Optimal Control and Dynamic Games in Epidemic Diffusion Processes - 11/12/2010

Arman Khouzani

Abstract: We investigate the optimal control of systems whose state varies with time in a non-stationary manner and where the transient behavior of the system is the subject of interest. Specifically, we consider a system whose evolution is governed by an epidemic diffusion. Epidemic behavior emerges whenever interactions between a large number of individual entities affect the overall evolution of the encompassing system. We first make a connection between the deterministic mathematical models based on nonlinear differential equations and the underlying stochastic processes. Next we present a general mathematical framework for calculating optimal controls of systems governed by epidemic evolution using Pontryagins Maximum Principle. Further, using simple analyses, we discuss how one can extract substantial information about the structure of optimum policies in the absence a closed-form solution. We illustrate the applicability of our model and analyses by considering the problem of optimal defense against malware outbreaks in a mobile wireless network. Specifically, we demonstrate how surprisingly simple policies prove to be optimal in containing the malware, while consuming the least bandwidth and energy resources. Dynamic resource management is significantly more challenging when independent entities can dynamically and strategically affect the evolution of the system as well. We draw tools from dynamic game theory to address such problems.

-- JiaxiJin - 13 Nov 2010