Modeling and forecasting the COVID-19 pandemic in India

Abstract

In India, 100,340 confirmed cases and 3155 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_qS_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.

Keywords: Basic reproduction number; COVID-19; Isolation; Mathematical model; Model prediction; Sensitivity analysis.

Fig. 1

Schematic representation of the model. The schematic flow diagram represents the interplays among different stages of infection in the model SARII_qS_q: susceptible or uninfected (S), asymptotic or mildly symptomatic (A), recovered or healed (R), infected or symptomatic (I), isolated infected (I_q) and quarantined susceptible (S_q) individuals.

Fig. 2

Parameter sensitivity. Partial rank correlation coefficients illustrating the sensitivity indices for the SARII_qS_q system for COVID-19 or SARS-CoV-2 model parameters with symptomatic or infected individuals (I) at different days with p < 0.01.

Fig. 3

Model estimation based on real data. The SARII_qS_q model fitted with the real data on daily new cases of COVID-19 in India and 5 provinces of India, namely Andhra Pradesh, Delhi, Gujarat, Haryana and Jammu & Kashmir. Observed data points are displayed in the red dot histogram and the blue curve represents the best fitting curve for the SARII_qS_q 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. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4

Model estimation based on real data. The SARII_qS_q model fitted with the real data on daily new cases of COVID-19 for 6 provinces of India, namely Karnataka, Kerala, Madhya Pradesh, Maharashtra, Punjab, and Rajasthan. Observed data points are shown in the red dot histogram and the blue curve represents the best fitting curve for the SARII_qS_q 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. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5

Model estimation based on real data. The SARII_qS_q model fitted with the real data on daily new cases of COVID-19 for 6 provinces of India, namely Tamil Nadu, Telangana, Uttar Pradesh, West Bengal, Bihar, and Odisha. Observed data points are shown in the red dot histogram and the blue curve represents the best fitting curve for the SARII_qS_q 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. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6

Basic reproduction number R₀. The bar-diagram represents the computation of basic reproduction number R₀ from the estimated parameter values in the Table 2 for the Republic of India and 17 different provinces of India.

Fig. 7

Effect of nation-wide lockdown. Epidemic evolution predicted by the SARII_qS_q model for SARS-CoV-2 epidemic in India for the time period 30 January 2020 to 16 August 2020. Before implemented 3rd Phase lockdown, the social distancing are not enforced, resulting a larger $R_0=2.0490$. After implementation of 3rd phase lockdown, the social distancing are enforced but not strong enough, as a result larger $R_0=1.6302$ and $R_0=1.0245,$ which shows a substantial outbreak of COVID-19. But after implementation of fully operational and strict lockdown, after 31 May 2020, resulting a smaller $R_0=0.4098,$ yields below 1. Bottom panel shows the trajectories of COVID-19 diseases for different values of the strengthen of the intervention ω.

Fig. 8

Contour plots of basic reproduction number R₀. Contour plots of R₀ for India and eight different provinces of India, with respect to the probability of disease transmission rate β_s and the quarantined rate ρ_s of susceptible individuals. All parameter values other than β_s and ρ_s 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 R₀.

Fig. 9

Contour plots of basic reproduction number R₀. Contour plots of R₀ for nine different provinces of India, with respect to the probability of disease transmission rate β_s and the quarantined rate ρ_s of susceptible individuals. All parameter values other than β_s and ρ_s 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 R₀.

Fig. 10

Model-Based Data-Driven prediction of SARS-CoV-2 life cycle. Model predictions based on the estimated data for the Republic of India and five provinces of India, namely Andhra Pradesh, Delhi, Gujarat, Haryana and Jammu & Kashmir. The blue curve represents the model prediction for the daily new confirmed cases of coronavirus diseases while the red dot histogram represents the actual cases. The turning and ending dates of COVID-19 in India and five provinces are displayed in the figure. The estimated parameter values are specified in the Table 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 11

Model-Based Data-Driven prediction of SARS-CoV-2 life cycle. Model predictions based on the estimated data for the five provinces of India, namely Karnataka, Kerala, Madhya Pradesh, Maharashtra, Punjab, and Rajasthan. The blue curve represents the model prediction for the daily new confirmed cases of coronavirus diseases while the red dot histogram represents the actual cases. The turning and ending dates of COVID-19 in India and five provinces are displayed in the figure. The estimated parameter values are specified in the Table 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 12

Model-Based Data-Driven prediction of SARS-CoV-2 life cycle. Model predictions based on the estimated data for the five provinces of India, namely Tamil Nadu, Telangana, Uttar Pradesh, West Bengal, Bihar, and Odisha. The blue curve represents the model prediction for the daily new confirmed cases of coronavirus diseases while the red dot histogram represents the actual cases. The turning and ending dates of COVID-19 in India and five provinces are displayed in the figure. The estimated parameter values are specified in the Table 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)