Optimal Control Applied to Resource Allocation in a PDE Ecosystem Model and a Zika Virus Epidemic Model

This dissertation focuses on developing two mathematical models applied to spatial ecology and epidemiology. More specifically, one model considers the spatiotemporal dynamics of a population with its resources while the second model studies the dynamics of a Zika Virus epidemic.

The effect caused by habitat heterogeneity on a diffusing population is a topic of great importance to understand population dynamics. We formulate a reaction-diffusion system of partial differential equations to analyze the effect of resource allocation in an ecosystem with resources having their own dynamics in space and time. We show \textit{a priori} estimates to prove the existence of state solutions given a control. We formulate an optimal control problem of our ecosystem model such that the abundance of a single species is maximized while minimizing the cost of inflow resource allocation. In addition, we show the existence and uniqueness of the optimal control as well as the optimal control characterization. We also establish the existence of an optimal diffusion rate. Moreover, we illustrate several numerical simulations with Dirichlet and Neumann boundary conditions with the space domain in 1D and 2D.

Zika virus (ZIKV) is a mostly non-lethal disease in humans transmitted by mosquitoes or humans that can produce severe brain defects such as microcephaly in babies and Guillain-Barr\'e syndrome in elderly adults. The use of optimal control strategies involving information campaigns about insect repellents and condoms alongside an available and effective vaccine can notably prevent the number of infected humans with ZIKV. A system of nonlinear ordinary differential equations is formulated for the transmission dynamics of ZIKV in the presence of three control strategies to evaluate the impact of various scenarios during a ZIKV epidemic. In addition, we estimate parameters using data from previous ZIKV outbreaks in Colombia and Puerto Rico to capture the dynamics of an epidemic in each country when control measures are available.