Impact of combined effects of vectors, protective measures, and vaccination under threshold policy control on the SISV model

This paper introduces a novel class of Filippov susceptible-infected-susceptible-vaccinated model that considers the combined effects of media coverage, protective measures, and vaccination. Unlike traditional models that focus on a single control measure, this study reveals more intricate dynamic behaviors arising from the synergistic impact of these three strategies. We use the scale of infected individuals and their rate of change as criteria for implementing control measures, conducting a comprehensive evaluation. By utilizing the characteristics of the Lambert W function, we effectively convert these criteria into a threshold value linked to the susceptible population, thereby facilitating the analysis of the dynamic behaviors of the two subsystems. Utilizing the theoretical framework of Filippov systems, we derive the conditions for the existence of sliding segments, sliding dynamics, various types of equilibria, and the occurrence of sliding bifurcations. Through qualitative analysis, and based on the critical thresholds R0i(i=1,2), we elucidate the complex dynamics of the proposed model, including scenarios of monostable, bistable, or even tristable coexistence. Numerical simulations further explore the effects of key parameters related to the treatment strategies, demonstrating that media coverage, protective measures, and vaccination play pivotal roles in controlling the spread of the disease. Our findings indicate that by selecting appropriate threshold values, it is possible to effectively limit the peak number of infected individuals and the overall scale of the outbreak to a desired level. This provides a robust control strategy for managing emergent infectious diseases that cannot be immediately eradicated.