AN ENHANCED MATHEMATICAL MODEL OF THE EFFECT OF TREATMENT IN THE CONTROL OF MALARIA IN A POPULATION WITH INFECTED IMMIGRANTS

ABSTRACT

In this work we modified a mathematical model to study the dynamics of the vector-host disease. In the enhanced model, we obtained seven classes from human population which includes the susceptible, infected, exposed, treated, non-treated, 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 k2-0.15 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  Disease-free equilibrium point                                                                        32

4.19.2  Local stability of DFE                                                                                                33

4.19.3  Generations is an epidemic                                                                             33

4.20     Basic Reproduction Ration (R₀)                                                                     34

4.21     Sensitivity Analysis of the Model Parameters                                                39

4.21.1  Sensitivity analysis of R₀                                                                                                           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.

Table 1.1 Estimates of malaria cases and deaths in Africa by WHO, 2000-2009.

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.

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;

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 vector-borne 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 5-30 days, Johansson et al (2010). This period is called the exo-erythrocytic 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 so-called 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 mosquito-repellent 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 Sulfadoxine-pyrimethamine 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 vector-host 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, non-treated, 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 Ross-Macdonald 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.

·         Anti-parasite 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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