Abstract
We develop a method that completely characterizes the global dynamics of models with multiple subpopulations that are weakly interconnected. The method is applied on two classes of models with multiple subpopulations: an epidemic model that involves multiple host species and multiple vector species and a patchy vector-borne model. The method consists of two main steps: reducing the system using tools of large scale systems and studying the dynamics of an auxiliary system related the original system. The developed method determines the underlying dynamics and the “weight” of each subpopulations with respect to the dynamics of the whole population, and how the topology of the connectivity matrix alters the dynamics of the overall population. The method provides global stability results for all types of equilibria, namely trivial, boundary or interior equilibria.
Highlights
Mathematical models in population dynamics that incorporate age, group, or spatial heterogeneities use network to describe the interactions between the units of the model
We presented a procedure that provides a complete analysis of a class of large dynamical systems that model a disease that involves the interactions of multiple populations
We provided a method that characterizes the global dynamics of these types of model regardless of the nature of the network configuration
Summary
Mathematical models in population dynamics that incorporate age, group, or spatial heterogeneities use network to describe the interactions between the units of the model. In most cases the graph that represents the connections between the subpopulations is not strongly connected and the dynamics is not well-understood It has long been assumed in the literature of Keywords and phrases: Large scale systems, network configuration, weakly connected populations, global stability, Lyapunov functions. When the patches are weakly connected, determining the sinks and sources of these vector-borne diseases allows informed intervention strategies that target the more prolific patches and a cost-effective control strategies The choice of these two types of models is motivated by the fact that the dynamics of these models have been well understood if the network is strongly connected [8, 11, 24, 29], a necessary step to determine the behavior of models with weakly connected subpopulations.
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