Direct and indirect optimal control applied to plant virus propagation with seasonality and delays

Abstract

In many applications of mathematical modeling to biology, economics, social sciences and engineering, the objective is to find optimal solutions. Usually we want to minimize an objective function depending on a number of functions subject to constraints given, for example, by systems of differential equations. Two main numerical approaches are used to solve these optimal control problems, depending on whether the problem is optimized first and then discretized, or vice versa. Each of these two approaches has its advantages and disadvantages. In this paper we describe both methods an apply them to a plant virus propagation model, where the virus is propagated through a vector that bites the infected plants. The model includes delays due to the time the virus takes to infect the plant and the vector, and seasonality due to the dependence of the behavior on the seasons. The objective function is the total cost to a farmer of having infected plants, and includes the actual cost of a plant plus the cost of the controls which are insecticides and a predator species that preys on the insects. Numerical simulations are done using both methods and comparisons are made.