Optimal Control and Numerical Optimization Applied to Epidemiological Models

The relationship between epidemiology, mathematical modeling and computational tools allows to build and test theories on the development and battling of a disease. This thesis is motivated by the study of epidemiological models applied to infectious diseases in an Optimal Control perspective, giving particular relevance to Dengue. It is a subtropical and tropical disease transmitted by mosquitoes, that affects about 100 million people per year and is considered by the World Health Organization as a major concern for public health. The mathematical models developed and tested in this work, are based on ordinary differential equations that describe the dynamics underlying the disease, including the interaction between humans and mosquitoes. An analytical study is made related to equilibrium points, their stability and basic reproduction number. The spreading of Dengue can be attenuated through measures to control the transmission vector, such as the use of specific insecticides and educational campaigns. Since the development of a potential vaccine has been a recent global bet, models based on the simulation of a hypothetical vaccination process in a population are proposed. Based on the Optimal Control theory, we have analyzed the optimal strategies for using these controls and respective impact on the reduction / eradication of the disease during an outbreak in the population considering a bioeconomic approach. The formulated problems are numerically solved using direct and indirect methods. The first discretize the problem turning it into a nonlinear optimization problem. Indirect methods use the Pontryagin Maximum Principle as a necessary condition to find the optimal curve for the respective control. In these two strategies several numerical software packages are used. Throughout this study, there was a compromise between the realism of epidemiological models and their mathematical tractability.