Asymptotic Profiles and Disease Prevalence at the Steady State for an SIS Patch Model

Infected individuals often display mobility patterns that differ significantly from those of healthy individuals—traveling less frequently, covering shorter distances, visiting fewer destinations, and altering their timing and modes of movement. In this paper, to explore the influence of changes in travel frequency and destination on the spatial spread of infectious diseases, we propose a susceptible–infectious–susceptible patch model in which susceptible and infected populations have different dispersal rates and connectivity matrices. We first establish the threshold dynamics in terms of the basic reproduction number (Formula presented.) and show the existence and uniqueness of endemic equilibrium (EE) when (Formula presented.). Then we examine the asymptotic profiles of the EE under small dispersal rate of the susceptible or infected population. In particular, we prove that as susceptible mobility tends to zero, the EE converges a disease-free equilibrium in the most general case. We find that asymmetric dispersal provides a new approach to eliminate infections than small susceptible mobility. Furthermore, we analyze both local and global disease prevalence to identify strategies for lowering endemic level. Variations in connectivity matrix can lead to high prevalence in low-risk patch, a failure of the order-preserving property on local prevalence. Numerical simulations are conducted to further reveal the role of heterogeneous mobility patterns. Overall, this study offers new insights into how human movement shapes the distribution of disease and generalizes many results in the literature.