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## ABSTRACT

In this project work, we formulated a mathematical model based on a system of ordinary differential equations to study the transmission dynamics of typhoid fever incorporating enlightenment campaign, treatment and vaccination as control strategies. The basic reproduction number and the effective reproduction number of the model without control and that with control were computed respectively using the next generation matrix approach. The Disease Free Equilibrium (DFE) states of the models were established and proved to be locally asymptotically stable using Routh-Hurwitz criterion for local stability. Endemic Equilibrium (EE) states of the models were also established and proved to be locally asymptotically stable using trace and determinant method. Result also shows that the Endemic Equilibrium (EE) is locally asymptotically stable whenever the basic reproduction number is greater than one. The global stability analysis of the Endemic Equilibrium point of the model with control was established to determine the condition that favoured the spread of the disease by the construction of a Lyaponov function and result shows that the Endemic Equilibrium point (EE) of the model is stable whenever the effective reproduction number is greater than one, else it becomes unstable. Sensitivity analysis was then carried out to determine which parameters that should be targeted by control intervention strategies of which the result shows that an increase in the enlightenment rate, treatment rate and rate of vaccination leads to a reduction in the prevalence of the disease. Finally, numerical simulation of the model was carried out and result shows that an increase in protection leads to low disease prevalence in a population.

Title page……………………………………………………………………. (i)

Certification…………………………………………………………………(ii)

Dedication…………………………………………………………………..(iii)

Acknowledgements…………………………………………………………(iv)

Abstract……………………………………………………………………..(v)

Table of content………………………………………………………………..(vi)

List of tables &figures………………………………………………………(ix)

CHAPTER ONE

• Introduction………………………………………………………………(1)
• Background of the study ……………………………………………….(1)
• Statement of Problem…………………………………………………….(2)
• Significance of the study………………………………………………..(3)
• Aims & Objectives of the Study………………………………………….(3)
• Scope of the study………………………………………………………..(4)

CHAPTER TWO

• Literature Review…………………………………………………………(5)

2.10    Literature Review of Typhoid Fever models…………………………..(5)

CHAPTER THREE

3.0      The Organism…………………………………………………………(16)

3.1      The Disease……………………………………………………………(17)

3.2      Symptoms…………………………………………………………….(18)

3.3      Contamination & Transmission………………………………….(18)

3.4     Diagnosis of Typhoid fever……………………………………….(19)

3.5     Treatment of Typhoid Fever………………………………………(20)

3.6     Prevention of Typhoid Fever……………………………………..(21)

CHAPTER 4

4.0     Model Formulation……………………………………………….(24)

4.1     Assumptions of the model………………………………………..(26)

4.2     Flow diagram for the model………………………………………(27)

4.3     Mathematical Model of Typhoid Fever & its control…………….(28)

4.4     Model Variables & Parameters……………………………………(29)

4.5    Analysis of the model without control…………………………..(30)

4.5.1  Disease Free equilibrium state (DFE) of the model………………(31)

4.5.2  The basic Reproduction Number () of the model………………(31)

4.5.3  Local Stability Analysis of the Disease Free Equilibrium ………..(36)

4.5.4  Endemic Equilibrium State (EE) of the model…………………….(38)

4.5.5  Local Stability Analysis of the Endemic Equilibrium…………….(42)

4.6    Analysis of the model with control………………………………(44)

4.6.1  Disease Free equilibrium state (DFE) of the model with control…(44)

4.6.2  Endemic Equilibrium State (EE) of the model…………………….(46)

4.6.3  The Effective Reproduction Number () of the Control model…(49)

4.6.4  Local Stability Analysis of the Disease Free Equilibrium ……………(53)

4.6.5  Local Stability Analysis of the Endemic Equilibrium…………………(57)

4.6.6  Global Stability Analysis of the Endemic Equilibrium……………….(63)

4.6.7  Parameter Estimation………………………………………………….(66)

4.6.8  Sensitivity Analysis……………………………………………………(67)

4.7     Model Simulation………………………………………………………(72)

CHAPTER FIVE

5.0 Conclusion ……………………………………………………………..(79)

5.1 Recommendations ……………………………………………………..(79)

References

## CHAPTER ONE

• INTRODUCTION

1.10   BACKGROUND OF THE STUDY

Typhoid fever was so named because its signs and symptoms resemble that of typhus. It is an endemic infectious disease caused by a highly virulent and invasive Salmonella enteric serovar Typhi(S.Typhi) that affects human. Nthiiri et al; (2016). It is spread through contaminated food, water or drink. The contaminated food or water that contains these bacteria causes illness by drinking. They travel in the human intestines, and then enter the blood, Muhammad et al; (2015).

Typhoid fever is a global health problem. Its real impact is difficult to estimate because the clinical picture is confused with those of many other febrile infections. The disease typhoid has a very high social and economic impact because of the hospitalization of patients with acute disease and the complications and loss of income attribute to the duration of the clinical illness, WHO (2003).

Abdominal pain, fever and general ill feeling are the symptoms of this disease. High fever (103F or 39.5c) or higher and severe diarrhea occurs as the disease gets worse. The incubation period is about 10-14 days, sometimes 3 days short or long for 21days. This disease is less severe when It is caused by S paratyphi A, B and sometimes C. It is endemic in Central America, Indian subcontinent, south east Asia and some parts of Africa, including Nigeria. In 2000, it was estimated that the disease caused 21.6 million illness and 216, 500 deaths globally, Muhammed et al; (2015).

Typhoid fever affects millions of people worldwide each year, where 20millions cases were reported and kills approximately 200, 000 annually. For instance, in Africa, it was estimated 50 per 100,000. It is believed that vaccinating high-risk populations is the best way to control typhoid fever disease. There are two types of typhoid fever vaccines namely, oral and inject-able vaccines. However they are not 100% effective. If one acquires drug resistant strain of typhoid fever and is not treated with effective antibiotics, a serious and prolonged illness may result, Nthiiri et al;(2016).

In many developing nations the public health goals that can help prevent and control the spread of typhoid fever disease through safe drinking water, improved  sanitation and adequate medical care may be difficult to achieve,Nthiiri et al;(2016). Health education is paramount to raise public awareness and induce behavioral change,WHO( 2003).

Typhoid fever is largely controlled in Europe and North America. Typhoid remains endemic in many parts of the world, notably Asia, where it is an important cause of febrile illness in crowded, low- income settings. A notable feature of typhoid is the carrier state- asymptotically infected individuals who continue to shed salmonella typhi in their stool or urine for many years, thereby sustaining transmission, Conall (2015).

Despite a recommendation by the World Health Organization in 2003 that typhoid vaccination be considered for the control of endemic disease and outbreaks, in the early twentieth- century, public health officers were debating the best methods of evaluating typhoid vaccine effectiveness, and whether vaccination was a distraction from improvements in sanitation and hygiene, These remain contemporary policy issues for ministries of health and other health partners who may be considering programmatic anti-typhoid vaccination as a counterpart to other anti-typhoid measures such as improvements in income distributions, sanitation, water supplies and hand washing with soap (post-defecation and before the preparation of food in the home or sold in the street) as well as identification and management of carriers. Hardy (2001).

1.11   STATEMENT OF PROBLEM

Typhoid disease remains a leading cause of infectious mortality in the world despite many decades of studies, widespread availability of vaccines and drugs and more recently a highly visible world Health Organization (WHO) effort to promote a unified global control strategy, WHO (2003). In a Communicable Disease Surveillance and Response Vaccines, Biological Journal declared Typhoid fever as a global health problem with an approximately 17 million cases and 600, 000 associated deaths which occur naturally. However, the estimates have been biased because studied populations have usually been in areas of high incidence.

The problem of Typhoid fever is ever becoming more alarming due to a very strong positive correlation between Typhoid and Malaria fever which has become an epidemic globally. The co- infection of Malaria and Typhoid are among the most endemic disease and thus, of major public health concerns in tropical developing countries. Misleading diagnosis due to similar symptoms of these diseases and incompetent testing methodologies is one of the biggest challenges for their control, Jones (et al; 2015).

As a result of this growing incidence of Typhoid fever, there is a very strong need to develop a mathematical model that will help establish conditions under which the spread of the disease can be controlled and eradicated.

1.12   SIGNIFICANCE OF THE STUDY

This study is significant because the incidence of Typhoid Fever is on the increase globally despite many decades of previous studies. There is therefore the need to intensify effort at controlling the spread of the disease so that epidemic does not occur.

To achieve this, the study on Typhoid Fever must take different dimensions compared to existing models. So many mathematical models in the past have assumed that susceptible individuals recovered with immunity against the disease, that is to say, there is no re-infection once an individual has recovered from the infection, Adetunde (2008). This assumption is not realistic as fully recovered individuals still stand the risk of re-infection if they are exposed to the bacteria again.

This present model assumes that the infection does not confer any immunity to return to the susceptible class and sufficient control measures are provided in this work and as such will give a necessary and sufficient condition that will be very effective in controlling the spread of Typhoid Fever in any population.

1.13   AIMS & OBJECTIVES OF THE STUDY

The study is to establish the role control intervention strategies which includes; Treatment, Vaccination and Enlightenment campaign, play on the transmission dynamics of typhoid fever with a view to bringing the disease under control and as such eradicating typhoid fever in our society.

In this direction, the study is aimed at achieving the following objectives.

• Build a mathematical model that can describe the transmission dynamics including the control strategies.
• To establish how the bacteria that causes Typhoid is transmitted.
• To find out the conditions that promotes the spread of the disease.

(iv)     To establish conditions that must hold if the disease is to be eradicated from              the society under Treatment, vaccination and Enlightenment Campaign as      Control Strategies.

• And to make recommendations that will help in the effort at controlling or

1.14 SCOPE OF THE STUDY

This study presents a deterministic mathematical model on Typhoid fever transmission dynamics with its control. Multiple control measures such as Educational enlightenment campaign, Treatment and Vaccination were considered with the aim of reducing the severity of the disease. Disease Free Equilibrium (DFE), Endemic Equilibrium (EE) states of Typhoid fever model without control and the model with control were analysed. Finally, Local Stability analysis, Global stability analysis and sensitivity analysis of both models were also respectively considered after which numerical simulation of the models was presented.

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