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 inter- 35 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 presented in the Table 3 . Table 7 : Data duration and their sources. The first column list the name of India and its provinces, the second column list the source of data, the third column list the duration of data and the fourth column list the web address of the data sources. 

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