A new stochastic threshold analysis for a spatial reaction-diffusion SVIR epidemic model with multiplicative noise

We study a spatially heterogeneous stochastic SVIR epidemic model with vaccination and two transmission pathways, posed as a reaction-diffusion system with homogeneous Neumann boundary conditions and driven by multiplicative Gaussian noise. We establish global well-posedness, boundedness, and positivity of solutions using a pathwise transformation and semigroup-based estimates. We then derive two explicit stochastic threshold indices: a persistence threshold ensuring that the time-averaged expected prevalence remains bounded away from zero, and an extinction threshold implying almost sure exponential die-out of the infection. We prove an ordering relation between these indices, which brackets the transition between persistence and elimination. Finally, we propose a positivity-preserving finite-difference Euler-Maruyama scheme and provide simulations illustrating the sharpness of the thresholds and the influence of spatial heterogeneity and noise intensity on long-time dynamics.