To research the combined impacts of mean-reverting white noise and spatial diffusion on the transmission of Kawasaki disease (KD), this paper introduces a stochastic reaction-diffusion model for KD incorporating the Ornstein-Uhlenbeck (O-U) process. Firstly, for the deterministic reaction-diffusion model, the local and global stability of the inflammatory factors-free equilibrium (IFFE) and inflammatory factors-existent equilibrium (IFEE) based on basic reproduction number \(R_0\) are discussed: when \(R_0<1\) , the IFFE is globally asymptotically stable and the IFEE is globally asymptotically stable when \(R_0>1\) and certain additional conditions are satisfied. Then, for the stochastic reaction-diffusion model, the existence of a unique solution is proven by constructing a Lyapunov function. Additionally, adequate conditions for the existence of the unique stationary distribution for the positive solution are provided. Finally, numerical simulations are conducted to explore the effects of noise intensity and spatial diffusion on KD, the results show that when the noise intensity is fixed in a certain interval and the diffusion coefficient is greater than a certain value, the disease is persistent. In addition, when the noise intensity is equal to zero, the dynamic behaviors of the stochastic reaction-diffusion system are consistent with the deterministic reaction-diffusion system.
