Migration-Contagion Processes

Abstract

Consider a migration process based on a closed network of N stations with K_N customers. Each station is a ./M/\infty queue with service (migration) rate mu. Upon departure, a customer is routed at random to another station. In addition to migration, these customers are subject to an SIS (Susceptible, Infected, Susceptible) dynamics: customers are either I for infected, or S for susceptible. They can swap their state either from I to S or from S to I only in stations. At any station, each S customer becomes I with rate alpha Y if there are Y infected customers in the station, and each I customer recovers and becomes S with rate beta. We let N tend to infinity and assume that lim_{N\to infty} K_N/N= eta>0. The main problem is about the set of parameters for which there exists a stationary regime where the epidemic survives in the thermodynamic limit. We establish several structural properties of the system, which allow us to give the phase transition diagram of this thermodynamic limit w.r.t. eta. The analysis of the SIS model reduces to that of a wave-type PDE for which we found no explicit solution. This SIS model is one among several companion stochastic processes with migration and contagion. Two of them are discussed as they provide some bounds and approximations to SIS. These two variants are the DOCS (Departure On Change of State) and the AIR (Averaged Infection Rate), which both admit closed-form solutions. The AIR system is a mean-field model where the infection mechanism is based on the empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. It leads to another wave-type PDE that admits a closed-form solution.