Vector-borne diseases models with residence times - a Lagrangian perspective

A multi-patch and multi-group modeling framework describing the dynamics of a class of diseases driven by the interactions between vectors and hosts structured by groups is formulated. Hosts’ dispersal is modeled in terms of patch-residence times with the nonlinear dynamics taking into account the $\mathit{\text{effective}}$ patch-host size. The residence times basic reproduction number ${\mathcal{R}}_0$ is computed and shown to depend on the relative environmental risk of infection. The model is robust, that is, the disease free equilibrium is globally asymptotically stable (GAS) if ${\mathcal{R}}_0\le 1$ and a unique interior endemic equilibrium is shown to exist that is GAS whenever ${\mathcal{R}}_0>1$ whenever the configuration of host-vector interactions is irreducible. The effects of $\mathit{\text{patchiness}}$ and $\mathit{\text{groupness}}$, a measure of host-vector heterogeneous structure, on the basic reproduction number ${\mathcal{R}}_0$, are explored. Numerical simulations are carried out to highlight the effects of residence times on disease prevalence.