Epidemics of random walkers in metapopulation model for complete, cycle, and star graphs.

We present the metapopulation dynamic model for epidemic spreading of random walkers between subpopulations. A subpopulation is represented by a node on a graph. Each agent or individual is either susceptible (S) or infected (I). All agents move by random walk on the graph; namely, each agent randomly determines the destination of migration. The reaction-diffusion equations are presented as ordinary differential equations, not partial differential equations. To evaluate the risk of each subpopulation (node), we obtain the solutions of reaction-diffusion equations analytically and numerically for small, complete, cycle and star graphs. If a graph is homogeneous, or if every node has the same degree, then the solution never changes for any nodes. However, when a graph is heterogeneous, the infection density in equilibrium differs entirely among nodes. For example, on star graphs, the hub seems to be a supply source of disease because the infection density at the hub is much higher than that at the other nodes. On every graph, the epidemic thresholds are identical for all nodes.