Mathematical exploration of a diffusive infection's dynamics and simulation

In this work, we construct Lyapunov functionals to analyze the global stability of the equilibria in reaction-diffusion systems arising in biological models. We employ Lyapunov functionals originally constructed for associated ordinary differential equation (ODE) models and extend them to partial differential equation (PDE) systems involving spatial diffusion. We analyze disease-free and endemic equilibrium stability in terms of the basic reproduction number [Formula: see text] a threshold parameter. Specifically, we show that when [Formula: see text] the disease-free equilibrium is globally asymptotically stable, while for [Formula: see text] the endemic equilibrium is globally stable under certain conditions. To make our methods more feasible, we supply some examples from epidemiology and good health, including spatially structured models with diffusion. Numerical simulations are provided to justify the theoretical results and to show the convergence behavior of the solutions.