A network SIS meta-population model with transportation flow

This paper considers a deterministic Susceptible-Infected-Susceptible (SIS) metapopulation model for the spread of a disease in a strongly connected network, where each node represents a large population. Individuals can travel between the nodes (populations). We derive a necessary and sufficient condition for the healthy equilibrium to be the unique equilibrium of the system, and then in fact it is asymptotically stable for all initial conditions (a sufficient condition for exponential stability is also given). If the condition is not satisfied, then there additionally exists a unique endemic equilibrium which is exponentially stable for all nonzero initial conditions. We then consider time-delay in the travel between nodes, and further investigate the role of the mobility rate that governs the flow of individuals between nodes in determining the convergence properties. We find that sometimes, increasing mobility helps the system converge to the healthy equilibrium.