A Partially Observable Stochastic Zero-sum Game for a Network Epidemic Control Problem

The mathematical theory of epidemics borrows its fundamental notions from epidemiology. These notions encompass the division of the population of individuals into compartments and thereafter the classification of epidemics in models like SIR (susceptible-infected-removed). Unlike in biology, a broad number of epidemics are spread under the action of rational and intelligent malicious individuals, called attackers. The centralized control of such epidemics implies that a network defender is in conflict with the attacker and therefore should be modeled by a zero-sum noncooperative game theoretic concept. We design a new zero-sum partially observable stochastic game (POSG) theoretic framework in order to capture the interactions between network defender and an attacker. More, our model takes into account random node state transitions. The particularity of our model is that even though the attacker knows the state of the network, which is not the case for the defender, she cannot infer the defender’s actions. The tractability of the solution of classical POSG is improved by the mean of value backup iteration. We prove that the value backup operator converges even in our particular POSG coupled with a compartmental epidemic model.