A diffusion time-changed stochastic SIS epidemic model: well-posedness, long-time behavior, and numerical approximation

In this paper, we propose and analyze a diffusion time-changed susceptible-infected-susceptible (SIS) epidemic model driven by time-changed Brownian motion. We prove that the proposed model admits a unique global positive solution for any initial value in $(0,N)$. The extinction and persistence of the disease are then investigated. To approximate the diffusion time-changed SIS model, we construct a positivity-preserving logarithmic Euler-Maruyama (LEM) method. Assuming that the time-changed is given by the inverse of a standard $\alpha$-stable subordinator with $\alpha \in (0,1)$, we prove that the numerical solution converges strongly to the exact solution with order $\alpha$. Finally, numerical experiments are provided to confirm the predicted convergence rates and illustrate the positivity-preserving property of the proposed method.