## Electronic Theses and Dissertations

6-2019

#### Level of Access Assigned by Author

Open-Access Thesis

#### Degree Name

Master of Arts (MA)

Mathematics

David Hiebeler

Jaehong Jeong

#### Third Committee Member

Peter Stechlinski

#### Abstract

Epidemiological models are an essential tool in understanding how infection spreads throughout a population. Exploring the effects of varying parameters provides insight into the driving forces of an outbreak. In this thesis, an SIS (susceptible-infectious-susceptible) model is built partnering simulation methods, differential equations, and transition matrices with the intent to describe how simultaneous recoveries influence the spread of a disease in a well-mixed population. Individuals in the model transition between only two states; an individual is either susceptible — able to be infected, or infectious — able to infect others. Events in this model (infections and recoveries) occur by way of a Poisson process. In a well-mixed population, individuals in either state interact at a constant rate where interactions have the potential to infect a susceptible individual (infection event). Recovery events, during which infectives transition from infectious to susceptible, occur at a constant rate for each infected individual. SIS models mimic the behavior of diseases that do not confer immunity to those previously infected. Examples of such diseases are the common cold, head lice, and many STIs [2]. This model describes the effects the scale of recovery events have on an outbreak. Thus, for each recovery event, k number of infectives recover. The rate at which recoveries occur is inversely proportionate to k in order to maintain the average per-capita rate of recovery. A system of ordinary differential equations (ODEs) is derived and supported by simulated data to describe the first and second moments (used to describe mean and variance) of the probability density function defining the number of infectious individuals in the population. Additionally, a Markov chain describes the process via transition matrices, which provide insight on extinctions caused by large-scale recoveries and their effect on the mean. The research shows as the values of k increase, there is a statistically significant decline in the average infection level and an increase in the standard deviation. The most extreme changes in the average infection level are observed under conditions that increase the probability of extinction. Even in small populations where the decreased infection level is not biologically significant, the results are beneficial. Because large-scale recovery events have no negative impact on average infection levels, treatment methods that may reduce costs and increase accessibility could be adopted. Healthcare professionals utilize epidemiological models to understand the severity of an outbreak and the effectiveness of treatment methods. A key feature of mathematically modeling real-world processes is the level of abstraction it offers, thus making the models applicable to many fields of study. For instance, those interested in agricultural development use these models to treat crops efficiently and optimize yield, cybersecurity experts use them to investigate computer viruses and worms, and ecologists implement them when studying seed dispersal.