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

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 R0 for the delayed $SEIQR$ model with diffusion. Subsequently, using the theory of partial functional differential equations, we established that if R0≤1 and μ/α>Λ and Λ > 1/α, the disease-free equilibrium is asymptotically stable, and no endemic equilibrium exists. In contrast, if R0> max ( 1,β Λ e-μ τ( μ +α Λ ) ηi d) and μ/α<Λ < 1/α, 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.