ABSTRACT
In this work we modified a mathematical model to study the dynamics of the vectorhost disease. In the enhanced model, we obtained seven classes from human population which includes the susceptible, infected, exposed, treated, nontreated, recovered, and protected classes. For the mosquito population, we have four classes, namely; class of mosquito larva, susceptible mosquitoes, infected mosquitoes and exposed mosquitoes. We assume free interaction between the vector and host populations. The mathematical analysis of the compartmental models leads us to eleven coupled systems of nonlinear ordinary differential equations. We calculated the basic reproduction of the model. We discussed the equilibrium analysis of the model including stability analysis. We also conducted sensitivity analysis of the parameters of the model using the method of next generation matrix. From the result, we discovered that some parameters have negative values example k1=0.25 k20.15 while some have positive values. For instance βm=+0.5 and βn=+0.5, this means that increasing or decreasing any of them by 10%increases or decreases by 10%. We also provided a graphical profile of each of the variables involved in the compartmental model.
TABLE OF CONTENTS
Title
Page i
Declaration ii
Certification
iii
Dedication iv
Acknowledgment v
Table
of Contents vi
Abstract ix
Chapter 1: Introduction
1.1 Background of the Study 1
1.2 Statement of the Problem 6
1.3 Aim and Objective of the Study 7
1.4 Motivations of the Study 7
1.5 Significance of Study 7
1.6 Scope and Limitations of the Study 7
Chapter 2
Literature
Review on Malaria Epidemic 11
Chapter 3: Methodology
3.1 Introduction 14
3.2 Nonlinear Systems 14
3.3 Theorem 15
3.4 Equilbria and Stability 16
3.5 Stability 16
3.6 Definition 17
3.7 Definition 17
3.8 Definition 17
3.9 Definition 17
3.10 Characterization of Equilibrium Points 20
Chapter 4: The Mathematics Model and
Results
4.1 Introduction 21
4.2 Construction of the Compartmental Model 22
4.2.1 Human Population 22
4.2.2 Mosquitoes Population 23
4.2.3 Remark 23
4.3 Invariant Region 24
4.3.1 Lemma 25
4.17 Mathematical analysis of the model 26
4.17.1 Modified human population 27
4.17.2 Modified mosquitoes 28
4.18 Positivity of Solutions 28
4.18.1 Theorem 28
4.19 Existence and Stability of Steady – STATE
solution 31
4.19.1 Diseasefree equilibrium point 32
4.19.2 Local stability of DFE 33
4.19.3 Generations is an epidemic 33
4.20 Basic Reproduction Ration (R_{0}) 34
4.21 Sensitivity Analysis of the Model
Parameters 39
4.21.1 Sensitivity analysis of R_{0 }39
4.22 METHCAD simulation of the model 41
4.23 Results and Discussion 47
4.3 Contribution to Knowledge 48
Chapter 5: Summary,
Conclusion and Recommendations
5.1 Summary 49
5.2 Conclusion 50
5.3 Recommendations 50
References 52
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND OF THE STUDY
Malaria
is a highly prevalent infectious disease especially in the tropical and
subtropical areas. Fig. 1.0 below is a map obtained from WHO Malaria Report
2010, depicting the countries where malaria was endemic in 2009 (shaded
region).
Fig 1.0 Malaria endemic countries 2009
In
addition to being widespread, malaria is also a deadly disease. This is because
statistics has shown that for Africa in particular, annually 145,000 million to
150,000 million infections are reported, among which, 800 to 850 cases result
in deaths as shown in Table 1.1 below. Most of the deaths are either children
under five or pregnant women.
Typical
symptoms of malaria infections start with headache, followed by periodic bouts
of fevers and chills, and sometimes even coma. The period of cyclical fevers
lasts several days, during which time a high probability of dying has been
observed for children, since their immune systems are weak. Such fever can also
lead to abortions in pregnant women.
Year

2000

2001

2002

2003

2004

2005

2006

2007

2008


Cases
(in
thousands)

173000

178000

181000

185000

187000

188000

187000

186000

181000


Deaths
(in thousands)

900

893

885

880

870

853

832

802

756


Table
1.1 Estimates of malaria cases and deaths in Africa by WHO, 20002009.
BRIEF
ANALYSIS OF MALARIA DATA
It is of interest to do a quick
statistical analysis of the data in Table 1.1, for the malaria cases in Africa
as provided by WHO.
A nonlinear regression analysis is
performed for both the reported cases (C) and deaths (D) against time (T). The
result follows from SPSS18.
Model Summary and Parameter Estimates

Dependent Variable:C
(Numbers of cases)

Equation

Model Summary

Parameter Estimates

R Square

F

df1

df2

Sig.

Constant

b1

b2

Quadratic

.981

180.044

2

7

.000

165283.333

7609.848

647.727

The independent variable is T.

Fig. 1.1 Curve
estimation of reported malaria cases
Observation:
It is quite clear from the WHO data, for the number of malaria cases reported
over the 10 year period that the incidence of malaria infection follows a
parabolic curve, rising sharply initially, to reach a maximum and then
declining sharply thereafter. The
equation of the parabola is given by;
Model Summary and Parameter Estimates

Dependent Variable:D
(Number of deaths)

Equation

Model Summary

Parameter Estimates

R Square

F

df1

df2

Sig.

Constant

b1

b2

Quadratic

.992

438.638

2

7

.000

882.883

12.070

2.890

The independent variable is T.

Fig. 1.2 Curve
estimation of malaria related deaths
Observation:
The number of malaria related deaths over the 10 year period as depicted in the
above graph, follows a parabolic curve, rising from a high value initially,
then reaching a maximum and then declining sharply thereafter. The equation of the parabola is given by;
LIFE
CYCLE OF MALARIA PARASITES
Malaria
is a vectorborne disease Aron et al (1982).
Malaria parasites are transferred between humans through mosquitoes. The
malaria parasite life cycle is divided into two parts, one is within host
(human) body and the other is within vector (mosquito) body.
Human
infection starts from a blood meal of an infectious female mosquito. The
parasites existing in the infectious mosquito’s saliva, called sporozoites at
this stage, enter the bloodstream of the human through mosquito bites and
migrate to the liver. Within minutes after entering in the human body,
sporozoites infect hepatocytes, and multiply asexually and asymptomatically in
liver cells for a period of 530 days, Johansson et al (2010). This period is called the exoerythrocytic stage. At
the end of this stage, thousands of merozoites (schizonts) emerge inside an
infected liver cell. These merozoites rupture their host cells undetectably by
wrapping themselves in the membrane of infected liver cells. Then, merozoites
escape into the bloodstream and get ready to infect red blood cells. Once
entering the bloodstream, free merozoites undergo the socalled erythrocytic
stage, in which merozoites invade red blood cells to develop ring forms before
experiencing asexual or sexual maturation. Within the red blood cells, a
proportion of parasites keep multiplying asexually and periodically break out
of infected old red blood cells to invade fresh red blood cells. Such
amplification cycles may cause the symptom of waves of fever. The remaining
parasites follow sexual maturation and produce male (micro) and female
(macro) gametocytes which may be taken up by bites of female mosquitoes.
Finally, when it has developed into an infectious form, it spreads the disease
to a new mosquito that bites the infectious human.
MALARIA CONTROL
AND TREATMENTS
According
to the transmission procedure of malaria, there are three conditions for the prevalence of the disease:
(i)
high density of Anopheles mosquitoes,
(ii)
high density of human population,
(iii)
large rate of transmission of parasites
between human beings and mosquitoes.
Obviously, not too much can be done in respect
to (ii). So, (i) and (iii) are naturally targeted. That is, either controlling
the population of Anopheles female mosquitoes at a lower level, or avoiding
biting by mosquitoes can reduce the chance of malaria becoming endemic. In the
middle of the last century, people in Africa have already knew how to remove or
poison the breeding grounds of mosquitoes Dietz et al. (1974) or the aquatic habitats of the larva stages, such as
filling or applying oil to places with standing water, to control the
population of mosquitoes. Killeen et.
al (2002). Later, pesticide was
widely employed to eliminate mosquitoes. On the other hand, mosquito nets,
bedclothes and mosquitorepellent incense (indoor residual spraying) also help
to keep mosquitoes far away from people and minimize the biting. This can
greatly reduce the chance of infection and transmission of malaria. There are
some effective drugs for malaria patients currently. For example, Chloroquine,
Quinuine, Primaquine and combinations of some other drugs like sulfadoxine and
pyrimethamine (SP) as may be prescribed by a doctor are effective medicines for
treating infections caused by the five major parasites. Although malaria is an
entirely preventable or curable disease thanks to these effective medicines,
there are still millions of people suffering from this disease, who are too
poor to afford full treatments.
Moreover,
insufficient treatments due to poor economic conditions, may result in drug
resistance and lead to emergence of new (drug resistant) strains of malaria
parasites. For instance, the first case of resistance to Chloroquine was
documented in 1957. Chloroquine, Quinine and Sulfadoxinepyrimethamine
resistance cases have been reported in almost all disease endemic areas.
Bloland (2001).
1.2 STATEMENT OF THE
PROBLEM
The
development of the model intended to reduce the spread of malaria infections
and eradication necessitates decisive measures to curb the malaria epidemic. In
particular, sustained minimization of the number of humans with incidence of
malaria as a result of adequate control, can be attained by developing a
suitable mathematical model which can enable us to understand better the
dynamics and control of the vectorhost endemic. In the past models studied
there is no protective measure. It is therefore our intension to modify the
previous model to incorporate protective measure in our model
In developing the model, the human population is
compartmentalized into seven classes including the susceptible, infected,
exposed, treated, nontreated, recovered, and protected classes. For the
mosquito population, we have four classes, namely; class of mosquito larva, susceptible
mosquitoes, infected mosquitoes and exposed mosquitoes. We assume free
interaction between the vector and host populations. The mathematical analysis
of the compartmental models leads us to eleven coupled systems of nonlinear
ordinary differential equations.
1.3 AIM AND OBJECTIVES OF THE STUDY
Our aim is
to enhance The
RossMacdonald Malaria Model by incorporating a protective measure into the
model.
OBJECTIVES:
·
To study the equilibrium analysis of the model.
·
To carry out the sensitivity analysis of the
parameters of the model.
·
To provide graphical profiles of the model.
1.4 MOTIVATIONS OF THE STUDY
We were
motivated by lack of protective measure in the past models studied.
1.5 SIGNIFICANCE OF STUDY
This study will help in the treatment and the
control of malaria, It is also significant to researchers modelling in other
areas. The present work is significant in that it
helps us to represent the underlying epidemiology. Generally, malaria models
help us to understand malaria epidemiology as well as the effect and targeting
of various interventions.
1.6 SCOPE AND
LIMITATIONS OF THE STUDY
The
applied mathematics involves synthesizing, comprehending and evaluating
problems so that a mathematical solution is tendered or offered as a solution
to the problems. In the same way, this
study focused on the modification of some models for the effect of treatment
and control of malaria in a population with infected immigrants.
This
study is limited to the modification of malaria model
1.7 DEFINITION OF TERMS
The
following are important terminologies and definitions related to the study of
the epidemiology of any given disease including malaria.
·
Immunity
wanes: Loss of immunity.
·
Latent
period: the period from the point of infection to the beginning
of the state of infectiousness. This is also known as period during which the
infected individuals stay in the exposed (E) class.
·
Dynamical
system: This is a system whose state evolves with time. Is
also a mathematical objects used to model physical phenomena whose state
changes over time.
·
Asymptomatic:
In some infections, symptoms do not appear in individual in spite of being a
carrier for a disease and this is called asymptomatic infection. The appearance
of symptoms is important for case diagnosis and treatment. Sometimes
asymptomatic infections are also known as subclinical infections.
·
Basic
reproduction number: Number of secondary cases which one
primary case introduced into a population that is wholly susceptibles.
·
Vectorial
Capacity: this is defined as the number of potentially
infective contacts an individual person makes through the vector population per
unit time.
·
Disease
generation time: Time from the moment one person becomes
infected until that person infects another person.
·
Enthomological
Inoculation Rate (EIR): this is the rate of infectious bites
per person.
·
Disease
free equilibrium (DFE): Is a state where the entire
population is susceptible.
·
Force
of infection: This is the per capita rate of
acquisition of infection by infectious bites.
·
Equilibrium
point: In a system of differential equation is a state where
all the equation equal to zero. This indicates that the state of the system is
not changing.
·
Clinical immunity: This is the immunity which
reduces the probability of clinical diseases.
·
Antiparasite
immunity: this is the immunity that is responsible for
clearance of parasite.
·
Mortality
rate: The total number of deaths in a population due to a
certain disease during a given period of time.
·
Morbidity
rate: Number of people afflicted with a certain disease
during a given period of time.
·
Incubation
period: The period from the point of infection to the
appearance of symptoms of the disease.
·
Surveillance:
Collecting,
analyzing and reporting data on rates of occurrence, mortality, and transmission
of infections.
·
Immunity:
Is the possession of sufficient resistance which protects a person against the
average infecting dose of the infection.
·
Endemic
equilibrium: Is a state where the disease persists in
the population.
·
Globally
asymptotically stable: Is a state where the behaviour of
the system at any point tends towards the equilibrium points as time tends to
infinity.
·
Herd
immunity: Immunity and protection of the entire community
achievable by vaccinating a proportion of the population.
·
Epidemiology:
The study of causes of occurrence and transmission of diseases in human
population.
·
Endemic:
Disease that exhibits a relatively steady state frequency over a long period of
time in a particular geographical area.
·
Sporadic:
When
occasional cases are reported at irregular intervals.
·
Effectiveness
of treatment: this is the ratio of the duration of the
infection for the untreated and treated sensitive parasites
·
Infectivity
rate; this is the ratio of number infected to number
exposed.
·
Pathogenity
rate; this is the ratio of number with symptoms and number
infected.
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