Multi-patch epidemic models with partial mobility, residency, and demography

The emergence and re-emergence of infectious diseases has been a global cause of concern in the past few decades. Previous research in the field has revealed that human connectivity and mobility behavior play a major role in the spreading of an infectious disease. In this work, we propose multi-patch models that take into account the effects of human mobility on the evolution of disease dynamics in a multi-population environment. In particular, we develop SEIRS multi-patch and multi-group epidemic models, extending the work of Bichara et al. (2015) and Bichara and Abderrahman (2018) to practically account for distinct epidemiological-status-dependent mobilities in each patch. We rigorously show that the disease free equilibria (DFE) for both models are stable when R₀≤1. We also prove that the models have a unique endemic asymptotically stable equilibrium when R₀>1. In addition, we introduce new local reproduction numbers from the point of view of the sub-populations, and establish some important relation between them and the global reproduction number. Various numerical simulations are conducted to study the effects of mobility and the residence time matrix on the evolution of the disease in individual patches and the overall environment.