








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 positive- 13 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 capi- 16 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 com- The 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 mod- rate 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 and En-380 gineering

html#/bda7594740fd40299423467b48e9ecf6, 2020

Clinical virology

A novel coronavirus outbreak 385 of global health concern

Wuhan wet market closes amid pneumonia outbreak

India COVID-19 Tracker

Report 9: 395 Impact of non-pharmaceutical interventions (npis) to reduce covid19 mortality and 396 healthcare demand

A realistic two-strain model 398 for mers-cov infection uncovers the high risk for epidemic propagation

An updated estimation 401 of the risk of transmission of the novel coronavirus (2019-ncov)

Effectiveness of airport screening at detecting trav-404 ellers infected with novel coronavirus (2019-ncov)

Modelling the epidemic trend of the 407 2019 novel coronavirus outbreak in china

Estimation 409 of the transmission risk of the 2019-nCoV and its implication for public health 410 interventions

Nowcasting and forecasting the potential 412 domestic and international spread of the 2019-ncov outbreak originating in wuhan, 413 china: a modelling study

Novel 415 coronavirus 2019-ncov: early estimation of epidemiological parameters and epidemic 416 predictions

Age-structured impact of social distancing on the covid-418 19 epidemic in india

A mathematical model 420 for simulating the transmission of wuhan novel coronavirus

Early dynamics of transmission and 423 control of COVID-19: a mathematical modelling study

Epidemic analysis of covid-19 426 in china by dynamical modeling

Modelling strategies 429 for controlling sars outbreaks

Prudent public health inter-433 vention strategies to control the coronavirus disease 2019 transmission in india: 434 A mathematical model-based approach

Statewise population data

Life expectancy at birth

An optimal 441 cost effectiveness study on Zimbabwe cholera seasonal data from

Dram: efficient adaptive mcmc

Mathematical analysis of a power-law form time depen-446 dent vector-borne disease transmission model

Convenient Functions for Ensem-449

ble Time Series Forecasts, 2020

World's biggest lockdown may have cost Rs 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.

