Table Of ContentDETERMINISTIC AND STOCHASTIC METAPOPULATION MODELS FOR
DENGUE FEVER
by
Carlos Alan Torre
A Dissertation Presented in Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
ARIZONA STATE UNIVERSITY
December 2009
UMI Number: 3391991
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DETERMINISTIC AND STOCHASTIC METAPOPULATION MODELS FOR
DENGUE FEVER
by
Carlos Alan Torre
has been approved
November 2009
Graduate Supervisory Committee:
C. Castillo-Chavez, Chair
RE. Greenwood
X. Wang
ACCEPTED BY THE GRADUATE COLLEGE
ABSTRACT
A spatial temporal data set of dengue in Peru from 1994-2008 was made available to us
by the Ministry of Health of Peru and the analyses of its spatio-temporal patterns motivated
the work in this dissertation. We found that aggregated reported data masked the spatio-
temporal patterns of dengue over this window in time. A series of models are presented in
this dissertation in order to identify mechanisms that capture observed patterns. We have in
fact identified a framework capable of capturing dengue outbreaks in Peru. Deterministic
and stochastic single and two-patch models are introduced and some of their properties
identified via some mathematical analyses complemented with extensive simulations. We
find that the asymptotics of the mean field model (final size epidemic), while useful, mask
critical details that are central to control and public health policies. We introduce a stochas
tic migration model that allows us to construct a family of distributions of time T, which
the first infected individual leaves the "home" patch, and estimate the variance along the
CDF. A two patch model, where each person has a positive probability of being in a patch
alternate to his home location, shows the effect coupling coefficients will have on the time
between the epidemic peaks. The inclusion of seasonality and human demographics to the
two patch model leads to reoccurring outbreaks, as seen in the data. The two approaches of
modeling migration, though different mathematically, complement each other in explaining
what factors affect the spread of dengue. Finally, optimal control methods are incorporated
into our models. We consider what strategy should be used if the objective is to minimize
the total number of infected individuals, at a minimal cost, during a fixed time interval.
When control is applied to a patch with the basic reproductive number i?o below a certain
threshold, it effectively stops the epidemic. Also, control will delay the spread of dengue
from one patch to another.
in
ACKNOWLEDGEMENTS
Special thanks to Carlos Castillo-Chavez for his infinite patience, Cindy Greenwood
for guidance, and Xiaohong Wang for advice. Thanks to Gerardo Chowell for opportunity
to do research at Los Alamos Laboratory, to Roxana Lopez-Cruz of Universidad Nacional
Mayor de San Marcos, and to Cesar Munayco of the Ministry of Health of Peru. All
this research would not be possible without the generous funding of the Alfred P. Sloan
Foundation, LSAMP, and AGEP. I am grateful for the continuos support of "Uncle" Leigh
Phoenix while at TAM, "cousin" Arun Subbiah, and wife Jing Shi. I am appreciative of the
insightful discussions and friendship with my colleague Daniel Rios-Doria. A great hug to
my family who supported me all these years. Last but not least, a salute to all the troops
who fight to protect freedom which allows me to conduct research and complete a PhD.
IV
To Myself
v
TABLE OF CONTENTS
Page
LIST OF FIGURES viii
LIST OF TABLES xvi
CHAPTER 1 INTRODUCTION 1
CHAPTER 2 DETERMINISTIC ONE PATCH MODEL 8
2.1 Model 8
2.2 The Basic Reproductive Number 10
2.3 Reduced System 10
2.4 Final Size Relation 12
2.5 Numerical Simulations 14
2.6 Discussion 17
CHAPTER 3 STOCHASTIC MIGRATION DENGUE MODEL 18
3.1 One Patch Deterministic Model Revisited 18
3.2 Stochastic Migration 20
3.3 Stochastic Migration Simulations 24
3.4 Spatial Stochastic Migration Model 26
3.5 Stochastic Epidemic 27
3.6 Stochastic Epidemic Model with Stochastic Migration 30
3.7 Discussion 33
CHAPTER 4 COUPLED TWO PATCH MODEL 34
4.1 Two Patch Model 34
4.2 The Basic Reproductive Number 37
4.3 Numerical Simulations 41
4.4 Stochastic Two Patch Model 43
4.5 Discussion 50
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Page
CHAPTER 5 TWO PATCH DENGUE MODEL WITH DEMOGRAPHICS FOR
HOST AND WITH SEASONALITY 53
5.1 The Model 54
5.2 Simulations 55
5.3 Stochastic Simulations of the Two Patch Model with Host Demographics
and Seasonality 60
5.4 Discussion 63
CHAPTER 6 OPTIMAL CONTROL OF ONE PATCH DENGUE MODEL . . .. 65
6.1 Dengue Model with Controls 66
6.2 Method 67
6.3 Numerical Results 70
6.4 A One Patch Model Incorporating Inefficiency of Implementation of Control 74
6.5 Optimal Control with Stochastic Migration 74
6.6 Discussion 76
CHAPTER 7 OPTIMAL CONTROL OF TWO PATCH DENGUE MODEL . . .. 78
7.1 Two Patch Dengue Model with Controls 78
7.2 Numerical Results 82
7.3 Discussion 87
CHAPTER 8 CONCLUSION 90
BIBLIOGRAPHY 94
vn
LIST OF FIGURES
Figure Page
1.1 Dengue Fever weekly data, 1994-2008 6
1.2 Dengue Fever weekly data by region, 1994-2008 7
2.1 Vector-human dynamics for dengue fever. Connection between model is in
rates per individual 9
2.2 Final epidemic size functions H(XJ) = 1 -X% and G(X») = j^T=JET (Z~ [XO° ~ x^~kl])
forJti > 1 13
2.3 Final epidemic size functions//(Xo) = 1 -A*« andG(X») = | y ^- (X&pko-Xi-*1])
foriki < 1 14
2.4 Comparison of final size result of analytic solution and numerical solution.
Analytic solution derived from simplified 3 dimensional model and numerical
solution from 5 dimensional model 16
2.5 Comparison of incidence curves for the three class and five class models. For
both models Ro = 5 16
3.1 Seven class vector-human model for dengue fever. Connection between model
is in rates per individual 19
3.2 The Probability Density Function of the time T, when the first infected migrant
moves to another patch if rate of migration per individual is m = 10~4, the
initial susceptible population 5/,(0)= 100,000, and RQ = 4 22
3.3 The family of cumulative distribution functions, as parameterized by m, of the
time T when the first infected migrant moves to the second patch. R$=3.6 . .. 23
3.4 The cumulative distribution functions of the time T as parameterized by RQ,
for a fixed small stochastic migration rate m 24
3.5 Infected hosts as a function of time for varying 7?o 25
viii
Figure Page
3.6 Histogram of the times of the first successful migration of an infected individ
ual to the second patch, for stochastic migration per individual rate m = 10~4,
Sfc(0)=100,000, and/? =4 25
0
3.7 Histogram of total arrivals in the second patch for stochastic migration per
individual rate m = 10~4, 5 (0)=100,000, and R = 4 26
/z 0
3.8 Stochastic migration along a linear path of n patches 26
3.9 Infected host epidemic curve for 1000 stochastic simulations, RQ = 4 30
3.10 Cumulative distribution functions for 1000 simulations, migration per individ
ual rate m = 10"6, R = 4 31
0
3.11 Variance of the value of the CDF of T, evaluated at t, as a function of t for
1000 simulations, with varying RQ 31
3.12 Variance of the value of the CDF of T, evaluated at f, as a function of t for
1000 simulations, with varying m 32
4.1 Simulation when epidemic starts in Patch 1 and moves to Patch 2 for strong
(left) and weak (right) coupling. There is movement from Patch 1 to Patch 2
only, i.e., P = 0.9999, P = .0001, P = 1, and P i = 0. Both patches have
n 12 22 2
local #o of 2 40
4.2 Simulation when epidemic starts in Patch 1 and moves to Patch 2. There is
movement from Patch 1 to Patch 2 only, i.e., P, i = 0.9999, P = -0001, P =
n 12
l,andP2i = 0. Both patches have local RQ of 2 41
4.3 Comparison of epidemic with no coupling triggered by a local infected indi
vidual and an epidemic that is triggered by infected individuals from another
patch entering due to coupling coeffient P\ = 0.9999. Both patches have lo
2
cal RQ of 4. Both patches have the same initial condition at the start of the
epidemic, 4(0) = 10"9 42
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