Corbitt_Thesis

APPROVAL SHEET
Title of Dissertation: Mathematical Modeling the Etiology of Schizophrenia and
White Matter Lesions
Name of Candidate: Paul Timothy Corbitt
Doctor of Philosophy, 2014
Dissertation and Abstract Approved: _________________________
(Ivan Kramer)
(Associate Professor)
(Physics)
Date Approved: 4/25/14

Curriculum Vitae
Paul Timothy Corbitt
Educational History
2014: Doctorate of Philosophy Applied Physics, University of Maryland, Baltimore
County
Thesis Topic: Application of Physical Mathematical Models to the Etiology of
Schizophrenia and White Matter lesions.
2014: Predoctoral Fellow: Intramural Research Training Award at the National Institute
on Deafness and Other Communication Disorders, Brain Imaging and Modeling
Section
2008 Master of Science Applied Physics, University of Maryland, Baltimore County
Non-Thesis Option, Qualifying Exams Full Pass at the Masters Level on First
Attempt
GPA: 3.6
2006 Master of Science Applied and Industrial Mathematics, Towson University
Thesis Project: National Institute of Justice Geographic Profiling Project
Mathematical model development to locate criminal anchor points by integrating
Geographic Information Systems and crime location information.
GPA: 3.92
2004 Bachelor of Science Washington College
Majors: Physics and Mathematics
Minor: Computer Science
Mathematics Thesis: Torus Links and the Bracket Polynomial (with Honors).
Published in non-refereed Online Journal of Undergraduate Papers in Knot
Theory
Physics Thesis: Modeling the Evolution of Radiation Toward Thermal
Equilibrium
GPA: 3.789

Teaching Experience
University of Maryland, Baltimore County Teaching Assistant Fall 2006 – Fall 2009
Instructor: Tutorial for First and Second Semester Calculus Based Physics, First
Semester Algebra Based Physics Laboratory
Grader: First and Second Semester Calculus Based Physics, Classical Mechanics,
Third Semester Introductory Calculus Based Physics
Research Assistant Spring 2010 – Spring 2013: National Institute of Health Living
Physics Project: Revising and enhancing education in introductory algebra-based physics
labs with the goal of adding genuine biological content.
Guest Lecturer: First Semester Introductory Calculus based Physics, First and Second
Semester Algebra Based Physics
Towson University
Part Time Graduate Assistant
Instructor: Pre-calculus Tutorial, Calculus Computer Lab
Grader: Basics Statistics, Pre-calculus, Multivariable Calculus Lab, Probability,
Mathematical Statistics
Guest Lecturer: Differential Calculus
Washington College
Physics Laboratory Assistant: 2000-2001 and 2003-2004 Academic Years
Math Center Tutor: 2000-2001 and 2003-2004 Academic Years
Presentations:
Corbitt, Paul. The Life of Willard Gibbs: An American Scientist. Joseph F. Mulligan
Memorial Lecture, University of Maryland, Baltimore County; May 11, 2011, Baltimore,
MD.
Corbitt, Paul. Applying Nuclear Decay Models to Schizophrenia. Presentation at:
33rd
Annual Graduate Research Conference at University of Maryland, Baltimore
County; April 29, 2011, Baltimore, MD.

Posters:
Corbitt, Paul, MalleTagamets, Peter Kochunov, Joanna Curran, Rene Olvera, John
Blangero, David Glahn. Toward a Normative Measure of White Matter
Abnormalities during the Adult Lifespan. Poster presented at: Neuroanatomy: White
Matter Anatomy, Fiber Pathways and Connectivity session at 18th
Annual Meeting of the
Organization for Human Brain Mapping; June 10-14, 2012, Beijing, China.
Corbitt, Paul, Eric Anderson, Lili Cui. Mathematical Modeling in Introductory
Physics for Biologists. Poster presented at: Reforming the Introductory Physics Course
for Life Science Majors IV Session at the AAPT (American Association of Physics
Teachers) 2011 Winter Meeting; January 8-12, 2011, Jacksonville FL.
Anderson, Eric ,Lili Cui, Amita Rajani, Paul Corbitt, Weihong Lin. Modeling the
Action Potential. Poster presented at: Reforming the Introductory Physics Course for
Life Science Majors IV Session at the AAPT (American Association of Physics
Teachers) 2011 Winter Meeting; January 8-12, 2011, Jacksonville FL.
Papers
Corbitt, Paul, Malle Tagamets, Peter Kochunov, Joanna Curran, Rene Olvera, John
Blangero, David Glahn. White Matter Lesion Evolution across the Adult Lifespan of a
Mexican American Population. In preparation.
Corbitt, Paul, Malle Tagamets, Ivan Kramer. Mathematical Models of Schizophrenia
Epidemiology: Towards an Etiology of Schizophrenia. In preparation.
Computer Skills:
MATLAB Software Proficiency
Mathematica Software Proficiency
Maple Software Proficiency
SPSS Software Proficiency
Blackboard Online Course Software Proficiency
Texas Instruments Graphing Calculator Proficiency
Java, C++, and Python Programming Languages

Achievements
November 2010: Advanced to Doctoral Candidacy
May 2009: Master Graduate Teaching Assistant
Assisted in the development of a course for improve graduate teaching
assistant training.
August 2008: Passage of Qualifying Exams at PhD Level
Summer 2007: Graduate Assistant in Areas of National Need (GAANN) Fellow
This program provides fellowships, through academic departments
and programs of IHEs, to assist graduate students with excellent
records who demonstrate financial need and plan to pursue the
highest degree available in their course study at the institution in a
field designated as an area of national need. Grants are awarded to
programs and institutions to sustain and enhance the capacity for
teaching and research in areas of national need.
Winter 2007: Passage of Qualifying Exams at Master's Level
Summer 2001: Research Experience for Undergraduates at Oregon State
University: Theoretical calculations related to magnetic
anisotropy.

Abstract: Mathematical Models for the Etiology of Schizophrenia and White Matter
Lesions
The thesis consists of two projects. The first project uses mathematical models
from nuclear physics to explore epidemiological data related to schizophrenia. These
models improve the state of the art understanding of the biological etiology of
schizophrenia, suggesting that regular internal biological events are responsible for
disease development. The schizophrenia project develops two families of mathematical
models that describe the course of schizophrenia. First, the models are applied to
schizophrenia prevalence data for different populations. Parameters from these models
are analyzed for trends relating to the parameters. The parameters are used to simulate
datasets showing the relationship of the models back to the observed parameters. These
models from theoretical physics can explain monozygotic twin discordance in
schizophrenia. The second project explores white matter lesions in a Mexican-American
population across the adult lifespan. A novel mathematical model is created to relate
white matter lesion development to aging, diabetes, and hypertension. The white matter
lesion project examined real data from a Mexican-American population. The model
revealed that diabetes, hypertension, and age are strongly associated with the
development of white matter lesions. The data revealed a transition from lower volume,
number, and average volume of lesions in the 36-45 to 46-55 decades of life. The novel
mathematical model uses a logistic differential equation and elements of probability
theory to recreate the data. Further analysis of the model showed that it not only fit the
Mexican-American data, but also fit data related to the Austrian Stroke Prevention Study.
It made predictions about the effects of diabetes and hypertension in a simulated

Mexican-American population. The totality of the projects show that physics is a fertile
ground for developing physically based mathematical models that can be applied to
diverse problems relating to medicine. Potential extensions to this work will also be
discussed.

Mathematical Models for the Etiology of Schizophrenia and White Matter Lesions
2014
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, Baltimore County, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2014

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ii
Dedication
This thesis is dedicated to my parents Myra and Dennis Corbitt who have been with me
through both thick and thin.

iii
Acknowledgements
As with any large scale project there are many people to thank who have
contributed either directly or indirectly. First and foremost I have my parents, Myra and
Dennis Corbitt, for supporting me and providing me with so many educational
opportunities. Without them I know I could not have succeeded. Second, I have had the
pleasure to have had many excellent teachers in my path leading to graduate school and
beyond. In high school I had a number of wonderful teachers including Mary Furlong
and Robert Sagedy. As an undergraduate at Washington College I had the opportunity to
learn from many excellent professors including Satinder Sidhu and Karl Kehm in the
physics department and Austin Lobo, Eugene Hamilton, Michael McLendon, and Louise
Amick in the mathematics department. Especially influential was Juan Lin who showed
me that physics techniques can be used to model biological systems. At Towson
University where I studied applied mathematics I had the good fortune to interact with
many excellent faculty members including John Chollet, Coy May, Michale O'Leary, and
Andrew Engel. Finally, at UMBC I relearned a great deal of physics working with Eric
Anderson and Lili Cui. I was privileged to learn from many of the great faculty members
including James Franson, Ivan Kramer, Kevin McCann, Todd Pittman, and Laszlo
Takacs. Credit also goes to the members of my PhD committee including Ivan Kramer,
Vanderlei Martins, Roy Rada, Malle A. Tagamets, and Laszlo Takacs. Thanks are
extended to the department chair Michael Hayden for allowing me to pursue this
particular line of inquiry. My advisor, Ivan Kramer, gave me unparalleled academic
freedom to pursue my own ideas and make my own mistakes. Malle Tagamets helped
me secure a position after graduation at the National Institutes of Health working with

iv
Barry Horwitz. At the NIH, fantastic colleagues Barry Horwitz, Iain DeWitt, Pearce
Decker, Jason Smith, and Antonio Ulloa provided encouragement in the final
preparations leading up to the final defense. Last, but certainly not least are the staff of
the interlibrary loan department of the UMBC Albin O. Kuhn Library. With the aid of
these individuals I located critical works without them this thesis would have been
untenable. If I forgotten anyone, please put it down to chance (p<.001) or my incomplete
memories.

v
Dedication ii
Acknowledgements iii
Table of Contents v
List of Tables ix
List of Figures xi
Chapter 1: Introduction
Introduction 1
Methods 2
Results 3
Conclusion 5
Thesis Organization 5
Chapter 2: Mathematical Modeling Background
Mathematical Modeling as a Physicist's Tool 7
Introduction to Differential Equations 9
Introduction to Difference Equations 12
Introduction to Probability 13
Mathematical Models 16
SIR Model 22
Mathematical Models in the Context of Physics History 27
Celestial Mechanics 28
Planck's Blackbody Radiation Formula 34
Conclusion 36
Chapter 3: Biological Background
Introduction 39
Brain Architecture and Communication 39
Genetic Background 45
Biological Models: Model Organisms 52

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Familial and Sporadic Disease 53
Chronic Disease 56
Magnetic Resonance Imaging 58
White Matter Lesions 65
Understanding P-Values 66
Chapter 4: Schizophrenia Modeling Project
Introduction 69
Schizophrenia is a Mental Disorder 70
Worldwide Incidence of Schizophrenia 73
Burden of Schizophrenia 74
Emerging Technologies 76
Genetic Component of Schizophrenia 77
Current Theories of Etiology 80
Evidence that Schizophrenia is Not Psychologically Contagious 82
Previous Work Quantifying Epidemiological Data 85
Mathematical Modeling Approach 86
Rarity of Early Onset 89
Twin Studies 89
Data Modeled 94
Mathematical Models of Schizophrenia 98
Derivation of Single Step Model 101
Independent Three Parameter Model 102
Independent Four Parameter Model 103
Ordered Model 103
Model Structure 111
Measures Derived from the Model 112
Identical Twin Data Application 118

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Model Fitting to the Data 119
Estimating the Uncertainties of Fitted Parameters 121
Kullback-Liebler Distance 126
Creating Synthetic Datasets 127
Model Results 131
Results: Comparing Populations 162
Native Populations Compared to Migrant Populations 162
Comparison of Rural and Urban Populations 167
Addressing Within Population Variability 173
Results: Aggregate Data 177
Results: Synthetic Datasets Results 179
Discussion: Heuristics 189
Discussion of Model Analysis 192
Discussion: Model Interpretation 194
Discussion: Model Analysis 208
Discussion: Synthetic Datasets 213
Conclusions 215
Chapter 5: White Matter Lesion Project
Overview 217
Introduction 218
Overview of Project 223
Materials and Methods 224
Data Results 229
Subcortical Results 229
Periventricular Results 238
Mathematical Model Methods 244
Simulations Conducted 263

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Modeling Results 268
Discussion 254
Conclusions 291
Chapter 6: Executive Summary and Recommendations for Future Research
Executive Summary 292
Recommendations for Future Research 293
Appendix A: Derivation of Equation 5.5 296
Bibliography 300

ix
List of Tables
Table 4.1: Best fit parameters for the Welham male data.
Table 4.2: Uncertainties calculated from the covariance matrices.
Table 4.3: Results of Monte Carlo simulation for N = 13 using the ordered four parameter
model listed in Table 4.1. Compare the means with the parameter values in
Table 4.1 and the standard deviations to the uncertainties in Table 4.2.
Table 4.4: Best fit parameters for the Malzberg non-Puerto Rican male models.
Table 4.5: Best fit parameters for the Malzberg Puerto Rican male models.
Table 4.6: Best fit parameters for the Malzberg non-Puerto Rican female models.
Table 4.7: Best fit parameters for the Malzberg Puerto Rican female models.
Table 4.8: Best fit parameters for Malzberg rural male models.
Table 4.9: Best fit parameters for Malzberg urban male models.
Table 4.10: Best fit parameters for Malzberg rural female models.
Table 4.11: Best fit parameters for Malzberg urban female models.
Table 4.12: Best fit parameters for Babigian 1970 male models.
Table 4.13: Best fit parameters for Babigian 1970 male models.
Table 4.14: Best fit parameters for Babigian 1970 female models.
Table 4.15: Best fit parameters for Babigian 1975 female models.
Table 4.16: Concordance data for identical twins: 18 male and 49 female as used in
Slater's 1953 study.
Table 5.1: Medical conditions found in the sample population as percentage of entire
population.
Table 5.2: Population age distribution.
Table 5.3: Maximum and average values for lesion volumes and numbers are reported
across the whole population. In all cases the minima were zero. Volumes are
reported as the number of cubic centimeters (cc).
Table 5.4: Lesions in terms of a semi-quantitative scale based on average lesion volume
across the whole population. The ratings are punctate < .524 cc < early
confluent < 4.189 cc < confluent.

x
Table 5.5: P-values for decadal comparisons of percentage of lobe occupied by
subcortical lesions: * Significant at p<.05, ** Significant at p<.005, ***
Significant at p<.001.
Table 5.6: P-values for decadal comparisons of number of subcortical lesions per cubic
centimeter of lobe volume: * Significant at p<.05, ** Significant at p<.005,
*** Significant at p<.001.
Table 5.7: P-values for decadal comparisons of average subcortical lesion volume: *
Significant at p<.05, ** Significant at p<.005, *** Significant at p<.001.
Table 5.8: Subcortical locations where the exponential regression was significant within
each decade. All were significant at FWE p < .05 corrected.
Table 5.9: P-values for decadal comparisons for volume of periventricular lesions. Key: -
implies that no lesions were present in either decade being compared, *
Table 5.10: P-values for decadal comparisons for number of periventricular lesions. Key:
- implies that no lesions were present in either decade being compared,*
Table 5.11: P-values for decadal comparisons for average periventricular lesion volume.
Key: - implies that no lesions were present in either decade being compared,
* Significant at p<.05
Table 5.12: Locations where the exponential regression was significant for
periventricular lesions. All were significant at FWE p < .05 corrected.
Table 5.13: Locations where the linear regression was significant for periventricular
lesions. All were significant at FWE p < .05 corrected.
Table 5.14: Lobe Volumes and Simulation Parameters.
Table 5.15: These are the parameters that specify the age of onset distributions for
diabetes and hypertension.
Table 5.16: The simulated maximum and average values recorded lesion volume and
number; comparable to Table 5.5. Volumes are reported in cubic
centimeters. In all cases the minima were zero.
Table 5.17: The simulated maximum and average values recorded lesion volume and
number; comparable to Table 5.5. Volumes are reported in cubic
centimeters. In all cases the minima were zero.

xi
List of Figures
Figure 2.1: An RC circuit with a switch; the flow of current once the switch is closed is
given by Equation 2.5.
Figure 2.2: Predator-prey model when approximated by Equations 2.14 and 2.15 with
parameters a = .12, b = .0001, c = .0003, and d = .039. The solid line is the
rabbit population and the broken line is the fox population.
Figure 2.3: SIR epidemic model with R0 = 19 (β = .95, ν = .05) with initial conditions of
95% of the population is susceptible while 5% is infected. The solid line is
the susceptible population, the broken line is the infected population, and
dashed line is the recovered population.
Figure 2.4: SIR epidemic model with R0 = 2 (β = .75, ν = .375) with initial condition of
45% of the population susceptible, 5% infected, and 50% recovered
(vaccinated). This shows the herd immunity property when sufficient levels
of immunity via vaccination prevent the disease from reaching some of the
susceptible population. The solid line is the susceptible population, the
broken line is the infected population, and dashed line is the recovered
population.
Figure 2.5: Picture of the masses m1 and m2 and the radius vector between them.
Figure 3.1: A drawing of a neuron where the cell body and nucleus are on the left and the
myelinated axon extends to the right. Dendrites that connect to other neurons
surround the cell body.
Figure 3.2: Picture of the axon terminal and synaptic cleft between the two neurons. The
left side is the presynaptic axon terminal; on the right is postsynaptic side
with receptors from the neurotransmitters. Vesicles will release
neurotransmitters that then diffuse across the synaptic cleft to receptors that
relay the message to the next cell.
Figure 3.3: This depicts time on the horizontal axis and shows the temporal applications
of the gradients and the radio frequency pulse. First slice selection gradient,
, is applied while the radio frequency pulse is emitted. This is followed by
the application of the phase encoding gradient, . Finally the frequency
encoding gradient, , is applied and the signal from the radio frequency
pulse is detected.
Figure 3.4: Picture of a head coil. Looks kind of claustrophobic.
Figure 4.1: The values and uncertainties in the parameter compared to the logarithm of
the population size. The circles indicate a weighted model fit while the
squares are the result of an unweighted model fit.

xii
Figure 4.2: This graph illustrates the variation in with respect to population size. The
circles indicate a weighted model fit while the squares are the result of an
unweighted model fit.
Figure 4.3: This graph illustrates the values of in the ordered four parameter model.
The circles indicate a weighted model fit while the squares are the result of
an unweighted model fit.
Figure 4.4: This graph illustrates the values of in the ordered four parameter model.
The circles indicate a weighted model fit while the squares are the result of an
unweighted model fit.
Figure 4.5: Comparison of prevalence functions derived from weighted datasets (solid
line) and the unweighted datasets (broken line).
Figure 4.10: The values and uncertainties in the parameter compared to the logarithm
of the population size. The circles indicate a weighted model fit while the
Figure 4.11: This shows the male models for the 29 analyzed datasets. Darker shading
means the models are closer.
Figure 4.12: This shows the agreement between the male consensus four parameter
models. The color coding is rescaled to reflect the smaller distances. The
first model in the each model block is the weighted followed by the
unweighted model.

xiii

xiv
Figure 4.27: Kullback-Liebler matrix plot for female models.
Figure 4.28: Kullback-Liebler distance matrix plot for female consensus models.
Figure 4.29: The dash-dot line is the independent three parameter model; the dashed line
is the independent four parameter model; the dotted line is the ordered three
parameter model; and the solid is the ordered four parameter model.

xv
Figure 4.43: Graphs of the male means plotted against the standard deviation.
Figure 4.44: Graphs of the female means plotted against the standard deviation.
Figure 4.45: Simulation using independent three parameter model male Babigian 1970
model (top), and the Babigian 1975 model (bottom).
Figure 4.46: Simulation using independent four parameter model male Babigian 1970
Figure 4.47: Simulation using ordered three parameter male Babigian 1970 model (top),
and the Babigian 1975 model (bottom).
Figure 4.48: Simulation using ordered four parameter model for the male Babigian 1970
Figure 4.49: Illustration of effects of using a model to generate data based on different
population sizes. The ordered four parameter 1975 Babigian model is used
to generate the data and plotted for reference.
Figure 4.50: Trajectories of events leading to onset of schizophrenia using the Babigian
1970 independent three parameter model, first 25 cases, and the Babigian
1975 independent three parameter models, second 25 cases.
1970 independent three parameter model, first 25 cases, and the Babigian
1975 independent three parameter models, second 25 cases.
1970 ordered three parameter model, first 25 cases, and the Babigian 1975
independent three parameter models, second 25 cases.
1970 ordered four parameter model, first 25 cases, and the Babigian 1975
independent four parameter model, second 25 cases.

xvi
Figure 4.54: Graph of independent model with k fixed and allowing m to vary; m
increases with graphs from left to right.
Figure 4.55: Graph of ordered model with k fixed and allowing m to vary; m increases
with graphs from left to right.
Figure 5.1: Top: mean +/- 1 standard deviation for observed WMLs as percentage of
lobe. Middle: mean +/- 1 standard deviation for number of observed WMLs
per cubic centimeter of lobe volume. Bottom: mean +/- 1 standard deviation
for average observed WML volume (Total Lesion Volume/Number of
Lesions). Key: * p<.05, ** p<.005, *** p<.001 when compared to previous
decade.
Figure 5.2: Red regions indicate where the exponential function was significant for the
subcortical white matter lesions.
Figure 5.3: Top: average gross volume of observed periventricular lesions in cubic
centimeters. Middle: Average number of observed periventricular lesions.
Bottom: Computed average volume of an observed periventricular lesion in
cubic centimeters. Key: * p<.05, ** p<.005 when, *** p<.001 when
compared to previous decade.
Figure 5.4: Locations where the exponential (red) and linear (green) functions were
significant for the probability of periventricular lesions. The linear function
is barely visible at this level of resolution, and appears on the lateral edges of
several of the regions.
Figure 5.5: Top: simulated lesion development trajectory of the percentage of frontal lobe
occupied by lesions in 25 individuals from birth to age 70. Bottom:
simulated lesion development trajectory for number of lesions per cubic
centimeter frontal lobe volume in 25 individuals from birth to age 70. All
individuals were free of diabetes and hypertension.
Figure 5.6: Top: mean +/- 1 standard deviation for simulated WMLs as percentage of
lobe. Middle: mean +/- 1 standard deviation for simulated number of WMLs
per cubic centimeter of lobe volume. Bottom: mean +/- 1 standard deviation
for average simulated WML volume (total lesion volume/number of lesions).
For each measure the model extrapolated to 86-95 decade. Note different
scales compared to observational data in Figure 5.1.
Figure 5.7: Simulations of the ASPS study. Top: Percentage of frontal lobe occupied by
lesions for 500 individuals who initially have two early confluent lesions at
age 60 compared to those with no lesions at age 60 compared to 500 with no
lesions at age 70, 500 at 60 and have diabetes, and 500 at 60 and have
hypertension. Middle: number of lesions per cubic centimeter of frontal lobe

xvii
volume for individuals in the same population. Bottom: average lesion
volume (total lesion volume/number of lesions) in the same population.
Figure 5.8: Simulation data and observed data comparing diabetes, hypertension, and
comorbid diabetes and hypertension. Top: fraction of frontal lobe occupied
by lesions for. Middle: number of lesions per cubic centimeter of frontal
lobe volume for the same population. Bottom: average lesion volume (gross
lesion volume/ number of lesions) of frontal lobe lesions for the same
population. These individuals were aged 46 to 75. Key: Stars are data points:
128 with no health problems, 60 with diabetes, 22 with hypertension, and 40
who have comorbid diabetes and hypertension. Open circles are simulated
data (drawn from populations depicted in Figure 5.6): 1914 healthy, 908 with
diabetes, 363 with hypertension, and 608 with comorbid diabetes and
hypertension.
Figure 5.9: Simulations comparing diabetes, hypertension, and comorbid diabetes and
hypertension. Top: fraction of frontal lobe occupied by lesions for each
population. Middle: number of lesions per cubic centimeter of frontal lobe
volume for the same population. Bottom: average lesion volume (gross
lesion volume/ number of lesions) of frontal lobe lesions for the same
population. Key: Filled circles: 1000 simulated individuals aged 70 that are
divided into four populations: 250 with no health problems, 250 with
diabetes, 250 with hypertension, and 250 who have comorbid diabetes and
hypertension.

1
1. Introduction
1.1 Introduction
This thesis develops mathematical models for two diseases: schizophrenia and
leukoaraiosis (White Matter Lesions, WMLs). The schizophrenia research uses
population level models to analyze schizophrenia epidemiological data. The
leukoaraiosis model is a novel technique for simulating the appearance and growth of
WMLs in healthy populations. Both branches of research focus on discovering the
diseases' biological etiology.
Modeling analysis can be conducted in two ways: a posteriori or a priori. These
approaches are succinctly explained by Ronald Ross, "Such computations may be either
deductive (a posteriori), or constructive (a priori). By the former we seek to deduce the
laws from the observed facts; in the latter we assume what we suppose are the laws, and
then try to verify them by inquiring whether they explain all the facts " [1]. Modeling is
done in the tradition of a priori reasoning guided by the literature, intuition, and extensive
knowledge of mathematical models in physics. A priori reasoning allows the models to
make predictions while treating statistical details as a secondary concern. Variables in
the models have specific physical interpretations. The chosen model is not the one that
fits the best, but the one that suggests plausible biological mechanisms. The physical
interpretation allows the exploration of causal relationships leading to the necessary and
sufficient conditions for disease progression.

2
1.2 Methods
The schizophrenia models are used to analyze age specific incidence and
prevalence data for clues to the causal biological processes. The models implicate
common developmental pathways, noting that some populations may have different
pathways. The models show that schizophrenia is a disease that progresses through a
number of stages. The interest in schizophrenia is driven by schizophrenia's high societal
burden. The unique contribution of this thesis is the analysis of schizophrenia in
geographically distinct populations, which confirms that schizophrenia is a heterogeneous
disease. The model explains discordance observed in identical twins and the fact that
much of the general population is immune to developing schizophrenia. The models used
to describe schizophrenia are motivated by radioactive decay chains. The differential
equations used in the models have an underlying probabilistic interpretation for
individuals, but are deterministic at the population level. The models supply the method
to simulate at the individual level and then recover the population level results. These
simulations can mark when developmental milestones are reached in the simulated
schizophrenia cases. Disorders such as schizophrenia often have a prodromal period
showing signs of pending onset of symptoms. These disease models are useful to
hypothesize a prodromal trajectory. Such models point to specific phases of disease
development and can lead to the discovery of potential points of intervention.
The model of leukoaraiosis focuses on the White Matter Lesions (WMLs) in a
Mexican-American population. Magnetic resonance imaging (MRI) images are analyzed
at the voxel (volumetric pixel) level using statistical methods. The WML model is based
on observations of voxel-based data to determine where WMLs are located at different

3
ages. Insights and results from this analysis were used to construct a model of subcortical
WML development across the human lifespan. These WMLs are important markers of
brain health and are associated with a number of conditions including bipolar disorder,
diabetes, hypertension, multiple sclerosis, myocardial infarction, sickle cell disease, and
stroke. The leukoaraiosis model enhances our understanding of normative WML
development. The model separates the process of lesion development and allows a priori
hypothesis for different conditions associated with WML development. This WML
model creates a new class of models which I provisionally call a Generalized Probability
Model (GPM). GPMs merge differential equations and probability theory to create a
modular modeling system. Differential equations are best at describing deterministic
dynamics while probability theory contains the inherent randomness found in many
biological systems. The differential equations have the ability to explain the times where
no major events are occurring while the probability elements account for abrupt changes.
GPM models are designed to be modular in construction and can easily be altered by
changing probability distributions.
1.3 Results
The schizophrenia model creates a mathematical framework for understanding
schizophrenia as the end result of a series of biological events. Multiple possible
biological processes could underpin the development of schizophrenia. Models are
selected based on the epidemiological data and biological plausibility. Physically based
models allow the exploration of a number of different aspects of schizophrenia. Using
the model's physical perspective illustrates the reasons for identical twin discordance.

4
The models of different groups such as those from different time periods, ethnicity, and
gender can be compared. The predictions of these models include specifying peak, mean,
and standard deviation of age of onset in a sample population. A universal model that
describes the development of schizophrenia in any population is a very useful tool.
Using the population based model, it is possible to simulate the timing of the discrete
biological events that lead to the development of schizophrenia. Ultimately, these models
can suggest developmental periods or even mechanisms to arrest or prevent the
development of schizophrenia. Simulating schizophrenia age of onset data by utilizing a
universal model can also explain the results of studies with only a small number of cases.
The WML model shows that stochastic events can explain the development of
WMLs. This model develops normative measures of these stochastic events.
Development of the model proceeded in three stages. The initial step was to analyze and
characterize WMLs in the Mexican-American population. The second step was to use
the model to replicate the Mexican-American data for subcortical WMLs in different
brain regions. The final result was to apply the model to explore the relative effects of
age, diabetes, and hypertension. The model results were compared to the observations
from the Austrian Stroke Prevention Study (ASPS) [2]. The model supports the ASPS
finding that existing WMLs are the most important predictors of future brain health when
compared to age and hypertension. The model found that diabetes, like age and
hypertension, was not as good a predictor of WML development. Thus the model
confirmed the ASPS results and predicts that diabetes is similar in effect to age and
hypertension. The model makes a novel prediction that comorbid diabetes and
hypertension create more WMLs than either condition alone. Four groups were

5
simulated: healthy with no diabetes or hypertension, those with diabetes, those with
hypertension, and those with comorbid diabetes and hypertension. This is of practical
importance because chronic conditions such as diabetes and hypertension are common in
the developed world. The model predicted that comorbid diabetes and hypertension had
a greater cumulative effect than either condition alone. The model was designed to be
modular so new findings can be incorporated allowing other researchers to make a priori
predictions and adapt the model to explain their findings.
1.4 Conclusion
Physical interpretations of the mathematical models are useful when analyzing
biological systems. The schizophrenia model brings a new perspective to understanding
the etiology of schizophrenia. The schizophrenia model emphasizes that schizophrenia is
a developmental process consisting of multiple events that occur over an extended period
of time. The WML model shows the explanatory power of a mathematical model in
understanding the development and progression of WMLs, and demonstrates a modular
model that can be readily expanded and modified. Development of these models serve as
a catalyst to apply mathematical and physical problem solving techniques to biological
systems.
1.5 Thesis Organization
The thesis is organized in the following manner. The second chapter examines
mathematical modeling of biological systems and presents several conceptual and
mathematical models in physics. This introduces the reader to the role of mathematical

6
modeling in physics. The third chapter reviews the literature regarding general biological
background thus setting the stage for the two projects. This chapter is broad in scope
because of the interdisciplinary nature of this thesis. To simplify the reading of the thesis
the entire schizophrenia and WML projects are encapsulated in chapters four and five
respectively. The sixth chapter is an executive summary of the findings and possible
extensions to future research programs.

7
2. Mathematical Modeling Background
2.1 Mathematical Modeling as a Physicist's Tool
Mathematics is one of the most important tools available to scientists, particularly
in chemistry, engineering, physics, and now even biology and medicine. The drive for
quantitative results led to the development of sophisticated mathematical tools ranging
from statistical analysis to mathematical models. This section focuses on the application
of mathematical models in the natural sciences. Physicists are mathematically skilled,
but see mathematics as more than solving equations and proving theorems. Physicists
give the mathematics an interpretation. The following anecdote illustrates the difference
between mathematicians and physicists:
A physicist and a mathematician were sleeping in a hotel room when a fire broke
out in one corner of the room and only the physicist woke up. He measured the intensity
of the fire, saw what material was burning, calculated the amount of water required to
extinguish the fire, filled a trashcan with the precise amount of water, and put out the fire;
the physicist went back to sleep. A little later another fire broke out in another corner of
the room. The mathematician woke up. He went over, looked at the fire, he saw that
there was a bucket and he noticed that it had no holes in it; he turned on the faucet and
saw that there was water available. He concluded that there was a solution to the fire
problem and he went back to sleep [3].
This story emphasizes that knowing a solution does exist is interesting but not
always practical. The physicist in the story went about measuring and quantifying the
parameters associated with the fire. He then took these quantities and turned them into

8
physical expressions. Making calculations and interpretations is the domain of the
physicist. There is no a priori reason to expect that mathematics be a language that is
useful for describing the universe.
The ability of mathematics to describe natural systems show what the Hungarian
physicist, Eugene Wigner, calls the 'unreasonable effectiveness' of mathematics in the
natural sciences [4]. An example of this 'unreasonable effectiveness' is sending a space
probe to a precise location in the distant reaches of a solar system. This 'unreasonable
effectiveness' led physicists to a distinct way of thinking about mathematics. Redish and
Bing summarize the physicist mode of thinking concisely, "Mathematics is an essential
component of university level science, but it is more complex than a straightforward
application of rules and calculation. Using math in science critically involves the
blending of ancillary information with the math in a way that both changes the way that
equations are interpreted and provides metacognitive support for recovery from errors"
[5]. Physicists use equations to think about and understand physical scenarios [6]. Any
mathematical explanation of a physical phenomenon must explain all available data. In
developing mathematical models of natural systems it is essential to recognize how the
system works. There may be a mathematical structure that describes the system under
consideration to a great extent, but contradicts an experimental finding, meaning that
particular mathematical model is inadequate to explain the phenomenon. The discussion
turns to physicists' most useful mathematical tools.
There is a plethora of mathematical tools that is available and it is necessary to
select those that are most useful. The foremost tool of applied mathematics is the

9
differential equation. A close partner of the differential equation is the difference
equation, which can approximate differential equations when no analytic solution is
available. The last tool discussed is probability theory. Probability forms the bedrock of
statistics and stochastic simulations. All three of these tools can be used to build
mathematical models.
2.2 Introduction to Differential Equations
Differential Equations (DE's) are a powerful mathematical modeling technique.
Differential equations are the natural tool for understanding time dependent interactions
between two or more quantities. Differential equations model the continuous change
between two variables, while the discrete analog is known as a difference equation.
Difference equations have a discrete step size and can be used to numerically
approximate differential equations. In the limit of a small step size a difference equation
is nearly identical to a differential equation. A discussion of differential equations and
difference equations as tools in the mathematical sciences follows.
For several centuries differential equations have been used for modeling of real
systems [7]. By 1700, solutions were known for most basic differential equations [7].
Differential equations may be either a single equation or a system of coupled equations.
Systems of DE's are coupled when the solution to one equation depends on the solution
of the other equations in the system. Classifying DE's is important for deciding what
types of solution techniques are most appropriate.

10
There is an infinite number of DE's; this multitude of equations can be classified
in several ways. A DE that describes the dependent variable in terms of multiple
independent variables is known as a Partial Differential Equation (PDE's). PDE's have
many applications in physics. In this thesis, the primary concern is Ordinary Differential
Equations (ODE's). ODE's relate one or more independent variables to a dependent
variable. The next major classification of ODE's (and PDE's) is the distinction between
linear and nonlinear equations. A differential equation is linear if it is a linear function of
the dependent variable and its derivatives [8]. Nonlinear DE's are much more difficult to
solve, requiring clever solution techniques such as similarity solutions. Many important
theorems have been proved for linear ODE's, particularly existence and uniqueness
theorems. DE's can also be classified according to the question posed as Initial Value
Problems (IVP), Boundary Value Problems, or a mixture of the two. IVP's are solved
using the initial conditions specified by the application. In this thesis the discussion is
confined to initial value problems of ODE's.
Methods of solving ODE's include separation of variables, integrating factors,
method of undetermined coefficients, series solutions, and the Laplace transform.
Summaries of these methods can be found in textbooks on differential equations [7-9].
Some of the aforementioned methods, such as the Laplace transform, are only applicable
to linear ODE's. Systems of ODE's are also solved using the methods such as Laplace
transforms or integrating factors.
Differential equations are the lifeblood of physics with applications in mechanics,
electricity and magnetism, and quantum mechanics. These will be discussed in the third

11
section of this chapter. DE's are the common language unifying the many diverse
applications of physics. The wide array of DE's found in physics means that analogs to
these equations may appear in areas outside of physics. These analogs are isomorphic
(having the same mathematical form or structure) to the DE's found in physics permitting
the application of methods of physical analysis. A quote which has been attributed to a
number of sources is that 'talent imitates, genius steals' is applicable in this case. Rather
than merely imitating the solution methods one uses the entirety of the physical analysis.
Using the isomorphic analysis deepens the understanding and interpretation of the
analogous system. However, often it is not possible to solve an ODE analytically and it
necessary to resort to numerical methods.
While some differential equations have analytic solutions, many do not. In these
cases it is necessary to resort to numerical methods. Numerical methods were of limited
application prior to the introduction of computers, but now even simple numerical
methods provide a high degree of accuracy. Numerical methods for solving IVP's
numerically integrate the differential equation. Numerical methods rely on solving first
order ODE's, requiring higher order equations to be rewritten as a system of first order
ODE's. This can be achieved by methods such as reduction of order. The simplest
numerical methods are based on Taylor series expansions [10, 11]. Euler's method uses
the first term in the Taylor series and is equivalent to numerically integrating the
differential equation using the trapezoidal rule. The Euler method works well and is
conceptually simple. Given a first order DE
(2.1) ( )

12
initial condition at x(t0), and selecting a time step t. The solution a time t after t0 is
obtained by
(2.2) ( ) ( ) ( ( ))
This solution at x(t0 + t) then becomes the new initial condition. This sequence
is iterated to obtain a series of points that approximates the solution of the differential
equation specified in Equation 2.1. This method of producing a series of points is known
as a difference equation which is covered in detail following this discussion. Numerical
methods work well, except in the case of stiff differential equations which require small
time steps to avoid errors. A discussion of stiff differential equations can be found in
texts concerning numerical analysis [10, 11]. One must be aware of stiff differential
equations in case numerical instabilities are found. This is a brief survey of numerical
methods and provides a natural transition to the topic of difference equations.
2.3 Introduction to Difference Equations
Difference equations are a method of generating a sequence of numbers. The
numerical methods mentioned above can be used to generate values of the solution of
DE. Difference equations find a wide variety of applications including epidemiology,
economics, and population biology [12]. Probability, electrical networks, and mechanical
applications can be modeled using difference equations [13]. Difference equations show
a remarkable degree of behavior for simple equations. Difference equations are good for
modeling systems that transition between discrete states. The best known difference

13
equations can be used to generate a Fibonacci sequence. Given two initial starting values
x1 and x2 the nth term can be computed according to the following equation
(2.3)
The relationship in Equation 2.3 is a recurrence relation. Recurrence relations are found
in the series solutions of differential equations.
Difference equations allow the exploration of complex phenomena using simple
tools. A famous difference equations is the logistic map which is defined by
(2.4) ( )
where μ is a parameter between zero and four and x lies on the interval zero to one. The
simple map in Equation 2.4 exhibits interesting behavior for different values of μ. In
some cases cyclic behavior is exhibited while in other cases chaotic dynamics emerge.
Extensive descriptions and analyses of this difference equation can be found in texts on
chaotic dynamics [12]. Awareness of this type of behavior is important because chaotic
dynamics is often the hallmark of sensitivity to initial conditions. Sensitivity to initial
conditions means that slight variation in the initial conditions of an IVP leads to
dramatically different results. Implications of sensitivity to initial conditions will be
related to the mathematical models found in this thesis.
2.4 Introduction to Probability
Probability is a key tool in mathematical modeling. In many cases only the
probability that an event will occur can be computed. The realization of a probability

14
function is a Random Variable (RV). RV can be either discrete (drawn from a countable
sample space) or continuous (drawn from a continuous interval). The probability of
selecting a particular discrete RV is given by a Probability Mass Function (PMF) also
known as discrete probability function [14]. Discrete RVs are used when there is an
integer quantity being described, (e.g. the number of heads when tossing a coin or the
number of cars passing through an intersection). Discrete probability functions include
the Bernoulli, binomial, and Poisson RVs. Continuous RVs are described by the
Probability Density Function (PDF) which, when integrated over an interval, is the
probability of selecting a random variable from that region. Continuous RVs are used
when the outcome may range over a given interval. Continuous RVs include the
uniform, normal (Gaussian), exponential, gamma, beta, and Weibull distributions.
Probability theory can define the relationship between events. Two events A and
B are said to be independent if the probability of A and B occurring is just the product of
either event occurring alone. Events can be correlated in a number of ways. In one case,
if A occurs then B occurs and likewise if B occurs then A occurs, implies the events are
one and the same. On the other hand if A occurs than B does not and likewise if B occurs
A does not occur, the events are said to be mutually exclusive. A variety of intermediate
possibilities are also possible. This is discussed at length in many probability texts [14].
Probability theory forms the foundation of statistical inference. In particular it is
desirable to obtain p-values, probability values, when comparing models with observed
data. P-values tell whether an observation can be attributed to chance alone. Statistical
inference requires precise language. The prototypical statistical hypothesis is the

15
comparison of two means for two different groups of observations. Two hypotheses are
advanced: the null hypothesis that there is no difference in the means and the alternative
hypothesis that there is a statistically significant difference between the means. To
compare the hypothesis a test statistic is chosen and a threshold for significance is
defined (e.g. p < .05). The value of test statistic determines the p-value. If the test
statistic does not reach the threshold for statistical significance then we fail to reject the
null hypothesis meaning that there is no statistically significant difference between the
two means. If the threshold of statistical significance is reached then the null hypothesis
is rejected meaning there is a statistically significant difference in the means. A large
body of literature has been written concerning statistical inference. Obsession with p-
values permeates the medical literature. P-values provide a statistical measure of whether
the results are due to chance alone or if they are truly the result of the hypothesis under
examination.
Monte Carlo and other stochastic simulations depend on probability theory;
random variables are the essential tools. Selection of appropriate random variables is
necessary to develop realistic mathematical models. Models summarize, explain, and
make predictions about particular scenarios [15]. Models act as guides for understanding
real world phenomena, even if the model is simple. It is always necessary to make
assumptions and simplifications. Real world systems are subject to random noise that
may hide the underlying mathematical structure. Once a model is developed, stochastic
simulation may be used to create simulated datasets allowing the exploration of
parameter space [15]. Stochastic simulation can be used in a wide variety of contexts
ranging from simple examples [16] to Monte-Carlo integration and use in statistical

16
inference [15]. Stochastic simulations depend on pseudorandom number generation.
High level mathematical programs such as MATLAB carefully control the generation of
random numbers to ensure statistical independence. It will be shown how stochastic
simulation can uncover interesting elements of the model that are not obvious from
analysis of equations. Simulations allow use of model for planning future studies. The
next topic shows several examples of mathematical models.
2.5 Mathematical Models
Models often take on a variety of forms; in this section the origin of mathematical
models is discussed and several examples are provided. The examples include sigmoidal
models, unrestricted predator-prey model, compartmental models, and simulation models.
The mathematical tools of the previous section are applied to these models. This section
provides a background into the application of various models and analysis thereof.
Prior to the development of a mathematical model there must be a precursor that
is conceptual in nature. The modeler then develops the underlying concepts often by
drawing figures and sketching out ideas. These ideas become the proto-model about
which the modeler then asks if the proto-model shares features or behavior with other
models. If possible, previously developed models can be used. The proto-model is cast
into a mathematical form. The modeler can now begin analysis of the model by writing
down and solving the equations. Solutions can be obtained analytically or using
numerical methods. The next step is to show that the model is capable of modeling or
reproducing real data. After careful analysis the model is now a tool for further
exploration. The ideal model uses a small set of parameters to describe and explain an

17
observed phenomenon. The model can be continually tested against emerging data and
modified to incorporate new findings. The first type of models examined is sigmoidal
models.
A sigmoidal function has an S shape which rises from an initial value to an
essentially unchanging maximum value. Many different mathematical equations exhibit
this kind of behavior. In particular when a continuous PDF is integrated (from left to
right) the Cumulative Distribution Function (CDF) has a sigmoidal shape. The
hyperbolic tangent and some other algebraic functions exhibit sigmoidal behavior. The
most extreme example occurs when there is an instantaneous change at one point, which
is the Heaviside step function. These equations are solutions to differential equations. In
these cases the differential equations are limited to a maximum possible value.
Sigmoidal equations occur in physics, population biology, and chemical kinetics.
In introductory physics, a sigmoidal function is found in the analysis of an RC
circuit with the configuration shown in Figure 2.1. Let the capacitance be C, the voltage
V, and the resistance R. The differential equation that describes the rate at which charge
is deposited on the capacitor when the circuit is first activated is:
(2.5) ⁄
This has the solution
(2.6) ( ) ( ⁄
)
Taking the long time limit of Equation 2.6 leads to the following result:
(2.7) ( )

18
In this case the charge that can be put on the capacitor reaches an absolute maximum that
is the product of the capacitance and voltage of the battery in the circuit. The capacitor is
filled and holds no more charge at the specified voltage.
Figure 2.1. An RC circuit with a switch; the flow of current once the switch is closed is
given by Equation 2.5.
A second example of a sigmoidal model is a population of animals that live alone
on an island, but the size of the population is limited by a particular resource, e.g. a
foraging area. If the animals are immortal the population grows according to the logistic
differential equation. If only K animals can live on the island, reproduces at a rate r, and
P(t) represents the population at time t the following differential equation approximates
the time dependent growth of the population:

19
(2.8)
( )
( ) (
( )
)
The population must have some positive initial value and grow to a maximum population.
When the maximum population is reached, the condition is said to have saturated or
reached the carrying capacity. Given an initial population size P0 then the population is
given by
(2.9) ( )
Taking the long time limit of Equation 2.9 gives:
(2.10) ( )
meaning the island eventually has the maximal number of animals living on it. In order
for this type of model to work, it is necessary to know the values of r and K. Two options
are open: first, values of r and K can be guessed or second, r and K can be determined
empirically from observation. In the first case, the assumption is that the given model is
correct for the estimated r and K, and makes an a priori prediction. If the selected values
of r and K are incorrect, post hoc analysis will determine the appropriate values. The
investigator can examine the data and discover why these parameters are correct. The
second approach is easier, but does not make an a priori prediction. However this
approach may be used to analyze one data set and then use the values of r and K to make
a priori prediction about a different population. The researcher should make educated
hypotheses about the model parameters, but ultimately the data is the ultimate arbiter of
truth.

20
The predator-prey model uses differential equations to describe the interaction
between a species of predators and a prey population. The predator-prey model was
developed by the Italian physicist Vito Volterra to explain the differences in the yearly
fish catch in the Mediterranean Sea [9]. Two versions of the predator-prey model are
possible: the first with populations that are not resource restricted and a second version
that has resource restrictions on the prey population. In the following examples we will
use rabbits (R(t)) and foxes (F(t)) to be the populations at time t. The population that is
not resource constrained will obey the following system of differential equations:
(2.11)
( )
( ) ( ) ( )
and
(2.12)
( )
( ) ( ) ( )
The parameter g represents the growth rate of the rabbit population and a represents the
decline in the fox population. The other two parameters b and c represent the value of the
interaction between the rabbits and foxes that are advantageous to the foxes (b) and
detrimental to the rabbits (c). The differential equations can be approximated by
difference equations. The first step to numerically solving a differential equation is to
select a time step Δt so that
(2.13) .
Using Equation 2.2 the approximate populations at time ti+1 are:
(2.14) ( ) ( ) ( ( ) ( ) ( ))

21
and
(2.15) ( ) ( ) ( ( ) ( ) ( ))
Using a particular set of parameters oscillation in the rabbit and fox populations appear.
Figure 2.2. Predator-prey model when approximated by Equations 2.14 and 2.15 with
parameters a = .12, b = .0001, c = .0003, and d = .039. The solid line is the rabbit
population and the broken line is the fox population.
As the rabbit population increases so does the fox population, but the rabbit
population peaks and declines; the fox population follows with a slight delay. This
shows that when prey are plentiful, the predator population increases and when prey are
scarce the predator population declines. This simple model gives insight into patterns
that are observed in real world cases. In the second model where the rabbit population is
constrained to a maximum size denoted by K as in the logistic equation described above.
The differential equation for the rabbits becomes

22
(2.16)
( )
( ) (
( )
) ( ) ( )
The fox population is still governed by Equation 2.12. Now converting the new
differential equation into a difference equation:
(2.17) ( ) ( (
( )
)) ( ) ( ) ( )
This equation yields different behavior; initially there are small oscillations but an
equilibrium state is reached for both rabbits and foxes. This model shows that if the size
of the prey population is limited the size of the predator population is also limited.
2.6 SIR Model
Beyond the sigmoidal and predator-prey models are compartmental models.
These are used to model a population in which different segments of the population are in
different states. For example children are in different developmental states
(compartments) depending on the action or inaction of growth hormones at a particular
time. Another compartmental model could consist of describing the portions of the
population in healthy or diseased states. The prototypical compartmental model for
infectious diseases is now examined.
The best known compartmental model is the classical Susceptible Infected
Recovered (SIR) model of epidemics and its related models. The SIR model is valid in
large populations. The SIR model has three compartments: susceptible, infected, and
recovered. In the SIR model recovery confers lifetime immunity. The usefulness of the
SIR model is its ability to demonstrate the phenomenon of herd immunity due to

23
vaccination. The Susceptible Infected Recovered Susceptible (SIRS) is a useful variant
of the basic SIR model where recovery imparts only temporary immunity. The SIRS
model can be used to model seasonal influenza epidemics. The SIRS model is
approximately isomorphic to an atomic three level system with three energy levels.
Nuclear decays can be thought of as compartmental models, with the isotope of a given
element representing a compartment.
In the SIR model disease spreads when there are interactions between susceptible
and infected individuals. Infections occur at a rate β per unit time and individuals recover
at a rate of ν per unit time, then this is modeled by a system of differential equations:
(2.18) ( ) ( )
(2.19) ( ) ( ) ( )
(2.20) ( )
Two assumptions made are: the population is static (no one dies) and recovered
individuals are no longer at risk of spreading the infection. The static population size is
embodied in the observation that
(2.21) ( ( ) ( ) ( ))
implying
(2.22) ( ) ( ) ( ) ,
where N is the size of the population. The equations can be normalized so that each of
the three possible states is the fraction of the population in the state. So

24
(2.23) ( )
( )
(2.24) ( )
( )
(2.25) ( )
( )
The system of equations is nonlinear so it does not admit a simple solution. The key
point is that it is possible to derive some important facts without solving the system of
differential equations. It is possible to find a time dependent solution for the susceptible
fraction of the population in terms of the recovered fraction:
(2.26) ( ) ( ) ( ( ) ( ))
At the end of an epidemic there is no further infection so:
(2.27) ( )
and
(2.28) ( ) ( ) ( ( ) ( ))
( )
At the end of the epidemic if the susceptible fraction is nonzero, then the number of
infected individuals has gone to zero. This means that we should have
(2.29)
We find that
(2.30) ( ( ) ) ( )

25
So if the quantities in parentheses are less than zero the rate of infection is decreasing. So
the term β over ν is critical to the disease dynamics; this quantity is known as the basic
reproduction number. Figure 2.3 shows the numerical solution to a simple set of
parameters of a disease that rapidly spreads in a population.
Figure 2.3. SIR epidemic model with R0 = 19 (β = .95, ν = .05) with initial conditions of
95% of the population is susceptible while 5% is infected. The solid line is the
susceptible population, the broken line is the infected population, and dashed line is the
recovered population.
This equation explains why vaccination, safely moving people to the recovered
group, is an effective strategy to control the spread of an epidemic. If the fraction of the
population that is susceptible is small, then the infected population will decline rapidly.
This is why if enough people are vaccinated the disease can be stopped in its tracks, even
if some individuals cannot be vaccinated. The utility of a vaccination program is

26
demonstrated in the Figure 2.4; vaccinating part of the population protects the whole
population. This models is extensively discussed in texts on infectious diseases [17].
Figure 2.4. SIR epidemic model with R0 = 2 (β = .75, ν = .375) with initial condition of
45% of the population susceptible, 5% infected, and 50% recovered (vaccinated). This
shows the herd immunity property when sufficient levels of immunity via vaccination
prevent the disease from reaching some of the susceptible population. The solid line is
the susceptible population, the broken line is the infected population, and dashed line is
the recovered population.
When populations are not large enough to model as a continuum, alternative
methods are microsimulation models. These models simulate every individual in the
population [17]. These models incorporate stochastic effects such as the probability of
contact between susceptible and infected individuals. Microsimulation models can be
divided into three types: individual-based models, discrete-time compartmental models,
and continuous time compartmental models [17]. Individual based models are

27
conceptually easy, but require following each individual in the population. This type of
model is employed later in the thesis to model the appearance and progression of white
matter lesions. The discrete-time compartmental models method keeps track of total
numbers within the given population without tracking of each individual. Continuous
time compartmental models required minimal calculation and only model when events
occur. Each of these methods has advantages and disadvantages; however computing
power is cheap, so the straight-forward individual based simulations are the tool of
choice.
2.7 Mathematical Models in the Context of Physics History
The development of models in physics has been central to the advancement of
new theories as evidenced by two historical vignettes. The first focus is on the
development of celestial mechanics beginning with the Copernican model and
culminating in Newton's derivation of Kepler's laws. These models show the iterative
process whereby models are created and refined. The second discussion centers on Max
Planck's development of his famous blackbody radiation formula. The significance is
that Planck interpolated between two different findings with a formula that was
eventually given physical meaning. There are many more examples of model building,
including the development of thermodynamics and statistical mechanics, the unification
of electricity and magnetism, and the development of quantum mechanics. Attention is
now turned to the development of celestial mechanics.

28
2.8 Celestial Mechanics
Nicolaus Copernicus, a Renaissance mathematician and astronomer, was the first
well known advocate of a heliocentric model of the universe. The heliocentric
Copernican model of the solar system challenged the prevailing geocentric model of the
solar system. Prior to the Copernican model, the Ptolemaic model of a geocentric
universe used epicycles to predict celestial events. Epicycles are a type of compound
motion where there is a circular orbit around the Earth and the planets move on circular
orbits that follows the first circular orbit. More epicycles were needed to accommodate
the observed retrogressive motion of the planets [18]. The addition of epicycles to fit
new observations was made on an ad hoc basis [19].
Copernicus eliminated five of the epicycles from the Ptolemaic model [20]. He
believed the planets traveled in circles; however his model disagreed with the
astronomical observations. The Ptolemaic model better fit the observed data than the
Copernican model [21]; the Ptolemaic model could be tuned to replicate any observations
[20]. In addition to putting the Sun at the center of the solar system Copernicus
postulated the rotation of the Earth as well as the precession of the Earth about its axis
[22]. The Copernican model, like the Ptolemaic model, was not based on observations
and Copernicus's criteria for model selection was the elegance of the model [23].
Copernicus did not effectively disseminate his views, though his ideas were published.
The Copernican model is an example that appeals to the principle of Occam's razor: the
simplest solution is the best. The Ptolemaic model became ever more complicated as
more epicycles were added; the Ptolemaic model could fit any data! An arbitrary degree
of complexity will make models of epicycles work, but add needless complexity. The

29
Ptolemaic model failed to advance new hypotheses, could be continually modified to
match observations, and thus lacked explanatory power. The key part of the Copernican
model is its utility as a conceptual model. Conceptual models are the necessary
precursors of mathematical models.
Copernicus's legacy continued in two lines of subsequent inquiry. The first line
of inquiry was led by Tycho Brahe and his student Johannes Kepler. Brahe and Kepler
both made their mark on science by interpreting astronomical observations. The second
line of investigation is associated with Galileo Galilei and his exposition of the
Copernican theory. Galileo was a pioneer experimenter who advanced observational
science. Precise astronomical observations revealed inconsistencies in the geocentric
model, and the Copernican model became a viable alternative. What is especially
amazing is that these observations were made with the naked eye. Brahe's geocentric
model had the mathematical benefits of the heliocentric model while retaining the
philosophical advantages of the geocentric model [24]. Brahe made systematic
observations and realized that random errors entered into his records [25].
Johannes Kepler used Brahe's observational data to empirically determine three
laws of planetary motion. Kepler's first law is that planetary orbits are ellipses, with the
Sun located at one focus. The second law is that the radius from the Sun to the planet
sweeps out equal area in equal time [26]. The third law states that the square of a planet's
orbital period is directly proportional to the cube of the semimajor axis of the planet's
elliptical orbit [27]. Discovering these three laws took Kepler twenty years using the
most accurate data to find the correct analytic curves (he tried at least 70) [28]. Kepler

30
never developed a coherent theory that explained his laws. Though contemporaneous
with Galileo, Kepler's writings are presented in mathematical terms and not as widely
read as the more accessible works of Galileo [28].
Galileo Galilei studied the motion of heavenly bodies making both experimental
and observational investigations. Observing the tides in the Adriatic Sea led Galileo to
hypothesize that the Earth's movement about the Sun could possibly account for the tides
[25]. Though this explanation of the tides is incorrect, it made Galileo consider the
Copernican model. Galileo's observations of the moons of Jupiter showed an example of
smaller bodies orbiting a larger body [29]. Galileo began the movement to elevate reason
and experiment as the arbiter in scientific matters. Galileo is reputed to have said, 'In
disputes about natural phenomena one must not begin with the authority of scriptural
passages, but with sensory experience and necessary demonstration' [23]. Galileo was
among the first to attempt to put descriptions of natural phenomena as simple
mathematical models [25]. This began the development of modern science with parallel
development of observation and mathematical theory.
The combined work of Galileo and Kepler made the Copernican model more
attractive [30]. This paved the way for Isaac Newton's heliocentric model [31]. Brahe
and Kepler's work and Galileo's ideas were reconciled by Isaac Newton who built the
theoretical edifice that would drive the development of the celestial mechanics from
observation to prediction. Newton's theory precisely fit the astronomical observations.
Newton's triumph was his three laws of motion: the law of inertia, the law of
acceleration, and the law of action and reaction [32]. The law of inertia states that an

31
object in motion remains in motion unless acted upon by an external force. The law of
acceleration, Newton's second law, states that the acceleration of an object is directly
proportional to the force that acts upon it. The constant of proportionality is the mass of
the object and is embodied in the equation
(2.31) ⃗⃗ ⃗⃗
Newton's third law states that if object A acts with a force on object B then B will act on
A with a force of the same magnitude, but opposite direction. Often this is stated as the
forces being equal and opposite, but this is misleading as there are two forces; the force
of object A on object B and the force of object B on object A. Newton knew of the
works of Galileo and Kepler, though perhaps not at the outset of his research [22].
Newton wondered if the same force that caused objects to fall to Earth could also be the
force that holds the moon in orbit [25].
Newton derived the universal law of gravity in 1665-1666 . This initial work was
put aside until 1679 [25]. Stimulated by discussions with Robert Hooke and Edmund
Halley, two prominent scientist of the time, Newton returned to his earlier analysis of the
universal law of gravitation [25]. The universal law of gravitation states that for two
point masses m1 and m2 with centers separated by a distance r the mass m1 experiences an
attractive force due to m2 along the axis separating the two masses and is given by:
(2.32) ⃗⃗ ̂

32
Here G is a universal constant. The term ̂ is a unit vector located at m1 directed toward
m2. The negative sign indicates that the force is always attractive. This unit vector can
be written as:
(2.33) ̂
⃗
|⃗ |
where ⃗ is the vector from m1 to m2. This is illustrated in Figure 2.5.
Figure 2.5 Picture of the masses m1 and m2 and the radius vector between them.
Newton derived Kepler's laws by applying his three laws of motion and the universal law
of gravitation [24]. Confirmation of the heliocentric model came from the discovery of
stellar aberration by James Bradley in the mid-1720s which showed the movement of the
'fixed' stars is due to the motion of the Earth [22]. These observations vindicated the
heliocentric model.

33
Copernicus, Galileo, Brahe, Kepler, and Newton contributed to the replacement of
the geocentric model, proving that no theory is impervious to the facts. In the natural
sciences, experiment and observation have a veto over theory. John Gribbin says this in a
particularly succinct manner: honest experiments always tell the truth [25]. The
development of mechanics established the concept that a theoretical framework is
necessary for scientific advances [31]. A good theoretical framework creates testable
hypotheses to complement experimental investigations. It will be a continuing theme in
all areas of science to develop immutable laws that explain all observations.
The Newtonian model is a useful first approach to mechanics. Many Newtonian
concepts, such as force and acceleration, are familiar from everyday life. Newtonian
methods may be difficult to apply in practice, but they will never fail. The Newtonian
model is intuitive and simple to understand, though increased complexity is inevitable. It
is worth noting that the Newtonian approach is powerful enough to explain much of
celestial mechanics. The Newtonian approach has been described as the 'sledgehammer'
of classical mechanics [27].
Newton's three laws plus the universal law of gravitation can be used to derive
Kepler's laws. Derivation of Kepler's laws can be found in any mechanics text such as
Fowles and Cassiday's [27]. The motion being described is that between a sun and the
planet. Kepler originally used geometric methods to deduce the functional form of the
equations based on empirical data. Newton's contribution was developing a theoretical
basis for deriving the equations, particularly the conservation of angular momentum and
the universal law of gravitation. The laws of mechanics were showcased in the discovery

34
of Uranus and Neptune. In the case of the discovery of Neptune, there were irregularities
in the orbit of Uranus that could not be explained. John Couch Adams in England and
Urbain Leverrier in France independently carried out calculations suggesting the possible
location of the unknown planet [23]. In 1846 the Berlin Observatory found the unknown
planet and named it Neptune [27]. Unlike the previous discovery of Uranus that was at
least partially attributable to luck [23], the discovery of Neptune was a triumph of
classical mechanics. Mechanics demonstrate the ability of a theory to make predictions
that advance experimental observations. Out of Newtonian mechanics arose several other
approaches to mechanics based on energy principles including the Lagrange and
Hamiltonian methods, used extensively in many areas of physics. Theories can often be
extended beyond the problem they were designed to solve.
2.9 Planck's Blackbody Radiation Formula
The next model is Planck's blackbody radiation formula. A blackbody is an
object that absorbs and emits radiation perfectly at all wavelengths. At the turn of the
twentieth century there was uncertainty in the explanation of blackbody radiation.
Scientists of the time wanted to develop a formula which would describe the energy per
unit volume per unit frequency ( ), ( ), at a given temperature (T). Early attempts to
mathematically model the black-body spectrum included the Wien radiation law and the
Rayleigh-Jeans formula. The Wien radiation law was first derived in 1893 and has the
form:
(2.34) ( ) ⁄

35
The Wien radiation law failed to hold at lower frequencies. The Rayleigh-Jeans formula
was first published in 1900 and had the form:
(2.35) ( )
The Rayleigh-Jeans law was able to explain the spectrum at low frequencies, but led to
the ultraviolet catastrophe: the prediction of infinite energy at high frequencies. This was
deftly explained by a mathematical trick invoked by Max Planck to create a mathematical
model of the black-body spectrum that explained the disparate data. In 1900 Planck
interpolated between these results and produced the formula [33]:
(2.36) ( ) ( ⁄ )
The ingenuity was quantizing the energy into packets of integral multiples of , where h
is Planck's constant. Planck's radiation law included both Wien's radiation law and the
Rayleigh-Jeans formula as limits. The Wien radiation law is satisfied when:
(2.37)
Then
(2.38) ( ⁄ ) ⁄
and the Wien radiation law can be written as:
(2.39) ( ) ⁄
The Rayleigh-Jeans formula is satisfied when:

36
(2.40)
In this case:
(2.41) ⁄
Substituting Equation 2.41 into Equation 2.36 leads to the recovery of the Rayleigh-Jeans
formula. The crucial idea introduced was that radiation only comes in integral multiples
of . Planck was unaware that this simple interpolation between these two results
would lead to a revolution in understanding microscopic physics. The key point is that
knowing the proper functional description can lead to new understanding of the
underlying phenomena.
2.10 Conclusion
There are a number of lessons to be learned from the history of physics and the
mathematics used to codify physical reality. The lessons from the development of the
heliocentric model of the universe is that it is necessary to see the universe as it is, not as
we think it should be. The development of classical mechanics argues that different
approaches to problems are needed. The formulations of mechanics suggest that theories
that have a strong mathematical basis may be used to model different systems. This
happens to be the case in the development of Maxwell's equations (a Lagrangian
formulation), statistical mechanics (Hamiltonian approach), and quantum mechanics (the
Hamiltonian formulation again). Maxwell's equations suggest that there should be an
overarching search for unity, linking disparate phenomena into a single theory. Like

37
Maxwell's addition of the displacement current, there should be no hesitation to a priori
add to a theory and then analyze the results. The development of thermodynamics and
statistical mechanics argues that model development is often an extended process.
Statistical mechanics was groundbreaking in its use of probability theory. Brownian
motion teaches the lesson that fictitious constructs do not produce real effects. Quantum
mechanics shows an interesting pattern of theoretical development. From explaining
inconsistencies in the experimental results, leads to theoretical advancements, that leads
to new experimental results. An example of the occasional over-emphasis on theory is an
anecdotal story about an encounter between an experimental physicist and a theoretical
physicist:
An excited experimentalist had just created a graph of his latest experimental results and
rushed down the hall to talk to his theorist friend. The experimentalist handed his graph
to the theorist. The theorist immediately began to explain how the graph fit exactly with
the theory. The experimentalist noticed that the graph was upside down and
communicated this to the theorist. The theorist turned the graph to the proper orientation
and immediately began to explain how this fit exactly with the theory.
This situation could be avoided if the theorist had derived an a priori model that made
predictions that would exclude other interpretations. An a posteriori analysis suffers
from the problem that it is easy to rationalize an interpretation of the data on an ad hoc
basis. This emphasizes the need to develop theory prior to experimental application. If
this is not possible the theory must be rigorously tested and scrutinized. These are the
important lessons to be learned from the history of physics.

38
In general, the scientific method is the tool to turn to when research in a particular
area has stalled or there appears to be an insurmountable problem. While it would be
nice if there were preexisting theories, often it is necessary to take a previous theory and
modify it so that it fits the facts. Often models must go through many iterations before a
becoming a useful theory. Modification to a theory should have a particular
interpretation. The present investigations use mathematical models describing nuclear
decays and population biology to synthesize novel mathematical models. A model
without an interpretation of parameters merely summarizes data, but a model with an
interpretation of the parameters is explanatory. Parameter interpretation is the key that
relates the model to the underlying science.

39
3. Biological Background
3.1 Introduction
The key biological background that is important to the proposed mathematical
models will be presented in eight parts as follows. First, the architecture of the brain and
associated communication networks are considered. Second, genetics and cellular
function are considered. Third is a description of the applications of model organisms.
Fourth, is the consideration of two possible modes of disease: familial and sporadic.
Fifth, issues related to chronic disease states are reviewed. Sixth, the technology of
magnetic resonance imaging is reviewed. Seventh, the existence and character of White
Matter Lesions (WMLs) is discussed. The final section reviews the role of statistical
significance in terms of p-values. This chapter is a general overview; specific details
relating to schizophrenia and WML's are included in the relevant discussions.
3.2 Brain Architecture and Communication
The brain is a complex organ and is divided into several lobes and other
structures. The human brain is composed of approximately 86 billion (uncertainty
plus/minus 8 billion) neurons and about 85 billion (uncertainty of 10 billion) non-
neuronal cells [34]. The brain can be considered on three levels of connectivity:
macroconnections, mesoconnections, and microconnections [35]. Macroconnections are
the connections between major anatomical regions and are largely genetically
determined; an example of a macroconnection is the gray matter regions and the white
matter tracts that connect them [35]. Mesoconnections are the connections between

40
different types of neurons in the central nervous system (CNS) [35]. At the lowest level
of connectivity are individual neurons [35]. The proper functions of cells that fall within
the neural circuit must be considered. As an organizing principle the macrostructures of
the brain are described first. Understanding of the mesoconnection level is an ongoing
project that is just beginning to bear fruit [36, 37]; discussion of mesoconnections will be
restricted to the specifics related to schizophrenia and WMLs. The level of
microconnections will be discussed at length because it is disruption of these systems that
is thought to underlie many of the symptoms of schizophrenia and other mental diseases.
Understanding of microconnections leads to the emergence of neuronal circuits that lead
to the emergence of mesoconnections. Microconnections are modifiable over the human
lifespan; these alterations are critical to learning [35].
The number of connections between neurons is much greater than the number
neurons in the brain. The frontal, parietal, and temporal lobes are the primary cognitive
areas of the brain. Each of the major lobes can be subdivided into sub-regions known as
Brodmann areas. Brodmann areas are specialized regions of the lobe that are typically
associated with a specific function. The occipital lobe primarily processes visual
information. The corpus callosum is a massive communication conduit linking the left
and right hemispheres. The limbic system is considered the fifth lobe of the brain that
forms an important region for information processing. The basal nuclei are the sixth
major structure in the cerebral hemispheres. The lower brain stem consists of the pons,
medulla, cerebellum, and the spinal cord. These are the major structures of the CNS.
The primary focus in this thesis is on the brain, so the spinal cord will not be considered.

41
The frontal lobe is the largest structure in the brain and is the location of many
higher functions. The frontal lobe in the human brain is much more developed than in
other animals; complexity arises from both internal connections and incoming
information from the other lobes. The eyes are part of the CNS and complex
information processing takes place at the retina [35]. The occipital lobe is subdivided
into a number of visual regions that are each specialized. The occipital lobe then
transmits the information to regions of the frontal, parietal, and temporal lobes. The
parietal lobes integrate sensory information making it responsible for awareness and
response to environmental stimuli. The temporal lobes contain substructures related to
perception, recognition, memory acquisition, and language comprehension. The limbic
system is primarily associated with emotional processes and memory formation. The
substructures of the limbic system include the amygdala, cingulate gyrus, hippocampus,
and hypothalamus. The hippocampus is the part of the brain that processes memories
before moving them into long term memory. The dentate gyrus, a component of the
hippocampus, plays a role in memory consolidation. The entorhinal cortex routes and
preprocesses information sent to the hippocampus. All these structures form the
cerebrum.
The brain stem is composed of the pons, medulla oblongata, and midbrain. The
pons regulates breathing and sensory analysis. The cerebellum regulates movement and
is related to learning and other cognitive functions. The pons connects the cerebrum, the
lobes and limbic system, to the cerebellum. The hypothalamus controls activities found
throughout the body: maintains homeostasis, controls hormonal secretions, and regulates

42
endocrine glands. The thalamus is a relay station for moving information between the
cortex, the brain stem, and various cortical structures.
Within the brain there are a number of cerebrospinal fluid (CSF) filled structures
known as ventricles. The lateral ventricles are in the cerebrum and are an important
landmark in the brain. The third ventricle is ventral and medial to the lateral ventricles.
The fourth ventricle is located near the pons and the medulla. All the ventricles are
connected and extend into the central column of the spinal cord. The CSF in and around
the brain protects it from mechanical injury [38].
This outlines the basic divisions within the human brain. Additional details are
provided as they relate specifically to schizophrenia and leukoaraiosis. There is a great
deal of diversity in human brains. There are standard regions such as the lobes, but there
is variation in the location of the folds of the cortical surface known as gyri and sulci
[39]. These folds in the cortical surface increase the surface area with additional gray
matter. The next topic is defining gray and white matter.
Gray matter is composed of the cell bodies of neurons. Gray matter can be
localized into Brodmann regions which are responsible for particular functions. Gray
matter in the isocortex (synonymous with neocortex), the outer layer of cerebrum, has a
laminated structure. This structure has six layers each composed of different types of
neurons. The particular layering varies with function; Brodmann distinguished 50
different patterns in his investigations [35]. It is believed that the different patterns of
lamination are responsible for the specific function of that region of the brain. Much of
laminar structure is brought about during neurodevelopment; abnormalities in

43
neurodevelopment may lead to anomalous function of the gray matter. Gray matter
nuclei are regions of gray matter that are not laminated and are found throughout the
brain. They can even lie deep within white matter. Gray matter is distributed throughout
the brain and is responsible for information processing.
Supporting the gray matter is white matter. White matter is composed of neuronal
axons and glial cells. In particular, white matter forms the communication links between
neurons, known as axons. Glial cells called oligodendrocytes compose the myelin
sheaths around the axon whose appearance gives white matter its name. White matter
forms the connections in the brain; these range from simple to complex. Some tracts of
white matter have been identified, but the small size of axons makes in vivo functional
imaging difficult. The supporting cells include astrocytes, glia, and other cells which
perform essential duties that support neuronal function. These tasks include
metabolically supporting neurons, regulating ionic concentrations in extracellular fluid,
and clearing neurotransmitters from synapses [40]. Emerging evidence suggests that
astrocytes, a type of glial cell, modulate signaling between neurons [40]. Microglia are
small cells that defend the brain during illness and injury by clearing cellular debris and
combating microorganisms [40]. White matter plays a critical role in communication
within the brain. The next topic to be discussed is intracellular and intercellular
communication.
Within the brain there are two distinct forms of communication: electrical and
chemical. Electrical communication is the primary form of intracellular communication,
though in some cases it is also used in intercellular communication. Neurons are

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