The Spread of Rabies in Raccoons: Numerical Simulations of a Spatial Diffusion Model Open Access

Keller, Joshua Parker (2011)

Permanent URL: https://etd.library.emory.edu/concern/etds/z603qx60f?locale=en
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Abstract

Abstract
The Spread of Rabies in Raccoons: Numerical Simulations of a Spatial Diffusion Model
By Joshua P. Keller
Raccoons, Procyon lotor, are a major carrier of the rabies virus in the Eastern United States.
Compartmental models of disease provide an effective means for simulating epidemics of rabies
and other diseases. However, most epidemiological models either do not consider individual
movement at all or model it between discrete patches. Modeling movement as a continuous
process provides several advantages, including the ability to incorporate spatially-oriented
geographic and environmental factors affecting disease. We develop an SEIR model for disease
spread with a diffusion component for continuous movement that is a system of partial
differential equations (PDEs). We perform numerical simulations of the PDEs using the finite
element method. We use the model to analyze a case study of the spread of rabies in raccoons in
New York State since 1990.

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Compartmental Models for Infectious Diseases . . . . . . . . . . . . 4
2.1 SIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 SIS, SEIR, SEI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Vital Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Malthusian dynamics and stability . . . . . . . . . . . . . . . . . . . 11
2.3.2 Non-Malthusian dynamics . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Spatial Movement in SEIR Models . . . . . . . . . . . . . . . . . . . . . 15
3.1 Discrete Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Diffusion: The continuous approach to movement . . . . . . . . . 18
3.3 The Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 The Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 Space discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.1 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Weak Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 28
4.3 Assembly of Stiffness and mass matrix . . . . . . . . . . . . . . . . . 30
4.4 Time Advancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5 Data Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.6 Unit Square Simulations: The Effect of Parameter Variation . . . 34
5 Case Study: Rabies in New York . . . . . . . . . . . . . . . . . . . . . . . 42
5.1 Numerical Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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