Global Analysis of Multi-Host and Multi-Vector Epidemic Models

We formulate a multi-group and multi-vector epidemic model in which hosts' dynamics is captured by staged-progression $SEIR$ framework and the dynamics of vectors is captured by an $SI$ framework. The proposed model describes the evolution of a class of zoonotic infections where the pathogen is shared by $m$ host species and transmitted by $p$ arthropod vector species. In each host, the infectious period is structured into $n$ stages with a corresponding infectiousness parameter to each vector species. We determine the basic reproduction number $\mathcal{R}_0^2(m,n,p)$ and investigate the dynamics of the systems when this threshold is less or greater than one. We show that the dynamics of the multi-host, multi-stage, and multi-vector system is completely determined by the basic reproduction number and the structure of the host-vector network configuration. Particularly, we prove that the disease-free \mbox{equilibrium} is globally asymptotically stable (GAS) whenever $\mathcal{R}_0^2(m,n,p)<1$, and a unique strongly endemic equilibrium exists and is GAS if $\mathcal{R}_0^2(m,n,p)>1$ and the host-vector configuration is irreducible. That is, either the disease dies out or persists in all hosts and all vector species.