we further assess the impact of lockdown during the outbreak. furthermore, we find that effective lockdown is very necessary to reduce the burden of diseases. an outbreak of 2019 coronavirus disease (covid-19) has resulted in 30,90,445 con10 firmed cases and 2,17,769 deaths as of april 30, 2020 according to who [6] . the out-11 break was first taken place in wuhan, china, in december 2019, with the majority of 12 early cases reported in the city. coronaviruses are single-stranded, positive-sense rna 13 viruses belonging to the coronaviridae family [9] . it has been confirmed that 27 people 14 have been infected due to viral pneumonia, including seven critically ill cases [30] , and 15 this epidemic has drawn tremendous attention worldwide. it causes variety of symptoms, 16 including dry cough, fever, weakness, trouble breathing, and bilateral lung infiltration, be the sixth public health emergency of international concern. since the first discovery 26 and identification of coronavirus in 1965, there are three major outbreaks occurred due to 27 coronaviruses and the outbreak was called 'severe acute respiratory syndrome (sars) 28 outbreak (2003) in china [23] . saudi arabia suffered from 'middle east respiratory 29 syndrome' (mers) outbreak (2012) [11] and south korea (2015) [10] . 30 the indian government reported that on 30 january 2020 in the state of kerala, across the world and an unprecedented threat to the community's health care, economy 44 and lifestyle. for all, there is a huge worry as to how long this condition can continue 45 and whether the epidemic can be handled. 46 we also study the cases of mexico, south africa and argentina as the 47 lockdown was carried out partially or in a less severe form in these countries. 48 in february 2020 the virus was confirmed to reach mexico. however, two recently lockdown measure has been used successfully to control covid-19 spread. the aim of this study is to investigate the qualitative effect of the imperfect lockdown 74 on the spread of disease dynamics. to achieve this goal, a mathematical model for 75 covid-19 with the lockdown is proposed and analyzed. in this model, we implemented 76 the imperfect lockdown, which means that the lockdown susceptible population also gets 77 infected during the lockdown period by unnotified infected individuals. we looked at in-78 dia's situation during the outbreak period and fitted our model with the newly daily cases 79 reported from march 14 th , 2020 to april 19 th , 2020. we also looked at the situation 80 of mexico, south africa and argentina during the outbreak period and fitted 81 our model with the new daily cases reported for a certain outbreak period. 82 we are providing a short-term prediction for india, mexico, south africa and 83 argentina of future cases of covid-19 using the estimated parameters for 84 the period march 14, 2020, to may 21, 2020, march 23 to july 9, march 17 85 to july 9 and march 14 to july 9 respectively. for the above-mentioned period we 86 aslo estimate the basic reproduction number. it is common for classical epidemic models 87 that a basic reproduction number is a threshold in the context that if the basic repro-88 duction number is greater than one, a disease will persist, and dies out if it is less than 89 one. in this case, for imperfect lockdown, the basic reproduction number does not rep-90 resent the required elimination effort; rather, the effort at the turning point is described the paper is organized as follows: our proposed mathematical model which incorporates 108 the lockdown of susceptible individuals and imperfect lockdown efficacy is described in 109 section 2. the model is analyzed specifically for the existence of backward bifurcation 110 in section 3. in section 4 we fitted our model to daily new cases. we provided a all the parameters and their biological interpretation are given in table 1 respectively. 146 proof. the system (2.1) can be written as follows .., f 6 (x)) denotes the right hand side functions. it is very obvious that for every j = 1, ..., the basic reproduction number r 0 is a threshold value that is epidemiologically significant and determines the potential of an infectious disease to enter a population. to obtain the basic reproduction number r 0 of the system (2.1), we apply the next generation matrix approach. the system has a disease-free equilibrium given by , 0, 0, 0, 0 . the infected compartments of the model (2.1) consist of (e(t), i(t), j(t)) classes. fol-153 lowing the next generation matrix method, the matrix f of the trransmission terms and 154 the matrix, v of the transition terms calculated at ε 0 are, so, the next generation matrix f v −1 is, calculating the dominant eigenvalue of the next generation matrix f v −1 , we obtain the basic reproductive number as follows [13; 33] the basic reproduction number r 0 is defined as the expected number of secondary cases 157 generated by one infected individula during its lifespan as infectious in a fully susceptible 158 population. the basic reproduction number r 0 of (2.1) given in 3.2. using theorem 2 in [33], the following result is established. lemma 3.1. the disease-free equilibrium ε 0 of system (2.1) is locally asymptotically 161 stable whenever r 0 < 1, and unstable whenever r 0 > 1. 3.3. existence of endemic equilibrium 163 we are now exploring the existence of endemic equilibrium. let ε * = (s * , l * , e * , i * , j * , r * ) be any endemic equilibrium of system (2.1). let us denote further, the force of infection be by setting the right equations of system (2.1) equal to zero, we have substituting the expression in 3.4 into 3.3 shows that the non-zero equilibrium of the model (2.1) satisfy the following quadratic equation, in terms of λ * h : the endemic equilibrium of the model (2.1) can be obtained by solving for λ * h from in order to check the possibility of backward bifurcation in (2.1), the discriminant b 2 − 4ac of the quadratic 3.5, is set to zero and the result solved for the critical value (denoted by r c 0 ) of r 0 . this gives: from which we have seen that backward bifurcation occurs for values of r 0 such that 184 r c 0 < r 0 < 1. we explore the details analysis of backward bifurcation in the next a stable endemic equilibrium co-exists with a stable disease-free equilibrium for r 0 < 1. clearly, this phenomenon has significant public health consequences, as it makes the 193 classical requirement of the associated basic reproduction number being less than unity 194 to be necessary, but not sufficient to eradicate the disease. in this section, we explore the phenomenon of backward bifurcation in system (2.1). first, we execute bifurcation analysis by using the center manifold theorem as follows: the jacobian of the system (3.7) at the dfe ε 0 is given as, choose β as the bifurcation parameter, then setting r 0 = 1 gives the system (3.7) at the dfe ε 0 evaluated for β = β * has a simple eigenvalue with zero 201 real part, and all other eigenvalues have negative real part. we therefore apply the center manifold theorem in order to analyze the dynamics of (3.7) near β = β * . the jacobian of (3.7) at β = β * , denoted by j ε 0 |β = β * has a right eigenvector (corresponding to the zero eigenvalue) given by w = (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 ) t , where similarly, from j ε 0 |β = β * , we obtain a left eigenvector v = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) t (corresponding to the zero eigenvalue), where (3.10) we calculate the following second order partial derivatives of f i at the disease-free equilibrium ε 0 to show the existence of a backward bifurcation and obtain since the coefficient b is always positive, system (2.1) undergoes backward bifurcation we have established the following conclusion. theorem 3.3. system (2.1) undergoes a backward bifurcation at r 0 = 1 whenever the 206 inequality 3.14 holds. furthermore, it should be noted that for the case when lockdown susceptible individuals do not acquire infection during lockdown period (i.e., r = 0), the bifurcation coefficient a becomes thus, since a < 0 in this case, it follows from theorem 4. globally-asymptotically stable (gas) under some certain conditions, as shown below. setting r = 0 in the model (2.1) gives the following reduced model: it can be shown that the reproduction number associated with the reduced model (3.15), is given by the model (3.15) has a dfe ε 01 = (s 1 , l 1 , 0, 0, 0, 0). theorem 3.4. the dfe (ε 01 ) of the reduced model (3.15), is globally asymptotically proof. consider the following lyapunov function d = γk 2 θk 3 (k 2 + l) e + k 2 θ(k 2 + l) i we take the lyapunov derivative with respect to t, since all the variables and parameters of the model (2.1) are non-negative, it follows that therefore by combining all above equations, it follows that each solution of the model 238 equations (2.1), with initial conditions ∈ ω , approaches ε 0 as t → ∞ for r * 0 ≤ θ < 1. the above result shows that, for the case when the lockdown efficacy in preventing tion of this technique for model fitting is given in [26] . the estimated parameters are 264 given in table 2 the estimated values of unknown initial conditions are given by table 3 . the fitting of the daily new hospitalized covid cases of this four country are displayed 266 in figure 4 . using these estimated parameters from table 2 and the fixed parameters 267 from table 4 , we calculate the basic reproduction numbers given in table 5 . in this section, the impact of lockdown is measured qualitatively on the disease transmission dynamics. a threshold study of the parameters correlated with the lockdown of susceptible individuals l is performed by measuring the partial derivatives of the basic reproduction number r 0 with respect to this parameters. we observe that so that ∂r 0 ∂l < 0 for all 0 < r < 1. we perform the sensitivity of model parameters with respect to the significant re-311 sponse variable and analyze different control parameters to limit covid cases for the 312 four countries. in order to get an overview of most influential parameters, we compute 313 the normalized forward sensitivity indices of the model parameters with respect to basic 314 reproduction number r 0 . we have chosen parameters transmission rate between hu-315 man population β, rate of transition from exposed to infected class γ, the rate at which 316 symptomatic infected become hospitalized or notified η, recovery rate for symptomatic 317 infected τ 1 , lockdown success rate l and lockdown efficacy r for sensitivity analysis. we 318 use the estimated parameters from table 2 for the baseline values. the rest of the pa-319 rameter values are the same as in table 4 . the bar diagram of the normalized forward 320 sensitivity values of r 0 against these parameters is depicted in figure 7 . however, the 321 mathematical definition of the normalized forward sensitivity index of a variable m with 322 respect to a parameter τ (where m depends explicitly on the parameter τ ) is given as: the normalized forward sensitivity indices of r 0 with respect to the parameters β, η, l and r for india are found to be the fact that x β r 0 = 1, means that if we increase 1% in β, keeping other parameters 324 be fixed, will produce 1% increase in r 0 . similarly, x η r 0 = −0.9415 means increasing argentina are given in the table 6 . we have seen that the transmission rate 332 between susceptible humans and lockdown efficacy is positively correlated 333 and the recovery rate of symptomatic infected and lockdown success rate is 334 negatively correlated with respect to basic reproduction number. reproduction number r 0 . we have seen a similar patteren for this four countries. in cases india, the contour plots in figure 8 show the dependence of r 0 on dif-339 ferent parameters. in figure 8 (a) and figure 8 table 2 and table 4 . from the above finding it follows that lockdown success rate and lockdown efficacy is table 1 and table 2. time point which is very slow(see the figure 12 ). that means extension of 375 lockdown for these two countries is not too much effective. table 1 and table 2. backward bifurcation phenomenon, where two stable equilibria, namely the disease-free 387 equilibrium and an endemic equilibrium coexist when the corresponding basic number 388 of reproduction is less than unity. this backward bifurcation phenomena of this study 389 is very important, and this occurs only under imperfect lockdown individuals. this is 390 basically telling us even if the basic reproduction number is less than one, but the disease 391 will persists which is against classical epidemiological theory. in such a situation, the 392 policy makers may stop surveillance, and the results will be disaster. our model exhibits 393 the non-existence of backward bifurcation when the lockdown is perfect (r = 0). we have 394 seen that the disease-free equilibrium is globally asymptotically stable whenever the as-395 sociated basic reproduction number is less than unity for the perfect lockdown model. table 1 and table 2 . table 1 and table 2 . covid-19 pandemic lockdowns coronavirus disease 2019 coronavirus disease 2019 applications of centre manifold theory dynamical models of tuberculosis and 465 their applications emerging coronaviruses: genome structure preliminary epidemiologic assessment of mers-cov outbreak in south 470 euro surveillance: bulletin europeen sur les maladies trans-471 missibles= sars and mers: recent insights into emerging coronaviruses mathematical epidemiology 476 of infectious diseases: model building, analysis and interpretation on the 479 definition and the computation of the basic reproduction ratio r 0 in models for 480 infectious diseases in heterogeneous populations backward bifur-483 cations in dengue transmission dynamics return of the coronavirus causes of backward bifurcations in some epidemiological models dram: efficient 490 adaptive mcmc the mathematics of infectious diseases clinical features of patients 495 infected with 2019 novel coronavirus in wuhan, china. the lancet the continuing 2019-ncov epidemic threat of novel coronaviruses to 500 global healththe latest 2019 novel coronavirus outbreak in wuhan, china. interna-501 tional 504 et al. early dynamics of transmission and control of covid-19: a mathematical 505 modelling study. the lancet infectious diseases the stability of dynamical systems angiotensin-converting enzyme 2 is a functional recep-510 tor for the sars coronavirus short-term predic-512 tions and prevention strategies for covid-2019: a model based study assess-515 ment of lockdown effect in some states and overall india: a predictive mathematical 516 study on covid-19 outbreak mathematical analysis of a power-law form time depen-518 dent vector-borne disease transmission model im-521 pact of intervention on the spread of covid-19 in india: a model based study influence of backward bifurcation on 524 interpretation of r 0 in a model of epidemic tuberculosis with reinfection. mathe-525 matical the theory of the chemostat: dynamics of microbial 527 competition estimation of the transmission risk of the 2019-ncov and its 530 implication for public health interventions mathematics in population biology pandemic potential of 2019-ncov. the lancet infectious diseases reproduction numbers and 536 sub-threshold endemic equilibria for compartmental models of disease transmission backward bifurcation of an epidemic model with 539 saturated treatment function preliminary estimation of 543 the basic reproduction number of novel coronavirus (2019-ncov) in china, from 2019 544 to 2020: a data-driven analysis in the early phase of the outbreak. international 545 journal of infectious diseases estimating the unreported 548 number of novel coronavirus (2019-ncov) cases in china in the first half of january 549 2020: a data-driven modelling analysis of the early outbreak the authors are grateful to the anonymous referees for their careful reading, valuable 441 the authors declare that they have no conflicts of interest. 442 appendix a the center manifold theory [7; 8] is used to determine the existence of the backward 553 bifurcation phenomenon of the model (2.1) theoretically. theorem 7.1. let us consider the following general system of ordinary differential equations with a parameter φwithout loss of generality, it is assumed that x = 0 is an equilibrium for system (a-1) 555 for all values of the parameter φ. (2) matrix a has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. let f k be the k-th component of f andthen, the local dynamics of system (a-1) around 0 are totally determined by the sign of (ii) a < 0, b < 0. when φ < 0, with |φ| 1, x = 0 is unstable; when 0 < φ 1, 565 x = 0 is locally asymptotically stable and there exists a negative unstable equilibrium; (iii) a > 0, b < 0. when φ < 0, with |φ| 1, x = 0 is unstable and there exists a 567 locally asymptotically stable negative equilibrium; when 0 < φ 1, x = 0 is stable and a 568 positive unstable equilibrium appears; (iv) a < 0, b > 0. when φ changes from negative to positive, x = 0 changes its 570 stability from stable to unstable. correspondingly, a negative unstable equilibrium becomes 571 positive and locally asymptotically stable. in particular, if a > 0, b > 0 then a backward bifurcation occurs at φ = 0. key: cord-301150-41lfsedz authors: sardar, tridip; nadim, sk shahid; rana, sourav; chattopadhyay, joydev title: assessment of lockdown effect in some states and overall india: a predictive mathematical study on covid-19 outbreak date: 2020-07-08 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110078 sha: doc_id: 301150 cord_uid: 41lfsedz in the absence of neither an effective treatment or vaccine and with an incomplete understanding of the epidemiological cycle, govt. has implemented a nationwide lockdown to reduce covid-19 transmission in india. to study the effect of social distancing measure, we considered a new mathematical model on covid-19 that incorporates lockdown effect. by validating our model to the data on notified cases from five different states and overall india, we estimated several epidemiologically important parameters as well as the basic reproduction number (r(0)). combining the mechanistic mathematical model with different statistical forecast models, we projected notified cases in the six locations for the period may 17, 2020, till may 31, 2020. a global sensitivity analysis is carried out to determine the correlation of two epidemiologically measurable parameters on the lockdown effect and also on r(0). our result suggests that lockdown will be effective in those locations where a higher percentage of symptomatic infection exists in the population. furthermore, a large scale covid-19 mass testing is required to reduce community infection. ensemble model forecast suggested a high rise in the covid-19 notified cases in most of the locations in the coming days. furthermore, the trend of the effective reproduction number (r(t)) during the projection period indicates if the lockdown measures are completely removed after may 17, 2020, a high spike in notified cases may be seen in those locations. finally, combining our results, we provided an effective lockdown policy to reduce future covid-19 transmission in india. • a new mathematical model on covid-19 that incorporates lockdown effect • several model parameters as well as the basic reproduction number are estimated • we provide ensemble model forecast under five different lockdown scenarios • correlation between important parameters with the lockdown effect are derived • combining all the results, we proposed an effective lockdown policy cov-2), were recorded worldwide [1] . coronaviruses are enveloped non-segmented positive13 sense rna viruses that belongto the coronaviridae family and the order nidovirales, and 14 are widely distributed among humans and other mammals [2] . the novel coronavirus, 15 covid-19 started in mainland china, with a geographical emphasis at wuhan, the capi16 tal city of hubei province [3] and has widely spread all over the world. many of the initial 17 cases were usually introduced to the wholesale huanan seafood market, which also traded 18 live animals. clinical trials of hospitalized patients found that patients exhibit symptoms 19 consistent with viral pneumonia at the onset of covid-19, most commonly fever, cough, 20 sore throat and fatigue [4] . some patients reported changes in their ground-glass lungs; 21 normal or lower than average white lymphocyte blood cell counts and platelet counts; has seriously disrupted the life, economy and health of citizens. this is a great concern 43 for everyone how long this scenario will last and when the disease will be controlled. 44 mathematical modeling based on system of differential equations may provide a comthe model we developed in this paper is based on the interaction of seven mutually ex population in the exposed compartment (e) increased by new infection coming from 100 susceptible compartment. a fraction κ of the exposed individuals become symptomatic 101 infected and remaining fraction (1 − κ) become asymptomatic infected after the disease 102 incubation period 1 σ . exposed population also decreased due to natural death at a rate 103 µ. asymptomatic infected compartment (a) increased due to a fraction (1 − κ) of infec-105 tion coming from exposed compartment. since, asymptomatic covid-19 cases are hard 106 to detect therefore, we assume that asymptomatic infection are not notified. population 107 in this compartment is decreased due to natural recovery and deaths at a rate γ 1 and µ, 108 respectively. population in the symptomatic infected compartment (i) increased due to a fraction 110 κ of infection coming from exposed compartment after the incubation period 1 σ . this 111 compartment decreased due to natural recovery at a rate γ 2 , natural death at a rate µ 112 and those infected population who are notified & hospitalized at a rate τ . notified & hospitalized infected population (c) increased due to influx of infection 114 coming from symptomatic infected class at a rate τ . this population decreased due to 115 natural death at a rate µ, disease related deaths at a rate δ, and recovery from covid-19 116 at a rate γ 3 . we assume that population of this compartment do not mix with the general model without lock-down a diagram of our model is provided in fig 1. information of our model parameters is 130 provided in table 1 . disease-free state is locally asymptotically stable whenever the corresponding basic re-136 production number (r 0 ) is less than unity (see supplementary appendix). by using a 137 nonlinear lyapunov function, it is also seen that the disease-free equilibrium is globally 138 asymptotically stable whenever r 0 < 1 (see supplementary appendix). in addition, the several important epidemiological parameters (see table 1 ) of our mathematical where, ∆t i is the time step length andθ is the set of unknown parameters of the modrate for our covid-19 mathematical model (see table 1 and table 1 and table 2 ) to obtained the forecast during the mentioned time period. forecast based on 20% reduction in current lockdown rate: we followed the same 206 procedure as previous two scenarios with 20% decrement in the estimate of lockdown rate 207 (see table 1 and table 2 ) to obtained the forecast during the mentioned time period. forecast based on 30% reduction in current lockdown rate: we followed the same 209 procedure as previous three scenarios with 30% decrement in the estimate of lockdown 210 rate (see table 1 and table 2 ) to obtain the forecast during the mentioned time period. are same and its expression is provided below: the effective reproductive number (r t ) is defined as the expected number of secondary 220 infection per infectious in a population made up of both susceptible and non-susceptible 221 hosts [33] . if r t > 1, the number of new cases will increase, for r t = 1, the disease 222 become endemic, and when r t < 1 there will be a decline in new cases. following [33] , the expression of r t is given as follows: where,ŝ is the fraction of the host population that is susceptible. r 0 can easily be estimated by plugin the sample values of the unknown parameters 226 (see table 2 ) of the model without lockdown (2.1) in the expressions of r 0 . following procedure is adapted to estimate r t during may 17, 2020 till may 31, 2020 228 under two lockdown scenarios: • using current estimate of the lockdown rate and different parameters of our mathe-230 matical model (see table 1 and • using different parameters (see table 1 and table 2) ically measurable parameters of our mathematical model (see fig 1) . there are several 239 important parameters of our mathematical model (see table 1 ) and among them there τ , respectively from their respective ranges (see table 1 ). partial rank correlation and 249 its corresponding p-value are examined to determine the relation between two mentioned 250 parameters with the lockdown effect and r 0 , respectively. table s3 ). that low percentage (about 11% to 20%) of symptomatic infected in the population (see 270 table 2 ). however, in tn and pj, relatively higher percentage (about 82% to 88%) of 271 symptomatic infection is found (see table 2 ). in overall india, our estimate shows that 272 currently about 62% of new infection are symptomatic (see table 2 ). except for gj, in 273 other five locations, estimate of the transmission rate (β 1 ) are found to be in same scale (see table 2 ). relatively higher value of β 1 is found in gujrat (see table 2 ). low value the estimates of ρ (below 50%) are found to be low (see table 2 ). this indicates small table 2 ). thus, lockdown is overall successful in those five states. however, this is not the case for 287 overall india, our estimate suggest that about 11% of the total susceptible population in 288 india maintained proper social distancing during the lockdown period (see table 2 ). our estimate of the basic reproduction number (r 0 ) (see table 3 ), in the six locations suggest that τ has a negative correlation on r 0 . thus, more testing will isolate more 296 infection from the community and therefore may reduce the covid-19 community trans-297 mission. furthermore, high positive correlation of κ with r 0 (see fig 3) indicates the table 4 in overall india (see table 4 ). these numbers are much higher than the total cumulative 306 cases between march 2, 2020 till may 15, 2020, in whole india. a global sensitivity analysis of κ and τ on the lockdown effect suggest that both of 308 these parameters have high positive correlation with the lockdown effect in all the six 309 locations (see fig. 5 ). therefore, lockdown will be effective in those region where higher 310 percentage of symptomatic infection is found in the population and also larger covid-19 311 mass testing will be required to isolate the cases. we may see a rise in the daily covid-19 cases in all of the six locations (see fig. 6 ). table 2 ), 15% reduction: daily notified case projection using 15% reduction in the estimated value of the lockdown rate (see table 2 ), 20% reduction: daily notified case projection using 20% reduction in the estimated value of the lockdown rate (see table 2 ), 30% reduction: daily notified case projection using 30% reduction in the estimated value of the lockdown rate (see table 2 ), and no lockdown: daily notified case projection based on no lockdown scenario, respectively. 1) . respective row subscripts are same as fig. 2 . all data are given in the format estimate (95% ci). fig. 2 . different lockdown scenarios are current rate: cumulative case projection using the estimated value of the lockdown rate (see table 2 ), 15% reduction: cumulative case projection using 15% reduction in the estimated value of the lockdown rate (see table 2 ), 20% reduction: cumulative case projection using 20% reduction in the estimated value of the lockdown rate (see table 2 ), 30% reduction: cumulative case projection using 30% reduction in the estimated value of the lockdown rate (see table 2 ), and no lockdown: cumulative case projection based on no lockdown scenario, respectively. all data are provided in the format estimate (95% ci). coronavirus covid-19 global cases by the center for systems science 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7-8 lakh crore to indian econ-452 omy modern epidemiology. lippincott the mathematics of infectious diseases population biology of infectious diseases: part i infectious diseases of humans: dy-502 namics and control the stability of dynamical systems the theory of the chemostat: dynamics of microbial 505 competition dynamical models of tuberculosis and their ap-507 plications conceptualization: tridip sardar, joydev chattopadhyay. data curation: tridip sardar, sk shahid nadim. formal analysis: tridip sardar, sk shahid nadim, sourav rana. investigation: tridip sardar, sk shahid nadim, sourav rana. methodology: tridip sardar, sk shahid nadim software: tridip sardar, sourav rana supervision: tridip sardar, joydev chattopadhyay. validation: tridip sardar writing original draft: tridip sardar, sk shahid nadim, sourav rana, joydev chattopadhyay. writing review & editing: tridip sardar, sk shahid nadim worlds-biggest-lockdown-may-have-cost-rs-7-8-lakh-crore-to-indian-economy/ all the authors declare that they have no conflicts of interest. key: cord-303030-8unrcb1f authors: gaeta, giuseppe title: social distancing versus early detection and contacts tracing in epidemic management date: 2020-07-16 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110074 sha: doc_id: 303030 cord_uid: 8unrcb1f different countries – and sometimes different regions within the same countries – have adopted different strategies in trying to contain the ongoing covid-19 epidemic; these mix in variable parts social confinement, early detection and contact tracing. in this paper we discuss the different effects of these ingredients on the epidemic dynamics; the discussion is conducted with the help of two simple models, i.e. the classical sir model and the recently introduced variant a-sir (arxiv:2003.08720) which takes into account the presence of a large set of asymptomatic infectives. different countries are tackling the ongoing covid-19 epidemics with different strategies. awaiting for a vaccine to be available, the three tools at our disposal are contact tracing, early detection and social distancing . these are not mutually exclusive, and in fact they are used together, but the accent may be more on one or the other. within the framework of classical sir [1] [2] [3] [4] [5] and sir-type models, one could say (see below for details) that these strategies aim at changing one or the other of the basic parameters in the model. in this note we want to study -within this class of modelswhat are the consequences of acting in these different ways. we are interested not only in the peak of the epidemics, but also in its duration. in fact, it is everybody's experience in these days that social distancing -with its consequence of stopping all kind of economic activities -has a deep impact on our life, and in the long run is producing impoverishment and thus a decline in living conditions of a large part of population. in the present study we will not specially focus on covid, but discuss the matter in general terms and by means of generalpurpose models. our examples and numerical computations will however use data and parameters applying to (the early phase of) the current covid epidemic in northern italy, in order to have realistic examples and figures; we will thus use data and parameters arising e-mail address: giuseppe.gaeta@unimi.it from our analysis of epidemiological data in the early phase of this epidemic [6, 7] 1 . unavoidably, we will also here and there refer to the covid case. some observations deviating from the main line of discussionor which we want to pinpoint for easier reference to them -will be presented in the form of remarks. the symbol marks the end of remarks. in the sir model [1] [2] [3] [4] [5] , a population of constant size (this means the analysis is valid over a relatively short time-span, or we should consider new births and also deaths not due to the epidemic) is subdivided in three classes: susceptibles, infected (and by this also infectives), and removed. the infected are supposed to be immediately infective (if this is not the case, one considers so called seir model to take into account the delay), and removed may be recovered, or dead, or isolated from contact with susceptibles. we stress that while in usual textbook discussions of the sir model [2] [3] [4] [5] the removed are either recovered or dead, in the framework of covid modeling the infectives are removed from the infective dynamics -i.e. do not contribute any more to the quadratic term in the eqs. (1) below -through isolation. this means in practice hospitalization in cases where the symptoms are heavy and a serious health problem develops, and isolation at home (or in other places, e.g. in some countries or region specific hotels were used to this aim) in cases where it is estimated that there is no relevant risk for the health of the infective. in this sense, the reader should pay attention to the meaning of r in the present context. the nonlinear equations governing the sir dynamics are written as d s/d t = − α s i d i/d t = α s i − βi (1) d r/d t = βi. these should be considered, in physicists' language, as mean field equations; they hold under the (surely not realistic) assumption that all individuals are equivalent, and that the numbers are sufficiently large to disregard fluctuations around mean quantities. note also that the last equation amounts to a simple integration, r (t) = r 0 + β t t 0 i(y ) dy ; thus we will mostly look at the first two equations in (1) . we also stress, however, that epidemiological data can only collect time series for r ( t ): so this is the quantity to be compared to experimental data [2] . in fact, as stressed in remark 1 , in the case of a potentially dangerous illness (as covid), once the individuals are identified as infective, they are effectively removed from the epidemic dynamic through hospitalization or isolation. according to our eqs. (1) , s ( t ) is always decreasing until there are infectives. the second equation in (1) immediately shows that the number of infectives grows if s is above the epidemic threshold γ = β/α. (2) thus to stop an epidemic once the numbers are too large to isolate all the infectives, we have three (non mutually exclusive) choices within the sir framework: (a) do nothing, i.e. wait until s ( t ) falls below the epidemic threshold; (b) raise the epidemic threshold above the present value of s ( t ) by decreasing α; (c) raise the epidemic threshold above the present value of s ( t ) by increasing β. in practice, any state will try to both raise β and lower α, and if this is not sufficient await that s falls below the attained value of γ . in order to understand how this is implemented, it is necessary to understand what α and β represent in concrete situations. the parameter β represents the removal rate of infectives; its inverse β −1 is the average time the infectives spend being able to spread the contagion. raising β means lowering the time from infection to isolation, hence from infection to detection of the infected state. the parameter α represents the infection rate , and as such it includes many thing. it depends both on the infection vector characteristics (how easily it spreads around, and how easily it infects a healthy individual who gets in contact with it), but is also depends on the occasions of contacts between individuals. so, roughly speaking, it is proportional to the number of close enough contacts an individual has with other ones per unit of time. it follows that -if properly implemented -social distancing results in reducing α. each of these two actions presents some problem. there is usually some time for the appearance of symptoms once an individual is infected, and the first symptoms can be quite weak. so early detection is possible only by fast tracing and laboratory checking of all the contacts of those who are known to be infected. this has a moderate cost (especially if compared to the cost of an intensive care hospital stay) but requires an extensive organization. on the other hand, social distancing is cheap in immediate terms, but produces a notable strain of the societal life, and in practice -as many of the contacts are actually work related -requires to stop as many production and economic activities as possible, i.e. has a formidable cost in the medium and long run. moreover, it cannot be pushed too far, as a number of activities and services (e.g. those carrying food to people, urgent medical care, etc.) can not be stopped. let us come back to (1) ; using the first two equations, we can study i in terms of s , and find out that as we know that the maximum i * of i will be reached when s = γ , this allows immediately to determine the epidemic peak . in practice, i 0 is negligible and for a new virus s 0 corresponds to the whole population, s 0 = n; thus note that only γ appears in this expression; that is, raising β or lowering α produces the same effect as long as we reach the same γ . on the other hand, this simple formula does not tell us when the epidemic peak is reached, but only that it is reached when s has the value γ . but if measures are taken, these should be effective for the whole duration of the epidemic, and it is not irrelevant -in particular if the social and economic life of a nation is stopped -to be able to evaluate how long this will be for. acting on α or on β to get the same γ will produce different timescales for the dynamics; see fig. 1 , in which we have used values of the parameters resulting from our fit of early data for the northern italy covid-19 epidemic [7] . this observation can be made more precise considering the scaling properties of (1) . in fact, consider the scaling numerically integrated and i ( t ) plotted in arbitrary units for given initial conditions and α, β parameters (solid), the maximum i * being reached at t = t * . then they are integrated for the same initial condition but raising β by a factor ϑ = 3 / 2 (dashed) with maximum i β = r i * reached at time t β = σ β t * ; and lowering α by the same factor ϑ = 3 / 2 (dotted) with maximum i α = i β reached at time t α = σαt * . time unit is one day, α = (4 / 3) * 10 −8 , β = 1 / 7 ; these parameters arise from our fitting of data from the early phase of covid epidemics in northern italy [7] ; the population of the most affected area in the initial phase is about 20 million, that of the whole italy is about 60 million. the numerical simulation is ran with n = 6 * 10 7 ; it results it is clear that under this scaling γ remains unchanged, and also the equations are not affected; thus the dynamics is the same but with a different time-scale . the same property can be looked at in a slightly different way. first of all, we note that one can write α = β/γ ; moreover, α appears in (1) only in connection with s , and it is more convenient to introduce the variable now, let us consider two sir systems with the same initial data but different sets of parameters, and let us for ease of notation just consider the first two equations of each. thus we have the two systems we can consider the change of variables ( λ > 0) with this, (8) becomes we can thus eliminate the factor λ in both equations. however, if we had chosen λ = β/β, we get ˆ β = β; if moreover γ = γ , the resulting equation is just but we had supposed the initial data for { s, i } and for { s , i } (and hence also for ϑ and ϑ ) to be the same. we can thus directly compare (10) with (7) . we observe that { ϑ , i } have thus exactly the same dynamics in terms of the rescaled time τ as { ϑ, i } in terms of the original time t . in particular, if the maximum of i is reached at time t * , the maximum of i is reached at τ * = t * , and hence at t * = λ τ * = λ t * . (11) analytical results on the timescale change induced by a rescaling of the α and β parameters have recently been obtained by m. cadoni [9] ; see also [10] . we have supposed infected individuals to be immediately infective. if this is not the case an "exposed" class should be introduced. this is not qualitatively changing the outcome of our discussion, so we prefer to keep to the simplest setting. (moreover, for covid it is known that individuals become infective well before developing symptoms, so that our approximation is quite reasonable.) one of the striking aspects of the ongoing covid-19 epidemic is the presence of a large fraction of asymptomatic infectives [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] ; note that here we will always use "asymptomatic" as a shorthand for "asymptomatic or paucisymptomatic", as also people with very light symptoms will most likely escape to clinical detection of covid -and actually most frequently will not even think of consulting a physician. 2 in order to take this aspect into account, we have recently formulated a variant of the sir model [7] in which together with known infectives i ( t ), and hence known removed r ( t ), there are unregistered infectives j ( t ) and unregistered removed u ( t ). note that in this case removal amounts to healing; so while the removal time β −1 for known infected corresponds to the time from infection to isolation, thus in general slightly over the incubation time t i (this is t i 5.1 days for covid), the removal time η −1 for unrecognized infects will correspond to incubation time plus healing time. in the model, it is supposed that symptomatic and asymptomatic infectives are infective in the same way. this is not fully realistic, as one may expect that somebody having the first symptoms will however be more retired, or at east other people will be more careful in contacts; but this assumption simplifies the analysis,and is not completely unreasonable considering that for most of the infection-to-isolation time β −1 the symptoms do not show up. the equations for the a-sir model [7] are note that here too we have a "master" system of three equations (the first three) while the last two equations amount to dithe parameter ξ ∈ [0, 1] represents the probability that an infected individual is detected as such, i.e. falls in the class i . in the absence of epidemiological investigations to trace the contacts of known infectives, this corresponds to the probability of developing significant symptoms. in the first (arxiv) circulated version [8] of our previous work [8] , some confusion about the identification of the class j was present, as this was sometimes considered to be the class of asymptomatic infectives, and sometimes that of not registered ones 3 . while this is not too much of a problem considering the "natural" situation, it becomes so when we think of action on this situation. actually, and unfortunately, this confusion has a consequence exactly on one of the points we want to discuss here, i.e. the effect of a campaign of chasing the infectives, e.g. among patients with light symptoms or within social contacts of known infectives; let us thus discuss briefly this point. if j is considered to be the set of asymptomatic virus carriers, then a rise in the fraction of these who are known to be infective, and thus isolated, means that the average time for which asymptomatic infectives are not isolated is decreasing. in other words, we are lowering η −1 and thus raising η. on the other hand, in this description ξ is the probability that a new infective is asymptomatic, and this depends only on the nature of the virus and its interactions with the immune system of the infected people; thus in this interpretation ξ should be considered as a constant of nature, and it cannot be changed. (this is the point of view taken in [7] ; however some of the assumptions made in its first version [8] were very reasonable only within the concurrent interpretation, described in a moment.) on the other hand, if j is the class of unknown infectives, things are slightly different. in fact, to be in this class it is needed ( a ) that the individual has no or very light symptoms; but also ( b ) that he/she is not traced and analyzed by some epidemiological campaign, e.g. due to contacts with known infected or because belonging to some special risk category (e.g. hospital workers). in this description, η is a constant of nature, depending on the nature of the virus and on the response of the "average" immune system of (asymptomatic) infected people, while effort s to trace asymptomatic infectives will act on raising the probability ξ . we want to discuss the effect of early detection of infectives, or tracing their contacts, within the second mentioned framework. note that a campaign of tracing contacts of infectives is useful not only to uncover infectives with no symptoms, but if accompanied by effective isolation of contacts with known infectives, and thus of those who are most likely to be infective, it will also reduce the removal time of "standard" (i.e. symptomatic) infectives, possibly to a time smaller than the incubation time itself. in this sense, we will look at an increase in ξ as early detection of infectives , and at an increase in both β and η (thus a reduction in the removal times β −1 and η −1 ) as tracing contacts of infectives . this should be kept in mind in our final discussion about the effect of different strategies. as mentioned above, one should also avoid any confusion between asymptomatic and pre-symptomatic infection. in our description, pre-symptomatic infectives -i.e. individuals which are infective and which do not yet display symptoms, but which will at a later stage display them -are counted in the class of "standard" infectives, i.e. those who will eventually display symptoms and hence be intercepted by the health system with no need for specific test or contact racing campaigns, exactly due to the appearance of symptoms. actually one expects that except for the early phase of the epidemics in the countries which were first hit in a given area (such as china for asia, or italy for europe), when symptoms could be attributed to a different illness, most infections by symptomatic people are actually pre-symptomatic , as with the appearance of symptoms people are either hospitalized or isolated at home; and even before any contact with the health system they will avoid contacts with other -and other people will surely do their best to avoid contacts with anybody displaying even light covid symptoms. in the case of asymptomatic infectives, instead, unless they are detected by means of a test or contact tracing campaign -see the forthcoming discussion -they remain infective until they recover, so that in this case removal is indeed equivalent to (spontaneous) recovery. this approach, indeed, was taken in one of the areas of early explosion of the contagion in northern italy, i.e. in vò euganeo; this had the advantage of being a small community (about 30 0 0 residents), and all of them have been tested twice while embargo was in operation. in fact, this was the first systematic study showing that the number of asymptomatic carriers was very high, quite above the expectations [23] . apart from its scientific interest, the approach proved very effective in practical terms, as new infectives were quickly traced and in that specific area the contagion was stopped in a short time. while testing everybody is not feasible in larger communities, the "follow the contacts" approach could be used on a larger scale, especially with the appearance of new very quick kits for ascertaining positivity to covid. the model will thus react to a raising of ξ by raising the fraction of i within the class of infectives, i.e. in k = i + j; but at the same time, as critical patients are always the same, i.e. represents always the same fraction of k , we should pay attention to the fact they will now represent a lower fraction of i . the chinese experience shows that critical patients are about 10% of hospitalized patients (i.e. of those with symptoms serious enough to require hospitalization); and hospitalized patients represented about half of known infected, the other being cured and isolated at home. similar percentages were observed in the early phase of the covid epidemic in italy; the fraction of infectives isolated at home has afterwards diminished, but it is believed that this was due to a different policy for lab exams, i.e. checking prioritarily patients with multiple symptoms suggesting the presence of covid rather than following the contacts. actually this policy was followed in most of italy, but in one region (veneto) the tracking of contacts and lab exams for them was pursued, and in there the percentages were much more similar to those known to hold for china. in our previous work [7] we have considered data for the early phase of covid epidemics in italy, and found that β −1 7 best fits them while the estimate η −1 21 was considered as a working hypothesis. this same work found as value of the contact rate in the initial phase α 1 . 13 * 10 −8 , and we will use this in our numerical simulations. 4 it should be stressed that the extraction of the parameter α from epidemiological data is based on the number s 0 n of susceptibles at the beginning of the epidemic, thus α and hence γ depend on the total population. the value given above was obtained considering n = 2 * 10 7 , i.e. the overall population of the three regions (lombardia, veneto and emilia-romagna) which were mostly affected in the initial phase. our forthcoming discussion, however, does not want to provide a forecast on the development of the covid epidemic in northern italy; we want instead to discuss -with realistic parameters and framework -what would be the differences if acting with different strategies in an epidemic with the general characteristics of the covid one. thus we will adopt the aforementioned parameters as "bare" ones (different strategies consisting indeed on acting on one or the other of these) but will apply these on a case study initial condition; this will be given by one important parameter is missing from this list, i.e. the detection probability ξ . following li et al. [24] we assumed in previous work that ξ is between 1/10 and 1/7. later works (and a general public interview by the head of the government agency handling the epidemic [25] ) suggested that the lower bound is nearer to the truth; moreover a lower ξ will give us greater opportunity to improve things by acting on it (we will see this is not the best strategy, so it makes sense to consider the setting more favorable to it). we will thus run our simulation starting from a "bare" value as for the total population, we set n = 2 * 10 7 . with these choices we get a projection of what could have happened if no action was undertaken. a note by an oxford group [26] , much discussed (also in general press [27] ) upon its appearance, hinted that in italy and uk this fraction could be as low as ξ = 1 / 100 . we have ascertained that with this value of ξ , and assuming α was not changed by the restrictive measures adopted in the meanwhile, the a-sir model fits quite well the epidemiological data available to the end of april. however, despite this, we do not trust this hypothesisat least for italy -for various reasons, such as (in order of increasing relevance): ( i ) a viral infection showing effects only in 1% of affected individuals would be rather exceptional; ( ii ) albeit in our opinion the effect of social distancing measures adopted in italy is sometimes overestimated, we trust that there has been some effect; ( iii ) if only 1% of infected people was detected, in some parts of italy the infected population would be over 100%. on the other hand, the main point made by this report [26] , i.e. that only a large scale serological study, checking if people have covid antibodies, will be able to tell how diffuse the infection is -and should be performed as soon as possible -is by all means true and correct. see also [28] . a look at eqs. (12) shows that i will grow provided where again γ = β/α, and we have introduced the ratio x ( t ) of known infectives over total infectives. in other words, now the epidemic threshold depends on the distribution of infectives in the classes i and j . note that if x = ξ (as one would expect to happen in early stages of the epidemic), then γ i = γ . needless to say, we have a similar result for j , i.e. j will grow as far as thus the epidemic threshold for unregistered infectives is it is important to note that x is evolving in time. more precisely, by the equations for i and j we get dx dt the behavior observed in fig. 2 , which displays x ( t ) and related quantities on a numerical solution of eq. (12) , can be easily understood intuitively. in the first phase of the epidemic, there is an exponential growth of both i and j ; due to the structure of the equations, they grow with the same rate, so their ratio remains constant; on the other hand, once the dynamics get near to the epidemic peak, the difference in the permanence time of the two (that is, the time individuals remain in the infect class) becomes relevant, and we see (plots (a2) and (b2) of fig. 2 ) that not only the peak for j is higher than the one for i , but it occurs at a slightly later time. moreover, descending off the peak is also faster for i , as β −1 < η −1 , and thus x further decreases, until it reaches a new equilibrium while both classes i and j go exponentially to zero. if we look at (19) we see that for fixed s the variable x would have two equilibria (one stable with 0 < x < 1 and one unstable with x > 1, stability following from β − η > 0 ), easily determined solving d x/d t = 0 . numerical simulations show that -apart from an initial transient -actually x ( t ) stays near, but in general does not really sticks to, the stable fixed point determined in this way. a relevant point should be noted here. if we consider the sum (20) of all infectives, the a-sir model can be cast as a sir model in terms of s, k , and q = r + u as as x varies in time, this average removal rate is also changing. on the other hand, the basic reproduction number (brn) ρ 0 (this is usually denoted as r 0 , but we prefer to change this notation in order to avoid any confusion with initial data for the known removed r ( t )) for this model will be in other words, not taking the asymptomatic infectives into account leads to an underestimation of the brn. if the standard sir model predicts a brn of ρ 0 , the a-sir model yields a brn ˆ ρ 0 given this means that the epidemic will develop faster, and possibly much faster, than what one would expect on the basis of an estimate of ρ 0 based only on registered cases, which in the initial phase are a subset of symptomatic cases as the symptoms may easily be leading to a wrong diagnosis (in the case of covid they lead to a diagnosis of standard flu). with our covid-related values β = 1 / 7 , η = 1 / 21 , and assuming that in the early phase there is thus a good reason for being surprised by the fast development of the epidemic: the actual brn is substantially higher than the one estimated by symptomatic infections [29] . more generally, one would wonder what is the effect of the "hidden" infectives j ( t ) on the dynamics of the known infectives i ( t ) -which, we recall, include the relevant class of seriously affected infectives -and it appears that there are at least two, contrasting, effects: 1. on the one hand, the hidden infectives speed up the contagion spread and hence the rise of i ( t ); 2. on the other hand, they contribute to group immunity, so the larger this class the faster (and the lower the i level at which) the group immunity will be reached. the discussion above shows that the balance of these two factors leads to a much lower epidemic peak, and a shorter epidemic time, than those expected on the basis of the standard sir model (albeit in the case of covid with no intervention these are still awful numbers). on the other hand, we would like to understand if uncovering a larger number of cases (thus having prompt isolation of a larger fraction of the infectives) by early detection , i.e. raising ξ , would alter the time-span of the epidemic. it appears that this effect can be only marginal, as it appears only past the epidemic peak. we stress that this statement refers to "after incubation" analysis; if we were able to isolate cases before they test positive -i.e. to substantially reduce β −1 -the effect could be different. we will discuss this point, related to contact tracing , later on. an ongoing epidemic is not a laboratory experiment, and apart from not having controlled external conditions, i.e. constant parameters, the very collection of data is of course not the top priority of doctors fighting to save human lives. there has been considerable debate on what would be the most reliable indicator to overcome at least the second of these problems. one suggestion is to focus on the number of deaths; but this is itself not reliable, as in many cases covid is lethal on individuals which already had some medical problem, and registering these deaths as due to covid or to some other cause depends on the protocol adopted, and in some case also on political choices, e.g in order to reassure citizens (or on the other extreme, to stress great care must be taken to avoid contagion). another proposed indicator, possibly the most reliable in order to monitor the development of the epidemic, is that of patients in intensive care units. this appears to be sufficiently stable over different countries, and e.g. the italian data tend to reproduce in this respect -at least in regions where the sanitary system is not overstretched -the chinese ones. in this case, ic patients are about 20% of the total number of hospitalized cases; in china and for a long time also in italy (when protocols for choosing would-be cases to be subject to laboratory analysis have been stable), hospitalized cases have been about half of the known infection cases, the other having shown only minor symptoms and been cured (and isolated) in their home. the other, more widely used, indicator is simply the total number of known cases of infection. in view of the presence of a large class of asymptomatic infectives, this itself is strongly depending on the protocols for chasing infectives. on the other hand, this is the most available indicator: e.g., the w.h.o. situation reports [30] provide these data. each of these indicators, thus, has advantages and disadvantages. we will just use the who data on known infected. in particular, in the case of covid we expect that with ξ 0 the "bare" constant describing the probability that an infection is detected, out of the class i ( t ) we will have a 50% of infected with little or no symptoms ( i l ), a 40% of standard care hospitalized infected ( i h ), and a 10% of ic hospitalized infected ( i ic ). needless to say, this class is the most critical one, also in terms of strain on the health system. more generally, we say that with ξ 0 the "bare" constant describing the probability that the infection under study is detected, there is a fraction χ 0 (of the detected infections) belonging to the i ic class; that is, i ic (t) = χ 0 i(t ) . we stress this depends on the protocol used to trigger laboratory tests; in our general theoretical discussion, this is any such protocol and we want to discuss the consequences of changing this in the sense of more extensive tests. we are now ready to discuss how modification of one or the other of the different parameters ( α, β, ξ ) on which we can act by various means will affect the a-sir dynamics. as it should be expected, this will give results similar to those holding for the sir model, but now we have one more parameter to be considered and thus a more rich set of possible actions. fig. 3 . the effect of a change in ξ on the i ic class. we have used β = 1 / 7 , η = 1 / 21 , and α = 1 . 13 * 10 −8 as in fig. 2 , with a total population of n = 2 * 10 7 , and ran simulations with ξ = 1 / 10 (solid curve) and with ξ = 1 / 4 (dashed curve). the substantial increase in ξ produces a reduction in the epidemic peak and a general slowing down of the dynamics, but both these effects are rather small. a more extensive test campaign will raise ξ , say from ξ 0 to ξ 1 ; but of course this will not change the number of the most serious cases, as these are anyway getting to hospital and detected as being due to the infection in question. thus the new fraction χ 1 of detected infections which need special care will be such that in order to describe the result of raising ξ , we should thus compare plots of this is what we do, indeed, in fig. 3 . raising ξ corresponds to having more infective detected, and has some advantages from the point of view of the epidemic dynamics. in practical terms, this means extending tests to a larger class of subjects, and be able to isolate a larger fraction of asymptomatic infectives with the same speed and effectiveness as symptomatic ones. a different strategy for rapid action is also possible, and it consists of rapid isolations of subjects who had contacts with people known to have been infected, or who have themselves been in contact with known infectives (and so on). in other words, the strategy would be to isolate would-be infection carriers before any symptom could show up. this means that β −1 could be even smaller than the usual infection-to-isolation time (about seven days for covid) for symptomatic infectives, and even shorter than the incubation time (about five days for covid). it should be stressed that as each of these "possible infected" might have a small probability of being actually infected (depending on the kind of contacts chain leading to him/her from known infectives), here "isolation" does not necessarily mean top grade isolation, but might amount to a very conservative lifestyle, also -and actually, especially -within home, where a large part of registered chinese contagions took place. (the same large role of in-home contagion was observed in italy in the course of lockdown.) we have thus ran a simulation in which ξ is not changed, but β is raised from β 0 = 1 / 7 to β = 1 / 3 ; the result of this is shown in fig. 4 . in this case we have a marked diminution of the epidemic peak, and a very slight acceleration of the dynamics. 4 . the effect of a change in β on the i ic class. we have used ξ = 1 / 10 , η = 1 / 21 , and α = 1 . 13 * 10 −8 as in fig. 2 , with a total population of n = 2 * 10 7 , and ran simulations with β = 1 / 7 (solid curve) and with β = 1 / 3 (dashed curve). the substantial increase in β produces a marked reduction in the epidemic peak and a very slightly faster pace in the dynamics. we have so far not discussed the most basic tool in epidemic containment, i.e. social distancing. this means acting on the parameter α by reducing it. direct measurement on the epidemiological data for northern italy show that this parameter can be reduced to about 20% of its initial value with relatively mild measures. in fact, albeit the media speak of a generalized lockdown in italy, the measures have closed schools and a number of commercial activities, but for the rest were actually more pointing at limiting leisure walk and sports and somewhat avoiding contacts in shops or in work environment than to a real lockdown as it was adopted in wuhan. this is a basic action to be undertaken, and in fact it is being taken by all nations. it is also the simplest one to be organized (albeit with high economic and social costs in the long run) and an action which can be taken together with other ones. no doubt this should be immediately taken when an epidemic is starting, and accompanied by other measures -such as those discussed above. but here we want to continue our study of what it means by itself in terms of modification of the epidemic dynamics. it is not clear what can be achieved in terms of reduction of social contacts. in fact, once the epidemic starts most of the dangerous contacts are the unavoidable ones, such as those arising from essential services and production activity (e.g. production and distribution of food or pharmaceutical goods), contacts at home, and above all contacts in hospitals. thus, after a first big leap downward corresponding to closing of schools and universities on the one side, and a number of unessential commercial activities on the other, and restrictions on travels, it is difficult to further reduce social contacts, not to say that this would have huge economic and social costs, and also a large impact on the general health in terms of sedentariness-related illness (and possibly mental health). a number of countries tried to further reduce social contacts by forbidding citizens to get out of their home; this makes good sense in densely populated areas, but is useless in many other areas. the fortunate slogan "stay home" risks to hide to the general public that the problem is not to seclude oneself in selfpunishment, but to avoid contacts . we point out that there is a further obstacle to reducing social contacts: as seen in the context of the simple sir model, reducing α will lower the epidemic peak, but it will also slow down the whole dynamic . while this allows to gain precious time to prepare fig. 5 . the effect of a change in α on the i ic class. we have used β = 1 / 7 , ξ = 1 / 10 , η = 1 / 21 , with a total population of n = 2 * 10 7 , and ran simulations with α = 1 . 13 * 10 −8 (solid curve) and with α = 8 . 47 * 10 −9 (dashed curve). the reduction in α produces a marked reduction in the epidemic peak and also a marked slowing down in the dynamics. hospitals to stand the big wave, there is some temporal limit to an extended lockdown, and thus this tool cannot be used to too large an extent. we have thus ran a simulation in which β and ξ are not changed, while α is reduced by a factor 0.75 (smaller factors, i.e. smaller α, produce an untenable length of the critical phase); the result of this is shown in fig. 5 . in this case we have a relevant diminution of the epidemic peak, and also a marked slowing down in the dynamics. an important remark is needed here. it may seem, looking at this plot, that social distancing is less effective than other way of coping with the epidemic. but these simulation concern a sir-type model; this means in particular that there is no spatial structure in our model [2] . the travel ban is the most effective way of avoiding the spreading of contagion from one region to the others; while the "local" measures of social distancing can (and should) be triggered to find a balance with other needs, travel ban is the simplest and most effective way of protecting the communities which have not yet been touched by the epidemic. we can thus compare the different strategies we have been considering. this is done in fig. 6 where we plot together i ic ( t ) for all our different simulations; and in table 1 where we compare the height of the epidemic peak -again for i ic ( t ) -and the time at which it is reached. fig. 6 . the effect of different strategies. we plot i ic ( t ) for n = 2 * 10 7 in the "bare" case, i.e. for α = 1 . 13 * 10 −8 , β = 1 / 7 , ξ = 1 / 10 , η = 1 / 21 , and in cases where (only) one of the parameters is changed. in particular we have the bare case (solid line), the case where ξ is changed into ξ = 1 / 4 (dotted), the case where β is changed to β = 1 / 3 (dashed), and that where α is changed to α = 8 . 47 * 10 −9 (solid, blue). we also plot a horizontal line representing a hypothetical maximal capacity of ic units. (for interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) table 1 epidemic peak (for i ic ) and time for reaching it (in days) as observed in our numerical simulations. all simulation were ran with n = 2 * 10 7 and η = in fig. 6 we have also drawn a line representing the hypothetical maximal capacity of ic units. this stresses that not only the different actions lower the epidemic peak, but they also -and to an even larger extent -reduce the number of patients which can not be conveniently treated. in looking at this plot, one should remember that the model does not really discuss permanence in ic units, and that i ic are the infected which when detected will require ic treatment; this may go on for a long time -which is the reason why ic units are saturated in treating covid patients. so the plots are purely indicative, and a more detailed analysis (also with real parameters) would be needed to estimate the ic needs in the different scenarios. it should be stressed that the strategies of contacts tracing and early detection are usually played together; but as confusion could arise on this point, let us briefly discuss it. we have tried to stress that these two actions are not equivalent: one could conduct random testing, so uncovering a number of asymptomatic infectives, and just promptly isolate them without tracing their contacts;or on the other extreme one could just isolate everybody who had a (direct or indirect) contact with a known infective, without bothering to ascertain if they are themselves infective or not. this strategy would be as effective in containing the contagion (and less costly in terms of laboratory tests) than that of tracking contacts, test them (after a suitable time for the infection to develop and test give positive if this happens), and isolate only those who really turn infective. the difference is that if we isolate everybody this would involve a huge number of people (e.g. all those who have been in the same supermarket the same day as an infective; and their families and contacts etc etc); so in this context early detection actually should be intended as early detection of non-infectives , so that cautionary quarantine can be kept reasonably short in all the cases where it is not really needed. finally we recall that it is a triviality, and it was already mentioned in the introduction, that in real situations one has not to choose between acting on one or the other of the parameters, and all kind of actions should be pursued simultaneously. the numerical computations of the previous subsections suggest that increasing ξ -that is, detection of a larger fraction of asymptomatic -is not a very efficient strategy to counter the diffusion of an infection with a large number of asymptomatic infectives, while a prompt isolation of infectives is a more effective action. it should be recalled, however, that in our computations -and in particular on fig. 6 , where their outcomes are compared -we are focusing on the number of patients needing ic support, i.e. the most critical parameter from the point of view of the health system. in order to substantiate our conclusions, it is worth considering also different ways to evaluate the effect of different strategies. we have thus considered also a different indicator, i.e. the total number of infectives we have run several simulations, with total population n = 2 * 10 7 and with parameters α = μ α α 0 β = μ β β 0 η = μ η η 0 ξ = μ ξ ξ 0 . the outcome of these simulations is displayed in fig. 7 ; see its caption for the parameter (that is, the modulation factor) values in different runs. we see from fig. 7 that action on α slows down substantially the epidemic dynamics 5 and reduce the epidemic peak, while action on ξ or on β alone produce only a moderate effect. on the other hand, actions affecting the value of η (alone or together with the value of β) reduce substantially the epidemic peak and slightly slow down the dynamics. it may be noted that the shapes of the i ic ( t ) (see fig. 6 ) and of the k ( t ) (see fig. 7 ) are different; in particular, the decay of i ic ( t ) after attaining its peak is faster than the decay of k ( t ). this corresponds to what is observed in the epidemiological data for italy. we have considered epidemic dynamics as described by "mean field" models of the sir type; more specifically, we have first considered the classical kermack-mckendrick sir model [1] [2] [3] [4] [5] and then a recently introduced modified version of it [7] taking into account the presence of a large set of asymptomatic -and thus most frequently not detected -infectives. these models depend on several parameters, and different types of measures can to some extent change these parameters and thus the epidemic dynamics. in particular, this action can effect two basic characteristics of it, i.e. the height of the epidemic peak and the time-span of the epidemic. while it is clear that in facing a real lethal epidemics (such as the ongoing covid epidemic) all actions which can contrast it should be developed at the same time, in this paper we have considered the result -within these models -of different tools at our disposal, i.e. (generalized) social distancing, early detection (of asymptomatic infectives) and contacts tracing (of symptomatic and asymptomatic infectives). it turns out that -both in the classical sir model and in the modified a-sir one -social distancing is effective in reducing the epidemic peak, and moreover it slows down the epidemic dynamics. on the other hand, early detection of asymptomatic infectives seems to have only a moderate effect in the reduction of the epidemic peak for what concerns critical cases, and also a very little effect on the temporal development of the epidemic. in contrast, contact tracing has a strong impact on the epidemic peak -also in terms of critical cases -and does not substantially alter the temporal development of the epidemic, at least for what concerns the curve describing the most serious cases. 6 remark 13. the conclusion that early detection of asymptomatic has only a moderate effect may appear to be paradoxical, and requires some further discussion. first of all we should remind that we are here actually talking about an increase of the parameter ξ (see remark 4 ) , while in a real situation early detection of asymptomatic will most likely go together with early detection of symptomatic, and hence a reduction in β as well. the increase of ξ per se means that some fraction of asymptomatic will be recognized as infective and be isolated on the same timescale β −1 as the symptomatic infectives, while the other asymptomatic will escape recognition and still be infective on a timescale η −1 . on the other hand, a realistic contact tracing campaign will lead to prompt isolation of symptomatic and asymptomatic alike, and thus correspond to a reduction in β −1 and in η −1 , and we have seen that this action is indeed the most effective one in terms of contrasting the spread of the epidemic. in other words, our result suggests that the key to fight covid is not so much in detection , but in prompt isolation of infectives, and most notably of asymptomatic ones. this can be achieved only by contact tracing -as already suggested by experienced epidemiologists. 7 slowing down the epidemic dynamic can be a positive or negative feature depending on the concrete situation and on the desired effects. it is surely positive in what concerns getting ready to face the epidemic peak, in particular in the presence of a faltering health system. on the other hand, it may be negative in that maintaining a generalized lockdown for a long time can have extremely serious economic and social consequences. balancing these two aspects is not a matter for the mathematician or the scientist, but for the decision maker; so we will not comment any further about this. it should also be recalled that our analysis was conducted in terms of very simple sir-type models, with all their limitations. in particular, we have considered no age or geographical or social structure, and only considered a population of "equivalent" indi-viduals. in particular, as we have noted above, in the early stage of an epidemic, which presumably develops in very populated areas, a generalized travel ban can simply stop the contagion to propagate to other (possibly less well equipped in medical terms) areas; moreover, social distancing measures can be implemented very simply -basically, by a government order (albeit if we look at the goal of these measures, i.e. reducing the occasion of exchanging the virus, a substantial role would be played by individual protection devices, such as facial masks; in many european countries, these were simply not available to the general public, and in some cases neither to medical operators, thus substantially reducing the impact of these measures) -and are thus the first action to be taken. in fact, in relation with the ongoing covid epidemics, one of the reproaches made to many governments is usually to have been too slow or too soft in stopping crowd gatherings, surely not the contrary. on the other hand, we hope that this study makes clear what are the consequences of different options. in particular, our study shows that contacts tracing , followed by prompt isolation of wouldbe infected people -is the only way to reduce the impact of the epidemic without having to live with it for an exceedingly long time. the veneto experience [23] shows that this strategy can be effectively im plemented without hurting privacy or personal freedom. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. we are now going to briefly discuss these matters; we point out that this appendix was inserted in the revised version of this paper, so it can make use of knowledge not available at the time of writing the first submitted version, nd mentions papers appeared after the first submittal. compartment models, i.e. sir-type ones in this context, are based on several implicit and explicit assumptions, which are not realistic in many cases and surely when attempting to describe the covid epidemic, in particular in a full country. that is, among other aspects, sir-type models are (in physics' language) mean field (averaged) models and as such describe the dynamics and the underlying system as if: • all individuals are equivalent in medical sense, i.e. they all have equivalent pre-existent health status and equivalent immune system and react in the same way to contact with the pathogen; • in particular, as we know that covid is statistically more dangerous for older people, we are completely disregarding the age structure of the population, as well as the existence of other high risk classes related to pre-existent pathologies: all these contribute to an average over the whole population; • all individuals are equivalent in social sense, i.e. they all have an equivalent social activity and hence the same number and intensity of contacts with other members of the group, thus the same exposure to (possible) infectives; • in particular, this means we are completely disregarding any geographical structure in the population, and consider in the same way people living in large cities or in remote villages, just considering them in the same global average; • similarly, we do not consider that work can cause some people to be specially exposed through contact with a large number of people (e.g. shop cashiers) or even with a large number of infected people (e.g. medical doctors or nurses). thus one cannot hope to retain, through such models, effects like the faster spreading of the infection in more densely populated areas or the specially serious consequences of the covid infection among older people. we stress that these could be obtained by including geographical, age or social structures into the model, i.e. increasing the number of considered classes; in principles this should provide a finer and more realistic description of the epidemic dynamic, and in fact it is done in cases for which there is a large set of data, e.g. for influenza. such structured models would of course loose the main attractive of the sir model, i.e. its simplicity -which also allows to understand in qualitative terms the mechanisms at work. in particular, a relevant intermediate class of models is that of sir-type models on networks: these take into account geographical and social structures and make use of known information about contacts between different groups of individuals and about different health characteristics of different groups. the problem with these networked or however more structured models is that the network should be inferred from data. in this respect, it could be objected that the influenza monitoring over many years could give us the relevant data for reconstruction of such network; but it is everybody's experience, by now, that the social behavior of people are completely different if dealing with a well known and not so serious (except for certain categories) illness like influenza or with an unknown and potentially lethal one like covid; this not to say that the restrictive measures put into effect by many governments have completely changed the interaction patterns among people, so that previously accumulated data cannot be used in the present situation. when thinking of covid; it should be kept in mind that even if the countries which were first hit by the epidemic, we only have data over some months; e.g. for italy we have about 100 days of data. if we were trying to give the model a geographical structure at the level of departments (which are themselves administrative units mostly with a very varied internal geographical structure), as there are 107 departments in italy this would require in the simplest form to evaluate a 107 × 107 interaction matrix, and i cannot see any way to reliably build this out of such a scarce set of data. moreover, the epidemiological data are to some extent not reliable, especially around the epidemic peak, in that they are collected in an emergency situation, when other priorities are present in hospitals (e.g. in italy the data show a weekly modulation, which appears to be due simply to the procedure of data collection); so an even larger amount of data would be needed to filter out statistical noise and random fluctuations. in this sense, the weak point of sir-type models, i.e. their being based on an average over the whole population, turns out to be an advantage: they contain few parameters (two for the sir, four for the a-sir) and are thus statistically more robust in that fluctuations are averaged efficiently with less data than for more refined models with a large number of parameters. similar considerations hold when one compares sirtype models to a purely statistical description or to an "emerging behavior" approach. these approaches are extremely powerful, but are effective when one has a large database to build on and to which compare the outcome of the "experiment" (in this case the epidemic) under consideration. when we deal with a completely new pathogen,like for covid, we simply don't have a database, and we can only rely on the very general features of infective dynamics -which are well coded by sir and sir-like models. in other words, we are not claiming the sir approach to be superior to others, but only that it is appropriate when we have few data -as for covid. within the sir-type class, the a-sir model is specially simple; from the theoretical point of view its appeal lies in that it is the simplest possible model taking into account the presence of a large class of asymptomatic infectives; thus it focuses on the effect of this fact without the complications of a more detailed model. but, of course, it makes sense to rely on this model only if it is able to give a good, or at least a reasonable, agreement with observed data. of course each infective agent has its own characteristics, and using only the general sir model would completely overlook them, apart from the different values of the α and β parameters. thus we have to do something more than just evaluating the sir parameters. in our study we have identified the presence of a large class of asymptomatic infectives as one of the key problems in facing the covid epidemic, and we have considered a simple model which allows to focus precisely on this aspect. one should be aware that the a-sir model is focusing on this and not considering other features of covid, and indeed other more detailed sir-type models for covid have been formulated and studied (see also below). here we are taking an approach which is classical in mathematical physics and mathematical modeling, i.e. try to build and study the simplest model describing the phenomenon of interest. this will give results which are quantitatively worse than a more detailed models, but which are qualitatively good in that the model is simple enough to see more clearly what are the mechanisms at work and to understand the qualitative features if the dynamics and the qualitative outcome of any intervention able to modify the parameters of the model. having said that, it remains true that -as mentioned aboveit makes sense to rely on this model only if it is able to give a reasonable agreement with observed data. this is not the argument of this paper, and it was discussed in a previous paper [7] ; the success of this model was the justification for this paper,i.e. for dishowever, the first version of this paper was submitted at mid-may, hence with two and half months of data available, while at the time of preparing this revised version we have four months of data; that represents a substantial increase in the available data, and it makes sense to wonder if the model is still describing the covid epidemic in italy. this is indeed the case, as shown in fig. 8 ; they represent epidemiological data as communicated by the italian health ministry and by who (and widely available online through the standard covid databases) against a numerical integration of the a-sir eqs. (12) . we refer to gaeta [7] for a discussion of the parameters and their determination. note that the contact rate α is assumed to vary in response to the restrictive measures (and to the availability of individual protection devices); as these measures were taken in different steps, we also have different values of α in different time intervals. more precisely, the equations were integrated for a total population of n = 6 * 10 7 for the period february 20, through june 26, with initial data at day 14 ( we stress that the parameter values are the same as in [7] , even for the most recent time, not considered in that paper: the model continues to reasonably well describe the development of the epidemic in italy. we focused on a specific sir-type model, but several models of this type have been considered in the context of covid modeling. here we give a very short overview of these, with no attempt to completeness -which cannot even be imagined in such a rapidly evolving field. first of all, we note that other researchers have considered, motivated by the ongoing covid epidemic, the temporal aspects of the standard sir dynamics. we mention in particular cadoni [9] (a related, but quite involved, approach had been considered by harko, lobo and mak [31] ) and barlow and weinstein [10] , who obtained an exact solution for the sir equations in terms of a divergent but asymptotic series [32] ; see also [33, 34] for a different approach to exact solution of sir and sir-type models. we also note that nonlinear modifications of the bilinear infection term of the standard sir model have been proposed -explicitly or implicitly -in the attempt to relate the standard sir model to covid dynamics [35, 36] . we find [36] of special interest, as this work introduces a model for the epidemic dynamics coupled to the immune system, and is thus able to take into account aspects related to the viral charge of infectives. extension of the sir model in the direction of allowing timedependence of the parameters -also to account for shifting public attitude -has also been considered [37] . as mentioned above, see remark 3 , considering the delay between infection and beginning of infectiveness would lead to consider seir-type models. the problem of temporal aspects of the dynamics for this class of models has been considered by becaer [38] . the role of asymptomatic transmission in this class of models has also been considered [39, 40] . the approach to sir by barlow and weinstein [10] leading to exact solution has been extended to seir model [41] . a generalization of the a-sir model, allowing for different infectiveness of symptomatic and asymptomatic infectives, has been considered by neves and guerrero [42] . more elaborated compartment models with a larger number of compartments have been considered by a number of authors. we would like to mention in particular two papers which we consider specially significant, i.e. the work by the pavia group, in which mathematicians, statisticians and medical doctors collaborated [43] , and the work by fokas, cuevas-maraver and kevrekidis [44] , in which such a model -involving five compartments like the present paper, but chosen in a different way -is used to discuss (as in the present paper) exit strategies from the covid lockdown. as mentioned in the main text, one could -and should -consider epidemic dynamics on networks [45] . attempts to analyze the covid epidemic in this way have of course been pursued, both on a small scale, with a network structure which can be determined by direct sociological study [46] , and on a nationwide scale [47] where the network structure has to be determined. this latter study [47] also attempted to evaluate the effect of the containment measures; such a matter is of course very relevant and has been considered by many authors in many countries; even a cursory mention of these is impossible, and we will just mention one study applying to italy [48] . we also stress that many of the papers mentioned above, see in particular [43, 44] aim at using the models they study to evaluate the effect of interventions and containment measures. finally we would like to end on a positive note, and mention that while on the one hand it was found that the presence of asymptomatic makes that the basic reproduction number of covid is higher than initially estimated [7, 29, 49] , the fact that the social contact rate is not uniform in the population makes that the herd immunity level should be lower than predicted on the basis of the standard sir-type models [50] ; this is a specially nice result of the analysis on networks,as it only depends on general -and very reasonable -properties of the network and not on its detailed structure, thus overcoming the low statistics problem mentioned in remark 14 above. contributions to the mathematical theory of epidemics mathematical biology. i: an introduction essential mathematical biology the mathematics of infectious diseases mathematical models in biology. siam arxiv: 2003.02062 ; 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zameer, aneela; muneeb, muhammad title: predictions for covid-19 with deep learning models of lstm, gru and bi-lstm date: 2020-08-19 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110212 sha: doc_id: 315676 cord_uid: y0qbkszx covid-19, responsible of infecting billions of people and economy across the globe, requires detailed study of the trend it follows to develop adequate short-term prediction models for forecasting the number of future cases. in this perspective, it is possible to develop strategic planning in the public health system to avoid deaths as well as managing patients. in this paper, proposed forecast models comprising autoregressive integrated moving average (arima), support vector regression (svr), long shot term memory (lstm), bidirectional long short term memory (bi-lstm) are assessed for time series prediction of confirmed cases, deaths and recoveries in ten major countries affected due to covid-19. the performance of models is measured by mean absolute error, root mean square error and r2_score indices. in the majority of cases, bi-lstm model outperforms in terms of endorsed indices. models ranking from good performance to the lowest in entire scenarios is bi-lstm, lstm, gru, svr and arima. bi-lstm generates lowest mae and rmse values of 0.0070 and 0.0077, respectively, for deaths in china. the best r2_score value is 0.9997 for recovered cases in china. on the basis of demonstrated robustness and enhanced prediction accuracy, bi-lstm can be exploited for pandemic prediction for better planning and management. and western pacific as (7, 515) ; while confirmed cases are (410,744), (6, 125, 802) , (1, 222, 070) , (2, 847, 887) , (1, 032, 167) , and (234,815) [1] . to be precise, covid-19 has followed specific patterns and these patterns are based on dynamic transmission of the epidemic. when it occurs, superseding measures of different methods are used to find and evaluate such infective diseases. any epidemic in a state or country has arisen with different aspect of magnitude with respect to time, particularly weather period changes and spread of virus over the time period, and exhibited as non-linear in nature. to capture these non-linear compelling changes, researchers have gained the attention and designed such non-linear systems to describe the abruptness of infective diseases [2] . therefore, mathematical models such as sir (susceptible-infective-removed) for analyzing the epidemics has been introduced [3] . a transmission model with incubation time for malaria [4] and a deterministic model to analyze the interaction between hiv and tuberculosis is successfully developed to solve the nonlinear behavior of parameters [5] . similar models of discrete time equations are used to control the infected population [6] . amid of physical and statistical methods, the difference is to learn the temporal behavior of data such as coronavirus and use of non-linear functions to predict the dynamics [7] [8] . usually statistical approaches are based on autoregressive integrated moving average (arima) model that is employed to predict the spread of epidemic trend covid-2019 [9] and seasonal autoregressive integrated moving average (sarima) model which estimates the fatality rate by use of time series analysis on influenza epidemic [10] . these models have also been used to monitor and predict the dengue hemorrhagic fever (dhf) cases in southern thailand [11] and hemorrhagic fever with renal syndrome (hfrs) cases in china to control diseases more effectively [12] . another popular statistical model in the field of health care system is known as artificial intelligence (ai) based which is used to learn and train the covid-19 dataset of hubei province in china to predict the epidemic peaks and trend size [13] . in numerous cases, these methods are not capable to fit actual data utterly and predicted accuracy is very low, while predicting the rise of covid-19 spread. in order to get better performance of statistical methods, machine learning (ml) models which cover several fields such as power and energy engineering [14] , technology [15] , psychology [16] , is used for early prediction and real-time spread of data. recently, one of ml approach namely, infection size aware random forest (isarf), observed by classification group has been proposed, which highlights the infection size and lung fields [17] . other models are multilayer perceptron (mlp) and adaptive network-based fuzzy inference system, (anfis) utilized for evaluating the complex variation behavior and predicting the covid-19 transmission [18] . hybrid approach of support vector regression (svr) and arima has been suggested to take the confirmed cases and give predictions related to the number of contaminated persons countrywide [19] . [24] . the novelty of the reported work lies in creating the three categories of confirmed cases, death cases and recovered cases from dataset and intelligently developing a covid-19 predictor to predict and analyze future trends of these three categories. this experiment is based on the data set of confirmed covid-19 cases available until june 27, 2020. additionally, owing to the dynamic nature of coronavirus, ml and dl models have been implemented for early predictions. the prominent features of the methodology are summarized in terms of highlights as follows:  statistical models as arima, ml technique of svr with polynomial and rbf kernels, and dl mechanisms of lstm, gru and bi-lstm are proposed to predict the covid-19 three categories, confirmed cases, deaths and recovered cases for ten countries.  accuracy of models is measured in terms of three performance measures, mae, rmse and r2_score.  bi-lstm time series model enhances the learning ability and memorizing the long sequence. dl techniques in general and bi-lstm in specific are proposed for smallest prediction error and higher accuracy. rest of the article is organized as follows: section ii describes the proposed methodologies, dataset and performance metrics; section iii includes detailed results of the designed scheme. while the conclusion are provided in the last section. in this work, two kind of methodologies, statistical model and machine learning models including simple and deep learning techniques are established for covid-19 predictions. in the first phase, design of arima and svr as simple ml algorithm are discussed, whereas in the next phase, description of various dl models are presented. the statistical performance in terms of three error measures, mae, rmse and r2_score are also specified in this section for performance evaluation. the graphical overview of the proposed scheme is illustrated in fig. 1 , in which three categories (confirmed case, deaths cases and recovered cases) of data is collected and after preprocessing, data is passed to respective models separately and performance of models are measured through error measures. furthermore, detail description of proposed models is provided below. arima model comprises three processes named as auto regression, integration and moving average which is data independent and employed for model architecture and parameter estimation that is linear function for past observations and arbitrary error [25] [26] . time series form of underlying process is: in equation (1) another effective time series implementation of support machine (svm) anticipated by vapnik is known as support vector regression [9] . both the svm and svr are used to minimize the error of margin and employ kernel functions for non-separable classes. the results can be improved by optimizing its parameters; in this regard grid and heuristic search are used to get best parameters [27] . svr for the multidimensional data is mathematically formulated as: here,  . this is useful to deal with nonlinear functions in which data is mapped into high dimension space known as kernel space for high accuracy results. finally, svr function is mathematically obtained as equation (6): here, primal formula of kernel function is represents the features in kernel space. various kernel functions such as rbf and polynomial kernels are used, and their mathematical formulae is given as: in equation (8, 9)  and d is the parameter of kernel that is tuned. rnn [28] has been employed for sequential time series applications with temporal dependencies. an unfolded rnn has the capability to process current data by use of previous data. meanwhile, rnn has the problem to train the long term dependencies data, which is solved by one of the variants of rnn. lstm anticipated by hochreiter and schmidhuber [29] , has been used as advance version of rnn network and has overcome the limitation of rnn by use of hidden layer unit known as memory cells. memory cells have the selfconnections that stored the network temporal state and controlled through three gates named as: input gate, output gate and forget gate [30] . the work of input gate and output gate is used to control the flow of memory cell input and outputs into the rest of network. in addition, forget gate has been added to the memory cell, which pass the output information with high weights from previous to next neuron. the information reside in memory depend upon the high activation results; if the input unit has high activation, the information is stored in memory cell. in addition, if the output unit has high activation then it will pass the information to next neuron. otherwise, input information with high weights resides in memory cell. lstm network is compute mapping between input sequence and output sequence, i.e. gru is the simple variant of lstm that has two gates, one is "update gate" which comprises of input, forget gates and "reset gate" [32] [33] . gru has no additional memory cell to keep information, therefore, it can only control information inside the unit. dataset of novel coronavirus is taken from the link [34]. the .csv file of confirmed cases, death cases and recovered cases of all countries is provided column wise. an individual file is created of these three categories from 22 january, 2020 to 27 june, 2020. covid19 dataset contains number of confirmed cases, deaths and recovered cases of 158 samples and we have taken cases from 1/22/2020 to 5/10/2020 for training purpose and to predict cases from 5/11/2020 to 6/27/2020. for each country, data comprises given cases for 110 days and have to predict for next 48 days. the data is preprocessed before it is given to ml models for training. three performance measures are used to evaluate the performance of the proposed model, these are mean absolute error (mae), root mean square error (rmse) and r 2 _score. c denotes the actual value and ĉ for estimated value. the expected values of mae is zero for the best model. rmse is well-defined as: (20) to demonstrates the variance between dependent and independent parameter. r 2 _score is presented as: the dataset comprises three features of confirmed cases, deaths and recovered cases. unscaled data slows down the convergence process. minmaxscaler subtracts the smallest value of feature and formerly divides by features range. the range is the difference between the original maximum and original minimum. minmaxscaler reserves the shape of the original distribution of data. it does not meaningfully change the information embedded in the original data and does not reduce the importance of outliers. parameters with their values of svr, arima and lstm is shown in table 1 , while results of actual and predicted cases in three categories in terms of performance measures are presented in table 2 . it can be observed from this table that none of the three models, arima, svr_poly, and svr_rbf fits the dataset very well and therefore does not generate consistent predictions. observing the values of rmse and mae, for some countries and even for some feature, one predicts better and for others, another model gives better results. in terms of r2_score, mostly the values are negative and thereby depicting poorer performance of the models than linear regressors. therefore, it can be inferred that none of these models is able to give reliable and accurate predictions. as a next step, deep learning techniques of lstm, gru and bi-lstm for three predicted categories are demonstrated in fig. 3 in terms of mae, rmse and r2_score. it is worth mentioning here that parameter optimization of all methods has been carried out through trial and error and values enlisted in table 1 have been used in generating all the results in this section. prediction errors in terms of performance measures are plotted as bar charts for comparison among dl techniques. the smallest value of mae is 20.79663 for israel among ten countries for confirmed case. as the number of cases is much more for usa and brazil, therefore the error measures are also higher for these countries as opposed to rest of other countries in actual figures. performance measure, r2_score, independently represent values very close to unity without any normalization and inverse transformation, which is a good sign of a consistent, efficient and accurate model for all countries and all cases. normalized values of mae and rmse closer to zero along with r2_score closer to unity are the main criteria to prefer one model on another with lowest prediction error for one country to others. it is noteworthy here that dl models generate normalized error measures which are then transformed corresponding to actual numbers through inverse scalar transformation for more understanding of these cases in real world figures. keeping all three performance measures in view, it can be safely concluded on the basis of results that after parameter tuning, bi-lstm performs as best model giving highest accuracy. predicted and actual plots of confirmed cases, death cases and recovered cases of bi-lstm are presented in fig. 4 . these scatter plots demonstrate a very good match of predicted cases against actual ones for all three techniques wit much better performance than baseline regressors. furthermore, among dl models, bi-lstm performs very well and its predicted values completely overlap number of actual cases. convergence of loss function for ten countries using gru has been plotted against number of days for confirmed, death and recovered cases in fig. 5 . these logrithmic graphs demonstrate a smooth evolutionary plot towards converged value of fitness function. for each country this convergence value differs, but overall congverges very well and remains stable and consistent. inferences on the performance of proposed scehmes are listed as follows:  covid-19 dataset has been modelled using various regressors including arima, svr with polynomial and rbf kernels, lstm, gru and bi-lstm for future predictions on confirmed cases, deaths and recovered case for ten countries across the globe.  performance measures of mae, rmse and r2_score have been used to compare various models.  arima and svr models are unable to follow the trend of these features with higher prediction error and negative values of r2_score. ☒ the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. coronavirus disease 2019 (covid-19) situation report-51 world health organization presumed asymptomatic carrier transmission of covid-19 containing papers of a mathematical and physical character global asymptotic properties of a delay sir epidemic model with finite incubation times mathematical analysis of the transmission dynamics of hiv/tb coinfection in the presence of treatment epidemic dynamics: discrete-time and cellular automaton models bridging the gap between evidence 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recurrent neural networks on sequence modeling gated recurrent unit (gru) for emotion classification from noisy speech ☐the authors declare the following financial interests/personal relationships which may be considered as potential competing interests: key: cord-354792-6ckgxn9l authors: ghosh, mousam; ghosh, swarnankur; ghosh, suman; panda, goutam kumar; saha, pradip kumar title: dynamic model of infected population due to spreading of pandemic covid-19 considering both intra and inter zone mobilization factors with rate of detection date: 2020-10-19 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110377 sha: doc_id: 354792 cord_uid: 6ckgxn9l most of the widely populated countries across the globe have been observing vicious spread and detrimental effects of pandemic covid-19 since its inception on december 19. therefore to restrict the spreading of pandemic covid-19, various researches are going on in both medical and administrative sectors. the focus has been given in this research keeping an administrative point of view in mind. in this paper a dynamic model of infected population due to spreading of pandemic covid-19 considering both intra and inter zone mobilization factors with rate of detection has been proposed. few factors related to intra zone mobilization; inter zone mobilization and rate of detection are the key points in the proposed model. various remedial steps are taken into consideration in the form of operating procedures. further such operating procedures are applied over the model in standalone or hybridized mode and responses are reported in this paper in a case-studies manner. further zone-wise increase in infected population due to the spreading of pandemic covid-19 has been studied and reported in this paper. also the proposed model has been applied over the real world data considering three states of india and the predicted responses are compared with real data and reported with bar chart representation in this paper. recent outbreak of corona virus disease or covid-19 global pandemic turns out to be one of the most severe threats to the mankind since first inception of the disease in 2019 at wuhan, china due to a novel virus whose specific source of origin is not yet identified [1] [2] [3] . also, after the inception, this virus spreads rapidly, first in china and then in more than 80 countries across the globe [4] . one of the preliminary reasons behind such rapid spreading of the disease has been identified as the contagious nature of the alleged virus which enables cumulative increase in the number of infections through daily anthropologic activities that require social interactions [5] . also the stability property of the disease free equilibrium of indicates that proper vaccination for cure from this virus is not yet developed [6] . therefore, social distancing and rapid detection test have been evolved as the most acceptable preventive measures in recent time [7, 8] . social distancing is attempted to achieve by different administrative bodies through local and global lockdown. such lockdowns are imposed through the restriction on daily human activities as well as population mobilization in both local and global level [9, 10] . on the other hand, rapid tests have been performed to detect the presence of the virus among the potential victims and if detected, sterner social distancing is being imposed on the infected persons through quarantine and isolation with proper medication and observatory procedure [11] . but such social lockdown and restriction on human mobilization can bring some severe and sweeping impact on economic sectors which in turn cause some detrimental effects on both social life and mental health of the human being [1] [2] [3] [4] . therefore, the prediction of probable duration of lockdown is absolute necessity and needs to be addressed as top priority which requires continuous monitoring of the spreading pattern and timeline of the covid-19 both locally and globally as well as the recovery rate and pattern of infected population [11] . but the continuously changing genetic structure of the respective virus make prior prediction of the disease difficult and ambiguous [5, 11] which in turn brings delay in devising plan on omitting lockdown fully or partially. under such scenario, the demand for precise model to predict the exact nature of the spread of covid-19 is ever increasing to envisage a proper protective strategy of preventing the aforementioned pandemic. all such predictive models can be broadly divided in two categories. in first category, samples are collected from one or more certain population considering different pandemic parameters such as doubling rate [12] , basic reproduction factor [13] , serial intermission [14] etc and then perform statistical analysis on collected samples to make required prediction of aforementioned pandemic. also based on such analysis, several statistical models have been proposed to detect actual inter country infected cases [15] as well as to trace unidentified cases [16] , to determine the effects of local and global migration of people [17, 18] etc. also different advanced statistical techniques have been used to predict the outbreak of corona virus in [19] [20] [21] [22] [23] . in second category, dynamic modelling has been used to assess the nature of covid-19 pandemic [24] more accurately. initially the final size and timeline of covid-19 pandemic was predicted based on dynamic sir model [25] . more advanced sier model was brought in use to predict different factors associated with the disease and possible measures [26, 27] . in such dynamic models, several factors like transmission process and risk [28] , effects of isolation and quarantine [29] etc., are also included to make prediction more accurate. an advanced version of sier model namely e-ishr model has also been proposed to introduce the effects of time delay in the existing models [30] . also, as the dynamics of covid-19 pandemic is inherently nonlinear in nature like other epidemics [24] [25] [26] [27] [28] [29] [30] , there will always be the provision of implementing some state of the art nonlinear dynamical methods proposed in recent times [31] [32] [33] [34] [35] [36] to make different forecast of spreading dynamics of covid-19. various recent studies in the field of pandemic covid-19 reflect the need of research in relation to determine spreading pattern of the pandemic and influence of different factors on it more accurately, which motivates the current studies performed in this paper. in view of these, a dynamic model to predict the pattern and volume of infected population due to the spread of covid-19 has been proposed in the present paper considering several real life factors such as intra and inter zone mobilization, lockdown on local and global activities before detection, rate of detection and the effects of quarantine after detection. also the zone-wise increase in infected population due to spreading of pandemic covid-19 has been given special emphasis in this paper. various remedial steps are taken into consideration in the form of operating procedures. further such operating procedures are applied over the model in standalone or hybridized mode and corresponding responses are reported considering several case studies to indicate that imposing restriction on intra and inter zone mobilization as well as proper quarantine leads to the flattening of pandemic curve. finally the proposed model is applied over the real data of few states of india and the predicted responses are reported in the appendix section of this paper compared with real data. also the proposed model has the provision of simulating various operating procedures as remedial steps in both standalone as well as hybridized mode to reduce the propagation or spreading strength of concerned pandemic in a particular geographical region which is useful in determining possible measure to be implemented based on the demographic properties of that region to counter the spreading of covid-19 pandemic. the proposed dynamic model of infected population due to spreading of pandemic covid-19 is represented in fig. 1 . this model considers three major factors such as intra zone mobilization; inter zone mobilization and rate of detection. in this model a country/state/territory is divided into n number of zones. at any point of time when it has been realized that such pandemic viral infection is spreading out and the time (day) has been taken as the initial time and total non-detected (implies nonquarantined) infected alive population on the day has been taken with zone wise distributions. the parameters which are marked in the proposed model are detailed as follows. number of alive non-detected infected population till time ( ) in zone (excluding death / detected with quarantined / cured). number of detected with quarantined infected population till time ( ) in zone (including death after detection / detected with quarantined / cured after detection). where, is the death factor and is the average death time delay in days. number of cured infected population belongs to detected with quarantined population till time ( ) in zone . ( ) number of cured infected population belongs to non-detected population till time ( ) in zone . where, and are the factors to become cure and is the average time delay to become cure in days. where, ( ) represents rate of detection of zone at time . ( ) enhancement factor of ( ) due to intra zone mobilization in zone at time . ( ) enhancement factor of ( ) due to inter zone mobilization from to zone at time and ( ) ( ) has been considered. total infected population (with/without detected including death) at the end of day. table 1 operating procedure op-1 standard lock-down but mobilization happens op-2 inter zone mobilization of few zones are stopped but intra zone mobilization happens op-3 both intra and inter zone mobilization of few zones are fully stopped op-4 rate of detection with quarantined increases in this model the update of alive non-detected infected population is done by equation (6) and further the distribution factors can be updated by equation (7). if mobilization of any zone is stopped then in this model the corresponding factors have to be zero. also in this model death factor, average death time delay, factors to become cure and average time delay to become cure have also been considered. the average death delay time indicates the delay between infected and death, whereas average time delay to become cure indicates the factor associated will be applied over the delayed infected population and the resulted population does not infect further. in this model the rate of detection also has been considered as a function of time. further clustering of zones has been considered based on the geographical locations to classify the zones where direct inter zone mobilization may happen. two adjacent clusters may have some common zones in the region of intersection. further various operating procedures have been presented in table 1 , which can be applied over the proposed model at any point of time to impose damping over the infected population response. considering the operating procedures various case studies are simulated and have been reported in the subsequent section. it is very much obvious that the community transmission of covid-19 is mostly due to the alive infected population which is not detected or quarantined till date. further how such infected population in each zone changes due the crucial realistic cause of spreading such as intra and inter zone mobilization factors with rate of detection has been estimated. also day wise death and cured/recovered are considered in the proposed model with various realistic coefficients such as death factor, average death time delay, factors to become cure, average time delay to become cure etc. this model is reported with equations (1)(8) and also with block diagram form as illustrated in fig. 1. the clustering of zones for the simulation is represented in fig. 2 and random initializations of population are reported in table 2 . the proposed dynamic model of infected population due to spreading of pandemic covid-19 has been simulated with case-1 parameters and the response of number of alive non-detected infected population ∑ ( ) has been represented in fig. 3 . also the total infected population (with/without detected including death) ( ) is represented in the fig. 3 . the responses indicate that the patterns are very much similar to the patterns of infected population of various countries. table 3 have been carried out and reported in fig. 4 . it is found that the rate of change of infected population are slowing down for case-2 to 6 compared to case-1 and in few cases the non-detected infected population are reducing. again simulations are carried out by applying op-4 with incremental rate of detection per day basis starting from a certain day and the responses are reported in fig. 5 . it seems that the responses are quite satisfactory compared to case-1. furthermore, for in-depth studies of the proposed model, the zone-wise surface maps have been reported in fig. 6 . when the model is operated with the parameters as per case-1, the zone-wise non-detected infected population are increasing day by day (fig. 6(a) ). but when op-3 applied to the model from day 110 for the zones 1 to 20 and 61 to 80 as per case-3, the zone-wise non-detected infected population are reducing for the said zones ( fig. 6(b) ). the proposed model is simulated with some initial population as per table 2 . various operating procedures have been applied as remedial steps in standalone or hybridized mode and the responses indicate the effectiveness. fig. 4 illustrates that if mobilization of population is reduces or test for detection increases or both then the increase in infected population will slowdown or may reduce. further fig. 5 clearly describes that increase in test for detection day by day with quarantine will slowdown or reduces the enhancement of non-detected infected population. further zone-wise surface map (fig. 6 ) patterns illustrate that how infected population profile changes in each zone if certain operating procedures have been imposed in few of the zones. it is obvious that remedial course of action with same intensity may not be possible to be imposed in all zones at same point of time. in view of these if certain zones are handled with rigid course of action then how the patterns may improve compared to other zones are studied and reported in fig. 6. fig. 3 . total infected population (with/without detected including death) ( ) and infected but not detected alive population ( ) with operating procedure op-1 (case-1). in view of the reported simulation responses it has to be admired that hybridization of various operating procedures may improve the situation by slowing down propagation of infected population. further to validate the proposed model, various real world data [37] have been taken into consideration and responses are reported in the appendix-a section. in this section three numbers of states of india such as west bengal, tamilnadu and gujarat, have been taken into consideration as zones for prediction of spreading of infected population. considering the data available in [37] upto 31-july-2020 the day-wise prediction of the entire aug-2020 has been simulated by using the proposed model. fig. a1(a) represents the factor to find the linear trend line which will be used to get the intra zone mobilization factor with rate of detection for the month of aug-2020. since in [37], migrated population is taken zero, the inter zone (in this case state to state) mobilization factor is assumed to be zero. a crucial factor behind the spreading of infection is the non-detected infected population roaming throughout the zone. in order to this it may be assumed that the infected population to be detected in next few days (here taken 07 days) are already infected at present but not detected or quarantined. this consideration is employed to find the trend line as reported in fig. a1(a) . further, factor ( ) to find the cured/recovered population from detected infected population has been determined taking = 5 with moving average basis (fig. a1(b) ). in fig. a1(c) , the day-wise predicted cumulative infected population for the month of aug-2020 has been reported with bar chart representation in comparison with real data [37] of west bengal, india. furthermore, in fig. a1(d) , the day-wise predicted cumulative cured/recovered population for the month of aug-2020 has been reported with bar chart representation in comparison with real data [37] of west bengal, india. in similar manner, predicted cumulative population both infected and recovered, compared with real data for the zones tamilnadu, india and gujarat, india, have been illustrated in fig. a2 -fig. a3 . in this paper a dynamic model of infected population due to spreading of pandemic covid-19 considering both intra and inter zone mobilization factors with rate of detection, have been proposed with various operating procedures. considering the operating procedures as followed in this paper, various case studies have been simulated and reported with adequate responses. the population responses obtained from the simulation of the proposed model are seen to have considerable similarities with the patterns of infected population of various countries as reported in the literatures which indicates the predictability of the proposed model. the coefficients or factors associated with the proposed model are needed to be tuned to get the pattern of infected population of any particular country. various operating procedures have been applied as remedial steps in standalone or hybridized mode after a certain day and the responses indicate the effectiveness. further this study also investigates that imposing various strict administrative protocols in certain zones by improving such factors may be achieved improved responses. further the proposed model is having various provisions to fed external inputs to realise the effects of imposing different remedial steps. in addition to this the proposed model has been applied over the real world data considering three states of india and the predicted responses are compared with real data and reported with bar chart representation in the appendix-a section. this proposed model can be tuned to validate the prediction for others states or country. the authors declare that they have no conflict of interest. declaration of interests  ☐the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 2020) who characterizes covid-19 as a pandemic novel coronavirus (covid-19) situation centers for disease control and prevention (cdc.gov/covid19) john hopkins hospital (jhh) (2020). coronavirus covid-19 global cases by the center for systems science and engineering (csse) at johns hopkins the novel coronavirus, 2019-ncov, is highly contagious and more infectious than initially estimated a recursive bifurcation model for predicting the peak of covid-19 virus spread in united states and germany lockdown may partially halt the spread of 2019 novel coronavirus in hubei province feasibility of controlling 2019-ncov outbreaks by isolation of cases and contacts effectiveness of airport screening at detecting travellers infected with novel coronavirus (2019-ncov) the impact of traffic isolation in wuhan on the assessing spread risk of wuhan novel coronavirus within and beyond china epidemic doubling time of the 2019 novel coronavirus outbreak by province in mainland china. medrxiv preliminary estimation of the basic reproduction number of novel corona virus (2019-ncov) in china, from 2019 to 2020: a data-driven analysis in the early phase of the outbreak serial interval of novel coronavirus (covid-19) infections epidemic size of novel coronavirus-infected pneumonia in the epicenter wuhan: using data of five-countries' evacuation action. medrxiv using predicted imports of 2019-ncov cases to determine locations that may not be identifying all imported cases the effect of travel restrictions on the spread of the 2019 novel coronavirus (covid-19) outbreak. science association between population migration and epidemic control of coronavirus disease 2019 statistics based predictions of coronavirus 2019-ncov spreading in mainland china modelling and prediction of the 2019 coronavirus disease spreading in china incorporating human migration data host and infectivity prediction of wuhan 2019 novel coronavirus using deep learning algorithm nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study early transmissibility assessment of a novel coronavirus in wuhan early transmission dynamics in wuhan, china, of novel coronavirus-infected pneumonia estimation of the final size of the coronavirus epidemic by the sir model incubation period of 2019 novel coronavirus (2019-ncov) infections among travellers from wuhan, china simulating the infected population and spread trend of 2019-ncov under different policy by eir model an updated estimation of the risk of transmission of the novel coronavirus (2019-ncov) real-time tentative assessment of the epidemiological characteristics of novel coronavirus infections in preliminary assessment of the covid-19 outbreak using 3-staged model e-ishr model predictive control of three-axis gimbal system mounted on uav for realtime target tracking under external disturbances. mechanical systems and signal processing digital currency forecasting with chaotic meta-heuristic bioinspired signal processing techniques lessons from being challenged by covid-19 casella f: can the covid-19 epidemic be controlled on the basis of daily test reports prediction of the epidemic peak of coronavirus disease in japan control strategies to curtail transmission of covid-19 key: cord-320262-9zxgaprl authors: asamoah, joshua kiddy k.; owusu, m.a.; jin, zhen; oduro, f.t.; abidemi, afeez; gyasi, esther opoku title: global stability and cost-effectiveness analysis of covid-19 considering the impact of the environment:using data from ghana date: 2020-07-10 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110103 sha: doc_id: 320262 cord_uid: 9zxgaprl covid-19 potentially threatens the lives and livelihood of people all over the world. the disease is presently a major health concern in ghana and the rest of the world. although, human to human transmission dynamics has been established, not much research is done on the dynamics of the virus in the environment and the role human play by releasing the virus into the environment. therefore, investigating the human-environment-human by use of mathematical analysis and optimal control theory is relatively necessary. the dynamics of covid-19 for this study is segregated into compartments as: susceptible (s), exposed (e), asymptomatic (a), symptomatic (i), recovered (r) and the virus in the environment/surfaces (v). the basic reproduction number [formula: see text] without controls is computed. the application of lyapunov’s function is used to analyse the global stability of the proposed model. we fit the model to real data from ghana in the time window 12th march 2020 to 7th may 2020, with the aid of python programming language using the least-squares method. the average basic reproduction number without controls, [formula: see text] is approximately 2.68. an optimal control is formulated based on the sensitivity analysis. numerical simulation of the model is also done to verify the analytic results. the admissible control set such as: effective testing and quarantine when boarders are opened, the usage of masks and face shields through media education, cleaning of surfaces with home-based detergents, practising proper cough etiquette and fumigating commercial areas; health centers is simulated in matlab. we used forward-backward sweep runge-kutta scheme which gave interesting results in the main text, for example, the cost-effectiveness analysis shows that, strategy 4 (cleaning of surfaces with home-based detergents) is the most cost-effective strategy among all the six control intervention strategies under consideration. that other optimal control model on covid-19 have been studied (see for example [27, 28, 29 , 30, 31, the model further assumes that, no exposed individual transmits the disease. the proportion of those in e class into both a and i classes is given as k 2 (1 − γ)e and k 1 γe respectively. 68 similarly, individuals recovering from the a class is v 2 (1 − φ)a and those joining the i class from the a 69 compartment is v 1 φa. though, not much had been said with regards to the possibility of the recovered 70 individuals joining the susceptible population. all the same we included it so to ascertain the impact short 71 and long-term immunity on the dynamics of covid-19, denoted as ρ, where ρ ≥ 0. table 1 progression rate from exposed to the symptomatic (severely infected) class k 2 progression rate from exposed to the asymptomatic class given that s ≥ 0, e ≥ 0, a ≥ 0, i ≥ 0, r ≥ 0 and v ≥ 0. we simplify equation (1) to get the total differential equation as; where n = s + e + a + i + r. non-negative such that the initial conditions are also given as the proof of theorem 1 can be obtained using the procedures presented in [33] , as shown below. proof. we let y = (s, e, a, i, r, v ) t and k 0 = βi, k 1 = βa, k 2 = β 1 v where t is transposition, then our differential equation (1) can be rewritten in a matrix form as dy now, using the third equation in model (1), thus and deploying the method of integration factor and change of variables [33] , yields t 0 e(s)e −(ω+v1φ+v2(1−φ))s ds . next, we consider the fourth, fifth, sixth equation of model (1) and using the same above process, we have written as where m = (m 1 + m 2 ). proof. from the first equation of (2) we have now, take m 2 to be a solution which is unique to the initial value problem which when solved gives hence, by the comparison principle (see for instant theorem 5 of [38]), it accompanies that also from the second equation of (2), we let m = (m 1 + m 2 ) with the assumption that 0 < a + i ≤ λ ω . then, now, let m 3 to be a solution which is unique to the initial value problem which when solved gives and, by the comparison principle (see for instant theorem 5 of [38]), it accompanies that from (5) and (8) here, we first focus on equilibrium points when there is no disease in the system. considering equation (1), we put e = a = i = r = v = 0. this indicates that, there is no disease in the system at this stage. therefore, solving for the stationary points, we have e = (s * , 0, 0, 0, 0, 0) where s * = λ ω . for the r 0 , we use the concept of the next generation approach. here, we seek to find the average number of new infections given that an infected individual is introduced into the population under study [39] . we let g be the next generation matrix which consists of is the rate at which a new infection occurs in compartment i. also, v + i and v − i are the rate of immigration into compartment i and the rate at which new individuals are transferred from compartment i respectively. we note that, all the functions are continuously differentiable at least twice [39] . now the next generation matrix is defined as; hence, the r 0 is given as the maximum absolute eigenvalue of the next generation matrix (g) given that, σ contains all the eigenvalues of g. this eigenvalue is known as the spectral radius (ρ). this is represented as among the infected classes (e, a, i, v), we have f i as finding the jacobian of the matrix f i gives considering the same compartments (e, a, i, v), we get the matrix v as finding the jacobian matrix of v i gives after computing for the eigenvalues of the matrix g, we have that the maximum absolute eigenvalue, r 0 is given as; secondary infection seeded by i state through direct contact secondary infection seeded by i state through the environment (9) which can be written as where r a is the secondary infections generated by asymptomatic persons through direct contact; r i is the secondary infections generated by symptomatic persons through direct contact; r ae is the secondary infection seeded by asymptomatic persons through the environment; and r ie is the secondary infection seeded by symptomatic persons through the environment. we can also express r 0 in terms of (t 1 , c 1 , q 1 , where now, expressing the other state variables in terms of e, it implies that we now substitute a, i and v into second equation (12) and factorizing e out gives from the initial hypothesis, e = 0 and this implies that, making s * in equation (14) the subject gives now, adding first equation and second equation of (12), we substitute s * from equation (15) and simplify e * we get where but, we know that the basic reproduction number is expressed as therefore, we express the endemic equilibrium points, (s * , e * , a * , i * , r * , v * ) in terms of r 0 as; based on the preliminary notes above, we now state the lyapunov stability theorem. theorem 3 (lyapunov stability theorem). the equilibrium, y * is globally stable if the function, l(y) is radially unbounded and positive definite globally such that it has globally negative time derivative, l(y) < 0 ∀y = y * . we say that; the function l(y) is a lyapunov function if it satisfies the above theorem, the proof can be 109 found in [41] . another important theorem which also plays a key role here is the kransovkii-lasalle theorem. this is an extension of lyapunov function. in summary, this theorem puts forward that; considering an autonomous system, y = f (y) which has equilibrium, y * and that f (y * ) = 0, we assume there is a continuously differentiable positive definite and radially unbounded function l : r n → r which meets the conditionl(y) ≤ 0 ∀ t, y ∈ r n . we then define the invariant set as proof. we employ the approach in [43] to analyze both the stability at disease free and endemic equilibrium. we define a lyapunov, l for the disease-free equilibrium point as follows; differentiating l with respect to t gives; we substituteė,ȧ,i̇,v from equation (1) intol gives; after further simplification we havė differentiating the function above gives; substituting equation (1) into equation (18) with further simplification gives; considering the expression we have that, h 1 = 1. this implies that the coefficients of x 1 x 4 , x 3 x 1 and x 6 x 1 are all 0. equating the coefficients of x 2 , x 3 , x 4 , x 5 and x 6 to 0 and solving for h 2 , h 3 , h 4 and h 5 gives; therefore, t can be rewritten as it then follows that, t ≤ 0 if x 1 = 1, x 2 = 1, x 3 = 1, x 4 = 1, x 5 = 1 and x 6 = 1. hence we may conclude that;l by lasalle theorem, the invariant set is defined as since the invariant set, ζ 1 only contains the endemic equilibrium (s * * , e * * , a * * , i * * , r * * , v * * ), then the 118 endemic equilibrium is said to be globally asymptotically stable under the given region d. in this section, our focus is to verify the validity of the model. this is achieved by fitting and comparing the proposed model with a real data to know its degree of accuracy. are shown in table 2 and figure 3 respectively. the blue points in figure 3 represent the cumulative number in computing for the normalized sensitivity index ( p r0 ) on the r 0 for each of the parameters p, we use the formulae below [41]; applying the formula above gives the parameters with their sensitivity index in the table 3 . from table 3 , asymptomatic class. the model shows that, asymptomatic individuals can join the severely infected class. however, the estimated parameter value shows that, only few people experience such situation, that is, 0.5%. considering the system x (t) = f (x(t)) with x(0) = x 0 , such that x 0 ∈ r n where f : r n → r n and x : [0, ∞) → r n . we introduce variables responsible for the control u i , i = 1, 2, 3...n ∈ n. we then have; the admissible control set is given as; u * = {u(t) ∈ l (t 0 , t f )|u(t) ∈ a}. the aim 198 here is to target the best control variables, u i , which can efficiently reduce the rate of secondary transmission 199 at a minimum cost of their implementation at any time t (0) ≤ t ≤ t (f ) [41] . that is, we seek to achieve a 200 reduction in the number of individuals in the susceptible, exposed, asymptomatic, severely infected classes 201 and also reduce the content of the virus in the system at a minimum cost simultaneously. to achieve the 202 above objective demands a lot of constructive considerations. for example, implementing total lock down for 203 about two months as a control strategy in this context might highly prove not to be feasible. the loop hole 204 here is; is the country adequately prepared both financially and technically to provide to the satisfaction of 205 its inhabitants the basic needs such as food, water and others throughout the whole period assigned for this 206 measure? it is highly probable the answer might be a big no. it is an undeniable fact that, this measure 207 might prove impractical in controlling the spread in this country. this is why there is a need to objectively 208 sort for more dense restrictive measures with flexible and feasible approaches to be employed in this setting 209 so as to control the disease. we rely on the pontryagin's maximum principle as applied in [48] for this 210 analysis. we base on the premises above to set below likely control strategies: the objective functional under discussion, q, which is to be minimized is given as; subject to the constraints; from equation (4) , we assume that, the weight constant 219 of the exposed, infected (a and i classes) and the virus in the system is 1. also, to better observe and 220 understand the influence of these control strategies on the model, we assumed that; no recovered individual 221 is vulnerable to be reinfected in equation (4). we accounted for the respective affiliated costs, b 1 u 2 1 , b 2 u 2 2 , 222 b 3 u 2 3 , b 4 u 2 4 , b 5 u 2 5 , which are possible to be incurred during implementation where the square denotes their 223 severity. it is very necessary to ensure that, the proposed optimal solution exists. for this reason, we employ filippovecesari theorem as used in [37] . in this case, we show that, the existence of the optimal control solution is 4. the convexity of the integrand of cost functional with respect to u on the set a [37] . 231 we now have the hessian matrix of the given cost functional as;  since the computed hessian matrix above is everywhere positive definite, it follows that, the objective functional, q(u 1 , u 2 , u 3 , u 4 , u 5 ) is strictly convex [37] . we also have that, given that the integrand of the objective functional, we now take into accounts the existence of the adjoint function λ i , i = 1, 2, 3...6, such that they satisfy the equations; with the transversality condition λ i (t f ) = 0 given that ∀u i where i = 1, 2, ..., 5, we have the optimal control strategies with respect to the befitting variation argument is given as; we progress with the numerical simulations on the optimal control by using the estimated parameters in 232 table 2 . dynamics of the subpopulations of exposed, asymptomatic, and symptomatic individuals and the number 254 of virus with control u 2 and without any control implementation is demonstrated by figure 12 . figure 255 13figure 15 shows the respective dynamics when control u 3 , u 4 and u 5 is used separately. it is revealed here, cost-effectiveness analysis is carried out based on the numerical implementation of the optimality system conducted in section 4. the cost benefits associated with the implementation of the control strategies can be compared through cost-effectiveness analysis. thus, following the approach used in several previous studies [53, 54, 55, 56], the incremental cost-effectiveness ratio (icer) is calculated to determine the most cost-effective strategy of all the different control intervention strategies considered in this work. most often, icer is employed to measure up the changes between the costs and the health benefits of any two different control intervention strategies i and j competing for the same limited resources. icer is defined mathematically as icer = difference in costs of control strategies i and j difference in infections averted by control strategies i and j . the numerator of icer in equation (21) table 4 . from table 4 , it is observed that the value of icer(6) is greater than that of icer(4). this indicates 285 that strategy 6 is more costly and less effective than strategy 4. for this reason, strategy 6 is excluded from 286 the list of alternative control interventions competing for the same limited resources and icer is recalculated 287 for strategies 4 and 3 as illustrated by table 5 . it can be seen in table 7 that icer(1) is greater than icer(4). this implies that the implementation 296 of strategy 1 is more costly and less effective than the implementation of strategy 4. hence, strategy 1 is 297 discarded from the list of alternative control intervention strategies competing for the same limited resources. now, the icer is finally recalculated for strategies 4 and 2 as shown in table 8. table 8 reveals that icer(2) is greater than icer(4). hence, strategy 2 is considered to be strongly domfound to be globally asymptotically stable. we found that, the major transmission parameters β, β 1 , m 1 308 and m 2 contributing to the basic reproduction number of 1.99 − 3.37 were all attributed to humans through 309 personal contact with the susceptible class or activities with the environs. it is further inferred from this 310 study that; applying optimal control strategy on the rate at which the virus is released into the system, m 1 311 and m 2 , and also on the relative transmission rate due to human behaviour will considerably strike down 312 covid-19 pandemic. it was also found that, it might be possible the recovered individuals can be reinfected, see figure 8b . when 314 this happens, then the number of the infected individuals will also increase. therefore, we highly recommend 315 that, drug manufacturers should aim at drug samples which will induce permanent immunity in the recovered 316 individuals so as to reduce the susceptible population. cost-effectiveness analysis was carried out based on 317 the numerical implementation of the optimality system conducted in section 4. this showed that, cleaning of 318 surfaces with home-based detergents is the most cost-effective strategy, followed by: the effective testing and 319 quarantine when boarders are opened, the combination of all the controls then that of intensifying the usage 320 of nose masks and face shields through education. it is highly guaranteed that, this study will help policy the authors declare that they have no known competing financial interests or personal relationships that 328 could have appeared to influence the work reported in this paper. the parameter values (data) used to support the findings of this study have been described in section 3. disclosure 338 the authors fully acknowledge that this paper was developed as a result of the first and second author's 339 thesis and project work. novel coronavirus (2019-ncov) situation report-1 novel coronavirus (2019-ncov) situation report-3 novel coronavirus (2019-ncov) situation report-5 novel coronavirus (2019-ncov) situation report-8 novel coronavirus (2019-ncov) situation report-11 novel coronavirus (2019-ncov) situation report-12 novel coronavirus (2019-ncov) situation report-14 novel coronavirus (2019-ncov) situation report-26 covid-19 coronavirus pandemic novel coronavirus (2019-ncov) situation report-54 schools close down and social gathering restrictions early dynamics of transmission and control 376 of covid-19: a mathematical modelling study the reproductive number of covid-19 is higher 380 compared to sars coronavirus mathematical modeling of the spread of the coro-383 navirus disease 2019 (covid-19) taking into account the undetected infections. the case of china analysis of a mathematical model for covid-19 population 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with digital contact tracing predicting optimal lockdown period with parametric 420 approach using three-phase maturation sird model for covid-19 pandemic a model based study on the dynamics 423 of covid-19: prediction and control optimal control design of impulsive 425 sqeiar epidemic models with application to covid-19 optimal policies for control of the novel coronavirus covid-428 19 outbreak a model based study on the dynamics 431 of covid-19: prediction and control modeling the impact of non-pharmaceutical interventions on the dynamics of 435 novel coronavirus with optimal control analysis with a case study a deterministic model for q fever transmission 440 dynamics within dairy cattle herds: using sensitivity analysis and optimal controls stability of a retrovirus dymanic model dynamical systems and population persistence qualitative theory of compartmental systems mathematical model of covid-19 spread in turkey and south africa: theory, 450 methods and applicationsdoi analysis of a siri epidemic model with 453 distributed delay and relapse analysis and optimal control of deterministic vector-borne diseases model sir epidemic model with mittag-leffler fractional derivative stability analysis of the generalized fractional differential equations with and without exogenous 461 inputs mathematical modelling of the potential role of supplementary feeding for 464 people living with hiv/aids mathematical modelling of the potential role of supplementary feeding for 467 people living with hiv/aids revision), total population life expectancy rate how long can the coronavirus that causes covid-19 survive on surfaces? modelling of rabies transmission dynamics 477 using optimal control analysis optimal control applied to biological models optimal covid-19 quarantine and testing policies controlling the transmission dynamics of covid-19 mathematical modelling of bacterial 486 meningitis transmission dynamics with control measures co-dynamics of pneumonia and typhoid fever diseases with 490 cost effective optimal control analysis optimal control strategies for dengue fever cost-effectiveness analysis of optimal control strategies for breast cancer 494 treatment with ketogenic diet modelling malaria dynamics with partial immunity 496 and protected travellers: optimal control and cost-effectiveness analysis key: cord-342591-6joc2ld1 authors: higazy, m. title: novel fractional order sidarthe mathematical model of the covid-19 pandemic date: 2020-06-13 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110007 sha: doc_id: 342591 cord_uid: 6joc2ld1 nowadays, covid-19 has put a significant responsibility on all of us around the world from its detection to its remediation. the globe suffer from lockdown due to covid-19 pandemic. the researchers are doing their best to discover the nature of this pandemic and try to produce the possible plans to control it. one of the most effective method to understand and control the evolution of this pandemic is to model it via an efficient mathematical model. in this paper, we propose to model the covid-19 pandemic by fractional order sidarthe model which not appear in the literature before. the existence of a stable solution of the fractional order covid-19 sidarthe model is proved and the fractional order necessary conditions of four proposed control strategies are produced. the sensitivity of the fractional order covid-19 sidarthe model to the fractional order and the infection rate parameters are displayed. all studies are numerically simulated using matlab software via fractional order differential equation solver. all the world states' governments introduce a big effort and vital measures to eliminate the outbreak of covid-19 [22] . covid-19 is a new progeny of coronavirus, sars-cov-2 and firstly detected in wuhan, china [52, 57] . in the few months after discovering it, the number of patients were increasing exponentially. the taken measures against covid-19 until the day of writing these words didn't prevent the growth of infected cases around the globe. the world health organization situation report published in 25 may 2020 informed that 5304772 cases as total cases and 342029 deaths around the globe [54] . using mathematical model to predict the epidemics is very useful in order to understand the nature of the epidemic and to design an efficient strategies to control it [4, 8, 14, 26, 27] . it is common to study the humanitarian diffusion of epidemics via sir or seir models [11, 13, 33, 48] . various models have been proposed to model and study covid-19 pandemic. taking into account the risk understanding and the accumulative issue of cases, covid-19 pandemic has been modeled by lin et al. via extending seir model [38] where s signifies the susceptible, e signifies the exposed, i signifies the infected and r signifies the removed cases. in [3] , anastassopoulou et al. have suggested the sir model in the discrete time mode taking into account the dead cases. in [9] by casella, sir model is expanded to study the delays effect and to compare the policies of containment. in [56] by wu et al., the covid-19 severity has been estimated using the dynamics of transition. in [25, 34] , random transition models have been studied. in [12] , the general multi-group seira model was represented and numerically tested for modelling the diffusion of covid-19 between a non-homogeneous population. the basic mathematical tool used to model several epidemics is differential equations in various modes (ordinary, fractional, with delay, randomly detected or partial) [27] [28] [29] . 4 many research efforts have been widely done to control the outbreaks of epidemics via optimal control [20, [44] [45] [46] [47] . the optimal control idea is to look for the utmost powerful plan that decreases the rate of infection to a possible minimum limit with optimal minimum cost of circulating a treatment or preventative inoculation [39, [44] [45] [46] [47] 51] . these plans may include treatments, inoculation with vaccines, social distances, educational programs [6, 10 ] . studying the epidemiological diseases mathematically become very important [6, 7, 16, 17, 37, 40] . the literature has several studies to control for example models of hiv [24] , dengue fever [2] , tuberculosis [50] , delayed sir [1] and delayed sirs [29, 58] . the fractional order differential equations add an extra dimensions in the study of dynamics of epidemiological models. therefore the fractional version of many epidemical models have been investigated as in [29] [30] [31] [32] , [39, 41] and [53] . here, a new epidemiological fractional mathematical model for the covid-19 epidemic is proposed as an extension of the classical sir model, similar to that introduced by gumel et al. for sars in [23] and as a generalization of sidarthe model proposed in [21] . as explained in section 3, in sidarthe model, the infected cases are classified into five different classes depending on the detection and the appearance of symptoms [21] . in this work, we consider the fractional order sidarthe model and then we derive the fractional order necessary conditions for existence of a stable solution. in addition, we study an optimal control plans for the fractional order sidarthe model via four control strategies that include the availability of vaccination and existence of treatments for the infected detected three population fraction phases. applying the fractional order differential equations numerical solver using matlab software, we show the dynamics of the state variables of the model and display the effect of changing the fractional derivative order on the system response. also, the effect of changing the infection rates on the fractional order sidarthe model's state 5 variables. we also implement the optimal control strategies numerically for the fractional order sidarthe model. the remaining parts of the paper are organized as follows. in section 2, preliminaries and basic definition of the fractional derivative are introduced. describing covid-19 epidemic sidarthe fractional mathematical model is introduced in section 3. the details of the optimal control strategy and its implementation are given in section 4. numerical simulations of the uncontrolled fractional order sidarthe model, the effects of changing the fractional derivative order on the system response and the effects of changing the infection rates are all given in section 5. numerical simulations of the controlled fractional order sidarthe model and the effects of applying the proposed control strategies are represented in section 6. the concluding remarks are put in section 7 followed by the list of cited references. many definitions of fractional order derivatives exist such as riemann-lioville's derivative, grunwald letnikov's derivative, caputo's derivative, caputo-fabrizio, atangana-baleanu, etc. the interested reader can consult for example [42, 43] and the references therein for more details about fractional order defintions with applications. we have used caputo's definition throughout the paper. caputo fractional derivative operator q  of order q (see [18, 30, 39, 42, 43, 49] ) is defined as: symbolizes the gamma function. for more details about the basic definitions and characteristics of fractional derivatives see [18, 30, 39, 49] . notation: for numerical simulations, the predictor-corrector pece method of adams-bashforth-moulton type described in details in [15, 19] has been utilized and programmed with matlab software. giulia giordano et al. in [21] modeled the covid-19 epidemic via sidarthe and compare its response with the real data in italy. sidarthe distinguishes between determined infected cases and undetermined infected cases and between various degrees of illness (doi). in sidarthe covid-19 epidemic model, the total population is partitioned into eight phases of malady as recorded in the table 1 . figure 1 shows the interaction graph (  ) between the eight phases of malady. in figure 1 , the susceptible population partition is partitioned into four sub-phases to show the hidden sub-phases of the susceptible population partition , namely . the detailed interaction digraph in figure 1 is named  that may be studied using graph theory tools to discover more features about the model. infected (symptomless, undetermined) population fraction. diagnosed (infected, symptomless, determined) population fraction. ailing (infected, with symptoms undetermined) population fraction. recognized (infected, with symptoms, determined) population fraction. threatened (infected, with life-menacing symptoms, determined) population fraction. healed (recuperate) population fraction. extinct (died out) population fraction. table 1 where the sub-phases si, sd, sa, sr are appeared. the covid-19 epidemic sidarthe is modeled mathematically by eight ordinary differential equations [21] . the deterministic characteristic is essential in modeling the epidemics transition phenomena; the fractional derivative is very useful in modeling the epidemics transition systems because they consider the memory effect and the universal properties of the system, that are primary in the deterministic feature. the system is said to have a memory effect if its future states depend on its current states and the history of the states, and the fractional operator has this memory effect feature so it is very helpful in modeling the covid-19 diffusion model. here, as recorded in equations (3.1) to (3.8) , the dynamics of the population in each phase with time is described with eight fractional order ( ) differential equations: [21] using the real data. figure 1 shows the impact of each different phases of epidemic graphically. the sidarthe covid-19 model parameters have the following real meaning:  a signifies the rate of infection as a result of contacting among a susceptible case and an infected case.  b signifies the rate of infection as a result of contacting among a susceptible case and a diagnosed case.  c signifies the rate of infection as a result of contacting among a susceptible case and an ailing case.  d signifies the rate of infection as a result of contacting among a susceptible case and a recognized case.  e signifies the detection probability rate of infected symptomless cases.  θ signifies the detection probability rate of infected with symptoms cases.  z signifies the rate of probability at which an infected case is not conscious of becoming infected.  h signifies the rate of probability at which an infected case is conscious of becoming infected.  m signifies the rate at which undetermined infected case develops life-menacing signs.  v signifies the rate at which the determined infected case develops life-menacing signs.  τ signifies the death rate (for infected cases with life-menacing signs).  g, k, x, r and σ signify the rate of healing for the five phases of infected cases. for more details about the model choices see [21] and the references cited there. 9 from equations (3.1) to (3.8) and since the states and are sink vertices in the model graph  ( see figure 1 ), then they are considered as accumulative state variables that rely only on their own starting situations and the other state variables. since summing up all equations (3.1) to (3.8) gives zero as a result then the system is compartmentalized and shows the conservation property of mass : as can be directly proved, which implies that the total population (the sum of all state variables) is constant. let (3.10) be the state variables vector. since the state variables signify the population fractions, we can suppose that ∑ , such that 1 signifies the total population, dead are included. ; ; ; ; 20) 3 3 3 3 31 32 3 3 2 33 3 34 3 3 3 3 35 3 36 3 37 3 3 38 0 ( ) ; ; ; where ( ) are all positive constants. then from (3.19) to (3.26) , each of the eight functions agree with the lipschitz condition [18, 39] . with respect to the eight arguments, then all eight functions are absolutely continuous. in this area, the existence of the fractional order sidarthe model's optimal control is investigated and then the hamiltonian of the optimal control problem is constructed in order to produce the optimal control necessary requirements. compute the optimal values of vaccination and treatment strategies that would maximize the healed population phase ( ) and minimize the determined infected phases ( ) and susceptible ( ) population phases. in addition, the charges of utilizing the vaccination and treatment methods are minimized. then the optimal control problem of the following form is considered (see for example [1, 2, 5, 6, 10, 20, 24, 25, 34, 37, [44] [45] [46] [47] . that obeys the constraints the control function represents the vaccination strategy applied on the susceptible the four constants 1 2 3 ,, c c c and 4 c are the cost correspond to utilizing each control function. the uncontrolled system (put 0 i u  in (4.2)). for certain initial conditions: that their sum equal one, it is obvious that the final values of the state variables approach to an equilibrium:  which means that the phenomenon of epidemic is finished (see [21] ). the possible equilibrium points of the system are given by ( , 0, 0, 0, 0, 0, , ) the incidence function of the system is: in the following result, the existence of the optimal control is proved. proof. in order to prove the optimal control quadruple                since 1 s i d a r t h e         this subsection records the needed conditions for the optimal control to be exist. the necessary conditions are computed here via constructing the hamiltonian (  ) and satisfying the maximum basis of pontryagin [5] . let us denote by is the optimal controls and be the related optimal population phases fractions. consequently, there exists such that the necessary conditions for the optimal control to be exist are produced by (see for example [5] ): from (4.4) and (4.6), the optimization constraints can be found as: from (4.7), we can find that from the state variables system given in (4.2), the co-state conditions are: which can be simplified to produce the following co-state system: the transversality conditions leads to the collection of permissible controls u is convex. 17 the effect of applying the different control strategies will be simulated numerically in section 6. in this fifth section, we solve the fractional order sidarthe model numerically utilizing the predictor-corrector pece method of adams-bashforth-moulton type described in details in [15, 19] . the parameters' values used for the numerical simulation are estimated from the italian real life statistics published in [21] where: the total population are taken 100 million and the initial values of the different population phases after normalization are (let be the total population): in the following, we display the results of the numerical simulation of the uncontrolled sidarthe model. figure 2 with different fractional derivative order . figure 9 displays the phase plane of state variables: total infected ( ) and susceptible cases (s(t)) with different fractional derivative order . figure 10 displays the phase plane of state variables: total infected ) and healed cases (h(t)) with different fractional derivative order . from the results and the figures mentioned here, we can state that decreasing the fractional derivative order decreases the number of each population phase (except susceptible 19 population fraction as expected) and flatten the curves also delays reaching the maximum in each population phase. 23 figure 10 . the phase plane of state variables: total infected ) and healed cases (h(t)) with different fractional derivative order . in this subsection, we show the effect of changing certain system parameters on different population phases at day 60 with various fractional derivative order. figure the mean rate of separation or contraction of tiny phase-space disturbances of a dynamical system beginning from near starting points is metered by the lyapunov exponents (les) [55] . thus, they can be utilized to study the stability of dynamical systems and to examine sensitive reliance on starting conditions, that implies, the presence of hidden chaotic dynamics. it is important to check the epidemic transition models if it is chaotic or not via calculating les. corresponding techniques for the les calculation and their distinction are studied, e.g., in [35, 36] . the studied system represented by equations: (3.1) -(3.8) is stable [21] . here, we confirm its stability for different values of fractional derivative order via plotting the relationship between the fraction derivative order and the eight lyapunov exponents of the system. figure 23 shows that all system eight lyapunov exponents are negative with different fraction derivative order as time approaches infinity (here, time is taken 1500). figure 24 shows the 31 dynamics of the system eight lyapunov exponents with time. in figure 24 -(a) to 24-(d) with different fraction derivative order, the system eight lyapunov exponents approaches negative end which confirm the system stability with different fractional derivative orders. for more details about lyapunov exponents see [55] . here, we use the method in [13] for calculating the fractional order lyapunov exponents. this research has been carried out to the analysis of an eight dimension fractional -order sidarthe covid-19 mathematical model. in this 8-d cvid-19 mathematical model, the infected population fraction is partitioned into five different population fractions: . it is the first time to study such model with fractional order. the existence of stable solution of the fractional order sidarthe model is proved. the fractional order necessary conditions for a four optimal control strategies are implemented. in addition, the system dynamics displayed via the fraction order numerical solver by matlab software with different fractional orders and the effects of changing the infection rates parameters are presented in this manuscript with different fractional orders. the effects of changing the fractional order on the system lyapunov exponent are also displayed. the dynamics of the system are presented before and after control. from our study, we can state that decreasing the fractional derivative order decreases the number of cases in all population fraction phases and delays the maximum plus changing the value of the fractional derivative order has no effect on the stability of the system since its all lyapunov exponents still negative. the proposed 38 fractional order covid-19 sidarthe model predicts the evolution of covid-19 epidemic and try to help in understanding the impact of different plans to limit the diffusion of this epidemic with different values of the fractional order. our results confirm the importance of decreasing the infection rates. decreasing the infection rates include taking various actions like insure the social distance, closing the airports, closing all teaching authorities, random testing the asymptomatic cases and contact tracing. the author hope that the covid-19 study using the proposed model continues. and via utilizing the real data the optimum fractional order can be estimated. no competing of interest. this research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. the hopf bifurcation analysis and optimal control of a delayed sir epidemic model an optimal control problem arising from a dengue disease transmission model data-based analysis, modelling and forecasting of the covid-19 outbreak infectious diseases of humans optimal control of a sir epidemic model with general incidence function and a time delays control of emerging infectious diseases using responsive imperfect vaccination and isolation the use of epidemic models can the covid-19 epidemic 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figure 29 shows the effect of the control strategies: (vaccination) and 33 (treatment of the diagnosed population phase ) on the different population phases at day 70 with fractional derivative order . figure 30 shows the effect of the control strategies:(vaccination) and (treatment of the diagnosed population phase ) on the different population phases at day 70 with fractional derivative order . figure key: cord-312120-xt5v3bjh authors: lahmiri, salim; bekiros, stelios title: the impact of covid-19 pandemic upon stability and sequential irregularity of equity and cryptocurrency markets date: 2020-05-28 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109936 sha: doc_id: 312120 cord_uid: xt5v3bjh we explore the evolution of the informational efficiency in 45 cryptocurrency markets and 16 international stock markets before and during covid-19 pandemic. the measures of largest lyapunov exponent (lle) based on the rosenstein's method and approximate entropy (apen), which are robust to small samples, are applied to price time series in order to estimate degrees of stability and irregularity in cryptocurrency and international stock markets. the amount of regularity infers on the unpredictability of fluctuations. the t-test and f-test are performed on estimated lle and apen. in total, 36 statistical tests are performed to check for differences between time periods (preversus during covid-19 pandemic samples) on the one hand, as well as check for differences between markets (cryptocurrencies versus stocks), on the other hand. during the covid-19 pandemic period it was found that (a) the level of stability in cryptocurrency markets has significantly diminished while the irregularity level significantly augmented, (b) the level of stability in international equity markets has not changed but gained more irregularity, (c) cryptocurrencies became more volatile, (d) the variability in stability and irregularity in equities has not been affected, (e) cryptocurrency and stock markets exhibit a similar degree of stability in price dynamics, whilst finally (f) cryptocurrency exhibit a low level of regularity compared to international equity markets. we find that cryptos showed more instability and more irregularity during the covid-19 pandemic compared to international stock markets. thus, from an informational efficiency perspective, investing in digital assets during big crises as the covid-19 pandemic, could be considered riskier as opposed to equities. since the financial crisis of 2008, a large number of studies has focused on the response of equity markets to such long and significant economic crises. for instance, scholars have examined the effect of the subprime mortgage crisis of 2008 upon the hierarchical structure of stock markets 1, markets contagion 2-5, crude oil markets 6,7, fertilizer markets 8,9, serial correlations in stocks 10-14 and currency markets 15, or upon the changes in market risk 16, volatility transmission 17-22, and markets' connectedness 23,24. since the world health organization has declared the covid-19 pandemic as global health emergency, the world economy has been drastically affected. sales declined, consumers changed their behaviour, the production was reduced, companies were in serious financial burden, and unemployment rates increased worldwide. such severe shifts in business and economy across the world are expected to affect equities as well as alternative investments such as cryptocurrency markets. to date, from an investment perspective, there is a need to assess how covid-19 pandemic affected efficiency in cryptocurrency and stock markets. the purpose of the current work is to examine any complexity traits in cryptocurrency and stock markets before and during indeed, this the first study to conduct a formal and robust empirical investigation on the effect of covid-19 pandemic upon the efficiency of cryptocurrency and equity markets, to the best of our knowledge. the covid-19 pandemic is not over yet. however, investors and traders still have time to adjust their decisions based on the current shifts and intrinsic dynamics of these markets. indeed, market agents and policy makers need a first assessment of current impacts of the covid-19 outbreak on equity markets to provide better decision making in the short run. the main purpose of this study is to assess complexity patterns in cryptocurrency and stock markets before and during the covid-19 pandemic based on the estimation of chaos and randomness in their respective prices. to this end, the largest lyapunov exponent (lle) 25 and approximate entropy (apen) 26 will be applied to price time series in order to estimate the degrees of chaos and randomness respectively. the lle describes the rate of divergence/convergence of initially close trajectories with time in a given non-stationary signal. it reveals the chaotic characteristic of a nonlinear signal and describes its stable/unstable state. the approximate entropy is suitable to measure the sequential and temporal regularity/irregularity of a nonlinear signal. hence, measuring both lle and approximate entropy in price time series allows to assess divergence/convergence and regularity/irregularity of cryptocurrency and stock time series before and during covid-19 pandemic. the main advantage of using the lle methodology in 25 and approximate entropy is that they provide efficient estimates even in case the length of original signals is small 26. for each category of equity markets, the difference in estimated lle between pre-and during pandemic periods will be tested via a student t-test and an f-test. similar tests will be applied to estimated approximate entropy measures. our contributions include a fist attempt to investigate the effect of covid-19 pandemic on the stability of a big data set composed of 45 cryptocurrencies and 16 international stock markets. next, we compare the effects of covid-19 pandemic across cryptocurrency and stock markets to shed light on which markets have been more affected by such outbreak. moreover, we enrich the relevant literature on the efficiency of cryptocurrency markets 27-33 and spillovers of stock markets due to financial crises 1-24. section 2 presents the largest lyapunov exponent (lle) and approximate entropy (apen) approaches. section 3 describes the big data sets and provides with the empirical results. finally, section 4 concludes. regularity and stability methodologies the largest lyapunov exponent describes the divergence/convergence of two trajectories with similar initial conditions. following the approach by rosenstein et al 25, let's define a time series x 1 ,x 2 ,…,x n  and x i =(x i ,x i+j ,…,x i+(m-t)j ) a system state at discrete time i, where j is the lag or reconstruction delay and m is the embedding dimension. thus, the reconstructed trajectory is x = (x 1 x 2 … x m ) t . afterwards, the method requires localizing the nearest neighbor of each point of the reconstructed trajectory x. the nearest neighbor x j to x i is determined by calculating the smallest distance d j (0) expressed as follows 25: the divergence is the average exponential rate characterized by the largest lyapunov exponent  1 as follows: ( ) (2) where d(t) is the average divergence at time t and c is a constant that normalizes the initial separation. if we assume that the j-th pair of nearest neighbors diverges approximately at a rate given by the lle, then we obtain: where t is the sampling rate and d j (i) the distance between the jth pair of nearest neighbors after i discrete-time steps. by taking the logarithm of both sides of eq.3, we obtain: then, eq.4 describes a set of almost parallel lines (for j = 0, 1 ,…, m-1), with slope roughly analogous to  1 [25] . finally, the lle is estimated by using a least-squares fit to the average line expressed as: where  denotes the average over all pairs of j. following the approach in [25] , it is assumed that the delay j corresponds to the lag before the first decline of the autocorrelation function, and the embedding dimension m is determined based on the smallest value that allows convergence. this way fast computation is enhanced. recall that a positive value of  1 indicates divergent trajectories (unstable system) whilst a negative value indicates convergent trajectories (stable system). accuracy, robustness to small and noisy data sets, and fast computation are the main advantages of using the approach by rosenstein et al [25] . the approximate entropy 26 was introduced to measure regularity/irregularity of a timeseries. consider a time series let m be a positive integer used to represent an embedding dimension and let r be a filter factor. then, let's form the m-vectors x(1), and i=1, nm+1. the distance between x(i) and x(j) is expressed as follows: the quantity  m (r) is computed as follows: in a similar way, the quantity  m+1 (r) is computed after increasing the dimension to m+1. finally, the apen value of the time series can be calculated by: hence, a large value of apen represents strong irregularity and unpredictability of the current time series as opposed to a low apen value which implies regularity. the covid-19 outbreak started in december 2019 in wuhan city in china and has been declared by the world health organization as pandemic on 30th january 2020. to investigate the impact of covid-19 pandemic upon the stability of daily prices of cryptocurrency and stock markets, we consider two different time periods. the pre-pandemic period spans september 2019 to december 2019 and the pandemic period from january 2020 to april 2020. the data was gathered from yahoo finance. there comprise respectively 123 and 120 samples in the prepandemic and pandemic periods. thus, we explore the chaotic and irregularity properties of 45 cryptocurrency markets and 16 international stock markets under a big data framework. for illustration purposes, fig.1 plots bitcoin price evolution before and during covid-19 pandemic and fig.2 depicts the s&p500 price evolution. as shown, the dynamics of each price data are different across pre-and during the covid-19 pandemic. in addition, fig.3 displays the boxplots of lle and apen for cryptocurrency markets, and fig.4 the boxplots of lle and apen for international equity markets. to check if the distributions of the lle and apen of each category of markets are significantly altered by the impact of the covid-19 pandemic, a student's t-test (test for equality of means) and an f-test (test for equality of variances) are both applied to the estimated sample populations of the lle and apen metrics. the results are provided in tables 1 and 2 respectively. table 3 provides the results from the t-test and f-test when performed to check if lle and apen metrics are different between cryptocurrency and international stock markets. in other words, the two statistical tests are applied at two different levels: firstly, checking differences between time periods and secondly checking for differences between markets. consequently, the total number of performed statistical tests is 36, all at 5% statistical significance level. a given null hypothesis is rejected when the test p-value is less than 5%. according to table 1 , in case of cryptocurrency markets, the null hypothesis of equality of means for the lle across the two time periods is rejected. the one-side t-test results show that the null that the mean of lles before covid-19 pandemic is larger than the one during the pandemic period is not rejected. on the contrary, the reverse null hypothesis is rejected. therefore, for cryptocurrency markets the average level of lles has decreased during the covid-19 period, hence stability has significantly diminished. similarly, according to table 1 , the null hypothesis of equality of the means of apen across the two time periods is rejected. interestingly, the null hypothesis that the mean of apen before the period in question is smaller than the mean of apen during covid-19 pandemic, is strongly not rejected. we see that the average level of apen has increased during the covid-19 pandemic and consequently irregularity in cryptocurrency markets significantly augmented. turning to international stock markets, we can see that the null hypothesis of equality of means of the lle across the two time periods is not rejected. the one-side t-test findings reveal that the null that the mean of the lles before the pandemic is larger than during this period is not rejected. likewise, the null that the mean before the covid-19 is smaller than the mean of lles during the pandemic is not rejected too. thus, for equity markets, the average level of the lle has not statistically altered during the pandemic and the level of stability remained unchanged during the pandemic period. similarly, the null of equality of means for the apen across the two time periods is rejected. also, the null hypothesis that the mean of the apen before the pandemic period is smaller than the mean of apen during the outbreak, is strongly not rejected. for international stock markets, the average level of apen has increased hence they demonstrated higher irregularity during the pandemic time period. next we analyze table 2 . in case of cryptocurrency markets, the two-side f-test shows that the variance of the lles for the pre-covid-19 period is not equal to the one during the pandemic. in this regard, an one-side f-test indicates that the variance of lles in cryptocurrency markets has significantly increased during the outbreak and we deduce that stability was severely perturbed across cryptocurrency markets during the pandemic period. furthermore, the variance of apen has significantly increased during the pandemic, thus the level of sequential irregularity increased across cryptocurrency markets. now, for international stock markets based on the ftests the variance of the lle as well as the variance of apen has not been altered during covid-19. in other words, the variability in stability and irregularity in stock markets has not been affected by the pandemic. according to the comparative table 3 , we show that the mean of the lles in cryptocurrency markets before the pandemic is higher than that of equity markets whilst during the outbreak it is statistically equal to that of stock markets during the pandemic. while the variance of lle for the cryptos before covid-19 is statistically lower than if stock markets before the pandemic, yet they are similar during the pandemic period. it also concluded from our testing that the mean of apen in cryptocurrency markets before the pandemic is equal to that of stock markets when the mean of apen in cryptocurrencies is lower compared to stock markets. finally, while we observe that the fluctuations of the apen variance in cryptocurrency markets before the pandemic are lower than in stock markets, during the pandemic the variance of apen in cryptocurrency markets is equal to that of equity indices. therefore, cryptocurrency and stock markets exhibit a similar degree of stability in price dynamics during the pandemic. in addition, during the pandemic time period, the cryptocurrency markets exhibit a low level of regularity compared to equity indices but with equal variability. our empirical findings can be summarized as follows:  the level of stability in cryptocurrency markets has significantly decreased during the pandemic whilst irregularity augmented.  while the level of stability in international stock markets has not been altered during the pandemic outbreak, yet equity markets embedded more irregularity.  stability became severely perturbed and volatile across cryptocurrency markets during the pandemic period and sequential irregularity presented higher volatility across cryptocurrencies rendering them more unpredictable and chaotic.  the variability in stability and irregularity in stock markets has not been affected by the pandemic.  both cryptocurrency and stock markets exhibit similar degree of stability in price dynamics during the pandemic, albeit cryptos exhibit lower level of regularity compared to international stock markets. table 3 . reported probability values of comparative testing of cryptocurrency vs. stock markets null hypothesis p-value t-tests applied to lle samples lle mean in cryptocurrency markets before pandemic = lle mean in stock markets before pandemic 0.0305 lle mean in cryptocurrency markets before pandemic > lle mean in stock markets before pandemic 0.9847 lle mean in cryptocurrency markets before pandemic < lle mean in stock markets before pandemic 0.0153 lle mean in cryptocurrency markets during pandemic = lle mean in stock markets during pandemic 0.3289 lle mean in cryptocurrency markets during pandemic > lle mean in stock markets during pandemic 0.8355 lle mean in cryptocurrency markets during pandemic < lle mean in stock markets during pandemic 0.1645 f-tests applied to lle samples lle variance in cryptocurrency markets before pandemic = lle variance in stock markets before pandemic 0.1103 lle variance in cryptocurrency markets before pandemic > lle variance in stock markets before pandemic 0.0551 lle variance in cryptocurrency markets before pandemic < lle variance in stock markets before pandemic 0.9449 lle variance in cryptocurrency markets during pandemic = lle variance in stock markets during pandemic 0.8294 lle variance in cryptocurrency markets during pandemic > lle variance in stock markets during pandemic 0.4147 lle variance in cryptocurrency markets during pandemic < lle variance in stock markets during pandemic 0.5853 t-tests applied to apen samples apen mean in cryptocurrency markets before pandemic = apen mean in stock markets before pandemic 0.3529 apen mean in cryptocurrency markets before pandemic > apen mean in stock markets before pandemic 0.1764 apen mean in cryptocurrency markets before pandemic < apen mean in stock markets before pandemic 0.8236 apen mean in cryptocurrency markets during pandemic = apen mean in stock markets during pandemic 1.1670×10 -08 apen mean in cryptocurrency markets during pandemic > apen mean in stock markets during pandemic 5.8348×10-09 apen mean in cryptocurrency markets during pandemic < apen mean in stock markets during pandemic 1.0000 f-tests applied to apen samples apen variance in cryptocurrency markets before pandemic = apen variance in stock markets before pandemic 0.0689 apen variance in cryptocurrency markets before pandemic > apen variance in stock markets before pandemic 0.0344 apen variance in cryptocurrency markets before pandemic < apen variance in stock markets before pandemic 0.9656 apen variance in cryptocurrency markets during pandemic = apen variance in stock markets during pandemic 0.6012 apen variance in cryptocurrency markets during pandemic > apen variance in stock markets during pandemic 0.3006 apen variance in cryptocurrency markets during pandemic < apen variance in stock markets during pandemic 0.6994 the covid-19 pandemic has seriously affected global economy causing major declines in sales, production, and employment rates. in this study we attempted to analyze the stability and sequential regularity detected within the prices of 45 cryptocurrencies and 16 stock markets prior and during the covid-19 pandemic by estimating the largest lyapunov exponents and the approximate entropy. a large number of robust statistical tests was employed to check any differences between prior and during covid-19 outbreak and across cryptocurrencies and stock markets. we concluded that both stability and regularity in these markets have been significantly altered during the pandemic time period. cryptocurrency fluctuations are found to be more affected by the pandemic than international stock markets. specifically, cryptocurrency markets revealed more instability and higher irregularity during the pandemic period compared to the equities. therefore, cryptocurrency markets are riskier and more unpredictable. although our empirical findings are preliminary in nature, our study constitutes a first attempt to assess the stability and regularity of fluctuations in both cryptocurrency and stock markets. our results could be essential to investors, traders and policy makers in order to take appropriate decisions. hopefully, our future work will consider a longer time period incorporating the aftermath of the pandemic for optimal comparative purposes. the hierarchical structure of stock market in times of global financial crisis financial contagion across major stock markets: a study during crisis episodes stock market contagion during the global financial crisis: a multiscale approach global financial crisis and emerging stock market contagion: a volatility impulse response function approach the impact of 2008 financial crisis on the efficiency and contagion of asian stock markets: a hurst exponent approach commodity 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modelling long memory volatility in the bitcoin market: evidence of persistence and structural breaks efficiency of the brazilian bitcoin: a dfa approach long-range behaviour and correlation in dfa and dcca analysis of cryptocurrencies market efficiency, liquidity, and multifractality of bitcoin: a dynamic study bitcoin at high frequency does bitcoin exhibit the same asymmetric multifractal cross-correlations with crude oil, gold and djia as the euro, great british pound and yen? chaos dynamic characteristic of bitcoin cryptocurrency in the reconstruction scheme key: cord-351343-zdh8ms1z authors: din, anwarud; khan, amir; baleanu, dumitru title: stationary distribution and extinction of stochastic coronavirus (covid-19) epidemic model date: 2020-06-24 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110036 sha: doc_id: 351343 cord_uid: zdh8ms1z similar to other epidemics, the novel coronavirus (covid-19) spread very fast and infected almost two hundreds countries around the globe since december 2019. the unique characteristics of the covid-19 include its ability of faster expansion through freely existed viruses or air molecules in the atmosphere. assuming that the spread of virus follows a random process instead of deterministic. the continuous time markov chain (ctmc) through stochastic model approach has been utilized for predicting the impending states with the use of random variables. the proposed study is devoted to investigate a model consist of three exclusive compartments. the first class includes white nose based transmission rate (termed as susceptible individuals), the second one pertains to the infected population having the same perturbation occurrence and the last one isolated (quarantined) individuals. we discuss the model’s extinction as well as the stationary distribution in order to derive the the sufficient criterion for the persistence and disease’ extinction. lastly, the numerical simulation is executed for supporting the theoretical findings. the 2019-novel coronavirus has been known to the virologists community as severe acute respiratory syndrome coronavirus-2 (sars-cov-2) [1] . the covid-19 refers to the virus associated syndrome. sars-cov-2 being previously unrecognized novel-strain of the coronavirus in humans [2, 3] . coronaviruses in general circulate among various animals with some being highly susceptible for infecting humans. among these animals, naturally bats are thought to be proven hosts of such novel coronaviruses, nevertheless, various species of other animals are also considered an active cause for such spreads [4] . at present, the middle east respiratory syndrome coronavirus (mers-cov)" which is much similar to covid-19 was spread from camels to humans, while the civet cats have been considered as source of "severe acute respiratory syndrome coronavirus-1 (sars-cov-1)" for transmission into human. bunch of information are presented in the ecdc factsheet on coronaviruses [4, 5] . though the animals are understood to be a proven source, however, currently, human-tohuman transmission is also considered as one of the spread source. at present, the epidemiological information are sparse for the determination of an effortless spread of this virus among the people, nonetheless, currently, on average, it is estimated that, infection in one person can cause the spread among 2-3 more people [1, 5] . the virus appears to be transferred mostly through narrow respiratory droplets by coughing, sneezing, or peoples interaction in close proximity (usually less than one meter) with each other for a certain time frame. these droplets can further be inhaled, or can stay on the surfaces being came in contact by the infected person, that can cause infection in others by touching their nose, mouth or eyes. the virus possesses ability to survive on various surfaces commencing several hours (e.g. copper, cardboard) up to a few days (e.g. plastic and stainless steel). nonetheless, the quantity of the viable virus certainly decays over a time span and might not be present in sufficient quantity for causing the infection. it is currently estimated that the appearance of symptoms and initial infections in case of covid-19 almost lies between 1-14 days [1, 5, 6] . moreover till today there is no proper treatment in term of vaccine etc. however, the scientists are working faster to develop vaccine for the novel covid-19, which will take enough time. therefore the only way to stop the spread of this disease is to quarantine or isolate the initially infected population as showed by the chinese govt and also the guide line of who. it could be also noted that most of the real world phenomenon are not simply deterministic, because in deterministic models, the output of the model is fully determined by the parameter values and the initial conditions. stochastic models possess some inherent randomness. the same set of parameter values and initial conditions will lead to an ensemble of different outputs or we can say in simple world a deterministic model is one that uses numbers as inputs, and produces numbers as outputs. a stochastic model includes a random component that uses a distribution as one of the inputs, and results in a distribution for the output. these distributions may reflect the uncertainty in what the input should be (e.g. a deterministic input plus noise), or may reflect a random process (i.e. a stochastic input) [7] [8] [9] . for describing the changing behavior of several epidemic diseases in a realistic sense, the mathematical modeling is considered as an influential tool. several epidemic models have been developed by various mathematicians and ecologists for comprehending and controlling various epidemic diseases in a region. in last twenty years, mathematical modeling is widely used for characterizing the communication of various infectious diseases (see e.g. [10, 11] . recently various comprehensions have been made to deepen the understanding about the novel coronavirus (covid-19) particularly grasping the valuable inferences through mathematical modeling [12] [13] [14] . the modelsdescribe the dynamics of infectious diseases, however, for modeling biological phenomenon, it is appropriate to use the stochastic differential equations due to its realistic approach. compared to deterministic models, the stochastic models can generally result in more valuable output, by several times execution, a distribution of the expected results can be built, such as the average infections at any time t, whereas the deterministic models result in a single predicted value [15] [16] [17] [18] . numerous approaches and methods exist for studying stochastic models (such as binomial moment equation etc.) [19, 20] . the most basic stochastic epidemic models are those involving global transmission, meaning that infection rates depend only on the type and state of the individuals involved, and not on their location in the population. how can a model be defined explaining the sometimes-observed scenario of frequent mid-sized epidemic outbreaks? how can evolution of the infectious agent transmission rates be modeled and fitted to data in a robust way? in this paper we understand the transmission mechanism of the covid-19 mathematically, we have formulated a model using the available literature on modeling epidemics, we propose a stochastic epidemic model for the transmission dynamics of covid-19 virus with a varying population environment for a longterm behavior. we categorize the total population into three different classes. the first class is the susceptible individuals in which the transmission rate is distributed by white noise. the second class includes the infected individuals in which the same transmission occurs. the third class consists of the quarantine individuals with white noise. in the recent study, we proposed a stochastic epidemic model for the transmission dynamics of the covid-19 with a changing environment considering long term behavior. the overall population has been divided into three exclusive classes: the susceptible individuals with white noise transmission rate distribution, the infected individuals in which the same perturbation occur and quarantined individuals. then, we will discuss the disease' extinction and stationary distribution and develop the sufficient condition for the covid-19. furthermore, sample simulations are find out with the help of stochastic runge-kutta method for supporting the theoretical results. the present section is devoted to formulation of a model based on stochastic theory for studying the transmissions dynamic of the novel virus i.e., covid-19 pandemic. we propose a susceptibleinfected-quarantined epidemic model as according to the characteristic of the disease. we also take the varying population environment to study the dynamics of the covid-19 particularly its long-term behavior. before to present the model, we put some assumption as given by the following assertions. (a 1 ). the total population at any time t is symbolized by n(t) and it is stratified into three exclusive groups of individual: the susceptible class s(t), the covid-19 infected people i(t) and the quarantined q(t), i.e., s(t) + i(t) + q(t) = n(t) which is changing with t. (a 2 ). the state variables and parameters included in the model are assumed to be nonnegative. (a 3 ) the initially infected individuals move to the quarantined class as performed by the chinese in wuhan city. (a 4 ). once the infection confirmed then the quarantined will go back to the infected compartment. in the light of the above assumption (a 1 ) − (a 4 ), the proposed model leads to the following stochastic epidemic problem which consist of three stochastic differential equations here in the model, λ represents the per capita constant fecundity rate. µ 0 , µ 1 and µ represents the natural mortality rate and disease-related mortality rate, respectively. γ 1 represents the constant rate at which people getting quarantined from covid-19 infected class. b(t) is considered to be the usual brownian motion with intensity η 1 ,η 2 and η 1 taken to be positive. let (ω, {f t } t≥0 , p) be the complete probability space with filtration {f t } t≥0 which satisfy the normal conditions, along with condition z(t 0 ) = z 0 ∈ r d , where b(t) denotes an m-dimensional usual brownian motion. define the operator l related to (4) by by operating l on v (a function from the space by generalized itô's formula, we have this section is about studying the existence and uniqueness of solution of the proposed stochastic covid-19 model (1). (1) is unique for t ≥ 0 with initial condition (s(0), i(0), q(0)) ∈ r 3 + . further, the solution will always remains in r 3 + with unit probability, that is, (s(t), i(t), q(t)) ∈ r 3 + ∀ t ≥ 0 almost surely (a.s). proof: as for initial value of the state variables (s(0), i(0), q(0)) ∈ r 3 + , the coefficients used in equations are continuous and locally lipschitz. thus, there must exists a local unique solution (s(t), i(t), q(t)) of the model over t ∈ [0, τ e ). for detail analysis of the explosion time τ e one must see the references [21, 22] . to prove the global nature of the solution, we must show that τ e = ∞ a.s. assume that we have a sufficiently large nonnegative number k 0 such that all of the initial conditions on the state lie within [ 1 k 0 , k 0 ]. let for each positive integer k ≥ k 0 , the finishing time be defined as throughout this manuscript, we must choose inf φ = ∞ where φ stand for the null set. definition of τ k force us to say that it is increasing as k tends to ∞. setting τ ∞ = lim k→∞ with τ e ≥ τ ∞ a.s. upon showing τ ∞ = ∞ a.s., we will declare that τ e = ∞ and hence (s(t), i(t), q(t)) will lies in r 3 + a.s. ∀t ≥ 0. thus, it is suffice to prove that τ e = ∞ a.s. in otherwise case, there must exists two positive constants from (0, 1) and t such that hence there is an integer k 1 ≥ k 0 , such that next, we will define a c 2 -function h : it is to be noted that the h is a nonnegative function, and it can be verified from the fact that 0 ≤ y − logy − 1, ∀0 < y. assume that k 0 ≤ k and 0 < t are arbitrary. upon applying itô formula to eq.(5) gives us in eq.(6), lh : r 3 + → r + is defined by the following equation thus, setting ω k = {τ k ≤ t} for k ≥ k 1 and by eq.(4), p(ω k ) ≥ . note that for each ω from ω k there must exist one or more than one s(τ k , ω), by using eq.(4) and eq. (8), we can write here 1 ω(ω) represent the indicator function of ω. approaching k to ∞ will lead us to the contra which implies that s(t) > 0. solving the second equation of the model (1) gives us which simply means that 0 ≤ i(t). it is handy to shown that 0 < q(t). hence (s(t), i(t), q(t)) ∈ r 3 + , for all t ≥ 0, which proves the conclusion. remark 1. clearly, theorem 1 and theorem 2 guarantees that for the initial data (s(0), by solving the differential inequality eq. (12) yields in upcoming study, we shall always assume that (s(0), i(0), q(0)) ∈ ω * unless otherwise stated. as for as the stochastic systems are concerned, they have no endemic equlibria. thus, the stability analysis cannot be used as a tool for studying the disease' persistence. as a result, one must turn his/her attention to the existence/uniqueness theory of the stationary distribution which in some sense, will work for persistence of the disease. for this purpose, we will cite a famous result from hasminskii [23] . let " lemma 2. [16, 17] " suppose that x(t) is a regular markov process (time-homogeneous) in r n + whose dynamics is given by "the diffusion matrix is of the form lemma 3. ( [16, 17] define a parameter theorem 3. the solution (s(t), i(t), q(t)) of the model (1) is ergodic as well as there is a unique stationary distribution π(.) whenever r s 0 > 1. in order to verify condition (2) of lemma (3), we need to develop a non-negative c 2 −function v : r 3 + → r + . for this, we will first define where c 1 , c 2 are the positive constant and need to be determined later on. by using the ito's formula and the proposed model (1), we obtain therefore, we have the above implies that let namely consequently in addition, we can obtain where the constant c 3 > 0 to be determined at later stages. it is handy to show that lim inf (s,i,q)∈r 3 + \u k v 2 (s, i, q) = +∞, as k → ∞, where . the next step is to prove that v 2 (s, i, q) has one and only one minimum value v 2 (s 0 , i 0 , q 0 ). the partial derivative of v 2 (s, i, q) with respect to s, i, q is as follow it could be easily obtain that v 2 have unique stagnation point (s 0 , i 0 , q 0 ) = 1+c 1 c 3 1+c 3 , c 3 c 2 1+c 3 , 1 1+c 3 . moreover, the hesse matrix of v 2 (s, i, r) at (s 0 , i 0 , q 0 ) is obviously, the hesse matrix is positive definite. thus, v 2 (s, i, q) has a minimum value v 2 (s 0 , i 0 , q 0 ). according to eq. (22) and from the continuity of v 2 (s, i, q), we can say that v 2 (s, i, q) has one and only one minimum value v 2 (s 0 , i 0 , q 0 ) inside r 3 + . next, we will define a non-negative c 2 −function v : r 3 + → r + as follows applying the itô's formula and using the proposed model, we get which leads to the following assertion where δ i > 0 for (i = 1, · · · , 6) are infinitesimally small constants to be determined later. for the sake of simplicity, we will divide the whole r 3 + \d into the following regions next, we shall prove that lv(s, i, q) < 0 on r 3 + \d which is the same as displaying it on the above-mentioned six regions. we can choose a small constant δ 1 > 0 in such a way that c 1 c 3 β + c 2 c 3 σ + β + 2µ 0 + η 2 2 +η 2 3 2 + µ + σ + λ − λ δ 1 ≤ 0 so we can get lv < 0 for each (s, i, q) ∈ d 1 . case 2. if (s, i, q) ∈ d 2 , then from eq.(24), we can obtain let δ 2 1 = δ 2 , we can select enough large c 3 > 0 and as small as possibleδ 1 by choosing small δ 3 > 0 such that c 1 c 3 β + c 2 c 3 σ + β + 2µ 0 + η 2 2 +η 2 we can select enough small δ 4 > 0 such that we can choose sufficiently small δ 5 > 0 such that c 1 c 3 βδ 1 + c 2 c 3 σ + β + 2µ 0 + η 2 2 +η 2 we can choose sufficiently small δ 6 > 0 such that thus, we reach to the conclusion that there exist a constant w > 0 such that assume that (s(0), and τ x is that time at which a path starting from x reach to the set d, τ n = in f {t : |x(t)| = n} and τ (n) (t) = min{τ x , t, τ n }. upon integration of both sides of the inequality (25) from zero to τ (n) (t), taking expectation, and then by applying dynkins formula, we obtain ev(s(τ (n) (t)), i(τ (n) (t)), q(τ (n) (t)))v(x) = e τ(n)(t) 0 lv(s(u), i(u), q(u))du, following the proof of theorem (3) we have p{τ e = ∞} = 1. alternatively, one can say that the system (1) is regular. thus, if we let t → ∞ and n → ∞ then we have τ(n)(t) → τ x almost surely. accordingly, with the help of fatou's lemma we get obviously, sup x∈k eτ x < ∞, where k being a compact subset of r 3 + . it directly proves the condition (ii) of lemma 3. moreover, the diffusion matrix for system (1) is given by it means, condition (1) of lemma (3) also holds. concluding the above discussion, we can say that lemma (3) guarantees that system (1) is ergodic as well as it has one and only one stationary distribution. hence the proof. (1) along with initial data (s(0), i(0), q(0)) ∈ r 3 + , then lim sup t→∞ (s(t) proof: from the proposed model (1) we can write the integration of both sides yields we define where , setting because of the quadratic variation, we can write by using lemma(1) (see for detail [16] [17] [18] ) and (31), we get similarly, we also get which proves eq. (26) and hence the lemma 4.1. for the purpose of disease' extinction, we have to state and prove the following theorem. (1) a.s., (i(t) approaches zero exponentially a.s., i.e., the covid-19 infection will dies out from the community with unit probability). moreover proof: to prove the theorem, we shall apply direct integration to the proposed stochastic covid-19 model (1) . first of all, we will apply the itô formula to the second equation of system (1) by integrating relation (35) from zero to t and dividing it by t leads to by using the theorem related to large number for local martingales, we obtain lim t→∞ b 2 (t) t = 0 a.s. by taking the limit superior of both sides in fig. 9 the total cases of covid-19 has been depicted from 9 th april to 2 nd june 2020, which becomes one month and 24 days. in fig. 10 we fitted the real data with the infected class of our covid-19 model which clearly shows the appropriateness of behavior of the infected class. fig. 11 shows long time behavior of the covid-19 cases vs time (months). we can see that the data is accurately fit to the model curve and further, one can observe that the cases with time on long term behavior grows exponentially. this case could be alarming that the incidence may increases further in the coming months if the government not applied the proper optimal strategies. covid-19 incidence data in the current section, we shall perform the numerical simulation of the developed coronavirus stochastic epidemic model. the well know stochastic runge-kutta (rk) method for the purposes of numerical findings will be used. this analysis will verify our derived analytical results and will show the influence and effect of noise intensity. we assume the numerical value of the parameters with biological feasibility to verify the extinction result are as: i.e., the infection of novel corona virus vanishes exponentially with increasing the value of white noise intensity. however there will be always susceptible population in the case of extinction. in a similar fashion, we assume the following parameter value and the strong effect of white noise to show the permanence or stationary distribution i.e., λ = 0.5, µ 0 = 0.2, β = 0.6, γ 1 = 0.3, µ 1 = 0.2, σ = 0.1, µ = 0.2, η 1 = 0.5, η 3 = 0.7, and η 2 = 0.6 while the initial population sizes will be taken as above. the simulation carried out for this are presented in fig.4b . again the three trajectories in fig.4b , which represent the dynamics of susceptible (red dashed), infected (purple solid) and quarantined population (red solid), which show that the model maintain the persistence i.e., there will be always susceptible, infected and quarantine individuals. hence it could be noted from the simulation analysis that the white noise intensity have a great influence on the dynamics of the disease: as when the value of the white noise intensity increases the infection will decreases, while on the other hand if the value of the white noise intensity decrease, the infection will increases. the novel covid-19 is one of the severe disease in the world and till today there is no proper treatment. it could be also noted that majority of real world phenomenon are not simply deterministic, and contain randomness. with the help of stochastic theory, we developed a model for the novel covid-19 keeping in view the characteristic of the disease to investigate the transmission dynamics with changing population environment. by adopting the idea of stochastic lyapunov functions theory, the existence and positivity are shown. we established a suitable stochastic lyapunov function to perform the above activity. the extinction as well as the stationary distribution have been further discussed to find the conditions that how to extinct the disease. it could be noted that the there is a great influence of noise intensity on the covid-19 transmission. clearly it has been observed, that the extinction of covid-19 infected individuals increases with increasing the noise strength, while decreases disease persisting. all the above analytical findings are supported graphically with the help of numerical simulation and therefore concluded that the work reveals stochastic analysis is a better approach to study the dynamics of infectious disease particularly novel covid-19 etc, because there are many factor which varies time to time and place to place. in future, the model can be further extended by adding an exposed class. one can also fractionalize the model by using atangana-baleanu, caputo or caputo-fabrizo operator. not only this can but researcher may apply optimal control technique to minimize the infected people by choosing suitable optimal control variables. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. severe acute respiratory syndrome coronavirus 2 (sars-cov-2) and corona virus disease-2019 (covid-19): the epidemic and the challenges commentary: middle east respiratory syndrome coronavirus (mers-cov): announcement of the 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survival and stationary distribution of a sir epidemic model with stochastic perturbations stability of sirs system with random perturbations threshold behaviour of a stochastic sir model detecting critical transitions in the case of moderate or strong noise by binomial moments linear approximations of global behaviors in nonlinear systems with moderate or strong noise a stochastic model for internal hiv dynamics thetransmissiondynamicsandcontrolofhepatitis b virus in the gambia alphen aan den rijn, netherlands, sijthoff and noordhoff(monographs and textbooks on mechanics of solids and fluids modelling the epidemiology of hepatitis b in new zealand classification of different hepatitis b infected individuals with saturated incidence rate the transmission dynamic and optimal control of acute and chronic hepatitis b multigroup sir epidemic model with stochastic perturbation the extinction and persistence of the stochastic sis epidemic model with vaccination the threshold of a stochastic sirs epidemic model with saturated incidence covid-19 outbreak: current scenario of pakistan this work was supported by the hec. adding respective sides of equations (40), we getcalculation leads towherefrom the last equation of system (40) we havewhich implieswe thus obtain lim key: cord-337275-phgfpzbt authors: andrew, jones; nikolay, strigul title: is spread of covid-19 a chaotic epidemic? date: 2020-10-20 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110376 sha: doc_id: 337275 cord_uid: phgfpzbt the covid-19 epidemic challenges humanity in 2020. it has already taken an enormous number of human lives and had a substantial negative economic impact. traditional compartmental epidemiological models demonstrated limited ability to predict the scale and dynamics of covid-19 epidemic in different countries. in order to gain a deeper understanding of its behavior, we turn to chaotic dynamics, which proved fruitful in analyzing previous diseases such as measles. we hypothesize that the unpredictability of the pandemic could be a fundamental property if the disease spread is a chaotic dynamical system. our mathematical examination of covid-19 epidemic data in different countries reveals similarity of this dynamic to the chaotic behavior of many dynamics systems, such as logistic maps. we conclude that the data does suggest that the covid-19 epidemic demonstrates chaotic behavior, which should be taken into account by public policy makers. furthermore, the scale and behavior of the epidemic may be essentially unpredictable due to the properties of chaotic systems, rather than due to the limited data available for model parameterization. the developing covid-19 pandemic challenges humanity in 2020. according to aggregated worldwide data posted at the international worldometer.info website, more than 15 million were infected by covid-19 and more than 600,000 people died in relation to this disease (as of july 20, 2020) . the pandemic has already caused a huge economic loss due to national lock-downs, travel restrictions, and global distractions of trade and manufacturing chains. as the pandemic developed around the globe, severe border control and travel limitations were implemented that led to unprecedented national isolation. different countries, while being isolated, implemented various strategies to limit epidemic spread: many countries implemented extended national-level lock-downs including closure of businesses, schools and stay-at-home orders (for example france, italy, germany, austria, hungary, south korea, and more), some introduced practically no such lock-down measures (for example, sweden, belarus, and brazil). many countries implemented only regional lock-downs (for example, china and the usa). the overall challenge for the development of scientifically-based covid-19 containment strategy is the apparent unpredictability of this pandemic [5] . traditional compartmental epidemiological models demonstrated quite limited ability to predict the scale and dynamics of this epidemic in different countries [5, 25, 11] . this apparent unpredictability of the covid-19 pandemic creates an additional challenge for governments around the globe, who need more accurate predictions to develop a reasonable containment strategies [16, 9] . this limitation of traditional modeling approaches can be partially explained by the novelty of the virus and limited relevant epidemiological data necessary for model parameterization [28] as well as statistical problems in model calibration [25] . in particular, until recently, we did not know the rate and mechanisms of the transmission of covid-19 virus as well as its biology and ability to survive and spread outside host organisms [19, 11] . in a previous study, [4] demonstrated that the coronavirus raw data in china's first two months of the disease suggest chaotic growth, similar to other epidemics like h1n1 and measles. mathematical chaos theory originated from studying deterministic systems represented as differential equations where initial conditions dictate their behavior. in these systems sensitivity to the initial conditions was so large that the systems were practically unpredictable. this was originally studied by poincaré in the 1880s in relation to the three-body problem in astronomy, an example of which is the earth-moon-sun celestial system [23] . as poincaré critically stated, "it may happen that small differences in the initial conditions produce very great ones in the final phenomena" [23] . in physical systems described by differential equations, systems which experience this effect are described as having "high sensitivity" to initial conditions. one example which demonstrates this concept is a bowling ball; imagine a very small spin on the ball at the beginning of its roll. this small spin in combination with the oiled bowling lane could cause it to move wildly off-course, making it "highly-sensitive" to initial conditions. due to the effect poincaré found, "measurements made on the state of a system at a given time may not allow us to predict the future situation" [3] . these systems are now termed "chaotic." unpredictability due to highly-sensitive reliance on initial conditions inspired the term "deterministic chaos." after poincaré's studies, the deterministic chaotic behavior was discovered in numerous dynamical systems and confirmed experimentally [15, 6, 2, 20] . the spread and "fade out" cycle of epidemic diseases often exhibits deterministic chaos. qualitative analysis of that deterministic chaos can offer more information than a "multifactorial stochastic paradigm of causation" [21] . in the late 80s, scientists noticed that measles outbreaks exhibited deterministic behavior. by studying this they hoped to quantify conditions for the maintenance of [an] infection in human populations" [7] . in 1991, grenfell published his study of the chaotic measles outbreaks in the paper "chance and chaos in measles dynamics." that study found that certain parameters in the chaotic disease system created deterministic behavior. for example, applying his model to the city of copenhagen and mapping with poincaré's methods produced drastic results: one example is that doubling the immigration rate into the city could completely eliminate the fade-out of measles [7] . similar findings in relation to covid-19 could have wide-ranging societal implications. in this work, we consider the covid-19 epidemic within the framework of complex dynamic systems [14] . this framework is broadly applied in modern ecology, social and medical sciences [21, 13, 6, 2, 20] . in particular, the covid-19 pandemic can be considered as a complex phenomena developing simultaneously at multiple temporal and spatial scales varying from individual level to the global scale. within this framework, particular disease dynamics patterns result from major self-organisation mechanisms within the system. the chaotic behavior is a common phenomenon in complex systems, and we hypothesise that the unpredictability of the epidemic scope in different countries is a fundamental property of this dynamic system which demonstrates chaotic behavior. in order to examine this hypothesis we have analysed covid-19 epidemic data from different countries. in particular, we examine if the covid-19 epidemic demonstrates a chaotic regime based on the analysis of observed data. we do not aim to derive or validate a correct dynamical system model. instead we consider epidemiological data collected in different isolated countries as an independent observations of the same dynamical system. in order to evaluate whether or not the spread of coronavirus is chaotic, we employ the following criteria, derived from poincaré's definition of chaos [23, 4] [23] . in the study, we find extremely high variation between countries, even when normalizing by patient-zero date and total population of each country. 2. sensitive -does the growth of the virus experience large regions of activity? are there real reasons for regions of activity, specifically changes in the physical system? this criterion has precedent in the aforementioned university of florida / wuhan university paper [4] . later in the study, we present multiple examples of high regions of sensitivity in the spread of covid-19, as revealed by the derivatives of the system. 3. numerically unpredictable -can the behavior of the system change unexpectedly from one point to another? this criterion was also introduced in the university of florida / wuhan university paper [4] . we found numerous examples of sets of countries which exemplified unpredictability by behaving similarly and then suddenly diverging, which we discuss later in the paper. 4. deterministic -what causes activity in the system? is the system random, or is it determined by some factors of the physical system, as required in order to be chaotic? this is a qualitative criterion derived from poincaré's definition of chaos [23] . we present cases where real-world changes such as mandated lockdowns had a profound impact on the spread of the disease, suggesting deterministic behavior. the study concluded that the spread of covid-19 exhibits the major qualitative characteristics of chaotic systems. most countries show a roughly logistic growth curve, but activity in the second derivative revealed great variability in system behavior. when examined under the context that infectious epidemics in the past have exhibited chaotic behavior, the study concluded that the spread of covid-19 suggests that the epidemic is a chaotic system. johns hopkins university has continuously collected data on the coronavirus epidemic from various sources such as the world health organization, including daily cases per country, which is the data source used for this study [10] . the data is organized by country and cumulative cases by date. a sample with the first five days of one country is shown in table 1 . we considered the data from the beginning of the global data set, january 22nd, 2020, to may 30th, 2020. each data-point tells the number of confirmed covid-19 cases in that country by that day. in the example shown in table 1 , the united states had five confirmed cases by january 26th, 2020. with almost 35,000 data-points corresponding to 267 countries or territories, the study required a dynamic analysis tool instead of analyzing static charts. in order to investigate deeper, a web-based interface was designed and developed to allow for close inspection of the data through interactive charts. the evolution of the covid-19 epidemic was mapped over time, in terms of total number of confirmed cases per country per day. data was adjusted to account for country population and date of first-confirmed-case in order to accurately compare the spread behavior between countries. the raw data was extracted as an csv file and analysed using original javascript software [12] , the graphing utility plotly [22] was employed to generate all the graphs. figure 1 demonstrated the raw data as percentages of each country's population. since the data only pertains to the total number of confirmed cases, not accounting for decreases like recovery and death, the total number of cases in every country/region increases or remains constant at all times. highlighted in figure 2 , almost all regions show 1 of 3 possible parts of the beginning of a logistic graph: a curved ramp-up (russia), or a curved ramp-up into roughly a line (us), or an upward curve into a line which curves toward horizontal (spain), which is a logistic curve. this is sensible for an infectious disease with a maximum number of infections, that maximum being the population of each country. per capita.png so far, these patterns don't directly suggest chaos: it still seems possible there might be some parameters which control the shape of the curves predictably, potentially population density, weather patterns, etc. figure 3 demonstrates the case data shifted to show the percent of the population that had been infected as a function of the number of days since the first case in each country, instead of the absolute date. this graph is more revealing about the relative behavior of the epidemic in each country. it is clear the virus ran its course faster in some countries, where the growth seems to have flattened in fewer days. for example, only 55 days after the first case in iceland, the country had almost completely stopped the spread. meanwhile, the us at 130 days in was still seeing almost linear positive spread. comparison of iceland and the us these differences confirm that there are a large number of potential solutions or equilibrium and varying behavior between countries over time. another key observation from this graph is that countries which seemed to follow similar growth curves can very rapidly diverge. for example, as shown in figure 5 , in the first 40 days of the epidemic in ireland, the netherlands, and turkey, the percentage of the population that was infected grew almost identically in each country. however, between days 40 and 42, the three countries' curves diverged greatly, to such an extent that by day 60, ireland had almost three times the infected population percentage as turkey, and turkey and the netherlands are 25% apart as well. this example exhibits unpredictability because, by poincaré's definition, the behavior following day 40 cannot be predicted by the state at day 40, since there are clearly multiple possible behaviors. this is one example confirming unpredictability in the spread of covid-19. it is also important to note that these three countries are not alone in displaying unpredictability after formerly-similar spread of covid-19. another example is kuwait, france, and sweden, shown in figure 6 . until approximately day 70, the infected percentage in all three countries had grew with similar shaped curves. however, within days after that point, the infected percentage in kuwait spiked drastically, while it entered linear growth in france and sweden. by 10 days later, even france and sweden diverged, with sweden's infected percent still growing linearly while the spread slowed down in france. by poincaré's definition of unpredictability, knowing the spread behavior in sweden from day 60 to 120 would not provide any accurate prediction of the behavior in kuwait or france, suggesting a chaotic nature.yet another example of unpredictability is cabo verde and new zealand, shown in figure 7 . in cabo verde and new zealand, the disease spread almost identically until day 25, at which point it continued to grow similarly in each country until day 40, and then the behaviors sharply diverge: cases grow steeply linearly in cabo verde, but level off almost entirely in new zealand. this example further demonstrate that spread of covid-19 is unpredictable, because at any given point there are multiple possible behaviors in the time following that point. the observation of unpredictability fulfills one of poincaré's key criteria for choatic systems. we looked at the rate of spread of covid-19, the derivative of the number of cases, adjusted similarly to figure 3 . this data, shown in figure 8 shows the spread rate as a function of the number of days since the first patient. negative derivatives represent decreasing numbers of cases, which shows corrections to data, since the data set does not account for real decreases in active cases (by recovery or death). while the raw data shows mostly similar shaped, roughly logistic trajectories with some small bumps and jumps, the daily rate of growth shows a lot of variation in behavior. for example, in the us from day 92 to day 99, just one week, the daily growth rate decreases by 30% and increases back to almost the same starting point (shown in figure 9 ). we also see drastically different behavior between different regions, unlike the roughly logistic graphs of the raw data. one example is the comparison of the spread rates of the us and spain. by their respective day 30s, the spread rate in spain started to decline, while the us spread rate was just starting to increase, as shown in figure 10 . the amount of unpredictable variation within a country and different possible models for each country suggest chaos. however, most countries still follow a very rough pattern of increasing and then leveling-off and decreasing spread rates. the second derivative helps to measure this sensitivity ( figure 11 ) because its behavior shows activity in the system [4] . the rate of change of the spread rate in each region fluctuates greatly. the us ( figure 12 ) seems to oscillate between roughly two values, spain and italy ( figure 13 ) appear to exhibit a single heart-beat-like pulse surrounded by ramp-up and ramp-down, and russia ( figure 14) and germany show clusters of high activity. in summary, the results reveal the following: 1. large number of solutions -while most countries have a roughly logistic curve of covid-19 growth, there is a huge amount of variation in: the time it takes to reach the same point in that logistic curve, the highest percent of the population which the logistic curve reaches, and the degree of curvature of the logistic model. 2. sensitive -the second derivative shows clusters, spikes, and oscillations, revealing that the system is highly sensitive due to the high amount of activity in the system instead of steady or constant behavior. 3. numerically unpredictable -while the raw data shows roughly predictable shapes, the extreme variation first and second derivatives of different countries reveal a high amount of variability, making the growth of the system unpredictable. furthermore, there are examples of growth curves appearing identical and then suddenly diverging. we may predict that spread in the us will level off logistically, but we cannot tell when that will occur based on the us data or other countries' curves. one important question remaining is what causes the activity: is the system random, or is it determined by some factors of the physical system, as required in order to be chaotic? qualitative investigation is required to answer that, and even then we cannot definitively determine the factors contributing to every change in every country. however, we investigated some of the countries and found good examples suggesting the system is deterministic. new cases take up to 14 days to show symptoms and therefore be reported [18] . in iceland, lockdown was initiated on march 24th [29] , day 25 when shifted. exactly two weeks after that change, there are few new cases reported ( figure 15 ) and almost all activity shown by the second derivative ceases (figure 16). this is also the day with the sharpest increase in testing in ireland [8] , which would explain the jump in the data and higher activity following that day. in bahrain, re-opening efforts began on april 9th [1] , which is the likely cause of a the immediately following spikes in number of cases ( figure 19) . these examples suggest that real changes in the physical conditions of the system are causing changes in the spread of the virus. the outcome, amount and rate of spread, is determined by these real factors. this suggests the behavior of the system is deterministic, not random, a key qualifier for a chaotic system. complex natural systems often demonstrate chaotic behavior. however, rigorous proof of chaos from empirical data or experimental observations alone is a substantial challenge. in this work, rather than attempt a rigorous mathematical proof, we investigate the hypothesis that the covid-19 pandemic exhibits chaotic behavior by mapping the disease over time from available epidemiological data. our results suggest that the covid-19 epidemic exhibits deterministic chaos. the overall predictions of the sir model demonstrate a typical sigmoidal curve. this functional response is characterised by an exponential growth stage, inflection point and slowdown growth phase towards a horizontal asymptote. the first wave of covid-19 pandemics in different countries demonstrates similar behavior ( figure 3 ). empirical sigmoidal models such as the logistic curve are also broadly employed for modeling of covid-19 and other epidemics [27, 26, 24] . the discrete counterpart of the logistic curve is a well-known logistic map model that demonstrates a chaotic behavior [17] . our examination of the epidemic dynamics in different countries reveals amazing similarity to the chaotic behavior known in this and many other dynamics systems. we conclude that the scale of the epidemic is essentially unpredictable due to fundamental reasons rather than due to the limited data available for model parameterization. we find that the chaotic behavior in the spread of covid-19 suggests that it is a deterministic chaotic system, which should be taken into account by public policy makers. through use of an interactive data map, it was shown that the spread of covid-19 exhibits the major characteristics of chaotic systems, namely, determinism, high sensitivity, large number of equilibria, and unpredictability. when examined under the context that infectious epidemics in the past have exhibited chaotic behavior, we conclude that spread of covid-19 is likely a chaotic system. we may be able to gain some insights into its behavior, such as the common logistic pattern, but we cannot assume that it will follow a logistic path in any one country and we cannot numerically predict the behavior of a particular logistic curve. ☒ the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐the authors declare the following financial interests/personal relationships which may be considered as potential competing interests: claration of interest statement coronavirus: 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recent confirmed covid-19 cases in china simple mathematical models with very complicated dynamics covid-19 information hub: what you need to know a systematic review of covid-19 epidemiology based on current evidence complexity, simplicity, and epidemiology chaos, population biology, and epidemiology: some research implications plotly graphing libraries: javascript sur leséquations de la dynamique et le probleme des trois corps estimation of covid-19 dynamics "on a back-of-envelope": does the simplest sir model provide quantitative parameters and predictions? chaos why is it difficult to accurately predict the covid-19 epidemic? real-time forecasts of the covid-19 epidemic in china from february 5th to february 24th covid-19 epidemic in italy: evolution, projections and impact of government measures modeling the epidemic dynamics and control of covid-19 outbreak in china hert samkomubann: ekki fleiri en 20 mega koma saman key: cord-330703-fbmy6osu authors: zhang, zizhen; jain, sonal title: mathematical model of ebola and covid-19 with fractional differential operators: non-markovian process and class for virus pathogen in the environment date: 2020-07-28 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110175 sha: doc_id: 330703 cord_uid: fbmy6osu differential operators based on convolution definitions have been recognized as powerful mathematics tools to help model real world problems due to the properties associated to their different kernels. in particular the power law kernel helps include into mathematical formulation the effect of long range, while the exponential decay helps with fading memory, also with poisson distribution properties that lead to a transitive behavior from gaussian to non-gaussian phases respectively, however, with steady state in time and finally the generalized mittag-leffler helps with many features including the queen properties, transitive behaviors, random walk for earlier time and power law for later time. very recently both ebola and covid-19 have been a great worry around the globe, thus scholars have focused their energies in modeling the behavior of such fatal diseases. in this paper, we used new trend of fractional differential and integral operators to model the spread of ebola and covid-19. very recently, the world have been surprised with an outbreak of a fatal disease called covid-19, [10, 11, 14] . severe acute respiratory syndrome coronavirus-2 (sars-cov-2) is a new type of virus family that has not been earlier identified in people. the virus seems to be transmitted mostly through the minute respiratory droplets via coughing, sneezing or when people interact with each other for some time in close proximity. these droplets can then be inhaled, or they can land on surfaces that others may come into touch with, who can then get contaminate when they contact their eyes, mouth, or nose. the novel corona virus can live on different surface like few days (stainless steel and plastic) and few hours (cardboard, and copper). however, the amount of viable virus declines over time and may not always be present in sufficient numbers to cause infection. in humans, the symptoms of this virus can be experienced in between 1 to 14 days from the day of infection. from then it has been spreading at the speed of knots, giving no time to prepare against a newly identified infectious and notorious virus which have compelled the who to declare covid-19 as a pandemic [1] due to its fast human to human transmission and people got infected in every continent and it had already taken so many lives. in applied mathematics, many new mathematical models have been suggested, some including fractional differential and fractal fractional operators [3, 4, 7, 12, 13] . ebola virus disease (evd), caused by infection with the filo virus. the virus cause hemorrhagic fever to both monkeys and human. the disease was first observed in 1976 in the ebola river valley in what is now the democratic republic of the congo, africa.1 since then, zaire ebola virus has caused a number of outbreaks over the past three decades and has culminated in the current largest outbreak, which has spanned a number of west african countries and spread throughout the world. the virus is transmitted via contact with mucous surfaces, non-intact skin, or through injury with contaminated needles. the disease course includes fever, aches, malaise, severe vomiting and diarrhea, as well as increased vascular permeability, which leads to profound intramuscular volume depletion. while the hemorrhagic manifestations can be as minor as petechiae and bruising, the disease can progress to include gastrointestinal hemorrhage, subsequent shock, and multi system organ dysfunction. swelling of the brain and kidneys can occur as well as necrosis of internal organs including the liver, testis, and ovaries.the recent outbreaks in liberia, guinea, sierra leon, and nigeria have been the largest to date. furthermore, with cases appearing in the united states and europe, concerns have been raised about the possibility of even further spread abroad. this article seeks to review the knowledge of the vascular, cardiac, and pulmonary effects of evd collected across medical specialties. disease in tropical regions of the world. ebola virus that causes severe bleeding, organ failure and can lead to death. humans may spread the virus to other humans through contact with bodily fluids such as blood. initial symptoms include fever, headache, muscle pain and chills. later, a person may experience internal bleeding resulting in vomiting or coughing blood. treatment is supportive hospital care. there are lots of bio-mathematical models have been proposed to recognize the transferral dynamics of these type of infectious diseases. recently, modeling has become a valuable tool in the analysis of ebola disease transferral dynamics and to determine the factors that influence the spread of disease to support control measures. many researchers have proposed epidemic model to study the transferral dynamics of ebola disease. there is no specific medicine to cure ebola disease. awareness programs can be helpful in reducing the prevalence of the disease. different bio-mathematical models have been proposed to study the impact of awareness in controlling ebola and these type diseases. the study showed that, for small portion of infected individuals, the whole country could die out in a very short period of time in case there is not good prevention. in this section, we recall some basic definitions and properties of fractional calculus theory which are useful in the next sections. definition 2.1. let f : r+ → r and β ∈ (n − 1, n), n ∈ n . the left caputo fractional derivative of order of the function f is given by the following equality; (2.1) then the caputo derivative of fractional order is given by (see in [9] ): where m (θ) is a normalization function such that m (0) = m (1) = 1 [8] . but, if the function u = h 1 (a, b) then, new derivative called the caputo-fabrizio fractional derivative can be defined as where b(θ) has the same properties as in the case of the caputo-fabrizio fractional derivative. here it should be noted that we do not recover the original function when θ = 0 except when at the origin the function vanishes. to avoid this kind of problem, the following definition is proposed. and not necessary differentiable then, the definition of the new fractional derivative (atangana-baleanu fractional derivative in riemann-liouville sense) is given as. definition 2.5. the fractional integral associate to the new fractional derivative with nonlocal kernel (atangana-baleanu fractional integral) is given as [2] : when alpha is zero we recover the initial function and if also alpha is 1, we obtain the ordinary integral. we begin to formulate the ebola epidemic disease by considering the human population in three compartments, that is, the susceptible individuals, s(t), individuals infected with ebola virus, i(t) and the individuals recovered from the ebola virus, r(t). the individuals infected with ebola and the deceased is d(t) and p (t) is the class for the ebola virus pathogen in the environment. the model that describes the dynamics of ebola disease modeled through differential equations is given by where λ = β 1 i + β 2 d + ψp and the appropriate initial conditions are given by the parameters of the model are given in the following where n = s + i + r denotes the total alive human population. it should be noted that ≤ (d + δ) which is an appropriate condition for the compartment d for which the model becomes relevant, otherwise the deceased human individuals will disappear and the model would be irrelevant. further, the model given by equation (3.1) is well posed and biologically feasible in the region given by where m = (s(t), i(t), r(t), d(t), p (t)). we present in this paper existence and uniqueness of system solution first we covert the model to volterra version (4.1) we defined the following norm will be used and in the same way finally we derive first the disease free equilibrium for the equation (3.1) now we derive the endemic equilibrium by setting the left hand to be zero we now present the reproductive number to achieve this, we consider the following equations: thus the reproductive number be expressed as we present the global asymptotic stability of disease-free and endemic. but first we prove the endemic possibility by assuming di dt > 0 and dd however, we have that such that at least we should have death from the disease thus thus one can conclude that r 0 > 1 we consider the following lyapunov we now consider the lyapunov function for endemic case since the solutions of this system describe a real world situation as they representing numbers as functions of time, it is worth showing that ∀t ≥ 0 those solution are positive or zero. λ is positive thus thus this leads to finally i(t), d(t) and r(t) are positive ∀t ≥ 0 thus therefore all solutions are positive. in this section, we construct a numerical scheme for fractional model based on the caputo fractional derivative, cf fractional derivative and atangana-baleanu fractional derivative [6] . on applying this scheme we first consider the following non-linear fractional ode: 7.1. numerical method for caputo fractional derivative. in this section, we concern with the following cauchy problem where the derivative is caputo fractional derivative. here we aim to present a numerical scheme to solve the above equation. for this, firstly we transform the above equation into at the point t n+1 = (n + 1)∆t, we have also we have replacing newton polynomial into the above equation, we have thus we can write the following and rearrange such as for the above integrals, we can have as follows if we put them into above equality, we obtain the following scheme so we can write the (3.1) with caputo derivative as s 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) − s 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) (s 1 (t n , s n , i n , r n , d n , p n )) − 2 s 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) (7.14) i n+1 = i 0 + (∆t) θ γ(θ + 1) n j=2 i 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) [(n − j + 1) θ − (n − j) θ ] + (∆t) θ γ(θ + 2) × n j=2 i 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) − i 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) (i 1 (t n , s n , i n , r n , d n , p n )) − 2 e 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) (∆t) θ γ(θ + 2) × n j=2 r 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) − r 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) (r 1 (t n , s n , i n , r n , d n , p n )) − 2 r 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) (7.16) d n+1 = d 0 + (∆t) θ γ(θ + 1) n j=2 d 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) [(n − j + 1) θ − (n − j) θ ] + (∆t) θ γ(θ + 2) × n j=2 d 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) − d 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) (d 1 (t n , s n , i n , r n , d n , p n )) − 2 d 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) (7.17) (∆t) θ γ(θ + 2) × n j=2 p 1 (t n−1 , s n−1 , i n−1 , rn − 1, d n−1 , p n−1 ) − p 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) (p 1 (t n , s n , i n , r n , d n , p n )) − 2 p 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) (7.18) 7.2. numerical method for caputo-fabrizio fractional derivative. in this section, we handle the following cauchy problem with caputo-fabrizio fractional derivative where the function f is non-linear. to present a numerical scheme for solution of our equation, we can reformulate the above equation as at the point t n+1 = (n + 1)∆t, we have at the point t n = n∆t, we have if we take the difference of this equations, we obtain tn f (τ, y(τ ))dτ. (7.23) and using the newton polynomial, we can write the approximation of the function f (t, y(t)) as follows f (t n , y(t n )) − 2f (t n−1 , y(t n−1 )) + f (t n−2 , y(t n−2 )) 2(∆t) 2 × (τ − t n−2 )(τ − t n−1 ). thus putting this polynomial into the above equation, we write the following (7.26) and reorder as follows we can have the following calculations for the above integrals if we replace them into above scheme, we obtain the following scheme [f (t n , y(t n )) − f (t n−1 , y(t n−1 ))] and we can rearrange as for simplicity, we write above equation (7.10) with caputo-fabrizio fractional derivative as follows; after applying caputo-fabrizio fractional derivative we have the following we can have the following solution for this model [s 1 (t n , s(t n ), i(t n ), r(t n ), d(t n ), p (t n )) −s 1 (t n−1 , (s(t n−1 ), i(t n−1 ), r(t n−1 ), d(t n−1 ), p (t n−1 ))] s 1 (t n−1 , s(t n−1 ), i(t n−1 ), r(t n−1 ), d(t n−1 ), p (t n−1 ))∆t + 5 12 s 1 (t n−2 , s(t n−2 ), i(t n−2 ), r(t n−2 ), d(t n−2 ), p (t n−2 ))∆t + 23 12 s 1 (t n , s(t n ), i(t n ), r(t n ), d(t n ), p (t n ))∆t (7.33) −i 1 (t n−1 , s(t n−1 ), i(t n−1 ), r(t n−1 ), d(t n−1 ), p (t n−1 ))] , d(t n−1 ), p (t n−1 ))∆t + 5 12 i 1 (t n−2 , s(t n−2 ), i(t n−2 ), r(t n−2 ), d(t n−2 ), p (t n−2 ))∆t −r 1 (t n−1 , s(t n−1 ), i(t n−1 ), r(t n−1 ), d(t n−1 ), p (t n−1 ))] r 1 (t n−1 , s(t n−1 ), i(t n−1 ), r(t n−1 ), d(t n−1 ), p (t n−1 ))∆t + 5 12 r 1 (t n−2 , s(t n−2 ), i(t n−2 ), r(t n−2 ), d(t n−2 ), p (t n−2 ))∆t + 23 12 r 1 (t n , s(t n ), i(t n ), r(t n ), d(t n ), p (t n ))∆t (7.35) , d(t n−1 ), p (t n−1 ))∆t + 5 12 d 1 (t n−2 , s(t n−2 ), i(t n−2 ), r(t n−2 ), d(t n−2 ), p (t n−2 ))∆t + 23 12 d 1 (t n , s(t n ), i(t n ), r(t n ), d(t n ), p (t n ))∆t (7.36) [p 1 (t n , s(t n ), i(t n ), r(t n ), d(t n ), p (t n )) −p 1 (t n−1 , s(t n−1 ), i(t n−1 ), r(t n−1 ), d(t n−1 ), p (t n−1 ))] 3 p 1 (t n−1 , s(t n−1 ), i(t n−1 ), r(t n−1 ), d(t n−1 ), p (t n−1 ))∆t + 5 12 p 1 (t n−2 , s(t n−2 ), i(t n−2 ), r(t n−2 ), d(t n−2 ), p (t n−2 ))∆t + 23 12 p 1 (t n , s(t n ), i(t n ), r(t n ), d(t n ), p (t n ))∆t (7.37) 7.3. numerical method for atangana-baleanu fractional derivative. now we deal with the following cauchy problem with atanganabaleanu fractional derivative in this section, we provide a numerical scheme to solve this equation. applying atangana-baleanu integral, we convert the above equation into at the point t n+1 = (n + 1)∆t, we have (7.40) at the point t n = n∆t, we have t j+1 j f (τ, y(τ ))dτ. (7.41) here, for the approximation of the function f (t, y(t)), we use the newton polynomial which is given by f (t n , y(t n )) − 2f (t n−1 , y(t n−1 )) + f (t n−2 , y(t n−2 )) 2(∆t) 2 × (τ − t n−2 )(τ − t n−1 ). thus, we can present the following scheme for numerical solution of our above equation as s 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) − s 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) (s 1 (t n , s n , i n , r n , d n , p n )) − 2 s 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) (7.49) (e 1 (t n , s n , i n , r n , d n , p n )) + θ(∆t) θ ab(θ)γ(θ + 1) n j=2 i 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) [(n − j + 1) θ − (n − j) θ ] + θ(∆t) θ ab(θ)γ(θ + 2) × n j=2 i 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) − i 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) (i 1 (t n , s n , i n , r n , d n , p n )) − 2 e 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) (7.50) r 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) − r 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) (r 1 (t n , s n , i n , r n , d n , p n )) − 2 r 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) (7.51) d n+1 = d 0 + 1 − θ ab(θ) (d 1 (t n , s n , i n , r n , d n , p n )) + θ(∆t) θ ab(θ)γ(θ + 1) n j=2 d 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) [(n − j + 1) θ − (n − j) θ ] + θ(∆t) θ ab(θ)γ(θ + 2) × n j=2 d 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) − d 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) (d 1 (t n , s n , i n , r n , d n , p n )) − 2 d 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) (7.52) (p 1 (t n , s n , i n , r n , d n , p n )) + θ(∆t) θ ab(θ)γ(θ + 1) n j=2 p 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) [(n − j + 1) θ − (n − j) θ ] + θ(∆t) θ ab(θ)γ(θ + 2) × n j=2 p 1 (t n−1 , s n−1 , i n−1 , rn − 1, d n−1 , p n−1 ) − p 1 (t n−2 , s n−2 , i n−2 , r n−2 , d n−2 , p n−2 ) (p 1 (t n , s n , i n , r n , d n , p n )) − 2 p 1 (t n−1 , s n−1 , i n−1 , r n−1 , d n−1 , p n−1 ) (7.53) now after obtaining the numerical solution of ebola model including (s n , i n , r n , d n , p n ). the number of new infected susceptible death and other are available in the literature, we can now suppose that the data function φ i follows a given distribution for example poisson distribution with parameters π i = p 1 i n . thus φ i = poisson(π i = p 1 i n ) the parameter p 1 is constant reflecting a combination of collected sampling efficiency also the detectability of infections, thus the likelihood can be defined as l(π i ) = π m t=0 f (φ t , π i ) = π m t=0 π φt i exp(−π i ) φ t using some collected data we can estimate ξ, ϕ, ν, υ, θ and σ. in this section using theoretical parameters, we present some numerical simulations for different values of fractional order. the obtained numerical solutions are therefore presented in figure 1, 2, 3, 4, 5, 6 and 7. ebola and covid-19 are very strange infectious diseases that put worry around the globe as they biological and mode of transmissions are not fully understood. white their effects have been fatal around the globe especially the covid-19, humans have taking serious measures to fight them back and possible eradicate them from the globe to insure security, peace and stability among humans. mathematical models have been used helpful to at least warn humans for the severity of these fatal disease. in this paper, we consider a mathematical model that take into account the ebola and covid-19 virus pathogen in the environment. stability analysis for disease-free and endemic equilibrium, we presented numerical solutions using a new trend of numerical scheme to derive approximate, additionally, we presented, the likelihood formula for poisson distribution was presented for parameters estimation. the author is grateful to the editor and the anonymous referees for their valuable comments and suggestions on the paper. this research was supported by the natural science foundation of the higher education institutions of anhui province (no. kj2020a0002). severe acute respiratory syndrome-related coronavirus: the species and its viruses, a statement of the coronavirus study group new fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model a new numerical approximation of the fractal ordinary differential equation fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system the role of power decay, exponential decay and mittag-leffler functions waiting time distributions: application of cancer spread new numerical method for ordinary differential equations: newton polynomial models of fluid owing in non-conventional media: new numerical analysis a new definition of fractional derivative without singular kernel numerical analysis for the fractional diffusion and fractional buckmaster's equation by two step adam-bashforth method covid-19 epidemic in switzerland: on the importance of testing, contact tracing and isolation a novel coronavirus from patients with pneumonia in china translated from the 1987 russian original, revised by the authors analysis of lassa hemorrhagic fever model with non-local and non-singular fractional derivatives a novel coronavirus genome identified in a cluster of pneumonia caseswuhan thus if we write this polynomial in (7.41), we have the following y n+1 = y 0 + 1 − θ ab(θ) f (t n , y(t n ))and we can reorganizethus we havewhen calculating the above integralsand putting this equalities into above scheme, we can obtain the following scheme key: cord-334413-h6n36jei authors: bhattacharyya, suvanjan; dey, kunal; paul, akshoy ranjan; biswas, ranjib title: a novel cfd analysis to minimize the spread of covid-19 virus in hospital isolation room date: 2020-09-17 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110294 sha: doc_id: 334413 cord_uid: h6n36jei the covid-19 is a severe respiratory disease caused by a devastating coronavirus family (2019-ncov) has become a pandemic across the globe. it is an infectious virus and transmits by inhalation or contact with droplet nuclei produced during sneezing, coughing, and speaking by infected people. airborne transmission of covid-19 is also possible in a confined place in the immediate environment of the infected person. present study investigates the effectiveness of conditioned air released from air-conditioning machines to mix with aerosol sanitizer to reach every point of the space of the isolation room so as to kill the covid-19 virus which will help to protect the lives of doctors, nurses and health care workers. in order to numerically model the laminar-transitional flows, transition sst k-ε model, which involves four transport equations are employed in the current study. it is found from the analysis that high turbulent fields generated inside the isolation room may be an effective way of distributing sanitizer in entire volume of isolation room to kill the covid-19 virus. the covid-19 is a severe respiratory disease originated from the devastating coronavirus family (2019-ncov) (zhou et al., 2020; zu et al., 2020) and has become a pandemic across the globe. it is a contagious virus and transmits by inhalation or contact with droplet nuclei of size φ < 5μm produced during sneezing, coughing and even speaking by infected persons. exhaled droplets from confirmed covid-19 patients or an active carrier of the virus can deposit in the mucosae of noses, mouths, and conjunctiva of eyes of people in close contact. the virus may be transmitted through personal contact with the covid-19 patient or indirect contact with fomites such as clothes, utensils, furniture, surface etc. used, touched or in the immediate environment of the infected person. generally, fever, breathlessness, cough, throat pain, weakness are traditional symptoms at the initial stage of the disease (huang et al., 2020) . the disease causes respiratory illness such as pneumonia and acute respiratory distress syndrome resulting in rapid death of those affected, depending on their age, condition of lungs, immunity and sociodemographic profile. a study (bukhari et al. 2020) showed that the covid 19 is highly harmful and transmitted the infection throughout the world. total of 212 countries and territories around the world are infected by the disease and became a pandemic as declared by the world health organization (who) (peng et al., 2020; who, 2020a; who, 2020b) . it has also been reported (who, 2020c ) that airborne transmission of covid-19 is also possible in specific circumstances like aerosol therapy performed during treatment of pulmonary critical illness such as asthma, bronchoscopy, chronic obstructive pulmonary disease (copd), tracheostomy and other related diseases. in addition to cough, the droplets produced by sneeze produced aerosols, the droplets containing virus present in the air also constitute a substrate for viruses which transmit the disease through air in an isolated space (setti et al., 2020; who, 2020c) . furthermore, relative humidity, temperature, rainfall etc. are recognized as factors affecting the infectivity of the virus in the respiratory system (ahmadi et al., 2020; bashir et al., 2020; prata et al., 2020; monserrate et al., 2020; taylor, 2019; xie and zhu 2020; qi et al., 2020; sahin 2020; shi et al., 2020) . as the characteristics of covid-19 virus is still not fully understood, there is no particular treatment, medication, therapy or vaccine approved by the medical authorities till date. the mortality rate of the patients affected by this virus is 2 to 3%, but it can be fatal for aged people and children and those with a prolonged illness with less immunity. the medical boards and health administration has implemented travel ban, complete lockdown, containment zone identification, home quarantine of all citizens, strict monitoring of movement of citizens etc. to combat the spread of covid-19. the transmission can also be reduced by controlling indoor dust level, temperature, humidity, ventilation, improved hygiene, sanitization, wearing mask, using personal protective equipment (ppe) by the healthcare and sanitary personnel etc. it is indispensable to accommodate the confirmed covid-19 patients and patients with symptoms in isolated rooms or separate icus in hospitals for treatment so that spread of this disease can be prevented. these isolated rooms are designated as "airborne infection isolation rooms (aii)". also, exhaust air released from aiis is likely to carry virus particles and hence an effective strategy should be employed to arrest the spread of infections. recent covid-19 outbreak in wuhan province of china found the evidence of covid-19 virus genetic material in the air about 4 metres from the affected persons in two icu wards of huoshenshan hospital in wuhan, risking to healthcare personnel (guo et al., 2020) . care should therefore be taken to disinfect the exhaust air through various available treatments such as hepa filtration, sanitization, heating, uv irradiation etc. (european centre for disease prevention and control, 2020; ministry of health & family welfare, government of india, 2020; who, 2006; who, 2020d ). very few researchers have however studied the design aspects of an effective ventilation system and influencing factors for room ventilation to reduce the spread of the virus (li et al., 2007) . the healthcare personnel working in hospitals are at greater risk of getting infected due to outbreak of airborne or droplet contact diseases. many researchers used computational fluid dynamics (cfd) based models (memarzadeh et al., 2000) to study air quality inside a room, comfort level, performance of hvac systems etc. in different types of buildings. it is a very robust and efficient tool to investigate the airflow and contaminant dispersion in rooms where many parameters involved. cfd analysis using complex particle tracking methodologies (cole et al., 1998; kowalski et al., 1998; memarzadeh, 2011; zhang et al., 2008) with dynamic process variables like velocity, movement, path lines followed by the air in order to determine control strategy for the trajectory of infectious particles moving in air, may be considered and simulated to control the spread the huge number of infectious droplets generated from patients cough, sneeze. as the medical treatments are often inaccurate, besides precautionary measures and supports, it is therefore reasonable to investigate the possibilities to sanitize the confined volume of air to mitigate the spread of covid-19 virus inside the airborne infection isolation rooms, and icus of a hospital. this can be accomplished by designing an aerosolized sanitization system which effectively can sanitize the air inside the room to be used for the treatment of confirmed covid-19 patients. this is essential to protect the lives of doctors, nurses and health care workers. however, no work is reported in literature on this topic till date. hence, the present paper depicts aerosol sanitizer delivery systems focusing the effectiveness of conditioned air released from air-conditioning machine to mix with aerosol sanitizer and enabling every corner of the isolation room killing the covid-19 virus, thereby achieving complete sanitization. also, results obtained from the research work could be used to help flatten the infection curve of covid-19. a cfd analysis is carried out considering factors affecting aerosol sanitizer delivery system such as temperature, turbulent kinetic energy and flow dynamics. tetrahedral and hexahedral elements are used to generate these meshes ( fig. 2 (b) ). hexahedral meshing is used to fill the intricate parts, while tetrahedral meshes are used in the remaining parts of the control volumes. on each computational element, governing equations, like mass, momentum and energy equations are solved using finite volume based cfd technique. boundary conditions constitute an important criterion for any cfd simulation and the present study is of no difference. inlet conditions are specified at the ceiling of the isolation room with a velocity of 3.91 m/s applied uniformly with an inlet temperature of 24c. with a mass flow rate of 29.97 kg/s. no-slip, no-temperature jump conditions are applied at the exit of the ducting system. similarly, inlet conditions are specified at the sanitizing machine of the isolation room with a velocity of 1.5 m/s applied uniformly with an inlet temperature of 30c, with a mass flow rate of 1.854 kg/s. numerical solution to any governing equation which is expressed in a partial differential equation requires discretization. second-order upwind schemes with boussinesq approximation (bhattacharyya et al. 2020 ) are used for this purpose. the purification of air in the isolation room with chemical diffusion requires simulating the diffusion as well as the natural ventilation process, which involves pressure forces, buoyant forces and elements of forced-convection, and conductive as well as convective heat transfer. an unsteady cfd analysis is carried out to get more insight into the flow physics of the system. in order to numerically model the laminar-transitional flows, transition sst k model, which involves four transport equations are employed in the current study (bhattacharyya et al. 2017 ). this special transition sst k model is formed by a combination of the sst k model along with two additional transport equations-one designed for the transition onset criteria while the other for flow separation induced transition, in terms of momentum thickness reynolds number (bhattacharyya et al. 2017 , rajnath et al., 2020 . (menter 1994) cfd simulation is carried out in a 16-core ibm hpc with 64 gb ram and 27 cpu seconds of processing time (51 days). the cfd models are well accepted and used to investigate thermal comfort, indoor air quality, load of the room, hvac performance, etc. in numerous buildings. cfd is one of the most resourceful and capable tools to study fluid flow (air as the working fluid) and pollutant spreading in the comfort area. so, computational technique is used in this particular study. this particular study is to computational investigate the flow characteristics of the sanitizer-laden conditioned air inside the room, which is essential for disinfection of the room air and thereby protecting the lives of doctors, nurses and healthcare workers. the inlet and outlet of the portions are shown in fig. 3 . before investigating the fluid dynamics/pattern of hospital isolation room geometry, the current numerical model and methodology are validated against published experimental and numerical works (chung and hsu 2001, jacob et al. 2019) . fig. 4 (a) and fig. 4 subsequently spreads towards the walls. another view (seen from the top) is presented in fig. 6 showing the ever-changing streamlines. it is understood from these figures that the fluid flow which influences the isolation hospital room has originated from the clean air openings (vents) positioned at the top of the isolation room (ceiling) irrespective of the direction of observation. together. it is evident from fig. 8 (refer to the right side the image) that the sanitizing machine releases sanitizer at relatively higher temperature, which mixes with the cool air coming from the air-conditioning vent. better mixing is ensured due to velocity as well as the temperature gradient available between the flows of the sanitizing machine and the air-conditioning vent. the cool air coming from the air-conditioning machine from the top of the isolation room exhibits asymmetric pattern due to the influence of the velocity and temperature gradient maintained by the sanitizing machine. an effect with fig. 8 is demonstrated in fig. 9 . the velocity vectors are shown in fig. 9 . with the other flow (sanitizer, horizontal flow). it is also found that both the flows slow a bit when striking the walls. flow circulation and large scale eddies found due to the mixing of flows between cool air and sanitizer and also due to bounding walls. due to thorough mixing between cool air and sanitizer it is expected that the entire isolation room air will be sanitized, and due to this sanitizing by volume, it is also expected that the isolation room is fully virus free and the occupants (patients) can stay comfortably. this results and novel idea could be used to help flatten the curve of covid-19. moreover, these results could be used to design the effective layout of the air-conditioning ducts. in the context of covid-19, since there is no specific treatment or established medical protocol, medication and vaccine, the objective should be to control and prevent the transmission of this virus as far as possible. it is absolutely essential to reduce the risk of airborne infection transmission to the lowest possible level in hospital isolation rooms to protect the lives of doctors, nurses and other health care workers, and simultaneously, flatten the curve of covidthis particular investigation was performed to offer understanding the airflow patterns in the isolation room. the study has been carried out to investigate the effectiveness of conditioned air released from air-conditioning machines to mix with aerosol sanitizer so as to reach every corner of the isolation room and kill the covid-19 virus. a cfd analysis has been carried out considering factors affecting aerosol sanitizer delivery systems such as temperature, turbulent kinetic energy and flow dynamics. in order to numerically model the laminar-transitional flows, transition sst k model, which involves four transport equations are employed in the current study. it is found from the analysis that high turbulent fields generated inside the isolation room may be an efficient way of distributing sanitizer in a volume of confined isolation room to kill or minimize the covid-19 virus. g k = generation of turbulent kinetic energy (k) comprehensive review and study of the buoyant air flow within positive-pressure hospital operating rooms investigation of effective climatology parameters on covid-19 outbreak in iran correlation 186 between climate indicators and covid-19 pandemic infuence of inlet turbulence intensity on transport phenomenon of modifed diamond cylinder: a numerical study computational investigation of heat transfer enhancement by alternating inclined ribs in tubular heat exchanger. progress in computational fluid dynamics characterization of infectious aero-sols in health care facilities: an aid to effective engineering controls and preventive strategies effect of ventilation pattern on room air and 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anaesthetists should know covid-19 transmission in mainland china is associated with temperature and humidity: a time-series analysis effects of flow control in the presence of asymmetric inflows in twin air-intake. aircraft engineering and aerospace technology impact of weather on covid-19 pandemic in turkey impact of temperature on the dynamics of the covid-19 outbreak in china evaluation of the potential relationship between particulate matter (pm) pollution and covid-19 infection spread in italy optimize occupant health, building energy performance and revenue through indoor-air hydration high temperature and high humidity reduce the transmission of covid-19 collecting, preserving and shipping specimens for the diagnosis of avian influenza a(h5n1) virus infection : guide for field operations who, 2020a. coronavirus disease 2019 (covid-19) situation report -51 who characterizes covid-19 as a pandemic modes of transmission of virus causing covid-19: implications for ipc precaution recommendations, scientific brief severe acute respiratory infections treatment centre: practical manual to set up and manage a sari treatment centre and sari screening facility in health care facilities association between ambient temperature and covid-19 infection in 122 cities from china study on biological contaminant control strategies under different ventilation models in hospital operating room the authors would like to gratefully acknowledge birla institute of technology and science, pilani, pilani campus and motilal nehru national institute of technology allahabad for this research. the authors declared that they have no conflict of interests. key: cord-353306-hwwswvi3 authors: zhu, bangren; zheng, xinqi; liu, haiyan; li, jiayang; wang, peipei title: analysis of spatiotemporal characteristics of big data on social media sentiment with covid-19 epidemic topics date: 2020-07-17 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110123 sha: doc_id: 353306 cord_uid: hwwswvi3 covid-19 blocked wuhan in china, which was sealed off on chinese new year's eve. during this period, the research on the relevant topics of covid-19 and emotional expressions published on social media can provide decision support for the management and control of large-scale public health events. the research assisted the analysis of microblog text topics with the help of the lda model, and obtained 8 topics (“origin”, “host”, “organization”, “quarantine measures”, “role models”, “education”, “economic”, “rumor”) and 28 interactive topics. obtain data through crawler tools, with the help of big data technology, social media topics and emotional change characteristics are analyzed from spatiotemporal perspectives. the results show that: (1) “double peaks” feature appears in the epidemic topic search curve. weibo on the topic of the epidemic gradually reduced after january 24. however, the proportion of epidemic topic searches has gradually increased, and a “double peaks” phenomenon appeared within a week; (2) the topic changes with time and the fluctuation of the topic discussion rate gradually weakens. the number of texts on different topics and interactive topics changes with time. at the same time, the discussion rate of epidemic topics gradually weakens; (3) the political and economic center is an area where social media is highly concerned. the areas formed by beijing, shanghai, guangdong, sichuan and hubei have published more microblog texts. the spatial division of the number of weibo social media texts has a high correlation with the economic zone division; (4) the existence of the topic of “rumor” will enable people to have more communication and discussion. the interactive topics of “rumors” always have higher topic popularity and low emotion text expressions. through the analysis of media information, it helps relevant decision makers to grasp social media topics from spatiotemporal characteristics, so that relevant departments can accurately grasp the public's subjective ideas and emotional expressions, and provide decision support for macro-control response strategies and measures and risk communication. with the development of covid-19 (corona virus disease 2019), the number of infected people worldwide has exceeded 8.9 million, and the epidemic has become the most serious public health event affecting humankind in the 21st century. in the course of human development, infectious disease has always been a life health problem that can't be ignored and has caused a certain impact on people's physiology and psychology. in december 2019, china's wuhan health commission reported 27 cases of pneumonia of unknown cause [1] . on february 11, 2020, the world health organization officially named the new coronavirus covid-19 in to mine people's true cognitive ideas [12] [13] [14] [15] . according to the analysis of emotional changes, we can understand which topics will have a positive and negative impact on emotions during emotional changes, and provide some social media emotional guidance opinions for relevant departments. with the widespread popularity of social media, the unique advantages of social media will provide the public with a rapid and convenient platform to obtain and communicate information, thereby improving people's ability to respond to emergencies [16] . latent dirichlet allocation (lda) is one of the most powerful technologies in social media semantic mining. the lda topic model has been widely applied, including semantic analysis of social media, topic extraction of articles, unordered text mining, etc. [17] . research shows that lda has obvious advantages in topics extraction and semantic mining [ 18 , 19 ] . for example, the subjective expression of customers to the hotel was extracted to analyze the key factors affecting customer satisfaction [20] . combined with the lda model, a case studied on the new york times report on nuclear technology, which proved that lda has the characteristics of fast speed in analyzing large-scale texts [21] . analysis of microblog dynamic information of china's dengue fever based on lda. meanwhile, the spatial aggregation characteristics and temporal evolution of dengue fever cases were explored in combination with spatial analysis [18] . the topic was extracted from tweets about drugs, and a new method of seasonal influenza monitoring was mined based on lda [22] . although these studies have made gratifying progress, for covid-19, and in the chinese new year of population migration, the online expression of people's concern about the epidemic may be significantly different from other epidemics. in order to obtain the public's social media expressions during this period, the lda model was chose to construct text topics and emotion recognition. with the help of crawler to obtain social media data. this study is to analyze the change of topics and mood during the epidemic from the perspective of time and space, as well as to answer the following research questions: rq1: what are the main topics of concern during the epidemic? rq2: what is the relationship between high-profile topics? rq3: what are the characteristics of social media topics and emotions changing with time under the background of major events? rq4: what are the characteristics of the spatial distribution of topics and emotions? as an unsupervised machine learning technology, lda uses the bag-of-words method to identify the topic information hidden in large-scale document sets or corpora. the principle is to project data in a low dimension. the projection points of each type of data to be as close as possible, and the distance between different types of data centers to be as far as possible. the purpose of the lda algorithm is to infer potential topics and build a comprehensive corpus [23] . in lda, a document is a set of words, and there is no order between words. a document can contain multiple topics, and each word in the document is generated by one of the topics. lda can distribute the topic of each document in the document set in the form of probability distribution [21] . the application of lda is based on three important classifications: corpus, documents in corpora, words in documents. there is a nested relationship among these three classes [23] . each document represents a probability distribution composed of some topics, and each theme represents a probability distribution composed of many words. based on lda, we can obtain the vocabulary of the topic and the frequency of vocabulary occurrence. document topic statistics show the probability that the text is associated with each topic in the original set topic [16] . finally, the semantic expression of the text was produced. the data processing flow is depicted in fig. 1 . the data for this study was obtained from qingbo big data agency ( http://www.gsdata.cn/ ). we defined "new crown" as the search keyword. with the help of python toolkit, crawled the data searched with the keyword "new crown" from january 24, 2020 to february 25, 2020. the data includes the publisher, title, publishing area, publishing time, likes, and comment segments. the data was acquired and stored according to the time. in order to improve the representativeness of text data, we removed short text. the "re" toolkit in python was used to remove special characters from text. in order to ensure the validity of the analysis statistics, we deleted the data with na value in the "title" field. because part of the text is longer, we only retained weibo with a length of less than 400 chinese characters as the analysis sample, and obtained 1,858,288 microblog data. the "jieba" package in python was used to segment weibo text. we have limited the part of speech of words, including 7 categories ("n", "nr", "ns", "nt", "eng", "v", "d"). the "gensim" package in python was used to implement the lda model. through more than 10 tests and some reference studies [ 16 , 24 , 25 ] , we set 10 themes every day, and each theme contains 15 words. baidu provides nlp platform ( https://ai.baidu.com/tech/nlp ) for sentiment analysis of text. we selected the "positive_prob" field returned by the api (application programming interface) as the sentiment score. the score is in the range of 0 to 1. the higher the value, the higher the passion of emotion. according to the vocabulary of daily topic extracted by lda, the microblog text related to covid-19 into 8 categories of topics as our analysis objects was summarized. the vocabulary contained in each category has been listed in table 1 . if the microblog text contains vocabulary of two topics, it is worth believing that the microblog also belongs to the interactive theme category. finally, we got 28 interactive topics. as shown in table 2 , "origin" is the topic with the largest number of microblogs (11.86%), with "role models" topic (0.6182) having the highest average popularity and "rumor" topic (0.2018) having the lowest average. in addition to expressing thoughts and needs directly by publishing the weibo text, people will also express their attention to topics through likes and comments. we also calculated the rate of discussion for each topic by summing the total number of comments and likes per topic divided by the sum of the total number of posts per topic. the topic of "host" had the highest discussion rate (19.70) . judging from the rate of topic discussion, there are three main areas of concern: (1) the appeal of intermediate host for the spread of the epidemic and the prohibition of the use of wildlife; (2) wide spread of the epidemic; (3) the situation in the place where the outbreak occurred. associated terms for each topic. selected search terms origin ["hubei", "wuhan"] host ["game", "wild animals", "bat", "pangolin", "intermediate host"] organization ["health committee", "health department", "red cross", "charity", "community"] quarantine measures ["in and out", "seal off", "prohibition", "separaten", "sealed type", "online", "remote", "mask"] role models ["front line", "angel", "white", "protector", "doctor", "assistance", "li lanjuan", "zhong nanshan", "li wenhong"] education ["ministry of education", "school", "teachers", "students", "teach", "internet courses"] economic ["resume work", "resume production", "economic", "company", "enterprise", "factory", "financial", "business", "work"] rumor ["rumor", "start a rumor", "fake news", "don't believe the rumors", "don't spread rumors"] table 3 shows that the "origin & model" contains the most data (2.07%), which is consistent with the results of the single topic analysis. although there are a lot of microblogs on the topic of "origin & model", the topic discussion rate (28.29) is not the highest. the topic discussion proportions were sorted in descending order, and 4 of the top 10 interactive topics were interactive with "rumor" topics: "economic & rumor" (143.69), "organization & rumor" (88.52), "quarantine measures & rumor" (45.78), "origin & rumor" (35.88). this shows that the existence of "rumor" will enable people have more communication and discussions. we sorted the sentiment values in ascending order. seven of the top 10 low-emotional topics were related to rumor. according to the analysis of the topic popularity, there are always high topic popularity and low emotion text expressions in the interactive topics of "rumor". in order to explore the changing characteristics of topics and emotions in the period, the research period was subdivided into daily as a unit of temporal level. as shown in fig. 2 (a) , after entering the new year, the number of microblog posts gradually decreased. after january 28th, the number of microblogs began to rise, and the peak of the epidemic topic data curve appeared around february 10. as shown in fig. 2 (b) , the topic discussion rate also showed a downward trend after the new year, and continued until january 31. the topic discussion rate fluctuated greatly from february 5 to february 13, and also fluctuated greatly from february 17 to february 25, but the volatility declined significantly. as shown in fig. 2 (c) , the emotion gradually showed an increasing trend. there was no abnormal fluctuation or reduction in the time interval. it can be seen that the emotion is relatively flat. (1) single topic as shown in fig. 3 , the number of microblogs began to increase after 7:00 a.m. and began to decrease after 10:00 p.m. as a result of the establishment of some embargo measures to prevent outbreaks of the epidemic, people generally reduced their travel. therefore, a large number of microblogs were published from 7:00 a.m. to 12:00 p.m. but it can still be seen that there were more microblog posts at 10:00 a.m. and 4:00 p.m. as shown in fig. 4 (a) , the topic of "origin" has long been ahead of other topics. however, after february 23, there was a decline and people were more inclined to publish texts on topics related to "quarantine measures" and "economic". as shown in fig. 4 (b) , the discussion rate on some dates is relatively high (february 6, february 12, and february 20). there are two major fluctuations in the interactive topic, including "host". it can be seen in fig. 4 (c) that the mood of the epidemic topic also changes with time. from january 24 to february 4, people had higher emotional expressions in the texts on the topic of "organizations" and "role models". from february 2 to february 25, the topics of "education" and "role models" alternately became the topics with the highest emotional value. weibo is time-sensitive, it will cause emotions to change with events in the period. but from the perspective of the emotional curve, the topics of "host" and "rumor" are in a low emotional state throughout the period. (2) interactive topic we chose the most popular interactive topic of the day as the research object. during the period from january 24 to january 30, the "origin & quarantine measures" released a larger amount of data, and from january 31 to february 25, the "origin & role models" released more data. the topic of "role models" in the interactive topic will bring more positive emotional expression, which has been reflected in the single topic analysis at the temporal level. among the 33-day high-level interactive topics, a total of 23 days contain "rumor" and there is a phenomenon of depression. the research area is subdivided into the province as a unit. although the epidemic has swept through all provinces and cities across the country, due to the differences in social conditions and the severity of the epidemic, there are some spatial heterogeneity in the topic of the epidemic. as can be seen in fig. 5 , the number of topic microblogs published in central china and coastal areas is higher. the area is formed around hubei with beijing, shanghai, guangdong and sichuan as its borders. as shown in fig. 6 , the areas with high topic discussion rate are mainly distributed in areas with convenient external contact. (1) single topic most regions still pay more attention to "origin". the discussion rate of popular topics in most areas of the country is between 11 and 40, and various provinces and cities also present different forms of regional interactive topics. people in beijing call for an end to the consumption of wildlife. this appeal led more people to participate in the discussion on the topic of "host". it is similar in hebei and chongqing. "origin" is the topic with the highest topic discussion rate in hubei province. as far as emotions are concerned, the topic with the highest emotions in most regions is "role models", while the topic with the lowest emotions is "rumor". (2) interactive topic the areas with the largest number of microblogs are mainly concentrated in the eastern part of the chinese mainland. although the theme with the largest amount of data in each province is "origin & role models", the number varies. there are more microblogs in beijing and hubei than other places. in terms of topic discussion rates, beijing ("host & quarantine measures", 1,391.35), chongqing ("origin & host", 537.55), yunnan ("origin & quarantine measures", 501.78), zhejiang ("organization & rumor", 150.68), gansu ("host & role models", 102.86) have higher discussion rates. the internet and social media have become a widespread, large scale and easy to use platform for real-time information dissemination. it has become an open stage for discussion, ideological expression, knowledge dissemination, emotions and sentiment sharing. social media has the characteristic of a double-edged sword. we should make full use of its advantages to serve the public, especially when dealing with emergency situations. we selected january 24, 2020 to february 25, 2020 as the study period. analyzing the opinions and emotions expressed by weibo users during the outbreak of the epidemic from a spatiotemporal perspective. the temporal and spatial changes of people's discussion topics and emotions are analyzed to provide decision support for relevant departments. on january 29, 31 regions of china initiated a first-level response to public health emergencies. the formation of medical teams all over china to support regions with severe epidemics, "academician zhong nanshan" and "academician li lanjuan" frequently appeared on the topic of "role models". at the same time, some doctors who once supported the sars epidemic have once again joined the fight against the covid-19 epidemic. people hold a positive attitude towards this example and it is easy to resonate. role models are a kind of spiritual carrier and embodiment, which affects people's behavior in a subtle way [26] . in the current society, it is necessary to publicize this topic, which will enhance people's determination and confidence in fighting the epidemic. the number of epidemic topics gradually decreased after january 24. however, compared with the general topics in daily life, its proportion is increasing constantly, and there is a "double peaks" phenomenon in the public opinion curve. we got the daily hot search list of microblogs as the total number of topics for each day. combining the topic phrases analyzed by this article and the phrases analyzed by scholars [27] as the characteristics of our judgment whether the hot search topic belongs to the epidemic topic. it is found that the epidemic topic search volume curve showed two peaks in a week, which appeared on january 29 and february 6 respectively. the epidemic topic searches accounts for 60% of all topic searches. since then, the proportion of popular topics has gradually decreased, but it remains above 30%. we count the public opinions of social media on some major disaster events. during the h1n1 flu, the search volume dropped sharply within one month after the peak search volume appeared on twitter [28] . there were two peaks of concern within one month during the h7n9 flu, but the second peak was about 50% lower than the first [29] . the public's attention to the ebola epidemic reached its climax, and the second peak of attention appeared three months later [30] . the current covid-19 epidemic shows different trends of concern, which have not been seen before, and is a new feature of this epidemic. as shown in fig. 7 , during the period from january 30 to february 3, although other hot topics appeared, the spring festival topic also received continuous attention. people published a large number of tweets about "origin" in the early stage, and gradually change to "quarantine measures" and "economic" topics in the later period. wuhan was sealed off on january 23. with the outbreak of the epidemic, every move of the city has attracted the attention of the whole nation. with the passage of time, the epidemic situation in wuhan has been controlled, and people's attention has gradually shifted to the topics of "quarantine measures" and "economic". this phenomenon is in line with the characteristic that online public opinion would change topics within a period of time [16] . the major media platforms have publicized the epidemic prevention measures, such as "stay at home", "wear masks", "keep distance", etc. at the same time, people's attention to the economy has gradually increased, and how to safely and orderly resume work. production is also the focus of discussion. beijing-tianjin-hebei, yangtze river delta, pearl river delta and chengdu-chongqing posted more microblogs. economic concentration also exists in these areas [16] . beijing, as the capital and political center of china, is geographically far from the outbreak area (wuhan), but still has more topics to participate in and discuss. first of all, beijing, as the transportation hub of the country, the impact of the changing epidemic on people may be clearly reflected on social media. second, beijing was greatly affected by sars in 2002. there are still some microblogs about sars in the topic (3.50%), and the mood was very low (0.2837). third, the economic, political, and cultural background has created people who can obtain information in time and participate in discussions [31] . the low-mood areas are not necessarily adjacent to hubei province. beijing, shandong, jiangsu, shanghai, guangdong and other regions are far away from hubei, but the mood for the epidemic is not high. this phenomenon may be related to the unique culture of china. this research found that there are usually more topics to discuss in interactive topics including "rumor". the existence of "rumor" can enable people to have more communication and discussion, and most of the low-emotional topics are related to "rumor". it can be seen that the interactive topics of "rumor" always have higher topic popularity and low emotional text expression, which will have a negative psychological impact on the prevention and management of the epidemic to a certain extent. this situation may be caused by the uncertainty of the development of major the purpose of this study is to analyze the public opinions of social media on the topic of covid-19 after the closure of wuhanand during the spring festival in china. we analyzed singel topic and interactive topic from the perspective of temporal and spatial. (1) the topic of "origin" and "quarantine measures" accounted for 21% of the total sample. this shows that the government's scientific report on the origin of the epidemic will help to stabilize public opinion in the early stages of the epidemic. the discussion of "economic and rumor" is more intensive. relevant departments should focus on controlling the spread of rumors and the economy in social media, and timely contain them to prevent further proliferation. (2) the "double peaks" appeared in the epidemic topic curve. several hot topics and chinese new year topics have led to two peaks in the search volume curve of popular topics within one week. after that, the epidemic topic search volume showed a significant downward trend, but the epidemic topic search volume accounted for more than 30% of the total topic search volume. (3) the topic gradually shifted from the epidemic itself to the potential impact of the epidemic over time, and continued to receive attention for a long time. people gradually shifting the topic from "origin" to potential topics such as "economic". potential topics have been concerned for a long time since the beginning of the new year. this shows that the prevention and control of the epidemic situation should be done well in a long period of time. (4) the political and economic center is a high-profile area of the epidemic network. with hubei province as the center, beijing-tianjin-hebei, the yangtze river delta, the pearl river delta, and chengdu-chongqing posted more microblogs. this is highly relevant to the division of economic regions. it is suggested that the government should strengthen public opinion response and prevention in and control of cities with better economic conditions. (5) "rumor" would enable people to have more communication and discussion. "rumor" will attract more attention. there are more exchanges on the topic of "rumor", which makes it spread faster than other topics. at the same time, "rumor" can also cause people to experience low emotions. however, this study also has some limitations. firstly, the location information contained in the data information is only used at the provincial level, so there is still room for improvement in spatial analysis. secondly, the data of this study does not collect statistical data on the gender and age of microblog users, therefore, when analyzing microblogs, some significant effects of gender and age are not reflected. finally, we only obtained data from sina weibo, but for people who did not use sina weibo to express their opinions, we could not collect their topic focus and topic emotions, so we should be more cautious about the generalization of results. with the popularization of covid-19, we should expand our data volume to provide more comprehensive public opinion prevention and control responses for relevant departments. origin of viruses: primordial replicators recruiting capsids from hosts social media analytics: extracting and visualizing hilton hotel ratings and reviews from tripadvisor mapping the anti-vaccination movement on facebook retrospective analysis of the possibility of predicting the covid-19 outbreak from internet searches and social media data, china social media analytics -challenges in topic discovery, data collection, and data preparaten combining machine-learning topic models and spatiotemporal analysis of social media data for disaster footprint and damage assessment social media mining for product planning: a product opportunity mining approach based on topic modeling and sentiment analysis social media in disaster risk reduction and crisis management the spreading of misinformation online researching mental health disorders in the era of social media: systematic review tweet for behavior change: using social media for the dissemination of public health messages. jmir public health surveill air pollution lowers chinese urbanites' expressed happiness on social media. nat hum behav infection breeds reticence: the effects of disease salience on self-perceptions of personality and behavioral avoidance tendencies pathogens, personality, and culture: disease prevalence predicts worldwide variability in sociosexuality, extraversion, and openness to experience a pox on the mind: disjunction of attention and memory in the processing of physical disfigurement using social media to mine and analyze public opinion related to covid-19 in china latent dirichlet allocation (lda) and topic modeling: models, applications, a survey use of social media for the detection and analysis of infectious diseases in china topic modeling and sentiment analysis of global climate change tweets mining meaning from online ratings and reviews: tourist satisfaction analysis using latent dirichlet allocation. tourism manage quantitative analysis of large amounts of journalistic texts using topic modelling enhancing seasonal influenza surveillance: topic analysis of widely used medicinal drugs using twitter data applying lda topic modeling in communication research: toward a valid and reliable methodology hpv vaccine coverage in australia and associations with hpv vaccine information exposure among australian twitter users top concerns of tweeters during the covid-19 pandemic: infoveillance study the motivational looking glass: how significant others implicitly affect goal appraisals chinese public's attention to the covid-19 epidemic on social media: observational descriptive study pandemics in the age of twitter: content analysis of tweets during the 2009 h1n1 outbreak chinese social media reaction to the mers-cov and avian influenza a(h7n9) outbreaks. infect dis poverty quantifying network dynamics and information flow across chinese social media during the african ebola outbreak building a national neighborhood dataset from geotagged twitter data for indicators of happiness, diet, and physical activity. jmir public health surveill this study was supported by the national natural science foundation of china (grant no. 71673256), and the fundamental research funds for the central universities (grant no. 2652020 0 01, no. 2652020 0 02). b.z and x.z. developed the original idea. all authors designed this study. b.z collected and analyzed the data and established the model and wrote the first paper. all authors read and approved the final manuscript. x.z. and h.l. provided project funding support. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. key: cord-308069-iydjrmhh authors: contreras, sebastián; biron-lattes, juan pablo; villavicencio, h. andrés; medina-ortiz, david; llanovarced-kawles, nyna; olivera-nappa, álvaro title: statistically-based methodology for revealing real contagion trends and correcting delay-induced errors in the assessment of covid-19 pandemic date: 2020-07-03 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110087 sha: doc_id: 308069 cord_uid: iydjrmhh covid-19 pandemic has reshaped our world in a timescale much shorter than what we can understand. particularities of sars-cov-2, such as its persistence in surfaces and the lack of a curative treatment or vaccine against covid-19, have pushed authorities to apply restrictive policies to control its spreading. as data drove most of the decisions made in this global contingency, their quality is a critical variable for decision-making actors, and therefore should be carefully curated. in this work, we analyze the sources of error in typically reported epidemiological variables and usual tests used for diagnosis, and their impact on our understanding of covid-19 spreading dynamics. we address the existence of different delays in the report of new cases, induced by the incubation time of the virus and testing-diagnosis time gaps, and other error sources related to the sensitivity/specificity of the tests used to diagnose covid-19. using a statistically-based algorithm, we perform a temporal reclassification of cases to avoid delay-induced errors, building up new epidemiologic curves centered in the day where the contagion effectively occurred. we also statistically enhance the robustness behind the discharge/recovery clinical criteria in the absence of a direct test, which is typically the case of non-first world countries, where the limited testing capabilities are fully dedicated to the evaluation of new cases. finally, we applied our methodology to assess the evolution of the pandemic in chile through the effective reproduction number r(t), identifying different moments in which data was misleading governmental actions. in doing so, we aim to raise public awareness of the need for proper data reporting and processing protocols for epidemiological modelling and predictions. since the outbreak of novel sars-cov-2 in late 2019, the spread of covid-19 has changed nearly every aspect of our daily life, challenging modern society to find a way to function under conditions never seen before. governmental plans on public health have played a crucial role in its control in the absence of an effective treatment to cure covid-19 or a vaccine to prevent it. typically reported variables in the covid-19 pandemic, available in public repositories (as worldometers.info, 02 jun, 2020; dong et al., 2020, among others) , are the incidence of new cases ∆t , total cases t , discharged/recovered cases r, and deaths d. these variables serve as input for the development and evaluation of governmental plans and to fit the vast variety of sir-like mathematical models recently proposed (see, e.g., contreras et al., 2020a; yang et al., 2020; pang, 2020; aguilar et al., 2020 , and references therein for a brief review of them). among the several factors conditioning the quality/reliability of the variables mentioned above are those related to the sensitivity and specificity of diagnostic tests, delays between sampling and diagnosing, and the delay between contagion, development of symptoms and testing. the latter varies from country to country, depending heavily on the local government's testing strategy and resources. even though sir-like models have been reported to be not-suitable to predict turning points or any other quantitative insight on this pandemic (castro et al., 2020) , to the best knowledge of the authors, delays have not been pointed out as the cause, as raw data has always been used to fit them. different parameters can be used to evaluate the evolution of the sars-cov-2 outbreak. among them we may find the documentation rate (wilder et al., 2020) , secondary infection rate, serological response to infection (who, 2020b), number of vacancies at icu (liew et al., 2020) , and the basic reproduction number r 0 (allieta et al., 2020; gevertz et al., 2020) , which is one of the most widely used. this parameter (r 0 ) represents the number of persons a single infected individual might infect before either recovering or dying, when the population is entirely susceptible (perasso, 2018) . traditional forms to estimate the r 0 are rather complex, heavily depending on the fitting of sir models to local data (heesterbeek, 2002; delamater et al., 2019; wang et al., 2020; ma, 2020) . in a previous work , we proposed a methodology to obtain real-time estimations of the effective reproduction number r t directly from raw data, which was satisfactorily applied to evaluate the panorama of the covid-19 spread in different countries and to forecast its evolution (medina-ortiz et al., 2020a) . nevertheless, its heavy dependence on reported data required the study of common error sources affecting this parameter, and the development of methodologies to control, correct, and quantify their impact . in this work, we analyze the sources of error in the typically reported epidemiological variables and their impact on our understanding of covid-19 spreading dynamics. we address the existence of different delays in the report of new cases, induced by the incubation time of the virus and testing-diagnosis time gaps, and provide a straightforward methodology to avoid the propagation of delay-induced errors to model-derived parameters. using our statistically-based algorithm, we perform a temporal reclassification of individuals to the day where they were -statistically-most likely to have acquired the virus, building a new smooth curve with corrected variables. we postulate this new temporally modified curve as the proper one for fitting purposes in sir-like models. we present an analogous methodology to estimate the number of discharged/recovered individuals, based on the reported evolution of the viral infection, the performance of the different tests for its diagnosis, and the case fatality, which can be easily adapted for a particular country. we used our methodology to assess the evolution of the pandemic in chile, identifying different moments in which the use of raw data was misleading governmental actions. different methods for diagnosing covid-19 have been developed and reported in the literature, with real-time rt-pcr being the standard applied globally (lippi et al., 2020) . nevertheless, techniques such as the igg and igm rapid tests, the chest computed tomography (chest ct), and crispr-cas systems are also being used. in this section we provide a brief analysis of them, highlighting the different characteristics of both the techniques and their basic approach to detect viral infection. real-time reverse transcription polymerase chain reaction (rt-pcr) is a mechanism for amplification and detection of rna in real time (see gibson et al., 1996 , for an exhaustive description of the technique). initially, rna obtained from samples is retrotranscribed to dna using a reverse transcriptase enzyme. by applying temperature cycles, the conditions are created for new copies of the dna to be synthesized from the initial one. the lower the initial dna concentration, the lower the probability of a synthesis reaction in a given cycle. it is assumed that the minimum time from contagion until testing positive in the rt-pcr test is 2.3 days (95 % ci, 0.8-3.0 days) before the onset of symptoms (he et al., 2020) , which typically appear 5.2 days (95 % ci, 4.1-7.0 days) after contagion . he et al. (2020) ; ganyani et al. (2020) showed that 44 % to 62 % of the total contagions occur in the pre-symptomatic period. it has been inferred that the viral load reaches a peak value before 0.7 days from the onset of symptoms (95 % ci, 0.2-2.0 days), when it starts falling monotonically together with the infectivity rate (he et al., 2020) . finally, the virus has been detected for a median of 20 days after the onset of symptoms (zhou et al., 2020) , but infectivity may decrease significantly eight days after symptom appearance. a high false-negative rate (li et al., 2020c; xiao et al., 2020) and a sensitivity of 71 to 83 % (fang et al., 2020; long et al., 2020) have been reported for the real-time rt-pcr technique, and several vulnerabilities of it have been identified and quantified (lippi et al., 2020) . considering sampling, handling, testing, and reporting, the total time necessary to get rt-pcr results for an individual may range between 2 to 3 days 4 (nguyen et al., 2020) . however, the time it takes to perform the rt-pcr experiment takes about 2 to 3 hours (li et al., 2020d) . part of the immune system response to the sars-cov-2 infection is the production of specific antibodies against it, including igg and igm (li et al., 2020b) . serological tests detect the presence of those antibodies and, unlike the other detection methods, take only 15 minutes to produce results (li et al., 2020d) . these tests have a sensitivity of 88.6 % and a specificity of 90.6 % (li et al., 2020d) . this technology was developed for the sars-cov epidemic, which was caused by a virus belonging to the same family of coronaviruses as sars-cov-2, providing satisfactory results after 2-3 days from the onset of symptoms (for igg), and after eight days (for igm) (woo et al., 2004) . the principle behind the chest computed tomography (chest ct) is the analysis of cross-sectional lung images to identify viral pneumonia features, like ground-glass opacity, consolidation, reticulation/thickened interlobular septa or nodules (ai et al., 2020) . chest ct has shown a sensitivity between 97 % and 98 % (long et al., 2020; fang et al., 2020) . however, due to the similarities between ct images accounting for covid-19 and ct images for other viral types of pneumonia, false-positives are likely to occur. compared to rt-pcr, chest ct tends to be more reliable, practical, and quick to diagnose covid-19 (ai et al., 2020) . nevertheless, requiring the presence of the potentially infected patient in a health center lacks the flexibility that rapid tests provide, and can backfire on movement restriction measures. covid-19 pneumonia manifests with abnormalities on computed tomography images of the chest, even in asymptomatic patients (shi et al., 2020) . in crispr-cas systems, a guide rna (grna) is designed to recognize a specific rna sequence, like any particular gene or partial arn sequence of sars-cov-2 coronavirus. endonuclease enzymes of the cas family and the specific grna will search for the sequence match. this match will deliver a signal that confirms the presence of sars-cov-2 rna in the sample (lemieux, 2020). detectr and sherlock are two examples of crispr-cas technologies for the detection of sars-cov-2, being able to obtain results in less than 1 hour at a significantly lower cost compared to the rt-pcr technique. detectr showed a 95% positive predictive agreement and 100% negative predictive agreement (broughton et al., 2020) , while sherlock has not been validated using real patient samples and is not suitable for clinical use at this time (jin et al., 2020) . our work aims to expose and quantify, both theoretically and in a case study, the impact of different sources of error in commonly reported data of the covid-19 spread, such as newly reported cases ∆t , total cases t , infected i, and recovered r fractions of the population. first, we define random variables associated with the delay in both sampling and diagnosing new cases, and by modeling their probability distribution functions, we derive a method to re-classify the newly reported cases accordingly. as reclassification occurs backwardly, for re-evaluating the current scenario through our methodology, we cast predictions on the reported new cases using an arima (autoregressive integrated moving average) model. having the corrected variables, we evaluate differences on the effective reproduction number r t , following the methodology presented by contreras et al. (2020b) . assuming the outbreak is well represented with a sir model in a given timeframe i, r t can be estimated as βi γi , which can be directly estimated from the discrete version of the sir differential equations. decoupling the "recovered" fraction in the sir model as clinically recovered individuals r and deaths d, r t would be given by equation 1: given the existence of a latent and incubation period for covid-19, which will be carefully described in the next section, sir models might not capture all the features of the outbreak and seir models would be more suitable. in such models, only infected individuals can propagate the infection and, equation-wise, estimations of r t remain more or less the same (chowell et al., 2007) . as data of exposed individuals is not widely available or entirely reliable, its contribution was not included in the estimations of r t to keep its straightforward use. autoregression models for the forecast of ∆t were implemented using the statsmodels python library (seabold and perktold, 2010) . all other calculations and visualizations were made using dmakit-lib python library (medina-ortiz et al., 2020b) and matlab r2018a. there exists an incubation period t c for the development of sars-cov-2-related symptoms, which average has been reported to range between 5.1 days (li et al., 2020a; he et al., 2020 ) and 6.4 days (backer et al., 2020) . he et al. (2020) also reported the existence of a latency period, which is the time required before an infected individual could spread the infection, which was extended until 2.3 days (95% ci, 0.8-3.0 days) before the onset of symptoms, which occurred t c days after contagion. the incubation time t c is especially relevant in the case of a symptoms-based testing strategy or when the spread has reached the non-traceability stage. to model it, we use τ 1 with log-normal distribution, as 6 reported in and suggested in nishiura (2007): according to the values reported in , µ = 1.63 = ln(t c ) and σ = 0.41 retrieve the expected values of 5.1 for the median, and 11.5 for the 98'th percentile. even though the required time for performing the test is short (li et al., 2020d) , delays between testing and diagnosis have been reported (nguyen et al., 2020) . we will sum up secondary delays, such as the symptom-to-testing and testing-to-diagnosis time gaps, into a random variable τ 2 , which, for the sake of simplicity, will be assumed to follow a uniform distribution between t min and t max : further knowledge on the nature of factors affecting τ 2 could lead to a different distribution. we may postulate a reclassification for obtaining the real number of new contagions occurred in a day t as the sum of contributions of cases reported with a delay of k days: where w i represents the fraction of patients that were notified at time t = t + k but had acquired the virus at t. note that the different delays are referred to the random variable z = τ 1 + τ 2 . the probability distribution function for z is obtained by the convolution method, assuming τ 1 and τ 2 are independent and combining equations 2 and 3: assuming that data is reported on a daily basis, we can calculate the probability associated to having a delay of k days: for practical reasons, we can define a threshold u < 1 for truncating the probability mass distribution (equation 6), which otherwise would assign a probability to every k ∈ n. let n 1 be the first positive integer for which equation 7 holds, we may rewrite equation 4 as: the magnitude of the total delay between infection and diagnosis can be estimated through the expected value of equation 5 (or equivalently, equation 6). a schematic representation of the proposed methodology is presented in figure 1 . assuming the lowest reported value for the average incubation time, t c = 5.2, and a conservative timeframe for the delay between the appearance of symptoms, testing and diagnosing, (2-5 days), the expected delay e (z) is about 7.5 days. algorithm 1: statistically-based temporal reclassification algorithm for correcting delay-induced errors in the report of new covid-19 infections result: corrected differential and cumulative total cases t . dt : officially reported new cases per day. dt ∈ r n ; λ, a, b : parameters of the τ 1 and τ 2 distributions; u : probability threshold for the maximum possible delay; calculate [p k, f ], respectively the probability and cumulative distribution functions for z = τ 1 + τ 2 ; idx = f < u ; logical indexes of the values of vector f smaller than u ; p k = p k(idx); p k ∈ r m ; define k, delay indexes of p k in idx. renormalize p k; forecast the next m values for dt , using an autoregression (arima) model; define an empty vector dt corr ∈ r 2m+n−1 to record the corrections; round dt corr values to the nearest integer; dt corr = round(dt corr); as discussed previously, errors in the amount of discharged/recovered patients r are likely to be greater only when no quantitative criteria are applied. in such cases, some countries (like chile) have adopted the following criteria (officially reported in the minsal (14 apr, 2020) report), possibly based on the recommendations published by the who (2020a). • if there were no previously existing pathologies, a patient should be discharged 14 days after testing positive for covid-19. • if there were previously existing pathologies, the patient should be discharged 28 days after testing positive for covid-19. this criterion turns out to be quite simplistic, especially considering the existence of uncertainties regarding the diagnosis and contagion days. if we try to model the probability of recovering from covid-19, some assumptions are necessary. let τ r be the random variable for the time of discharge/recovery such that: • p (τ r ≤ 14) = 0 • p (τ r ≤ 28) ≈ 1 further assumptions are necessary to estimate the probability distribution function f τr , as it depends on local diagnosis criteria, testing strategy, and the fraction of the population having preexisting pathologies. in particular, depending on the test applied for diagnosis and its sensitivity/specificity -which were carefully described in section 2-the probability profile would change. the simplest form that can be assumed, and which, for clarity reasons, is adopted herein, is a triangular distribution: further knowledge on the nature of factors affecting τ r could lead to a different distribution. algorithm 2: statistical estimation of dr, daily patients that have been discharged/recovered from result: estimated discharged/recovered cases per day r. dt corr : corrected new cases per day. dt corr ∈ r n ; dd : officially reported daily deaths due to covid-19. dd ∈ r n ; θ : parameters of the probability distribution function for the time of discharge/recovery τ r .; t min , t max : time frame for discharging an infected patient.; w = t max −t min +1, width of the time-frame of discharge; k = linspace(t min , t max , w); calculate the probability distribution function for τ r ; p kr = f ( θ, w); define an empty vector drcorr ∈ r w+n to record the corrections; subtract the daily deaths due to covid-19 to obtain the final estimator, round to the nearest integer. confirm the result is positive or zero.; drcorr = max(round(drcorr − dd), 0) ; the spread of covid-19 in chile is far from being controlled, as shown by the exponential growth that new cases have had in recent weeks (worldometers.info, 02 jun, 2020) . in order to apply our methodology, we need to cast predictions on the trends of ∆t . figure 2 presents the current and forecast trends, using an autoregression arima model. these models were chosen over others because of the great success they have had on representing covid-19 dynamics in other relevant studies (chakraborty and ghosh, 2020; benvenuto et al., 2020) . (02 jun, 2020), while the red curve was generated using an autoregression arima model. as these reported data are likely to be affected by several exogenous factors (ribeiro et al., 2020) , the best performance metric for the generated forecast is trend consistency. first, we performed a temporal reclassification of new cases to obtain ∆t corr , presented in figure 3 . it can be seen that our methodology, besides exhibiting an horizontal semi-displacement, generates a smooth curve, which is due to the inclusion of a probability function that spreads the influence of daily informed cases in previous days (where they were most likely to have occurred). the last part of the red curve is dashed because it partially contains contributions of the forecast of ∆t , and therefore might change in the upcoming days, when the required data for completing the reclassification of cases would be available. after obtaining statistically-based corrections of both ∆t and ∆r by following algorithms 1 and 2, and knowing the daily deaths due to covid-19 ∆d, we proceed to calculate the variation on the active cases. based on the closure of the population-balance of the different considered classes, the cumulative new cases t will be the sum of active i, recovered/discharged r, and dead d, therefore their differences would follow: ∆t = ∆i + ∆r + ∆d ⇐⇒ ∆i = ∆t − (∆r + ∆d). (10) in the context of seir models, it is not possible to explicitly include exposed individuals e, as they are not being recorded separately from covid-19 positive new cases. after obtaining ∆i using equation 10, we proceed to calculate r t , using equation 1 with raw data, mobile averages of raw data, and the methodology proposed herein. as shown in figure 4 , an abrupt growth in r t was evidenced around april 22nd, consistently with the relaxation of restrictive measures that were applied in santiago, the capital and most populated city in the country, and the apogee of the governmental plan for a "safe return to work". even though different trends seem to stabilise again in the second week of may, a lowering of the trend is not totally clear. the different iconic dates highlighted in figure 4 were obtained from the chronology presented in wikipedia.org (23 may, 2020) and references therein. from this plot, we can assess both successful and unfortunate effects that the different governmental actions have had on the spreading dynamics of covid-19 in chile, and specifically in santiago, by analyzing corrected r t trends and comparing them with daily reported cases. the various public-health actions were strongly based on locally-observed values from (raw) daily reported cases, yet appear to have been taken too late, or too early, according to the statisticallycorrected trend. in particular, the apogee of the governmental plan for a safe return to work occurred in a temporal window where raw-data-driven r t values were at a minimum, but the corrected contagion rates anticipated a steep growing trend for the following days. can be smoothed through mobile averages (dark red curve), but the trends are the same. a significantly different scenario is shown by the statistically-corrected rt trend (green curve). highlighted dates associated with iconic governmental actions in chile: march 31st (second week of sectorised quarantine for the high-income districts of santiago, capital of chile), april 7th (compulsory use of facemask in public transport), april 24th (governmental call for a "safe return to work"), april 30th (sectorised quarantine -low-income districts of santiago-), may 15th (total quarantine in santiago). we have presented an exhaustive assessment of error sources in reported data of the covid-19 pandemic and provided a methodology to minimize -and correct-their effect on both reported variables and modelderived parameters by applying a statistically-driven reclassification of newly reported cases, and corrections to discharge/recovery criteria. by using the corrected variables, sir-like models could be fitted directly, as every value would represent the real contagion dynamics. we present the methodology as a general framework, aiming to provide a useful tool for researchers and decision-making actors looking to adapt it for their particular interests. in a case study on the spreading dynamics of covid-19 in chile, we observed the effects that different iconic public health actions taken by the government had on sars-cov-2 spread rate r t , readily calculated using our previously reported approach, and discussed the reasons behind these effects interpreted under the light of raw data and corrected daily contagion rates, calculated with the methodology presented in this manuscript. the delay-induced error in raw data slowed-down the reaction time, so the actions were taken too late for restrictive actions or too early for the comeback to normality. our statistically-driven method corrected such reporting errors, exposing the real contagion dynamics at a given time. the proposed methodology also serves as a non-invasive smoothing process, as it only temporally re-sorts daily reported 13 cases according to their most likely report delay from the real contagion day. we expect our methodology to serve as a valuable input for researchers and public health practitioners trying to add statistical value to their calculations and predictions in order to vanquish the current sars-cov-2 pandemic. finally, we also expect this work to contribute in raising public awareness on the need for a proper (and standardized) strategy for report and curation of data in the covid-19 pandemic and other highly-contagious diseases. the authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. investigating the impact of asymptomatic carriers on covid-19 transmission correlation of chest ct and rt-pcr testing in coronavirus disease 2019 (covid-19) in china: a report of 1014 cases covid-19 outbreak in italy: estimation of reproduction numbers over two months toward the phase 2. medrxiv the incubation period of 2019-ncov infections among travellers from wuhan. china. medrxiv application of the arima model on the covid-2019 epidemic dataset crispr-cas12-based detection of sars-cov-2 predictability: can the 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infection diagnosis preparing for covid-19: early experience from an intensive care unit in singapore potential preanalytical and analytical vulnerabilities in the laboratory diagnosis of coronavirus disease 2019 (covid-19) diagnosis of the coronavirus disease (covid-19): rrt-pcr or ct? estimating epidemic exponential growth rate and basic reproduction number country-wise forecast model for the basic reproduction number r 0 in the covid-19 outbreak. under review in dmakit: a user-friendly web platform for bringing state-of-the-art data analysis techniques to non-specific users criteria for discharging a covid-19 infected individual (criterios que se consideran para un paciente covid-19 sin riesgo de contagio) novel coronavirus disease (covid-19): paving the road for rapid detection and point-of-care diagnostics early efforts in modeling the incubation period of infectious diseases with an acute course of illness public health policy: covid-19 epidemic and seir model with asymptomatic viral carriers an introduction to the basic reproduction number in mathematical epidemiology short-term forecasting covid-19 cumulative confirmed cases: perspectives for brazil statsmodels: econometric and statistical modeling with python radiological findings from 81 patients with covid-19 pneumonia in wuhan, china: a descriptive study. the lancet infectious diseases zhonghua liu xing bing xue za zhi = zhonghua liuxingbingxue zazhi considerations for quarantine of individuals in the context of containment for coronavirus disease (covid-19): interim guidance protocol for assessment of potential risk factors for coronavirus disease 2019 (covid-19) among health workers in a health care setting cronología de la pandemia de enfermedad por coronavirus de 2020 en chile the role of age distribution and family structure on covid-19 dynamics: a preliminary modeling assessment for hubei and lombardy longitudinal profile of immunoglobulin g (igg), igm, and iga antibodies against the severe acute respiratory syndrome (sars) coronavirus nucleocapsid protein in patients with pneumonia due to the sars coronavirus official numbers for the coronavirus outbreak in chile false-negative of rt-pcr and prolonged nucleic acid conversion in covid-19: rather than recurrence modified seir and ai prediction of the epidemics trend of covid-19 in china under public health interventions clinical course and risk factors for mortality of adult inpatients with covid-19 in wuhan, china: a retrospective cohort study the authors gratefully acknowledge support from the chilean national agency for research and development through anid pia grant afb180004, and the centre for biotechnology and bioengineering -cebib (pia project fb0001, conicyt, chile). dm-o gratefully acknowledges conicyt, chile, for phd fellowship 21181435. key: cord-355689-mo4mvwch authors: huang, jiechen; wang, juan; xia, chengyi title: role of vaccine efficacy in the vaccination behavior under myopic update rule on complex networks date: 2019-09-06 journal: chaos solitons fractals doi: 10.1016/j.chaos.2019.109425 sha: doc_id: 355689 cord_uid: mo4mvwch how to effectively prevent the diffusion of infectious disease has become an intriguing topic in the field of public hygienics. to be noted that, for the non-periodic infectious diseases, many people hope to obtain the vaccine of epidemics in time to be inoculated, rather than at the end of the epidemic. however, the vaccine may fail as a result of invalid storage, transportation and usage, and then vaccinated individuals may become re-susceptible and be infected again during the outbreak. to this end, we build a new framework that considers the imperfect vaccination during the one cycle of infectious disease within the spatially structured and heterogeneous population. meanwhile, we propose a new vaccination update rule: myopic update rule, which is only based on one focal player’s own perception regarding the disease outbreak, and one susceptible individual makes a decision to adopt the vaccine just by comparing the perceived payoffs vaccination with the perceived ones of being infected. extensive monte-carlo simulations are performed to demonstrate the imperfect vaccination behavior under the myopic update rule in the spatially structured and heterogeneous population. the results indicate that healthy individuals are often willing to inoculate the vaccine under the myopic update rule, which can stop the infectious disease from being spread, in particular, it is found that the vaccine efficacy influences the fraction of vaccinated individuals much more than the relative cost of vaccination on the regular lattice, meanwhile, vaccine efficacy is more sensitive on the heterogeneous scale-free network. current results are helpful to further analyze and model the choice of vaccination strategy during the disease outbreaks. over the past two decades, the outbreak of infectious diseases has been threatening the safety of human lives and properties, such as the severe acute respiratory syndrome sars [1] , h1n1 [2] , ebola [3] and so on. thus, how to prevent the extensive outbreaks of epidemics has become a challenging topic in the field of public health [4] [5] [6] . meanwhile, the difference of population distribution, religious belief and regional differences may greatly affect the spread of infectious diseases, for example, refs. [7] [8] [9] [10] explore the impact of various topological structures within the population on the infectious diseases spread, and it is convincingly found that heterogeneous networks may quicken the disease spreading within the population, even lead to the absence of epidemic threshold [11] . meanwhile, the individual reactions to infectious diseases may also substantially influence the diffusion processes of epidemics. one of the most striking cases regarding the outbreaks was h1n1 pandemic in 2009 [12] , which induced around 15 , 0 0 0 deaths. during the outbreaks of h1n1, the suppression of epidemic processes can not only be attributed to the public measures, but also through personal and uncoordinated responses, that is, the human behavior has noticeably interfered with the epidemic spreading. in the long run, human behavior has been intricately correlated with the contagion of infectious diseases. in medieval ages, the deadly bubonic plague rendered many people to avoid and flee away from the sick and their close contacts so that their own immunity can be secured [13] . similarly, the villagers of yorkshire in eyam tried to voluntarily quarantine themselves to stop the spread of the plague from that village [14] . in 2003, during the outbreak of sars, many citizens spontaneously wear the face masks, some schools are temporarily closed and the students are imperatively required to stay at home so as to avoid the further epidemic infection as much as possible [15] . in addition, protective behavior when confronting the epidemics has also been observed in many other contexts, such as measles-mumps-rubella(mmr) [16] , tuberculosis(tb) [17] and hiv [18] etc. while the impact of human behaviors on the epidemic spreading process has often been mentioned anecdotally, the accurate modeling or quantitative models are relatively fewer regarding their nature, property, or the effect they may have on the spread of the disease. at present, mathematical models have been put forward to study the role of human behavior in the context of social population, such as escape panic [19] , pedestrian trails [20] , but effort s to quantitatively explore the role of human behavior in the large-scale epidemics generally focus on assessing the effectiveness of various public health measures including the social distancing, school closure etc. in the recent years, there are many fields and methods to help us to study infectious disease [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] , however, there is an increasing attention about the effect of spontaneous individual action or response strategy on the progression of an infectious disease, in which this kind of spontaneous actions may highly restrain from the further diffusion of epidemics and even change the fate of outbreaks. thus, it is of great significance to fully understand the interacting mechanisms between the human behavior and disease dynamics within the specified population. it is worth mentioning that funk et al. [33] systematically summarized the related works and provided a taxonomy framework of behavior-disease models. on the one hand, they classify these models according to source and type of information that individuals base their neighbors on, in which source of information may be local or global and the type of information that individuals change their behaviors are prevalence-based or belief-based; on the other hand, they classify the previous works based on the impact of individual behavior changes on the disease dynamics, which include the following three aspects: (i) the disease state; (ii) model parameters (infection or recovering rate); and (iii) the network contact structure relevant for the spread of epidemics. in particular, for some preventable infectious diseases with the help of vaccines, the epidemic outbreaks are intricately linked with the individual vaccination behavior since the vaccines can help the vaccinators not to be infected by a specific disease. meanwhile, these vaccinators may indirectly protect their nearest neighbors with whom they contact, and then these neighbors may choose not to vaccinate again (that is, free-ride the vaccinators) so as to avoid the necessary vaccine fees or other potential risk and side effects. henceforth, the vaccination behavior may dominate the evolutionary process of vaccine preventable diseases. among them, bauch and earn [34] seminally utilized the game theory to model the dilemmatic situation for an individual facing with the epidemics, and they proposed a class of vaccination game to denote the individual decision making and found that, for the well-mixed population, the nash equilibrium is never to vaccinate if the vaccination cost is higher than that of being infected; but there exists a nash equilibrium yielding a suboptimal vaccinated fraction if the vaccine cost is lower than that of being infected. as a further step, complex networks, beyond the well-mixed topology, provide a unified platform to characterize the topology of real-world populations, where the nodes represent individuals and links mimic the contacts among them [35] . thus, under framework of game theory, many works are devoted to exploring the interplay between contact patterns, behavioral responses and disease dynamics. as an example, fu et al. [36] found that heterogeneous networks, such as scale-free ones, can induce a broad range of vaccinating actions of many individuals since highdegree hubs with many neighbors become voluntary vaccinators more probably in order to reduce the risk of being infected. after that, zhang et al. [37] demonstrated that the hubs may largely inhibit the outbreaks of infectious diseases under the voluntary vaccination policy. in the meantime, various subsidy policies on controlling the epidemic spreading have been determined from the socioeconomic perspectives within the well-mixed and networked population [38] [39] [40] [41] [42] [43] . furthermore, most previous works [44] [45] [46] [47] [48] [49] [50] [51] [52] often assume that the vaccine has a perfect efficacy, which will endow the complete immunity for the inoculated individuals, but an interesting topic is on how the epidemic spreads when the vaccine is not fully effective for the disease (i.e., 100% efficacy)? besides, in the vaccination game, whether an individual decides to vaccinate the vaccine or not will be often determined by the estimated payoff, and the vaccinating decision may be transferred from one player to another one inside the population according to a specific role model during the outbreaks. nevertheless, under some real-world scenarios, some individuals are not willing to imitate the behaviors of others as a result of special belief, religion, opinion and even awareness of a disease. according to the above description, in the real world, when a new epidemic starts to outbreak, people want to take some measures to protect themselves immediately. generally, inoculating the vaccine is considered as an effective measure, but the vaccine may fail as result of invalid storage, transportation and usage, and then rendered that the vaccinated individuals may confront the risk of being infected. henceforth, some individuals make a decision to vaccinate just judge by themselves and don't consider their neighbors. thus, in order to deeply analyze the role of vaccine efficacy and spontaneous individual decision mechanism in the outbreak of vaccine preventable diseases, we propose a vaccination game model to explore the impact of imperfect vaccine efficacy and myopic update rule in the spatially structure and heterogeneous populations. the rest of this paper is structured as follows. firstly, we depict the new vaccination game model in section 2 in detail. then, section 3 provides extensive numerical simulation results, which are obtained in the regular lattice and heterogeneous scalefree networks, respectively. lastly, in section 4 , we end this paper with some conclusions and point out the potential works in the future. as mentioned above, we consider the vaccination game for a class of emerging epidemics within the structured population, where the vaccine can be obtained after the epidemic spreads. thus, in the current model, all individuals have no chances to vaccinate at the initial time step ( t = 0 ). after that, each time step ( t ≥ 1) is divided into two elementary sub-steps: one step is for the decision of inoculating the vaccine; the other one is used to model the process of epidemic spreading. among them, for the epidemic sub-steps including t = 0 , we leverage the frequently used susceptible-infective-recovery (sir) compartment model to characterize the evolution of epidemics, where each individual may lie in the susceptible (s), infective (i) or recovery (r) state. during the vaccination decision sub-steps, each susceptible individual needs to assess the risk of being infected, and then make the decision whether he will vaccinate or not. regarding the sir model, any susceptible individual may be infected through the contact with infective neighbors and the transmission rate along each infective link is assumed to be β. meanwhile, the infected individual can be cured with the recovering rate μ, and the recovered one will not be infected again or infect any other healthy ones. hence, the probability that the susceptible player i without inoculating the vaccine will be infected by all possible infective neighbors can be written as follows, where k i in f denotes the total number of infected neighbors of the focal player i . the epidemic continues until there are no more newly infected individuals. as for the individual vaccination decision, each susceptible one will evaluate the risk of being infected and compare the difference between the vaccine cost ( c v ) and the potential expenses once he has been infected. without loss of generality, we fix the infection cost c i = 1 , while the vaccine cost is usually lower than c i and then its relative cost can be re-scaled as 0 < c = c v /c i < 1 . in order to quantitatively perform the decision, by borrowing from the terms in game theory, we assume that the decision process is based on the comparison between the perceived payoffs of vacci-nation i v and the perceived payoffs of being infected i nv if he is not vaccinated, which can be expressed as follows, respectively, where λ i denotes the potential infection probability that can be calculated according to eq. (1) . then, the susceptible agent i will independently decide to inoculate the vaccine with the following fermi-like probability, where k represents the impact of the noise or its reverse 1/ k means the strength of strategy selection, which reflects the uncertainty of vaccination strategy adoption. here, we term this vaccination decision as the myopic update rule just based on one player's own perception, which is different from imitating the vaccination strategy of others in many previous works [37, 38, [43] [44] [45] [46] [47] . when k → 0, agent i is totally rational and whether he will vaccinate or not is fully determined by comparing i v and i nv , on the contrary, when k → ∞ , agent i will randomly perform the vaccination choice. in addition, we consider the imperfect vaccination program, that is, the vaccine efficacy is not 100% and the vaccination may fail as a result of incorrect transportation, storage, and usage of vaccine. thus, we introduce an independent parameter θ to characterize the vaccine failure rate, which implies that the susceptible individual to choose the vaccination is still kept in the susceptible state with the probability θ , while the probability changing from s into v ( s → v ) state is set to be ( 1 − θ ). once the healthy individual decides to vaccinate, he will have a chance to enter into the vaccinated ( v ) state, the sir epidemic dynamics will evolve into the sirv model as illustrated in fig. 1 , in which the successfully vaccinated individual will be equivalent to the r-type one at the next time step, that is, the successfully vaccinated (v-type) agent does not get the infection or infect others, either. we do not know when does vaccine fails, so that at each propagating time step, inoculation individuals are judged whether the vaccine is fails or not. to be noted, the vaccinated healthy individuals will not be vaccinated again during this epidemics even if the vaccine lost its effect. in order to further explore the impact of network topology on the evolutionary process under the imperfect vaccination, we simulate the current mechanism on l × l regular lattices and scale-free networks with n = l 2 nodes, respectively. initially, we stochastically choose i 0 = 10 individuals as the infective seeds within the population, and other ( n − i 0 ) individuals are kept in the susceptible state, and thus there have no vaccinators. at t = 0 step, the system starts the evolution according to the sir model. after that, from t = 1 , each susceptible agent has the opportunity to receive the vaccination and then the system carries out the evolution of epidemics based on the above-mentioned two elementary steps. the system continuously evolves until there are no infective individuals. in addition, the current results are averaged over 10 0 0 independent runs so that the large fluctuations can be removed. in all the numerical simulations, the population size is fixed to be n = 10 0 0 0 . in the homogenous topology, we use the regular lattice satisfying the periodic boundary with the size l = 100 as the underlying networks, and each individual has 4 nearest neighbors (that is, the von-neumann neighborhood). as for the heterogeneous topology, we generate the scale-free network by using the configuration model, in which the average degree is fixed to be < k > = 4 and the power exponent is set to be γ = 3 . in this section, we conduct extensive numerical simulations to demonstrate the vaccination behavior on the regular lattices and scale-free networks, respectively. among them, we mainly discuss the influence of vaccine cost c and failure rate θ on the collective vaccination level within the population. without loss of generality, we set the value of parameters in the sir model as β = 0 . 46 and γ = 1 / 3 , which are identical with those in ref. [36] . first, we investigate the equilibrium fraction of both vaccinated ( ρ v ) and recovered ( ρ r ) state individual size for different values of relative cost of vaccination c and vaccine failure rate θ . fig. 2 plots 0.06 (yellow triangle), respectively. on the one hand, for a specific vaccine failure rate θ , ρ v declines with the increase of the relative vaccination cost c , while ρ r increases as c augments, which means that the vaccination cost will markedly affect the willingness of individuals to inoculate the vaccine. as an example, when c ≤ 0.5, ρ v and ρ r can almost keep the similar vaccination level as c increases; however, c > 0.5 leads to the substantial reduction of ρ v and the continuous rising of ρ r since the vaccination cost is comparable to the infection cost; in particular, ρ v will be dramatically reduced when c is up to 0.95, even tends to zero as the vaccination cost is too high, especially for c = 0 . 95 . on the other hand, under the same vaccination cost c, ρ v decreases as the vaccine failure rate becomes higher, for instance, the fraction of adopting the vaccination strategy under c = 0 . 06 is much less than that with c = 0 . 05 , which implies that the vaccinated fraction within the whole population will be a little more sensitive to the vaccine failure rate. then, we discuss the influence of the noise factor k on the vaccination behavior within the population in fig. 3 , where we set σ = 1 k , termed as the strength of selection (0 < σ < ∞ ), as 2 and 2.5, which are slightly different from that in fig. 2 . likewise, it can be clearly shown that ρ v declines and ρ r increases slowly when the relative cost of vaccination c lies between 0 and 0.4. afterwards, when the relative cost of vaccination c is more than 0.5, the greater the noise selection strength, the more rapidly the varying trend of ρ v and ρ r . in fact, as the strength of selection increases, individuals become much more rational and will not tend to take the vaccination strategy since they will take their own economic cost and the related interests, say, free-riding behavior, into account. in particular, the relative cost c is beyond 0.5, or the vaccine loss rate is higher (i.e., θ = 0 . 06 ), the un-vaccination behavior of rational individuals become much more prominent, and thus it is unable to prevent the outbreaks of epidemics, which can be observed from the larger ρ r as c > 0.5 or θ = 0 . 06 . next, in order to fully check the impact of relative vaccination cost c and the vaccine failure rate θ on the vaccination behavior, fig. 4 illustrates the evolution of ρ v and ρ r within the broader ranges of c and θ . it is clearly indicated that at the lower vaccine failure rates (say, θ < 0.04), the fraction of vaccinated individuals is often more than half of the total population, even if relative cost of vaccination c is large (e.g., 0.9); meanwhile, a plethora of vaccithus, creating the high quality vaccine is significant, which greatly determines the individual vaccination inclination. furthermore, to deeply understand individual state change in the lattice as sirv model evolves, we record the evolutionary snapshots of individual states at various time steps for θ = 0 . 03 , c = 0 . 3 and θ = 0 . 08 , c = 0 . 8 in fig. 5 . among them, the upper eight panels denote the snapshots under θ = 0 . 03 and c = 0 . 3 , while the lower eight panels represent the ones for θ = 0 . 08 and c = 0 . 8 . at time step t = 0 , there are no vaccinated individuals on the lattice and only i 0 = 10 randomly infected seeds, and then the epidemic starts to propagate at this time. after that ( t ≥ 1), the susceptible individuals have the opportunity to determine whether they will inoculate the vaccine or not. it is clearly observed that most individuals choose to vaccinate under these two cases when epidemic begins to spread. however, when θ = 0 . 03 is lower, there is fewer vaccinated individuals to become susceptible, and then most of vaccinated individuals are immunized, in which the epidemic is hard to spread and finally tends to be extinct. reversely, for the higher vaccine failure rate (i.e., θ = 0 . 08 ), the vaccine is easy to be invalid, many vaccinated individuals become susceptible due to the loss of vaccine efficacy. therefore, the epidemic can be pandemic and then most of individuals enter the recovered state in the end. all these results again demonstrate that the vaccine efficacy plays the significant role in the evolution of vaccination behavior of epidemic outbreaks within the structured population. in the real world, many networks are often heterogeneous, and thus it is necessary to understand the mechanics of myopic update rule better on heterogeneous topology. to this end, we formulated the game of taking the vaccine on the scale-free network. here, we generate the scale-free network with 10,0 0 0 node under the configuration model, where the average degree of the whole network is equal to 4 and the power exponent 3. after the fundamental networks are created, the system evolves according to the sirv model, which is identical with the iteration procedure on regular lattices, and the epidemic continues until there are no more newly infected individuals. first of all, we plot the time courses of fraction of susceptible, vaccinated and recovered individuals for different the relative cost of vaccination c and vaccine failure rate θ in fig. 6 . in all panels, the red, blue and yellow lines denote the evolution of susceptible, recovered and vaccinated individuals, respectively. it can be found that the vaccinated individuals increase rapidly in a very short time, and then reach a peak. vaccinated individuals are rarely become susceptible because of the vaccine failure θ is lower (as shown in fig. 6 a,c) so that the number of recovered ones increases a little and arrives at the equilibrium quickly, which states clearly that the epidemic is eliminated and has not become pandemic. due to the vaccine failure rate, the fraction of vaccinated individuals goes down and then tends to be zero after reaching the peak. however, when the value of vaccine failure rate θ is raised (as shown in fig. 6 b,d) , even though the vaccinated individuals increase rapidly, they become susceptible quickly due to the high vaccine failure rate θ , it can't prevent the epidemic spread so that the number of recovered individuals increases. additionally, we found that for the value of vaccine failure rate θ = 0 . 03 , whatever the values of relative cost of vaccination c , the number of recovered agents is the same as that at the equilibrium. generally, when the epidemic starts to spread, many susceptibles take the vaccine in the population at a short due to the perception of infection risk. also, this vaccination behavior is almost widespread regardless of the values of relative cost of vaccination c and vaccine failure rate θ . at the lower vaccine failure rate, vaccinated individuals are hard to become susceptible, which leads to the disease propagates difficultly and be eliminated as soon as possible. these results are also consistent with the work of zhang et al. [37] , since the hub nodes are often vaccinated immediately after the disease starts to spread. but for the higher vaccine failure rate, the vaccinated individuals become susceptible quickly, the disease can outbreak. we also consider the equilibrium fraction of both vaccinated ( ρ v ) and recovered ( ρ r ) state individual size for different values of it can be found that in the figs. 7 and 8 , whatever the ways of vaccination, the epidemic can outbreak and vaccinated individuals are more sensitive to the vaccine failure rate. therefore, except β = 0 . 46 , we consider the epidemic evolution of sirv model under the lower transmission rate β = 0 . 3 . meanwhile, we set the i 0 = 10 initial infective seeds as the top 10 largest degree nodes, which is here termed as the hub infection scheme. correspondingly, we call the randomly selecting one for the generous case as the random infection. in summary, based on the sir epidemic dynamics, we investigate the imperfect vaccine immunity under the myopic update rule in different foundation topology including the regular lattice and scale-free networks, where the focal player makes the vaccination decision just according to his own judgement about the epidemic situation. extensive numerical simulations show that most unvaccinated susceptible individuals are willing to inoculate the vaccine under the myopic update rule, whatever the type of network is, in particular for the lower vaccine cost ( c ≤ 0.5) and failure rate ( θ ≤ 0.06). after the epidemic starts to propagate, and most of individuals change their strategies to adopt the vaccine in a short time since the individual can estimate the infection risk at the early stage, which leads the epidemics to be hard to spread within the population. however, due to the failure of vaccine or the free-riding behavior of susceptible individuals, vaccinated individuals become susceptible again and then confront the risk of being infected, which creates the potential epidemics situation. to be of great interest, we find that the impact of vaccine failure rate on the vaccination coverage becomes much higher, when compared to the role of the relative cost of vaccine. for example, the value of vaccine failure rate θ is usually assumed to be no more than 10%, or else most vaccinated individuals become re-susceptible again. at a fixed θ , the fraction of vaccinated individuals almost keep unchanged when the relative vaccine cost is not beyond c = 0 . 5 , but this value will become lower and lower after c is more than 0.5, which is basically consistent with the reality of vaccine usage. on the contrary, on the scale-free networks, the number of vaccinated individuals is more sensitive to the effect on vaccine failure rate θ . hub nodes have a stronger inclination to adopt the vaccine under the myopic update rule, which can effectively prevent the diffusion of epidemics. hence, the disease will be eliminated quickly in heterogeneous topology. however, when the values of vaccine failure rate increase a little bit, hub nodes become susceptible and cannot prevent the epidemic spread, which leads to the epidemic outbreak. meanwhile, when the relative cost of vaccination c increases, unvaccinated susceptible individuals are not willing to choose the vaccination strategy, which can not stop the outbreak of infectious diseases. when the vaccine failure rate further increases, such as 0.03, the number of vaccinated individuals decays to zero before the epidemic is eliminated, which is almost equivalent with the classic sir model. anyway, current results are conducive to better understanding the individual vaccination behaviors when confronting the real epidemics. the authors declare that they do not have any financial or nonfinancial conflict of interests. transmission dynamics of the etiological agent of sars in hong kong: impact of public health interventions novel influenza a (h1n1) pregnancy working group. h1n1 influenza virus infection during 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approach based on evolutionary-game dynamics a susceptible-infected epidemic model with voluntary vaccinations group interest versus self-interest in smallpox vaccination policy imitation dynamics predict vaccinating behavior impact of committed individuals on vaccination behavior the impact of imitation on vaccination behavior in social contact networks the influence of social norms on the dynamics of vaccinating behavior for paediatric infectious diseases risk assessment for infectious disease and its impact on voluntary vaccination behavior in social networks can influenza epidemics be prevented by voluntary vaccination? this project is financially supported by the national natural science foundation of china (nsfc) (grant nos. 61773286 and 71401122 ). key: cord-327544-7ws2kleo authors: hammoumi, aayah; qesmi, redouane title: impact assessment of containment measure against covid-19 spread in morocco date: 2020-08-22 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110231 sha: doc_id: 327544 cord_uid: 7ws2kleo since the appearance of the first case of covid-19 in morocco on march, 02,2020, the cumulative number of reported infectious cases continues to increase and, up to date, the peak-time of infection is not reached yet. in this study, we propose a susceptible-asymptomatic-infectious deterministic model to evaluate the impact of compulsory containment imposed in morocco on march, 21 on the spread of covid-19 epidemic across the country. the model takes account of the unconfined individuals that continue to work or to leave their home for urgent needs and the existence of infectious asymptomatic and unreported individuals within susceptible population. furthermore, the model is able to predict the peak-size, peak-time, final size and epidemic duration according to different rates of containment. advanced knowledge of these details will be of great interest to establish an optimal plan-of-action to control or eradicate the epidemic. indeed, mitigating and delaying the epidemic peak allow the official health authorities to anticipate and control the spread of covid-19. moreover, prediction of the epidemic duration can help the government to predict the end time of containment to avoid consequent social-economic damages as well. using our model, the basic reproduction number r(0) is estimated to be 2.9949, with [formula: see text] reflecting a high speed of spread of the epidemic. the model shows that the compulsory containment can be efficient if more than 73% of population are confined. in the absence of other efficient measure of control, even with 90% of containment, the end-time is estimated to happen on july, 4,2020 with 7558 final cumulative cases. furthermore, a threshold value of containment rate, below which the epidemic duration is postponed, has been determined. finally, the sensitivity analysis is performed and showed that the covid-19 dynamics strongly depends on the asymptomatic duration as well as the contact and containment rates. our previsions can help the government to adjust its plan-of-action to fight the disease and to face the social-economic shock induced by the containment. • the model is able to predict the peak-size, peak-time, final size and epidemic duration. • we estimated the basic and control reproduction numbers. • the model shows that the compulsory containment can be efficient if more than 73% of population are confined. • we performed sensitivity analysis which shows that covid-19 depends strongly on the asymptomatic duration as well as the contact and containment rates. morocco on march, 21 on the spread of covid-19 epidemic across the country. the model takes account of the unconfined individuals that continue to work or to leave their home for urgent needs and the existence of infectious asymptomatic and unreported individuals within susceptible population. furthermore, the model is able to predict the peak-size, peaktime, final size and epidemic duration according to different rates of containment. advanced knowledge of these details will be of great interest to establish an optimal plan-of-action to control or eradicate the epidemic. indeed, mitigating and delaying the epidemic peak allow the official health authorities to anticipate and control the spread of covid-19. moreover, prediction of the epidemic duration can help the government to predict the end time of containment to avoid consequent social-economic damages as well. using our model, the basic reproduction number r 0 is estimated to be 2.9949, with ci (2.6729 − 3.1485) , reflecting a high speed of spread of the epidemic. the model shows that the compulsory containment can be efficient if more than 73% of population are confined. in the absence of other efficient measure of control, even with 90% of containment, the end-time is estimated to happen on july, 4, 2020 with 7558 final cumulative cases. furthermore, a threshold value of containment rate, below which the epidemic duration is postponed, has been determined. finally, the sensitivity analysis is performed and showed that the covid-19 dynamics strongly depends on the asymptomatic duration as well as the contact and containment rates. our previsions covid-19 is an infectious disease caused by a novel betacoronavirus that primarily targets the human respiratory system [12] . severity of covid-19 symptoms can range from very mild to severe. indeed, some people generally have mild to moderate respiratory illness and recover without requiring special treatment [32] . others, like older adults and people with existing chronic medical conditions, are more likely to develop serious illness and have a high risk of death [32] . human-to-human transmission of disease occurs via direct contact with the droplets saliva, discharge from the nose of an infectious person's coughs, sneezes or through contaminated objects and surfaces [32, 12] . the novel coronavirus was first emerged in the chinese city of wuhan in december 2019. in a few months, the virus rapidly spread throughout china and was exported to 114 countries around the world [12, 35] . the worldwide number of humans diagnosed with covid-19 has surpassed 118,000 and nearly 4,300 people have died [1] . on march 11, 2020, the world health organization (who) has officially declared the novel coronavirus outbreak a pandemic [2] . to the best of our knowledge, no specific vaccine or antiviral exist for covid-19 up to date, but there are many ongoing clinical trials evaluating potential treatments [14] . meanwhile, classical public health measures such as isolation and quarantine, social distancing and community containment were adopted by many countries to prevent dissemination of the disease within their populations and to curb the epidemic [1, 35] . the containment measure, defined as an intervention applied to an entire community in order to lower intermixing of unreported infectious individuals with susceptibles as well as the spread of the virus [35] , ranges from social distancing to community-use of face masks, including locking entire cities or areas. morocco is one of the countries affected by covid-19. as of april 09, 2020, the total number of confirmed cases in morocco was reported to be 1374 infectious and 97 deaths [26] . becoming ill are identified and tested [17, 18, 19, 20, 21, 22] . since seven imported cases have emerged, moroccan government decided to suspend flights to neighboring countries affected by covid-19 such as italy, spain, algeria and france [29] . however, reported cases continued to grow with eight new imported cases and a first confirmed local case appeared on march 14 [23] . the epidemic spreads to other moroccan cities such as tetouan, rabat and khouribga and, therefore, morocco rapidly responded to the covid-19 infection and the moroccan authorities implemented a series of control measures to limit person-toperson transmission. the ministry of interior has decided to close all schools of the country and promotes online interactive learning as an alternative [28] . movements of moroccan people are restricted to their living space and all gathering of more than fifty persons has become prohibited [29] . mosques and non-essential common areas are closed except for pharmacies and stores selling necessary goods to citizen. moreover, all international flights are suspended and all public events are canceled [29] . on march 19, the government decreed a state of health emergency and the compulsory containment of population has been declared [27] . however, a part of the population continued to work to supply the basic necessities to the confined population. every day, infectious individuals are reported by the ministry of health and all individuals that have a close contact with them are identified and quarantined. unfortunately, the unreported and asymptomatic infectious cases may exist within confined or working people and constitute a source of contamination of susceptible individuals like family members, co-workers, sellers, etc. despite being the most affected country in the world by the disease, china is the first country that has succeeded in mitigating the progression of covid-19 by a stringent containment of the population [33, 36] . others countries such as morocco still continue their fight to curb the growth in the number of daily infectious cases. up to date, the number of confirmed cases of covid-19 continues to increase and changes on a daily basis. the efficacy of containment and how long it must be maintained is still unknown. the longer the period of containment, the more serious consequences on the socioeconomic situation of the country will be. in addition, the short peak period of the epidemic and large notified cases prevent health care teams from adequately preparing for and responding to the huge load of patients. thus, lowering the peak size and the postponing the peak time of the reported cases are of major interest. in this study, using the reported cumulative confirmed cases in morocco from march 2nd, 2020 to april 9, 2020 (see table 1 the data of reported symptomatic infectious cases is collected each day at 11 pm from the official coronavirus portal of morocco [26] . the model without containment the population considered in our basic model , as shown in fig here, we assume that reported symptomatic infectious individuals are hospitalized and can not contact susceptibles anymore. we assume also, as confirmed by rothe et al. [13] , that asymptomatic individuals can infect susceptible individuals through an effective contact. furthermore, macintyre in [15] proved that asymptomatic and symptomatic infectious individuals share the same infection probability. taking account of the previous assumptions, the dynamics of covid-19 can be described as follows: susceptibles (s) contacted with either unreported symptomatic (ĩ u ) or asymptomatic infectious individuals (ã), at an effective contact rate, c, are infected with infection probability, β, and move to the asymptomatic infectious class (ã). after an average period 1/δ days the asymptomatic infectious individuals (ã) become symptomatic and proceed either to the unreported symptomatic infectious (ĩ u ), at rate δ 1 , or to the reported symptomatic infectious (ĩ r ) at rate δ 2 with δ = δ 1 + δ 2 . once becoming symptomatic, individuals of classĩ u andĩ r remain asymptomatic for 1/µ days on average before they are recovered. since the containment measure started 19 days since the first reported case then the model equations without containment is defined for 0 ≤ t < t 0 := 19 as follows here, we assume that confined asymptomatic and confined unreported individuals can still spread the virus to their families. furthermore, since the moroccan government starts to impose public major measures, and taking account the fact that moroccan individuals gradually began to reduce their contact with the people nearby, we assume that the contact rate changes from a constant rate to an exponentially decreasing rate, cl(t), with time. it is meaningful to assume that the only subpopulation that will not be confined is the reported symptomatic infectious subpopulation. the model with containment control will given, for t ≥ t 0 days, by the following equations is the total asymptomatic infectious population and i u (t) = i n u (t) + i c u (t) is the total unreported symptomatic infectious population. moreover, if the containment rate of susceptible, asymptomatic infectious and unreported infectious subpopulations is denoted by p, then the new initial data for system (2.2) shall be given by the basic reproduction number, r 0 , is the average number of secondary infections produced when one infectious individual is introduced into a host susceptible population. this quantity determines whether a given disease may spread, or die out in a population. to compute this number, we apply the next generation matrix method in [31] . computation method of r 0 is presented in the appendix. here, r 0 can be explained as follows: assume that one asymptomatic infectious individual is introduced into the susceptible population. this asymptomatic individual produces, on average, βcs 0 1 δ asymptomatic individuals during his average lifespan 1/δ. these asymptomatic individuals then become unreported symptomatic infectious individuals over their lifespan 1/δ at a rate δ 1 and then each infectious symptomatic produces, on average, βcs 0 1 µ asymptomatic individuals during his lifespan 1/µ. the control reproduction number, r c , is an important value, used to determine whether a control policy, such as containment in our case, will be efficient to decrease the number of secondary infections to be less than one. here, we compute the control reproduction number related to the early stage of the containment (during the first 20 days starting from the first day of containment). computation method of r c is presented in the appendix. the parameter estimation is a crucial step of our study since the epidemiologically relevant choice of the parameters must establish and confirm the observed dynamics of the infection during the onset of the epidemic. table 2 gives the values of the parameters used in the model. in our simulations, the the total population of the moroccan kingdom,s(0), is chosen based on estimates from [5] , the asymptomatic duration, 1/δ, based on estimates from [26] and the symptomatic duration, 1/µ, based on estimates from [34] . note that the data of reported cases used to estimate the model parameters was carried out before containment and during the epidemic period. thus, we assume that the contact rate, c, since the first reported case and before the first day of containment is around 10 contacts per person in average [9] . furthermore, we assume that a likely confined individual can only go out once every 2 weeks on average to do the necessary shopping while an individual who cannot respect containment, due to a job of paramount importance, must go 5 days a week to work. thus, it is meaningful to assume that, during the short time from march 21 to april 9, the average contact rate satisfy the relation 1 39−20´3 9 20 cl(t)dt ≈ c/10. thus, we assume that α ≈ 0.078. the other parameters and initial data are estimated as follows: since the first and the only infectious symptomatic individual is reported on march 2nd, 2020, which corresponds to t = 0, thenĩ r (0) = 1. for the estimation of β,ã(0) andĩ(0) we will use the data of cumulative reported cases collected from march 2nd to march 20 (before the start of containment) in table ( 1) and we follow the procedure by [8] . the cumulative reported infectious population is given, for t ≥ 0, by cr(t) = δ 2´t 0ã (s)ds + 1. it is obvious that cumulative reported infectious population increases slowly and then accelerates rapidly with time. hence, we will use exponential regression with 95% of confidence level to find an exponential function that best fits the data, from march 2nd to march 20th, in table ( from the second equation of system (2.1) and using (2.3) we obtain the government have deployed a series of severe control measures to limit transmission of covid-19 across morocco. the moroccan people entered compulsory containment 19 days since the first infectious case has been confirmed. in this study we adopted a deterministic mathematical model of covid-19 dynamics which took into account the moroccan containment strategy used to control and eradicate the disease in the country. we used reported infectious case data, from march 2nd to april 9th, 2020, provided by the health ministry of morocco to parameterize the model. on the one hand, as shown in fig. 3 .1, our simulations shows that our model fit well the cumulative data of reported infectious cases giving in table 1 , under the absence of compulsory containment conditions. on the other hand, to be as close as possible to reality, we used the cumulative reported case data from march 21th to april 09th and we applied the same method in section 2.4 to estimate the containment rate p during these first 20 days of containment (see appendix 4). the fig. 3 .2 showed that our model is able to predict the provided real reported data during the first 20 days past after the beginning of containment. these results are horrifying and no country in the world could control a major wave of infectious cases. for practical reasons, it is imperative to reduce the peak-value so as not to exceed the limit capacity of hospital beds and lead the country towards an uncontrollable dramatic situation. it is also better to delay the peak appearance time to allow official health structures to prepare for such a high level of emergency. increasing the containment rate p from 0.73 to 0.90 lead to a rapid saturation of cumulative number of reported infectious cases, ranging from 16950 to 7558 (fig. 3.4) . furthermore, as shown in fig. 3 .4, the peak times of reported infectious cases decreased slightly while the epidemic duration decreased significantly from 180 to 125 days and the peak-sizes decreased from 338 to 188 cases (see table 3 ). if moroccan government succeeded to contain more than 90% of population, then the peak size of infectious cases would be less than 188 reported cases on april, 27th, 2020. however, these results could be overestimated or underestimated since many epidemiological factors are still unknown and are subject of study by researchers. up to date, morocco still did not reach the epidemic peak yet. furthermore, if the containment period will last so long, morocco must combine its actual strategy with the mass-testing or the contact-tracing strategies which allow finding and controlling the suspected cases through the susceptible population [10, 6] . for example, to the best of our knowledge, south korea is the only country that has succeeded to greatly slow its epidemic without resorting to drastic containment [3] . the increase of the containment rate does not significantly affect the peak-time but acted differently on epidemic duration (see fig. 3.4) . indeed, we found that, by increasing the containment rate, there exists a threshold control reproduction number r c = 1 corresponding to a threshold containment rate p * = 0.73, below which the epidemic end-time is postponed ( fig. 3 .5) and above which the epidemic end-time is advanced (fig. 3.4) . this phenomenon can be explained as follows: in the absence of containment, the whole population becomes infected within a short time. however, by increasing the rate of containment gradually to the threshold p * , the daily number of infectious individuals becomes lower and continues to significantly infect the susceptible individuals, which increases the duration of the epidemic. moreover, if the containment rate passe the threshold, the containment becomes efficient since the daily number of infectious individuals significantly diminish, which in turn decrease the number of new infected and, consequently, shorten the duration of the epidemic. finally, the study of sensitivity analysis supported the results above and found that the containment rate has significant impact with correlation coefficient prcc=−0.8. in general, the only three parameters playing a major role on the model dynamic are the contact rate, the containment rate and the asymptomatic duration ( fig. 3.6 ). however, our sensitivity analysis study show that the asymptomatic duration have a greater impact, with correlation coefficient prcc=0.863, and consequently, can play a major role in covid-19 spread across morocco. indeed, it has been shown that younger age individuals are the most contributor of silent transmission of covid-19 to older family members [15] , which could be the case in morocco since 26% of the moroccan community are less then 15 years old [4] . fortunately, the moroccan government have decided to close schools earlier to avoid such a critical epidemic consequences. the covid-19 infection was primary imported to morocco from european countries. and the associated jacobian matrix is given by  and r 0 is its spectral radius which is given following the remark on the contact behaviour and the estimation given in section 2.4 we will assume that, in average, cl(t) = c/10. consequently, the control reproduction number is given by r c = (βcps(t 0 )/10 + βc(1 − p)s(t 0 )) 1 δ + δ 1 δµ . estimation of p between march 21 and april 9 the cumulative reported cases for t ≥ t 0 is given by cr(t) = δ 2´t t 0 a(s)ds+i r (t 0 ) = be at . here we will use once again the exponential regression with 95% of confidence level to find an exponential function that best fits the data, from march 21 to april 9, in table (1) . we found that exponential model given by be at with a = 0.139 with confidence interval ci (0.125 − 0.153) and b = 7.8303 with ci (5.176 − 11.858) fits well the data with a correlation coefficient given by r = 0.981. differentiating the both terms of cr(t) leads to since the initial susceptible population is not dramatically affected in the early phase of the epidemic, we will assume that s(t) ≈s(t 0 ). adding the third and fourth equation of system and solving equations (4.5) and (4.6) for p lead to p = 1 9βcs(t 0 )/10 βcs(t 0 ) − (a + µ) (a + δ) a + µ + δ 1 . evaluation and treatment coronavirus (covid-19) [update who declares covid-19 a pandemic strategies to control covid-19 and future pandemics in africa and around the globe population et développement au maroc : vingt-cinq ans après la conférence du caire de 1994 dynamic modeling to identify mitigation strategies for covid-19 pandemic 2020 the reproductive number of covid-19 is higher compared to sars coronavirus understanding unreported cases in the 2019-ncov epidemic outbreak in wuhan, china, and the importance of major public health interventions social contacts and mixing patterns 366 relevant to the spread of infectious diseases covid-19 mass testing facilities could end the epidemic rapidly covid-19 and italy: what next? health policy the epidemiology and pathogenesis of coronavirus disease (covid-19) outbreak transmission of 2019-ncov infection from an asymptomatic contact in germany world health organization declares global emergency: a review of the novel coronavirus (covid-19) global spread of covid-19 and pandemic potential the ministry of health of morocco, communiqué n°10: morocco announces 1st covid-19 case, www.covidmaroc the ministry of health of morocco, communiqué n°11: morocco registers 2nd covid-19 case, www.covidmaroc the ministry of health of morocco, communiqué n°12: morocco registers third covid-19 case. www.covidmaroc the ministry of health of morocco, communiqué n°14: morocco announces two new cases infected with covid-19 the ministry of health of morocco, communiqué n°15: morocco registers sixth covid-19 case. www.covidmaroc the ministry of health of morocco, communiqué n°16: morocco registers seventh covid-19 case. www.covidmaroc the ministry of health of morocco, communiqué n°18: morocco registers eighth covid-19 case. www.covidmaroc the ministry of health of morocco, communiqué n°19: morocco announces 9 new cases infected with covid-19. www.covidmaroc the ministry of health of morocco, communiqué n°20: morocco announces 11 new cases infected with covid-19. www.covidmaroc 2020) communiqué n°22: the epidemiological situation of covid-19. www.covidmaroc the ministry of health of morocco, the official coronavirus portal of morocco. www.covidmaroc the ministry of interior the ministry of interior the ministry of interior, map-anti-corona mapanticorona.map covid-19 r0: magic number or conundrum reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission report of the who-china joint mission on coronavirus disease (covid-19) who, news briefing on isolation, quarantine, social distancing and community containment: pivotal role for old-style public health measures in the novel coronavirus (2019-ncov) outbreak a systematic approach is needed to contain covid-19 science bulletin the authors thank the anonymous referees, whose careful reading, insights, valuable comments, and suggestions significantly enabled us to improve the quality of the paper. the linearized equation related to infectious individuals of system (2.1) is given by the author declare that there is no conflict of interest 25 key: cord-344252-6g3zzj0o authors: farooq, junaid; bazaz, muhammad abid title: a novel adaptive deep learning model of covid-19 with focus on mortality reduction strategies date: 2020-07-21 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110148 sha: doc_id: 344252 cord_uid: 6g3zzj0o we employ deep learning to propose an artificial neural network (ann) based and data stream guided real-time incremental learning algorithm for parameter estimation of a non-intrusive, intelligent, adaptive and online analytical model of covid-19 disease. modeling and simulation of such problems poses an additional challenge of continuously evolving training data in which the model parameters change over time depending upon external factors. our main contribution is that in a scenario of continuously evolving training data, unlike typical deep learning techniques, this non-intrusive model eliminates the need to retrain or rebuild the model from scratch every time a new training data set is received. after validating the model, we use it to study the impact of different strategies for epidemic control. finally, we propose and simulate a strategy of controlled natural immunization through risk based population compartmentalization (pc) wherein the population is divided in low risk (lr) and high risk (hr) compartments based on risk factors (like comorbidities and age) and subjected to different disease transmission dynamics by isolating the hr compartment while allowing the lr compartment to develop natural immunity. upon release from the preventive isolation, the hr compartment finds itself surrounded by enough number of immunized individuals to prevent spread of infection and thus most of the deaths occurring in this group are avoided. o an effective strategy to minimize the number of deaths through controlled natural immunization in absence of availability of vaccination at mass level. covid-19 is a highly contagious epidemic disease caused by novel coronavirus (sars-cov-2) that originated in wuhan, hubei province of china in late december 2019. world health organization (who) declared covid-19 as a pandemic on 12th march 2020 [1] . researchers are policy makers are working round the clock to find solutions and design strategies to control the pandemic and minimize its impact on human health and economy. the transmission of sars-cov-2 in humans is mostly through respiratory droplets (sneezing, coughing and while talking) and through contaminated surfaces [2] . the most significant property of sars-cov-2 is that it can persist on a variety of surfaces from hours to 9 days at room temperature which makes its transmission more rapid [3] . this virus can cause acute respiratory distress syndrome (ards) or multiple organ dysfunction, which may lead to physiological deterioration and death of an infected individual [4] . mathematical modeling of infectious diseases and epidemics has been employed as an important tool for analysis of disease characteristics and investigation of disease spread ever since the ground breaking work of kermack and mckendrick in 1972 [5] . it plays a useful role in efficient decision making and optimal policy framing. different models have been developed to analyse the transmission dynamics of many infectious diseases like malaria (ronald ross model) [6] , cholera (capasso and pareri-fontana model, 1979) [7] , gonorrhea (hethcote and yorke model, 1984) [7] , ebola [8] , h1n1 [9] etc. in this work, we employ deep learning to propose an artificial neural network (ann) based real-time online incremental learning technique to estimate parameters of a data stream guided analytical model of covid-19 to study the transmission dynamics and prevention mechanism for sars-cov-2 novel coronavirus in order to aid in optimal policy formulation, efficient decision making, forecasting and simulation. modeling and simulation of such problems poses an additional challenge of continuously evolving training data in which the model parameters change over time depending upon external factors. our main contribution is that in a scenario of continuously evolving training data, unlike typical deep learning techniques, this model eliminates the need to retrain or rebuild the model from scratch every time a new training data set is received. using a data science approach, model parameters are intelligently adapted to the new ground realities. to the best of our knowledge, this paper develops for the first time a deep learning model of epidemic diseases with data science approach in which parameters are intelligently adapted to the new ground realities with fast evolving infection dynamics. the covid-19 data from india has been taken as the case study. the first case of covid-19 in india was reported on 30th january 2020 originating from wuhan, china [10] . as on 13 june 2020, the total number of cases reported in india is 308,993 with 154,330 recoveries and 8,884 deaths [11] . hence the number of active cases is 145,779. the government of india imposed a country wide complete lockdown on 24th march 2020 with strict restrictions on the movement of people while allowing only the essential services to operate under the supervision of administration and health officials. the lockdown was renewed thrice on 14 april, 3 may and 18 may 2020 and has been relaxed since 8 june 2020 [12] . india is the second largest populated country in the world with a total population of around 1.35 billion. the health care facilities in india are considered poor with 0.55 hospital beds per thousand people of the population [13] . therefore, the covid-19 pandemic has emerged as a major challenge for the people, health workers and policy makers of the country. using a control theory approach, we analyze the stability of different disease prevention strategies. finally, we propose a strategy of controlled natural immunization of the population by dividing it in low risk and high risk compartments based on various risk factors like age and comorbidities. the two groups are treated separately and subjected to different disease mechanics with the aim to minimize the total number of deaths given the fact that the probability of death is very high in the high risk group as compared to the low risk group. the two compartments are isolated from each other for a certain period of time. the low risk compartment is allowed to fully brace the infection with maximum speed and develop immunity owing to its very low death rate while as the high risk compartment is put under preventive isolation till the infection growth curves fatten for the low risk group. upon release from the preventive isolation, the high risk group finds itself surrounded by enough number of immunized individuals to prevent spread of infection and thus most of the deaths occurring in this group are avoided. we simulate this strategy in matlab environment to establish its effectiveness in significant reduction in the number of deaths while demonstrating the usefulness of the deep learning based mathematical model. to the best of our knowledge, such an approach to reduce mortality has not been modeled and simulated earlier by the scientific community. sirvd refers to susceptible, infected, recovered, vaccinated and deceased states of individuals in a population going through an epidemic. the pioneer work in development of mathematical models for infectious diseases was carried out by by [5] known as the susceptible-infectious-recovered (sir) model. as one of the most classical models, it has been used by many researchers to study and analyse many infectious diseases like seasonal flu [14, 15] , pandemic flu [16, 17] , hiv/aids [18] , sars [19, 20] etc. these studies have shown that sir models are reliable for analysis of the infectious disease spread and evaluation of the impact of prevention schemes in different scenarios. the basic sir model is described by the following differential equations: where s , i and r are functions of time representing the number of susceptible, infected and recovered individuals in a population of size n at time t. β is the rate of transmission and γ is the rate of recovery of infected individuals. it is assumed that those recovered develop immunity and do not catch the infection again in the time span of interest. the basic sir model can be modified in various ways to accommodate different scenarios. a modified sir model known as sird (susceptible-infected-recovered-deceased) model is of our interest here and is based on the following assumptions: (i) this model is fatal unlike a typical non-lethal sir model which means that there is a positive probability of an infected person dying, p(death) = δ and δ ≥ 0. (ii) a typical sir model assumes that the recovered group gains full immunity from reinfection. however, this model accommodates the possibility of a recovered person being reinfected with probability of reinfection, p(rein f ection) = σ and σ ≥ 0. (iii) the impact of new births and unrelated deaths in ignored and the total population remains constant, n = constant ∀ t. (iv) the population is distributed randomly over the area. therefore, there are four classes of individuals: susceptible (s), infected (i), recovered (r) and deceased (d) as described by the following equation: where β = rate of infection, σ = rate of susceptibility, γ = rate of recovery and δ = rate of death. since the final cure for covid-19 pandemic is the successful discovery and optimal administration of the vaccine in the population, therefore we introduce the effect of vaccination with a given rate of vaccination under resource limited settings. this is achieved by adding a new class of individuals called vaccinated (v) in the population. it can be fairly assumed that there is no limit on the total number of vaccines produced as all the available resources for vaccine production are employed to eliminate the epidemic. however, the vaccine production capacity will have some limit based on availability of resources and facilities. therefore, there will be a limited number of vaccines available at a particular point of time. thus, it is assumed that the per capita rate of vaccination, α < where is a constant. it is assumed that the vaccination imparts long term immunity against the disease in the vaccination individuals. based on these assumption, we propose a final model as described by the following first order ordinary differential equations (ode's) and it illustrated by figure 1 : it must be noted here that the process of mass action transmission is described by the non-linear term βs i/n where β = number of contacts per unit time by a person in group i required to transmit the disease to a person in group s, n − 1 ≈ n = total number of possible contacts of a person, s /n = the fraction of possible contacts of a person that are from group s, i = the number of infected persons at time t, therefore βs i/n = number of people transmitted from group s to group i per unit of time. the next task is to to learn the model parameters which can be quite challenging in an epidemic scenario like covid-19 as the the model parameters are supposed to change with time. this section proposes an artificial neural network (ann) based adaptive incremental learning technique (annail) for online learning of the sirvd model parameters with the following assumptions: (i) the rate of vaccination α as a function of time (t) is set by the vaccine production capacity decided by availability of skilled labour, resources and facilities and by vaccination policy as well. maximum vaccine production capacity α max has been assumed to be constant indicating that there is no change in vaccine production infrastructure, technology or facilities. (ii) the rate of infection β as a function of time (t) is the major challenge for parameter learning. it is affected by external factors like degree of social distancing, lockdown etc. in case of a lockdown β decreases exponentially. therefore, in order to take into account both the lockdown and no lockdown scenarios, β has been modelled as: where t l is the time when the lockdown begins. therefore, the learning algorithm has to learn 3 parameters (β 0 , β 1 , τ β ) to find β. (iii) the rate of reinfection σ has been assumed to be zero for covid-19 disease as the human body develops antibodies to prevent re-infections in future against such a virus [21] . (iv) rate of recovery γ and rate of death δ are affected by factors like change in health care facilities, possible overcrowding of hospitals, development of new drugs to manage or treat the disease etc. both of these have been assumed to be constant in this paper. for a typical neural network or any other technique of model parameter estimation, the training data is required first to train the model before applying it on future scenarios. however, in case of an epidemic like covid-19, the training data is continuously evolving with time and the model needs to be trained and executed at the same time as the model parameters may change over time based on different external factors like government policies, social distancing etc which can be known only from newly arriving data sets. therefore, we propose a technique for the model to learn these parameters from new data sets in an adaptive manner while continuously updating the old models without the need to build the model from scratch every time a new training data set is received. deep learning and other machine learning techniques stand out in solving problems of data based model parameter estimation due to their state-of-the-art results. however, they face the problem of catastrophic forgetting which reduces their performance as new training data becomes available with time. this is because the typical neural networks require the entire dataset to update the model each time a new training data set becomes available as in case of an epidemic modelling problem where the training data becomes available incrementally with time. to address such issues, different incremental learning algorithms have been suggested in the literature [22, 23, 24] . incremental learning refers to an online learning technique of continuous model adaptation under a scenarios of continuously evolving training data. therefore, storage or access to the previously observed data is not required each time a new data set is received, as in case of an epidemic like covid-19. in order to adapt the model parameters in light of new data, it is not needed to use all the previously accumulated data for developing the model from scratch. rather, the learning network modifies the previous hypothesis to adapt to the new data chunk. in this paper, we propose hypothesis generation via an artificial neural network (ann). let d j−1 be the data set received between time t j−1 and t j , and h j−1 be the hypothesis generated on this data set. the hypothesis h j for a new data set d j received between time t j and t j+1 is a function of d j and h j−1 only as under: the experience gained from this step is stored and integrated to support in future adaptation process. thus the objective here is to integrate the previously learned knowledge into the new raw data set to adapt the model parameters accordingly; and to accumulate this experience over time to increase the model efficiency, accuracy and flexibility. the proposed framework for the above problem is shown in figure 2 . the ann is based on a non-linear activation function for successful regression analysis. the hidden layers are represented by the function f nn . with the continuous data stream, the weight distribution functions are generated to describe the learning capability of the ann where the decision boundary is adjusted to focus especially on the hard to learn data examples. the algorithm for this framework is given in algorithm 1. , where x i represents the input vector and y i represents the output. -associated mapping function output: where f nn is the ann defined mapping function. (ii) apply hypothesis h t−1 to d t , and find the pseudo-error (vi) repeat the above procedure for d t+1 . the final hypothesis is: where t is the set of incrementally developed hypotheses in the learning life. this algorithm is run in a top-down and horizontal signal flow, as shown in figure 2 . the adaptive nature of this algorithm is due to the mapping function based on ann which estimates the initial distribution functionφ t−1 for d t while providing a quantitative approach to indicate the learning power of the new data set based on previously trained model.φ t−1 is applied to the new data set to find pseudo-error, υ. thus a hard to learn example will have higher υ in step (iii) of the learning procedure, and will in turn receive higher weight in step (iv). this ensures the adaptive nature of the algorithm. mapping function connects the past experiences to the new data in an adaptive fashion. there can be many ways to design the mapping function. however, we implemented the nonlinear regression by an ann based approximation of mapping function owing to its flexibility. any such neural network based function approximation technique can be used. as in illustration we take the multilayer perceptron (mlp) in this paper. this is shown in figure 3 . the input is an n-dimensional vector (for example, number of infections, deaths, recoveries in an epidemic) of example i. the distribution function is currently estimated as j t−1 , w represents the weights of a layer. backpropagation is used to tune the weights w of different layers, where error function is defined as where k is the training epoch of the backpropagation. the neural network gives the following output: where h f represents the input to fth hidden node while as g f represents its output, υ is the input to the final node, n h is number of hidden neurons, and n is the total number of inputs. weights on the ann are updated by applying the above defined backpropagation strategy as explained below. backpropagation: weight adjustment for the hidden to out similarly, weight adjustments for the input to hidden layer is described as: where α(k) describes the learning rate. estimation of initial distribution functionφ t for d t required only the feedforward path of the mlp. this model was validated for covid-19 in india where the first 80% of data was used for training and the remaining 20 % was used for testing as shown in figure 4 . it is clearly evident from the plots shown in the figure that the results given by the model during testing are very close to the actual data. the inputs and outputs in this algorithm were: inputs : number of new infections, deaths and recoveries; rate of vaccination (α). outputs : β 0 , β 1 , τ β , δ, γ these outputs are fed to the analytical sirvd model at every time instant when a new set of input data is received to simulate and forecast different scenarios. in this section, we analyse three possible strategies to combat an epidemic like covid-19: (i) herd immunity, (ii) complete vaccination, (iii) complete lockdown and finally our proposed (iv) controlled natural immunization through risk based population compartmentalization is discussed in the next section. however, following definitions are needed beforehand: definition 1. (stability) if the jacobian matrix j for a system of n differential equations has eigenvalues λ 1 , λ 2 , ..., λ n , for a trivial steady state equilibrium at (0,0,...,0), then the stability of the solution is determined as following: .., n then the system has uniform and asymptotic stability (uas). .., n and the algebraic multiplicity equals the geometric multiplicity whenever λ i = 0 for any i, then the system has uniform stability (us). (iii) if re(λ i ) > 0 for any i and the algebraic multiplicity is greater then the geometric multiplicity whenever λ i = 0 for any i, then the system has instability. since the impact of new births and unrelated deaths has been fairly ignored for the purpose of this study and it has been assumed that the total population n stays constant throughout the epidemic, therefore the system of odes (8 -12) describing the model satisfies the following two conditions: this shows that a state of equilibrium always exists in the system. definition 2. (dfe) a disease free equilibrium (dfe) is defined as a state of equilibrium according to (25, 26) in which the number of infected and recovered individuals equal to zero such that there are no further deaths (i → d), infections (s → i) or reinfections (r → i): which gives us the following from the system of odes (8 -12) : we take the jacobian matrix of the above system to evaluate its stability. however, the fifth differential equation (representing d) is uncoupled from the first four differential equations and it can be derived from the first four equations using (25, 26) , therefore we consider the jacobian for the first four equations only which is given as: given below is the investigation of dfe stability for different epidemic control strategies. herd immunity is the idea that a virus cannot spread easily after enough people develop immunity against it. this reduces the chances of the virus transmitting from person to person and infecting those who haven't been infected yet [25] . such an immunity can be induced artificially by a vaccine, however it can also be developed naturally by the infection itself as the immune system of the body develops antibodies against the virus which prevent reinfection in future. this is based on the fact that the human body produces a non-specific innate response to a viral infection initially using neutrophils, macrophages, and dendritic cells. however, this is followed by a more specific adaptive response in the form of development of proteins called immunoglobulins which act as antibodies specifically binding to the virus. this is coupled with the formation of t-cells which generate cellular immunity by identifying and eliminating the cells that are infected with the virus. generally, sufficient presence of such antibodies in collaboration with cellular immunity prevents reinfection after recovery. although every recovered patient may not develop complete immunity, but that is the case for the most of them [21] . in case of covid-19, although research is till going on to reach a conclusive opinion, promising studies suggest that nearly all the recovered patients develop such antiviral immunoglobulin-g (igg) antibodies and are immune to reinfection [26] . this lays the basis for mass serological testing for covid-19 being practised by many governments across the world in which the blood samples of people are tested for the presence of these antibodies indicating present or past covid-19 infection. further, the data on reinfection, even if rare, is not available. therefore, the idea of herd immunity is to let the infectious disease take its natural course of action and let the population naturally develop immunity against the disease after most of the population gets infected. the dfe after herd immunity has the following form: where d 0 is the total number of deaths at the time of dfe and thus r = n − d 0 . since there is no vaccination, therefore α = 0 and v = 0. further, the rate of reinfection σ is zero as well, meaning recovered people cannot catch the infection again. using these values in (32), the jacobian at the dfe after herd immunity is given as: equating characteristic equation for this jacobian to zero gives: therefore, the eigenvalues of this system are: thus, all the eigenvalues have real parts less or equal to zero: re(λ i ) ≤ 0 ∀ i = 1, 2, ..., n and d 0 ∈ r ∩ [0, n]. hence, according to definition 1, the dfe for herd immunity possesses uniform stability (us). in this case, there are no chances of retriggering of the disease after the dfe has been reached. however, this approach has been criticised as dangerous as it will result in a large number of deaths [27] . to develop herd immunity for covid-19 in the population, roughly 70 % or more of the population needs to have gone through the infection [28] . the current death rate in india due covid-19 is around 3 % which means that if this death rate is maintained, nearly 2.1 % of the population will die till 70 % gets infected. this leads to nearly 28 million deaths. the actual number of deaths would be more than this, because with rapid growth of the infectious disease, hospitals would be over-flooded and health care infrastructure will not be able to cater to the demands of high number of patients leading to increase in the death rate. the growth of disease in india is shown in figure 5 if the disease is allowed to take its natural course without any intervention and control. the strategy of complete lockdown focuses on minimization of mobility and contact among the population with maxi-mum possible social distancing. therefore, it minimizes β (the rate of infection). the dfe under complete lockdown has the following form: where d 0 is the total number of deaths at the time of dfe and thus s = n − d 0 . since there is no vaccination, therefore α = 0 and v = 0. using these values in (32), the jacobian at the dfe under complete lockdown is given as: furthermore, equating characteristic equation for this jacobian to zero gives: therefore, the eigenvalues of this system are: thus, real part of one of the eigenvalues is always grater than zero: re(λ 3 ) > 0 ∀ d 0 ∈ r ∩ [0, n]. hence, according to definition 1, the dfe under complete lockdown is always unstable. thus, whenever a dfe is reached in this case, the tendency for the disease to be triggered again is always there. even a single case of infection can restart the disease. apart from the chances of reemergence of the disease, the time taken by the system to reduce the number of active infections to zero may be large enough to make the total lockdown unsustainable and practically impossible. the impact of lockdown on the disease growth in india is shown in figure 6 . it has been assumed that the lockdown ends on day 250 of the disease after it starts on day 55. it can be compared with figure 5 which shows the disease growth in case no preventive measures are taken. these results confirm that the infection plots shoot up as soon as the lockdown is lifted and there is no significant difference in terms of total number of infections or deaths. the only benefit of lockdown, as can be seen from these plots, is that the peak of infection is delayed which can be useful for the administration to buy some time to prepare the healthcare infrastructure of the country to brace for the full blown impact of the disease. researchers and health industries worldwide are working around the clock to discover a vaccine against sars-cov-2 [29]. moderna, a pharmacological company had started clinical testing of its mrna-based vaccine (mrna-1273) just 2 months after the complete sequencing of covid 19 was done and published by different research groups [30] . the most potential candidate for the vaccine development would be to induce our immune system to synthesize neutralizing antibodies against the viral spike protein which block its entry via ace2 receptors [31] . the dfe with vaccination of all living members of the population n has the following form: where d 0 is the total number of deaths at the time of dfe and thus v = n − d 0 as the rest of population has been vaccinated and is now immune to the disease. using these values in (32), the jacobian at the dfe with full vaccination is given as: furthermore, equating characteristic equation for this jacobian to zero gives: therefore, the eigenvalues of this system are: thus, all the eigenvalues have real parts less or equal to zero: members of the population is the way to completely eliminate the disease and its chances of re-triggering as well. however, successful development of a vaccine for sars-cov-2 has still a long way to go. further, a vaccination rate of 10 million per day is highly ambitious for a country like india. therefore, elimination of the current wave of covid-19 epidemic and minimization of number of deaths by vaccination is practically impossible as the successful development and mass administration of a vaccine is expected to take more than a year, at least. however, vaccination may still be necessary to prevent new waves of the disease in future. the strategy of herd immunity discussed in the previous section aims at minimizing the number of s (susceptible) and maximizing the number of r (recovered) people who are supposed to have developed the immunity. this will minimize the the factor β s i n in the sirvd model ode's discussed in section 2. however, this strategy results in the maximum number of deaths. the strategy of complete lockdown aims to minimize the rate of infection (β) by reducing the mobility of people. however, this strategy is not sustainable in the long term as discussed. the strategy of complete vaccination is an ideal solution to the problem. it aims at maximizing the number of people falling in v (vaccinated group) through artificial immunization. however, as discussed, the successful development of a vaccination and its administration in the population is expected to take enough time to cause a large number of deaths. in this section we propose a strategy of minimizing the number of deaths by controlled natural immunization by compartmentalization of the population in two groups: low risk and high risk. high risk group comprises of the people having co-morbidities or are aged above 50 years and as a result have high probability of death under covid-19 infection. the aim of this strategy is to minimize the number of deaths caused by covid-19 disease: the death rate (δ) is very high in the high risk group while as it is very low in the low risk group. the most prevalent comorbidity for covid-19 is hypertension, followed by diabetes with mean age of around 48.9 years [32]. in india, the percentage of population above 50 years of age is 19.28% while as the percentage above 60 is nearly 8% [33] . most of the people having comorbidities are expected to fall in above 50 age group. in this study, the high risk group has been assumed to be 20% of the population. it is proposed that these two population groups be subjected to different disease mechanics. the high risk group is subjected to a preventive quarantine or isolation wherein they are isolated from the low risk group by placing them in separate rooms or sections in homes with minimum contact with the low risk group. whatever necessary contact is required, it should be done with maximum preventive measures like wearing of face masks, sanitization etc. very high risk individuals may be placed in designated care centers where their health needs are met by medical professionals. therefore, for the high risk compartment, β is reduced to minimum as in the case of a lockdown. meanwhile, the low risk group is subjected to maximum mobility and contact among its members effectively increasing β to the maximum possible value. there should be no social distancing and other preventive measures for this group. as a result, the infection will spread very quickly in the low risk group and its members shall develop the immunity and transfer from s to r having gained the immunity naturally with a very low death rate. once this is achieved or in other words once the disease curves are flattened for the low risk group, the high risk group is released from the preventive isolation and allowed to mix with the low risk group. since most of the population (70 -80 %) is already immune and the value of s and i is low, the chances of the high risk group receiving infection from the low risk is very low because the factor β s i n has already been reduced to minimum. during pc, β is minimum for high risk group while as after pc, s i is minimum. after pc, the mobility and the contact in the population, represented by β, should be kept moderate. since, α and σ are zero, therefore: here, β represents the rate of infection which depends upon the mobility and contact among the population. due to the lockdown, this can be considered the lowest possible β in india. we have simulated the pc strategy with different values of β l ranging from 2 to 5 times of β given above to account for high mobility and contact during pc. β h = 0.2β to assume that the isolation of high risk individual is at least 5 times stronger than than the social distancing practised by the whole population during the nation wide lockdown. δ and γ are complementary in the sense that an individual either recovers or dies from the infection. to distribute the current death and recovery rate in low and high risk groups: integrating the equations of sirvd model for this case: where s (t 0 ) and i(t 0 ) are initial states. this is true for both low and high risk groups. however, for high risk group, s h (t 0 ) = i h (t 0 ) = 0 while as for low risk group; where t f is the time when pc ends and the two groups are allowed to remix. after pc: δ and γ have been restored to original values while as β has been multiplied by a factor of 2 to signify higher level of mobility and contact than lockdown albeit with social distancing. the stability analysis and eigenvalues for this case are same as that of herd immunity i.e, it posses uniform stability. this strategy was simulated using the model proposed in section 1 and 2. the results of the simulation are shown in figure (9 -10) . in all these figures, black line represents the number of deaths, blue is the total number of cases, red is the active number of infections while as green line represents the number of recovered people. in figure 9 , the preventive quarantine or compartmentalization of the population is done from day 130 after the start of the population till day 200 while as in in figure 10 the same is continued till day 300. the rate of infection β is shown as a multiple of the average rate of infection during the nation wide lockdown from 24 march to 08 june, 2020 which was the minimum possible. increase of β is achieved by increase of contact and mobility among the population, for example β = 5x means five times mobility and contact as compared to the days of nation wide lockdown. if no preventive measures are taken, then as per the current trend in the disease growth we may expect more than 55 million deaths in the country if all the population gets infected. this is shown in figure 8 . 55 million deaths is not surprising even if linear growth of the disease is assumed with 2.8% death rate for a population of 1.35 billion. different hypothetical experiments were simulated and their results are given in table 1 . as discussed earlier, β is supposed to be kept on lower side after the end of pc and social distancing is advised. these results show that the total number of deaths can be reduced to 1.3 million from 55 million, if mobility and contact is made 5 times to that of the lockdown period and pc is ended on day 300 of the pandemic while as after the end of pc, the mobility is reduced to two times. the growth of disease in such a scenario is shown in figure 11 . in this work, we developed an analytical epidemiological model for covid-19 pandemic where model parameters are continuously updated to intelligently adapt to new data sets using an ann based adaptive online incremental learning technique. in a scenario of continuously evolving training data, unlike typical deep learning techniques, the model eliminates the need to retrain or rebuild the model from scratch every time a new training data set is received. the model was validated and different scenarios were simulated to demonstrate its usefulness and significance. india was taken as a case study. however, this model can be applied to any population in the world and would be a useful tool for policy makers, health officials and researchers in improving decision making efficiency, policy formulation and forecasting. the simulation work was carried out in matlab environment. using this model, we simulated preventive measures like lockdown, vaccination and herd immunity to study their impact on the evolution of covid-19 disease. finally we proposed an effective method to significantly reduce the number of deaths caused by the pandemic in case a vaccine is not available at the mass level. this technique aims to develop natural immunity in the low risk group of the population by subjecting them to the full blown impact of sars-cov-2 virus while as subjecting the high risk group to preventive isolation during this time period. once the low risk group develops natural immunity and its disease curves are flattened, the high risk group is released from the preventive isolation. upon release, the high risk group doesn't find enough infected or susceptible people in the environment to catch the infection at a high rate and in this way the maximum number of deaths are avoided in the high risk group. the impact of this strategy has been simulated and it has been shown the the number of deaths can be reduced from 55 million to 1.3 million if the population compartmentalization starts tomorrow and ends on day 300 of the pandemic in india. during this period, the mobility and contact in low risk group has to be made five times as compared to the lockdown period and upon remixing of the two groups the mobility and contact should be reduced to 2 times from 5. the novelty of this paper lies in the use of real-time online incremental learning technique in epidemic disease modeling. many machine learning techniques have been used in epidemic disease modeling [36], however this paper is the first instance of development of an incremental learning algorithm as a real-time adaptive deep learning technique for parameter estimation of an epidemiological model thus providing the model with the capability to work online i.e, unlike typical machine learning techniques, it doesn't require to rebuild or retrain the model from scratch every time a new data set is received but intelligently adapts the model to ever changing infection dynamics. since the model is non-intrusive, adaptive, intelligent, real-time and online in nature, therefore it can be employed to monitor, forecast and simulate the growth of any infectious disease over a large sized population without losing accuracy, fidelity or com-putational performance due to limitations like run-time duration, size of training data, computational complexity, change in transmission dynamics due to mutations in virus or bacteria, change in prevention mechanisms or government policies. even if the epidemic continues for decades in the whole world, the model will keep working efficiently on daily basis without any decay in performance or rte (run-time environment). further, to the best of our knowledge, population compartmentalization to achieve natural immunity against an infectious disease while significantly reducing the mortality has been modeled and simulated for the first time in this paper. these findings could be highly useful to policy makers around the world to reduce the number of deaths in any country in case a vaccine is not readily available and lockdown is not sustainable economically. further, this is a demonstration of the usefulness and efficiency of deep learning based incremental learning algorithm in model parameter estimation and simulation of different epidemic scenarios. the doctoral research funding from ministry of human resource development, government of india, in favor of the first author is duly acknowledged. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. detail/who-director-general-sopening-remarks-at-the-mission-briefing-on-covid aerosol and surface stability of sars-cov-2 as compared with sars-cov-1 covid-19 ards: clinical features and differences to "usual" pre-covid ards contributions to the mathematical theory of epidemics mathematical biology: 1. an introduction modeling influenza epidemics and pandemics: insights into the future of swine flu (h1n1) covid-19 covid-19 pandemic 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impact of non-pharmaceutical interventions on covid-19 in 11 european countries covid-19 vaccine tracker draft landscape of covid-19 candidate vaccines immune responses in covid-19 and potential vaccines: lessons learned from sars and mers epidemic ☒ the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.☒the authors declare the following financial interests/personal relationships which may be considered as potential competing interests: key: cord-320980-srpgcy4b authors: aldila, dipo; khoshnaw, sarbaz h.a.; safitri, egi; anwar, yusril rais; bakry, aanisah r.q.; samiadji, brenda m.; anugerah, demas a.; alfarizi gh, m. farhan; ayulani, indri d.; salim, sheryl n. title: a mathematical study on the spread of covid-19 considering social distancing and rapid assessment : the case of jakarta, indonesia date: 2020-06-28 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.110042 sha: doc_id: 320980 cord_uid: srpgcy4b the aim of this study is to investigate the effects of rapid testing and social distancing in controlling the spread of covid-19, particularly in the city of jakarta, indonesia. we formulate a modified susceptible exposed infectious recovered compartmental model considering asymptomatic individuals. rapid testing is intended to trace the existence of asymptomatic infected individuals among the population. this asymptomatic class is categorized into two subclasses: detected and undetected asymptomatic individuals. furthermore, the model considers the limitations of medical resources to treat an infected individual in a hospital. the model shows two types of equilibrium point: covid-19 free and covid-19 endemic. the covid-19-free equilibrium point is locally and asymptotically stable if the basic reproduction number [formula: see text] is less than unity. in contrast, covid-19-endemic equilibrium always exists when [formula: see text]. the model can also show a backward bifurcation at [formula: see text] whenever the treatment saturation parameter, which describes the hospital capacity, is larger than a specific threshold. to justify the model parameters, we use the incidence data from the city of jakarta, indonesia. the data pertain to infected individuals who self-isolate in their homes and visit the hospital for further treatment. our numerical experiments indicate that strict social distancing has the potential to succeed in reducing and delaying the time of an outbreak. however, if the strict social distancing policy is relaxed, a massive rapid-test intervention should be conducted to avoid a large-scale outbreak in the future. coronavirus disease 2019, or covid-19, is an infectious disease caused by a new type of coronavirus named severe acute respiratory syndrome coronavirus 2 (sars-cov-2), which is known to have originated in the city of wuhan, china in december 2019 [1] . this virus is transmitted from human to human and has spread widely across china and 214 other countries and territories. it is spread through droplets 5 that exit the nose or mouth when a person infected with covid-19 coughs or exhales. these droplets then land and settle on surrounding objects and surfaces. if a person who touches any of those objects or surfaces then touches their eyes, nose, or mouth, they may become infected. transmission can also occur if a person inhales droplets from a cough or breath of a person infected with covid-19 [2]. on march 12, 2020 , the world health organization (who) declared covid-19 as a pandemic. till may 16, according to the who [3] , covid-19 generally has an incubation period of 5-6 days, with a range of 1-14 days. the symptoms of covid-19 are nonspecific and vary widely, ranging from no symptoms to severe pneumonia and eventual death. based on specific cases, some of the symptoms of covid-19 include fever, dry cough, fatigue, nasal congestion, diarrhea, and headache. people who are at high risk for severe illness include those aged 60 years and over and those with hypertension, diabetes, cardiovascular problems, cancer, or a chronic respiratory disease. the mortality rate increases with age, with the highest death rate among people aged over 80 years. in children, the disease is relatively rare. each country is at a different stage of the epidemic. in most countries where the spread of the 20 virus has caused outbreaks with exponential growth, governments have called for physical distancing and movement restrictions, commonly known as lockdown, to slow the spread of the covid-19 outbreak [4] . some of these countries include china, italy, and the united kingdom. moreover, there are some countries that have managed to handle the covid-19 outbreak without a lockdown; one such country is south korea. based on a report in [5] , south korea conducted massive numbers of polymerase chain 25 reaction tests, which reached 726,747 by may 15, 2020 [6] . in addition to death, this pandemic has also had negative psychological, economic, and social impacts globally. the covid-19 pandemic has resulted in changes in the work environment, indirectly affecting gender inequality, and an increased risk of suicide owing to lockdown, social distancing, and economic crisis [7] . panic in communities may be reduced by educating the public regarding the predictions of the 30 covid-19 epidemic and the interventions that must be conducted. communities must also unite and work together to help governments overcome the epidemic by following the guidelines provided by the relevant authorities [7] . several mathematical models have been proposed by several authors to understand the spreading mechanism of covid-19. in [8, 9, 10, 11, 12, 13, 14] , the authors proposed a modified susceptible-exposed35 infected-recovered (seir) model to understand the effects of undetected infection, hospitalization, and quarantine. the model was analyzed to determine the equilibria and the basic reproduction number. parameter estimation was conducted by the authors in [11, 12, 15, 16] using a statistical approach involving a bayesian or markov chain monte carlo method. in addition to compartmental deterministic modeling, the spread of covid-19 can be predicted statistically using a time-series approach [17, 18, 19] . a time-series model is effective owing to its ability to accommodate the factors influencing the spread of covid-19 that cannot be calculated using other statistical approaches [17] . a frequently used time-series model is the autoregressive integrated moving average (p, d, q). in this study, we propose a modified seir model that considers asymptomatic cases. these asymptomatic cases describe a hidden case in the field. as an intervention, rapid testing has been implemented 45 by many countries to detect infected individuals in this asymptomatic group. therefore, we categorize our asymptomatic individuals into two groups: detected and undetected. furthermore, we also accommodate the limitation of resources for medical treatment. this is an extremely important factor that will have an essential role in a successful eradication of covid-19. many countries are not ready for an exponential growth of covid-19 cases because many hospitals will become overwhelmed by the high 50 number of patients. to validate the model, covid-19 incidence data from the city of jakarta are used for parameter estimation. the basic reproduction number is calculated, and a sensitivity analysis of the model is conducted. the model is then used to predict the effects of social and physical distancing to stop the spreading of covid-19, and to monitor the effects of the plan to gradually ease social distancing guidelines of the government in jakarta. the remainder of this paper is organized as follows. in section 2, the construction of the sea u a d ir compartmental model is described. next, the mathematical properties of the model, such as the equilibrium points, basic reproduction number, and existence of backward bifurcation, are detailed in section 3. in section 4, we explain the real-world problem using the incidence data of jakarta, indonesia. a discussion on the basic reproduction number and the results of the sensitivity analysis are provided in section 6. finally, some conclusions are presented in section 7. the objective of our study is to analyze the effect of rapid testing and self-monitored isolation, and to predict the long-term dynamics of the incidence data of jakarta, indonesia. to achieve these purposes, our model should consider asymptomatic cases, as well as a parameter describing a rapid test intervention. to achieve this, let us divide the human population into six categories based on their health status: (i) susceptible population (s(t)) := group of susceptible individual. (ii) exposed population (e(t)) := group of individual, who already infected by covid-19, but not yet infective. (iv) asymptomatic undetected population (a u (t)) := infected population, have a capability to transmit covid-19, do not show any symptoms, and undetected by the government. this individual has a larger probability of spreading the disease compared with i and a d . (v) asymptomatic detected population (a d (t)) = individuals in this category are similar to those in a u 75 in terms of their health status, but have already been detected by the government through a swabtest, rapid test, or other tests. although this group of individuals has the capability of spreading the disease, they do not isolate themselves in a hospital owing to a limited hospital capacity. therefore, a monitored self-isolation is applicable to them. (vi) recovered population (r(t)) := recovered individuals, with a short-term immunity to covid-19. therefore : we make the following assumptions for the formulation of the model. using the transmission diagram given in figure 1 and mentioned assumptions, the model which describes the transmission of covid-19 considering rapid testing and asymptomatic cases is given by the following systems of equation. here, λ, β, µ, and φ are the natural recruitment rate, infection rate, natural death rate, and death rate from covid-19, respectively. ξ i and ξ a represent the reduction of β for i and a d , respectively, owing to isolation at home or in a hospital. furthermore, α, γ 0 , and δ denote the progression rate of covid-19 based on its incubation period, natural recovery rate, and disappearance of temporal immunity, respectively. note that p describes the proportion of exposed individuals who have progressed into asymptomatic individuals, γ 1 is the enhancement of natural recovery rate owing to treatment in a hospital, η is the rate of hospitalization from a d to i, and ν is the effort required for early detection of covid-19 infection. note that system (1) is supplemented with an initial non-negative condition: it is easy to prove that all variables in system (1) remain nonnegative for all time t ≥ 0 as long as the initial conditions are nonnegative. next, we show that our model is well-posed in biological meaning. let us consider the possible region summing all rate of change for each variable in system (1) yield clearly, if n > λ µ , then we have that dn dt < 0. since dn dt is bounded by λ − µn , we have that n (t) ≤ n (0)e −µt + λ µ (1 − e −µt ). furthermore, we have that n (0) ≤ λ µ → n (t) ≤ λ µ . also, it can be seen that 100 every solution of our covid-19 model in (1) with initial condition in d will remains in d for all t > 0. therefore, we have that d is positively invariant and attracting. hence, the covid-19 model in (1) is well-posed. taking the right-hand side of system (1) equal to zero, we have two types of equilibrium points for system (1) . the first equilibrium is the covid-19 free equilibrium point which given by using the next-generation matrix approach [20] , the basic reproduction number of system (1) is given by (see appendix a for the derivation of the r 0 ) from results in [21] , we have the local stability criteria of e 0 depend on r 0 in the following theorem. theorem 1. the covid-19 free equilibrium e * 0 is locally asymptotically stable if r * 0 < 1, and unstable otherwise. threshold quantity r 0 presents the expected number of new covid-19 infections generated from one primary infection into an absolute susceptible population during a single infection period. from the results of theorem 1, it can be shown that covid-19 can be eliminated from the population if r 0 < 1. please note that r 0 in (4) is a basic reproduction number when a rapid test is implemented. when a rapid test is not implemented into the model, then r 0 in (4) is reduced to the following: this shows the multiplication among the total human population when no covid-19 exists, ratio between the infection rate and exposed/incubation period of category e, and infection period of categories a u 110 and i. it is easy to see that reducing r * 0 is highly related to reducing β, which can be implemented by reducing the contact probability through lockdown or social distancing, and reducing ξ i by conducting a proper quarantine procedure in a hospital, such that ξ i → 0. another way to reduce r * 0 is by increasing the recovery rate owing to hospitalization (γ 1 ). further discussion on the complete r 0 is provided in section 5. the endemic equilibrium point of system (1) is given by where s * , e * , a * u , a * d , r * i * as a function of i * can be seen in appendix b, while i * is taken from the positive roots of the following third order polynomial : here, a 2 and a 1 have significantly long expressions and are therefore omitted in this study. because a 3 is always positive, and a 0 < 0 if r 0 > 1, we have the following theorem. theorem 2. system (1) has always a covid-19 endemic equilibrium pint whenever r 0 > 1. proof. let p(i) = p 1 (i) + a 0 , where p 1 (i) = a 3 i 3 + a 2 i 2 + a 1 i. let us assume that a 0 = 0 ⇐⇒ r 0 = 1. thus, p 1 (i) has i = 0 as one of the roots, whereas the other two roots can be positive, negative, or even imaginary. let us consider the most extreme case in which we have no positive roots. because we have a 3 > 0, we then have lim i→−∞ = −∞ and lim i→∞ = ∞. therefore, when a 0 < 0 ⇐⇒ r 0 > 1, we have p i as being translated downward and providing one positive root. this completes the proof. theorem 1 and 2 indicate that r 0 become the endemic threshold, since when r 0 < 1, we have that the covid-19 free equilibrium stable, but when r 0 > 1, then we have at least one positive covid-19 endemic equilibrium. furthermore, since p i is a third-order polynomial, we have at most three covid-19 endemic equilibrium. next, we analyze the possibility of having a covid-19 endemic equilibrium when r 0 < 1. the results stated in the following theorem. and proof. for the proof of the theorem, please see appendix c. as a direct consequence of theorem 3 and 2, we have the following corollary. we close this section with the following theorem, which explained the condition of backward bifurca-135 tion of system (1). proof. for the proof of theorem, please see appendix d the case study on the city of jakarta, indonesia jakarta is the capital of indonesia, with a population of 10,374,235 people in 2019, with a population growth rate of 0.94% per year. jakarta consists of 6 sub-governments, namely south, east, west, north and central jakarta, and the thousand islands. the largest population is in east jakarta at 2,916,020 people. the first covid-19 case in jakarta occurred on march 3, 2020, from a patient who had contact with japanese citizens (whom later on was confirmed to be positive for covid -19) . the social distancing rule is the most preferred action by the jakarta city government to reduce the spread of covid-19. this rule was applied from april 10, 2020, which is based on jakarta governor regulation no 33/2020. the regulation requires the closure of schools, public facilities such as malls, and many other places that 150 might potentially gather people in the same place. the data used in this article is covid-19 incident data in jakarta, from march 3 to april 10, 2020. this data can be divided into two types of active cases, namely active cases that must be treated in the hospital (i) and independent isolation at home (a d ). incident data used are given in table 1 . to conduct the parameter estimation for data-driven in indonesia, due to the short-term of the data, we consider model (1) but neglect the natural newborn, natural death rate, and the drop out rate from recovered compartment due to temporal immunities running out. based on this assumption, model (1) now read as : to conduct the parameter estimation, we divided our data based on the date of strict social distancing in jakarta first time implemented. the first interval is from march 3 until april 10, 2020. in this interval, we find the best fit parameters are : with these parameters, we have that r 0 in jakarta during this interval is 1.75. this indicates that 160 covid-19 has a big potency to be endemic in jakarta if accurate, precise, and fast policies are not carried out soon. in the second interval, after the strict social distancing implemented, we set other parameters constant, while β and η should be re-estimated. in this second interval, we have β = 0.94 × 10 −7 and η = 0.06. this data gives r 0 = 1.22 in the second interval. it can be seen that β was reduced by 34%, which indicates the effect of social distancing. on the other hand, η reduced by 68.4%, since the number 165 of infected individual keep increasing in the hospital. therefore, the transition from a d to i need to be reduced because of the limitation on the hospital capacity. the result of parameter estimation for system (1) respect to incidence data in jakarta given in figure 2 . we conduct numerical experiments in this section in three scenarios. the first experiment is to analyze 170 the elasticity of r 0 . from theorem 1, 2, 3 and 4, it is clear to see that our model is depending on the size of r 0 . understand how r 0 may be changed when parameters changed will help a more effective intervention to control the spread of covid-19. the second experiment is to see how the estimated parameter from incidence data in jakarta might exhibit a backward bifurcation when the quality and size of handled patients in the hospital getting worse. the last experiment is the sensitivity analysis to 175 determine the most significant parameters in determining the dynamics of each variable. to perform the elasticity analysis on r 0 , we calculate the normalized forward sensitivity index of r 0 using the following recipe : where ω is the set of parameters in system (1). for example, the elasticity index of r 0 respect to α is given by in a similar way, we can find all elasticity indices of r 0 for the rest of parameters. substituting parameters value found from section 4 and assuming λ = 10467629/(65 × 365), µ = 1/(65 × 365) and δ = 1/90, the elasticity value of all parameters in system (1) respect to r 0 is given in table 2. positive and negative sign in 2 indicates increasing and decreasing of r 0 respect to the parameters, respectively. therefore, we can say that r 0 increase when λ, β, ξ i , ξ a , α or p increase. in the other hand, whenever µ, η, ν, γ 0 , γ 1 , or φ increase, then r 0 will decrease. furthermore, for an example, since γ r0 ν = −0.2174486541, we have that increasing ν for 10% will reduce r 0 2.174486541%. a similar interpretation is given for the rest parameters in table 2 . it is interesting to see that the saturation 185 parameter b, which describes the capacity of the hospital or the number of a medical officer do not affect the size of r 0 since γ r0 b = 0. however, b plays an important role in determining whether the system (1) undergoes a forward or a backward bifurcation when r 0 = 1. from table 2 , it can be seen that β is the most positive significant parameter that can be used to control r 0 . therefore, social/physical distancing is a very reasonable intervention to control the spread of covid-19. furthermore, it can be seen that 190 reducing ξ i and ξ a will reduce r 0 , which indicates that the better the reduction of contact from infected humans being isolated in the hospital or at home is able to reduce r 0 . the most negative indices of r − 0 is given by µ. however, this parameter can not be changed in the field. it can be seen that the most negative indices parameters that controllable in the field is ν, which describe the rate of the rapid test. therefore, we can conclude that more massive the government to find the asymptomatic individual, 195 and then ask them to do independent isolation at home will reduce r 0 . furthermore, we can see that increasing the additional recovery rate caused by hospitalization (γ 1 ) will reduce r 0 . figure 3 presents an area for a combination between ν and β that will determine the size of r 0 . it can be seen that the increased value of β will increase r 0 , while an increased value of ν will reduce r 0 . the area of β can differ into three intervals. the first interval is when β ∈ (1.010006455 × 10 −7 , ∞). in this interval, r 0 is always larger than unity for all value of ν between 0 and 1. this means that the intervention of rapid tests will not make the covid-19 free equilibrium stable. the second interval is when β ∈ (5.421388711 × 10 −8 , 1.010006455 × 10 −7 ). in this area, a combination of ν and β should be considered carefully to reach the condition of a stable covid-19 free equilibrium point. for more precisely, for a random β = β 0 in the second interval, it needs ν > ν * where . for an example, if β 0 = 0.9 × 10 −7 , then it requires ν > 0.5630546276 to achieve a stable covid-19 free equilibrium point. the last interval is when β ∈ [0, 5.421388711 × 10 −8 ). in the third interval, the intervention of rapid test is not needed to reach a stable covid-19 free equilibrium point. however, 200 giving a positive value of ν will accelerate the time needed to achieve a stable covid-19 free equilibrium point. in the other hand, when b > b * , then system (1) undergoes forward bifurcation in r 0 = 1. using estimated parameters value 205 from previous section, we have that r 0 = 1 when β = 8.599684570 × 10 −8 , and b * = 1/573. therefore, we have that our system (1) undergoes forward bifurcation when r 0 = 1 since b = 1/10000, which illustrated in figure 4 (a) . on the other hand, backward bifurcation appears when we choose b = 1/100, which describes a low capacity of the hospital to take care of the infected individual. this is illustrated in figure 4 we compute the local sensitivity for our suggested model equations of the covid-19 in system (1). computational results here are obtained using three different techniques: non-normalizations, half normalizations and full normalizations using simbiology toolbox for matlab ; see results show that the exposed infected, asymptomatic undetected individuals, asymptomatic detected individuals, symptomatic infected and recovered individuals are more sensitive to the set of parameters {β, p, α, ν, γ 0 } while they are less sensitive to the other model parameters, see figures 7a and 7b . this gives us how public health partners pay more attention priority on interventions for such groups. as a result, identifying critical model parameters in this study based on computational simulations 225 is an effective way to further study the model practically and theoretically and give some suggestions for future improvements of the covid-19 transmissions, interventions and controlling the spread of disease. it can be concluded that the contact between person-to-person, transmission rate between exposed and asymptomatic, progression rate of incubation period, the effort for early detection test and natural recovery rate may have a great role in controlling this disease. the other factors have also role to infect 230 people in different levels, this is clearly occurred in our computational simulations. 6. analyzing the plan for gradual relaxing of strict social distancing in the city of jakarta, indonesia in figure 8 , we present estimation and actual incidence data of covid-19 in jakarta, from march 3 until may 10, and then the simulation continues for a longer period of time. note that policy from the government to conduct physical and social distancing conducted on april 10, 2020. since april 10, the reduction of incidence occurs significantly. the physical distancing in jakarta called psbb (in indonesian: pembatasan sosial berskala besar (large-scale social restriction)). it can be seen that if social distancing intervention maintained for a longer period of time, then the outbreak of covid-19 in jakarta will be reduced significantly, and delayed. with this intervention, the hospital can treat infected 240 individuals maximally. recently, the jakarta government planning to relax the strict social distancing policy. this policy will be conducted in five-phase: based on this description, it is assumed that at the transmission rate will increasing step by step from β = 1.05 × 10 −7 (when social distancing implemented in april 10) to β = 1.44 × 10 −7 (early infection period of data). to handle this, we assume β as a step function as follows : using above β(t), the dynamic of i and a d (t) is given in figure 9 . it appears that the policy of relaxing the strict social distancing can result in an increasing number of new infections. however, it will not be as it was before social distancing was implemented. the possible explanation is because of the policy to relaxing the social distancing is too early to take place. based on the elasticity analysis of r 0 in the previous section, the rapid test is one of a promising alternative for the eradication of covid-19. therefore, we simulate how if the policy to relaxing the strict social distancing combined with more rapid test intervention. to simulate this scenario, we use the same β as in (11) , but increasing the rapid test and hospitalization rate twice larger. the results are shown in figure 10 . it can be seen that increasing rapid tests and hospitalization as a tolerance policy of relaxing 260 the social distancing success to reduce the number of the infected population. unfortunately, when the social distancing completely stops (after july 27), then the effect of rapid test and hospitalization no longer able to compensate for the impact of relaxing the strict social distancing in a purpose to reduce the spread of covid-19. therefore, the number of the infected population start to re-increase and produce a new outbreak. 265 figure 10 : long-time simulation for prediction of incidence of covid-19 in jakarta with easing the social distancing policy combined with more massive rapid test and hospitalization. a new deterministic compartmental model was constructed in this study to evaluate the spreading of covid-19 among the human population. the model considers many important factors, such as hidden cases, rapid testing to trace hidden cases, limitation of medical resources, social distancing, quarantine/isolation, and parameter estimation for the incidence date from the city of jakarta, indonesia. 3. the model undergoes a backward bifurcation phenomenon when the associated r 0 is less than unity, and the saturation parameter for hospitalization (b) is larger than a specific threshold (b * ). this means that whenever the medical resources are insufficient (larger b), the risk of the appearance of a 280 backward bifurcation increases; this is related to the existence of the covid-19-endemic equilibrium despite r 0 < 1. therefore, the success of the intervention also depends on the initial condition when the response is implemented. the study shows that increasing the capacity of a hospital or providing a considerably better quality of treatment in the hospital increases the probability of avoiding a backward bifurcation. numerical experiments of the model based on the incidence data of the city of jakarta suggest the following. 1. the basic reproduction number in jakarta during the early spread of covid-19 is 1.75, which is larger than unity. this means that covid-19 will persist in the population if no intervention is implemented by the government or the community. 2. from an analysis of the elasticity of r 0 , we observe that the infection rate (β) is the most significant controllable parameter to reduce r 0 , followed by the effectiveness of self-isolation and quarantine. smaller values reduce r 0 , thereby increasing the chance of eradicating covid-19 from the community. 3. the government must be careful when relaxing the policy of strict social distancing, particularly in 295 terms of when it should be initiated. mistakes in the prediction of when to start relaxing the social distancing policy can affect the emergence of a second outbreak. a rapid test-based intervention has been proven to have potential in reducing r 0 as an alternative approach, instead of relying solely on a lockdown or strict social distancing. significantly better results might be obtained if these interventions can be implemented simultaneously. during this pandemic, it is important to avoid overconfidence in the capabilities of the model for the long term prediction of the data. many assumptions were made in the study to simplify the model without compromising the main objective. although many important qualitative features were found from the model applied in this study, several limitations can still be found, and an alternative way to improve the model should be developed. one of the limitations in this study is that the applied model 305 does not include the spatial spread of covid-19 and the possibility of a relapse for recovered individuals. further research is required in this field to address this limitation and a better modeling is needed to understand and anticipate the outcome of the covid-19 pandemic. the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. the basic reproduction number r 0 is calculated here with taking e(t), a u (t), a d (t), i(t) as the infected compartments and then using notation in [20] we define t as the transmission matrix and σ as the transition matrix. the transmission and transition matrices of the corresponding linearized subsystem are four-dimensional, with it can be seen that t has three zero rows in row 2, 3 and 4. therefore, the auxiliary matrix e is given 320 by e = [1, 0, 0, 0] t . hence, we have the next-generation matrix is given by the basic reproduction number as the spectral radius of k is given by the covid-19 endemic equilibrium point is given by , , and 325 k s = p (ξ a ν + η + µ + γ 0 ) (i * bµ + i * bφ + i * bγ 0 + µ + φ + γ 0 + γ 1 ) + i * bη µ ξ i + η µ ξ i , k r = η µ γ 0 + η ν γ 1 + η γ 0 2 + η γ 0 γ 1 + µ 2 γ 0 + µ ν γ 0 + 2 µ γ 0 2 + ν γ 0 2 + ν γ 0 γ 1 + γ 0 3 + γ 0 2 γ 1 + ν γ 0 . where i * is taken from the positive roots of p(i). to show the possible existence of the covid-19 endemic equilibrium point when r 0 < 1, we will analyze the sign of ∂i ∂r0 when r 0 = 1 and i = 0. if the sign is negative, then we have at least one covid-19 endemic equilibrium point when r 0 < 1 but close to 1. first step, let rewrite each coefficient of a i in p(i) as a function of r 0 . to do this, let substitute this β * into a i , then taking the implicit derivative of i respect to r 0 from p(i), we get : . is always negative , then we have that ∂i ∂r0 < 0 ⇐⇒ a 1 (r 0 ) < 0, or equivalently b < b * where b * = β((1 − p) ξ i (µ + γ 0 ) (η + µ + ν + γ 0 ) − p (µ + φ + γ 0 + γ 1 ) (ν ξ a + η + µ + γ 0 ) − η ν ξ i )k b1 (µ + δ) ((1 − p) (µ + γ 0 ) (η + µ + ν + γ 0 ) − η ν) k b2 and β = β * . with k b1 = (η + µ + ν + γ 0 ) ((1 − p) α δ φ γ 0 + α δ µ γ 1 p) + (µ + φ + γ 0 + γ 1 ) (µ + ν + γ 0 ) η µ 2 + µ 2 (µ + γ 0 ) + α η µ + γ 0 (φ + γ 0 + γ 1 ) (ν + γ 0 ) + α δ µ 2 + µ ν + µ φ + µ γ 0 + ν φ η + µ (µ + φ + γ 0 ) (µ + ν + γ 0 ) + (µ + ν + φ + 2 γ 0 + γ 1 ) η + µ 2 µ + η (φ + γ 0 + γ 1 ) (ν + γ 0 ) + α 2 µ 5 γ 0 2 γ 1 + α µ 3 ν + α µ 3 φ + 3 α µ 3 γ 0 + α µ 3 γ 1 + α µ γ 0 3 + α µ 2 ν φ + 2 α µ 2 ν γ 0 + α µ 2 ν γ 1 + 2 α µ 2 φ γ 0 + 3 α µ 2 γ 0 2 + 2 α µ 2 γ 0 γ 1 + α µ ν φ γ 0 + α µ ν γ 0 2 + α µ ν γ 0 γ 1 + α µ φ γ 0 2 k b2 = 2 (1 − p) λ α ξ i (µ + γ 0 ) (η + µ + ν + γ 0 ) − λ α (2 µ + 2 φ + 2 γ 0 + γ 1 ) (ν ξ a + η + µ + γ 0 ) p + µ (2 µ + 2 φ + 2 γ 0 + γ 1 ) (µ + ν + γ 0 ) (µ + η + γ 0 ) (µ + α) . we have ω 1 = ω 5 α δ (µ + η + ν + γ 0 ) (µ pγ 1 + (1 − p) φ γ 0 ) + c ((p − 1) (µ + γ 0 ) (µ + η + ν + γ 0 ) − ν η) (δ + µ) α µ , ω 2 = (γ 0 + γ 1 + µ + φ) (µ + ν + γ 0 ) (η + µ + γ 0 ) ω 5 α (ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 )) , ω 3 = (η + µ + γ 0 ) ω 5 p (γ 0 + γ 1 + µ + φ) ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 ) , ω 4 = ω 5 ν p (γ 0 + γ 1 + µ + φ) ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 ) , − ω 5 ((µ + η) (µ + ν) γ 0 − η ν γ 1 − ((+µ + ν + η) γ 1 ) γ 0 ) ν η + (1 − p) (µ + γ 0 ) (η + µ + ν + γ 0 ) (δ + µ) where c is long expression with positive sign, and 340 v 1 = v 6 = 0, v 2 = (1 − p) pξ i (µ + γ 0 ) (µ + η + ν + γ 0 ) + η ν ξ i + p (γ 0 + γ 1 + µ + φ) (ν ξ a + η + µ + γ 0 ) ξ i (µ + ν + γ 0 ) (η + µ + γ 0 ) (α + µ) v 3 = γ 0 2 + (ν ξ a + η + 2 µ + φ + γ 1 ) γ 0 + µ 2 + (ν ξ a + η + φ + γ 1 ) µ + (ν ξ i + φ + γ 1 ) η + ν ξ a (γ 1 + φ) v 5 next we will calculate the values of a and b. since v 1 = v 6 = 0, we only need to compute the partial derivatives of f 2 , f 3 , f 4 , f 5 at the dfe. for system (d.1) the associated non-zero partial derivatives of f 2 , f 3 , f 4 , f 5 are given by ∂ 2 f 2 ∂x 1 ∂x 3 = ∂ 2 f 2 ∂x 3 ∂x 1 = β, ∂ 2 f 2 ∂x 1 ∂x 4 = ∂ 2 f 2 ∂x 4 ∂x 1 = β ξ a , ∂ 2 f 2 ∂x 1 ∂x 5 = ∂ 2 f 2 ∂x 5 ∂x 1 = β ξ i , ∂ 2 f 5 ∂x 5 ∂x 5 = 2 γ 1 b. = v 2 ω 1 2 ν p (γ 0 + γ 1 + µ + φ) β ξ a ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 ) + 2 (η + µ + γ 0 ) p (γ 0 + γ 1 + µ + φ) β ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 ) + v 2 ω 1 2 β ξ i + 2 γ 1 b. for the sign b, the association non-vanishing partial derivatives of f 2 , f 3 , f 4 , f 5 are it also follows that = v 2 (γ 0 + γ 1 + µ + φ) (µ + ν + γ 0 ) (η + µ + γ 0 ) λ α (ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 )) µ + ν p (γ 0 + γ 1 + µ + φ) λ ξ a (ν η + (1 − p) (µ + γ 0 ) (µ + η + ν + γ 0 )) µ coronavirus disease 2019: tinjauan literatur terkini report of the who-china joint mission on coronavirus disease 2019 (covid-19) covid-19 strategy update testing on the move south koreas rapid response to the covid-19 pandemic coronavirus disease-19, republic of korea covid 2019-suicides: a global psychological pandemic, brain, behavior, and immunity mathematical modeling of covid-19 transmission dynamics with a case study of wuhan mathematical modeling of the spread of the 365 coronavirus disease 2019 (covid-19) taking into account the undetected infections. the case of china a covid-19 epidemic model with latency period epidemic trend and transmission risk of sars-cov-2 after 370 government intervention in the mainland of china: a mathematical model study estimation of the transmission risk of the 2019-ncov and its implication for public health interventions an updated estimation of the risk of transmission of the novel coronavirus a model based study on the dynamics of covid-19: prediction and control healthcare impact of covid-19 epidemic in india: 380 a stochastic mathematical model prediction of the covid-19 spread in african countries and implications for prevention and controls: a case study in south africa, egypt, algeria, nigeria, senegal and kenya estimation of covid-19 prevalence in italy, spain, and france, science of the total envi-385 ronment covid-19 disease outbreak forecasting of registered and recovered cases after sixty day lockdown in italy: a data driven model approach real-time forecasts and risk assessment of novel coronavirus (covid-19) 390 cases: a data-driven analysis the construction of next-generation matrices for compartmental epidemic models credit author statement dipo aldila conceptualization, methodology, analysis, software, investigation, validation, writing, review and editing salim investigation, data collection the coefficient b is clearly positive; the presence of backward bifurcation in the model (1) is determined by the sign of coefficient a. therefore, to conduct a backward bifurcation a should be positive, which gives us b > b * where b * is given in theorem 3. the backward bifurcation is shown using the concept of center manifold theory on system (1) . 330 to use the center manifold theory, we make the following changes in variables. let s = x 1 , e = x 2 , a u = x 3 , a d = x 4 , i = x 5 and r = x 6 and let β the bifurcation parameter. denote x = (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) t and dx dt = (f 1 , f 2 , f 3 , f 4 , f 5 , f 6 ) t as given belowthe jacobian of system (d.1) evaluated at the disease-free equilibrium, is given byconsider the case r 0 = 1. suppose that β is chosen as a bifurcation parameter. setting r 0 = 1 and solving for β givesthe jacobian matrix d x f evaluated at β = β * has a simple zero eigenvalue and the other eigenvalues 335 having negative real parts. computing the right and the left eigenvector of d x f . the right and the left eigenvector associated with zero eigenvalue denoted respectively by ω = (ω 1 , ω 2 , ω 3 , ω 4 , ω 5 , ω 6 ) key: cord-355267-ndzgxk0k authors: kassa, semu m.; njagarah, john b.h.; terefe, yibeltal a. title: analysis of the mitigation strategies for covid-19: from mathematical modelling perspective date: 2020-06-05 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109968 sha: doc_id: 355267 cord_uid: ndzgxk0k in this article, a mathematical model for the transmission of covid-19 disease is formulated and analysed. it is shown that the model exhibits a backward bifurcation at [formula: see text] when recovered individuals do not develop a permanent immunity for the disease. in the absence of reinfection, it is proved that the model is without backward bifurcation and the disease free equilibrium is globally asymptotically stable for [formula: see text]. by using available data, the model is validated and parameter values are estimated. the sensitivity of the value of [formula: see text] to changes in any of the parameter values involved in its formula is analysed. moreover, various mitigation strategies are investigated using the proposed model and it is observed that the asymptomatic infectious group of individuals may play the major role in the re-emergence of the disease in the future. therefore, it is recommended that in the absence of vaccination, countries need to develop capacities to detect and isolate at least 30% of the asymptomatic infectious group of individuals while treating in isolation at least 50% of symptomatic patients to control the disease. a novel coronavirus, named a severe acute respiratory syndrome coronavirus 2 (sars-cov-2; previously known as 2019-ncov), was identified as the infectious agent causing an outbreak of viral pneumonia in wuhan, china, in december 2019 [47] . the world health organization (who) medical team codenamed the new outbreak caused by sars-cov-2 as "coronavirus disease 2019 (covid19) ". the infection is in the same category as the severe acute respiratory syndrome (sars) which emerged in southern china in 2002, spreading to up to 30 countries, with a total of 8,098 cases and claiming 774 lives [35] . covid-19 is also in the same category as the middle east respiratory syndrome (mers) which was first identified in saudi arabia in 2012, and ended up spreading to 27 countries around the world, reaching a total of 2,519 cases confirmed and claiming up to 866 lives [45] . since january 2020, an increasing number of cases confirmed to be infected with covid-19 have been detected outside wuhan, and currently the infection has spread all over the world. as of may 28, 2020, (09:09 gmt), covid-19 had affected all continents including island nations (213 countries and territories as well as 2 international conveyances), with the total number of cumulative infections globally standing at 5,807,012 cases and 357,800 deaths [52] , and the numbers still increasing. the major portal of entry of the virus into the body is the tissue lining the t-zones of the face (including the nose, eyes and mouth). the infection is characterised by loss of the sense of smell (a condition referred to as hyposmia/anosmia), taste and poor appetite. although, such conditions have been observed in covid-19 patients, many carriers of the infection may not show any severe symptoms like fever and cough but have hyposmia, loss of taste and loss of appetite. whereas knowledge of the virus dynamics and host response are essential for formulating strategies for antiviral treatment, vaccination, and epidemiological control of covid-19, estimation of changes in transmission over time can provide insights into the epidemiological situation and help to identify whether public health control measures are having a measurable effect [5, 39] . the analysis from mathematical models may assist decision makers to estimate the risk and forecast the potential spread of the disease in the population. understanding the transmission dynamics of the infection is crucial in designing alternative intervention strategies [29] . in general, by approaching infectious kassa) mechanisms and adherence to the advice given by public health agencies. in particular, the people in some parts of asia have fully embraced the measures while many in other parts of the world were very much hesitant to use them. for example, wearing a face mask everyday in public appearances is like a ritual in most of the countries in southeast asia, while the same is considered as a bad gesture in many of the other parts of the world [50] (even if it is now becoming a "new normal" also everywhere in the globe). one key difference between these societies and the people in the west is that, the communities in the southeast asia have experienced similar disease outbreaks before and the memories are still fresh and painful [50] . that means, recent history of a similar event plays a role in behavioural change of the population especially at the beginning of the outbreak in addition to the perceived threat from the disease. therefore, in this paper, we consider a mathematical model that takes into account 1. the transmission dynamics of covid-19 similar to the sirs model, 2. the contribution of the asymptomatic infectious individuals in the transmission dynamics of the disease in the population, 3. the effect of indirect transmission of the disease through the environment, 4. behavioural change of individuals in the society to apply self-protective measures, and 5. the intensity of historical events from recent similar outbreaks. by analysing the proposed basic mathematical model, the effect of each of these factors is investigated in terms of their contribution to the control strategies of the disease. moreover, the use of isolation or quarantining and strict social distancing measures are also considered as mitigation strategies, and a comparative study is made for different scenarios. the layout of this paper is as follows: the model is described and formulated in the next section. its qualitative analysis is presented in section 3. estimation of the parameters and the sensitivity analysis of the reproduction number of the model with respect to involved parameters are discussed in section 4. numerical simulations of the model with some assumed intervention scenarios are also presented and analysed in this same section. concluding remarks of the study are given in sections 5. in this section, we present a mathematical model for the transmission dynamics of covid-19 which spreads in a population. the susceptible individuals can be infected through either direct contact with infectious individuals or indirect contact with the novel coronavirus infected environment. the population under consideration is grouped into disjoint compartments. individuals who are susceptible to the disease and without formal awareness about the prevention mechanisms or who did not decide to use any protective mechanisms are grouped in the class. individuals who are susceptible but are aware of and decide to apply any of the existing protective mechanisms after receiving public health information on how to protect themselves from the novel coronavirus infection are placed in the class. covid-19 infected individuals who are asymptomatic and symptomatic are grouped in classes and , respectively. some studies consider the asymptomatic class as the "exposed" class (see for instance [9] ). but since the individuals in this group are known to be infectious and some of them also recover from the disease without going through the group [8] , we used a name "carrier" to avoid confusion. the class contains the recovered individuals from covid-19. finally, denotes the amount of the novel coronavirus pathogen that contaminates the environment due to shedding by covid-19 infectious individuals. in the analysis of the model, we intentionally excluded the actual exposed class for mathematical simplicity. however, a 5 days incubation period is taken into consideration in the numerical simulation part of this paper. by combining the direct and indirect ways of transmission, the force of infection will have the form where = + + + + and is the concentration of the novel coronavirus in the environment which increases 50% chance of triggering the disease transmission. the proposed flow diagram for the transmission dynamics of covid-19 is depicted in figure 1 while the description of each of the state variables is given in table 1 . pathogen concentration on contaminated surfaces or objects in the environment. table 1 description of the model variables the dynamics of the pandemic is described by using the following system of differential equations (see table 2 for the description of the involved parameters): where with 0 is the value of the force of infection corresponding to the threshold infectivity in which individuals start reacting swiftly (that means, the point at which the behaviour change function changes its concavity). we append the following nonnegative initial conditions to the system (2): (0) = 0 , (0) = 0 , (0) = 0 , (0) = 0 , (0) = 0 , and (0) = 0 . rate of disease transmission from the environment the pathogen concentration in the environment that yields 50% of chance for a susceptible individual to catch the viral infection from the environment modification parameter (transmission of relative to ) rate of loosing immunity after recovery natural death rate fraction of recovered individuals moving into the class after loosing immunity table 2 description of the model parameters in this section, we study the quantitative and qualitative analysis of the model system eq. (2). we begin by determining the biologically feasible set for the model (2) . the following theorem implies that the solutions of (2) are nonnegative and bounded from above, provided that the initial conditions are nonnegative. proof: the proof of this theorem is outlined in appendix a. to determine the equilibrium solutions, we set the right-hand-side of eq. (2) equal to zero and obtain then, the disease-free equilibrium (dfe) is found to be the basic reproduction number, which is very important for the qualitative analysis of the model, is determined here below by using the method of the next generation matrix used in [11, 42] . for the model under consideration, using the notation = ( , , ), we have the vector functions with 1 = + + and 2 = + + represent the rates at which the disease compartments increase and decrease in size due to the infection, respectively. then the next generation matrix is where the basic reproduction number denoted by  0 is defined as the average number of secondary cases produced in a completely susceptible population by a typical infected individual during its entire period of being infectious [11, 42] . mathematically,  0 is the spectral radius of in eq. (7) and after further simplification, we obtain the next result is a direct application of theorem 2 in [42] . (2) is locally asymptotically stable whenever  0 < 1 and unstable if  0 > 1. the epidemiological implication of theorem 3.2 is that the transmission of covid-19 can be controlled by forcing the dynamics through its parameter values so that  0 < 1 if the initial total numbers in each of the subpopulation involved in eq.(2) are in the basin of attraction of  0 . to ensure that elimination of the disease is independent of the initial size of the subpopulation, the disease-free equilibrium must be globally asymptotically stable when  0 < 1. this is what we present here below. where = 1 1 the proof of theorem 3.3 is carried out using the center manifold theory in [6] and is given in appendix b. in the above theorem (theorem 3.3), if the parameter describing the waning of immunity, , is zero, we can observe that both and are negative. hence, eq. (9) fails to be true. in this case, we give below a theorem which asserts the global stability of the dfe of the model. the disease-free equilibrium of system (2) in the case when = 0 is globally asymptotically stable for  0 < 1. to prove the theorem for the case = 0, we use kamgang-sallet stability theorem stated in [24] . let = ( 1 , 2 ) with 1 = ( , , ) ∈ r 3 and 2 = ( , , ) ∈ r 3 . then the system (2) can be written aṡ and analysis of the mitigation strategies for covid-19 we show that the five sufficient conditions of kamgang-sallet theorem (in [24] ) are satisfied as follows. 1. the system (2) is a dynamical system on ω. this is proved in theorem 3.1. 2. the equilibrium * 1 is gas for the subsysteṁ 1 = 1 ( 1 , 0)( 1 − * 1 ). this is obvious from the structure of the involved matrix. 3. the matrix 2 ( ) is metzler (i.e., all the off-diagonal elements are nonnegative) and irreducible for any given ∈ ω. again this is straight forward from the formulation of matrix 2 ( ). 4. there exists an upper-bound matrix̄ 2 for the set this can also be verified by checking the eigenvalues of̄ 2 under the condition of  0 < 1. hence, by the kamgang-sallet stability theorem [24] , the disease-free equilibrium is globally asymptotically stable for  0 < 1. □ the reality behind theorem 3.4 is that, if immunity is permanent ( = 0), coronavirus will be effectively controlled in the community by reducing  0 effectively to a value less than unity. the novel coronavirus being a new strain of coronaviruses, information about the dynamics of the infection is still evolving. biological studies of parameter values describing the vital dynamics of the infection are still ongoing as more laboratory tests become available. although some studies have been done on the early dynamics of the disease most especially on data from wuhan, extensive reading reveals that some of the disease dynamics parameters are highly variable and some processes are not fully explored. in this work, we use new cases data from hubei province of china extracted from who situation reports 1-57 [47] , i.e. for the period january 21, 2020 to march 17, 2020. we ought to fit the proposed model to the extracted data and estimate the unknown parameters. the total population of hubei province was estimated as 59.2 million. the life expectancy of hubei province varies depending on the area of dwelling (i.e urban or rural) as well as gender [48] . for urban dwellers, the average life expectancy is estimated to be 75.68 years (with an average being 73.72 years for men and 77.79 years for women). the life expectancy of china of the year 2019 was estimated to be 76.79 years whereas that for the year 2020 is estimated as 76.96 years [32] . owing to the negligible difference in the provincial and countrywide value, we use the country life expectancy for the year 2019 which gives an average mortality rate of = 3.57 × 10 −5 per day. the recruitment rate is thus given as = × 0 , where 0 is taken to be the total population size, 59.2 million. the average time period taken for symptoms to appear after exposure is observed to vary considerably, with ranges between 2-14 days [7] , 2-24 days [51] , and some outliers going to up to 27 days. the observed median incubation period was nearly 5 days [16] . the time to recovery from the onset of symptoms varies depending on the seriousness of the infection. individuals presenting mild illness were observed to recover in an average period of 2 weeks while those presenting serious/critical illness recovering in about 3 to 6 weeks. for our parameter estimation and simulations, we consider a nominal value of 0.0476 day −1 (corresponding to 3 weeks) estimated from an interval (0.0238, 0.0714). we note that a patient is considered recovered: (1) if two swab tests taken in a time interval of at least 24 hours both test negative, (2) if the time taken after the end of respiratory symptoms and fever is at least 72 hours. the waning of the immunity after recovery is estimated to range between 4 months to 1 year, which gives an interval for as (0.00274, 0.00824) day −1 . for our simulation, we consider a nominal value of = 0.00274 day −1 (approximately one year). we propose that at the end of the epidemic, at least 20 − 65% of the recovering population will learn from the experiences during the infection and even when the acquired immunity wanes, such individuals will become susceptible individuals with past history/knowledge of the disease. the rate of recovery for the symptomatic individuals ( ) in wuhan varied considerably but majority of individuals who recovered from the virus were discharged from hospital after 2 1 2 weeks [4] . however, the patients in wenzhou-china stayed in hospital for 27 days (0.037 per day) on average. in the model fitted on the early trends data from wuhan-china [28] , the recovery rate obtained for symptomatic cases was 0.0897 per day (accounting for 10 days to recovery). the rate of recovery ( ) for carrier individuals is expected to be higher [9] . according to the who situation report [46] , it is estimated that up to 80% of covid-19 cases are asymptomatic or show mild symptoms, 15% show severe symptoms and up-to 5% end up with critical infection and require oxygen or a ventilator. the proportion of individuals who do not show symptoms or have mild symptoms can be as high as 94% [18] . for our model fitting, we use a range of (0.65, 0.86) for both and with selected initial values within the prescribed interval. although hubei province was put on a lockdown on january 23, 2020, the first major decline in the number of new confirmed cases was only observed on february 20, 2020 (situation report 31 [47] ), approximately 1 month after the lockdown was imposed. from february 14, 2020, the method of identification of new cases was revised to include both cases confirmed through laboratory tests and clinical observations. as such, there was an observed spike in the number of new cases on february 14, 2020 to 4823 compared to 1508 cases (reported on february 13, 2020) and 2420 cases (reported february 15, 2020). applying the above described set of assumptions in the bounds for some of the parameters, we optimized the model output to fit the daily new cases data reported from the hubei province, china. the parameter values for which the model best fits to the incidence data are given in table 3 . figure 2 shows the plot of the reported new-cases data together with the incidence of the disease obtained from the model. as we can observe from the graph, the model analysis of the mitigation strategies for covid-19 slightly overestimates the reported data except for the two highest points. in addition, since our model does not assume any control measure at this stage while the reported data after the 31st day may represent the effect of the strict lockdown measure taken by the authorities, the parameters estimated give a good result. we note however that, in some recent work [37, 36] , the use of fractional-order calculus is recommended to get better data fit. when we calculate the value of  0 from eq. (8) using the estimated parameters given in table 3 , we obtain  0 ≈ 2.91, which is within the range of values reported in [10, 53] . we examine the sensitivity of  0 to variations in parameter values and establish the significance of the sensitivity indices. we used the latin hypercube sampling (lhs) scheme, which is an efficient stratified monte carlo sampling that allows for simultaneous sampling of the multi-dimensional parameter space [19] . for each run, 1000 simulations were done and partial rank correlation coefficients (prccs) [1] calculated between each of the selected input parameters and the disease threshold. the prccs indicate the degree of effect each parameter has on the outcome, which in this case is the disease threshold. the sign of the prcc identifies the specific qualitative relationship between the input parameter and the output variable. the positive value of the prcc of the variables implies that when the value of the input parameter increases, the future number of cases will also increase. on the other hand, processes underlying the parameters with negative prccs have a potential to contain of the number of cases when enhanced. the results of sensitivity analysis are indicated in figure 3 (a) and the box plot (figure 3(b) ) gives the five-number summary for the computed disease threshold value from the sampled parameter space. the processes described by parameters 1 , 2 , and with the greatest positive prccs have the greatest potential of worsening pandemic when increased. on the other hand, parameters (ℎ and ) with negative prccs have the greatest potential in helping contain the infection when maximised. in this respect, we note that increasing social/physical distancing directly reduces 1 as this lowers the likelihood of a susceptible individual getting in contact with a potentially infected individual. in addition, practising good hygiene (such as regularly washing hands, using sanitisers to disinfect the infected environment and avoiding touching the t-zones of the face) is associated with lowering the likelihood of contracting the virus from infected surfaces. anything contrary to the above increases the likelihood of getting the infection through the two aforementioned routes. we further note that practising good hygiene also involves the infected individuals reducing the shedding of the virus into the environment. it is evident from the results in figure table 3 . the processes underlying the parameters 1 , 2 , and have the greatest potential of making the epidemic worse if increased, whereas processes described by and ℎ have the greatest potential of containing the epidemic when enhanced. table 4 that reducing the rate at which the virus is shed into the environment is significant in reducing the severity of the problem. from the five number summary of the results in figure 3 (b), the lower quartile of the computed values of  0 is about 2, the median around 2.9 and the upper quartile of about 4. the obtained value of  0 is within the range of 3.11(95% , 2.39 − 4.13) obtained in the early studies in [38] . we note that for a selected combination of underlying processes much higher values of  0 can be obtained, which is an indication of possible worsening of the situation. in a similar way, we observe that for particular underlying processes (a selected combination of parameters) the value of  0 can be reduced to values below one. as indicated in [1] , we note that although some parameters in the model may have very small magnitudes of prccs (non-monotonically related to the disease threshold output), they may still produce sizeable changes in the disease burden. to identify the most important parameters in containing or aggravating the epidemic, we computed p-values for the simulated parameters using fisher's transformation [1] . we note that the computed prccs are bounded between the interval [−1, 1]. for this matter, some sampling distribution of variables that are highly correlated is skewed. the fisher's transformation ( ) = 0.5 log 1+ 1− is used to transform the skew distribution to a normal distribution and then compute p-values for each of the parameters based on the prccs [1] . the prccs for the parameters together with their corresponding p-values are indicated in table 4 . we carry out pairwise comparison of the significant parameters (whose p-values are less than 0.05, see table 4 ) to ascertain whether the processes described by the compared parameters are different. we computed the p-values for the different pairs of significant parameters while accounting for the false discovery rate (fdr) adjustment and the results are given in table 5 . the major question posed at this point is: are the different pairs of significant parameters different after fdr adjustment? based on the fdr adjusted p-values in table 5 , the compared pair of parameters are rendered to be different if their p-value is less than 0.05 and not different otherwise. we summarise our results in table 6 , where "true" indicates that the compared parameters are significantly different and "false" indicating otherwise. table 6 are the parameters different after fdr adjustment? we observe that the more sensitive parameters are also significantly different (see table 6 ) except for the 1 − pair which may not necessarily be related. we examine effect of variation of the sensitive parameters on the reproduction number ( 0 ). the results of the variation of parameters with more negative prccs are indicated in the bar graphs in figure (4) . from figure 4 , it is evident that the decay of the virus from the environment (figure 4(b) ) which can be accelerated by disinfecting surfaces reduces the value of  0 and consequently the disease burden. in addition, we observe that an increased proportion of individuals with knowledge of similar infections from the past that are practising selfprotection and preventive measures (see figure 4 (a)) is important in slowing down the infection at the initial stage. such proportions of individuals would normally have knowledge about prevention and control mechanisms of the infection just at the onset of the disease. we observe in figure 5 that the increase in person-to-person contact, 1 (figure 5(a) ), poor personal hygiene, 2 ( figure 5(b) ), and the rate of shedding of the virus into the environment by both carriers (figure 5(c) ) and symptomatic individuals ( figure 5(d) ) increase the value of  0 and therefore the disease burden. it is evident that the most effective way of containing the infection is by minimizing contact, which is why in most cases imposing a lockdown becomes an effective way of slowing the spread of the infection. in addition, good hygiene practices by all individuals are twofold: (1) avoiding touching surfaces, always washing hands with soap and water, or using alcohol based hand sanitizer, which reduced the likelihood of contracting the pathogen from the environment; (2) those who are sick with symptoms like cough and flu, ought to use masks, when they cough or sneeze, must do so in a sanitary tissue which must then be properly disposed off. we also note that hygienic practices without social/physical distancing may not significantly slow down the infection. in summary, we observe that it is possible to reduce the value of  0 to a value less than unity by reducing only the value of 1 below 0.1 (see figure 5(a) ). this observation is in direct agreement with mitigation approaches that are aimed at minimising human-to-human contact (such as social distancing and imposing a lockdown). therefore, the table 3 paremeter 1 is more influential in the model and can also play a significant role in eradication of the disease. the other parameters (see figures 4, 5(b) , 5(c) and 5(d)) may reduce the value of  0 significantly when applied in combination but not as independent mitigation processes. there are various intervention mechanisms for covid-19 that are being implemented in different parts of the world. the strategies differ from country to country depending on the scientific information available to decision makers. to investigate the outcomes of the mitigation strategies, we include the isolated and/or quarantined classes to the model system (2) . we assume that the individuals in the asymptomatic class ( ) are detected at a rate of and placed in isolation 1 class, while the individuals in the symptomatic class ( ) are identified at a rate of and quarantined in 2 class. moreover, the system is formulated as a mixed-delay system of differential equations, where the time delay is assumed to account for the incubation period of the disease. then the system becomes where the force of infection is now modified to with representing the average percentage of contacts reduced due to the social distancing measures and representing the total average rate (or percentage) of disinfecting the environment. in the modified model system (12) , the parameters analysis of the mitigation strategies for covid-19 table 3 . for the simulation purpose of this study, we considered five different cases or scenarios of how to apply the interventions. the strategies described in each of the cases below are in addition to the awareness creation for voluntary self-protective mechanisms which are widely communicated through various media outlets. here, we assume that the average effectiveness of the self-protective measures is 15% (as estimated from the data and reported in table 3) , and the individuals who decided to use any one of them are strict in following the appropriate rules. here below, we consider each of the scenarios for mitigation strategies case by case. in this case, we assume that about 40% of the symptomatic infectious individuals and only 1% of the asymptomatic infectious individuals are detected and quarantined. this scenario is based on the assumption that among the people in the class only about 40% show "above mild" symptoms and hence visit health care facilities, while the remaining individuals in this class (nearly 60% of them) remain at home or at large in the society. then, through contact tracing mechanisms corresponding the hospitalized individuals, some people will be traced and tested, thereby about 1% of the total asymptomatic individuals can be detected. a similar scenario is being applied currently in some sub-saharan african countries. the time profile in figure 6 shows the situation described in case 1. from this graph we can observe that the infection stabilizes around its endemic equilibrium, which is nearly at 5000 cases. (this number depends on the initial conditions and the demographic variables of the population under study.) this shows that the disease persists in the population. analysis of the mitigation strategies for covid-19 figure 6 : dynamics of the disease with no additional intervention is applied. in this case, we assume that strict and longer time (6 weeks) of social distancing rules are enforced by the government nearly 4 weeks after the first positive case of covid-19 is reported in the community. we assume for simulation purpose that the implementation of the intervention strategy is divided into the following 4 time phases. phase 1: the first phase in this case, is 24 days long (measured starting from the time the first positive case of covid-19 is reported). during this phase, because of lack of information and the nature of the infection, we assume (as in case 1) that only 40% of symptomatic infectious individuals and 1% of the asymptomatic infectious individuals are detected and quarantined. phase 2: the second phase is assumed to last for 6 weeks (42 days). during this period, it is also presumed that; * 80% of the symptomatic class and 30% of the asymptomatic class are detected and quarantined, * a mandatory strict social distancing rule is imposed, which is assumed to have a 70% reduction of effective contacts of individuals in the society, * environmental disinfection is widely carried out, which is assumed to result in a 50% reduction in the rate of infection from the environment, and to contribute about the same percent impact in increasing the rate of decay of the pathogen from the environment. phase 3: the third phase is assumed to be 4 weeks (28 days) long, and is characterised by the partial lifting of the 'lockdown' imposed in phase 2. during this period, it is assumed further that; * 70% of the symptomatic class and 25% of the asymptomatic class are detected and quarantined, * a relaxed social distancing rule is exercised, which is assumed to have a 25% reduction of effective contacts of individuals in the society, * environmental disinfection is partially carried out, which is assumed to have an impact of reducing the rate of infection from the environment by 30% and increasing the rate of decay of the pathogen from the environment by the same 30%. phase 4: the last and fourth phase is the time when the social distancing rule is fully lifted. due to the lesson learnt from the previous phases, we assume that the following interventions will continue during this period as well; * 70% of the class and 10% of the class are detected and quarantined, analysis of the mitigation strategies for covid-19 possible to significantly reduce the infection to a level where it cannot be a public threat. otherwise, any lower proportion of this effort will imply the resurgence of a second wave of the infection in the community. therefore, to contain covid-19 in every given community, public health authorities need to work more on the detection and quarantining of the asymptomatic infectious individuals. in this case, we assume that there is no lockdown imposed but only a large number of testing is applied to detect and quarantine a larger proportion of infected cases. if it is possible to intensify the effort of tracing the asymptomatic infectious individuals and be able to quarantine at least 35% of them continuously and effectively, our simulation shows that there is a possibility for the disease to be contained without imposing the strict lockdown rule on the total population. the plot in figure 10 shows the time profile of the count of the infected groups while about 50% of the individuals from class are effectively quarantined (for example inside appropriate health facilities). we can observe that this intervention strategy can also produce the required result in containing the outbreak as some countries (like south korea) are currently following this pattern. in this case, we assume that the length of the lockdown period is nearly twice to the scenario in case 2. however, the effort in detecting the asymptomatic infectious individuals is kept at the minimum level. this scenario is more applicable in highly resource constrained countries as the current cost of testing is high. in this case, it is assumed that the length of the duration of each phase (except for phase 2) is the same as that given in case 2. more still, it is supposed that; 1. the conditions in phase 1 remain the same, 2. phase 2 lasts 11 weeks with 50% of the symptomatic individuals and 5% of the asymptomatic individuals detected and quarantined. moreover, strict social distancing rules are still in place with an effect of reducing 70% of human contacts and 50% of environmental variables, 3. phase 3 lasts 4 weeks (the same as in case 2), but with 50% of individuals in the class and 10% of individuals in the class detected and quarantined, while partial social distancing rules are in place with an effect of reducing 25% of human contacts and 25% of environmental variables, 4. phase 4 continues with detecting and quarantining 50% of members in the class and 10% of members in the class, while the other mandatory intervention are lifted. the time profile of the infection following the scenario in case 5 is plotted in figure 11 . the simulation for this scenario shows that even if we increase the length of lockdown period to 11 weeks (like it was practised in the hubei province, china) the disease may re-emerge after some period of time. however, the heights of the peaks in the subsequent waves of the disease are much lower than the first peaks. therefore, once again, unless the authorities apply some kind of strict contact tracing mechanism and conduct enough testing to detect and isolate up to 30% of the asymptomatic infectious individuals, the disease persists in the community with multiple subsequent waves. in general, from the simulations, we can observe that in all of the above scenarios a transition from one phase to the other intervention phase is characterised by a surge in new cases. however, the number will eventually go down if the intervention in the immediate next phase is effective, or else the disease re-emerges in the population. analysis of the mitigation strategies for covid-19 figure 11 : dynamics of the disease with at most 10% of the population in the class and at least 50% of the class are detected and quarantined just after phase 1 period, with strict social distancing rule imposed for 11 weeks. we presented a mathematical model for the dynamics of covid-19 whose first cases were reported in december 2019 in wuhan-china. the model incorporates a behaviour change function to account for the proportion of individuals who decided to use any of the self-protective measures and adhered to them. in addition, it also considers a proportion of individuals with a history/knowledge of similar infections from the past and practice necessary protective measures right from the onset of the epidemic. the model also accounts for asymptomatic carriers of the infection as well as the concentration of the pathogen within the environment. the basic properties of the model including well-posedness, the disease free equilibrium and its stability, model basic reproduction number as well as the existence of backward bifurcation were examined. to estimate the parameter values, the model was fitted to the data on daily new cases reported in who situation reports 1-57 [47] , which accounts for the period from january 21, 2020 to march 17, 2020. from the nominal values from the data fitting, we obtained a reproduction number,  0 ≈ 2.9 (2.1-4) which compares well with the values of  0 obtained in other researches, for instance, (2.24 − 3.58) [55] and 3.11(95%ci, 2.39 − 4.13) [38] . from our sensitivity analysis simulations, we observed that for some given parameter combinations, the value of  0 can be reduced to below 1, and similarly for values much higher than 4. we observed that if the recovering individuals do so with permanent immunity ( = 0), then reducing the reproduction number to a value below unity is enough to contain the infection. on the other hand, if recovering individuals do so with temporary immunity ( ≠ 0), the proposed model exhibits backward bifurcation, which implies that reducing the value of  0 below 1 is not enough to contain the infection. by applying the latin hypercube sampling scheme, we observed that if the disease is to be easily contained, measures such as; physical/social distancing (which reduces the rate of disease transmission directly from human to human), improved personal hygiene (which reduces the rate of disease transmission from the environment to humans), and minimal shedding of the pathogen into the environment by both asymptomatic and symptomatic individuals, have the greatest potential of slowing the epidemic when enhanced. we further observed that increased decay of the pathogen from the environment (achieved by disinfecting surfaces) alone is less significant in reducing/curbing the number of cases. we further observed that having high numbers of people with knowledge from previous similar infections, that are practising the prescribed self-protective measures can delay/slow down the otherwise potentially explosive outbreak. consequently, the daily number of cases is kept at low manageable levels. in addition, increasing the average effectiveness of the self-protective measures and adherence to such measures is vital in realising low peaks of the number of cases. furthermore, due to the absence of vaccination or any approved medication, developing capacity to detect carrier groups is very important. from our results, it is recommended that countries should develop capacities to identify and quarantine at least 30% of carriers as well as at least 50% of symptomatic cases if the infection is to be controlled. our model predicts a possible resurgence of the number of cases, if the asymptomatic cases are still many by the time disease spread curbs/lockdown measures are lifted. in addition, we observe from simulations that although disease spread curbs (such as a lockdown measure) may be imposed, their real impact on the number of new cases may only be realised after approximately 21 days, and the reduction (when it appears) could be sharp in the case of a strict lockdown measure with high impact in reducing effective contact between individuals in the population. when providing the mitigation strategies, we did not account for the delay between the actual incidence and the point when cases have been confirmed since the actual parameters describing such a delay are not known. in health systems where testing of suspected cases is done after individuals show symptoms or on demand, it is likely to have a big gap between the actual incidence and confirmation of new cases. the impact of the delay between actual incidence and confirmation of case can be explored in future work. in addition, we assumed that all individuals who recover, do so with the same level of immunity. however, this may not necessarily be the case since immunity of individuals is affected by a number of factors, including age, cortisol levels and nutrition among others. the impact of differentiated levels of immunity on the disease dynamics and potential resurgence of the epidemic can be explored in the future when relevant data becomes available. our model did not include the possibility of vaccination or treatment. we, however acknowledge their importance in controlling the infection. therefore, optimal control of the infection in the presence of these mitigation strategies can be explored in future works when relevant data becomes available. since the disease has been observed to affect age groups differently, it is plausible to consider age-structured models to better understand the effect of the disease in the respective age groups. likewise, from the fourth equation of (2), we obtain ′ = (1 − ) + (1 − )(1 − ) + − ( + + ) ≥ −( + + ) . solving (15) similarly, using the last two equations of eq. (2), we have and because ( ), ( ), ( ), and ( ) are nonnegative for ≥ 0. solving eqs. (17) and (18) gives and respectively. thus, any solution of eq. (2) is nonnegative for ≥ 0 and any initial condition in ω. finally, the total number of the population ( ) at time is governed by thus, for the initial data 0 ≤ (0) ≤ , by gronwall inequality, we obtain moreover, for the environmental variable , we have because ( ) and ( ) are less than for all ≥ 0. applying again the gronwall inequality, for 0 ≤ (0) ≤ ( + ) , leads into combining the above two steps and theorem 2.1.5 in [11] for the existence of unique bounded solution, we infer that any solution of eq. (2) is nonnegative and bounded. hence, eq. 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coronavirus: why some countries wear face masks and others don't coronavirus incubation period covid-19 coronavirus pandemic nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study duration of antibody responses after severe acute respiratory syndrome preliminary estimation of the basic reproduction number of novel coronavirus (2019-ncov) in china, from 2019 to 2020: a data-driven analysis in the early phase of the outbreak the analysis of the mitigation strategies for covid-19 figure 7 : dynamics of the disease with a mandatory 6-weeks lockdown and a 4-weeks of partial social distancing is imposed as described in case 2 * environmental disinfection is partially carried out, which is assumed to have an impact of reducing the rate of infection from the environment by 20% and increasing the rate of decay of the pathogen from the environment by the same 20%.the time profile of the disease dynamics after implementing the above described interventions strategy is plotted in figure 7 . the figure shows that the count of the infected individuals decreases down to nearly zero in phases 2 and 3, but the disease resurges back into the society soon after. however, the peak of the second wave looks to be much smaller than the first one. that means, the intervention mechanisms described in the above 4 phases of this case are not enough to contain the disease, and unless some additional intervention mechanisms are developed the disease persists in the society. in this case, we assume that early action with shorter time social distancing rule is applied. in this scenario, it is assumed that the interventions described in case 2 started half way through the time that phase 2 was implemented in case 2. that means, the implementation of the interventions described in the four phases of case 2 is assumed to be followed, but the length of the time in phases 1 and 2 is reduced as described below. otherwise, all the details of the interventions in case 2 are kept the same. the time profile for this set of interventions is given in figure 8 . the general behaviour of the graph in figure 8 is the same as that of figure 7 . however, this strategy has an advantage in significantly reducing the height of the first peak. the height of the subsequent peaks are found to be the same unless some additional measures are taken after phase 3.unfortunately, the strategies in both of the above two scenarios (case 2 and case 3) do not help to fully contain the disease once it spreads in the population. as it can be seen from figures 7 and 8 , another wave of outbreaks of the disease emerge at a later stage. here, we can see that the asymptomatic infectious individuals play the greater role in becoming the major source for the second wave. therefore, if there is a possibility to track and detect people with asymptomatic infection, and if they can be effectively quarantined for the required period of time, then there is a possibility for the disease to be contained. as it can be observed from figure 9 , if we can increase the rate of detecting and quarantining the asymptomatic individuals to a proportion of about 30%, it is appendix a. proof of theorem 3.1the proof of theorem 3.1 is outlined here below based on the following two steps. first, we show that all solutions of eq.(2) are nonnegative as required in [3, 41] . to show that the state variables and of the model are positive for all ≥ 0, we use proof by contradiction. we suppose that a trajectory crosses one of the positive cones at times 1 or 2 such that:using the first equation of eq. (2), the first assumption leads towhich contradicts the first assumption that ′ ( 1 ) < 0. thus, ( ) remains positive for all ≥ 0. here, 1 is chosen so that our point to be on the positive axis of ( ) so that ( 1 ) is positive.using the second equation of eq. (2), ′ ( 2 ) = ℎ + + > 0, which also contradicts the assumption ′ ( 2 ) < 0. hence, ( ) remains positive for all ≥ 0. based on the third equation of eq. (2),analysis of the mitigation strategies for covid-19 proof: the theorem is the direct application of theorem 4.1 in [6] . to check the existence of backward bifurcation of the model eq. (2) at  0 = 1, we use the center manifold theorem [6] . for this purpose, we introduce the following change of variables.so that = 1 + 2 + 3 + 4 + 5 , = 1 ( 4 + 3 ) + 2 6 6 + , and ( ) = 0 + .moreover, by using the vector notation = ( 1 , 2 , 3 , 4 , 5 , 6 ) , the model eq.(2) can be written in the form ′ ( ) = = ( 1 , 2 , 3 , 4 , 5 , 6 ) as follows: where, 1 = + + , 2 = + + , and 3 = + . when  0 = 1 and 1 is considered as a bifurcation parameter, from (8) we get further more, 1 < * 1 if and only if  0 < 1 and 1 > * 1 whenever  0 > 1. the jacobian of the system (26) at the associated ( 0 ) isthe transformed system eq. (26), with 1 = * 1 , has a non-hyperbolic equilibrium point such that the linear system has a simple eigenvalue with zero real part and all other eigenvalues have negative real parts. hence, the centre manifold theory [6] can be used to analyse the dynamics of the model eq. (26) near 1 = * 1 . by using the notation in [6] , the following computations are carried out.the right-eigenvectoranalysis of the mitigation strategies for covid-19 associated with the zero eigenvalue of ( 0 ) such that ( 0 ). = 0 at 1 = * 1 is given byof ( * ) such that the right-eigenvector and the left-eigenvector need to satisfy the condition . = 1.the bifurcation coefficient at the dfe ( 0 ) is given by analysis of the mitigation strategies for covid-19thus, the bifurcation coefficient , can be positive for the right choice of the parametric values that satisfy the condition in eq. (9) .the second bifurcation coefficient is given by clearly, > 0 because and are positive. when = 0, and in (30) are negative and in (32) is negative as well. hence, by theorem 4.1 in [6] , the model will not exhibit a backward bifurcation at  0 = 1. ☒ the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.☐the authors declare the following financial interests/personal relationships which may be considered as potential competing interests: semu mitiku kassa, hatson j. b. njagarah, yibeltal a. terefe. key: cord-322862-dcb237an authors: bekiros, stelios; kouloumpou, dimitra title: sbdiem: a new mathematical model of infectious disease dynamics date: 2020-04-23 journal: chaos solitons fractals doi: 10.1016/j.chaos.2020.109828 sha: doc_id: 322862 cord_uid: dcb237an a worldwide multi-scale interplay among a plethora of factors, ranging from micro-pathogens and individual or population interactions to macro-scale environmental, socio-economic and demographic conditions, entails the development of highly sophisticated mathematical models for robust representation of the contagious disease dynamics that would lead to the improvement of current outbreak control strategies and vaccination and prevention policies. due to the complexity of the underlying interactions, both deterministic and stochastic epidemiological models are built upon incomplete information regarding the infectious network. hence, rigorous mathematical epidemiology models can be utilized to combat epidemic outbreaks. we introduce a new spatiotemporal approach (sbdiem) for modeling, forecasting and nowcasting infectious dynamics, particularly in light of recent efforts to establish a global surveillance network for combating pandemics with the use of artificial intelligence. this model can be adjusted to describe past outbreaks as well as covid-19. our novel methodology may have important implications for national health systems, international stakeholders and policy makers. the world health organization (who) reported on december 31, 2019 cases of pneumonia of undetected etiology in the city of wuhan, hubei province in china. a novel coronavirus (covid19) was identi…ed as the source of the disease by the chinese authorities on january 7, 2020. eventually, the international committee on taxonomy of viruses on 11 february, 2020 named the severe acute respiratory syndrome coronavirus as sars-cov-2 [1] . concerns on public health were dispersed on a global scale about potentially infected countries. the virus might have been generated by animal populations and transmitted via the huanan wholesale market [2] [3] [4] albeit not proven, while clinical …ndings demonstrated that international spread was caused mainly by commercial air travel [4] [5] [6] [7] . the who declared sars-cov-2 a pandemic on march 11, 2020 . throughout the globe, huge e¤orts were in progress to limit the spread of the virus and …nd medications and vaccines. however, the scienti…c community could not fully comprehend the dynamics of the spread [8] [9] [10] . several outbreaks of infectious diseases have occurred in the past with immense impact on public health. for instance, the severe acute respiratory syndrome (sars) occurred in 2003, the swine ‡u in 2009 and the middle east respiratory syndrome coronavirus (mers) in saudi arabia in 2012, which still survives at a sub-critical level causing some peaks [11] [12] [13] [14] . additionally, the ebola epidemic emerged between 2014 and 2016 and caused over 28,000 cases in west africa [15] . its temporal decline coincided with the outbreak of zika virus in brazil [16] . consequently, the outbreak of severe pathogens such as the aforementioned, require global interdisciplinary e¤orts in order to decode key epidemiological features and their transmission dynamics, and develop possible control policies. insights from mathematical modelling can be extremely bene…cial. indeed, dealing with infectious diseases from a mathematical angle could reveal inherent patterns and underlying structures that govern outbreaks. stakeholders utilize available data from current and previous outbreaks in order to forecast infection rates, identify how to restrict the spread of diseases, and eventually introduce vaccination policies that will be most e¤ective. epidemiology is essentially a biology discipline concerned with public health and as such, it can be heavily in ‡uenced by mathematical theory. most phenomena observed at population level are often very complex and di¢ cult to decode just by observing the characteristics of isolated individuals [17] . statistical analyses of epidemiological data help to characterize, quantify and summarize the way diseases spread in host populations. interestingly, mathematical models appear as e¢ cient ways to explore and test various epidemiological hypotheses, mostly due to the existence of ethical and practical limitations when deducting experiments on living populations. models provide conceptual results on e.g., the basic reproduction number, threshold e¤ects or herd immunity. one additional element of epidemiological modeling is the link with data via statistical methods. although simple epidemiological models are often used, viral and bacterial infections commonly require increased complexity. there are many models in the literature on single epidemics, endemic diseases and spatiotemporal disease dynamics. the aim is to develop robust public health policies in de…ning optimal vaccination strategies. our study presents for the …rst time a new stochastic mathematical model for describing infectious dynamics and tracking virus temporal transmissibility on 3-dimensional space (earth). this model can be adjusted to describe all past outbreaks as well as covid19 . as a matter of fact, it introduces a novel approach to mathematical modelling of infectious dynamics of any disease, and sets a starting point for conducting simulations, forecasting and nowcasting investigations based on real-world stereographic and spherical tracking on earth. in short, a single epidemic outbreak as opposed to disease endemicity occurs in a time span short enough not to have the demographic changes perturbing the dynamics of contacts among individuals. the most popular mathematical model in this category is the susceptible-infected-recovered (sir) epidemic model, in which all individuals of a …nite population interact in the same manner. individuals at time t are susceptible (s), infected (i) or recovered (r). the …nal size of the epidemic will strongly depend upon the initial conditions of the number of susceptible and infected individuals as well as the infection parameter. the …nal size distribution of the simple sir model in most cases is bimodal presenting two local maxima. this bimodal feature is caused by two likely scenarios; either the epidemic dies out quickly infecting few individuals, or it becomes long-lasting and substantial. however, stochasticity in the form of random walk transmission mechanisms related to spreading processes has never been explored in epidemiology widely [18] [19] [20] . for example, in computer science, some arti…cially created viruses propagate randomly by a plethora of online communication channels. to the best of our knowledge, we are the …rst to scrutinize extensively the role of random walks in epidemic spreading and provide the proper mathematical arsenal to model it robustly. interestingly, random walk paths converge in distribution to brownian motions [21] . in this work, we assume that a biological carrier of virus y is at position x(t) at any given time t. we call this the inaugural contamination focal point on earth. the path de…ned by its motion is considered infectious. x t ; t 0 is supposed to follow a brownian motion on a 2-dimensional sphere s 2 of radius a, i.e the sphere in r of dimension 3. we consider this a proxy for earth, spreading via spherical and stereographic coordinates. next, using the laplace-beltrami operator we construct the brownian motion infectious process on the 2-dimensional sphere, using spherical and stereographic coordinates as local coordinates. we evaluate explicitly certain quantities related to generated di¤usion processes. in what follows, we compute the transition and transmission density for the x t ; t 0, and we derive the stochastic di¤erential equations that govern the infectious disease dynamics for x t ; t 0 in those local coordinates. we continue with the calculation of expectations of outbreak exit times in time and space of speci…c domains, possessing certain symmetries. moreover, the moment generating functions are produced. in mathematical terms, we derive the stochastic re ‡ection principle on s 2 for the infectious disease transmission process. re ‡ection points can be extremely useful to calculate the distribution functions of certain temporal quantities related to the dynamics. additionally, we evaluate boundary local times of …rst hitting of the outbreak for an epidemic or a hybrid endemicepidemic model. hence, biological carrier(s) of a virus (infectious individuals) are tracked at any given time on earth coordinates, and the path(s) de…ned by each infectious dynamical motion. in the following two chapters we present a thorough literature review and a state-of-the-art analysis in order to pose clearly our novel approach optimally among the various methodologies followed thus far. the rest of paper is organized as follows: section 2 provides a brief literature review, past and recent, of mathematical epidemiology. section 3 presents the state-of-the-art, and focal concepts and term de…nitions required to introduce our novel model. it also states which category the new model falls into, according to the o¢ cial taxonomy of the various methodologies already utilized so far in the relevant literature. next, section 4 exposes in detail the mathematical formulation of the model. lastly, section 5 discusses proposed policies and future paths of research, and concludes. the beginning of mathematical modeling in epidemiology dates back to 1766, when bernoulli developed a mathematical model to analyze the mortality of smallpox in england [22] . bernoulli used his model to show that inoculation against the virus would increase the life expectancy at birth by about three years. a revision of the main …ndings and a presentation of the criticism by d'alembert, appears recently in dietz and heesterbeek [23] . lambert in 1772 as well as laplace in 1812 extended the bernoulli model by incorporating age-dependent parameters [24, 25] . however, further systematic research was absent until the beginning of the twentieth century with the pioneering work of ross in 1911, which is considered the inaugural study of modern mathematical epidemiology [26] . ross used a set of equations to approximate the discrete-time dynamics of malaria via a mosquito-based pathogen transmission [27] . importantly, the past century has witnessed the rapid emergence and development of substantial theories in epidemics. in 1927, kermack and mckendrick [28] derived the celebrated threshold theorem, which is one of the key results in epidemiology. it predicts -depending on the transmission potential of the infectionthe critical fraction of susceptibles in the population that must be exceeded if an epidemic is to occur. kermack and mckendrick published three seminal papers, establishing what is called the deterministic compartmental epidemic modelling [29] [30] [31] , wherein they addressed the mass-action incident in disease transmission cycles, assuming that the probability of infection of a susceptible is analogous to the number of its contacts with infected individuals. this deterministic representation was in line with the law of mass action [32] introduced by guldberg and waage in 1864 and renders the basic most commonly used sir model, which assumes homogeneous mixing of the contacts and conservation of the total population and low rates of interaction. macdonald extended ross's model to explain in depth the transmission process of malaria. utilizing modern computer power, the mathematical model for the dynamics and the control of mosquito-transmitted pathogens provided robust results in real-word applications. overall, the family of models they introduced is known by now as ross-macdonald models [33] . moreover, the classic work of bartlett [34] examined models and data to explore the factors that determine disease persistence in large populations. arguably, a landmark book on mathematical modelling of epidemiological systems was published by bailey [35] and highlighted the importance of public health decision making [36] . given the diversity of infectious diseases studied since the middle of the 1950s, an impressive variety of epidemiological models have been developed. in addition, we should highlight the 19th century works by enko [37] [38] [39] , who …rst published a probabilistic model for describing the epidemic of measles, yet in discrete time. this model is the precursor of the popular reed-frost chain binomial model introduced by frost in 1928 in biostatistics'lectures at johns hopkins university [40] . it assumes that the infection spreads from an infected to a susceptible individual via a discrete time markov chain, and set the basis of contemporary stochastic epidemic modelling, on which we will also focus in our present work. moving to the 21st century, we mention some interesting works; xing et al., [41] introduced a mathematical model on h7n9 in ‡uenza among migrant and resident birds, domestic poultry and humans in china. in this study they concluded that temperature seasonality might be a source of the disease, yet they suggested for the …rst time that controlling markets could help controlling outbreaks. lee and pietz [42] developed a mathematical model for zika virus using logistic growth in human populations. sun et al., [43] proposed a transmission model for cholera in china and observed that reducing the spread requires extensive immunization coverage of the population. nishiura et al. [44] developed a zika mathematical model which exhibited the same dynamics as dengue fever, and khan et al. [45] introduced a model whereby a saturation function describes well the typhoid fever dynamics. gui and zhang [46] , developed a modi…ed sir model demonstrating nonlinearities in recovery rates. their model exhibited a backward bifurcation phenomenon, which in turn implied that a plain reduction of the reproduction number less than one, was not rendered su¢ cient to stop the disease spread. li et al. [47] constructed a multi-group brucellosis model and found out that the best way to contain the disease is to avoid cross infection of animal populations. moreover, yu and lin [48] identi…ed complex dynamical behaviour in epidemiological models and particularly the existence of multiple limit cycle bifurcations using a predictor-prey model. shi et al. [49] proposed an hiv model with a saturated reverse function to describe the dynamics of infected cells. additionally, bonyah et al. [50] developed a sir model to study the dynamics of buruli ulcer and suggested policy measures to control the disease. lastly, zhang et al. [51] developed a model with a latent period of the disease wherein the person is not infectious with saturated incidence rates and treatment functions, called seir epidemic model. the sir model is the basic one used for modelling epidemics. kermack and mckendrick created the model in 1927 [29] in which they considered a …xed population with only three compartments, susceptible (s), infected (i) and recovered (r). there are a large number of modi…cations of the sir model, including those that include births and deaths, the sir without or with vital dynamics, a model where upon recovery there is no immunity called sis and where immunity lasts for a short period of time, called sirs model. furthermore, a model where there is a latent period of the disease and where the person is not infectious is indenti…ed as seis and seir respectively, or where infants can be born with immunity is named msir. also, we mention the herd immunity model [52, 53] . overall, the transmission mechanism from infective populations to susceptibles is not well-comprehended for many infectious diseases. interactions in a population are very complex, hence it is extremely di¢ cult to capture the large scale dynamics of disease spread without formal mathematical modeling. an epidemiological model uses microscopic e¤ects -the role of an infectious individual -to forecast the macroscopic behavior of disease spread via a population. deterministic models do not incorporate any form of uncertainty and as such, they can be thought to account for the mean trend of a process, alone. on the other hand, stochastic models describe the mean trend as well as the variance structure of the underlying processes. two basic types of stochasticity are commonly used: demographic and environmental. within the context of demographic stochasticity, all individuals are subject to the same potential events with the exact same probabilities but di¤erences in the fates of population individuals. disease propagation in large populations obeys to the weak law of large numbers, thus e¤ects of demographic stochasticity can be decreased signi…cantly, and many times a deterministic model becomes more suitable. however, random events cannot be neglected and a stochastic model can be equally appropriate. environmental stochasticity involves variations in the probability associated with an exogenous event. model parameters of stochastic models are characterized by probability distributions, whilst for …xed parameter values deterministic models will always produce the same results, except when chaotic behaviour emerges. in the classic sir model it is assumed that the individuals leave the infectious class at a constant rate and even if this assumption seems most intuitive, it is not always the most realistic, regarding the duration individuals stay infective [54] [55] [56] . usually, random variables describe the time of recovery since infection. for discrete random variables (e.g., number of individuals) it is easy to de…ne a probability distribution, whilst for continuous variables the time of recovery since infection is modelled. often, in this last category it is not possible to …x a probability as there is in…nity of such times. hence, we …rst de…ne a cumulative distribution and then express a probability density function from this cumulative distribution. infectious periods are exponentially distributed with a mean infectious duration, however as frequently real data does not back up this assumption, we rather use constant duration. to account for such more realistic distributions, the assumption that the probability of recovery does not depend on the time since infection, is often relaxed. then, a common method of stages can be used to replace the infective compartment by a series of successive ones, each with an exponential distribution of the same parameter, leading to a total duration of the infectious period modelled by a gamma distribution [17] . epidemic models presented above describe rapid outbreaks during which normally the host population is assumed to be in a constant state. for longer periods, deaths and births feed the population with new susceptibles, possibly allowing the disease to persist at a constant prevalence. this state renders an endemic state in the population [17] . in this case, we account for birth and death rate of the host population, whereby a good approximation is that the population size n=s+i+r is constant. when deterministic dynamics prevail a threshold on the value of the basic reproduction number exists. conditions regarding this number guarantee the disease persistence, but in epidemic models such persistence can be dependent upon the magnitude of the stochastic ‡uctuations around the steady-state equilibrium. furthermore, many times diseases are in an endemo-epidemic state. as endemic models exhibit damped oscillations which converge toward an endemic equilibrium, this equilibrium can be weakly stable with perturbations (intrinsic or extrinsic), which excite and sustain the inherent oscillation behaviour [57] . this behaviour is due to heterogeneity that is added temporally to the coe¢ cient of transmission, spatially in the context of meta-populations, or by cohorts for age-structured models. lastly, heterogeneity can be added statistically in case of stochastic versions. for example, a stochastic version of the endemic sir model can utilize a markov process, in which the future is independent of the past given the present, with a state space de…ned by the number of individuals in each of the three classes, and changes in the state space characterized by probabilistic transition events. and as future events are independent on past events, the time to the next event follows a negative exponential distribution. over the years, a vast number of mathematical modeling approaches has been proposed, tackling the problem from di¤erent perspectives. the prevailing taxonomy proposed by siettos and russo (2013) [58] encompasses three general categories: (1) statistical methods of outbreaks and their identi…cation of spatial patterns in real epidemics, (2) state-space models of the evolution of a "hypothetical"or on-going epidemic spread, and (3) machine learning methods, all utilized also for predictability purposes vis-à-vis an ongoing epidemic. in particular, the …rst category includes i) regression methods [59] [60] [61] [62] [63] [64] , ii) times series analysis, namely arima and seasonal arima approaches [65] [66] [67] [68] , iii) process control methods including cumulative sum (cusum) charts [69] [70] [71] [72] [73] [74] and exponentially weighted moving average (ewma) methods [75, 76] , as well as iv) hidden markov models (hmm) [77, 78] . the second category incorporates i) "continuum"models in the form of di¤erential and/or (integro)-partial di¤erential equations [79] [80] [81] [82] , ii) discrete and continuous-time markov-chain models [83] [84] [85] , iii) complex network models which relax the hypotheses of the previous stochastic models that interactions among individuals are instantaneous and homogeneous [86] [87] [88] [89] [90] [91] , and iv) agent-based models [92] [93] [94] [95] . lastly, the third category includes well-known machine learning approaches widely used in computer science, such as i) arti…cial neural networks [96] , ii) web-based data mining [97, 98] and iii) surveillance networks [99] , to name a few. for the …rst time in the relevant literature, we introduce a new stochastic model laying in the intersection of categories (1) and (2), called "stereographic brownian di¤usion epidemiology model (sbdiem)". figure 1 presents a graphical overview of the models utilized so far, and the "positioning" of our novel approach for modelling infectious diseases. insert figure 1 here let n 2 n = f1; 2; 3; : : :g. the n-dimensional sphere s n with center (c 1 ; :::; c n+1 ) and radius a > 0 is (de…ned to be) the set of all points x = (x 1 ; x 2 ; :: de…nition 4.2. we consider r n r n+1 to be the hyperplane given by x n+1 = 0. for convenience, we will let (x 1 ; x 2 ; :::; x n ; x n+1 ) be coordinates on r n+1 and ( 1 ; 2 ; ::: the stereographic projection coordinates of s n is the map : s n f0; 0; : : : ; 2ag ! r n given by this map de…nes coordinates ( 1 ; 2 ; :::; n ) on s n so that the point (x 1 ; x 2 ; :::; x n ; x n+1 ) of s n has coordinates ( 1 ; 2 ; :::; n ) ; where the inverse map is given by the points of the 2-sphere with center at the origin and radius a may also be described in spherical coordinates in the following way: x 1 = a cos sin ' where 0 < 2 and 0 ' : ; 3. the family fu ; x g is maximal relative to conditions 1 and 2. each pair (x ; u ) is called a coordinate chart on m: (for more details see [101] ) let g = [g ij ] be the riemmanian metric tensor on a riemmanian manifold m . this means that, in any coordinate chart (x 1 ; x 2 ; :::; x n ) on m , the length element can be computed by given local coordinates (x 1 ; : : : ; x n ); we can easily compute the matrix g = [g ij ] by the inner product (see [101] ). we denote by g ij the elements of the inverse matrix g 1 . de…nition 4.5. the laplace-beltrami operator m associated with the metric g is de…ned by where f is a c r function on m: in this work we are interested in the case where m = s 2 , i.e., the 2-dimensional sphere. we will denote the corresponding laplace-beltrami operator of s 2 by 2 or just using the spherical coordinates. we have x = @x @ = ( a sin sin '; a cos sin '; 0) x ' = @x @' = (a cos cos '; a sin cos '; a sin ') hence the laplace-beltrami operator of a smooth function f on s 2 is where x 1 = and x 2 = ': thus in case where the function f is independent of the laplace-beltrami operator of f is generally the laplace-beltrami operator of a smooth function f on s n is we have x k = @x @ k = = 8a 2 1 k ( 2 1 + + 2 n + 4a 2 ) 2 ; : : : ; ; g ij = 0; if i 6 = j; and p det(g) = (4a 2 ) n 2 1 + + 2 n + 4a 2 n : therefore, the laplace beltrami operator of a smooth function f on s 2 , using stereographic projection coordinates is de…nition 4.6. let m be a riemannian manifold (see de…nition 1.5) and its corresponding laplace-beltrami operator. any function p (t; x; y) on (0; 1) m m satisfying the di¤ erential equation @p @t where x is acting on the x-variables and the initial condition p (t; x; y) ! x (y) as; t ! 0 + (4.10) (where x (y) is the delta mass at x 2 m ) is called a fundamental solution of the heat equation (4.9) on m . the smallest positive fundamental solution of the heat equation (4.9) and (4.10) is the heat kernel on m . it has been proved by j. dodziak [102] , that the heat kernel always exists, and is smooth in (t; x; y). moreover the heat kernel possesses the following properties. de…nition 4.7. a process x t ; t 0 is a markov process if for any t; s 0, the conditional distribution of x t+s , given the information about the process up to time t, is the same as the conditional distribution of x t+s , given x t . de…nition 4.8. the brownian motion x t , t 0, on a riemannian manifold m is a markov process with transition density function p (t; x; y) the heat kernel associated with the laplace-beltrami operator. remark 4.1. in the case where m = s n , n 2, the transition density function p (t; x; y) of the brownian motion x t depends only on t and d(x; y), the distance between x and y. thus in spherical coordinates it depends on t and the angle ' between x and y. hence, the transition density function of the brownian motion can be written as p (t; x; y) = p(t; '); (4.13) where p(t; ') is the solution of and lim t!0 + aa n 1 p(t; ') sin n 1 (') = ('): here ( ) is the dirac delta function on r and a n denotes the area of the n-dimensional sphere s n with radius a. it is well known that [103] a n = 2 n+1 2 a n ( n+1 2 ) ; where ( ) is the gamma function. more precisely a n = 2 n+1 2 a n ( n 1 2 )! for n odd (4.17) a n = 2 n ( n 2 1)! n 2 a n (n 1)! for n even (4.18) remark 4.2. the fact that s n is a compact and smooth manifold implies that (4.14) -(4.15) has a unique positive solution which also satis…es z s n p (t; x; y)d (y) = 1: furthermore, as t ! 1, p (t; x; y) approaches the uniform density on s n , i.e. p (t; x; y) ! c; where c = 1 a n : in the sequel for typographical convenience we will write x t instead of fx t g t 0. in this section we shall represent the transition density function p(t; ') of the position x(t) of a biological carrier (infected individual) of virus y at any given time t. for the next sections we suppose that the infected individual is at position x(t) at any given time t, namely the path de…ned by its motion is considered infectious. x t ; t 0 describes a brownian motion on a 2-dimensional sphere s 2 of radius a. the solution of the di¤usion equation is given by the function see [104] . here p 0 n ; n = 0; 1; 2; : : : is the associated legendre polynomials of order zero, i.e. proof. first we prove that p(t; ') satis…es the di¤erential equation we have that where k(t; ') is given by the (4.24), therefore however from the (4.22) @k @t i.e. @p(t; ') @t = 1 2a 2 sin ' cos ' @p(t; ') @' + sin ' @ 2 p(t; ') @' 2 : we recall the following well-known fact be such that a(x) = (x) t (x) is positive de…nite. if x t is the ito di¤ usion process (4.28) then, its generator a is given by the formula conversely, the operator a given above is the generator of di¤usion (4.28). for the proof see [105] . the generator of brownian motion on s 2 in spherical coordinates is therefore, the brownian motion on s 2 in spherical coordinates is the solution of the stochastic di¤erential equation where x t = ( (t); '(t)) : case of sterographic projection coordinates expressed in stereographic projection coordinates, the generator of brownian motion on s 2 is hence, the brownian motion on s 2 in stereographic projection coordinates is the solution of the stochastic di¤erential equation where x t = (x 1 (t); x 2 (t)) : we recall some basic de…nitions. de…nition 4.9. a measurable space f ; fg is said to be equipped with a …ltration {f t }, t 2 [0; +1), if for every t 0 {f t } is a -algebra of subsets of such that f t f and for every t 1 ; t 2 2 [0; +1) such that t 1 < t 2 , we have that f t1 f t2 . (i.e. {f t } is an increasing family of sub -algebras of f). let x t be the brownian motion in s n and d s n a domain. then is a stopping time with respect to f t = f x s j 0 s tg, called the exit time on @d . and then the expectation of t is given by proof. based on [106] , we have the unique solution of the di¤erential equation thus therefore, however (see [105] ) consequently, finally, and then the expectation of t is given by proof. according to [106] , e ' [t] satis…es the poisson equation on d with dirichlet boundary data. by uniqueness is the unique solution of the di¤erential equation (4.31), i.e., hence from (4.34) however u( ; ' 1 ) = u( ; ' 2 ) = 0; i.e. thus 1 sin x dx and c 2 = 0: consequently, namely, proposition 4.4. we consider the 2-dimensional sphere s 2 of radius a: let two circles pass through the north pole, such that in stereographic coordinates are represented by the parallel lines 2 = b and 2 = c; where b; c 2 r; say b < c: we consider the set d in s 2 ; whose stereographic projection is if x t is the position of the carrier of virus y at a given time t starting at the point a; where the stereogrpaphic projection coordinates of a are then, and g( ; t) = 2a 2 ln (c b) 2 ln 2 j j 2 + t 2 + 4a 2 (4.41) as we have seen the function satis…es the di¤erential equation here, 2 is the laplace-beltrami operator on s 2 expressed in stereographic projection coordinates. hence, the di¤erential equation takes the form (4.42) however the function satis…es the di¤erential equation (4.42) . with boundary conditions if we take the transformation of variables x = 1 and y = 2 b and set the function (x; y) = f ( 1 ; 2 ); then (x; y) we satisfy @ 2 @x 2 + @ 2 @y 2 = 0; with boundary conditions (x; 0) = 2a 2 ln( and where = c b: now let z = x + yi and w = exp z ; i.e. z = ln w : thus, if w = u + vi; u; v 2 r then u = exp x cos y and v = exp x sin y : (4.43) introducing the function (u; v) = (x; y): it follows that (u; v) satis…es with boundary conditions this is the standard dirichlet boundary value problem for the half line, and it is well known that (see e.g. [107] ) its solution is given by the poisson integral formula for the half-plane: where g( ; t) = 2a 2 ln 2 ln 2 j j 2 + t 2 + 4a 2 : notice that g( ; t) = g( ; t): hence, where u; v are given in (4.43). therefore exp 2 x sin 2 y + exp x cos y 2 d ; i.e. and in case t 1 = inf ft 0 j x t = 2 d 1 g ; then the probabilities p r a ft = t 1 g and p r a ft = t 2 g are given by and : proof. it is known that (see [21] ), u( ; ') = p r a ft = t 1 g is the unique solution of the di¤erential equation in we set f (') = du d' ; hence from (4.47) however, u(' 1 ) = 1 and u(' 2 ) = 0; hence : of course, @d 1 = f ( 1 ; 2 ) j 1 2 r and 2 = bg and @d 2 = f ( 1 ; 2 ) j 1 2 r and 2 = cg : let x t be the position of the carrier of virus y at a given time t starting at the point a; and the stereographic projection coordinates of a are proof it is known that (see [21] ) the function u( 1 ; 2 ) = p r a ft = t 1 g is the unique solution of the di¤erential equation here, 2 ; is the laplace-beltrami operator on s 2 expressed in the stereographic projection coordinates. hence from (4.8) the di¤erential equation (4.53) takes the form and then the expectation of exp( t ) is given by where is such that ( + 1) = 2a 2 and p ( ) is the legendre function where the multiple-valued function (z + p z 2 1 cos ') is to be determined in such a way that for ' = 2 it is equal to (the principal value of ) z (which is, in particular, real for positive z and real ). if > 1 2 ; where 1 is the …rst dirichlet eigenvalue of d s 2 ; then it satis…es the di¤erential equation with boundary condition u(' 0 ) = 1: (4.58) here 2 is the laplace-beltrami operator on s 2 . by the symmetry of d, it follows that the expectation of exp[ t ] is independent of . hence u is independent of . from (4.4) the di¤erential equation (4.57) takes the form if we set z = cos '; and (4.59) transforms to this is legendre's di¤erential equation. however, u(') is bounded for all ' 2 [0; ] and u(' 0 ) = 1. therefore (see [107] ), the solution of (4.59) is where is such that ( + 1) = 2a 2 : where then p r a ft < tg = 2p r a fx t = 2 dg : (4.60) proof. p r a ft < tg = p r a ft < t; x t = 2 dg + p r a ft < t; x t 2 dg : however, if x t = 2 d then of course t < t: on the other hand, if we setx then by the strong markov property of x t therefore from (4.61), (4.62) and (4.63) we obtain that p r a ft < tg = 2p r a fx t = 2 dg : the re ‡ection principle can help to calculate the distribution functions of certain exit times. let i.e. where p(t; ') is the transition density function of the brownian motion on s 2 of radius a: hence from (4.26) it is known that (see [107] ) however, p 0 n (1) = 1 for every n 2 r: it is also known that for every n 2 n p 0 2n (0) = ( 1) n (2n)! 2 2n (n!) 2 and p 0 2n+1 (0) = 0: thus, if n is even then i n = 0: if n is odd, i.e. n = 2k + 1; then i.e. i n = ( 1) n (2k)! (k + 1)(k!) 2 ( 1) n exp (2n + 1)(2n + 2) p t a (2n)!(4n + 3) 2 2n+1 (n!) 2 (n + 1) : furthermore, if s(0; ) namely the south pole of s 2 , then p r s fx t = 2 dg = p r n fx t = 2 dg = p r n fx t 2 dg = 1 p r n fx t = 2 dg: therefore p r s fx t = 2 dg = 1 2 ( 1) n exp (2n + 1)(2n + 2) p t a (2n)!(4n + 3) 2 2n+1 (n!) 2 (n + 1) : ( 1) n exp (2n + 1)(2n + 2) p t a (2n)!(2n + 3) 2 2n+1 (n!) 2 (n + 1) : is a subset of s 2 : the re ‡ected brownian motion in d 1 is the di¤ usion y t whose generator is n in d 1 with neuman boundary condition at @d 1 : roughly speaking y t behaves like x t inside d 1 but when it reaches the boundary, it is re ‡ected back in d 1 : we de…ne the boundary local time l t of y t ; as it can be shown that the limit exist in the l 2 sense. proposition 4.8. let ' 0 ; ' 1 2 (0; ), such that ' 0 < ' 1 , both …xed. we consider the sets d; 0 in s 2 ; such that d = f ( ; ')j 2 [0; 2 ) and ' 2 (' 0 ; ' 1 )g : and let y t be the re ‡ected brownian motion in 0 starting at the point and l t is the boundary local time of y t ; then , as long as the function z is positive (see [109] ). here 2 is the laplace-beltrami operator on s 2 . by the symmetry of d it follows that e a [ exp ( l t ) ] is independent of : from (4.2) the di¤erential equation takes the form we have shown that the solution of (4.71) is however, z( ; ' 0 ) = 1 and @z @' ( ; ' 1 ) + z( ; ' 1 ) = 0: hence and c 2 = 1: thus however, therefore, : a worldwide multilevel interplay among a plethora of factors ranging from micro-pathogens and individual interactions to macro-scale environmental, socio-economic and demographic conditions, necessitate the development of highly sophisticated mathematical models for robust representation of contagious dynamics of infectious diseases that would lead to the establishment of e¤ective control strategies and prevention policies. ethical and practical reasons defer from conducting enormous experiments in public health systems, hence mathematical models appear to be an e¢ cient way to explore contagion dynamics. a key aspect of epidemiological models is their link to real data, which is of particular utility toward the design of vaccination policies. two major vaccination strategies exist currently, i.e., the mass vaccination, which is most applied, and the recently developed pulse vaccination which is used in an increasing number of countries. however, most vaccination strategies are imperfect in the sense that they decrease the number of cases, without however eradicating the disease. public-health organizations in the world use the epidemiological models that fall in the three categories already presented in this work, to evaluate disease outbreak policies for epidemics. as we pointed out, many shortcomings exist for those models. all the models already used in the literature assume that the host population has constant size. however, this excludes diseases in exponentially growing populations as in most developing countries, or disease-induced mortality as childhood diseases in developing countries e.g., malaria. modeling infectious dynamics in non-stationary host populations requires explicit modeling of the host population as well as of the disease per se. models sometimes can be highly complicated in order to improve best …t to real data. nonetheless, very complex models do not always perform optimally in real-world applications or in simulations. real-world models allow for swift decision making, and suitable quanti…cation of the spatiotemporal dynamics of an outbreak. multidisciplinary research e¤orts are speeding up, integrating the advances in epidemiology, molecular biology, computational science and applied mathematics. mathematical modeling allows better understanding of the transmission process of infectious diseases in space and time, by setting forth rigorously the proper assumptions, the variables, the equations and their parameters. due to the complexity of the underlying complex interactions, either deterministic or stochastic epidemiological models are built upon incomplete information about e.g., the basic reproduction number, threshold e¤ects, intensity of spread, precise data of infected versus susceptible individuals, and other inaccuracies regarding the entire infectious network. simulations or brute-force computational techniques have been implemented in that direction to provide approximate solutions with encouraging results. nevertheless, some of the underlying generating processes of the outbreaks, such as the virus pathogenicity or variant social network topologies, ethnological characteristics and other quantities, may in ‡uence the spread of an outbreak. simulations often prove to be ine¢ cient for the systematic analysis of an emergent epidemic. new rigorous mathematical modeling methodologies, such as the one presented in this work for the …rst time, can be used to address inherent incomplete data structure and hidden nonlinear complex dynamics, with an aim to enhance forecastability in combating epidemic outbreaks. in the present study we introduced a novel approach for surveillance and modeling of infectious disease dynamics, called sbdiem. we explicitly described the mathematical framework underpinning the implementation and conceptualization of our new-age epidemiological model. our goal is to contribute to the arsenal of models already developed so far. it can be of particular interest, in light of a recent intensive worldwide e¤ort to speed up the establishment of a global surveillance network for combating pandemics of emergent and re-emergent infectious diseases. toward this aim, mathematical modeling will play a major role in assessing, controlling and forecasting potential outbreaks. we have to better understand and model the impact of numerous variables on contagious dynamics, ranging from the microscopic hostpathogen level, to individual and population interactions, as well as macroscopic environmental, social, economic and demographic factors all over the world. as a path for future research, we intend to conduct simulations, and empirical analyses based on real-time spatiotemporal datasets, in case of past outbreaks of infectious diseases as well as for covid-19. furthermore, we plan to convey an extensive comparative evaluation investigation of sbdiem vis-à-vis the three major categories set forth by the taxonomy of siettos and russo (2013) , and more speci…cally versus (1) statistical methods for epidemic surveillance, (2) state-space models of epidemic spread and (3) machine learning methods. in this way, the forecasting and nowcasting capabilities of the new model will be thoroughly explored. we also intend to investigate embedding the proposed analytical model into integrated arti…cial intelligence systems in the near future. our novel methodology apart from o¤ering a much better understanding of the complex and heterogeneous infectious disease dynamics could enhance predictability of epidemic 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