Investigation on stochasticity-induced persistence

The spread of epidemics is inevitably influenced by a wide range of random factors, resulting in complex and diverse dynamics. This study develops a stochastic epidemic model driven by the Black-Karasinski process to investigate the impact of environmental noise on disease extinction and uniform persistence. Two key quantities are formulated to characterize the conditions for disease persistence and extinction, with rigorous analysis of their relationship. Sufficient conditions for the existence of a stationary distribution and disease extinction are established. Additionally, an optimal control problem is proposed for the stochastic model to achieve a dynamic balance between disease control and socio-economic requirements, with the optimal control derived. Notably, our findings reveal that the persistence-determining quantity for the stochastic system is higher than the basic reproduction number of the deterministic system. Most intriguingly, we find that noise favors to the persistence of the disease. Such a stochasticity-induced persistence effect challenges the conventional conclusion that large noise suppresses disease prevalence. Numerical studies further examine the effect of random factors on the quantities and the dynamic behavior especially when unity is between the two quantities. Additionally, the effect of noise intensity on extinction probability and the noise distributions associated with the transition from extinction to persistence is explored through numerical simulations.