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Mathematical modelling ofmalaria transmission and

pathogenesis

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A Doctoral Thesis. Submitted in partial fulfilment of the requirementsfor the award of Doctor of Philosophy of Loughborough University.

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Mathematical Modelling of Malaria Transmission and

Pathogenesis

by

Aniayam Bernard Okrinya

Doctoral thesis

submitted in partial fulfilment of the requirements

for the award of the degree of

Doctor of Philosophy

of

Loughborough University

December, 2014

cA B Okrinya 2014

DEDICATION

I wish to dedicate this thesis, to my father (Chief Bernard Okrinya of Blessed memory) who

apart from initiating my early education gave me a positive direction to life, to the

Petroleum Technology Development Fund (PTDF) Nigeria for funding this project and to

Senator Clever Ikisikpo, without whom this thesis would not exist. May God receive the

Glory.

i

ACKNOWLEDGEMENT

I am grateful to God for His protection and mercy. I would like to acknowledge the contribu-

tions made by various people to the successful completion of this thesis. First, I particularly

thank my supervisor, Dr. John Ward for introducing me to mathematical biology, enhancing

my interest in the field, and encouraging me through the process of application that illumi-

nated my preconception of a project of this sort. His humility, patience, unique interest and

vast experience in mathematical modelling of problems in biology and medicine had created

many of the techniques, ideas and directions for the research, including suggested methods

for the analysis, result verification and proofreading of numerous drafts. I thank the Director

of Research Degree Programmes for the Department of Mathematical Sciences, Dr Maureen

MacIver for her academic advice and monitoring of my progress.

I wish to also express my appreciation to the Petroleum Technology Develelopment Fund

(PTDF) Nigeria for sponsoring this project and the particular contributions of Senator Clever

Ikisikpo, an honourable senator of the Federal republic of Nigeria for the role he played in

securing the sponsorship. I would also like to thank Mrs Namitanighe Clever Ikisikpo for her

moral surport.

My special thanks goes to Professor Samuel Bankole Arokoyu and his wife, Dr (Mrs)

ii

Abosede Samuel Arokoyu for their cares and support to my family. Finally, I would like

to thank the department of mathematical sciences, Loughborough University for creating an

enabling environment characterised by its rich research and support facilities for the successful

completion of this project.

iii

CONTENTS

Dedication i

Acknowledgement ii

Abstract viii

1 Introduction 1

1.1 The biology of malaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Life cycle of malaria parasite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Intervention strategies and immunity to malaria . . . . . . . . . . . . . . . . . 8

2 Modelling background 11

2.1 Infectious disease models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 A survey of mathematical models in malaria epidemiology . . . . . . . . . . . 14

2.2.1 Transmission models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Summary from the survey . . . . . . . . . . . . . . . . . . . . . . . . . 17

iv

3 Transmission model 19

3.1 Derivation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Establishing the basic reproduction number of the transition model . . . . . . 30

3.5 Steady state solution and model analysis . . . . . . . . . . . . . . . . . . . . . 32

3.6 Stability analysis of the transition model . . . . . . . . . . . . . . . . . . . . . 33

3.7 Time scale analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7.1 t = O(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7.2 t = O(4/3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7.3 t = O(5/4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.7.4 t = 54 ln(1/2/y0)/K0 + O(

54 ) . . . . . . . . . . . . . . . . . . . . . 43

3.7.5 t = O() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.7.6 t = ln(1/)/ + O() . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.7.7 Conclusion from the analysis . . . . . . . . . . . . . . . . . . . . . . . . 49

3.8 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Pathogenesis of malaria 63

4.1 In-host pathogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 The immune system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.1 Review of within host models . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.2 Summary from the survey . . . . . . . . . . . . . . . . . . . . . . . . . 73

5 Description and analysis of within-host mathematical models 75

5.1 Model development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

v

5.1.1 Initial and history conditions . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Existence and uniqueness of solution . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 The basic reproduction number . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.6 Steady state solution and stability analysis . . . . . . . . . . . . . . . . . . . . 91

5.7 Asymptotic analysis of in-host model . . . . . . . . . . . . . . . . . . . . . . . 93

5.7.1 t = O() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.7.2 t = b0

ln(1) + O() . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.7.3 t = O(1), t < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.7.4 t = + O(), Rc > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7.5 t = + O(1), Rc > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.7.6 t = + 2Rln(1

) + O(1), R > 0 . . . . . . . . . . . . . . . . . . . . . 101

5.7.7 t = O (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.7.8 Conclusion from this analysis . . . . . . . . . . . . . . . . . . . . . . . 108

5.8 Numerical simulations of in-host model . . . . . . . . . . . . . . . . . . . . . . 111

5.8.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Conclusion 120

6.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.1.1 Limitations of the models . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.1.2 Suggestion for future work. . . . . . . . . . . . . . . . . . . . . . . . . . 124

A Appendix 127

A.1 Expressions for important constants in the stability analysis of transition model 127

A.2 Demonstrating the effect of inequalities obtained in 3.6.7 on R0 . . . . . . . . 128

vi

B Appendix 130

B.1 Time-scale analysis (transition model) . . . . . . . . . . . . . . . . . . . . . . 130

B.1.1 Time scale 1: t = O(2) . . . . . . . . . . . . . . . . . . . . . . . . . . 131

B.1.2 Time scale 2: t = O(4/3) . . . . . . . . . . . . . . . . . . . . . . . . . 132

B.1.3 Time scale 3: t = O(5/4) . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.1.4 Time scale 4: t = 54 ln(1/2/y0)/K + O(

54 ) . . . . . . . . . . . . . 134

B.1.5 Time scale 5: t = O() . . . . . . . . . . . . . . . . . . . . . . . . . . 135

B.1.6 Time scale 6: t = ln(1/)/ + O() . . . . . . . . . . . . . . . . . . 136

C Appendix 137

C.1 Asymptotic analysis (in-host model) . . . . . . . . . . . . . . . . . . . . . . . 137

C.1.1 Time scale 1: t = O() . . . . . . . . . . . . . . . . . . . . . . . . . . 138

C.1.2 Time scale 2: t = b0

ln(1) + O() . . . . . . . . . . . . . . . . . . . . 138

C.1.3 Time scale 3: t = O(1), t < . . . . . . . . . . . . . . . . . . . . . . 139

C.1.4 Time scale 4: t = + O(), Rc > 1 . . . . . . . . . . . . . . . . . . . 139

C.1.5 Time scale 5: t = + O(1), Rc > 1 . . . . . . . . . . . . . . . . . . . 140

C.1.6 Time scale 6: t = + 2Rln(1

) + O(1), R > 0 . . . . . . . . . . . . . 140

C.1.7 T