An optimal control problem for in vitro virus competition

Competition between organisms that occupy the same physical space is an important phenomenon that has been studied extensively through mathematical models, primarily in ecology. Using a relatively benign pathogen with a competitive advantage to contain a more harmful one is an important and relatively new concept in infectious disease management. But many questions remain about effective strategies for the implementation of this concept. This paper presents a method to formulate instances of these questions in the language of dynamical systems theory, amenable to methods from optimal control theory. Specifically, the method begins with a stochastic simulation of in vitro competition of two different virus strains that reproduces competition dynamics observed in experiments. After transforming this simulation into a deterministic simulation, the method proceeds by interpolating a state space model of this dynamical system from time series of system states. That is, a closed form model is constructed from "experiments" performed on the simulation. The interpolation is done using methods from computational algebraic geometry, resulting in a (typically nonlinear) polynomial dynamical system over a finite field. Based on available biological methods, a cost function is defined and a control component is introduced, which provides input to enhance the competitive fitness of one of the two strains. The resulting system can be expressed as a discrete event dynamical system. One can now employ methods developed for optimal control of polynomial dynamical systems over a finite field to solve the control problem so defined. An example of a controller is given, that has been implemented in the laboratory and successfully verified.