Traveling waves in a delayed reaction-diffusion SVIR epidemic model with generalized incidence function and imperfect vaccination

This paper is concerned with traveling wave solutions for a delayed reaction-diffusion SVIR epidemic model that includes both general incidence function and imperfect vaccination. In the model, the spread of infection in space is explicitly taken into account by using a heterogeneous environment; it takes into consideration the delay in immune response and inefficiency in vaccinations. The analysis carried out below shows that the basic reproduction number Ro will be a critical value for determining the existence of traveling waves. More precisely, when Ro > 1 there exists a minimal wave speed ρ* > 0 such that the system admits nontrivial traveling wave solutions for ρ ≥ ρ* whereas no such solutions exist for ρ < ρ*. On the other hand, if Ro ≤ 1, there are no traveling wave solutions. The introduction of delays and imperfect vaccination adds richness and complexity to the dynamics, such as possible wave speed adjustments and pattern formations, which are hallmarks of complex systems. This work develops a theoretical framework that shall guide the understanding of how delays, spatial spread, and control measures interact in epidemic systems and offers insights applicable to real-world infectious disease dynamics. Numerical simulations for some typical nonlinear incidence functions are given in the last to illustrate the existence of traveling waves.