Stochastic analysis of a generalized reaction–diffusion epidemic model with latency and Black–Karasinski process

As population mobility and environmental noise are crucial for pathogen spread, this paper constructs stochastic partial differential equations to study the dynamic evolution of an epidemic model with latency and general incidence rates, which is the first attempt to consider the Black–Karasinski mean-reverting process as the perturbation in a reaction–diffusion system. Based on the unique existence verification of the global classical solution that guarantees the availability of Ito^ formula, we present two main contributions. Firstly, by formulating appropriate functions, the time average of the illness population within the bounded area is analyzed to obtain sufficient criteria for the disease persistence or extinction. Secondly, from an innovative decomposition perspective, we prove that the solution asymptotically follows the normal distribution near the quasi-endemic equilibrium, and the specific density function is calculated. From the presented results, which are also supported by computer simulation, we emphasize that the mean-reverting process has a significant impact on the epidemic, that is, diseases that would become extinct in a deterministic environment may persist. More importantly, the corresponding deterministic model is not only generalized and optimized in form but also in conclusion.