On the Analysis and Resolution of Certain Optimal Control Models in Biomedical Sciences

Abstract

This thesis focuses on the application of mathematical models to better understand and control infectious diseases. First, it uses a classic model (SEIR) to show how a disease moves through a population. The result is that if the reproduction number R0 is less than 1, the outbreak will die out. In order to model diseases, which have a sort of “memory” of past infections, a more advanced type of math called fractional calculus is used. This new model gives a more realistic view of how diseases like COVID-19 really behave. The second part of this thesis is concerned with determining optimal methods for halting an outbreak. Two major mathematical approaches to finding these optimal control strategies are compared. The results demonstrate that one approach—the Nonsmooth Newton Method is faster and better at handling real-world, complicated limits on resources than its predecessor, Pontryagin’s Maximum Principle. A new neural network-based approach is also proposed for solving these control problems. In other words, this work develops better mathematical tools for the simulation of epidemics and offers more efficient means of calculating the optimal public health actions like vaccinations required to control such outbreaks