Recent Advances in Modeling and Control of Epidemics using a Mean Field Approach

Modeling and control of epidemics such as the novel Corona virus have assumed paramount importance at a global level. A natural and powerful dynamical modeling framework that has emerged in this context is a continuous time Markov decision process (CTMDP) that encompasses classical compartmental paradigms such as the Susceptible-Infected-Recovered (SIR) model. Using a CTMDP based model poses certain technical and computational challenges. These challenges motivate the need for a more efficient approach and the mean field approach offers an effective alternative. The mean field approach computes the collective behavior of a dynamical system comprising numerous interacting nodes (where nodes represent individuals in the population). This paper provides a state-of-the-art update on recent advances in the mean field approach to epidemic modeling and control. Our discussion in this paper proceeds in two threads. The first thread assumes that the individual nodes faithfully follow a socially optimal control policy prescribed by a regulatory authority. The second thread allows the individual nodes to exhibit independent, strategic behavior. In this case, the strategic interaction is modeled as a mean field game and the control is based on the associated mean field Nash equilibria. In this paper, we start with a discussion of modeling of epidemics using an extended compartmental model - SIVR and provide an illustrative example. We then provide a review of relevant literature, using a mean field approach, on optimal control of epidemics, dealing with how a regulatory authority may optimally contain epidemic spread in a population. Following this, we provide an update on the state-of-the-art literature on the use of the mean field game based approach in the study of epidemic spread and control. We conclude the paper with some future research directions.