St. Lawrence University
Abstract
Epidemiology is a field of science that has made many
significant advances in studies of the spread of disease. Studies
of epidemics and disease spread have vast mathematical components.
Epidemiological models are based on differential equations that
provide the foundations for modeling change over time. These
are useful in modeling rates of infection, rates of recovery, contact
rates, birth and death rates, etc. Such models can
predict the impact of a disease on a population and can suggest
valuable strategies for its control.
This paper will first introduce the most basic epidemiological
model and examine the common uses and implications of certain features.
In the following chapters, we will look at several more complex
models and learn how to modify a disease model according to the
characteristics of a disease. Lastly, we will
examine how researchers have modeled SARS, one of the most recent
disease outbreaks. Our basic knowledge of disease
models will provide the necessary background for such analysis.
We will study three published articles and use this information
to further understand some applications of epidemiological models.
Contents
|
Introduction |
1 |
|
1 What is Epidemiology? |
3 |
|
2 A Simple Model: SIR |
6 |
|
2.1 Law of Mass Action . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . |
6 |
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2.2 Introduction to SIR . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . |
7 |
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2.3 Threshold Quantity . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . |
10 |
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2.4 A Second Method for Estimating R0 . . . . . . . . . . . . . . . . . . |
15 |
|
2.5 Sensitivity of Parameter Estimates . . . . . . . . . .
. . . . . . . . . . |
16 |
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2.6 Maximum Number of Infectives
. . . . . . . . . . . . . . . . . . . . . . |
17 |
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2.7 Immunization . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . |
17 |
|
2.8 Examples . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . |
18 |
|
3 Alternative Approach to Simple Model |
21 |
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3.1 Alternative Approach . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . |
21 |
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3.2 Examples . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . |
22 |
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4 Diseases with no Immunity: SIS |
23 |
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4.1 Introduction to SIS . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . |
23 |
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4.2 Example . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . |
24 |
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5 Demographic Effects |
25 |
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5.1 Birth Rate and Death Rate . . . . . . . . . . . . . .
. . . . . . . . . . . . . |
25 |
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5.2 Model with Demographic Effects . . . . . . . . . . . .
. . . . . . . . . |
25 |
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5.3 Herd Immunity . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . |
28 |
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5.4 Age at Infection . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . |
29 |
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6 SARS |
32 |
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6.1 Introduction to SARS . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . |
32 |
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6.2 Modeling SARS: Detailed Analysis and Case Studies . .
. . . |
34 |
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6.3 Evaluating the Impact of Public Health Measures: SEIR Model . |
47 |
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6.4 Dealing with Superspread
Events: SEIR Model . . . . . . . . . . . . . |
51 |
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6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . |
54 |
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7 Conclusion |
56 |
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References |
58 |