Modeling the effects of the contaminated environments on COVID-19 transmission in India

Abstract

COVID-19 is an infectious disease caused by the SARS-CoV-2 virus that caused an outbreak of typical pneumonia first in Wuhan and then globally. Although researchers focus on the human-to-human transmission of this virus but not much research is done on the dynamics of the virus in the environment and the role humans play by releasing the virus into the environment. In this paper, a novel nonlinear mathematical model of the COVID-19 epidemic is proposed and analyzed under the effects of the environmental virus on the transmission patterns. The model consists of seven population compartments with the inclusion of contaminated environments means there is a chance to get infected by the virus in the environment. We also calculated the threshold quantity $R_0$ to know the disease status and provide conditions that guarantee the local and global asymptotic stability of the equilibria using Volterra-type Lyapunov functions, LaSalle's invariance principle, and the Routh-Hurwitz criterion. Furthermore, the sensitivity analysis is performed for the proposed model that determines the relative importance of the disease transmission parameters. Numerical experiments are performed to illustrate the effectiveness of the obtained theoretical results.

Keywords: 00A71; 34D20; 37M05; 65P40; 92B05; 92Bxx; COVID-19 dynamics; Contaminated environments; Mathematical model; Numerical simulations; Sensitivity analysis; Stability analysis.

Fig. 1

Model diagram representing transmission routes for COVID-19 epidemic.

Fig. 2

Parameter estimation and fitting active cases $(I\left(t\right)+C\left(t\right)+H\left(t\right))$ to actual data for India using lmfit. The plot shows actual data and best fit by minimizing sum of the squares of the errors.

Fig. 3

PRCCs of ${\mathcal{R}}_0$ performed on model parameters.

Fig. 4

Temporal variation of the different population classes.

Fig. 5

Dynamics of ${\mathcal{R}}_0$ for $\mu$, ${\gamma }_1$, $\alpha$ and $\theta$.

Fig. 6

Dynamics of ${\mathcal{R}}_0$ for ${\beta }_V$, ${\beta }_E$, ${\beta }_I$ and ${\beta }_A$.

Fig. 7

3D dynamics of ${\mathcal{R}}_0$ for ${\alpha }_1$ and $\alpha$ along with contour plot.

Fig. 8

3D dynamics of ${\mathcal{R}}_0$ for ${\beta }_V$ and $\theta$ along with contour plot.

Fig. 9

3D dynamics of ${\mathcal{R}}_0$ for ${\beta }_I$ and ${\gamma }_1$ along with contour plot.

Fig. 10

3D dynamics of ${\mathcal{R}}_0$ for ${\beta }_E$ and $\mu$ along with contour plot.