Dynamics of a Spatially Heterogeneous Stochastic SVIR Epidemic Model

This paper investigates a spatially heterogeneous stochastic reaction–diffusion SVIR epidemic model incorporating vaccination, waning immunity, and environmental fluctuations. The proposed system extends the classical deterministic epidemic framework by introducing diffusion effects, spatially dependent transmission rates, and multiplicative Gaussian white noise perturbations. The model is formulated as a system of stochastic partial differential equations subject to homogeneous Neumann boundary conditions. First, the well-posedness of the model is established by proving the existence, uniqueness, positivity, and global boundedness of strong solutions through semigroup theory, Itô calculus, and comparison techniques. The stochastic transformation approach is employed to convert the original system into an equivalent random partial differential system, enabling rigorous analytical treatment. Furthermore, the qualitative behavior of the model is analyzed in terms of disease extinction and persistence under the combined influence of spatial diffusion and stochastic perturbations. The effects of vaccination, immunity loss, and environmental noise intensities on epidemic propagation are investigated theoretically and numerically. Numerical simulations, including two-dimensional and three-dimensional surface plots, heat maps, stochastic time-series profiles, and phase portraits, are presented to validate the theoretical findings. The results demonstrate that diffusion and stochasticity significantly affect disease transmission patterns, whereas vaccination plays an essential role in suppressing infection spread. The proposed framework provides a more realistic and comprehensive description of epidemic dynamics in heterogeneous environments and may contribute to the development of effective disease control strategies.