a delay differential equation model for dengue fever transmission in ...

A DELAY DIFFERENTIAL EQUATION MODEL FOR DENGUE FEVER
TRANSMISSION IN SELECTED COUNTRIES OF SOUTH-EAST ASIA
MR.WERAPONG SAKDANUPAPH
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF MASTER OF SCIENCE IN APPLIED MATHEMATICS
DEPARTMENT OF MATHEMATICS
GRADUATE COLLEGE
KING MONGKUT'S UNIVERSITY OF TECHNOLOGY NORTH BANGKOK
ACADEMIC YEAR 2007
COPYRIGHT OF KING MONGKUT'S UNIVERSITY OF TECHNOLOGY NORTH BANGKOK

Name
: Mr.Werapong Sakdanupaph
Thesis Title : A Delay Differential Equation Model for Dengue Fever
Transmission in Selected Countries of South-East Asia
Major Field : Applied Mathematics
King Mongkut’s University of Technology North Bangkok
Thesis Advisor : Dr. Elvin James Moore
Academic Year : 2007
Abstract
Dengue Fever is a dangerous virus infection caused by the Dengue virus.
Dengue Fever is a viral disease transmitted by female Aedes mosquitoes.
Mathematicians have created models of Dengue Fever to study the causes of the
spread of the disease and to try to develop methods for reducing the spread of the
disease. In this research we study a model for Dengue fever consisting of a system of
four nonlinear differential equations with time delays. The model includes infected
humans, infectious humans, infected mosquitoes and infectious mosquitoes. The
equilibrium points and asymptotic stability of the equilibrium points are studied
analytically. The Matlab computer program is used to obtain numerical solutions of
the model of Dengue Fever for both zero and nonzero time delays for a range of
parameter values.
The results obtained from the model are compared with actual data of Dengue
Fever in Thailand, Malaysia and Singapore.
(Total 79 pages)
Keywords: Dengue fever, Delay Differential Equations
______________________________________________________________Advisor
ii

ชื่อ : นายวีรพงศ ศักดานุภาพ
ชื่อวิทยานิพนธ : แบบจําลองสมการเชิงอนุพันธที่มีการหนวงเวลาสําหรับ
การแพรเชื้อ ของโรคไขเลือกออกในบางประเทศของ
เอเชียตะวันออกเฉียงใตที่เลือกมา
สาขาวิชา
: คณิตศาสตรประยุกต
มหาวิทยาลัยเทคโนโลยีพระจอมเกลาพระนครเหนือ
อาจารยที่ปรึกษาวิทยานิพนธหลัก : Dr. Elvin James Moore
ปการศึกษา : 2550
บทคัดยอ
โรคไขเลือดออกเปนโรคติดตอที่อันตรายซึ่งมีสาเหตุมาจากไวรัสเดงกี (Dengue Virus)
และเปนโรคติดตอที่มียุงลายเปนพาหะ นักคณิตศาสตรไดสรางแบบจําลองของไขเลือดออกเพื่อ
ศึกษาสาเหตุของการแพรเชื้อของโรคไขเลือดออก และ ไดพัฒนาวิธีการลดการแพรระบาดของ
โรคไขเลือดออก ในงานวิจัยนี้ไดศึกษาแบบจําลองของโรคไขเลือดออกที่อธิบายดวยระบบ
สมการเชิงอนุพันธซึ ่งประกอบดวยสี่สมการซึ่งเปนสมการไมเปนเชิงเสนที่มีการหนวงเวลา ใน
แบบจําลองประกอบไปดวย คนที่ไดรับเชื้อแตยังไมเปนโรค คนที่เปนโรค ยุงที่ไดรับเชื้อแตไม
สามารถแพรโรคและยุงที่รับเชื้อจนเชื้อฟกตัวสามารถแพรโรค ในงานวิจัยนี้เราศึกษาจุดสมดุล
และ เสถียรภาพของจุดสมดุลโดยใชวิธีเชิงวิเคราะหและ หาคําตอบเชิงตัวเลขของแบบจําลอง
โรคไขเลือดออกทั้งชนิดที่มี การหนวงเวลา และ ไมมีการหนวงเวลาโดยใชโปรแกรม Matlab
ผลที่ไดจากแบบจําลองจะนํามาเปรียบเทียบกับขอมูลจริงของโรคไขเลือดออกในประเทศ
ไทย มาเลเซีย และ สิงคโปร
(วิทยานิพนธมีจํานวนทั้งสิ้น 79 หนา)
คําสําคัญ : โรคไขเลือดออก สมการเชิงอนุพันธที่มีการหนวงเวลา
iii
อาจารยที่ปรึกษาวิทยานิพนธหลัก

ACKNOWLEDGEMENTS
I would like to thank my advisor, Dr.Elvin James Moore, for helpful
discussions and advice during the preparation of this thesis. I would also like to thank
the other lecturer in the Department of Mathematics who have taught me during my
study for the degree of Master of Science at King Mongkut’s University of
Technology North Bangkok and who have also made helpful suggestions for my
thesis research. Appreciation is extended to the Graduate College of King Mongkut’s
University of Technology North Bangkok for the award of a scholarship.
Lastly, I want to express my gratitude to my parents, my brother, and my friends
who have always been supportive of what I do. Thank you for all that you have
provided for me during these years.
Werapong Sakdanupaph
iv

TABLE OF CONTENTS
Page
Abstract (in English)
ii
Abstract (in Thai)
iii
Acknowledgements
iv
List of Tables
vii
List of Figures
viii
Chapter 1 Introduction 1
1.1 Background and General Statement of the Problem 1
1.2 Purpose of the Study 5
1.3 Scope of the Study 5
1.4 Method 5
1.5 Utilization of the Study 6
Chapter 2 Literature Review 7
2.1 Background for Dengue fever 7
2.2 Mathematical Models for Malaria 9
2.3 Mathematical Models of Dengue Fever 16
2.4 Mathematics of Equilibrium Point and Stability of Dynamical
Systems 18
2.5 Basic reproduction rate 19
2.6 Linearization Stability Analysis 19
2.7 Routh - Hurwitz Criteria 23
2.8 Dynamical Systems with Time Delays 25
Chapter 3 Methodology 29
3.1 The mathematical model 29
3.2 Existence of Steady States 31
3.3 Asymptotic stability of steady states 33
3.4 Numerical Solution of Delay Differential Equations 42
Chapter 4 Numerical Results 43
4.1 Asymptotically Stable Disease-Free Equilibrium State 43
4.2 Asymptotically Stable Endemic-Disease Equilibrium State 48
v

TABLE OF CONTENTS (CONTINUED)
Page
4.3 Numerical results for Dengue fever in Thailand, Malaysia and
Singapore 54
Chapter 5 Conclusion and Recommendations 65
5.1 Conclusion 65
5.2 Suggestions for Future Work 66
References 68
Appendix A 70
Biography 79
vi

LIST OF TABLES
Table
Page
2-1 Macdonald’s stability index for several regions where malaria is
indigenous 12
4-1 Parameters for disease-free equilibrium state 44
4-2 Parameters for endemic-disease equilibrium state 49
vii

LIST OF FIGURES
Figure
Page
1-1 A diagram of a basic model for malaria 2
2-1 Dengue Fever in South-East Asia from 1895 to 2005 8
2-2 Dengue fever in Singapore from 2005 to 2007 8
2-3 Dengue Fever in Malaysia from 1991-2000 9
2-4 Transmission cycle of malaria and dengue fever including time delays 10
2-5 Graph of Numerical methods for Delay Differential Equations 28
4-1 Numerical solution with zero time delays for disease-free
equilibrium state 45
4-2 Phase plane plots with zero time delays for disease-free equilibrium
state 46
4-3 Numerical solution with nonzero time delays for disease-free
equilibrium state 47
4-4 Phase plane plots with nonzero time delays for disease-free equilibrium
state 48
4-5 Numerical solution with zero time delays for endemic-disease
equilibrium state 51
4-6 Phase plane plots with zero time delays for endemic-disease equilibrium
state 52
4-7 Numerical solution with nonzero time delays for endemic-disease
equilibrium state 53
4-8 Phase plane plots with nonzero time delays for endemic-disease
equilibrium state 54
4-9 Numerical solution with zero time delays for Thailand data 56
4-10 Phase plane plots with zero time delays for Thailand data 56
4-11 Numerical solution with nonzero time delays for Thailand data 57
4-12 Phase plane plots with nonzero time delays for Thailand data 58
4-13 Numerical solution with zero time delays for Singapore data 59
4-14 Phase plane plots with zero time delays for Singapore data 60
4-15 Numerical solution with nonzero time delays for Singapore data 61
4-16 Phase plane plots with nonzero time delays for Singapore data 61
viii

LIST OF FIGURES (CONTINUED)
Figure
Page
4-17 Numerical solution with zero time delays for Malaysia data 63
4-18 Phase plane plots with zero time delays for Malaysia data 63
4-19 Numerical solution with nonzero time delays for Malaysia data 64
4-20 Phase plane plots with nonzero time delays for Malaysia data 64
ix

CHAPTER 1
INTRODUCTION
1.1 Background & General Statement of the Problem
Dengue fever is a dangerous disease which is common in South East Asia.
Dengue fever is a viral disease transmitted by Aedes mosquitoes, usually Aedes
aegypti. There are four dengue viruses (DEN-1 through DEN-4) which are
immunologically related, but they do not provide cross-protective immunity against
each other [2,3,4,5].
Dengue Fever epidemics first occurred almost simultaneously in Asia, Africa,
and North America in the 1780s. The disease was identified and named in 1779. A
global pandemic began in Southeast Asia in the 1950s and by 1975 the disease
appeared frequently in this region. In Thailand Dengue Fever almost always appears
in the rainy season between May-August. In 2005 there were about 31,000 reported
cases of Dengue fever. In the same year there were about 12,000 reported cases of
Dengue fever in Singapore, and about 32,000 reported cases of Dengue fever in
Malaysia [5,16,17].
The method of transmission of Dengue fever is similar to the method of
transmission of Malaria, i.e., an infectious female mosquito bites a human who might
become infected. The disease develops in the infected human who then becomes
infectious. A female mosquito bites an infectious human and it becomes infected.
The disease develops in the infected mosquito which then becomes infectious. The
cycle then repeats [1]. A diagram of the transmission cycle of malaria and Dengue
fever is shown in Figure 2-4.

2
A summary of some of the mathematical models developed for malaria has been
given by Anderson and May [1]. They describe two basic models. The first model,
which is based on the work of Ross [15] and Macdonald [15,20], is a two-equation
differential equation system with no time delay. The second model, which is based on
the work of Macdonald [15,20], is a four-equation differential equation system with
two time delays. A diagram his model is shown in follow
acy(1 − yˆ
)
y
ŷ
γ
Nˆ
abyˆ (1 − y)
N
u
Figure 1-1 A diagram of a basic model for malaria [1]
Ross’s basic model consisted of 2 differential equations:
No time delay:
where
y is proportion of infected humans
dy
= (abN ˆ / N)y(1 ˆ −y) −γy
dt
dyˆ
= acy(1 −y) ˆ −uyˆ
dt
ŷ is proportion of infected mosquitoes
a is the rate of biting on humans by a single mosquito
b is the proportion of mosquito bites on humans that produce infectious in
humans
c is the proportion of mosquito bites on humans that produce infectious in
mosquitoes
N is the size of human population

3
ˆN is the size of the female mosquito population (only females transmit the
disease)
γ is the rate per human of human recovery from infection
u is the mortality rate for mosquitoes
The equilibrium populations of humans and mosquitoes have been obtained for
the model with no time delay (see, [1], 392-399). It is found that there are 2
equilibrium points which correspond to a disease-free equilibrium and an endemic
disease equilibrium. The asymptotic stability of the two equilibrium points has also
been analyzed. The asymptotic stability of the disease-free equilibrium has been
2
ma bc
ˆN
investigated and found to depend on a parameter R
0
= , where m = is the
uγ
N
ratio of the female mosquito and human populations. R 0 is called the basic
reproduction rate. The disease-free equilibrium is asymptotically stable if R0
< 1 and
the endemic disease equilibrium exists and is stable for R0
> 1.
As stated above, Anderson and May [1] also describe a 4-equation model for
malaria which includes the time delays τ
1
and τ
2
shown in Figure 1-1. Their model
with time delay is:
dh(t)
= abmy(t)(1 ˆ −y(t)) −u1h(t) −abmy(t ˆ −τ1)(1 −y(t −τ1))
dt
dy(t)
= abmy(t ˆ −τ1)(1 −y(t −τ1)) −u1y(t) −γy(t)
dt
dh(t) ˆ
= acy(t)(1 −y(t)) ˆ −u ˆ
ˆ
2h(t) −acy(t −τ2)(1 −y(t −τ2))
dt
dy(t) ˆ
= acy(t −τ ˆ
ˆ
2)(1 −y(t −τ2)) − u2y(t)
dt
where
h(t) is proportion of humans who are infected but not yet infectious
(0

4
ŷ(t) is proportion of mosquitoes that are infectious
u
1
is mortality rate in the human population
u
2
is mortality rate in the mosquito population
ˆN
m is number of female mosquitoes per human ( m = ) N
τ
1
is time delay in humans from infected to infectious stage
τ
2
is time delay in mosquitoes from infected to infectious stage
Anderson and May [1] also give an analysis of the equilibrium points of this 4-
equation model and their asymptotic stability. They find 2 equilibrium points
corresponding to a disease-free and an endemic disease equilibrium point. The
asymptotic stability of the disease-free equilibrium point is again described in terms
of a basic reproductive rate R
0
with R0
< 1 corresponding to asymptotic stability.
In Chapter 2, we discuss previous mathematical models which have been used
to describe malaria and Dengue Fever. We also adapt the model for malaria given in
[1] to describe Dengue Fever. The new model is a modified version of the four
equation system of time delay differential equation with two time delays given above
and describes the movement of humans from infected to infectious stage and of
mosquitoes from infected to infectious stage. In Chapter 3, an analysis is given of the
equilibrium populations and their asymptotic stability for our modified four-equation
model. We again find disease-free and endemic disease equilibrium points. We have
analyzed the asymptotic stability of the equilibrium points using a linearization
method (Liapunov’s first method) [7,14] to convert the models into linearized systems
and have found the characteristic equation for these linearized systems. The Routh-
Hurwitz criteria [7,13] are then used to determine the asymptotic stability of all of the
equilibrium points. In Chapter 4, we use numerical methods to solve the model
equations for typical values of the parameters and compare the results with the
available data for prevalence of Dengue fever in Thailand, Singapore and Malaysia.
In Chapter 5, we discuss the results and draw conclusions about the usefulness of the
models in understanding the occurrence and transmission of Dengue fever in these
three countries.

5
A summary of the purpose, scope, methods and utilization of the research in this
thesis is as follows:
1.2 Purpose of the Study
1.2.1 To study models for Dengue fever transmission in Selected Countries of
South-East Asia.
1.2.2 To use numerical methods to compute approximate solutions to the models
for Dengue fever transmission in South-East Asia and to examine the behavior of
solutions for reasonable values of parameters in the model.
1.2.3 To create a Matlab program to show the behavior of the solutions of the
models for Dengue fever transmission in Selected Countries of South-East Asia.
1.2.4 To examine actual data for Dengue fever transmission in Selected
Countries of South-East Asia.
1.3 Scope of the Study
We shall study the model for Dengue Fever Transmission in Selected Countries
of South-East Asia and create a program to show the behavior of the solutions. We
shall examine the effects of time delays and compare solutions with actual disease
data.
1.4 Method
1.4.1 Study relevant background of delay differential equations.
1.4.2 Study methods for computing solutions of systems of delay differential
equations.
1.4.3 Analyze the equilibrium points and their asymptotic stability for the 4-
equation model.
1.4.4 Use numerical methods to show the behavior of the solution of the models
for Dengue fever transmission in Selected Countries of South-East Asia.
1.4.5 Create computer Matlab programs to show the behavior of solutions of
models of Dengue fever transmission in Selected Countries of South-East Asia.
1.4.6 Compare the numerical solutions from the models with actual data on
Dengue fever in selected countries of South-East Asia.

6
1.5 Utilization of the Study
1.5.1 We learn about delay differential equations and their solutions.
1.5.2 We learn about Dengue fever.
1.5.3 We learn numerical methods for finding solutions of delay differential
equations.
1.5.4 We use numerical methods to find solutions of the model for Dengue Fever
Transmission in Selected Countries of South-East Asia.
1.5.5 We get a Matlab program to find behavior of solutions of the model for
Dengue Fever Transmission in Selected Countries of South-East Asia.
1.5.6 The Matlab program could be useful for exploring possible methods for
reducing the level of this dangerous disease in Selected Countries of South-East Asia.

CHAPTER 2
LITERATURE REVIEW
In this chapter we will talk about the background of Dengue fever, the
occurrence of Dengue fever in Singapore, Thailand, and Malaysia, and some of the
mathematical models that have been proposed for malaria and Dengue fever. We
will also discuss mathematical methods that are required to analyze and solve the
mathematical models. These methods include solution of ordinary differential
equations and delay differential equations.
2.1 Background for Dengue fever
Dengue fever [10,11] is a dangerous disease, afflicting mainly older children
and adults and often remaining unapparent in young children. The sudden onset of
fever and a variety of non-specific signs and symptoms characterize Dengue Fever.
The high fever lasts for two or three days, followed by additional symptoms. Its
clinical presentations are similar to those of several other diseases, meaning thereby
that many of the reported cases of Dengue fever could be due to other febrile illnesses
and also that many dengue infections are not recognized.
2.1.1 Occurrence
The first reported epidemics of Dengue fever occurred in 1779-1780 in Asia,
Africa, and North America [11]; the near simultaneous occurrence of outbreaks on
three continents indicates that these viruses and their mosquito vector have had a
worldwide distribution in the tropics for more than 200 years. During most of this
time, Dengue fever was considered a benign, nonfatal disease of visitors to the
tropics. Generally, there were long intervals (10-40 years) between major epidemics,

8
mainly because the viruses and their mosquito vector could only be transported
between population centers by sailing vessels.
Dengue fever is now a common disease in South-East Asia. In Thailand in May
2005 Dengue fever infected 31000 people and 12 people died. A graph of the
frequency of Dengue fever in South and South-East Asia from 1985 to 2005 is shown
in Figure 2-1.
Figure 2-1 Dengue Fever in South-East Asia from 1985 to 2005 [16]
In Singapore in 2003 Dengue Fever infected 4,788 people and in 2004 it
infected 9,460 people. In 2005 it infected 12700 people. In 2007 it infected 4,029
people and 8 people died. A graph of the frequency of Dengue fever in Singapore
from 2005 to 2007 is shown in Figure 2-2.
Figure 2-2 Dengue fever in Singapore from 2005 to 2007 [5]

9
In Malaysia in 2005 Dengue Fever infected 32,950 people. A graph of the
frequency of Dengue fever in Malaysia from 1991 to 2000 is shown in Figure 2-3.
Figure 2-3 Dengue Fever in Malaysia from 1991-2000 [17]
These data show that Dengue fever is a common disease in South-East Asia. As
stated in Chapter 1, the method of transmission of Dengue fever is similar to the
method of transmission of malaria and mathematical models of malaria can also be
applied to Dengue fever. We will now give a survey of some of the mathematical
models developed for malaria.
2.2 Mathematical Models for Malaria
In 1911 Ronald Ross [1, :392-9] created a basic model for malaria describing
the interaction between the number of infected humans at a time t (y(t) ) and the
number of infected mosquitoes at time t ( ŷ(t) ). A diagram of his model is shown in
Figure 1-1.
Ross’s basic model consisted of 2 differential equations
dy
= (abN ˆ / N)y(1 ˆ −y) −γ y
(2-1)
dt
dyˆ
acy(1 y) ˆ uyˆ
dt = − − (2-2)

10
where y(t) is proportion of infected humans, ŷ(t) is proportion of infected
mosquitoes, a is the rate of biting on humans by a single mosquito, b is the proportion
of mosquito bites on humans that are infectious, c is the proportion of bites by
susceptible mosquitoes on infected people, N is the size of human population, ˆN is
the size of the female mosquito population (only females transmit the disease), γ is
the rate per human of human recovery from infection, u is the mortality rate for
mosquitoes.
Equation (2-1) represents the proportion of infected humans. The term
(abN ˆ / N)y(1 ˆ − y) represents the rate at which humans are infected by mosquitoes.
The term γ y represents the rate at which the infected humans recover and return to
the uninfected class. Equation (2-2) represents the proportion of infected mosquitoes.
The term acy(1 − y) ˆ represents the rate at which mosquitoes are infected. The term
ˆ uy represents the death rate of infected mosquitoes.
For a disease model, the basic reproduction rate R
0
is the expected number of
secondary cases directly caused by an infected individual which is introduced into an
otherwise susceptible population.
For the Ross model, the basic reproduction rate ( R
0
) is as follows [1]
2
ma bc
R
0
=
(2-3)
uγ
Equation (2-3) is usually derived algebraically by analysis of the stability
properties of the differential equations (2-1) and (2-2). However, in 1982 Aron and
May [1,16] used a geometric “phase-plane” analysis of the dynamical behavior of the
model to derive the equation which they said gave a more transparent and
generalizable derivation of the equation.
It has been shown that there are two equilibrium solutions of equations (2-1)
and (2-2). The first equilibrium is a disease-free equilibrium with
y= yˆ
= 0 which is
stable for R0
< 1. The second equilibrium solution is an endemic-disease equilibrium
with the equilibrium proportion of infected humans (the prevalence of infection)
being

11
y
(R −1)
=
R + (ac/u)
* 0
and the corresponding prevalence of infection for mosquitoes being
0
(2-4)
ŷ
⎛R −1⎞⎛ ac / u ⎞
= ⎜ ⎟⎜ ⎟
* 0
⎝ R0
⎠⎝1+
ac/u⎠
(2-5)
It can be seen that the endemic equilibrium corresponds to positive proportions when
R0
> 1 (i.e., when the disease-free equilibrium becomes unstable).
Macdonald [15] concluded that ac / u is an index of stability where ac / u
represents the average number of bites on human host made by a mosquito in its
lifetime. If this number is high, then changes in mosquito populations or biting rates
produce only a small change in the values of
*
y and
*
ŷ . In this case, the number of
malaria cases is relatively stable. If the number is low, then changes in mosquito
populations or biting rates can produce a large change in the values of
*
y and
*
ŷ. In
this case, the number of malaria cases can have large changes with time. A table of
the values of Macdonald’s stability index for several regions in which malaria is
indigenous is given in Table 2-1.
Table 2-1 Macdonald’s stability index ac / u for several regions where malaria is
indigenous
Anopheles spp Location/time period Stability Reference
index
A. punctulatus Maprik, New guinea 2.9 Peters and Standfast (1960)
(1957-8)
A. balabacensis Khmer(1960) 4.9 Slooft and Verdrager (1972)
A. minimus Bangladesh (1966-7) 4.4 Khan and Talibi (1972)
A. gambiae Kankiya, Nigeria
(1967)
3.4 Garrett-Jones and Shidrawi
(1969)
A. gambiae Garki, Nigeria (1972) 3.9 Molineaux et al. (1979)
A. gambiae Khashm, El Girba,
Sudan (1967)
0.47 Zahar (1974)

12
Macdonald used the stability index to make broad geographical comparisons
between the stable malaria of Africa and the unstable malaria of parts of India.
However, the values of the stability index are difficult to determine [1], especially in
areas where the stability index is small and the malaria is unstable. In 1982 Aron and
May [15] described the dynamical behaviour of the model both when the transmission
rate is very high and when the transmission rate is just above a threshold.
A modification of the basic 2-equation model [1] is the incorporation of the
latent periods during which infected hosts are infected but not yet infectious.
A
diagram of the model has been given in Figure 1-1. The model is a system of
nonlinear delay differential equations for the infected human population (h(t) ), the
infectious human population ( y(t) ), the infected mosquito population ( ˆ (h(t) ) and the
infectious mosquito population ( ˆ (y(t)). The model is given by (see Chapter 1) A
diagram of the transmission cycle of malaria and Dengue fever is shown in Figure
2-4.
Infected Human
h(t)
1
Infectious
mosquito
ŷ(t)
Infectious
human
y(t)
2
Infected
mosquito
ĥ(t)
Figure 2-4 Transmission cycle of malaria and dengue fever including time delays

13
dh(t)
= abmy(t)(1 ˆ −y(t)) −u1h(t) −abmy(t ˆ −τ1)(1 −y(t −τ1))
dt
dy(t)
= abmy(t ˆ −τ1)(1 −y(t −τ1)) −u1y(t) −γy(t)
dt
dh(t) ˆ
= acy(t)(1 −y(t)) ˆ −u ˆ
ˆ
2h(t) −acy(t −τ2)(1 −y(t −τ2))
dt
dy(t) ˆ
= acy(t −τ ˆ ˆ
2)(1 −y(t −τ2)) − u2y(t)
dt
where all symbols have already been defined in chapter 1.
In 1957 Macdonald analyzed the equilibrium solutions of this model and solved
the ensuing set of algebraic equations. He found that the basic reproduction rate of
malaria in this model is now
2
⎛ma cb ⎞
R
0
= ⎜ ⎟exp( −u1τ1−u 2τ2)
⎝ γu
2 ⎠
Here b is now the proportion of bites by sporozoite-bearing mosquitoes that
result in infection and c is the proportion of bites by susceptible mosquitoes on
gametocyte-bearing people that result in the mosquito infection.
A more detailed model of Plasmodium Vivax malaria has been given by
Kammanee et al [8]. This model includes susceptible ( S′ h
), infected ( I′ h
) and dormant
human populations ( D′
h
) and susceptible ( S′ v
) and infected ( I′ v
) mosquito
populations. The dormant human population consists of humans who have recovered
from malaria and have some immunity but could still get the disease again. The
dynamic equations describing the density of the human populations are as follows:
dS′
h
dt
dI′
h
dt
dD′
h
dt
=λ N + (1 −α )rI′ + rD ′ −( γ ′ I ′ +µ )S′
(2-6)
h 1 h 3 h h v h h
=γ ′ I′ S′ + r D ′ − (r +µ )I′
(2-7)
h v h 2 h 1 h h
=αrI ′ − (r + r +µ )D′
(2-8)
1 h 2 3 h h
dN′
h
= (h γ −µ
h)Nh
(2-9)
dt

14
where all parameters in the model are assumed positive. S′ h is the susceptible human
population, I′ h is the infected human population, D′
h is the dormant human population,
N h is the total human population, λ is the natural birth rate of human population, µ
h
is the natural mortality rate of human population which will be same for all classes,
1
r −
1
is the mean life time for the parasite to remain infectious in the human, α is the
percentage of individuals leaving the infected state and entering the dormant state, r 2
is the relapse rate, and r 3
is the recovery rate.
Equation (2-6) represents the proportion of susceptible population. The term
λ N h
represents the rate of natural birth rate of total population. The term (1 −α )r1I′
h
represents the death rate of infected population. The term rD′ 3 h
represents the
recovery rate of dormant population and the term ( γ ′ h
I ′ v
+µ h
)S′
h
represents the
mortality rate of human susceptible population.
Equation (2-7) represents the proportion of infected population, The term
γ ′′ IS′
represents the rate at which the susceptible population becomes infected. The
h v h
term rD′
2 h
represents the relapse rate of dormant population. The term (r 1
+µ h
)I′ h
represents the mortality rate of infected population.
Equation (2-8) represents the proportion of dormant population. The term
α rI′
1 h
represents the mortality rate of infected population. The term (r 2
+ r 3
+µ h
)D′ h
represents relapse and recovery rate of dormant population.
Equation (2-9) represents the proportion of total population. The term
(h γ −µ )N represents the mortality rate of total population. The transmission rate
h
h
for malaria is given by
βh
γ ′
h
= b N + p
where b is the species-dependent biting rate of mosquitoes, p is the population of
other animals the mosquitoes can feed on and β
h
is the probability that the P.vivax is
passed on by the mosquito to the human.
The dynamic equations for the mosquito population are as follows:
h

15
dS′
v
= A−γ′ vI′ hS′ v−µ vS′
v
dt
(2-10)
dI′
v
= γ′ vIS
′ ′
h v−µ vI′
v
dt
(2-11)
dN′
v
= A−µ vNv
dt
(2-12)
where S′
v
is the susceptible mosquitoes population, I′ v
is the infected mosquitoes
population,
N′
v
is the total mosquitoes population. A is the recruitment rate, λ
v
is the
mosquitoes lay eggs which give rise to larvae stage of the mosquitoes.
Equation (2-10) represents the proportion of susceptible mosquitoes. The term
γ ′′ IS′
represents transmission rate for infected human and susceptible mosquitoes.
v h
v
And the term µ
vS′
v
represents the mortality rate of susceptible mosquitoes.
v h
Equation (2-11) represents the proportion of infected mosquitoes. The term
γ ′′ IS′
represents transmission rate for infected human and susceptible mosquitoes.
v
And the term µ
vI′
vrepresents the mortality rate of infected mosquitoes.
Equation (2-12) represents the proportion of total mosquitoes. The term µ
vNv
represents the mortality rate of total mosquitoes.
Kammanee et al [8] found the equilibrium populations for the model in
Equations (2-6)-(2-9) and analyzed the local stability using the linearization method
and the Jacobian matrix. From the analysis they obtained the basic reproduction rate
R
0
such that if R0
< 1 then the malaria becomes extinct whereas if R0
> 1 then the
disease-free equilibrium point is not asymptotically stable and an endemic state
occurs.
As stated previously, the method of transmission of Dengue fever is similar to
that of malaria and the models developed for malaria have been adapted to Dengue
fever. We will now give a review of some Dengue fever models.

16
2.3 Mathematical Models of Dengue Fever
Derouich and Boutayeb [6] have analyzed a model for Dengue fever. They
divide the human population into a susceptible group ( S h
), an infected group ( I h
) and
a removed group ( R ). They assume that the removed group cannot get the disease
h
because of immunity obtained either by recovering from the disease or by
immunization. They divide the mosquito population into a susceptible group ( S v
)
and an infected group ( I v
). Their model is as follows:
Human population
dS
dt
h
dIh
dt
dR
dt
h
=Λ − ( µ + p+ C I /N )S
(2-13)
h h vh v h h
= (C I / N )S − ( µ +γ +α )I
(2-14)
vh v h h h h h h
= pS +γ I −µ R
(2-15)
h h h h h
dNh
=Λh −µ
hNh −α
hIh
(2-16)
dt
Vector (mosquito) population
Here
N
h
and
dS
dt
v
dIv
dt
=µ N − ( µ + C I /N )S
(2-17)
v v v vh v h v
= (C I / N )S −µ I
(2-18)
vh v h v v v
N
v
denote the human and vector population sizes. In this model µ
h
is
the proportional death rate of human population, µ
v
is the proportional death rate of
vector population, and
Λ
h
is a population increase due to births and immigrations. S
h
is the susceptible human population, I h
is the infected human population,
removed human population.
R
h
is the
S
v
is the susceptible vector population, I v
is the
infected vector population. p is fraction of susceptible humans P hv
is the average
transmission probability of disease from infected human to a susceptible vector, P
vh
is
the average transmission probability of disease from infected vector to human and I v
is the infected vector population,
C
hv
is the rate of adequate contact of humans to

17
vectors,
C
vh
is the rate of adequate contact of vectors to humans. Equation (2-13)
represents the proportion of susceptible human population. The term
( µ
h
+ p+ CvhI v
/N
h)Sh
represents proportional death rate of susceptible population.
Equation (2-14) represents the proportion of infected human population. The term
(CvhI v
/ N
h
)S
h
represents contact rate of infected human population. The term
( µ
h
+γ
h
+α
h)Ih
represents death rate of infected population. Equation (2-15)
represents the proportion of removed human population. The term µ h
R h
represents
death rate of removed population. Equation (2-16) represents the proportion of the
human populations. The term µ h
N h
represents death rate of human population.
Equation (2-17) represents the proportion of susceptible vector population. The term
µ
vNv
represents the death rate of vector population. The term ( µ
v
+ CvhI v
/N
h)Sv
represents the adequate contact death rate of susceptible vector. Equation (2-18)
represents the proportion of infected vector population. The term (CvhI v
/ N
h
)S
v
represents the adequate contact rate of susceptible vector. The term µ
vIv
represents
the death rate of infected vector.
Derouich and Boutayeb used their model to analyze the equilibrium
populations, the stability of the equilibrium populations and the dynamics of Dengue
Fever for one epidemic of the disease. They also considered the case of two epidemics
following each other. In the second epidemic the human population would include
people who had acquired immunity during the first epidemic. They compared the
proportions of the human populations in the three groups for the two epidemics and
found the typical behavior of the solutions for the two epidemics. They found that the
rate of susceptible, infectious and removed populations for the two epidemics
approached each other asymptotically. They were particularly interested in
understanding of the dynamics of Dengue fever and its evolution to the haemorrhagic
form.
Derouich and Boutayeb conclude that by nature, Dengue fever is a complex
disease resulting from the interaction of human, biological, environmental,
geographical and socio-economic factors. They also concluded that their model shows
that environmental management alone is not sufficient as a means of vector control

18
and that it can only delay the outbreak of the epidemics. They state that the
eventuality of a vaccine protecting simultaneously against the four different serotypes
of Dengue fever remains a hope for the future, but that meanwhile, partial vaccination
could be part of a preventive strategy based on the control of environmental and
socio-economic factors.
As stated in chapter 1, we want to examine a model containing delay times.
This model will be a system of four differential equations containing two constant
delay times. In the following sections of this chapter, we will review the methods
used to analyze the equilibrium points, the stability of the equilibrium points and the
numerical methods of solving differential equations.
2.4 Mathematics of Equilibrium Point and Stability of Dynamical Systems [7,14]
In this section we will review the mathematical methods for analyzing
equilibrium points and their stability for a system of first-order nonlinear differential
equations with no time delays. The methods for systems with time delays are similar
and will be considered in chapter 3.
The definitions of equilibrium point and stability are as follows:
Definition A point
Xe
∈ R
n
is an equilibrium point (or stationary point or singular
point or critical point or rest point) of the differential equation
dX f(t,X)
dt =
*
if there exists a finite time t* such that f(t,X )=0 for all t ≥ t
Note: In the special case of an autonomous system in which f is a function of X only,
i.e., f(t,X) = f(X), then if X
e
is an equilibrium point of dX f(X)
dt = at t * , then it is an
*
equilibrium point for all t ≥ t
Definition An equilibrium point
X
e
of
any t0
∈ R + there is a ωδ (,t)
0
> 0such that
e
dX
= f(t,X) is stable if for every δ> 0 and
dt
u(t,t
0, γ) − X
e

19
whenever γ− X
e
0 such that
lim u(t, t , γ ) = X whenever γ− X
e
1 then the number of
secondary infections is greater than the number of initial infections and the number of
people infected with the disease will increase and an epidemic may occur. The basic
reproduction rate also gives a measure of the stability of any disease-free equilibrium
point of the mathematical model of the disease.
2.6 Linearization Stability Analysis [7,14]
Although it is usually not easy to determine the stability of an equilibrium point
of a system of differential equations, the determination of the asymptotic stability is
usually quite easy. The method involves linearization of the equations about the
equilibrium point and the determination of the stability of the linearized equations.

20
The linearization method is often called Liapunov’s first method. As the model of
Dengue fever that we will examine will be a system of four autonomous first-order
differential equations, we will consider a system of four equations here. The system
is as follows:
dh(t)
= F(h, y,h,y) ˆ ˆ
(2-19)
dt
dy(t)
= G(h, y, h, ˆ y) ˆ
(2-20)
dt
dh(t) ˆ
= H(h, y,h, ˆ y) ˆ
(2-21)
dt
dy(t) ˆ
= I(h, y, h, ˆ y) ˆ
(2-22)
dt
* * * *
where F,G,H and I are nonlinear function. We let (h ,y ,h ˆ ,y ˆ ) be the equilibrium
point and then
* * ˆ* * * * ˆ* * * * ˆ* * * * ˆ* *
F(h,y,h,y) ˆ = G(h,y,h,y) ˆ = H(h,y,h,y) ˆ = I(h,y,h,y) ˆ = 0 (2-23)
The linearization method examines the behavior of the system close to an
equilibrium point. We define:
*
h(t) = h + h
(2-24)
*
y(t) = y + y
(2-25)
*
ˆ ˆ ˆ
y(t) = y + y
(2-26)
ˆ ˆ*
ˆ
h(t) = h + h
(2-27)
This method is called perturbation of equilibrium point. We substitute h(t), y(t),
ĥ(t) and ŷ(t) in (2-24),(2-25),(2-26) and (2-27) into (2-19),(2-20),(2-21) and (2-22),
*
d(h + h) * * * ˆ*
ˆ
dt
= F(h + h, y + y, yˆ
+ y, ˆ h + h)
(2-28)
*
d(y + y) * * * ˆ*
ˆ
ˆ
dt
ˆ
= G(h + h,y + y,yˆ
+ y,h ˆ + h)
(2-29)
*
d(h + h) * * * ˆ*
ˆ
dt
= H(h + h,y + y,yˆ
+ y(,h ˆ + h)
(2-30)

21
*
d(yˆ
+ y) ˆ * * * ˆ*
= I(h + h, y + y, yˆ
+ y,h ˆ + h) ˆ
(2-31)
dt
We then expand F,G,H and I in a Taylor series about the equilibrium point
* * ˆ * *
(h ,y ,h ,y ˆ ) and obtain
where
*
dh dh * * * * * * * * * * * *
+ = ˆ ˆ + ˆ ˆ ˆ ˆ
h
+
y
F(h,y,y,h) F(h,y,y,h)h F(h,y,y,h)y
dt dt
+ F (h , y , y ˆ , h ˆ )yˆ + F (h , y , y ˆ , h ˆ )hˆ + term of order h , y , y ˆ , h ˆ , hy (2-32)
ŷ
* * * * * * * * 2 2 2 2
hˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
, hy, hh, yy, yh, yh and higher
*
dy dy * * * * * * * * * * * *
+ = ˆ ˆ + ˆ ˆ ˆ ˆ
h
+
y
G(h,y,y,h) G(h,y,y,h)h G(h,y,y,h)y
dt dt
+ G (h , y , y ˆ , h ˆ )yˆ + G (h , y , y ˆ , h ˆ )hˆ + term of order h , y , y ˆ , h ˆ , hy (2-33)
ŷ
* * * * * * * * 2 2 2 2
hˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
, hy, hh, yy, yh, yh and higher
*
dhˆ
dhˆ
* * * * * * * * * * * *
+ = H(h,y,y,h) ˆ ˆ + H(h,y,y,h)h ˆ ˆ ˆ ˆ
h
+ H(h,y,y,h)y
y
dt dt
+ H (h , y , y ˆ , h ˆ )yˆ + H (h , y , y ˆ , h ˆ )hˆ + term of order h , y , y ˆ , h ˆ , hy (2-34)
ŷ
* * * * * * * * 2 2 2 2
hˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
, hy, hh, yy, yh, yh and higher
*
dyˆ
dyˆ
* * * * * * * * * * * *
+ = I(h,y,y,h) ˆ ˆ + I(h,y,y,h)h ˆ ˆ ˆ ˆ
h
+ I(h,y,y,h)y
y
dt dt
+ I (h , y , y ˆ ,h ˆ )yˆ + I (h , y , y ˆ , h ˆ )hˆ + term of order h , y , y ˆ ,h ˆ ,hy (2-35)
ŷ
* * * * * * * * 2 2 2 2
hˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
,hy,hh, yy, yh, yh and higher
* * * *
F(h,y,y,h)
h
is
ˆ
ˆ
∂F
calculated at
∂h
ˆ
ˆ
* * * *
(h ,y ,h ,y ) and similarly for
F (h ,y ,y ˆ ,h ˆ ),F (h ,y ,y ˆ ,h ˆ ),F (h ,y ,y ˆ ,h ˆ ),G (h ,y ,y ˆ ,h ˆ )
* * * * * * * * * * * * * * * *
y yˆ
hˆ
h
* * * * * * * * * * * * * * * *
ˆ ˆ
y yˆ
ˆ ˆ
hˆ
ˆ ˆ ˆ ˆ
h
* * * ˆ* * * * ˆ* * * * ˆ* * * * *
ˆ
y yˆ
ˆ
hˆ
ˆ h ),I ˆ ˆ
h
(h ,y ,y ,h )
,G (h ,y ,y ,h ),G (h ,y ,y ,h ),G (h ,y ,y ,h ),H (h ,y ,y ,h )
,H (h ,y ,y ,h ),H (h ,y ,y ,h ),H (h ,y ,y ,
,I (h ,y ,y ˆ ,h ˆ ),I (h ,y ,y ˆ ,h ˆ ),I (h ,y ,y ˆ ,h ˆ ) and other terms
* * * * * * * * * * * *
y yˆ
hˆ
By the definition of the equilibrium point we have
ˆ ˆ ˆ ˆ ˆ ˆ
* * * * * * * * * * * *
F(h,y,h,y) = G(h,y,h,y) = H(h,y,h,y)
* * * ˆ *
dh dy dyˆ
dh
= = = = 0
dt dt dt dt
= ˆ ˆ = .
* * * *
I(h,y,h,y) 0
We consider only linear terms. Thus, from (2-32), (2-33), (2-34) and (2-35) we obtain

22
dh
= a11h+ a12y+ a13hˆ + a14yˆ
dt
dy
= a21h+ a22y+ a ˆ ˆ
23h+
a24y
dt
dhˆ
= a ˆ ˆ
31h+ a32y+ a33h+
a34y
dt
dyˆ
= a ˆ ˆ
41h+ a42y+ a43h+
a44y
dt
This system can be written in vector form as dx = J(x*)x where x is column vector
dt
of (h,y,h,y) ˆ ˆ
We now define the Jacobian matrix J of equation (2-19),(2-20),(2-21) and (2-22) to be
* * * * *
J(x ) J(h ,y ,h ,y )
⎡∂F ∂F ∂F ∂F⎤
⎢∂h ∂y hˆ
∂yˆ
⎥
⎢
∂
⎥
⎡a a a a ⎤ ⎢∂G ∂G ∂G ∂G⎥
11 12 13 14
⎢
a21 a22 a23 a
⎥ ⎢
24 h y hˆ
yˆ
⎥
⎢ ⎥
∂ ∂ ∂
⎢
∂
⎥
⎢a31 a32 a33 a ⎥
34 ⎢∂H ∂H ∂H ∂H⎥
⎢ ⎥
a h y ˆ yˆ
41
a42 a43 a
⎢ ⎥
⎣ ∂ ∂
44 ⎦ ∂h
∂
⎢ ⎥
= ˆ ˆ = =
⎢ ∂I ∂I ∂I ∂I
⎥
⎢
∂h ∂y hˆ
∂yˆ
⎥
⎣ ∂ ⎦
* * ˆ* *
(h ,y ,h ,y ˆ )
(2-36)
The linear system in dX = f(t,X) has an equilibrium point at x*=0. In the
dt
theory of equilibrium points of linear systems of the form dx J(x*)x
dt = , it is known
that x = 0 is an equilibrium point and that solutions have the time dependence e λt ,
where λ is an eigenvalue of J(x*) [7,14,21]. Therefore the equilibrium point 0 is
asymptotically stable if the real parts of all eigenvalues of J(x*) are negative and not
asymptotically stable if the real part of some eigenvalue is greater than zero. A
critical value for asymptotic stability is therefore that the real part of some eigenvalue
is zero and the real parts of all eigenvalues are less than or equal to zero.
Using the linear results, the linearized test for the equilibrium point of a
nonlinear system will be asymptotically stable if the real parts of all eigenvalues of
the Jacobian are negative and not asymptotically stable if the real part of some
eigenvalue is positive. The test fails if the real part of any eigenvalue is zero and the

23
real parts of all eigenvalues are less than or equal to 0. Therefore the equilibrium
point ˆ dX
(h,y,h,y) ˆ of = f(t,X) will be asymptotically stable if the real parts of all
dt
eigenvalues of the Jacobian matrix in Eq. (2-36) are negative.
2.7 Routh - Hurwitz Criteria [7,13]
Although the numerical calculation of eigenvalues of matrices can now easily
be carried out with many mathematical software packages (e.g., Matlab, Maple,
Mathematica), the analytical calculation of eigenvalues can usually only be carried
out for very small systems with less than or equal to 4 equations.
One method of testing if all eigenvalues of a matrix have negative real parts is
through the Routh-Hurwitz criteria.
An eigenvalue λ of a matrix A must be a solution of the characteristic equation:
Det( λI − A) =λ + b λ + ... + b = 0
(2-37)
k k−1
1 k
The stability of the equilibrium point can be determined without solving the
characteristic equation for the actual values of the eigenvalues by using the Routh-
Hurwitz criteria.
The Routh-Hurwitz criteria for asymptotic stability
Given the characteristic equation (2-37), define k matrices as follows :
H
H
H
=
[ b ]
1 1
⎡b 1 ⎤
1
2
= ⎢
b3 b ⎥
2
⎣
⎡b1
1 0⎤
H3 =
⎢
b3 b2 b
⎥
⎢
1 ⎥
⎢⎣
b5 b4 b ⎥
3⎦
⎡b1
1 0 0 ⎤
⎢
b b b 1
⎥
⎢
⎥
3 2 1
4
= ⎢ b5 b4 b3 b2
⎥
⎢⎣
b7 b6 b5 b4⎥⎦
⎦

24
H
⎡ b1
1 0 0 0⎤
⎢
b b b 1 0
⎥
⎢
⎥
3 2 1
j
= ⎢ b5 ⎢
b4 b3 b2
0⎥
⎥
H
where the term (l,m) in the matrix
⎢ ⎥
⎢b2j −1 b2j −2 b2j −3 b2j −4 b ⎥
⎣
j⎦
k
⎡ b1
1 0 0 ⎤
⎢
b3 b2 b1
0
⎥
⎢
= ⎥
⎢ ⎥
⎢
⎥
⎢⎣
b2j −1 b2j −2 b2j −3 bk⎥⎦
H
j
is
b
2l− m
for 0 < 2l − m
1 for 2l = m
0 for 2l < m
If all of the determinants of the Routh-Hurwitz matrices are positive, then all
eigenvalues have negative real parts. This means that the equilibrium point
X
e
is
asymptotically stable if and only if the determinants of all Routh-Hurwitz matrices are
positive which is
Det H
j
> 0 for j = 1,2,3,...,k
For the special case of k=4, the Routh-Hurwitz criteria for case k = 4 we need
to show that Det H
j
> 0 for j = 1,2,3 and 4 . Since coefficients b
5,b 6
and b
7
in a
order characteristic polynomial equation are equal to zero, we have the conditions
H
H
[ b ]
= ; Det H1 = b1
1 1
⎡b 1 ⎤
1
2
= ⎢
b3 b ⎥
2
⎣
⎦
; Det H2 = b1b2 − b3
rd
4
⎡b1
1 0⎤
H3 =
⎢
b3 b2 b
⎥
⎢
1 ⎥
; Det
⎢⎣0 b4 b ⎥
3⎦
H = b b b −b − b b
2 2
3 1 2 3 3 1 4

25
H
⎡b1
1 0 0 ⎤
⎢
b b b 1
⎥
⎢
⎥
2 2
; Det H4 = b
4(b1b2b3 −b3 − b1b 4)
⎢
⎥
⎣0 0 0 b4
⎦
3 2 1
4
= ⎢ 0 b4 b3 b2
⎥
So, the four conditions which correspond to Det
H
j
> 0 for j = 1,2,3 and 4 are
b > 0, bb − b > 0, b (bb −b ) − bb > 0 and b (bbb −b − bb ) > 0
2 2 2
1 1 2 3 3 1 2 3 1 4 4 1 2 3 3 1 4
Note that if det H3
> 0, then det H4
> 0if and only if b4
> 0 The four conditions then
reduce to the simpler form
b1
> 0 , b1b2 − b3
> 0,
2
b
3(b1b2 − b
3) − b1b4
> 0 , b4
> 0
Therefore the three conditions of Routh-Hurwitz criteria for local asymptotical
stability for a
i) b1
> 0
rd
4 order characteristic polynomial equation are
ii) b1b2 − b3
> 0
iii)
b (b b −b ) − b b > 0
2
3 1 2 3 1 4
iv) b4
> 0
2.8 Dynamical Systems with Time Delays
In a time-delay system, the derivative of the unknown function y(t) ′ at a certain
time t is given in terms of the values of the functions at previous times. A typical
first-order time-delay differential equation is of the form:
( )
y ′(t) = f t, y(t), y(t −τ ), y(t −τ ),..., y(t −τ )
(2-38)
1 2 k
and this equation has to be solved on a time interval a≤ t≤ b with given history
y(t)
= S(t) for t ≤ a. The delays are such that τ = min( τ1,..., τ
k) > 0 . Although delay
differential equations with variable delays (i.e., the delays might be functions of y or
t) do occur in applications, the delays that appear most frequently in the modeling
literature are constants [9]. In this thesis we will only consider systems with constant
delays. Although the behavior of time-delay nonlinear systems can be very
complicated, the behavior of the model we are considering is reasonably simple. We

26
will only require results for equilibrium points, their asymptotic stability and
numerical programs for solving the equations.
2.8.1 Equilibrium points and asymptotic stability
The derivation of the equilibrium points of a time-delay system is the same as
the derivation for systems without time delays, because at an equilibrium point y,
*
we must have
*
y = y(t) = y(t −τ ) However, the analysis of the asymptotic stability is
more complicated. As we have seen in sections (2.7) and (2.8), the analysis of the
asymptotic stability of an equilibrium point for a system with zero time delays can be
carried out by applying the Routh-Hurwitz criteria to the characteristic polynomial of
the Jacobian of the system. However, for systems with time delays the characteristic
function is not a polynomial but contains factors such as e −λτ . We will discuss
asymptotic stability of equilibrium points for time-delay systems in chapter 3.
2.8.2 Numerical Programs for Delay Differential Equations [9]
Numerical programs for solving delay differential equations are available in
Matlab as well as in some other packages such as Maple and Mathematica. In this
thesis we will use Matlab which has a function dde23 that is able to solve delay
differential equations.
A popular approach to solving delay differential equations is to extend one of
the methods used to solve ordinary differential equations. The Matlab program dde23
uses this approach by modifying a Runge-Kutta order 2 solver (ode23 in Matlab) to
include time delays [9]. Solving a delay differential equation with dde3 is much like
solving an ordinary differential equation with ode23, but there are some notable
differences because of the need to store and access the past history of the system at
each point of the integration.
We will now give an example of the use of dde23 in solving a system of delay
differential equations. However, it is not unusual for equations to have different
forms in different circumstances, which leads to discontinuities in low-order
derivatives of the solution when the circumstances change. This matter is more
serious for delay differential equation because discontinuities propagate and
discontinuities can occur in the history [9].

27
Example
We illustrate the straightforward solution of a delay differential equation by
computing and plotting the solution of the following equation:
y(t) ′
1
= y(t
1
−1)
y ′
2(t) = y
1(t − 1) + y
2(t −0.2)
y(t) ′ = y(t)
3 2
on the interval [0,5] with history y
1(t) = 1, y
2(t) = 1, y
3(t) = 1 for t ≤ 0
A typical invocation of dde23 has the form
sol = dde23(ddefile,lags,history,tspan);
The input argument tspan is the interval of integration, here [0, 5]. The history
argument is the name of a function that evaluates the solution at the input value of t
and returns it as a column vector. Here exam1h.m can be coded as
function v = exam1h(t)
v = ones(3,1);
Quite often the history is a constant vector. A simpler way to provide the history then
is to supply the vector itself as the history argument. The delays are provided as a
vector of delays, here [1, 0.2]. ddefile is the name of a function for evaluating the
right hand sides of the delay differential equation. Here exam1f.m can be coded as
function v = exam1f(t,y,Z)
ylag1 = Z(:,1);
ylag2 = Z(:,2);
v = zeros(3,1);
v(1) = ylag1(1);
v(2) = ylag1(1) + ylag2(2);
v(3) = y(2);
The input t is the current t and y, an approximation to y(t) .
The input array Z
contains approximations to the solution at all the delayed arguments. Specifically,
Z(:, j) approximates
y(t −τ
j)
for the delays τ
j
given as the vector component lags(j).
It is not necessary to define local vectors ylag1, ylag2 as we have done here, but often
this makes the coding of the delay differential equaiton clearer. The ddefile must
return a column vector.

28
The function dde23 does not actually assume that terms like
y(t −τ
j)
appear in
the equations. Because of this, it is possible to use dde23 to solve ordinary differential
equations. This can be useful to check the computer programs.
The input arguments of dde23 are much like those of ode23, but the output
differs formally in that it is one structure, here called sol, rather than several arrays as
in the ode23 call:
[t,y,...] = ode23(...
The field sol.x corresponds to the array t of values of the independent variable
returned by ode23 and the field sol.y, to the array y of solution values. So, one way to
plot the solution is
plot(sol.x,sol.y);
After defining the equations in exam1f.m, the complete program exam1.m to compute
and plot the solution is
sol = dde23(’exam1f’,[1, 0.2],ones(3,1),[0, 5]);
plot(sol.x,sol.y);
title(’Figure 1. Example 3 of Wille’’ and Baker.’)
xlabel(’time t’);
ylabel(’y(t)’);
Note that we must supply the name of the ddefile to the solver, i.e., the string
’exam1f’ rather than exam1f. Also, we have taken advantage of the easy way to
specify a constant history. A plot of the solution is shown in Fig. 2-5.
Figure 2-5 Graph of Numerical methods for Delay Differential Equations

CHAPTER 3
METHODOLOGY
In this chapter, we first present the model that we will use to study the effect of
time delays in the transmission of Dengue fever. The model is an extension of the
four equation model for malaria originally discussed by Macdonald [15] and
Anderson and May [1] that we have already given in section 2.2 and Figure 1-1. We
then analyze the equilibrium points and their stability for the two cases in which the
time delays are zero and the time delays are nonzero. We then discuss the methods
for numerical solution of these equations using Matlab.
3.1 The mathematical model
The model consists of two delay differential equations for four populations
consisting of infected humans (h(t)), infectious humans (y(t)), infected mosquitoes
( ĥ(t) ) and infectious mosquitoes ( ŷ(t) ). The model contains two time delays for
transition from infected to infectious stage in humans ( τ 1
) and from infected to
infectious stage in mosquitoes ( τ 2
). The equations of the model are:
dh(t)
dt
−u 1 τ
= abmy(t)(1 ˆ −h(t) −y(t)) −u ˆ
1
1h(t) −abmy(t −τ1)(1−h(t −τ1) −y(t −τ
1))e
(3-1)
dy(t)
−u 1 τ
= abmy(t ˆ −τ 1
1)(1−h(t −τ1) −y(t −τ1))e −u1y(t) −γ y(t)
(3-2)
dt
dh(t) ˆ
acy(t)(1 h(t) ˆ y(t)) ˆ u h(t) ˆ acy(t )(1 h(t ˆ ) y(t ˆ ))e
dt
−u 2 τ
= − − − 2
2
− −τ2 − −τ2 − −τ
2
(3-3)
dy(t) ˆ
ˆ −u 2 τ
= acy(t −τ ˆ 2
ˆ
2)(1−h(t −τ2) −y(t −τ2))e − u2y(t)
(3-4)
dt
where h(t) is the proportion of humans who are infected but not yet infectious
(0

30
proportion of mosquitoes that are infected but not yet infectious, ŷ(t) is the
proportion of mosquitoes that are infectious, u 1
is mortality rate in the human
population, u
2
is mortality rate in the mosquito population, γ is the recovery rate of
infectious humans from the disease, m is the number of female mosquitoes per human
ˆN
( m = , where ˆN is the mosquito population and N is the human population), τ
1
is a
N
time delay in humans from infected to infectious stage, and τ 2
is time delay in
mosquitoes from infected to infectious stage.
Eq. (3-1) represents the rate of change of the infected human population. In Eq.
(3-1) the term abmy(t)(1 ˆ −h(t) − y(t)) represents the rate of infection of humans by
infectious mosquitoes at time t. The factor 1-h(t)-y(t) in this term represents the
proportion of the human population who do not have the disease (i.e., who are not
infected or infectious) at time t. The original model given in [1] and section 2.2 has a
factor 1-h(t) which we have replaced by 1-h(t)-y(t), i.e., in the original model it was
assumed that infectious humans can become infected. The term
abmy(t ˆ )(1 h(t ) y(t ))e − τ
u
−τ 1 1
1
− −τ1 − −τ
1
represents the rate at which humans move
from the infected to the infectious stage after a latency period of time τ 1
. The factor
u
e − τ 1 1
allows for the death rate of humans during the period τ 1
. In the original model
u1 1
given in [1] and section 2.2, this e − τ factor is not included, i.e., no allowance is
made for death of infected humans in the period τ
1
. The term − uh(t)
1
represents the
death rate of infected humans at time t.
Eq. (3-2) represents the rate of change of the infectious human population. In
u1 1
Eq.(3-2) the term abmy(t ˆ −τ )(1−h(t −τ ) −y(t −τ ))e − τ again represents the rate at
1 1 1
which infected humans move to the infectious stage after a delay time τ 1
. The term
− uy(t)
1
represents the death rate of infectious humans at time t and the term −γ y(t)
represents the recovery rate of infectious humans from the disease
Eq. (3-3) represents the rate of change of the infected mosquito population. In
Eq. (3-3) the term acy(t)(1−h(t) ˆ − y(t)) ˆ represents the rate at which mosquitoes
become infected by biting an infectious human. The factor 1−h(t) ˆ −y(t)
ˆ represents

31
the proportion of the mosquito population that do not carry the disease (i.e., that are
not infected or infectious) at time t. The original model given in [1] and section 2.2
has a factor
1− h(t) ˆ which we have replaced by 1−h(t) ˆ − y(t) ˆ , i.e., in the original
model it was assumed that infectious mosquitoes can become infected.
acy(t )(1 h(t ˆ ) y(t ˆ ))e − τ
The term
u
−τ 2 2
2
− −τ2 − −τ
2
represents the rate at which mosquitoes move
from the infected to the infectious stage after a latency period τ 2
. The factor
u
e − τ 2 2
allows for the death rate of mosquitoes during the period τ 2
. In the original
u2 2
model given in [1] and section 2.2, this e − τ factor is not included, i.e., no allowance
is made for death of infected mosquitoes in the period τ
2
.The term
−uh(t)
represents
the death rate of infected mosquitoes at time t.
Eq. (3-4) represents the rate of change of the infectious mosquito population.
u2 2
The term acy(t −τ )(1 −h(t ˆ −τ ) −y(t ˆ −τ ))e − τ again represents the rate at which
2 2 2
mosquitoes move from the infected to the infectious stage after a time τ 2
. The term
− uy(t) represents the death rate of infectious mosquitoes at time t.
2 ˆ
We now examine the steady states of the above system of equations.
2 ˆ
3.2 Existence of Steady States
We will set time derivatives of equation (3-1)-(3-4) equal to zero and look for a
steady state solution
*
= −τ
1
= ,
y(t) y(t ) y
h(t) ˆ h(t ˆ ) hˆ
ˆ
ˆ
* * * *
(h ,y ,h ,y ) such that
*
= −τ
2
= ,
y(t) ˆ y(t ˆ ) yˆ
*
= −τ
1
= ,
h(t) h(t ) h
*
= −τ
2
= We obtain steady states
by solving the equations for all time derivatives equal to zero. The equations then
become:
ˆ
− − − − ˆ − − = (3-5)
* * * * * * * −u 1 τ
abmy(1 h y) uh 1
1
abmy(1 h y)e 0
abmy ˆ (1 −h −y )e −u y −γ y = 0
(3-6)
* * * −u 1 τ 1 * *
1
− ˆ − ˆ − ˆ − − ˆ − ˆ = (3-7)
* * * * * * * −u 2 τ
acy(1 h y) uh 2
2
acy(1 h y)e 0
acy (1 − hˆ −y ˆ )e − u y ˆ
= 0
(3-8)
* * * −u 2 τ 2 *
2

32
We notice that (0,0,0,0) is always a steady state of the system. This steady
state is called the disease-free equilibrium.
Eq. (3-5)-(3-8) can be rewritten in the simpler form:
abmy ˆ (1 −h −y )(1 −e ) − u h = 0
(3-9)
* * * −u 1 τ 1 *
1
abmy ˆ (1 −h −y )e −u y −γ y = 0
(3-10)
* * * −u 1 τ 1 * *
1
acy(1−hˆ
−y)(1 ˆ −e ) − uh ˆ = 0
(3-11)
* * * −u 2 τ 2 *
2
acy (1 −hˆ −y ˆ )e − u y ˆ = 0
(3-12)
* * * −u 2 τ 2 *
2
We will now examine the equilibrium states and their stability, first for the case of
zero time delays and then for the case with nonzero time delays.
3.2.1 Steady states for zero time delays
For the special case of τ
1
=τ
2
= 0 Eqs. (3-9-3-12) reduce to the form
* *
Therefore we must have h = 0,hˆ
= 0and
*
u1h 0
* * * * *
ˆ − − −
1
−γ =
*
u ˆ
2h 0
* ˆ * * *
− − ˆ − ˆ
2
=
− =
abmy (1 h y ) u y y 0
− =
acy (1 h y ) u y 0
abmy ˆ (1 −h −y ) −u y −γ y = 0
(3-13)
* * * * *
1
acy (1 −hˆ −y ˆ ) − u y ˆ = 0
(3-14)
* * * *
2
There are two solutions for Eqs. (3-13) and (3-14). The first is
*
y = 0,
*
ŷ = 0. This
corresponds to a disease-free equilibrium state.
The second solution is a nonzero solution which is obtained as follows. From
acy (1 −y ˆ ) − u yˆ
= 0 we get
* * *
2
ŷ
*
acy − acy
=
*
acy + u
* *
and then substituting this result into Eq. (3-14) we obtain
2
(3-15)
y
2
− abcm+ uu +γu
abcm−
uu
1 2
−γu2
= =
− − − γ
2
a bcm + acu + acγ
2
* 1 2 2
2
abcm acu1
ac
Then substituting this value for y* into Eq. (3-15), we obtain
1

33
ŷ
*
=
( abmc 2 − uu
1 2
−γu2)
2
a bmc
+ abmu
We have therefore found a disease-free equilibrium state (0,0,0,0) and an endemicdisease
equilibrium state
2
2 2
* * ˆ * *
⎛ abcm−uu 1 2
−γu2 abmc−uu 1 2
−γu
⎞
2
(h , y ,h , y ˆ ) = ⎜0, ,0,
2 2
⎟
⎝ abcm+ acu1+ acγ abmc+
abmu2
⎠
(3-16)
3.2.2 Steady states for nonzero time delays
We first note that (0,0,0,0) is a solution of Eqs. (3-9)-(3-12) and therefore there
exists a disease-free equilibrium. Eqs. (3-9)-(3-12) are very complicated to solve by
hand and therefore we have used Maple to solve the equations. The Maple solution
gives only two equilibrium solutions, the first is the disease-free equilibrium (0,0,0,0)
and the second is a nonzero endemic-disease equilibrium. This shows that the
existence of nonzero time delays does not change the number of equilibrium
solutions. The endemic-disease solution obtained from Maple is:
h = {(u a bmc + a cbm γ)e −(a cbmγ+
a cbmu )e
* 2 2 ( −u1τ1−u 2τ2) 2 2
( −2u1τ1−u 2τ2)
1 1
+ (u u + 2u u γ+ u γ )e −u u −2u u γ−u γ }
2 2 −u1τ1
2 2
1 2 1 2 2 1 2 1 2 2
{1/(ace (u + u γ+ (abmu + abm γ)e −abmγe ))}
−u1τ1 2
−u 2τ2 ( −u1τ1−u 2τ2)
1 1 1
(3-17)
y
ĥ
ŷ
(a bcme −u u −u γ)u
a(u u (abmu abm )e abm e )c
2 ( −u1τ1−u 2τ2)
* 1 2 2 1
=
2
−u 2τ2 ( −u1τ1−u 2τ2)
1
+
1γ+ 1+ γ − γ
(a bcme − a bcme + u u e + u γe −u u −u γ)u
2 ( −u1τ1−u 2τ2) 2 ( −u1τ1−2u 2τ2) −u2τ2 −u2τ2
* 1 2 2 1 2 2 1
=
−u2τ2 −u1τ1 −u1τ1
abme (u1u2 + u2γ+ acu1e −u 2γe )
u(acbme −uu −u γ)
abm(u u u u ace u e )
2 ( −u1τ1−u 2τ2)
* 1 1 2 2
=
−u1τ1 −u1τ1
1 2
+
2γ+ 1
−
2γ
(3-18)
(3-19)
(3-20)
3.3 Asymptotic stability of steady states
We will analyze the asymptotic stability of the steady states by linearization of
the system Eq. (3-1)-(3-4) about a steady state
ˆ
ˆ
* * * *
(h ,y ,h ,y ), where
ˆ
ˆ
* * * *
(h ,y ,h ,y )
will be any one of the four equilibrium states we have found in section (3.2) for the
zero and nonzero time delay cases.

34
The method is similar to the linearization method described in section 2.7, but
with modifications due to the presence of the time delays. We first define a function
for each equation in the system
dh(t)
−u 1 τ
= abmy(t)(1 ˆ −h(t) −y(t)) −u ˆ
1
1h(t) −abmy(t −τ1)(1 −h(t −τ1) −y(t −τ1))e
dt
= F(h, y, h, ˆ y) ˆ
(3-21)
dy(t)
−u 1 τ
= abmy(t ˆ −τ 1
1)(1 −h(t −τ1) −y(t −τ1))e −u1y(t) −γy(t)
dt
= G(h, y,h, ˆ y) ˆ
(3-22)
dh(t) ˆ
ˆ ˆ ˆ
−u 2 τ
= acy(t)(1 −h(t) −y(t)) ˆ −u ˆ
2
2h(t) −acy(t −τ2)(1−h(t −τ2) −y(t −τ2))e
dt
= H(h, y,h, ˆ y) ˆ
(3-23)
dy(t) ˆ
ˆ −u 2 τ
= acy(t −τ ˆ 2
ˆ
2)(1 −h(t −τ2) −y(t −τ2))e −u2y(t)
dt
= I(h, y, h, ˆ y) ˆ
(3-24)
then change variables by defining
ˆ ˆ ˆ
ˆ ˆ ˆ
* * * *
h(t) = h + h(t) , y(t) = y + y(t) , y(t) = y + y(t) , h(t) = h + h(t)
where
h, y, y, ˆ h ˆ are deviations from the steady state value. Eq.(3-21)-(3-24) can then
be rewritten as
*
d(h + h(t)) * * * ˆ*
ˆ
dt
= F(h + h(t),y + y(t),yˆ
+ y(t),h ˆ + h(t)) (3-25)
*
d(y + y(t)) * * * ˆ*
ˆ
ˆ
dt
ˆ
= G(h + h(t),y + y(t),yˆ
+ y(t),h ˆ + h(t)) (3-26)
*
d(h + h(t)) * * * ˆ*
ˆ
ˆ
dt
ˆ
= H(h + h(t),y + y(t),yˆ
+ y(t),h ˆ + h(t)) (3-27)
*
d(y + y(t)) * * * ˆ*
ˆ
dt
= I(h + h(t),y + y(t),yˆ
+ y(t),h ˆ + h(t)) (3-28)

35
* * * *
dh dy dyˆ
dhˆ
At steady state on the left hand side = = = = 0 since
dt dt dt dt
ˆ
ˆ
* * * *
(h ,y ,h ,y )
are constant. On the right hand side, we expand F,G,H and I in a Taylor series about
* * * *
the equilibrium point (h , y ,h ˆ , y ˆ ) . By definition of the equilibrium point
ˆ ˆ ˆ ˆ ˆ ˆ
* * * * * * * * * * * *
F(h,y,h,y) = G(h,y,h,y) = H(h,y,h,y)
can rewrite the system in the new form
ˆ
ˆ
* * * *
= I(h , y ,h , y ) = 0 so that we
dh * * * * * * * * * * * *
= F(h,y,h,y)h ˆ ˆ ˆ ˆ
h
+ F(h,y,h,y)y ˆ
y
+ F(h,y,h,y)y
yˆ
ˆ ˆ
dt
+ F (h , y ,h ˆ , y ˆ )hˆ + h (h , y ,h ˆ , y ˆ )
* * * * * * * *
hˆ 1
dy * * * * * * * * * * * *
= G(h,y,h,y)h ˆ ˆ ˆ ˆ
h
+ G(h,y,h,y)y ˆ
y
+ G(h,y,h,y)y
yˆ
ˆ ˆ
dt
+ G (h , y ,h ˆ , y ˆ )hˆ + h (h , y ,h ˆ , y ˆ )
* * * * * * * *
hˆ 2
dhˆ
* * * * * * * * * * * *
= H(h,y,h,y)h ˆ ˆ ˆ ˆ
h
+ H(h,y,h,y)y ˆ
y
+ H(h,y,h,y)y
yˆ
ˆ ˆ
dt
+ H (h , y ,h ˆ , y ˆ )hˆ + h (h , y ,h ˆ , y ˆ )
^
h
* * * * * * * *
3
(3-29)
(3-30)
(3-31)
where
dyˆ
* * * * * * * * * * * *
= I(h,y,h,y)h ˆ ˆ ˆ ˆ
h
+ I(h,y,h,y)y ˆ
y
+ I(h,y,h,y)y
yˆ
ˆ ˆ
dt
+ I (h , y ,h ˆ , y ˆ )h ˆ + h (h , y ,h ˆ , y ˆ )
* * * * * * * *
hˆ 4
ˆ
ˆ
* * * *
F(h,y,h,y)
h
is the partial derivative of function F with respect to h at
ˆ
ˆ
* * * *
(h , y ,h , y ) and similarly
ˆ
ˆ
(3-32)
* * * *
F(h,y,h,y)
n
is the partial derivative of function F
with respect to n
The linearized system of equations (3-29)-(3-32) is the linear system of delay
differential equations given by
dh
dt
= ˆ + − ˆ + − − ˆ + ˆ
* * * * * −u1τ1
-abmy u1h ( abmy )y (1 y h )abmy (abmy e )hτ1
ˆ
* −u1τ1 * *
−u1τ1
(abmy e )y
τ
+ (1 −y −h )( −abme )y
1
τ1
ˆ
(3-33)
dy
* −u1τ1 * * −u1τ1
= ( − abmyˆ
e )y (u ˆ
τ
+
1 1
−γ )y + (1−y − h )abme yτ
(3-34)
1
dt

36
ˆ
dh * ˆ* −u2τ2 * ˆ* * −u2τ2
= (1−yˆ
− h )acy + ( −ace (1−yˆ
− h ))y acy e hˆ
τ
+
2
τ2
dt
+ ˆ −
* −u2τ2
acy e yτ
u
2 2h
ˆ
(3-35)
where
dyˆ
−u2τ2 * ˆ* * −u2τ2ˆ
* −u2τ2
= ace (1 −yˆ −h )y u ˆ ˆ
τ
−
2 2y −acy e hτ
− acy e y
2 τ
(3-36)
2
dt
yτ = y(t −τ
1 1)
,
hˆ
h(t ˆ
τ
= −τ )
2 2
yˆ
y(t ˆ
τ
= −τ ),
1 1
yτ = y(t −τ
2 2)
,
yˆ
y(t ˆ
τ
= −τ ),
2 2
hτ = h(t −τ
1 1)
,
For convenience in finding the characteristic equation of the above system we
rewrite Eq.(3-33)-(3-36) in the matrix form
⎡dh⎤
⎢ ⎥
⎢dt
⎥
h ⎡hτ
⎤ ⎡
1
hτ
⎤
2
⎢dy⎥ ⎡ ⎤
⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎢ ⎥
dt y ⎢yτ
⎥ ⎢y
1 τ ⎥
2
⎢ ⎥
⎢ ⎥
= A ⎢ ⎥+ A ⎢ ⎥+
A ⎢ ⎥
ˆ ˆ
0 1 2
⎢dhˆ ⎥ hˆ ⎢h ⎥ ⎢h
⎥
τ1 τ2
⎢ ⎥
⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢dt ⎥ ⎢⎣ŷ ⎥⎦ ⎢
⎣
yˆ yˆ
τ
⎥ ⎢
1⎦ ⎣ τ
⎥
2⎦
⎢dyˆ
⎥
⎢⎣dt
⎥⎦
(3-37)
* * * *
⎡-abmyˆ
-u1
-abmyˆ
0 abm(1 −y −h ) ⎤
⎢
⎥
0 −u1
−γ 0 0
A0 =
⎢
⎥
⎢ * ˆ *
0 ac(1−yˆ
−h ) −u2
0 ⎥
⎢
⎥
⎢⎣
0 0 0 −u2
⎥⎦
* −u1τ1 * −u1τ1 * * −u1τ1
⎡abmyˆ
e abmyˆ
e 0 −abm(1 −y −h )e ⎤
⎢
* −u1τ1 * * −u1τ
⎥
1
0 −abmyˆ
e 0 abm(1 −y −h )e
A1
= ⎢
⎥
⎢ 0 0 0 0 ⎥
⎢⎣
0 0 0 0 ⎥⎦
A
⎡0 0 0 0 ⎤
⎢
0 0 0 0
⎥
=
⎢
⎥
⎢ 0 −ac(1 −yˆ
−h ˆ )e acy e acy e ⎥
⎢
⎥
* ˆ * −u2τ2 * −u2τ2 * −u2τ
⎢
2
⎣0 ac(1 −yˆ
−h )e −acy e −acy e ⎥⎦
2 * * −u2τ2 * −u2τ2 * −u2τ2

37
We assume that solutions of the system (3-37) are in the form
h(t)
t
= c1e λ ,
y(t)
t
= c2e λ ,
ĥ(t)
t
= c3e λ and
ŷ(t)
t
= c4e λ , where
1
constants. Substituting these into Eq. (3-37) we obtain
c, c
2
, c
3
and c
4
are arbitrary
λt λt λt −λτ1 λt
−λτ2
⎡λce ⎤ ⎡
1
ce ⎤ ⎡
1
ce
1
e ⎤ ⎡ce 1
e ⎤
⎢ λt⎥ ⎢ λt⎥
⎢ λt −λτ ⎥ ⎢
1 λt
−λτ ⎥
2
⎢λce 2 ⎥ ce
2
ce
2
e ce
2
e
= A ⎢ ⎥
t 0
+ A ⎢ ⎥
t 1
+ A ⎢ ⎥
t −λτ 2
⎢
λ λ λ 1 λt
−λτ2
λce ⎥ ⎢
3
ce ⎥ ⎢
3
ce
3
e ⎥ ⎢ce 3
e ⎥
⎢
λt⎥ ⎢
λt⎥
⎢
λt −λτ
⎥ ⎢
1 λt
−λτ
⎥
2
⎢⎣λ
ce
4 ⎥⎦ ⎢⎣ce 4 ⎥⎦ ⎢⎣ ce
4
e ⎥⎦ ⎢⎣ce 4
e ⎥⎦
(3-38)
⎡c1
⎤
⎢
c
⎥
⎢ ⎥
−λτ1 −λτ2
2 λτ
= (A0 + A1e + A2e ) e
⎢c
⎥
3
⎢⎣
c4
⎥⎦
On dividing by
t
e λ
, we can rewrite Eq. (3-38) in the form:
λ c = (A + A e + A e )c (3-39)
−λτ1 −λτ2
0 1 2
The characteristic equation associated with (3-39) is given by
det( λI − A -A (e )-A (e )) = 0
(3-40)
−λτ1 −λτ2
0 1 2
From matrix A
0
, A,
1
A
2
we change variables in term
ˆ
*
Q= abmy ,
* *
Q1
abm(1 y h )
= − − ,
= − ˆ − ˆ ,
* *
Q2
ac(1 y h )
Q3
= acy
From Eq.(3.40) we get the characteristic equation (we used Maple for the calculation)
λ + a λ + b λ + b λ e + b λ e + b λ e + cλ+ c λ e
4 3 2 2 −2 λτ1 2 −λτ1 2 −λ( τ 1+τ2)
−λτ1
1 1 2 3 4 1 2
+ cλ e + cλ e + cλ e + de + de + de + de
−2 λτ1 −λ( τ 1+τ2) −λ(2 τ 1+τ2) −λτ1 −2 λτ1 −λ( τ 1+τ2) −λ(2 τ 1+τ2)
3 4 5 1 2 3 4
+ d5
= 0
(3-41)
*
where
a1 =γ+ 2u2 + 2u1+
Q
b = 4u u + 2Qu + Qγ+ 2γ u + u + u + Qu + u γ
2 2
1 1 2 2 2 1 2 1 1
2 2u
b 1 1
2
=− Q e − τ
2 u
b 1 1
3
= ( −Qγ+
Q )e − τ

38
b =− Q Q e − τ − τ
( u2 2 u 1 1)
4 2 1
c = Qu + 2Qu u + 2u u + 2u u +γ u + 2u u γ+ 2Qγ
u
2 2 2 2
1 2 1 2 1 2 1 2 2 1 2 2
c = (2Q u − 2Qru )e − τ
2
2 2 2
c =− 2Q u e − τ
2
3 2
2u 1 1
u 1 1
c = (Q Q Q −Q Q u −QQ Q − u Q Q )e − τ − τ
4 3 1 2 2 1 2 2 1 1 2 1
c = QQ Qe − τ − τ
( 2u1 1 u 2 2)
5 2 1
d = (Q u −Qγ
u )e − τ
2 2 2
1 2 2
d =− Q u e − τ
2 2
2 2
2u 1 1
u 1 1
( u2 2 u 1 1)
d = (QQ Q Q − QQ Q u + u Q Q Q − u Q Q u )e − τ− τ
3 3 1 2 2 1 2 1 3 1 2 1 2 1 2
d = ( − QQ Q Q + QQ Q u )e − τ− τ
4 3 1 2 2 1 2
d = u γ u + Qγ u + u u + Qu u
2 2 2 2 2
5 1 2 2 1 2 1 2
( 2u1 1 u 2 2)
( u1 1 u 2 2)
The local asymptotic stability of a steady state
ˆ
ˆ
* * * *
(h , y ,h , y ) can be analyzed by
looking at the signs of the real parts of zeroes of Eq. (3-41).
* * * *
We recall (h , y ,h ˆ , y ˆ ) is locally asymptotically stable if and only if all roots of
Eq.(3-41) have negative real parts, and its stability will be lost when roots cross the
vertical axis and any root has positive real part. The critical values are those values
when the real part of some root is zero.
3.3.1 Asymptotic stability of the disease-free steady state for zero time delay and
existence of endemic-disease state with positive populations
For the asymptotic stability of the disease-free state we substitute
ˆ
ˆ
* * * *
(h ,y ,h ,y ) (0,0,0,0)
= and τ
1
=τ
2
= 0 into Eq. (3-41). We obtain:
λ + f λ + f λ + f λ+ f = 0
(3-42)
4 3 2
1 2 3 4
where after substituting for Q, Q
1,Q 2,Q
3
f1 = a1
=γ+ 2u + 2u
2 1

39
f = b + b + b + b
2 1 2 3 4
2 2
= 4u1u2 + 2γ u2 + u1 + u2 + u1γ−Q2Q1
2 2 2
=
1 2
+ γ
2
+
1
+
2
+
1γ−
4u u 2 u u u u a bcm
f = c + c + c + c + c
2u u 2u u u 2u u (Q Q u u Q Q )
3 1 2 3 4 5
2 2 2
=
1 2
+
1 2
+γ
2
+
1 2γ− 2 1 2
+
1 2 1
2 2 2 2
= 2u1u2 + 2u1u2 +γ u2 + 2u1u 2γ− (u2 + u
1)a bcm
f = d + d + d + d + d
4 1 2 3 4 5
2 2 2
= u1γ u2 + u1u2 −u1Q2Q1u
2
2 2 2 2
=
1γ 2
+
1 2
−
1 2
u u u u u u a bcm
From the Routh-Hurwitz criterion, all eigenvalues of Eq.(3-42) have negative
real parts if and only if
i) f 1
> 0,ii) ff 1 2
− f 3
> 0, iii)
f(ff − f) − ff > 0, iv) f4
> 0
2
3 1 2 3 1 4
From the equations it is clear that f 1
> 0, since r > 0, u1
> 0 and u2
> 0 The testing
of the conditions (ii) and (iii) is difficult in general. Condition (iv) gives us f 4
> 0.
We can rewrite
f = u u ( γ+ u )(1− R ), where
2
4 1 2 1 0
R
0
2
ma bc
=
(u +γ)u
1 2
is called the basic
reproductive rate (section 2.6). From the condition f 4
> 0 we have that a necessary
condition for asymptotic stability of the disease-free equilibrium is R0
< 1.
If we substitute the value of R 0
into the endemic-disease equilibrium state
given in Eq. (3-16), we obtain:
*
h = 0,
y
(R − 1)(u +γ)u
=
+ + γ , *
ĥ = 0,
* 0 1 2
2
abmc acu1
ac
ŷ
(R − 1)(u +γ)u
=
* 0 1 2
2
a bmc + abmu
2
Therefore
*
y = 0 and
*
ŷ = 0 only if R0
> 1.
Therefore we find that the disease-free equilibrium state is asymptotically stable
only for R0
< 1and that a positive endemic-disease equilibrium state exists only for
R0
> 1.

40
3.3.2 Asymptotic stability of the disease-free steady state for non-zero time delay
and existence of a positive endemic-disease equilibrium state
For this case we substitute
(3-41). We obtain:
ˆ
ˆ
* * * *
(h ,y ,h ,y ) (0,0,0,0)
= and τ1 ≠0, τ2
≠0
into Eq.
λ + a λ + b λ + b λ e + cλ+ c λ e + d e + d = 0 (3-43)
4 3 2 2 −λ( τ 1+τ2) −λ(2 τ 1+τ2) −λ( τ 1+τ2)
1 1 4 1 5 3 5
It is not possible to use the Routh-Hurwitz criterion on (3-43) because it is not a
polynomial equation. However, we can obtain a condition for asymptotic stability as
follows. We note that the condition for the equilibrium point to become unstable is
that the real part of some eigenvalue changes from negative to positive, i.e. a critical
condition is that the real part of an eigenvalue is zero with the real parts of all
eigenvalues less than or equal to zero. There are two possibilities. One possibility is
that the critical eigenvalue is real, the other possibility is that the critical eigenvalues
are a complex conjugate pair. In the second case, we would be looking for a Hopf
bifurcation [18]. In this thesis, we will not examine the Hopf bifurcation case.
We will look at the condition for the real case. If we substitute λ = 0 into Eq. (3-43),
we obtain
2 2 2 2
( − u 1τ1− u 2τ2
d )
3
d5 u1 u2 u1u2 u1a cbmu2e 0
form
+ = γ + − = . This can also be written in the
d
(1−
R
0)
+ d = = 0, where
u(r+
u)
3 5
2 1
R
0
=
2 −(u1τ1−u 2τ2)
ma bce
(u +γ)u
1 2
is called the basic reproductive number for the system. We can argue that the
condition for asymptotic stability should be R0
< 1 as follows. For λ close to zero
the left-hand side of (3-43) can be approximated by a polynomial. We could then
look at the Routh-Hurwitz conditions for this polynomial. One of the important
Routh-Hurwitz conditions is that the constant term in the polynomial should be
positive. The constant term for the approximating polynomial in Eq. (3-43) is d3 + d5
which leads to the condition d 3
+ d 5
> 0, i.e., R0
< 1.
We now look at Eqs. (3-17)-(3-20) for the endemic-disease state with nonzero
time delays and we substitute the value of R
0
into the endemic-disease state with
nonzero time delays and obtain

41
{ [ ]}
h = u (R − 1)(u +γ )u + R (u +γ)u ) γ−(a cbmγ+
a cbmu )e
* 2 2
1 0 1 2 0 1 2 1
+(u u + 2u u γ+ u γ )e −u u γ−u γ }
{1/(ae (u u (abmu abm )e abm e )c)}
2 2 −u1τ1
2
1 2 1 2 2 1 2 2
−u1τ1 2
−u 2τ2 ( −u1τ1−u 2τ2)
1
+
1γ+ 1+ γ − γ
2
(( −u 1τ1) −u 2τ2)
⎧
* ⎪ ⎡ ⎛
(u
1+γ)u
⎞⎤⎫
2
⎪
y = ⎨u 1⎢(R0 −1)
⎜ 2
−u 2τ2 ( −u1τ1−u 2τ2)
⎟⎥⎬
⎪⎩
⎣ ⎝a(u1 + u
1γ+ (abmu1+ abm γ)e −abmγe )c ⎠⎦⎪⎭
( −u 1τ 1+ ( −u 2τ2) {[ ]
}
2 ) −u2τ2 −u2τ2
0 1 2 1 2 2
* 2
ĥ (R 1)(u )u a bcme u u e u e
= − +γ − − − γ
⎧
u
⎫
1
⎨ −u2τ2 −u1τ1 −u1τ
⎬
1
⎩abme ( −u1u 2
−u 2γ− acu1e + u2γe ) ⎭
⎧
* ⎪ ⎡ ⎛
(u
1+γ)u
⎞⎤⎫
2
⎪
ŷ = ⎨u 1⎢(R0 −1)
⎜
−u1τ1 −u1τ
⎟⎥⎬
1
⎪⎩
⎣ ⎝abm(u1u 2
+ u
2γ+ u1ace −u2γe ) ⎠⎦⎪⎭
Therefore
> ˆ > > and
* * *
h 0,h 0,y 0
*
ŷ > 0 only if R0
> 1
Therefore we find that the disease-free equilibrium state is asymptotically
stable only for R0
< 1 and that a positive endemic-disease equilibrium state exists
only for R0
> 1.
3.3.3 Asymptotic stability of the Endemic Disease State for zero time delay
For this case we substitute
2 2
* * ˆ * *
⎛ abcm−uu 1 2
−γu2 abmc−uu 1 2
−γu
⎞
2
(h ,y ,h ,y ˆ ) = ⎜0, ,0,
2 2
⎟
⎝ a bcm + acu1+ acγ a bmc + abmu
2 ⎠
(3-41). We obtain:
4 3 2
1 2 3 4
where after substituting for Q, Q
1,Q 2,Q
3
and τ
1
=τ
2
= 0 into Eq.
λ + f λ + f λ + f λ+ f = 0
(3-44)
f = a =− Q+γ+ 2u + 2u =− abmyˆ
+γ+ 2u + 2u
*
1 1 2 1 2 1
f = b + b + b + b
2 1 2 3 4
2 2
= 4u1u2 + 2Qu
2
+ 2γ u2 + u1 + u2 + Qu1+ u1γ−Q2Q1
* 2 2 *
4u ˆ
ˆ
1u2 2abmy u2 2 u2 u1 u2 abmy u1 u1
2 * *
= + + γ + + + + γ
−a bcm(1−y )(1−y
ˆ )

42
f = c + c + c + c + c
Qu 2Qu u 2u u 2u u u 2u u Q Q Q Q Q (u u )
3 1 2 3 4 5
2 2 2 2
=
2
+
1 2
+
1 2
+
1 2
+γ
2
+
1 2γ+ 3 1 2
−
2 1 2
+
1
* 2 * 2 2 2
= abmyˆ
u
2
+ 2abmyˆ
u1u 2
+ 2u1u 2
+ 2u1u 2
+γ u
2
+ 2u1u
2γ
3 2 * * * 2 * *
+ a bc my ˆ (1−y )(1−y ˆ ) −a bcm(1−y )(1− y ˆ )(u2 + u
1)
f = d + d + d + d + d
4 1 2 3 4 5
2 2 2 2 2 2
=−Qu γ
2
+ uQQQ
1 3 1 2
− uQQu
1 2 1 2
+ u1γ u2 + Qu γ
2
+ uu
1 2
+ Quu
1 2
* 2 3 2 * * * 2 * *
=−abmyˆ γ u2 + a bc my ˆ (1 −y )(1 −y ˆ )u ˆ
1−a bcm(1 −y )(1 −y )u1u2
2 * 2 2 2 * 2
+ u ˆ
ˆ
1γ u2 + abmy γ u2 + u1u2 + abmy u1u2
From the Routh-Hurwitz criterion, all eigenvalues of Eq.(3-44) have negative
real parts if and only if
i) f 1
> 0,ii) ff 1 2
− f 3
> 0, iii)
2
f(ff
3 1 2
− f)
3
− ff
1 4
> 0, iv) f4
> 0
The formulae for the Routh-Hurwitz criterion are very complicated and it is
extremely difficult, if not impossible, to obtain any general results from them. We
will look at some numerical results for reasonable values of the parameters in Chapter
4.
3.3.4 Asymptotic stability of the Endemic Disease State for non-zero time delay
It is possible to obtain general analytic expressions for the characteristic
equation for the endemic disease state for non-zero time delay by substituting the
* * * *
values of (h , y ,h ˆ , y ˆ ) in Eqs. (3-17)-(3-20) into the characteristic equation Eq. (3-
41). However, the expressions are too difficult to analyze in general. We will look at
numerical results in chapter 4 for reasonable values of the parameters.
3.4 Numerical Solution of Delay Differential Equations
In chapter 4 we will use Matlab to obtain numerical solutions for the
equilibrium states and asymptotic stability conditions discussed in this chapter. We
will also use Matlab to obtain numerical solutions for the differential equations of the
model given in Eqs. (3-1)-(3-4) for a range of parameter values that are reasonable
estimates for the transmission of Dengue fever in selected countries of South-East
Asia. We will use the Matlab function dde23 which is designed to give numerical
solutions of systems of first-order equations that include time delays.

CHAPTER 4
NUMERICAL RESULTS
In this chapter we will use numerical methods to study the mathematical model
for Dengue fever developed in Chapter 3. We will find numerical solutions of the
equations for a range of parameter values that have been suggested by previous
workers and for parameter values that appear reasonable for Thailand, Malaysia and
Singapore. Some of these parameter values correspond to asymptotic stability of the
disease-free equilibrium state and some values correspond to asymptotic stability of
the endemic disease equilibrium state. For each set of parameter values we will first
examine the asymptotic stability of equilibrium points by computing eigenvalues of
the Jacobian matrix and checking the Routh-Hurwitz criteria. We will then use the
Matlab delay-differential equation solver dde23 to compute the numerical solution of
the differential equations for nonzero time delays and the Matlab ordinary differential
equation solver ode45 to compute the numerical solution for zero time delays.
4.1 Asymptotically Stable Disease-Free Equilibrium State
4.1.1 Parameter values
The parameter values listed in Table 4-1 have been selected from the work of
Tumwiine et al. [15] and Torres-Sorando and Rodrigues [20]. Tumwiine et al studied
the occurrence of malaria in Uganda. The assumption that abm = ac was made by
Torres-Sorando and Rodriques and they based it on the assumption that the proportion
of bites that result in infections is the same in humans and mosquitoes. They argued
that this proportion should be less than or equal to 1 and proposed the value abm = ac
−5
= 8.33× 10 . Tumwiine et al were interested in the progress of short-term malaria
epidemics in Uganda. We will use these parameters to examine the numerical
solutions of our model for a disease-free equilibrium state.

44
TABLE 4-1 Parameters for disease-free equilibrium state [15]
Parameter name Values used Unit
abm = ac
8.33×
10 −5
u
1
0.333
u
2
0.071
γ 0.143
day −1
day −1
day −1
day −1
4.1.2 Study of solutions for zero time delays for parameters in Table 4-1
For the disease-free equilibrium state with zero time delay
ˆ
ˆ
* * * *
(h ,y ,h ,y ) (0,0,0,0)
= and τ
1
=τ
2
= 0 . For the parameter values given in Table 4-
1, we can use Matlab to find the coefficients of the characteristic equation (3-41), i.e.,
in the equation
λ + a λ + bλ + b λ e + b λ e + b λ e + cλ+ c λ e + c λ e
4 3 2 2 −2 λτ1 2 −λτ1 2 −λ( τ 1+τ2) −λτ1 −2λτ1
1 1 2 3 4 1 2 3
+ c λ e + c λ e + d e + d e + d e + d e + d = 0
−λ ( τ 1+τ2 ) −λ (2 τ 1+τ2 ) −λτ1 − 2 λτ1 −λ ( τ 1+τ2 ) −λ (2 τ 1+τ2
)
4 5 1 2 3 4 5
The values of the coefficients are:
−
a1 = 0.951, b1 = 0.278427, b2 = 0, b3 = 0, b4
=− 6.93889×
10
−9
c1 = 0.026586, c2 = 0, c3 = 0 , c4 =− 2.80331× 10 , c5
= 0
−
d = 0, d = 0, d =− 1.64056× 10 , d = 0, d = 7.9904×
10
10 −4
1 2 3 4 5
The value for the basic reproductive rate is:
−7
R
0
= 2.05317× 10 < 1
For these coefficients, the characteristic equation reduces to (see Eq. 3.42)
λ + f λ + f λ + f λ+ f = 0
4 3 2
1 2 3 4
The Routh-Hurwitz criteria for this characteristic equation (see section 3.3.1) are:
f1
= 0.9510 > 0 , f 1
f 2
− f 3
= 0.23820 > 0 ,
−4
f4
= 7.99× 10 > 0
f (f f − f ) − f f = 0.00561 > 0
2
3 1 2 3 1 4
Since these values are all positive they satisfy the conditions of the Routh-Hurwitz
criteria and therefore all eigenvalues have negative real parts
We have also used Matlab to compute the eigenvalues of the Jacobian matrix
for the disease-free equilibrium state for zero time delays. We found four eigenvalues
given by:
9

45
λ
1
=−0.47600001713306
λ
2
=−0.33300000000000
λ
3
=−0.07100000000000
λ =−0.07099998286694
4
It can be seen that the real parts of all eigenvalues are negative, in agreement
with the results from the Routh-Hurwitz critera. Therefore the steady state solution
ˆ
ˆ
* * * *
(h ,y ,h ,y ) (0,0,0,0)
= is asymptotically stable for R0
< 1.
From Eq. (3-16), we find that the endemic-disease equilibrium state
corresponding to these parameters is (h * , y * ,h ˆ * , y ˆ
* ) = (0, −28.25,0, − 183.89) and
therefore the endemic-disease equilibrium state does not exist since negative values of
populations are not allowed.
We have used the Matlab differential equation solver ode45 to obtain a
numerical solution of the equations for the parameter values in Table 4-1 with zero
time delays. The initial values for the 4 variables were taken as:
(h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , i.e., proportions of 0.1 of the human and mosquito
populations in the infected stage, and proportions of 0.1 of the human and mosquito
populations in the infectious stage. The solutions are shown in Figure 4-1. Phase
plane plots of the solutions for a range of initial values are shown in Figure 4-2.
Figure 4-1 Numerical solution with zero time delays for disease-free equilibrium
state for initial state
(h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1)

46
Phase plane plots
Figure 4-2 Phase plane plots with zero time delays for disease-free equilibrium
state. The black solid line shows solutions for the initial values of
Figure 4-1.
The numerical solutions show that the disease dies out and approaches the
disease-free equilibrium state, as expected from the asymptotic stability of the
disease-free state.
4.1.2 Study of solutions for nonzero time delays for parameters in Table 4-1
For time delays we use τ 5 days for the time delay for humans to change
1 =
from the infected to the infectious stage and τ
2
= 5 days for the mosquitoes to
change from infected to infectious. These values have been suggested in Tumwiine et
al [16] as reasonable values.
Because the characteristic equation Eq. (3-43) is not a polynomial equation, the
Routh-Hurwitz criteria cannot be used directly to check stability. However, as
explained in section 3.3.2, the basic reproduction rate
R
0
=
2 −(u1τ1−u 2τ2)
ma bce
(u +γ)u
1 2
can still

47
be used as a test for asymptotic stability of the disease-free equilibrium state. For the
given values of time delays, the value is
0
−8
R = 2.7326× 10 < 1 and therefore the
disease-free equilibrium state is asymptotically stable for the given nonzero time
delays. It is also possible to calculate eigenvalues of Eq. (3-43) numerically using
Maple or Matlab. The results from Maple are:
λ
1
=−0.47601016086605
λ
2
=−0.33300000000000
λ
3
=−0.07100000000000
λ =−0.07099998286694
4
These are the same values as for the zero time delay, since the Jacobian is
independent of the time delay. Since all real parts are negative, the disease-free
equilibrium state is asymptotically stable.
For nonzero time delays we have computed the solutions of the 4 equations
using the Matlab delay-differential equation solver dde23. The results of the
numerical integration using dde23 are shown in Figure 4-3. The initial values were
again assumed to be
(h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) . Phase plane plots of the solutions
for a range of different initial values are shown in Figure 4-4.
Figure 4-3 Numerical solution with nonzero time delays for disease-free equilibrium
state for initial state (h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , τ 5 , τ
2
= 5
1 =

48
Phase plane plots
Figure 4-4 Phase plane plots with nonzero time delays for disease-free equilibrium
State. The black solid line shows solutions for the initial values of
Figure 4-3.
It can be seen that the disease again dies out, but at a faster rate than for the zero
time delays, e.g., after 25 days the infectious mosquito population is approximately
0.05 for zero time delays and approximately 0.01 for the time delay case.
4.2 Asymptotically Stable Endemic-Disease Equilibrium State
4.2.1 Parameter values
Wyse et al [15] have estimated parameters for malaria for the Brazilian Amazon
region using 1 month as time unit. They chose parameter values from a combination
of direct observations of malaria and values based on data in Anderson and May [1].
These parameter values are shown in Table 4-2.

49
TABLE 4-2 Parameters for endemic-disease equilibrium state (Wyse etal. [19])
Parameter name Values used Unit
a 5.974
month −1
b 0.3
month −1
c 0.3
month −1
m 11.57 −
u
1
0.00139
month −1
u
2
0.00238
month −1
γ 1.5
month −1
4.2.2 Study of solutions for zero time delays for parameters in Table 4-2
We first examine the asymptotic stability of the disease-free equilibrium state
with zero time delay
ˆ
ˆ
* * * *
(h , y , h , y ) (0,0,0,0)
= and τ 1
=τ 2
= 0 . For the parameter
values given in Table 4-2, we can use Matlab to find the coefficients of the
characteristic equation (3-41). The values are:
a = 1.508, b = 0.0092, b = 0, b = 0, b =−37.1626
1 1 2 3 4
c = 1.845× 10 , c = 0, c = 0 , c =− 0.1401, c = 0
−5
1 2 3 4 5
d = 0, d = 0, d =− 1.229× 10 , d = 0, d = 1.182×
10
−4 −8
1 2 3 4 5
The value for the basic reproductive rate is:
4
R0
= 1.04× 10 > 1
The Routh-Hurwitz criteria for the coefficients f 1
,f 2
,f 3
,f 4
defined in Eq (3-42) are as
follows;
f1
= 1.508 > 0 , f 1
f 2
− f 3
=− 55.870 < 0 ,
−4
f4
= − 1.229× 10 < 0
f (f f − f ) − f f = 7.827 > 0
2
3 1 2 3 1 4
Since the value of ff 1 2
− f 3
< 0 and f 4
< 0, the Routh-Hurwitz criteria show that the
real part of at least one eigenvalue is positive and that the disease-free equilibrium
point is not asymptotically stable.
We have also used Matlab to compute the eigenvalues of the Jacobian matrix
for the disease-free equilibrium state for zero time delays. We found four eigenvalues
given by:

50
λ
1
=−6.89390226339360
λ
2
= 5.39013226339360
λ
3
=−0.00238000000000
λ =−0.00139000000000
4
It can be seen that the real part of one eigenvalue is positive. Therefore the
* * * *
steady state solution (h ,y ,h ˆ ,y ˆ ) = (0,0,0,0) is not asymptotically stable for R 1.
We then examined the asymptotic stability of the endemic-disease equilibrium
state:
2 2
* * ˆ * *
⎛ abcm−uu 1 2
−γu2 abmc−uu 1 2
−γu
⎞
2
(h , y ,h , y ˆ ) = ⎜0, ,0,
2 2
⎟
⎝ abcm+ acu1+ acγ abmc+
abmu2
⎠
From Matlab we find that
* * * *
h = 0, y = 0.9324, h = 0, y = 0.9986
and therefore this solution exists with nonnegative populations. Using Matlab, we can
find the coefficients of the characteristic equation (3-41)
2 2
a1 = 22.214, b1 = 31.196, b2 =− 4.287× 10 , b3 = 3.977× 10 , b4
=−0.00357
c1 = 0.14812, c2 = 1.893, c3 =− 2.0408 , c4 =− 0.068, c5
= 0.07399
d = 0.00225, d =− 0.0024286, d = 0.12347, d =− 0.12345, d = 1.761×
10 −
1 2 3 4 5
The Routh-Hurwitz criteria for the coefficients f 1
,f 2
,f 3
,f 4
defined in Eq (3-42) are
then as follows:
f1
= 22.2138 > 0 , f 1
f 2
− f 3
= 2.9486 > 0 ,
−6
f4
= 8.4629× 10 > 0
4
R
0
= 1.040× 10 > 1
ˆ
ˆ
f (f f − f ) − f f = 0.01419 > 0
2
3 1 2 3 1 4
The coefficients satisfy the conditions of the Routh-Hurwitz criteria and all
eigenvalues have negative real parts. We have also used Matlab to compute the
eigenvalues of the Jacobian matrix for the endemic-disease equilibrium state for zero
time delays. We found four eigenvalues given by:
λ
1
=−22.20782578277732
λ
2
=−0.00229348492224 - 0.01639805026737i
λ
3
=−0.00229348492224 + 0.01639805026737i
λ =−0.00139000000000
4
0
>
4

51
As expected from the Routh-Hurwitz criteria, all eigenvalues have negative real parts.
Therefore the endemic-disease equilibrium state
2 2
* * ˆ * *
⎛ abcm−uu 1 2
−γu2 abmc−uu 1 2
−γu
⎞
2
(h , y ,h , y ˆ ) = ⎜0, ,0,
2 2
⎟
⎝ abcm+ acu1+ acγ abmc+
abmu2
⎠
is asymptotically stable for R0
> 1.
We have used the Matlab differential equation solvers ode45 and dde23 to
obtain numerical solution of the equations for the parameter values in Table 4-2 for
both zero and nonzero time delays. The initial values for the 4 variables were taken
as:
(h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , i.e., 0.1 as initial values for infectious humans and
mosquitoes and 0.1 as initial values for infected humans and mosquitoes. The
solution for zero time delays is shown in Figure 4-5. Phase plane plots of the
solutions for a range of initial values are shown in Figure 4-6. It can be seen that the
solution is approaching the endemic-disease equilibrium state
* * ˆ * *
h = 0, y = 0.9324, h = 0, yˆ
= 0.9986
However, it can be seen from Figure 4-5 that it takes approximately 900 months for
the system to reach its equilibrium values.
Figure 4-5 Numerical solution with zero time delays for endemic-disease
equilibrium state for initial state (h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , τ = 1
0 ,
τ = 0 2

52
Phase plane plots
Figure 4-6 Phase plane plots with zero time delays for endemic-disease equilibrium
state. The black solid line shows solutions for the initial values of
Figure 4-5.
4.2.3 Study of solutions for nonzero time delays for parameters in Table 4-2
For time delays we use τ
1
= 5 days (0.167 months) for the time delay for
humans to change from the infected to the infectious stage and τ
2
= 5 days (0.167
months) for the mosquitoes to change from infected to infectious. These values have
been suggested in Wyse et al [15] as reasonable values.
As before, because the characteristic equation Eq. (3-43) is not a polynomial
equation, the Routh-Hurwitz criteria cannot be used directly to check stability.
However, as explained in section 3.3.2, the basic reproduction rate
R
0
=
2 −(u1τ1−u 2τ2)
ma bce
(u +γ)u
1 2
can still be used as a test for asymptotic stability of the
disease-free equilibrium state. For the given values of time delays, the value is
4
R
0
1.039 10 1
= × > and therefore the disease-free equilibrium state is not

53
asymptotically stable. It is also possible to calculate eigenvalues of Eq. (3-43)
numerically using Maple or Matlab. The results from Matlab are:
λ =−0.002380000012867
1
λ = 2.000095177646546
2
λ =−0.002380000000000
3
λ =−0.001390000000000
4
Due to numerical difficulties, the first eigenvalue is a repeat of the third eigenvalue.
However, the second eigenvalue is positive showing that the disease-free equilibrium
state is not asymptotically stable for nonzero time delays.
The results of the numerical integration using dde23 are shown in Figure 4-7. It
can be seen that the solution is approaching the endemic-disease equilibrium state
= = ˆ = ˆ = and
* * * *
h 0.1895, y 0.7557, h 0.00396, y 0.9978
4
R
0
= 1.039× 10 > 1
The initial values are assumed to be the same as for Figure 4-5. As in the case of zero
time delays, the system takes approximately 900 months to approach the equilibrium
solutions. Phase plane plots for a range of initial values are shown in Figure 4-8.
Figure 4-7 Numerical solution with nonzero time delays for endemic-disease
equilibrium state for initial state (h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) ,
τ
1
= 0.167 , τ
2
= 0.167

54
Phase plane plots
Figure 4-8 Phase plane plots with nonzero time delays for endemic-disease
equilibrium state. The black solid line shows solutions for the initial
values of Figure 4-7.
4.3 Numerical results for Dengue fever in Thailand, Malaysia and Singapore
4.3.1 Thailand
As stated in section 2.1, there were 31,000 cases of Dengue fever in Thailand in
2005 out of a total population of 65.4 million. This gives an estimate of
4.7368×
10 −4
of the population infected. From the graph given in Figure 2-1 of Dengue fever cases
in Thailand from 1985 to 2005, Dengue fever appears to be endemic in Thailand. As
estimates for the parameters in the model, we will use the estimates given in Table 4-2
for a,b,c,u
1,u 2,γ as a guide. As stated in Section 4.3, these values were estimated
for malaria-carrying mosquitoes in Brazil [19]. However, as we have been unable to
find reliable date for Dengue fever-carrying mosquitoes in South-East Asia we have
used them as an indication of the results that might be expected for Dengue fever. We

55
will then try to estimate the parameter m (the ratio of the Dengue carrying mosquito
population to the human population) by comparing the number of observed cases with
the population of infectious humans in the endemic disease equilibrium state:
2 2
* * ˆ * *
⎛ abcm−uu 1 2
−γu2 abmc−uu 1 2
−γu
⎞
2
(h , y ,h , y ˆ ) = ⎜0, ,0,
2 2
⎟
⎝ abcm+ acu1+ acγ abmc+
abmu2
⎠
The estimated value we obtain is m = 0.00151 From Matlab we find that the endemic
disease equilibrium state for zero time delays is:
= = × ˆ = ˆ = and the basic reproductive rate is
* * −4 * *
h 0, y 4.7368 10 , h 0, y 0.2629
R
0
= 1.3573 > 1.
We have also used Matlab to compute the eigenvalues of the Jacobian matrix
for the endemic-disease equilibrium state for zero time delays. We found four
eigenvalues given by:
λ
1
=−1.50448685348144
λ
2
=−0.001189890853106 −0.000785908455457i
λ
3
=− 0.001189890853106 + 0.000785908455457i
λ =−0.00139000000000
4
It can be seen that that all eigenvalues have negative real parts. Therefore the
endemic disease state * * ˆ * * −
(h , y ,h , y ˆ ) = (0,4.7368× 10 4 ,0,0.2629) is asymptotically
stable for R
0
> 1. The fact that R
0
= 1.3573 > 1 is consistent with our assumption that
Dengue fever is endemic in Thailand. However, for these parameter values, the value
of R 0 is close to 1, i.e., to the value at which the disease-free state would become
asymptotically stable.
For zero time delays, we have used the Matlab ordinary differential equation
solver ode45 to obtain a numerical solution of the equations for the parameter values
for Thailand given in Table 4-2 for a,b,c,u
1,u 2,γ and with ratio of mosquito to
human population m = 0.00151 for Thailand. The initial values for the 4 variables
were taken as:
(h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , i.e., 0.1 for infected and infectious
humans and for infected and infectious mosquitoes. The solution is shown in Figure
4-9. It can be seen that the solution approaches the equilibrium solution, but at a very

56
slow rate. Phase plane plots of the solutions for a range of initial values are shown in
Figure 4-10.
Figure 4-9 Numerical solution with zero time delays for Thailand data for initial
state
(h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , τ
1
= 0 , τ
2
= 0
Phase plane plots
Figure 4-10 Phase plane plots with zero time delays for Thailand data. The black
solid line shows solutions for the initial values of Figure 4-9.

57
We have used Matlab to compute the endemic-disease equilibrium state state for
nonzero time delays τ
1
= 0.167, τ
2
= 0.167 . This gave an estimate of
4.7368× 10 −4 of
the population infected. The estimated value we obtain for m is m = 0.00151 and the
endemic-disease equilibrium is:
ˆ
ˆ
* * * * −4 −4 −4
(h , y ,h , y ) (1.188 10 ,4.737 10 ,1.0477 10 ,0.2628)
= × × × . The basic
reproductive rate is R 0
= 1.35766 > 1 showing that the disease-free state is not
asymptotically stable. However, this value is close to the value of 1 at which the
disease-free state would become asymptotically stable.
We have used Matlab to compute the solution of the differential equations for
non zero delays τ 1
= 0.167, τ 2
= 0.167 with the same initial conditions as for the zero
time delay case shown in Figure 4-9. The results are shown in Figure 4-11. The
solutions can be seen to approach the equilibrium solutions, but at a very slow rate.
Phase plane plots are shown in Figure 4-12 for a range of initial conditions. The main
difference that can be seen between the zero delay solutions and nonzero delay
solutions is that in the nonzero delay solutions there is a nonzero population of
infected humans and infected mosquitoes.
Figure 4-11 Numerical solution with nonzero time delays for Thailand data for initial
state (h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , τ
1
= 0.167 , τ
2
= 0.167

58
Phase plane plots
Figure 4-12 Phase plane plots with nonzero time delays for Thailand data. The black
solid line shows solutions for the initial values of Figure 4-11.
4.3.2 Singapore
As stated in section 2.1, there were 12,700 cases of Dengue fever in Singapore
in 2005 out of a total population of 4.35 million. This gives an estimate of
2.917× 10 −3 of the population infected. From the graph given in Figure 2-2 of
Dengue fever cases in Singapore from 2005 to 2007, Dengue fever appears to be
endemic in Singapore. As estimates for the parameters in the model, we will use the
estimates given in Table 4.2 for a,b,c,u
1,u 2,γ as a guide. As explained in Section
4.3.1, we have used the data shown in Table 4-2 estimated for malaria-carrying
mosquitoes in Brazil [19] as an estimate for Dengue fever-carrying mosquitoes in
South-East Asia. We will then try to estimate the parameter m (the ratio of the
Dengue carrying mosquito population to the human population) by comparing the

59
number of observed cases with the population of infectious humans in the endemic
disease equilibrium state:
2 2
* * ˆ * *
⎛ abcm−uu 1 2
−γu2 abmc−uu 1 2
−γu
⎞
2
(h , y ,h , y ˆ ) = ⎜0, ,0,
2 2
⎟
⎝ abcm+ acu1+ acγ abmc+
abmu2
⎠
The estimated value we obtain is m=0.00357. From Matlab we find that
= = × ˆ = ˆ = and R
0
= 3.2059 > 1
* * −3 * *
h 0, y 2.917 10 , h 0, y 0.68716
We have also used Matlab to compute the eigenvalues of the Jacobian matrix
for the endemic-disease equilibrium state for zero time delays. We found four
eigenvalues given by:
λ
1
=−1.50818348861158
λ
2
=−0.00118935000632 − 0.00332324855962i
λ
3
=− 0.00118935000632 + 0.00332324855962i
λ =−0.00139000000000
4
It can be seen that all eigenvalues have negative real parts. Therefore the
* * * * 3
endermic state solution ˆ
−
(h , y ,h , y ˆ ) = (0,2.917× 10 ,0,0.68716) is asymptotically
stable for R0
> 1. The fact that R
0
= 3.2059 > 1 is consistent with our assumption
that Dengue fever is endemic in Singapore. The value for R 0 for Singapore is greater
than the value for Thailand.
The results of numerical solutions of the differential equations are shown in
Figure 4-13 for zero time delays. The slow rate of convergence to the endemicdisease
steady state can again be seen in this figure. Phase plane plots of the solutions
for a range of initial conditions are shown in Figure 4-14.
Figure 4-13 Numerical solution with zero time delays for Singapore data for initial
state (h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , τ
1
= 0 , τ
2
= 0

60
Phase plane plots
Figure 4-14 Phase plane plots with zero time delays for Singapore data. The black
solid line shows solutions for the initial values of Figure 4-13.
For nonzero time delays τ 1
= 0.167, τ 2
= 0.167 , we estimate the value of m to
be m=0.00358. From Matlab we find that the endemic-disease equilibrium state is:
ˆ
ˆ
* * * * −4 −3 −3
(h , y ,h , y ) (7.315 10 ,2.917 10 ,2.38 10 ,0.6869)
= × × × and the basic
reproductive rate is R
0
= 3.21057 > 1. The numerical solutions obtained from dde23
are shown in Figure 4-15. The slow rate of convergence to the equilibrium state can
again be seen in Figure 4-15. Phase plane plots of the solutions for a range of initial
conditions are shown in Figure 4-16.

61
Figure 4-15 Numerical solution with nonzero time delays for Singapore data for
initial state (h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , τ
1
= 0.167 , τ
2
= 0.167
Phase plane plots
Figure 4-16 Phase plane plots with nonzero time delays for Singapore data. The
black solid line shows solutions for the initial values of Figure 4-15.
4.3.3 Malaysia
As stated in section 2.1, there were 32,950 cases of Dengue fever in Malaysia in
2005 out of a total population of 23.95 million. This gives an estimate of
1.3756× 10 −3 of the population infected. From the graph given in Figure 2-3 of

62
Dengue fever cases in Malaysia from 1991 to 2000, Dengue fever appears to be
endemic in Malaysia. As estimates for the parameters in the model, we will use the
estimates given in Table 4.2 for a,b,c,u
1,u 2,γ as a guide. As explained in Section
4.3.1, we have used the data shown in Table 4-2 estimated for malaria-carrying
mosquitoes in Brazil [19] as an estimate for Dengue fever-carrying mosquitoes in
South-East Asia. We will then try to estimate the parameter m (the ratio of the
Dengue carrying mosquito population to the human population) by comparing the
number of observed cases with the population of infectious humans in the endemic
disease equilibrium state:
2 2
* * ˆ * *
⎛ abcm−uu 1 2
−γu2 abmc−uu 1 2
−γu
⎞
2
(h , y ,h , y ˆ ) = ⎜0, ,0,
2 2
⎟
⎝ abcm+ acu1+ acγ abmc+
abmu2
⎠
The estimated value we obtain is m=0.00227 From Matlab we find that
= = × ˆ = ˆ = and R
0
= 2.03867 > 1. This value for R 0
* * −3 * *
h 0, y 1.376 10 , h 0, y 0.5088
is between the values for Thailand and Singapore.
We have also used Matlab to compute the eigenvalues of the Jacobian matrix
for the endemic-disease equilibrium state for zero time delays. We found four
eigenvalues given by:
λ
1
=−1.505848304357893
λ
2
=−0.001189687389878 −0.002114123714916i
λ
3
=− 0.001189687389878 + 0.002114123714916i
λ =−0.00139000000000
4
It can be seen that that all eigenvalues have negative real parts. Therefore the
* * * * 3
endermic state solution ˆ
−
(h , y ,h , y ˆ ) = (0,1.376× 10 ,0,0.5088) is asymptotically
stable for R0
> 1. The fact that R
0
= 2.03867 > 1 is consistent with our assumption
that Dengue fever is endemic in Malaysia.
The results of numerical solutions of the differential equations are shown in
Figure 4-17 for zero time delays. The slow rate of convergence to the endemicdisease
steady state can again be seen in this figure. Phase plane plots of the solutions
for a range of initial conditions are shown in Figure 4-18.

63
Figure 4-17 Numerical solution with zero time delays for Malaysia data for initial
state (h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , τ
1
= 0 , τ
2
= 0
Phase plane plots
Figure 4-18 Phase plane plots with zero time delays for Malaysia data. The
black solid line shows solutions for the initial values of Figure 4-17.
For nonzero time delays τ 1
= 0.167, τ 2
= 0.167 , we estimate the value of m to
be m=0.00227. From Matlab we find that the endemic-disease equilibrium state is:
* * ˆ * * −4 −3 −4
(h , y ,h , y ˆ ) = (3.449× 10 ,1.3756× 10 ,2.022× 10 ,0.5086) , and the basic

64
reproductive rate is R 0 =2.04007 > 1.
The results of the numerical solution of the differential equations using dde23
for nonzero delays τ 1
= 0.167, τ 2
= 0.167 are shown in Figure 4-19. The slow rate of
convergence to equilibrium can again be seen in this figure. Phase plane plots of the
solutions for a range of initial conditions are shown in Figure 4-20.
Figure 4-19 Numerical solution with nonzero time delays for Malaysia data for initial
state (h, y,h, ˆ y) ˆ = (0.1,0.1,0.1,0.1) , τ
1
= 0.167 , τ
2
= 0.167
Phase plane plots
Figure 4-20 Phase plane plot with nonzero time delays for Malaysia data. The
black solid line shows solutions for the initial values of Figure 4-19.

CHAPTER 5
DISCUSSION AND CONCLUSIONS
5.1 Discussion and Conclusions
We have extended a model for malaria originally discussed by Macdonald [15]
and Anderson and May [1] and used it as a model for transmission of Dengue fever in
Thailand, Singapore and Malaysia. The mathematical model used in this thesis is a
system of four nonlinear delay differential equations which include infected humans,
infectious humans, infected mosquitoes and infectious mosquitoes. The model
contains two time delays which are associated with the transitions from the infected to
infectious stage in humans and from the infected to the infectious stage in mosquitoes.
We have found that the model has two equilibrium points, a disease-free equilibrium
point and an endemic-disease equilibrium point. We have analyzed the asymptotic
stability of the equilibrium points by two methods. In the first method, we have used
an analytical method in which we linearized the nonlinear equations about the
equilibrium points. In the second method we used Matlab to compute numerical
solutions for the original nonlinear delay differential equations. We have examined
two cases corresponding to zero time delays and nonzero time delays.
For the case when all time delays are zero, we found an analytic formula for the
endemic-disease equilibrium point and analytic formulas giving necessary and
sufficient conditions for asymptotic stability for both the disease-free and the
endemic-disease case. These asymptotic stability conditions were obtained by using
the Routh-Hurwitz criteria. We found that the basic reproductive rate parameter gave
necessary conditions for asymptotic stability of the disease-free equilibrium and for
the existence of the endemic-disease equilibrium.
For the second case with nonzero time delays, we obtained an analytical
formula for the endemic-disease equilibrium point and necessary conditions for the
asymptotic stability of the disease-free equilibrium point and the existence of the
endemic-disease equilibrium point.

66
We have used Matlab to numerically integrate the four nonlinear delay
differential equations both for zero time delays and for a range of nonzero values of
the time delays. We have used a selection of parameter values in our model obtained
from previous authors [15, 19]. In one case, these values corresponded to an
asymptotically stable disease-free equilibrium point and in other cases these values
corresponded to an endemic-disease equilibrium point. In each case we compared the
results obtained from the analytic conditions for asymptotic stability with the results
from the numerical integrations and found good agreement.
Finally, we examined the data for occurrence of Dengue fever in Thailand,
Malaysia and Singapore. In each country the data suggest that Dengue fever is
endemic. For all parameters except the number of mosquitoes, we used values that
appeared to be reasonable from Wyse et al [19]. The values of Wyse et al. were
estimated for malaria-carrying mosquitoes in Brazil [19]. However, as we have been
unable to find reliable date for Dengue fever-carrying mosquitoes in South-East Asia
we have used them as an indication of the results that might be expected for Dengue
fever. Using the Wyse et al. parameter values we then estimated the ratio m of
number of female mosquitoes to human population for Thailand, Malaysia and
Singapore. For each country, we found that the mathematical model had an endemicdisease
equilibrium point and that the disease-free equilibrium point was not
asymptotically stable. We also found differences between the three countries. The
basic reproduction rates for the three countries increased from approximately 1.36 for
Thailand to 2.04 for Malaysia to 3.21 for Singapore. A noticeable feature of our
numerical results was the long time period of approximately 900 months before the
system reached equilibrium. If our model accurately reflects the actual time scale to
reach equilibrium, then an equilibrium point analysis of the model would appear to
have limited value and the short-term behavior of the model would be of greater
interest. The existence of the time delays will affect this short-term behavior.
5.2 Suggestions for Further Study
In this thesis we have not attempted to model the quite complicated variation in
disease levels shown in the data in Figures 2-1, 2-2 and 2-3. In our analysis we have
assumed that the parameters in the model are constants. In particular, we have

67
assumed that the population of female Dengue-carrying mosquitoes is constant, that
the immunity of the human population does not change and that the death rate of
mosquitoes does not change. A more accurate model would have to include
variations in each of these factors. The use of Brazil data for parameter values or
malaria-carrying mosquitoes should also be replaced in a more accurate model by
obtaining estimates of the parameter values for the Dengue fever-carrying mosquitoes
of South-East Asia.

REFERENCES
1. Roy M. Anderson and Robert M. May. Infectious Diseases of Humans:
Dynamics and Control. Oxford University Press, 1992.
2. “Bureau Of Epidemiology, Department of Disease Control, Ministry of Public
Health.” Available from: http://epid.moph.go.th/dssur/vbd/df.htm.
3. “Dengue Fever Statistics in Malaysia.” Available from :
http://www.chhs.com.my/denguestat.htm.
4. “2005 Dengue Outbreak in Singapore.” Available from :
http://en.wikipedia.org/wiki/2005_dengue_outbreak_in_Singapore.
5. “Compaign Against Dengue.” Available from : http://www.dengue.gov.sg.
6. M. Derouich, A. Boutayeb.. “Dengue fever: Mathematical modelling and computer
Simulation.” Journal of Applied Mathematics and Computation . 177
(2006) : 528–544.
7. Aekabut Sirijampa. A Mathematical Model of Periodic Chronic Myelogenous
Leukemia with Delay Diffferential Equations. A thesis submitted in partial
fulfillment of the requirements for the Master of Science in Applied
Mathematics. King Mongkut’s University of Technology North Bangkok,
2006.
8. A. Kammanee,Y. Lenbury and I.M. Tang. Transmission of Plasmodium Vivax
Malaria A thesis submitted in partial fulfillment of the requirements for the
Master of Science in Mathematics. Mahidol Unoversity, 2006.
9. L.F. Shampine. Solving Delay Differential Equations with DDE23. Mathematics
Department, Southern Methodist University, 2000.
10. M Sriprom, et al. Dengue Haemorrhagic Fever in Thailand, 1998-2003: Primary
or Secondary Infection. A thesis submitted in partial fulfillment of the
requirements for the Master of Science in Mathematics. Mahidol University,
2003.
11. “Dengue fever.” Available from: http://en.wikipedia.org/wiki/Dengue_fever.
12. J.Guardiola, A. Vecchio. Basic Reproduction number for Infectious Dynamics

69
Models and the Global Stability of Stationary Points. Napoli – Italy, 2003.
13. J.D. Murray. Mathematical Biology Springer-Verlag New York Berlin
Heidelberg, 2002.
14. Dennis G. Zill, Michael R. Cullen. Differential Equations with Boundary-Value
Problems. Thomson Brooks/Cole a division of Thomson Learning, 2005.
15. J. Tumwiine, L.S. Luboobi & J.Y.T. Mugisha. Modelling the Effect of
Treatment and Mosquito Control on Malaria Transmission. Department of
Mathematics & Statistics. Makerere University, Uganda, 1998.
16. “Dengue, Reported cases in SEAR from 1985-2005.” Available from:
http://www.searo.who.int/EN/Section10/Section332_1101.htm.
17. Ang Kim Teng and Satwant Singh. Epidemiology and New Initiatives in the
Prevention and Control of Dengue in Malaysia. Ministry of Health, Kuala
Lumpur, Malaysia, 2001.
18. Charoen Kaewpradit, Wannapong Triampob and I-Ming Tang. “Limit Cycle in a
Herbivore-Plant-Bee Model Containing a Time Delay.” Department of
Mathematics and Physics. Mahidol University, 2005.
19. Ana Paula P. Wyse, Luiz Bevilacqua, Marat Rafikov. “Simulating malaria model
for different treatment intensities in a variable environment” Journal of
Ecological Modelling. 206 (2007) : 322-330.
20. Torres-Sorando, L. and Rodriguez, D.J. “Models of spatio-temporal
dynamics in malaria”. Journal of Ecological Modelling. 104 (1997) :
231-240.
21. Luenberger, D.G., Introduction to Dynamical Systems: Theory, Models and
Applications, John Wiley and Sons, New York, 1979.

APPENDIX A
MATLAB AND MAPLE PROGRAMMING

71
A1. Characteristic Equation.
Maple program to find Characteristic Equation (3-41).
> restart;
> with(LinearAlgebra):
> A[0]=Matrix([[-a*b*m*yhat-u[1],-a*b*m*yhat,0,(1-y-
h)*a*b*m],[0,-u[1]-r,0,0],[0,a*c*(1-yhat-hhat),-
u[2],0],[0,0,0,-u[2]]]);
A 0
⎡− a b m yhat − u 1
−a b m yhat 0 ( 1 − y − h)
a b m⎤
0 − u =
1
− r 0 0
0 ac( 1 − yhat − hhat ) −u 2
0
⎢
⎥
⎢ 0 0 0 −u ⎥
⎣
2 ⎦
> A[1]=Matrix([[a*b*m*yhat*exp(-
u[1]*tau[1]),a*b*m*yhat*exp(-u[1]*tau[1]),0,-a*b*m(1-y-
h)*exp(-u[1]*tau[1])],[0,-a*b*m*yhat*exp(-
u[1]*tau[1]),0,(1-y-h)*a*b*m*exp(-
u[1]*tau[1])],[0,0,0,0],[0,0,0,0]]);
A 1
⎡
a b m yhat e
=
⎢
⎢
⎣
( −u τ )
( )
1 1
a b m yhat e
−u τ 1 1
0 −a b m ( 1 − y − h)
e
0
( −u τ ) 1 1
−a b m yhat e 0 ( 1 − y − h)
a b m e
0 0 0 0
0 0 0 0
> A[2]=Matrix([[0,0,0,0],[0,0,0,0],[0,-a*c*(1-yhat-
hhat)*exp(-u[2]*tau[2]),a*c*y*exp(-
u[2]*tau[2]),a*c*y*exp(-u[2]*tau[2])],[0,a*c*(1-yhat-
hhat)*exp(-u[2]*tau[2]),-acy*exp(-u[2]*tau[2]),-
a*c*y*exp(-u[2]*tau[2])]]);
A 2
( −u τ ) 1 1
( −u τ ) 1 1
⎡0 0 0 0 ⎤
0 0 0 0
=
( −u τ )
( −u τ )
( −u τ )
2 2 2 2 2 2
0 −a c( 1 − yhat − hhat ) e acye acye ⎢
( −u τ )
( −u τ )
( −u τ ) ⎥
⎢
2 2 2 2 2 2
⎥
⎣0 ac( 1 − yhat − hhat ) e −acy e −acye ⎦
⎤
⎥
⎥
⎦

72
> A[lambda] :=
Matrix([[lambda,0,0,0],[0,lambda,0,0],[0,0,lambda,0],[0,0
,0,lambda]]);
A λ
⎡λ 0 0 0⎤
:=
0 λ 0 0
0 0 λ 0
⎢
⎥
⎣0 0 0 λ⎦
> sol:=A[lambda]-(A[0]+A[1]*exp(-lambda*tau[1])+A[2]*exp(-
lambda*tau[2]));
Change variable in to
⎡λ 0 0 0⎤
( −λ τ ) ( −λ τ )
1 2
sol := − A 0
− A 1
e − A 2
e +
0 λ 0 0
0 0 λ 0
⎢
⎥
⎣0 0 0 λ⎦
> Q:=a*b*m*yhat;Q[1]:=a*b*m*(1-y-h);Q[2]:=a*c*(1-yhathhat);Q[3]:=a*c*y;
> sol1:=Determinant(k);
( e )
( e )
Q := a b m yhat
Q 1
:= abm( 1 − y − h )
Q 2
:= ac( 1 − yhat − hhat )
Q 3
:= acy
2
2 Q 2 λ 2 Q 2 2
u 2 Q 2 λ 2
2 λ u 1
u 2
+ Q λ u 2
+ + −
( u τ ) ( λτ ) ( u τ ) ( λτ )
2
2
1 1 1 1 1 1 ( u τ ) ( λτ )
e e e e 1 1 1
( e ) ( e )
Q 2 2
u 2
− + 2 Qu
2
2 1
λ u 2
+ 2 Qrλ u 2
+ 2 u 1
r λ u 2
+ λ 4 + 4 u 1
λ 2 u 2
( u τ ) ( λτ )
1 1 1
2 r λ 2 2
u 2
2 u 1 λ u2 2 λ 3 u 2
λ 2 2
u 2
2 u 1
λ 3 r λ 3 Q λ 3 2
u 1 λ
2 2 2
+ + + + + + + + + u u2 1
Q 2
Q 1
λ 2
2 Q 2 λ u 2 Qrλ 2
2
− + − + λ ru
( u τ ) ( λτ ) ( u τ ) ( λτ ) ( u τ ) ( λτ ) ( u τ ) ( λτ ) 2
+ 2 Q λ 2 u 2
2 2 2 1 1 1 1 1 1 1 1 1
e e e e e e e e
Qu 1
λ 2 2
Qu 1
u 2
Qrλ 2 2
Qru 2
u 1
r λ 2 2
2 Q r λ u 2
+ + + + + + u 1
ru 2
−
e
( u τ ) ( λτ )
1 1 1
e

73
2
Qru 2
2 Q 2 λ u 2
QQ 3
Q 1
Q 2
− − −
( u τ ) ( λτ )
2
2
2
2
1 1 1 ( u τ ) ( λτ ) ( u τ ) ( λτ )
e e 1 1 1 1 1 1
( e ) ( e ) ( e ) ( e )
QQ 2
Q 1
λ
QQ 2
Q 1
u 2
+ +
2
2
2
2
( u τ ) ( λτ ) ( u τ ) ( λτ ) ( u τ ) ( λτ )
1 1 1 2 2 2 1 1 1
( e ) ( e ) e e ( e ) ( e )
λ Q 3
Q 1
Q 2
λ Q 2
Q 1
u 2
+ −
( u τ ) ( λτ ) ( u τ ) ( λτ ) ( u τ ) ( λτ ) ( u τ ) ( λτ )
2 2 2 1 1 1 2 2 2 1 1 1
e e e e e e e e
QQ 3
Q 1
Q 2
QQ 2
Q 1
λ
+ −
( u τ ) ( λτ ) ( u τ ) ( λτ ) ( u τ ) ( λτ ) ( u τ ) ( λτ )
2 2 2 1 1 1 2 2 2 1 1 1
e e e e e e e e
QQ 2
Q 1
u 2
u 1
Q 3
Q 1
Q 2
− +
( u τ ) ( λτ ) ( u τ ) ( λτ ) ( u τ ) ( )
2 2 2 1 1 1 2 2
e e e e e e
u 1
Q 2
Q 1
λ
u 1
Q 2
Q 1
u 2
−
−
( u τ ) ( λτ ) ( u τ ) ( λτ ) ( u τ ) ( )
2 2 2 1 1 1 2 2
e e e e e e
A2 Check Routh-Hurwitz Criteria
λτ ( u τ ) ( λτ )
2 1 1 1
e e
λτ ( u τ ) ( λτ )
2 1 1 1
e e
( u τ ) ( λτ )
2 2 2
e e
( u τ ) ( λτ )
2 2 2
e e

74
A3 Numerical Solutions
First is function used in ode23 command when the model has no time delay.

75
Second is function used in dde23 command when the model has 2 delay.
Third is m-file to show numerical solutions.
This m-file for solve numerical for non delay

76
And this m-file for solve numerical for delay
A4 Find Eigenvalue
This m-file to find the Eigenvalue of Eq.(3-41)

79
BIOGRAPHY
Name
Thesis title
: Mr.Werapong Sakdanupaph
: A Delay Differential Equation model for Dengue Fever
Transmission in Selected Countries of South-East Asia
Major Field : Applied Mathematics
Biography
Born 23 october 1982
Education
Bachelor Degree in Science (Mathematics) at Silpakorn
University
Address 197/1 M.6 T.Kuring A.Thasae Chumphon 86140