Metapopulation dynamics on double graphs for mediated infectious disease in patchy environment

We study the metapopulation dynamics of such mediated infectious disease as mosquito-borne infection in a patchy environment. The action range and habitat of mediators (e.g. mosquito) are different from those of humans. We focus on the effect of the difference on epidemic spreading in a patchy environment. The patchy environment and migration paths are represented by double graph: the first graph is for human and the second graph for mediators. We present a metapopulation dynamic model on homogeneous and heterogeneous double graphs; a subpopulation (patch) is represented by a node on a graph and a link represents a migration path between patches. Mobile individuals move by random walk through a link between nodes. Each individual is either susceptible human (S), infected human (I), susceptible mediator (M) or infected mediator (D). We extend SIS epidemic model to taking into account mediated infection. The reaction–diffusion equations on double graphs are presented for four species of S, I, M, and D. To evaluate the infection risk of each patch (node), we obtain the solutions of reaction–diffusion equations on double graphs numerically. It is shown that the densities of infected individuals depend highly on both action range and graph structure. The infection threshold is controlled by the density of mediators and their living area.