#### Year of Award

2016

#### Document Type

Dissertation

#### Degree Type

Doctor of Philosophy (PhD)

#### Degree Name

Mathematics

#### Department or School/College

Department of Mathematical Sciences

#### Committee Chair

Emily Stone

#### Commitee Members

Jonathan Bardsley, Leonid Kalachev, Greg St. George, Erin Landguth

#### Keywords

Vector-Borne Relapsing Diseases, disease modeling, qualitative dynamics

#### Publisher

University of Montana

#### Subject Categories

Ordinary Differential Equations and Applied Dynamics

#### Abstract

We begin this dissertation with a review of the relevant history and theory behind disease modeling, investigating important motivating examples. The concept of the fundamental reproductive ratio of a disease, $R_0$, is introduced through these examples. The compartmental theory of disease spread and its results are introduced, particularly the next-generation method of computing $R_0$. We review center manifold theory, as it is critical to the reduction of the dimension of our problems. We review diseases that have a relapsing character and focus in on relapsing diseases that are spread by vectors in a host population. The primary example of such a disease is Tick-Borne Relapsing Fever (TBRF). Motivated by TBRF we establish a general model for the spread of a vector-borne relapsing disease. We then compare our model to current literature.

With a model in hand we confirm that it meets the required hypotheses for the use of compartmental theory. A technical computation then leads to an explicit form of $R_0$ that is given in terms of the number of relapses. Further technical computations then allow us to describe the bifurcation at $R_0=1$, finding that it is always transcritical regardless of the number of relapses. We also show the existence of a unique endemic equilibrium for all values of $R_0$ greater than 1.

Variations of the simple model are explored. Adding in removal to the recovered compartment, in which individuals leave an earlier relapse state and recover, we find how this changes $R_0$ and show that the bifurcation at $R_0$ is still transcritical. We investigate the addition of latent infective compartments and describe how they affect $R_0$. We also find the reproductive ratio when there are two host species that undergo the same number of relapses.

We establish a continuity result between the reproductive ratios of systems with differing numbers of compartments. This allows us to state the reproductive ratio of a smaller system as a limit of the reproductive ratio of a larger system. This result is then used to compute the reproductive ratio for a coupled host-vector system where the hosts undergo a different number of relapses. We close with some conclusions and directions for future work.

#### Recommended Citation

Palmer, Cody, "The Dynamics of Vector-Borne Relapsing Diseases" (2016). *Graduate Student Theses, Dissertations, & Professional Papers*. 10652.

http://scholarworks.umt.edu/etd/10652

© Copyright 2016 Cody Palmer