SIR model with random diffusion, reinfection, and random transmission: exponential attractors and spread of the disease

We introduce a stochastic SIR-type partial differential equation model incorporating random diffusion, reinfection, vital dynamics, and a randomly varying transmission rate. For the associated random dynamical system, we prove the existence of both random and exponential attractors. We construct a non-stationary, random disease-free global solution, which serves to localize the random attractor. Furthermore, we analyze the mean value of the random transmission coefficient to establish conditions under which the disease may either be eradicated or persist in an endemic state, depending on the system’s parameters.