Dynamical behaviors of a stochastic SIR epidemic model with reaction–diffusion and spatially heterogeneous transmission rate

Much effort has been paid to epidemic models built by ordinary differential equations (ODEs), partial differential equations (PDEs), or stochastic differential equations (SDEs) and received remarkable achievement. Different from these models, we establish and analyze a SIR epidemic model by using stochastic partial differential equations (SPDEs) in this paper, which incorporates the influence of inevitable population diffusion, spatial heterogeneity, and environmental perturbation. For this model, the existence and uniqueness of the global positive solution is first proved through an innovative variable transformation approach. Then, we establish the sufficient condition for the existence of the Infected class by constructing suitable Lyapunov functions. The exponential extinction of disease is also investigated. More importantly, the exact expression of the probability density function near the equilibrium is obtained by theoretical analysis and matrix calculation. Further, we perform several numerical simulations to illustrate theoretical results. Finally, the corresponding conclusions and prospects are discussed.