Applications of Optimization in Runner and Mosquito Models

Optimization is a tool used across mathematics for problems of varying nature. In this dissertation we present 2 models where optimization was applied to obtain solutions. The first model describes a runner competing in a race, in particularly in a marathon where they take nutrition throughout the race. As nutrition is an integral part of successfully running long distance races, such as a marathon, it needs to be included in models of running strategies. We formulate a system of ordinary differential equations to represent the velocity, fat energy, glycogen energy, and nutrition for a runner competing in a long-distance race. The energy compartments represent the energy sources available in the runner’s body. We allocate from which energy source the runner draws from, based on how fast the runner is moving. The food consumed during the race is a source term for the nutrition differential equation. With our model, we are investigating strategies to manage the nutrition and propulsion force in order to minimize the running time in a fixed distance race. We obtain simulation results for different levels of runners completing different lengths of races. Our results confirm the belief that the most effective way to run a race is to run approximately the same pace the entire race without letting ones energies hit 0. Our other model considers the La Crosse virus in a mosquito population. In Appalachia, La Crosse virus (LACV) is a leading pediatric arbovirus and public health concern for children under 16 years. LACV is transmitted via the bite of an infected \emph{Aedes} mosquito. Thus, it is imperative to understand the dynamics of the local vector population in order to assess risk and transmission. Using entomological data collected from Knox County, Tennessee in 2013, we formulate an environmentally-driven system of ordinary differential equations to model mosquito population dynamics over a single season. Further, we include infected compartments to represent LACV transmission within the mosquito population. Findings suggest that the model, with dependence on degree days and accumulated precipitation, can closely describe field data. This model confirms the need to include these environmental variables when planning control strategies.