key: cord-199107-01w8fk5f
authors: Savi, Pedro V.; Savi, Marcelo A.; Borges, Beatriz
title: A Mathematical Description of the Dynamics of Coronavirus Disease (COVID-19): A Case Study of Brazil
date: 2020-04-07
journal: nan
DOI: nan
sha: 
doc_id: 199107
cord_uid: 01w8fk5f

This paper deals with the mathematical modeling and numerical simulations related to the coronavirus dynamics. A description is developed based on the framework of susceptible-exposed-infectious-recovered model. Initially, a model verification is carried out calibrating system parameters with data from China, Italy, Iran and Brazil. Afterward, numerical simulations are performed to analyzed different scenarios of COVID-19 in Brazil. Results show the importance of governmental and individual actions to control the number and the period of the critical situations related to the pandemic.

Coronaviruses is related to illness that vary from a common cold to more severe diseases related to respiratory syndromes. Coronavirus disease 2019 (COVID-19) was discovered in

A frame-by-frame description of the reality can be represented by a set of differential equations. By assuming only time evolution of state variables, ∈ ℜ , spatial aspects are not of concern, allowing to establish a governing equation of the form: ̇= ( ), ∈ ℜ . The description of coronavirus disease 2019 dynamics defines its propagation considering animals and humans transmission. Different kinds of populations need to be defined in order to have a proper scenario of the disease propagation. Lin et al. (2020) proposes a susceptible-exposed-infectious-removed (SEIR) framework model to describe the coronavirus disease 2019 . This model was inspired on the original model of He et al. (2013) for influenza. Essentially, the description considers a total population of size N that contains two classes: D is a public perception of risk regarding severe cases and deaths; and C represents the cumulative infected cases. In addition,

S is the susceptible population, E is the exposed population, I is the infectious population and R is the removed population that includes both recovered and deaths. A simplified version of the model considers only person-to-person transmission, and therefore, zoonotic effect is neglected. This scenario assumes the second stage of the Wuhan -China case, after the close of the Huanan Seafood Wholesale Market. Emigration effect is also neglected in order to simplify the original model. Therefore, the simplified version of the governing equations considers the interaction among all these populations, being expressed by the following set of differential equations

where the following parameters are defined: is the mean infectious period; is the adjusted removed period, defining the relation between removed population and the infected one; is the mean latent period; is the proportion of severe cases; is the mean duration of public reaction. It should be pointed out that the parameter − defines the evolution of the nonreported removed population, which means that, if = , populations are restricted to the classical SEIR case.

The function = ( ) represents the transmission rate that considers governmental action, represented by (1 − ); and the individual action, represented by the function

(1 − ) . Therefore, the transmission rate is modeled as follows,

where ̂0 = 0 ( ) ( − Step function employed to consider parameter variations through time.

Note that, it is assumed that if 0 ( ) does not exist, the term 0 ( +1) does not exist as well, considering that index m is bigger than 1. Based on that, this general function can represent constant values, or different step functions.

Using the same strategy, it is defined the governmental action as follows:

where different steps are considered defined by time instants ( ) .

In addition, individual action is represented by

which the intensity of responses is defined by parameter . These parameters need to be adjusted for each place, being essential for the COVID-19 description.

In general, the parameter definitions depend on several issues, being a difficult task. In this regard, it should be pointed out that real data has spatial aspects that are not treated by this set of governing equations. Hence, this analysis is a kind of average behavior that needs a proper adjustment to match real data. Besides, R Li et al. (2020) evaluated Wuhan situation concluding that undocumented novel coronavirus infections are critical for understanding the overall prevalence and pandemic potential of this disease. The authors estimated that 86% of all infections were undocumented and that the transmission rate per person of undocumented infections was 55% of documented infections. This aspect makes the description even more complex.

The use of step functions to define some parameters allows a proper representation of different scenarios, especially the transmission rate. It is also important to observe that either governmental or individual actions have a delayed effect on system dynamics. Virus mutations are another relevant aspect related to the description of coronavirus dynamics that can dramatically alter the system response, but are not treated here.

Numerical simulations are performed considering the fourth-order Runge-Kutta method. The next sections treat the COVID-19 dynamics considering two different objectives.

Initially, the next section performed a model verification using information from China, Italy, Iran and Brazil. Afterward, the subsequent section evaluates different scenarios for the Brazilian case, using the parameters adjusted for the verification cases.

As an initial step of the developed analysis, a model verification is carried out using information available on Worldometer (https://www.worldometers.info/coronavirus/), considering different countries (Last updates: China -March 26, Italy -Mar 21; Iran -Mar 26; Brazil -Mar 24). The fundamental hypothesis of the analysis is that average populations of the country is of concern. Therefore, it is assumed that each country has a homogeneous distribution, without spatial patterns.

Basically, information from China, Italy, Iran and Brazil are employed. This information is useful to calibrate the model parameters, evaluating its correspondence with real data. Table   1 presents parameters employed for all simulations. They are based on the information of the Lin et al. (2020) that, in turn, is based on other references as He et al. (2010) and Breto et al. (2009) . For more details, see other citations referenced therein. Mean duration of public reaction 11.2 days

In addition, susceptible population initial condition is assumed to be 0 = 0.9 − 0 − 0 − 0 . In addition, it is assumed that there is no recovered population initially, i.e., 0 = 0. Another information needed for the model is the number exposed persons for each infected person. It is adopted that each infected person has the potential to expose 20 persons, 0 = 20 0 .

Transmission rate considers specific parameters for each case. Nevertheless, the reference values are presented in Table 2 . Other parameters are adjusted depending on the case. In the sequence, the dynamics of four different countries is analyzed in order to promote a model verification.

The first scenario for the model verification is based on China results. It should be pointed out that this analysis considers all cases in China, not restricted to Wuhan. Parameters presented in Table 3 are employed for simulations with a population of N = 1.43109 and an initial state with 554 infected persons ( 0 = 554). It should be highlighted again that these parameters are average ones since they are valid for the whole country. Of course, reaction time is different from the distinct parts of the country, which makes necessary to estimate this parameter based on the real data in an average way. Figure 2 presents infected population evolution showing a good agreement between simulation and real data. Due to chronological issues, Chinese case is the one with a large number of real data, which makes it useful to establish a comparison of the model prediction error. Figure 3 presents daily errors from china, highlighting the average and maximum errors. Note that the maximum error is less than 28%, with an average error of 13.58%. For the following three cases, Italy, Iran and Brazil, it is assumed that the second stage of governmental action has not been reached yet. Therefore, it is represented by a step function = [0, 0.4239], which means that (2) is neglected and 3 does not exist.

Italian case is now in focus considering parameters presented in Table 4 with a population of N = 60.48106 and an initial state with 20 infected persons ( 0 = 20). A step function is considered to define the nominal transmission rate, 0 , due to extreme governmental actions that have not been effective until present days. Figure 4 presents the infected population simulation compared with real data, showing a good agreement. Figure 5 presents daily errors from Italy, highlighting the average and maximum errors. For this case, the maximum error is less than 19%, with an average error of 10.60%. 

(1) Figure 4 : Italy -infected population through time. Iran case is now treated considering the parameters presented in Table 5 with a population of N = 81.16106 and an initial state with 20 infected persons ( 0 = 20). Results are presented in Figure 6 showing a good agreement with real data. Figure 7 presents daily errors, highlighting the average and maximum errors. Although the average error is 15.46%, the maximum error is around 42%, which is a large value. Nevertheless, it should be observed that the big values are related to the beginning of the predictions, probably due to problems with the original data. 

(1) 24 days Figure 6 : Iran -infected population through time. Brazilian case is now of concern considering parameters presented in Table 6 with a population of N = 209.3106 and an initial state with 10 infected person ( 0 = 10). Figure 8 presents the infected population evolution showing that the same trend of the other cases is followed, being enough to have a general scenario. It should be highlighted that Brazilian outbreak is in the beginning, with information that is not enough for a better calibration. 

(1) 17 days Figure 8 : Brazil -infected population through time.

This section has the objective to investigate different scenarios related to COVID-19 dynamics in Brazil. Parameters adjusted on the previous section are employed to evaluate different scenarios varying governmental and individual reactions. It should be pointed out that this adjustment does not have enough information, but it is possible to perform, at least a qualitative analysis of the COVID-19 dynamics in Brazil.

Initially, two different transmission rates are defined: naive scenario, without intervention ( = = 0); and with governmental and individual actions ( ≠ 0; ≠ 0). Figure 9 presents numerical simulations together with the real data that is presented just for the first days. The same parameters presented in Table 6 are employed assuming

(2) = 37 days. A logarithm scale is adopted since the naive scenario has a dramatic increase of the infected cases. Besides the big difference between both cases, it is clear the huge impact of variations on the transmission rate function that represents governmental and individual actions. It is noticeable that the effective actions tend to reduce the infected population, reducing the final crisis period as well. Nowadays, one of the most relevant issue to be discussed in terms of propagation is the governmental and individual actions. A parametric analysis is of concern considering distinct scenarios related to intervention. Scenarios defined by the variation of the intervention moments is initially treated. The moment of the governmental action start, represented by parameter (1) (day), is analyzed in Figure 11 , considering the following values: 17, 22, 27 and 32 and (2) is assumed to be 20 days after (1) . Note that the delay to the start of the governmental action dramatically alters the response, increasing the number infected population and its duration. The same conclusion can be established considering the second governmental action, represented by (2) (day), presented in Figure 12 that shows the same trend considering a different set of start instants: 37, 42, 47 and 52. 

A mathematical model based on the susceptible-exposed-infectious-recovered framework is employed to describe the COVID-19 evolution. A verification procedure is performed based on the available data from China, Italy, Iran and Brazil. Afterward, different scenarios from Brazil is analyzed. Results clearly show that governmental and individual actions are essential to reduce the infected populations and also the total period of the crisis.

The mathematical model can be improved in order to include more phenomenological information that can increase its capability to describe different scenarios. Nevertheless, it should be pointed out that the mathematical model and its numerical simulations are important tools that can be useful for public health planning.

Time series analysis via mechanistic models

A mathematical model for simulating the phase-based transmissibility of a novel coronavirus

Inferring the causes of the three waves of the 1918 influenza pandemic in England and Wales

Plug-and-play inference for disease dynamics: measles in large and small populations as a case study

Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia

China with individual reaction and governmental action

Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2)

Dynamics of coronavirus infection in human

Pattern of early human-to-human transmission of Wuhan

Chaos and order in biomedical rhythms

Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study

The authors would like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES and FAPERJ. Since this work was developed during a quarantine period, the authors would like to express their gratitude to familiar support that, besides the patience, helps to collect available information. Therefore, it is important to acknowledge: Raquel Savi, Rodrigo Savi, Antonio Savi and Bianca Zattar.
