Optimal control of a discrete time stochastic model of an epidemic spreading in arbitrary networks

Preparedness for any future epidemic has become an urgent need. Epidemic modeling and simulation are at the core of the healthcare efforts that are underway to assert some level of control over the spreading and the treatment of a pathogen. In this milieu, this paper describes a stochastic dynamic model to simulate the spreading of infectious diseases. We present the equations that describe the system dynamics, their adjoint systems, and their optimal control characterization by means of the discrete-system extension of Pontryagin's Maximum Principle. This derivation is presented in two different cases: a vaccination policy and a combined vaccination-treatment approach. We show the behavior of such models via numerical simulations using the forward-backward sweep procedure. While somewhat speculative, this paper provides insights into how to evaluate different theoretical optimal healthcare policies during an epidemic, either at the individual or metapopulation resolution level.