Mathematical Modeling of Malaria: A Susceptible, Infected, Recovered Model (SIR MODEL)

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ABSTRACT

Malaria, a life-threatening disease transmitted primarily through the bite of infected female mosquitoes, remains a critical global health concern. In this study, we investigate the transmission dynamics of malaria within the framework of an SIR model. This model is tailored to capture the fundamental aspects of malaria transmission.The SIR model involves compartments representing the human population, divided into three key epidemiological classes: Susceptible, Infectious, and Recovered. Susceptible individuals are vulnerable to malaria, while those who recover gain lasting immunity. Infectious individuals can transmit the disease to susceptible ones.Our research centers on the mathematical analysis and stability assessments of the SIR model for malaria. We examine the transitions of individuals between these compartments, offering insights into the disease’s progression within the human population.Furthermore, we compute the critical reproduction number, R₀, of the model. R₀ is a pivotal parameter indicating the potential for an epidemic. When R₀ falls below one, the disease is anticipated to diminish, while R₀ surpassing one implies the potential for an epidemic outbreak.This investigation advances our comprehension of malaria transmission dynamics among human populations, providing valuable insights that can inform strategies for effective disease control and prevention.

TABLE OF CONTENTS

DECLARATION.. ii

CERTIFICATION.. iii

DEDICATION.. iv

ACKNOWLEDGEMENTS. v

ABSTRACT. vi

CHAPTER ONE. 1

INTRODUCTION.. 1

1.1 Background of the Study. 1

1.2 Statement of the Problem. 3

1.3 Aim and Objectives of the Study. 3

1.4 Scope and limitation of the Study. 4

1.5 Definitions of terms. 4

1.5.5 Mosquito. 4

1.5.6 Anopheles Mosquito. 5

1.5.7 Model 5

1.6 Characteristics of a good Model 5

1.6.1 Simplicity. 5

1.6.2 Validity. 6

1.6.3 Robustness. 6

1.7 How to create a Model 7

1.7.1 Descriptive Modeling. 7

1.7.2Rule–based Modeling. 7

1.8 Clinical aspects. 9

LITERATURE REVIEW… 10

2.1  Symptoms of Malaria. 11

2.2 Characteristics of Malaria. 12

2.3 Temporal Variation of Malaria. 13

2.4 History about Model development 14

2.5 History of the SIR Model 15

CHAPTER THREE. 18

METHODOLOGY.. 18

3.1 Formulation of the Model 18

CHAPTER FOUR.. 21

MALARIA MODEL AND ITS QUALITATIVE ANALYSIS. 21

4.0 Mathematical Analysis of the Model 21

4.1 Positivity of the solutions. 21

4.1.1 Positivity of infected human population. 21

4.1.2  Positivity of susceptible human population. 22

4.1.3 Positivity of recovered human population. 22

4.2 Boundedness of the Solution region. 23

4.2.1 Boundedness of total human population. 23

4.3 Disease Free Equilibrium.. 24

4.4 Basic Reproduction Number 25

4.5 Stability Analysis of the Disease Free Equilibrium point 25

4.6 Results and Discussion. 26

CHAPTER FIVE. 28

SUMMARY, CONCLUSION AND RECOMMENDATIONS. 28

5.1 Summary. 28

5.2 Conclusion. 28

5.3 Recommendations. 29

REFERENCES. 30

 

 

CHAPTER ONE

INTRODUCTION

1.1 Background of the Study

Malaria is a mosquito-borne infectious disease that affects humans and other animals.  Malaria causes symptoms that typically include fever, tiredness, vomiting, and headaches. In severe cases it can cause yellow skin, seizures, coma, or death. Symptoms usually begin ten to fifteen days after being bitten by an infected mosquito. If not properly treated, people may have recurrences of the disease months later. In those who have recently survived an infection, reinfection usually causes milder symptoms. This partial resistance disappears over months to years if the person has no continuing exposure to malaria. It is caused by single-celled microorganisms of the Plasmodium group. The disease is most commonly spread by an infected female Anopheles mosquito. The mosquito bite introduces the parasites from the mosquito’s saliva into a person’s blood. The parasites travel to the liver where they mature and reproduce. Five species of Plasmodium can infect and be spread by humans. Most deaths are caused by P. falciparum because P. vivax, P. ovale, and P. malariaegenerally cause a milder form of malaria. Malaria is typically diagnosed by the microscopic examination of blood using blood films, or with antigen-basedrapid diagnostic tests. The risk of disease can be reduced by preventing mosquito bites through the use of mosquito nets and insect repellents, or with mosquito control measures such as spraying insecticides and draining standing water. The disease is widespread in the tropical and subtropical regions that exist in a broad band around the equator. This includes much of Sub-Saharan Africa, Asia, and Latin America. In 2016, there were 216 million cases of malaria worldwide resulting in an estimated 445,000 to 731,000 deaths. Approximately 90% of both cases and deaths occurred in Africa. Rates of disease have decreased from 2000 to 2015 by 37%, during which there were 198 million cases. Malaria is commonly associated with poverty and has a major negative effect on economic development. In Africa, it is estimated to result in losses of US$12 billion a year due to increased healthcare costs, lost ability to work, and negative effects on tourism. Malaria is currently affecting more people in the World than any other disease. It is currently endemic in over 100 countries and is one of the 10 most prevalent and deadly diseases in the world. The disease is caused by tropical parasite that kills people more than any other communicable disease except tuberculosis. Between 300 to 500 million clinical cases occur every year with over 1.2 to 2.7 million deaths, of which 90% occur in sub-Saharan Africa. Malaria menace has become an economic burden in tropical Africa. According to the report of the American Association for the Advancement of Science (AAAS) Washington, D.C 1991 on Malaria and Development in Africa, pregnant women and children under the age of five are at high risk of Malaria morbidity and mortality. The World Health Organization (1994)stated that some 90% of the World’s Malaria occurs in Africa because the World wide eradication programme of 1960s which successfully remove Malaria from North America and Europe, exclude sub-Saharan African altogether, due to the lack of technological capability in individual countries and because Malaria was so huge that eradication was considered not feasible. It is also reported that 255 children in Africa die every 2.5Hours, while about 2173 children under the age of 5 die daily in the continent from malaria. Indeed the African region lies in areas where the population is at risk of getting malaria since 74% of the population live in highly endemic areas where malaria transmission is intense. It is responsible for about 20-30% infant mortality, 10% of hospitals admissions, and 20-30% outpatient case in Africa. In Nigeria alone, 60 million people experience Malaria attack at least twice in a year, with no less than 80% of the population exposed to the disease. Scott (2000) ascribed 90% of health problem caused by Malaria to environmental conditions. To corroborate this, Paul, (1997) emphasized the role of temperature on the range, development, timing and intensity of Malaria outbreak. He described mosquito as hot weather insects that have fixed thresholds for survival. For instance, Anopheles mosquito and Falciparum malaria transmission are sustained only where the winter temperature is kept above 160 C.It was observed that P. falciparum transmission was limited by low temperature in areas of high altitude. In Kano metropolis, a detailed study of 278 households made up of 3071 individuals that inhabit around ten non-water outlet ponds from various segments of the metropolis revealed that Malaria is the most common sickness among them. On the average about two members of a household suffered from malaria fever monthly, with females and children having high frequencies of and vulnerable to malaria attack .

1.2 Statement of the Problem.

Malaria is a mosquito-borne infectious disease that affects humans and other animals. It is characterized by symptoms such as fever, fatigue, vomiting, and headaches. In severe cases, it can lead to yellow skin, seizures, coma, or even death. Typically, symptoms manifest around ten to fifteen days after an individual has been bitten by an infected mosquito. Without appropriate treatment, individuals may experience recurring bouts of the disease several months later.

The SIR (Susceptible-Infectious-Recovered) model, originally proposed by W.O. Kermack and A.G. McKendrick in 1927, has been instrumental in studying the dynamics of malaria transmission.

1.3 Aim and Objectives of the Study

The primary aim and objective of this research is to mathematically model malaria transmission using the SIR (Susceptible-Infectious-Recovered) model and is achieved through the following:

1. Behavior Study: Investigate the behavior of both the Anopheles mosquito and humans.
2. Transmission Dynamics: Examine the dynamics of disease transmission, between mosquitoes and humans and vice versa.
3. Mathematical Formulation: Formulate and solve an Ordinary Differential Equation (ODE).
4. Analysis: Present a analysis of the SIR model.

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