  The relationship between epidemiology, mathematical modeling and
computational tools allows to build and test theories on the development and
battling of a disease. This PhD thesis is motivated by the study of
epidemiological models applied to infectious diseases in an Optimal Control
perspective, giving particular relevance to Dengue. Dengue 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 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
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.

Optimization and Control;Populations and Evolution