On the dynamics of a class of multi-group models for vector-borne diseases

TL;DR

This paper analyzes the dynamics of multi-group models for vector-borne diseases, establishing conditions for disease extinction or persistence based on the basic reproductive number, and extends classical models to more complex contact networks.

Contribution

It introduces a generalized class of multi-group models for vector-borne diseases, including various infection functions and network structures, and characterizes their stability properties.

Findings

The model has only two equilibria: disease-free and endemic.

The basic reproductive number $\\mathcal{R}_0$ determines stability.

Global stability results depend on whether $\mathcal{R}_0$ is above or below 1.

Abstract

The resurgence of vector-borne diseases is an increasing public health concern, and there is a need for a better understanding of their dynamics. For a number of diseases, e.g. dengue and chikungunya, this resurgence occurs mostly in urban environments, which are naturally very heterogeneous, particularly due to population circulation. In this scenario, there is an increasing interest in both multi-patch and multi-group models for such diseases. In this work, we study the dynamics of a vector borne disease within a class of multi-group models that extends the classical Bailey-Dietz model. This class includes many of the proposed models in the literature, and it can accommodate various functional forms of the infection force. For such models, the vector-host/host-vector contact network topology gives rise to a bipartite graph which has different properties from the ones usually found in directly transmitted diseases. Under the assumption that the contact network is strongly connected, we can define the basic reproductive number $\mathcal{R}_0$ and show that this system has only two equilibria: the so called disease free equilibrium (DFE); and a unique interior equilibrium---usually termed the endemic equilibrium (EE)---that exists if, and only if, $\mathcal{R}_0>1$. We also show that, if $\mathcal{R}_0\leq1$, then the DFE equilibrium is globally asymptotically stable, while when $\mathcal{R}_0>1$, we have that the EE is globally asymptotically stable.