A Mathematical Modelling and Analysis of COVID-19 Transmission Dynamics with Optimal Control Strategy

Abstract

We proposed a deterministic compartmental model for the transmission dynamics of COVID-19 disease. We performed qualitative and quantitative analysis of the deterministic model concerning the local and global stability of the disease-free and endemic equilibrium points. We found that the disease-free equilibrium is locally asymptotically stable when the basic reproduction number is less than unity, while the endemic equilibrium point becomes locally asymptotically stable if the basic reproduction number is above unity. Furthermore, we derived the global stability of both the disease-free and endemic equilibriums of the system by constructing some Lyapunov functions. If $R_0\le 1$ , it is found that the disease-free equilibrium is globally asymptotically stable, while the endemic equilibrium point is globally asymptotically stable when $R_0>1$ . The numerical results of the general dynamics are in agreement with the theoretical solutions. We established the optimal control strategy by using Pontryagin’s maximum principle. We performed numerical simulations of the optimal control system to investigate the impact of implementing different combinations of optimal controls in controlling and eradicating COVID-19 disease. From this, a significant difference in the number of cases with and without controls was observed. We observed that the implementation of the combination of the control treatment rate, $u_2$ , and the control treatment rate, $u_3$ , has shown effective and efficient results in eradicating COVID-19 disease in the community relative to the other strategies.