Qualitative Analysis of a COVID-19 Mathematical Model with a Discrete Time Delay

The aim of this paper is to investigate the qualitative behavior of a mathematical model of the COVID-19 pandemic. The constructed SAIRS-type mathematical model is based on nonlinear delay differential equations. The discrete-time delay is introduced in the model in order to take into account the latent stage where the individuals already have the virus but cannot yet infect others. This aspect is a crucial part of this work since other models assume exponential transition for this stage, which can be unrealistic. We study the qualitative dynamics of the model by performing global and local stability analysis. We compute the basic reproduction number R0d, which depends on the time delay and determines the stability of the two steady states. We also compare the qualitative dynamics of the delayed model with the model without time delay. For global stability, we design two suitable Lyapunov functions that show that under some scenarios the disease persists whenever R0d>1. Otherwise, the solution approaches the disease-free equilibrium point. We present a few numerical examples that support the theoretical analysis and the methodology. Finally, a discussion about the main results and future directions of research is presented.