Dynamics of two reaction-diffusion SIS epidemic models with spatio-temporal periodic control strategies

In the practice of infectious disease prevention and control, the regional and cyclical interventions (e.g., seasonal sub-regional disinfection, localized lockdowns, or phased vaccination campaigns) are key factors in regulating the dynamic characteristics of disease transmission. In this paper, we investigate the dynamics of two reaction-diffusion susceptible-infected-susceptible (SIS) epidemic models incorporating spatio-temporal periodic control strategies. The models simulate the application of two strategies: one involves establishing local spatial control zones where the disease transmission rate is reduced to zero; the other entails population-wide control (e.g., large-scale vaccination, mask-wearing, or voluntary quarantine) to achieve an overall reduction in the transmission rate. By analyzing the impact of control cycles on disease spread, we find that longer durations and higher proportions of control measures enhance the likelihood of disease eradication, and further derive the critical period value that determines the outbreak and extinction of infectious diseases. In particular, we obtain the optimal and worst positions of the control zone in one-dimensional periodic space. We establish the threshold dynamics of the ordinary differential equation SIS model with multi-stage temporal control and the partial differential equation SIS model with multi-stage population-wide control. Finally, numerical simulations indicate that the optimal control zones should be prioritized for establishment in high-risk areas rather than low-risk areas. These insights may provide theoretical guidance for designing effective disease control policies.