A mathematical model for COVID-19 transmission dynamics with a case study of India

Abstract

The ongoing COVID-19 has precipitated a major global crisis, with 968,117 total confirmed cases, 612,782 total recovered cases and 24,915 deaths in India as of July 15, 2020. In absence of any effective therapeutics or drugs and with an unknown epidemiological life cycle, predictive mathematical models can aid in understanding of both coronavirus disease control and management. In this study, we propose a compartmental mathematical model to predict and control the transmission dynamics of COVID-19 pandemic in India with epidemic data up to April 30, 2020. We compute the basic reproduction number R ₀, which will be used further to study the model simulations and predictions. We perform local and global stability analysis for the infection free equilibrium point E ₀ as well as an endemic equilibrium point E* with respect to the basic reproduction number R ₀. Moreover, we showed the criteria of disease persistence for R ₀ > 1. We conduct a sensitivity analysis in our coronavirus model to determine the relative importance of model parameters to disease transmission. We compute the sensitivity indices of the reproduction number R ₀ (which quantifies initial disease transmission) to the estimated parameter values. For the estimated model parameters, we obtained $R_0=1.6632,$ which shows the substantial outbreak of COVID-19 in India. Our model simulation demonstrates that the disease transmission rate β_s is more effective to mitigate the basic reproduction number R ₀. Based on estimated data, our model predict that about 60 days the peak will be higher for COVID-19 in India and after that the curve will plateau but the coronavirus diseases will persist for a long time.

Keywords: COVID-19; Isolation; Mathematical model; Reported and unreported cases; Sensitivity analysis.

Fig. 1

The schematic flow diagram represents the susceptible or uninfected (S), asymptomatic (A), reported symptomatic infectious (I) and unreported symptomatic infectious (U) individuals for novel coronavirus disease that persuades the formulation of the SAIU model (1).

Fig. 2

The figures shows the model fitting of daily reported symptomatic infectious individuals (upper panel) and the reported cumulated symptomatic infectious individuals (lower panel) for the SARS-CoV-2 or COVID-19 pandemic in India. The epidemic turning point of the daily reported symptomatic and cumulated cases data from January 30, 2020 to April 30, 2020 (day 1 = January 30, 2020). The observed data points are shown in the red circle and the solid blue line portrays the model simulations. We use the initial size of the population $S_0=100,$$A_0=10,$$I_0=1,$$U_0=5$ and $t_0=1.0$.

Fig. 3

The figure shows the basic reproduction number R₀ when β_s (probability of disease transmission rate) and γ_a (rate of transition from asymptomatic to symptomatic infectious class) varies. The other parameter values are listed in the Table 1.

Fig. 4

The figures shows the prediction of our SAIU model (1) for the Republic of India. Here, ${\Lambda }_s=2500,$$q_i=0.58$ and the initial values are $S_0=4000,$$A_0=3000,$$I_0=10,$$U_0=1000$ rest of the parameter values are listed in the Table 1. The model simulation demonstrates that about 60 days the peak will be higher for the COVID-19 in India and after that the curve will be flatten but the coronavirus diseases will be continued for a long-time with lesser magnitude.

Fig. 5

The figure shows the sensitivity indices of the basic reproduction number R₀ with respect to the each of the system parameters related to R₀ for the SAIU model system (1). The baseline parameter values are taken from the Table 1. The simulation exhibits that the most influential parameter is the probability of disease transmission rate (β_s), and the least influential parameter is the fraction of asymptomatic infected individuals become reported symptomatic infected individuals at the rate q_i. The list of sensitivity indices are given in the Table 2.

Fig. 6

Contour plots for the SAIU model system (1). Contour plots of basic reproduction number R₀ with respect to the most effective parameters, β_s (probability of disease transmission) versus (a) modification factor for the reported symptomatic infected individuals (α_u), (b) modification factor for the asymptomatic infected individuals (α_a), (c) transition rate (γ_a) from asymptomatic individuals to symptomatic infected individuals, and (d) a portion of pre-symptomatic infected individuals become symptomatic infected individuals at the rate q_i. All the parameter values are listed in the Table 1 except the varied parameters.