## Physics & Astronomy ETDs

#### Publication Date

6-9-2016

#### Abstract

This dissertation is a report on an interdisciplinary investigation consisting of an application of random walk techniques to problems in ecology, particularly to the spread of Hantavirus epidemic among rodents that live on an open terrain. The population of mice that we consider is made up of infectious disease-carrying mice and susceptible mice that are disease-free, and each mouse has its own home range around which it executes a random walk. We describe an event of infection transmission in such a population via reaction-diffusion theory. Our simple model consists of two mice, one infected and the other susceptible, the disease being passed upon encounter as the two mice move on the terrain. The existence of home ranges of the mice is included in the model by representing each mouse to be a Smoluchowski random walker. Such a simple model is appropriate for a dilute population where only one infected-susceptible mice pair is considered to meet at a time. However the calculation helps the understanding of underlying microscopic processes of an epidemic outbreak in an arbitrary population density. The two-mice model is formulated in an arbitrary number of dimensions and explicit calculation in 1-dimension is performed first. We uncover an interesting effect of the home ranges on the characteristics of infection-transmission event. We find that there is an optimal configuration of the home ranges for which infection-transmission occurs most efficiently. Furthermore, the practical application of our model to higher dimensions requires an extension of the theory to circumvent a seemingly well-known problem in reaction-diffusion theory that the `reaction' site cannot be a 0-dimensional object for problems considered in higher dimension than 1. We develop a detailed resolution and present a practical extension with an explicit calculation demonstrated in 2-dimensions. Our work is, thus, useful in two ways. One is the further development of reaction-diffusion theory to tethered random walkers and dimensions higher than 1. The other is to gain insights into the practical problem of the spread of the Hantavirus epidemic.

#### Degree Name

Physics

#### Level of Degree

Doctoral

#### Department Name

Physics & Astronomy

#### First Advisor

Kenkre, V. M.

#### First Committee Member (Chair)

Dunlap, D.

#### Second Committee Member

Lidke, K.

#### Third Committee Member

Moses, M.

#### Project Sponsors

Program in Interdisciplinary Biological & Biomedical Sciences

#### Language

English

#### Keywords

Physics, Non-equilibrium Statistical Mechanics, Epidemic spread, Random walk, Smoluchowski

#### Document Type

Dissertation

#### Recommended Citation

Sugaya, Satomi. "The Smoluchowski equation in population dynamics and the spread of infection." (2016). https://digitalrepository.unm.edu/phyc_etds/66