A mathematical model of COVID-19 transmission dynamics with a case study in Ethiopia

Abstract

COVID-19 is an infectious diseases caused by SARS-CoV-2, a betacoronavirus. Globally, it has been affecting many country including Ethiopia. This study aims to develop and analyze a mathematical model of COVID-19 transmission dynamics and apply it as case study in Ethiopia. For this, we formulated a deterministic mathematical model of seven compartments to describe the transmission dynamics of COVID-19 infection using a system of non-linear ordinary differential equations with optimal control. We investigated the dynamical behavior of this model and performed the qualitative analysis of the model. The system has two equilibrium points, namely the disease free equilibrium point and the endemic equilibrium point which exists conditionally. The basic reproduction number R0 was calculated using the next-generation matrix and the stability of the equilibrium points were analyzed. From the qualitative analysis, the disease free equilibrium point is both locally and globally stable if R0 < 1, and the endemic equilibrium point is also both locally and globally stable if R0 > 1 under some conditions on the system parameters. Furthermore, sensitivity analysis of the model equation was performed on the key parameters to find out their relative significance and potential impact on the transmission dynamics of COVID-19 diseases. Also bifurcation analysis have been performed to reveal the transmission dynamics of corona virus diseases. Then we extended the proposed model to minimize the number of exposed and symptomatic individual with taking into account the cost of implementation. We proved the existence of optimal controls with the help of Pontryagin’s Minimum Principle. Finally, numerical simulations of the model equations were carried out using MATLAB. The simulations result show that, it is more efficient when triple(u1, u2&u3) or all the four controls are considered in simulation at time to reduce the transmission dynamics COVID-19 infection.