Complex dynamics and pattern formation in a diffusive epidemic model with an infection-dependent recovery rate

A diffusive epidemic model with an infection-dependent recovery rate is formulated in this paper. Multiple constant steady states and spatially homogeneous periodic solutions are first proven by bifurcation analysis of the reaction kinetics. It is shown that the model exhibits diffusion-driven instability, where the infected population acts as an activator and the susceptible population functions as an inhibitor. The faster movement of the susceptible class will induce the spatial and spatiotemporal patterns, which are characterized by k-mode Turing instability and (k₁, k₂)-mode Turing-Hopf bifurcation. The transient dynamics from a purely temporal oscillatory regime to a spatial periodic pattern are discovered. The model reveals key transmission dynamics, including asynchronous disease recurrence, spatially patterned waves, and the formation of localized hotspots. The study suggests that spatially targeted strategies are necessary to contain disease waves that vary regionally and cyclically.