Threshold Dynamics of an Emergent Respiratory Infectious Disease Model With Semi‐Markovian Switching

ABSTRACT Outbreaks of emergent respiratory infectious diseases pose a significant threat to the daily functioning of human society, causing profound disruptions to life and massive loss of life. To effectively address current and potential future infectious disease emergencies, global academia has actively engaged, utilizing highly specialized epidemic dynamic models to analyze the spread and inherent patterns of diseases. This paper delves into the construction and examination of an innovative semi‐Markov switching diffusion model, which comprehensively considers the dual effects of hospital bed capacity constraints and vaccination strategies on disease transmission. Specifically, we first theoretically ensure the positivity of solutions in our stochastic epidemic model, a fundamental prerequisite for its validity. Subsequently, we ingeniously construct a triplet Markov process that meets specific conditions and rigorously proving its Feller property. These derivations not only deepen our understanding of the model's internal mechanisms but also provide a solid theoretical foundation for establishing the basic reproduction number as the pivotal threshold parameter that determines the future trajectory of the disease (i.e., eradication or persistence). Specifically, an below 1 signifies effective disease control and eventual eradication, whereas an above 1 indicates sustained transmission within the population, accompanied by the existence of a stationary distribution in the stochastic model, posing a severe challenge for public health.