Consider the following migration process based on a closed network of N queues with K_N customers. Each station is a ./M/infty queue with service (or migration) rate mu. Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to an SIS (Susceptible, Infected, Susceptible) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can only swap their state either from I to S or from S to I in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate alpha.Y if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate beta. We let N tend to infinity and assume that \lim_{N\to infty} K_N/N= eta := lambda/mu, where \eta is a positive constant representing the customer density.The main problem of interest is about the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We establish several structural properties (monotonicity and convexity) of the system, which allow us to give the structure of the phase transition diagram of this thermodynamic limit w.r.t. \eta.

The analysis of this SIS model reduces to that of a wave-type PDE for which we found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds. These two variants are the SIS-DOCS (Departure On Change of State) and the SIS-AIR (Averaged Infection Rate), which both admit closed form solutions. The SIS-AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. SIS-DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed form solution. In this text, the main focus is on the closed systems and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest.

Joint work with S. Foss and S. Shneer.