Epidemic-Like Stochastic Processes with Time-Varying Behavior: Structural Properties and Asymptotic Limits

The mathematical analysis of epidemic-like behavior has a rich history, going all the way back to the seminal work of Bernoulli in 1766 [5]. More recently, mathematical models of epidemic-like behavior have received considerable attention in the research literature based on additional motivation from areas such as communication and social networks, cybersecurity systems, and financial markets; see, e.g., [6]. The types of viral behaviors exhibited in many of these applications tend to be characterized by epidemic-like stochastic processes with time-varying parameters [12, 13]. In this paper we consider variants of the classical mathematical model of epidemic-like behavior analyzed by Kurtz [8],[7, Chapter 11], extending the analysis and results to first incorporate time-varying behavior for the infection and cure rates of the model and to then investigate structural properties of the interactions between local (micro) and global (macro) behaviors within the process. Specifically, we start by formally presenting an epidemic-like continuous-time, discretestate stochastic process in which each individual comprising the population can be either in a non-infected state or in an infected state, and where the rate at which the noninfected population is infected and the rate at which the infected population is cured are both functions of time. We established that, under general assumptions on the timevarying processes and under a mean-field scaling with respect to population size n, the stochastic processes converge to a continuous-time, continuous-state time-varying dynamical system. Then we study the stationary behavior of both the original stochastic process and the mean-field limiting dynamical system, and verify that they, in fact, have similar asymptotic behavior with respect to time. In other words, we establish that the following diagram is commutative.