Mathematical models, analyses, and simulations of the spread of disease

Abstract

Models in mathematical biology have been evolving for centuries and have had a significant impact on our understanding of biological processes. This work includes three new models. We present an extension of the SIR (Susceptible-Infected-Recovered) model, which allows hospitalization rates and infection rates to vary with time as system variables, and we present two cancer treatment models. The first of the two cancer treatment models explores the effects of combining chemotherapy and immunotherapy in the treatment of a tumor. The second cancer treatment model considers the combination of chemotherapy, immunotherapy, and virotherapy in the treatment of a tumor. For each of these models, we establish that they are positive invariant and bounded, prove that a stable solution exists, and use numerical methods to explore the behavior of the solutions. Each model produced results that encourage further study. Thus, this work is a contribution to the discipline of mathematical epidemiology.