| While the immune system is extraordinarily complex and powerful, and medical advancements are more spectacular than ever, in recent history we have seen the unfortunate failure of both processes (immune system and drugs) in the increasing levels of persistent infections. This work is an example of a collaborative effort to study multiple forms of persistent infections using mathematical tools and analyses. We will discuss the biological backgrounds for the immune system's components and functions, the bacterial and viral resistance mechanisms for methicillin-resistant Staphylococcus aureus and the human immunodeficiency virus, respectively, and some of the current methods for treating these diseases. Then, using ordinary and partial differential equations we present the results of models that were created to study specific infections---namely, methicillin-resistant Staphylococcus aureus osteomyelitis, Staphylococcus aureus nasal carriage, and human immunodeficiency virus prophylactic gel. These models are shown to be in good agreement with the biology by looking at, when possible, their analytical solutions and numerical results when compared to experimental evidence. We further explore these models using several different computational analyses that can be classified as at least one of the following methods: uncertainty quantification, sensitivity analysis, and data assimilation. We give an overview of each of these topics and delve into the technicalities and caveats of each of the analyses we apply. We show that all of the methods presented, individually and in concert, are valuable tools for not only revealing details about the model structure and verifying model robustness, but they can also bring to light elements of the biological phenomena that the model represents. While considering all these details, throughout the manuscript we consider the philosophical perspective of biological modeling and modeling in general. |
