ABSTRACT
The formulation of mathematical models using differential equations has become crucial in predicting the evolution of viral diseases in a population in order to take preventive and curative measures. In December 2019, a novel variety of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, China, which causes a severe and potentially fatal respiratory syndrome. Since then, it has been declared a pandemic by the World Health Organization and has spread around the globe. A reaction−diffusion system is a mathematical model that describes the evolution of a phenomenon subjected to two processes: a reaction process, in which different substances are transformed, and a diffusion process, which causes their distribution in space. This article provides a mathematical study of the Susceptible, Exposed, Infected, Recovered, and Vaccinated population model of the COVID-19 pandemic using the bias of reaction−diffusion equations. Both local and global asymptotic stability conditions for the equilibria were determined using a Lyapunov function, and the nature of the stability was determined using the Routh−Hurwitz criterion. Furthermore, we consider the conditions for the existence and uniqueness of the model solution and show the spatial distribution of the model compartments when the basic reproduction rate R0<1 and R0>1. Thereafter, we conducted a sensitivity analysis to determine the most sensitive parameters in the proposed model. We demonstrate the model's effectiveness by performing numerical simulations and investigating the impact of vaccination, together with the significance of spatial distribution parameters in the spread of COVID-19. The findings indicate that reducing contact with an infected person and increasing the proportion of susceptible people who receive high-efficacy vaccination will lessen the burden of COVID-19 in the population. Therefore, we offer to the public health policymakers a better understanding of COVID-19 management.
ABSTRACT
COVID-19 pandemic represents an unprecedented global health crisis which has an enormous impact on the world population and economy. Many scientists and researchers have combined efforts to develop an approach to tackle this crisis and as a result, researchers have developed several approaches for understanding the COVID-19 transmission dynamics and the way of mitigating its effect. The implementation of a mathematical model has proven helpful in further understanding the behaviour which has helped the policymaker in adopting the best policy necessary for reducing the spread. Most models are based on a system of equations which assume an instantaneous change in the transmission dynamics. However, it is believed that SARS-COV-2 have an incubation period before the tendency of transmission. Therefore, to capture the dynamics adequately, there would be a need for the inclusion of delay parameters which will account for the delay before an exposed individual could become infected. Hence, in this paper, we investigate the SEIR epidemic model with a convex incidence rate incorporated with a time delay. We first discussed the epidemic model as a form of a classical ordinary differential equation and then the inclusion of a delay to represent the period in which the susceptible and exposed individuals became infectious. Secondly, we identify the disease-free together with the endemic equilibrium state and examine their stability by adopting the delay differential equation stability theory. Thereafter, we carried out numerical simulations with suitable parameters choice to illustrate the theoretical result of the system and for a better understanding of the model dynamics. We also vary the length of the delay to illustrate the changes in the model as the delay parameters change which enables us to further gain an insight into the effect of the included delay in a dynamical system. The result confirms that the inclusion of delay destabilises the system and it forces the system to exhibit an oscillatory behaviour which leads to a periodic solution and it further helps us to gain more insight into the transmission dynamics of the disease and strategy to reduce the risk of infection.