A Mathematical Model for the Transmission Dynamics of Malaria in Eastern Uganda: A case Study of Butaleja District

Abstract A deterministic mathematical model for studying the transmission dynamics of malaria in Butaleja district was developed using ordinary differential equations (ODEs) where hinnans and mosquitoes interact and infect each other. fhe model has live non - linear differential equa tions with two state variables for mosquitoes (S,~anc I,, ) and three state variables for humans (S~ I~ and A,~) The available literature on previous work in this area was reviewed. Susceptible humans (S, ) are infected when they are bitten by in fectious mosquitoes (Im ). They then progress through infectiousand asyrnptornatic classes before re-entering susceptible class. Susceptible mosquitoes (Sm) become infected when they bite Infectious (1,) and Asymptomatic (A,,) Immans. They move to infectious class but (10 not recover due to their short life span. Following ideas advanced by Ross. [Chapter 2] (ii]. the model can be applicable to other infectious diseases of humans such as yellmv fever. typhoid. sleeping sic ness, cholera etc: using specific niodel parameters. Model Analysis was clone, equilibrium points analyzed to establish their local and global stability. The important threshold in this re search called the basic reproduction number (R0) was obtained using the niethod of next—generation rrmatrix to determine whether the (us— ease dies out or persists. The rule of thumb is that: the disease— free equilibrium is locally asymptotically stable if R0 < I and the endemic equilibrium exist provided that R0 > 1. Using parameter values, R0 fbr Butaleja district was ft)und to be = 0.00000315 < i; an indication that malaria will be rolled out of the district after a certain period of time. Numerical snnulations show that there is a strong positive relation ship between the number (proportion) of infected mosquitoes and in— fectecl humans in tIme same locality. Reducing the current rate of fe male anopheles mosquito bites could assist Butaleja district to achieve malaria free status by the year 2030 [26], [25]. Therefore. I recommend control methods such as ITNs and IRS the t increase mosquito death rate and reduce mosquito birth rate/mosquito bites, as well as treat ing asvniptonmatic hosts using ACTs. and IPT. Hence. the formulated model. provides a franimvork for studying and designing effective inter vention strategies for prevention audI control of malaria in the district. 

Contents

Declaration A ii

Declaration B iii

Dedication iv

Acknowledgements v

list of tables ix

listoffigures x

Chapter 1 1

1 Introduction 1

1.1 Background 1

1.2 The Research Problem 5

1.3 Objectivew 5

1.3.1 Major objective 5

1.3.2 Specific Objectives 5

1.4 Scope 6

1.5 Methods of the study 8

1.6 Significance 8

1.7 Structure of the dissertation 9

Chapter 2 10

2 Literature Review 10

2.1 Management Strategies 10

2.2 Reduction of Malaria endemic 12

2.3 Modeling and control of Malaria 13

VI

Chapter 3 16

3 Mathematical Models for Malaria 16

3.1 The Mathematical Model 16

3.1.1 Model Variables 17

3.1.2 Model Parameters 18

3.1.3 Model Assumptions 18

3.1.4 Compartmental Diagram 19

3.1.5 Model Equations 20

3.2 The Basic Reproduction Number (I?~) 20

3.3 Model Analysis at Disease Free Equilibrium (DEE) 26

3.3.1 Disease Free Equilibrium 28

3.4 Establishing and Analyzing Stability Conditions 29

.3.4.1 Local Stability Analysis 29

3.4.2 Global Stability Analysis 33

Chapter 4 35

4 Numerical Simulations 35

4.1 Introduction 3.5

4.2 Model Parameter Values 35

4.3 Numerical Simulation Results 37

4.3.1 Tlic effect of varying b on R 37

1.3.2 The effect of varying b on S1~ 38

1.3.3 The effect of varying p on S, 39

4.3.4 The effect of Ab with time 40

4.3..5 The effect of I,~ on 11

Chapter 5 42

5 Discussion, Conclusion, Recommendations 42

5.1 Introduction 42

.5.2 Discussion 12

5.3 Conclusion . 43

5.4 Recommendations 44

5.5 Areas for further model development 45

6 Appendices 53

6.1 MATLAB codes 53

6.1.1 The effect of varying b on R~, 53

6.1.2 The effect of varying b on 53

6.1.3 The effect of varying p on 54

6.14 The effect of Ak with time 54

6.1.5 The effect of I,,, on Si 55