key: cord-0479380-heux67zz
authors: Sarkar, Kankan; Khajanchi, Subhas
title: Modeling and forecasting of the COVID-19 pandemic in India
date: 2020-05-13
journal: nan
DOI: nan
sha: f7ed51444c210f58c010f7d6a8e8ff454520a796
doc_id: 479380
cord_uid: heux67zz

Background: The ongoing COVID-19 epidemic dilated rapidly throughout India. To end the global COVID-19 pandemic major behavioral, social distancing, contact tracing, and state interventions has been undertaken to reduce the outbreak and avert the persistence of the coronavirus in humans in India and worldwide. In absence of any vaccine or therapeutics, forecasting is of utmost priority for health care planning and control the transmission of COVID-19. Methods: We have proposed a mathematical model that explain the transmission dynamics of COVID-19 in India. Based on the estimated data our model predicts the evolution of epidemics and the end of SARS-CoV-2 and aids to evaluate the influence of different policies to control the spread of the diseases. Findings: With the real data for infected individuals, we find the basic reproduction number $R_0$ for 17 states of India and overall India. A complete figure is given to demonstrate the estimated pandemic life cycle along with the real data or history to date. Our study reveals that the strict control measures implemented in India substantially mitigated the disseminate of SARS-CoV-2. Importantly, model simulations predict that 95% reduction of outbreak on June 26, 2020 and 99% reduction of outbreak on July 26, 2020 in India. Interpretation: Our model simulation demonstrates that the community-wide elimination of SARS-CoV-2 is possible by mitigating the social distancing and use essential precautions. Lockdown can be implemented strictly to prevent the human-to-human transmission of COVID-19. The model-based and parameter estimation of epidemic life cycle, and end dates, if can be done precisely, may decrease distress and over optimism and develop the mentality for all of us for the next stages of the outbreak evolution.

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] .

According to the WHO report dated April 09, 2020 reported 3,855,812 total confirmed cases and 2,65,862 confirmed deaths worldwide [3] . COVID-19 or SARS-CoV-2 has already surpassed the earlier history of two coronavirus outbreaks, namely SARS-CoV (Severe Acute Respiratory Syndrome coronavirus) and MERS-CoV (Middle East respiratory syndrome coronavirus), posing the substantial endanger to the world-wide public health as well as global economy after the 2nd world war [4] . SARS-CoV-2 exhibits distinctive epidemiological traits collated with coronavirus epidemics of SARS-CoV and MERS-CoV.

The outbreak was first announced by the "Health Commission of Hubei province", China, a cluster of unexplained cases of pneumonia of unknown etiology (unknown cases) [5] , which is lethal, was first identified in Wuhan city of Hubei province, China, on December 31, 2019 [28, 7] . After that an exceptionally large number of patients were diagnosed with SARS-CoV-2 in mainland of China, prodding Chinese Governments to initiate stringent measures to control the epidemic [5] . In spite of these precautions, SERS-CoV-2 pandemic evolved in the following months. The confirmed cases of COVID-19, the symptoms like sneeze, fever, or even a runny nose, dry cough, fatigue, breathing problem and bilateral lung infiltration to severely ill and dying. Due to the human mobility, this communicable disease has now spread throughout the world, making USA and Europe as new epicenters [4] .

The first indigenous case of COVID-19 in India was reported on January 30, 2020 in Thrissur district of Kerala and the patient, a student of Wuhan University, China [8] . The authorities recommend the level of infection could be assload as the India's testing rates are very poor among the world [9] . The rate of infection of SARS-CoV-2 in India is chronicled to be 1.7 (that is, one coronavirus positive infects 1.7 in India), materially lower than in hot zones [10] . The estimated basic reproduction number R 0 for COVID-19 ranges from 2.0 to 3.5 [18, 19] , that seems similar, or perhaps higher than that of SARS-CoV and MERS-CoV. Higher level of viral loads for COVID-19 were observed in upper respiratory specimens of symptomatic patients resulting little or no symptoms, with a viral shedding pattern similar to that of influenza viruses [20] . Thus, uncertain viral transmission may take part a crucial and underestimated role in sustaining the epidemic.

India has suspended all the tourist visas as the majority of confirmed coronavirus cases were connected to other countries [11] . Governments will not be competent to reduce both fatalities from SARS-CoV-2 epidemic and the economic impact of viral outbreak. Maintaining the fatality rate as low as possible will be the utmost importance for the populations; therefore the Governments must put in place measures to mitigate the unavoidable economic downturn. Due to absence of any specific pharmaceutical interventions, 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) [12] . On March 19, 2020 the Govt. of India suspended all the international flights till March 22, 2020 [16] , and on March 23, 2020 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 [13] .

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 [14] , and the lockdown has been extended to May 03, 2020 to prevent stage-III spreading of the virus or human-to-human transmission [15] .

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 effective vaccine [21, 22, 23] . In the year 1927, Kermack & McKendrick [24] 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 the transmission dynamics of COVID-19 pandemic. A Bats-Hosts-Reservoir-People network model has been developed by Chen et al. [25] to study the transmission dynamics of novel coronavirus. Lin et al. [26] 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. [38] studied an extended SEIR model to study the transmission dynamics of COVID-19 and perform a short-term prediction based on the data from India. A discrete-time SIR (susceptible-infectious-removed) model introducing dead compartment system studied by Anastassopoulou et al. [27] [30] . Here, we developed 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. [31] 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 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 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. In our model, quarantine describes the separation of coronavirus infected populations from the susceptible individuals before progression of 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 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 contact 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 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 individuals. Asymptomatic population develop to infected population at the rate γ a , so the average time 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 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).

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' [32, 33] . 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 equilibrium. 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 [32, 33] . In order to do this, we consider the nonnegative matrix F and the 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 next generation matrix F V −1 . Thus, the basic reproduction number of the system (1) is 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 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.

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 disease transmission (β s ), quarantined rate of susceptible individuals (ρ s ), contact rate of entire individ-uals (ǫ 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 ). 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 for cumulative cases, the parameters are not estimated from confirmed cumulative COVID-19 cases to optimize the error in parameter estimation [39] and errors are listed in the Table 5 .

Initially 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 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 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 7 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 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.

To investigate the influence of intervention policies in reducing the SARS-CoV-2 viruses, we modified Figure 6 shows the effect of lockdown in three different time periods. Figure 6 predict that on 17

May 2020, the cumulative number of COVID-19 infected population in India without Phase-3 lockdown is 70948 but after strictly enforcing nation-wide Phase-3 lockdown in India the cumulative number of infected population reduced and it will be 64838. Another interesting outcome can be noticed as an effect of nation-wide lockdown is the appearance of epidemic peak. The effect of lockdown not only reduce the infected population but also causes delay in an appearance of the peak, which has shown the bottom panel of the Figure 6 . The bottom panel of Figure 6 exhibits that the infected population reaches its peak at a different time.

The 2020. So, if more observed data are available, the parameters values will alter and the long time prediction 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 respectively. That is the 99% end days deviates 10 days from the 99% end day of India.

We have investigated a methodology in estimating the crucial epidemiological model parameters as well as the modeling and prediction of the spread of SARS-CoV-2 outbreak in 17 different provinces of India and the overall India by evaluating publicly accessible real data till April 30, 2020.

Our model simulations provide an important tool to evaluate the consequences of possible policies, incorporating social distancing and lockdown. Our model findings, obtained by combining the SARII q S q model with the accessible data regarding the coronavirus outbreak in India, indicate that imposing social distancing is crucial, essential and effectual, in line with other details in the literature [28] . If the strong lockdown can be imposed earlier, the effective results can be achieved. We accept these indications can be handy to control the outbreak in India, together with the countries that are still in the early phases of epidemic.

The future is always unpredictable and we must keep this in mind when we read any kind of predictions. 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 . Table 3 . All parameter values other than βs and ρs are listed in the Table 2 . The contour plots demonstrates that higher disease transmission probability of the COVID-19 virus will remarkably increase the reproduction number R 0 . Table 2 . Table 2. such pharmaceutical interventions will remarkably change the results.

In the modern and developed world, data and detailed information regarding the COVID-19 or SARS-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 the equivocal outcomes and inappropriate predictions by orders of size, as also identified by the other researches [35] . 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. 

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