Stability analysis of a delayed SEIQR epidemic model with diffusion and elementary saturated incidence rate

Abstract

The present study delves into the impact of delay and spatial diffusion on the dynamical behavior of the $SEIQR$ epidemic model. The inclusion of delay in this model renders it more realistic, modeling the latency period of the disease. Additionally, introducing diffusion into the $SEIQR$ model aims to provide better insight into the effects of spatial heterogeneity and individual mobility on disease persistence and extinction. Initially, we derived a threshold value $\mathcal{R}{0}$ for the delayed $SEIQR$ model with diffusion. Subsequently, using the theory of partial functional differential equations, we established that if $\mathcal{R}{0}\leq1$ and $\dfrac{\mu}{\alpha}>\Lambda$ and $\Lambda > \dfrac{1}{\alpha} $, the disease-free equilibrium is asymptotically stable, and no endemic equilibrium exists. In contrast, if $\mathcal{R}{0}> \max \left( 1,\dfrac{\beta \Lambda e^{-\mu \tau}}{\left( \mu +\alpha \Lambda \right) \eta{i} d}\right)$ and $\dfrac{\mu}{\alpha}<\Lambda < \dfrac{1}{\alpha} $, a unique, asymptotically stable endemic equilibrium is present. Next, by constructing an appropriate Lyapunov function, we determined the threshold parameters that ensure the global asymptotic stability of the disease-free equilibrium. Finally, we illustrated the theoretical results through numerical simulations.