# Patch Models and Applications on the Spread of Avian Influenza

5-2009

Thesis

## Degree Name

Master of Science (MS)

## College/School

College of Science and Mathematics

## Department/Program

Mathematical Sciences

Baojun Song

Arup Mukherjee

Diana Thomas

## Abstract

The avian influenza virus (AIV) is an infectious disease that predominantly affects birds. Economic losses due to large-scale deaths of domestic poultry as a result of past outbreaks have been devastating. Additionally, there is major concern about the spread of the virus to humans. The virus has spread to humans in the past, but has not yet been known to spread beyond one human. Since influenza viruses are known to mutate easily, there is serious concern that the virus could mutate into a strain that can be transmitted easily to and among humans.

There has been much speculation that migratory birds may play a role in the spread of avian influenza. In particular, many types of migratory waterfowl have been shown to carry the disease without having any severe symptoms, suggesting the possibility that they may be ideal vectors for carrying the disease between their breeding and wintering grounds. For this reason, we seek to use mathematical modeling to study the spread of avian influenza, and, in particular, investigate the impact that migratory birds may have.

First we work with a continuous model with three populations. Each population is called a patch. Some birds stay local to each patch, but there are also migratory birds that travel between patches. From the continuous model, we seek to find and interpret the basic reproductive number, Ro. We then estimate parameter values and run simulations to confirm that the disease returns to disease free equilibrium when Ro < 1 and that the disease may invade the populations when Ro > 1.

Then we construct a discrete model for the spread of avian influenza. In the model we consider two separate populations of birds as well as a third population that seasonally travels between the first two populations. We estimate parameter values for a particular species, the mallard duck, in North America. We then use the model and parameter values to run simulations to observe the changes in the prevalence of AIV. We compare our results to the data and also observe long term trends in the prevalence of the disease. The 2-cycles in the observed data are confirmed by simulations of our model. We then do theoretical work on the model to determine under what conditions we will have a periodic solution. In particular, we determine for which types of birth functions and which types of functions for the probability of infection, the periodic solution will hold.

PDF

COinS