Integrating the heterogeneity of contact structures and stochastic factors, we establish a continuous-time Markov SIRS epidemic model with vaccination on networks. Based on this framework, we derive both ordinary differential equations and stochastic differential equations, serving as first- and second-order approximations for the Markov chain, respectively. Additionally, we investigate the quasi-stationary distribution and the time to extinction to characterize the long-term dynamical behavior of the Markov chain. The results reveal that the time to extinction from the quasi-stationary distribution follows an exponential distribution. By using the diffusion approximation, we further obtain approximations for both the quasi-stationary distribution and the expected time to extinction, which are given by a multivariate normal distribution and an explicit formula, respectively. Finally, numerical simulations illustrate that vaccination can accelerate disease extinction and that a greater proportion of individuals with higher degrees leads to a longer time to extinction.
Chen et al. (Fri,) studied this question.