Design and analysis of a discrete method for a time‐delayed reaction–diffusion epidemic model

Abstract

In this work, we propose a time‐delayed reaction–diffusion model to describe the propagation of infectious viral diseases like COVID‐19. The model is a two‐dimensional system of partial differential equations that describes the interactions between disjoint groups of a human population. More precisely, we assume that the population is conformed by individuals who are susceptible to the virus, subjects who have been exposed to the virus, members who are infected and show symptoms, asymptomatic infected individuals, and recovered subjects. Various realistic assumptions are imposed upon the model, including the consideration of a time‐delay parameter which takes into account the effects of social distancing and lockdown. We obtain the equilibrium points of the model and analyze them for stability. Moreover, we examine the bifurcation of the system in terms of one of the parameters of the model. To simulate numerically this mathematical model, we propose a time‐splitting nonlocal finite‐difference scheme. The properties of the model are thoroughly established, including its capability to preserve the positivity of solutions, its consistency, and its stability. Some numerical experiments are provided for illustration purposes.

Reaction–Diffusion Equations in Mathematical Models Arising in Epidemiology

“…It should be noted that there are some papers, see, e.g., [73,74], in which the models based on reaction-diffusion equations with time delays are suggested for describing epidemic processes. Actually, such types of models are natural extensions of the models based on ODEs with time delays (see, e.g., Chapter 10 in [1]).…”

A review of commonly used compartmental models in epidemiology

“…The reader should know that the ongoing and future works are in the aim of developing more complex models combining the reviewed types (2), ( 3), ( 4) and ( 5). We refer for instance to the following recent works Khan, Ikram, Din, Humphries & Akgul (2021), Macías-Díaz et al (2020), Chinnathambi et al (2019), Ge & Chen (2021). Knowing all these types, it can be concluded that, while modeling a disease, the pertinence of the chosen model is indeed based on its capacity of maximizing the interpretation of reality but also on its ability to minimize the difficulty of the required mathematical and numerical tools for its analysis.…”