Estimation of Epidemic Model Parameters: A Spatial Analysis using Bayesian Techniques Público

Abstract

Infectious disease models attempt to evaluate the effects on the spread and trans-
mission of disease. One particular model, the susceptible-infected-recovered (SIR)
model, places individuals into classes of disease progression, where a series of dif-
ferential equations tracks the rates of transmission and recovery for a given disease
through a susceptible population. Two parameters, the transmission parameter and
the recovery parameter, drive the dynamics of the model, and their ratio, R0, is the
average number of cases caused by one infectious individual within a completely sus-
ceptible population. R0 is seen as one of the most important quantities in the study of
epidemics, and signals how quickly a particular disease can spread amongst a suscep-
tible population. Previous analyses have focused primarily on tracking these epidemic
disease parameters over time, and classifying individuals due to baseline differences
which reflect heterogeneity within the population. For example, these differences can
be based on age, gender, vaccination status, or behavior.

However, we choose to quantify the spatial heterogeneity that exists in spatially-
referenced data in an effort to define core areas of disease rates and transmission.
We first consider geographically weighted regression (GWR) models in an effort to
assess the spatial variability that exists between disease rates and baseline tract-
level characteristics which can define core disease areas. Next, we build hierarchical
Bayesian models which incorporate random effects structures, inducing correlation
in local estimates of disease transmission with exchangeable random effects, which
smooth local estimates based on global averages, and conditionally autoregressive
(CAR) random effects, which smooth local estimates based on neighboring estimates.
We extend a chain binomial model to predict the spread of disease, while considering
two different parameterizations of the chain binomial model, and simulate outbreaks
to assess model performance. In addition, we extend a general epidemic model, which
incorporates aspects of frailty models in assessing heterogeneity within the popula-
tion. Through our modeling approaches, we are able to identify cores areas for the
transmission of sexually transmitted infections (STIs) in Baltimore, Maryland from
2002-05.

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 SIR Disease Modeling . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Parameters of interest . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Differential equations . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 R_O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Analysis and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Mathematical vs. Statistical Modeling . . . . . . . . . . . . . . . . 12

2.1 Calculation of R_O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Basic calculation . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2 Survival function . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.3 Multitype model . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.4 Next-generation operator . . . . . . . . . . . . . . . . . . . . . 18

2.2 Statistical Estimation of R_O . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Epidemic curve estimation . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Final outbreak size . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.3 Least squares estimation . . . . . . . . . . . . . . . . . . . . . 20

2.2.4 Chain binomial models . . . . . . . . . . . . . . . . . . . . . . 21

3 Bayesian Inference for Epidemic Modeling . . . . . . . . . . . . . 23

3.1 Chain binomial models . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Transitional approach . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.2 Reed-Frost approach . . . . . . . . . . . . . . . . . . . . . . . . . .30

3.2 General epidemic model . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Introduction to Conceptual Epidemic Models . . . . . . . . . . . . . . . . . 35

4.1 Initial Spatial Analysis: Tracking Spatial Patterns in Prevalence . . . . 35

4.2 Methods and Model Descriptions: A Geographically

Weighted Regression Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Application to Baltimore STI Data . . . . . . . . . . . . . . . . . . . . . . 45

5 Extending the SIR Model to Spatial Analysis . . . . . . . . . . . . . . 51

5.1 Chain Binomial: A Transitional Approach . . . . . . . . . . . . . . . . . 53

5.1.1 Random effects - Exchangeable, Conditionally Autoregressive,

and Convolution Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Transmission Estimation in Chain Binomial Models . . . . . . . . . . 60

5.3 Reed-Frost Chain Binomial Model . . . . . . . . . . . . . . . . . . . . 62

5.4 Chain Binomial Model Overview . . . . . . . . . . . . . . . . . . . . . . . 64

5.5 A Spatial Approach to the General Epidemic Model . . . . . . . . . . 66

5.6 Results: Chain Binomial - Spatial Model . . . . . . . . . . . . . . . . 70

5.6.1 Estimation of Transmission Probability . . . . . . . . . . . . . . . . 70

5.6.2 Estimation of R_O - Transition Chain Binomial Model . . . . . . . 72

5.6.3 Estimation of R_O - Reed-Frost Chain Binomial Model . . . . . 78

5.6.4 Chain Binomial Model Comparison . . . . . . . . . . . . . . . . . . . 82

5.7 Results: General Epidemic Model - Spatial Estimation . . . . . . . . 93

5.8 Assessing Model Performance through Simulations

of the Chain Binomial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.9 Discussion - R_O Estimation Models . . . . . . . . . . . . . . . . . . . 108

6 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.1 Extensions to Existing Models . . . . . . . . . . . . . . . . . . . . . . 110

6.2 Spatially-varying Coefficient Models . . . . . . . . . . . . . . . . . . . 112

6.3 Identifiability Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

About this Dissertation

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School	Laney Graduate School
Department	Biostatistics
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Biology, Biostatistics
Palavra-chave	MCMC Bayesian inference Spatial analysis Infectious disease modeling STI
Committee Chair / Thesis Advisor	Waller, Lance, Emory University
Committee Members	Real, Leslie, Emory University Haber, Michael J, Emory University Hill, Andrew , Emory University

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