Mathematical modelling on interaction between malaria parasites and the host immune system

immune system

by

Theresia Marijani

Dissertation presented in full fullment of the

a ademi requirements for the degree of

Do tor of Philosophy in Mathemati s

at the Stellenbos h University

Promoter: Prof. Edward M. Lungu (University of Botswana)

Co-Promoter: Prof. John Hargrove (University of Stellenbos h)

De laration

I, the undersigned, hereby de larethat the work ontained inthis dissertationis my own

originalworkand has not previously,in itsentirety or inpart, been submitted atany

university for a degree.

--- ---

-Theresia Marijani Date

Copyright ©

Abstra t

Malaria is a deadly tropi al disease aused by protozoa of the genus plasmodium. The

malaria parasite life y le involves three y les namely the sporogony (mosquito stages),

exo-erythro yti s hizogony (humanliverstages),andthe erythro yti s hizogony(human

blood stage). We onsider a mathemati al model for malaria involving, sus eptible red

blood ells, latent infe ted red blood ells, a tive infe ted red blood ells, intra ellular

parasites, extra ellularparasitesand ee tor ells. Weextend themodeltoin ludeallthe

threestagesofthe malarialife y le. Theee tof treatmentontheprognosis ofmalariais

alsointrodu edinthesemodels. Themodelsareanalysedmathemati allyandnumeri ally.

One of the question addressed in our study is: what repli ative hara teristi s oer the

parasite opportunities to evade the host immune system? The results showed that the

longer it takes to produ e the parasites, the higher the han e that an infe ted red blood

ellwillbeidentied and apoptosised by the ee tor ells. Oursensitivity analysis results

show that poor parametri estimation has serious impli ations on the prognosis of the

disease. Treatment results suggest that a high drug e a y an stop the development of

the disease. The study has revealed thatthe parasiterepli ative hara teristi senablethe

parasitetoevadetheimmuneresponseduringtheredbloodstagemalaria. Firstly,wehave

found that the parasite has a strategy of infe ting older red blood ells as a strategy to

evade immunesurveillan e. Se ondly,wedis overedthatthe administrationofanee tive

drug anpreventmalariainallstagesdespitethe urrentbeliefthatonlyamalariava ine

an reliably prote t against all stages malaria infe tion. We re ommend treatment to be

used in areas where anti-malarialdrugs do not show resistan e to the parasites. We also

re ommend that individuals with malaria or showing some symptoms should be treated

Opsomming

Malariais'ndodeliketropiesesiektewatveroorsaakword deurdieprotosoëvandiegenus

Plasmodium. Die malariaparasietlewensiklus bestaan uitdrie siklusse naamlik die

sporo-gony (muskiet stadiums), exo-erythro yti s hizogony (menslike lewer stadiums), en die

erythro yti s hizogony (menslike bloed stadium). Ons kyk na 'n wiskundige model vir

malaria, vatbaar rooibloedselle, latente besmet rooibloedselle, aktief besmette

rooibloed-selle,intrasellulêreparasiete, ekstrasellulêreparasieteen eektorselle. Onsbrei diemodel

om al die drie fases van die lewensiklus malaria te sluit. Die eek van behandeling op

die voorspelling van malariais ook in hierdie modelle ingevoer. Die modelle is wiskundig

en numeries ontleed. Een van die vraag in ons studie is: watter repli ative eienskappe

bied die parasietgeleenthede om diegasheer seimmuunrespons te ontduik? Die resultate

het getoon dat hoe langer dit neem om die parasiete te produseer, hoe groter die kans

dat besmette rooibloedselle sal geïdentiseer word en deur die eektor selle apoptosised.

Ons sensitiwiteitsanalise resultate toon dat die arme parametriese beraming het ernstige

implikasiesvirdievoorspellingvandiesiekte. Behandelingresultatedui daaropdat 'nhoë

dwelmdoeltreendheidkandieontwikkeling vandiesiekte stop. Diestudiehet getoondat

dieparasiet repli ative eienskappe dieparasiet instaat stel omdieimmuunreaksie tydens

dierooibloedstadiummalariateontduik. Eerstens, hetons gevinddat dieparasiethet 'n

strategievanbesmetomouerrooibloedselle as'nstrategieomimmuuntoesigteontduik.

Tweedens, het ons ontdek dat die administrasie van 'n doeltreende middel kan malaria

in allestadiumsvoorkom ten spytevandiehuidigeoortuigingdat slegs' nmalaria-entstof

betroubaarkanbeskermteenallestadiamalariainfeksie. Onsraaidatbehandelinggebruik

word ingebiedewaar dieanti-malariamedisynenieweerstandtoonaan dieparasiete. Ons

beveelookaandat individuemetmalariaofwatsekere simptomehet,behandelmoetword

Dedi ation

Idedi atethis dissertationtothegreater glory ofGod, tohisgra eandpower,to thepeople

A knowledgments

I would likethank almighty Godfor hisgra e, inspiration,strength and guidan e,to God

be the glory!

I would liketo thank my beloved parents for their en ouragement and prayer through all

the time.

I would like give thank my promoter Prof. Edward Lungu, for his valuable guidan e

and supervision. He has been an ex ellent supervisor, a wise mentor, an inspiring and

motivationaltea heravailableinmomentsofneed. Hishelpduringthetime inneedisreal

appre iated. My sin erethanks go toMrs. ElizabethLungu.

I deeply appre iate my o-promoterProf. John Hargrove for hisvaluablesuggestions and

insightful omments onthis dissertation.

Irealappre iatethedire tionandhelpbytheDire torofSouthAfri anCentreof

Epidemi-ologi alModellingandAnalysis(SACEMA) Dr. AlexWelte,resear hmanagerLynnemore

S heepers, assistantdire tor trainingDr. GavinHit h o k, administratorNatalie Roman,

Dr. RashidOuifki,allthestaatSACEMA,UniversityofStellenbos h,Internationalo e

atStellenbos hUniversity. UniversityofBotswana, mathemati sdepartmentatUniversity

ofBotswana,O eofInternationalEdu ationandPartnershipsatUniversityofBotswana

during these years of my study. My spe ial thanks to them for all o ial assistan e they

gaveme and the personal on ern they showed me.

I would like to thank all my fellow students at SACEMA and at University of Botswana

for their help and support inpersonallyand a ademi ally.

v

during the time of ex hanging my studies from Stellenbos h University toBotswana

Uni-versity.

This thesisissupportedby SACEMA andOWSD (OrganizationforWomen inS ien e for

the Developing World). International o e at Stellenbos h University, O e of

Interna-tional Edu ation and Partnerships at University of Botswana. I would like to thank all

these sponsors.

To my familymembers, my sisters, my brothers' in-law,my brother, sister'sin-law, thank

you for being patient with me during this time of my study, espe ially for my nephews

(Godwin and Crispin) and my nie es (Winlove, Imma ulata and Agnes). Thank you so

mu h my family for your support, en ouragement and prayers I real appre iate.

Last, but not least, my deepest a knowledgement to all my friends, Angelina Lutambi,

Asha Kalula, Sara Mkango, Rose Kibe hu, Doreen Mbabazi, Joseph Ssebuliba, Amani

Lusekelo, Maggie Goosen,BoitumeloMogaleemang, Malebogo,Christina Meela andFazia

Glossary

Abbreviation Meaning

RBCs Red Blood Cells

LVCs LiverCells

MGCs Midgut Cells

CD in CD

4

Clusterof Dierentiation

4

CD in CD

8

Clusterof Dierentiation

8

HIV Human immunode ien y virus

AIDS A quired Immunode ien y Syndrome

ODE Ordinary DierentialEquation

T in T- ell Thymus

ACT Artemisin-basedCombinationTherapy

WHO World Health Organization

DDT Di hloro-Diphenyl-Tri hloroethane

UNICEF United Nations Children's Fund

SIV SimianImmunode ien y Virus

Contents Abstra t i Opsomming ii Dedi ation iii A knowledgements iv Glossary vi 1 Introdu tion 1 1.1 Ba kground . . . 1

1.2 Statement of the problem . . . 5

1.3 Obje tives of the study . . . 6

1.4 Organizationof the work . . . 6

2 Literature review 8 3 Mathemati al tools 16 3.1 The denition and omputationof

R

0

. . . 16

Contents viii

3.3 Sensitivity analysis . . . 19

4 A within host model of blood stage malaria 22 4.1 Introdu tion . . . 22

4.2 Methodology . . . 24

4.2.1 Model formulation . . . 24

4.2.2 Mathemati al analysis of the model . . . 32

4.2.3 A within host treatmentmodelof bloodstage malaria. . . 36

4.2.4 Simulationsof the within host modelof bloodstage malaria . . . . 37

4.2.5 Simulationsofthe withinhost treatmentmodelof bloodstage malaria 39 4.3 Results of the withinhost modelof blood stagemalaria . . . 41

4.4 Results of withinhost treatment modelof blood stage malaria . . . 51

4.5 Dis ussion . . . 54

5 A within host treatment model with three stages of malaria life y le 56 5.1 Introdu tion . . . 56

5.2 Methodology . . . 58

5.2.1 Model development . . . 58

5.2.2 Mathemati al Analysisof the model . . . 65

5.2.3 Simulations . . . 70

5.3 Results . . . 73

Contents ix

5.4 Dis ussion . . . 85

6 Con lusion 86

6.1 Limitationsand re ommendations . . . 87

6.2 Futurework . . . 87

Appendix 88

List of Figures

1.1 A diagramshowing malariaendemi area inAfri a. . . 2

2.1 A diagramshowing ampliedrelationship between HIV and malaria . . . . 9

2.2 Malaria life y les, opied from Parasite image library [10℄ . . . 10

3.1 Representation of

F

and

V

. . . 17

4.1 A diagrammati representation of withinhost malaria model.. . . 25

4.2 A diagram showing sensitivity of various parameters on the reprodu tion

number. . . 38

4.3 Shows a diagramof parasite-free equilibriumwith

R

02

= 0.8327.

. . . 42

4.4 A diagramof parasite-present equilibriumwith

R

02

= 1.3165.

. . . 43

4.5 A diagram showing population of intra ellular parasites for

n

1

< 16

the

parasite-present equilibrium ases and

n

1

≥ 16

the parasite-freeequilibrium

ases.. . . 44

4.6 A diagram showing population of extra ellular parasites for

n

1

< 16

the

parasite-present equilibrium ases and

n

1

≥ 16

the parasite-freeequilibrium

ases.. . . 45

4.7 Representsrelativeimpa tofthetwoparasiteprodu tionme hanisms

10∗P

1

Listof gures xi

4.8 A diagramshowing the evolution of RBCs with time. . . 47

4.9 A diagramshowing the evolution of a tive infe ted RBCs with time. . . 47

4.10 Contour plots for

n

1

= 12

and

n

1

= 15

and

n

1

= 16.

. . . 48

4.11 Shows the population ofa tivelyinfe ted RBCs atfor

n

1

= 24.

. . . 48

4.12 Shows the population oflatently infe ted RBCsat for

n

1

= 24.

. . . 49

4.13 Diagram showing the population of a tively infe ted RBCs population for dierent values of

m

and

k

tp

.

. . . 49

4.14 Diagram showing the populationof sus eptible RBCspopulationfor dier-entvalues of

m

and

k

tp

.

. . . 50

4.15 A diagramshows malaria withouttreatment. . . 52

4.16 Diagramshowingonetypeofdrugintreatmentofmalariaafter

32

daysand

ǫ

1

= 0 =⇒ R

04

= 1.3723, ǫ

1

= 0.4 =⇒ R

04

= 1.2114, ǫ

1

= 0.6 =⇒ R

04

=

1.0932, ǫ

1

= 0.95 =⇒ R

04

= 0.5074.

. . . 53

5.1 Diagram shows sensitivity analysis of

R

07

.

. . . 72

5.2 Diagramshowstheparasite-freeequilibrium(DFE)atliverstagewith

R

07

=

0.0063.

. . . 73

5.3 Diagram shows the parasite-free equilibrium (DFE) at blood stage

R

07

=

0.0063.

. . . 74

5.4 Diagramshowstheparasite-freeequilibrium(DFE)atmosquitostage

R

07

=

0.0063.

. . . 75

5.5 Diagram shows the parasite-present equilibrium point (EEP) at liver stage

R

07

= 4.7265.

. . . 76

5.6 Diagramshows theparasite-presentequilibriumpoint(EEP) atbloodstage

R

07

= 4.7265.

. . . 77

Listof gures xii

5.7 Diagram shows the parasite-present equilibrium point (EEP) at mosquito

stage

R

07

= 4.7265.

. . . 78

5.8 Showsthe ontourplotof

R

07

asafun tionofanaveragenumberofs hizonts

release from an infe ted liver ells that die naturally

(n

1

)

and the rate of

lossof s hizontsinside liver ells that are killedby ee tor ells

(k

tp

).

. . . 79

5.9 Shows the ontour plot of

R

07

as a fun tion of the natural death of an

infe tedliver ells

(µ

il

)

and naturaldeath of sus eptible midgut ells

(µ

mc

).

80

5.10 Shows the ontour plot of

R

07

asafun tionof the growth ratedue to

infe -tion of RBCs

(k

r

)

the rate of killingof merozoites by ee tor ells

(k

7

).

. . . 81

5.11 Diagram shows anappli ation of treatmentafter

30

days at liver stageand

ǫ

1

= 0 =⇒ R

06

= 4.7265, ǫ

1

= 0.3 =⇒ R

06

= 4.0980, ǫ

1

= 0.7 =⇒ R

06

=

2.9200, ǫ

1

= 0.99 =⇒ R

06

= 0.7491.

. . . 82 5.12 Diagramshowsanappli ationoftreatmentafter

30

daysatbloodstageand

ǫ

1

= 0 =⇒ R

06

= 4.7265, ǫ

1

= 0.3 =⇒ R

06

= 4.0980, ǫ

1

= 0.7 =⇒ R

06

=

2.9200, ǫ

1

= 0.99 =⇒ R

06

= 0.7491.

. . . 83 5.13 Diagramshows an appli ationof treatmentafter

30

days atmosquito stage

and

ǫ

1

= 0 =⇒ R

06

= 4.7265, ǫ

1

= 0.3 =⇒ R

06

= 4.0980, ǫ

1

= 0.7 =⇒

List of Tables

1.1 The table that shows the drug resistan e for anti-malarialdrug . . . 5

2.1 The table shows plasmodiumspe ies and hara teristi s. . . 10

3.1 The Routh-Hurwitztable showing the hara teristi equation . . . 18

4.1 The table with the variables,des riptions and units. . . 25

4.2 The table that shows parameters and their des riptions. . . 26

4.3 The table that shows the parameter values of the model. . . 37

4.4 The table that shows the parameter values of the model. . . 40

5.1 The table des ribing the variables and units of variables . . . 58

5.2 The table des ribing the parameters and units of parameters . . . 59

5.3 The table des ribing the parameters and units of parameters . . . 60

5.4 The table with the parameters, values and sour e . . . 70

5.5 The table with the parameters, values and sour e . . . 71

A.1 The table that shows the initialvariables that used inFIG. (4.3,4.4) . . . 88

Listof gures xiv

A.3 The table with parametersvalues used inFIG. (4.4) . . . 89

A.4 The table that shows the initialvariables that used inFIG. (4.5,4.6) . . . 89

A.5 The table with parametersvalues used inFIG. (4.5,4.6) . . . 89

A.6 The table that shows the initialvariables that used inFIG. (4.7) . . . 89

A.7 The table with parametersvalues used inFIG. (4.7) . . . 89

A.8 The table that shows the initialvariables that used inFIG. (4.8,4.9,4.10) . 89 A.9 The table with parametersvalues used inFIG. (4.8,4.9,4.10) . . . 90

A.10ThetablethatshowstheinitialvariablesthatusedinFIG.(4.11,4.12,4.13,4.14) 90 A.11The table with parametersvalues used inFIG. (4.11,4.12, 4.13,4.14) . . . 90

A.12The table that shows the initialvalues used in FIG.(4.15,4.16) . . . 90

A.13The table with parametersand values used inFIG. (4.15,4.16) . . . 90

A.14The table that shows the initialvalues used in FIG.(5.2,5.3,5.4) . . . 90

A.15The table with parametersand values used inFIG. (5.2,5.3,5.4) . . . 91

A.16The table that shows the initialvalues used in FIG.(5.5,5.6,5.7) . . . 91

A.17The tablewith parametersand values used inFIG.(5.8,5.9, 5.10,5.5,5.6,5.7, 5.11,5.12, 5.13) . . . 91

Chapter 1

Introdu tion

1.1 Ba kground

Malaria is a mosquito borne infe tious disease aused by protozoan of genus

plasmod-ium [73℄. The four spe ies that an infe t humans are; Plasmodium fal iparum, whi h

auses severe disease and possibly death [41, 92℄ if not diagnosed and treated promptly,

the other three plasmodium vivax, plasmodium ovale and plasmodium malariae

gener-ally ause milderdisease that is rarely fatal. [50℄. The parasitewas dis overed in

1880

by

CharlesLaveran[45℄,whowasworkinginthemilitaryhospitalinConstantine,Algeria. He

observedthe parasites inabloodsmeartaken fromapatientwhohad justdiedofmalaria.

ButLaveranlinkedthe auseofmalariawiththemonkeys[8℄. In

1902

SirRonaldRoss

dis- overed the malaria parasitein the gastrointestinal tra t of the anopheles mosquito. This

ledto the realizationthat malariawas transmitted by the Anopheles mosquito [98℄. This

study [98℄ laid the foundation for ombating the disease su h as the use of the pesti ide

Di hloro-Diphenyl -Tri hloroethane(DDT)for the ontrolofmosquitoesduringworld war

Chapter 1. Introdu tion 2

FIG.1.1. A diagramshowing malariaendemi area inAfri a.

Malaria remains a burden in terms of morbidity and mortality for developing ountries

(FIG. 1.1) with tropi al and subtropi al limates. Malaria is prevalent in these regions

be auseofheavy rainfall,warm onsistenttemperaturesandhighhumidity onditionsthat

are ondu ive to the development of the larvae. Temperature determines ve tor survival,

Chapter 1. Introdu tion 3

breeding [23,77℄.

Itisestimated thathalfof theworld's population,over

3.3

billionlivesinmalariaendemi

areas. Thereare about

300

to

500

million ases of lini almalariareported[24,67,80,96℄,

resulting in

1.5

to

2.7

million deaths annually. sub-Sahara Afri a is the region with the

highestinfe tionrate [96℄. Inthis regionalone,the diseasekillsatleast onemillionpeople

ea h year and is responsible for as many as half of the deaths in Afri an hildren under

the age of 5[96℄, and a ountsfor

20%

of all hildhooddeathsglobally. Malariade reases

the grossdomesti produ tbyasmu has

1.3%

in ountries withhighdiseaserate [70,96℄.

This ontributesto poverty and underdevelopment in most sub-Sahara ountries.

TheMalariaparasites areintrodu edintothehumanbloodstreamafterabitebyafemale

anopheles mosquito. Parasitesinthe formofsporozoites, enter the liverwherethey divide

several times before maturing into s hizonts whi h rupture and release merozoites whi h

ompleting the initialparasiterepli ation in the liver(exo-erythro yti s hizogony).

Dur-ing this initial stage, two spe ies, namely, plasmodium vivax and plasmodium ovale an

remaindormant(hypnozoites) inthe liverand ause repla eby invading thebloodstream

weeks, or even years later. After this initialrepli ation in the liver, the merozoites enter

the blood stage where they undergo asexual multipli ation in the erythro ytes

(erythro- yti s hizogony). Mature merozoites in the blood stream are apableof invading the red

blood ell. At this point the symptoms of disease will start to manifest, in the form of

fever,heada he, vomiting, hills,weaknessandsweating. Thesesymptomsareintermittent

depending on the immunity of the host. Some merozoites dierentiate and develop into

sexual forms of the parasite, alled male and female gameto ytes, that ir ulate in the

bloodstream [2, 11,41,45, 101℄.

When a mosquito bites aninfe ted human, it ingests the gameto ytes whi h initiate

par-asite multipli ation repli ation in the mosquito known as the sporogoni y le. In the

mosquito's stoma h,male and femalegametes fuse toform diploidzygotes whi h develop

intoookinetesthat invade the mosquitomidgut walland formoo ysts. Theoo ysts grow,

rupture, and release thousands of sporozoites. Sporozoites invade the mosquito salivary

glandsfromwherethey an beinje ted intohumanhosts to ontinue the infe tionpro ess

Chapter 1. Introdu tion 4

low immunity to malaria like young hildren, pregnant woman and visitors that travel to

tropi alareas from malariafree areas. Malaria an be prevented by preventing bites from

infe tedmosquitoes. This anbea hievedbywearinglong lothesthat overasmu hofthe

skinaspossible,treatingexposed partsofthebodywithinse trepellent,using

inse ti ide-impregnatedbed netswhilesleepingand sprayingindoorswithinse ti idearoundsleeping

areas. People visiting malaria endemi areas are advised to take anti-malarial drugs

be-fore entering these area. It is advisable for all residents of malaria endemi areas to be

diagnosed for malariaroutinely and be treatedif they test positive.

Despite the intense resear h and number of lini al trial, urrently there is no ee tive

va ine [23, 64℄and therehas been very littlesu ess in produ ingsu h va ines [64℄.

A urate diagnosisofmalariais anintegral partof treatmentof malariapatientsand

pre-ventionof furtherspreadof malariainthe ommunity. Treatingmalariadepends onmany

fa tors in ludingdiseaseseverity, the spe ies of malariaparasite ausing the infe tionand

the part of the world in whi h the infe tion was a quired. The use of a simple,

inexpen-sive and rapid diagnosis tests for malaria may be of in reasing importan e as ountries

inAfri a shiftfromlow- ostanti-malarialtomore expensive drugsArtemisin-based

Com-bination Therapy (ACT). ACT is the most ee tive strategy for plasmodium fal iparum

infe tion re ommended by WHO inthe fa eof wide spread drug resistan e [63, 96℄.

Parasites have be ome resistan e due to usual anti-malarial drug like hloroquine whi h

wasthe drugof hoi e totreatmalariaforde ades followingWorld WarII;itwasstopped

after parasites be ame resistant to it. Also the mosquito has be ame resistant to most

inse ti ide [6, 54℄, DDT whi h was a very ee tive ve tor ontrol pesti ide was stopped

be ause the mosquito be ame resistant to it [63℄. Malaria an be managed with proper

diagnosisand prompttreatment. Early diagnosisand prompt treatmentare the prin iple

te hni al omponents of the global strategy to ontrol malaria and is highly dependent

on the e a y, safety, availability, aordability and a eptability of anti-malarial drugs

[91℄. An ee tive anti-malarial drug not onlyredu es mortality and morbidity of malaria

but also redu es the risk of drug resistan e. Due to drug resistan e of parasites towards,

Chapter 1. Introdu tion 5

TABLE. 1.1. The table that shows the drug resistan e for anti-malarialdrug

Anti-malarialdrug Pla es that shows drug resistan e

Chloroquine Plasmodium fal iparumspe ies allareas

of the world ex ept the following:

North Afri a; the Middle East

(though ases have been reported

in Oman,Yemenand Iran);Haiti; Domini an

Republi ; ruralareas of Mexi o;

and CentralAmeri a,north and west

of the Panama anal[86℄.

Fansidar South East Asia;the Indian

sub- ontinent;the Amazonbasin;

many ountries inAfri a south

of the Sahara;and O eania[90℄.

Meoquine South East Asia espe iallyin Thailand;

parts of Afri a and South Ameri a;

the Middle East; and O eania[90℄.

Quinine South East Asia;parts of Afri a;

Brazil; and O eania[90℄.

Halofantrine Thailand and shows ross resistan e

with meoquine, fansidar and

sulfadoxine-pyrimethamine [90℄.

1.2 Statement of the problem

In-host mathemati almodels are important and ne essary to enhan e our understanding

of the dynami s of the Malaria parasites [58℄. Su h models an also be used to give an

insight into the ee tiveness of treatment drug and other intervention strategies. In this

study,weinvestigatethedynami softhemalariaparasiteduringthe redblood y le,then

extend themodel tolookatallthe stagesof the malarialife y le, namely;liverstage,red

blood ell stage and mosquito stage. These models, whi h also in lude the ee tor ells

willbe used to addressthe followingquestions:

(i) What repli ative hara teristi s oer the parasite opportunities to evade the host

immune system?

ex-Chapter 1. Introdu tion 6

(iii) What are the ee t of therapy onthe prognosis of the malariadisease?

(iv) What are these models' ontributiontothe publi health?

1.3 Obje tives of the study

Malaria remains one of the world's worst problem. It is shown that more people are

lini ally ill with malaria than any other disease [96℄. A ording to WHO estimates, in

2008 alone, there o urred

190 − 311

million lini al ases of malaria [11, 96℄. Hopesthat

malaria might be eradi ated have proved impossible to realize. In many tropi al areas,

the threat of epidemi malaria is in reasing and the ontrol measures are be oming less

ee tive. For this malaria asso iated burden to be redu ed, we ondu t a study whose

obje tives are summarizedas follows;

ˆ To understand biologi al pro esses that enable the parasites to evade the immune

response.

ˆ Togaininginsightsintothe dynami sbetween themalariaparasitesandtheimmune

system

ˆ To investigatethe ee t of anti-malarialtherapy onthe prognosis of the disease.

ˆ To ontribution poli yre ommendations onmanagementof malaria.

1.4 Organization of the work

Thiswork isorganizedasfollows: Chapter2reviews variousstudiesabout malaria.

Chap-ter 3 provides mathemati al and numeri al tools ne essary for this study. Chapter 4

presents the model with the following lasses; sus eptible RBCs, latent RBCs, a tive

in-fe tedRBCs,intra ellularparasites,extra ellularparasitesandee tor ells. Alsopresents

the ee ts of treatment. The modelis analysed mathemati ally and numeri ally, and the

Chapter 1. Introdu tion 7

(LVCs), infe tedliver ells (LVCs), sporozoitesand s hizonts inside the liver ells (LVCs),

(ii) the blood stage whi h onsist of; sus eptible red blood ells (RBCs), infe ted RBCs,

merozoites,trophozoites,ands hizontsinsidetheredblood ells(RBCs),(iii)themosquito

stage whi h has;sus eptible midgut ells (MGCs),infe ted midgut ells (MGCs),

gameto- ytes. The model alsohas an ee tor ell lass. The mathemati al and numeri al

simula-tionsaredone,theresultsarealsodis ussed. Chapter 6 on ludeswhathas beendis ussed

in the resultsand suggests limitations,re ommendations,and possibilities offuture work.

Chapter 2

Literature review

Itisoverone hundred years sin emalariawasre ognizedasadiseaseinhumans. Initially,

malaria was known as mal

′

aria implying that the disease was aused by bad or spoiled

air. The fever lini alsymptoms were relatedtoswaps and lowlyingwater. In

1880,

Lav-eran dis overed parasites insidered blood ells of asi k personand mistakenlyrelatedthe

disease tomonkeys following hisearlier study whi h identied the same parasitesinside a

monkey's red blood ells [8℄.

The study by Ross (see [8℄and the referen es therein) onmalariahas be omethe basis of

epidemiologi alstudies of malaria in luding ve tor spread, treatment et . and these have

led toimmunologi alstudies of the disease [8, 41℄. To demonstrate the seriousness of the

malaria epidermi , we have analysed available re ords for the trend. Re ords regarding

malaria lini al ases and mortalitybefore theyear

1950

are not available. The treatment

drug hloroquine was dis overed in

1950

and at almost the same time the pesti ide DDT

was dis overed. From available re ords depi ted by FIG. 2.1, it is evident that malaria

lini al ases were in de line between

1950

and

1975

due to ee tive ve tor ontrol

pro-gramsand the ee tivenessof the treatmentdrug hloroquine [66, 76℄. Overalong period

ofadministrationof hloroquine, the parasitesdeveloped resistan e tothedrug [6,54℄and

urrently, hloroquine is not re ommended for treatment of malaria in sub-Sahara Afri a

[66℄. The trend for malaria lini al ases has been on the in rease sin e

1975

(FIG. 2.1).

There appears to be a link between malaria and HIV/AIDS (FIG. 2.1). During the late

Chapter 2. Literature review 9

FIG.2.1. A diagramshowing ampliedrelationship between HIVand malaria

HIV/AIDS exploded malaria lini al ases alsostarted toin rease rapidly.

We want to exer ise aution about the link mentioned above and to note that the rapid

in rease inmalaria lini al ases duringthe late

1980

's ould alsoberelated to other

fa -tors su hasthe ollapseofthe ve tor ontrolprograms followingthe deteriorationinmost

e onomies insub-Sahara Afri a,but itmay alsobedue toweakened immuneresponses in

patients o-infe ted with HIV and malaria.

Therearefourspe iesofplasmodiumthatareknowntoinfe thumansnamely,Plasmodium

fal iparum, Plasmodium malariae, Plasmodium vivax, and Plasmodium ovale. The

dis-tribution of the various malaria parasitesare indi ated in the [TABLE 2.1℄. Plasmodium

fal iparum, whi h auses more deaths in humans, is found mainly in tropi al and

sub-tropi al areas of the world whi h in lude sub-Sahara Afri a and most of the poor regions

ofAsia. Theseparasitesdevelop througha y ledepi tedin(FIG.2.2)anddis ussed in

Chapter 2. Literature review 10

TABLE. 2.1. The table shows plasmodium spe ies and hara teristi s.

Spe ies Globaldistribution Re ur Type of RBCs Infe tion

P. fal iparum Tropi aland

sub-tropi alworldwide Re rudes en e All Severe anaemia

P. malariae Worldwide Re rudes en e Older RBCs Milder disease

P. ovale Afri a Relapse Young RBCs Normalinfe tion

P. vivax Asia,Latin Ameri a

some part ofAfri a Relapse Young RBCs Normalinfe tion

FIG.2.2. Malaria life y les, opied fromParasite image library [10℄

Chapter 2. Literature review 11

eradi ated, it is important to understand the fa tors that inuen e malaria pathogenesis

[1, 31℄. It is evident from (FIG.2.2) that the important questions for the mathemati al

study of malaria are both epidemiologi al and immunologi al and span several levels of

biologi alorganization.

Knowingthefa tors involved inpathogenesis,however, isonlythe rststep towards

quan-tifying how ea h fa tor inuen es the result [51℄. As a sequel to the study by Anderson

et al. [1℄, several studies Hetzel and Anderson [31℄, Antia et al. [2℄, et ., have studied

the within-host ellulardynami s of bloodstage malaria. Hetzel and Anderson [31℄

inves-tigated the properties of a mathemati al modelof blood stage infe tion of malaria. The

analysis [31℄, done in the absen e of the host immune response to demonstrate the

rela-tionship between host and parasites parameters, led to the determination of parameters

ne essary for the su essful invasion and persisten e of the parasites. The parameters in

Hetzel and Anderson [31℄ are used in this study as a rst approximation in our model to

understand the role of treatment and innateimmune response to malariapathogenesis.

A more re ent study by Antia et al. [2℄ has onsidered a ute malaria infe tions with a

viewtodeterminethe dynami sof parasiteandanaemiaduringa ute primarymalaria

in-fe tions andwhy somestrains ofmalariarea hhigherdensitiesand ausegreateranaemia

thanothers. Whilemoststudiesagreethat spe i immunitydoesnotplayamajorrolein

the initialdynami s of pathogenesis, there is onsiderable ontroversy over whi h fa tors

drive the dynami s shortly afterinfe tion [1,30, 31,55℄.

Despite these numerous studies on malaria pathogenesis, the relative signi an e of

dif-ferent fa tors inuen ing malaria pathogenesis ited in [1, 2, 31℄, the development of the

disease is far from lear [60℄. So far, many studies fo us on the innate immune response

[31, 34℄ to understand the RBC-parasites intera tion and a few on adaptive immune

re-sponse. However the fa t that resear hers have su essfully ategorized the immune

re-sponsesdoesnotimplyunderstandingofthesepro esses. Forexample,StevensonandRiley

[82℄ have dis ussed how several adaptive immune response omponents like ma rophages

andnatural killer ells areinvolved intheinnateimmuneresponse and on luded thatthe

intera tionof those ellsremainsspe ulativeand oni tingexperimentaldatahasopened

un-Chapter 2. Literature review 12

Several studies onthe innate immune response to malariainfe tion have been formulated

[13, 65,84, 89℄. A study by Su et. al [84℄on the syn hronizationof parasiterepli ation in

dierentred blood ells, onsiders an age-stru turalhuman malariainfe tionof red blood

ells. The numeri alsimulationresultsinthis study show thatsyn hronization with

regu-larperiodi os illationo urs whenthe repli ation rates in rease. Amore re ent study by

NigerandGumel[65℄hasinvestigatedtheinnateimmuneresponsetomalariainfe tionand

the ee t of imperfe tva ines. The simulationresults[65℄ showthat ava ine e a yof

atleast

87%

isne essary toeliminateInfe ted Red Blood Cells (IRBCs)in vivo.

The study by M Queen and M Kenzie [55℄ onsiders the sus eptibility of red blood ells

and the dynami s of malariainfe tion. The authors [55℄ assume a predator-prey type

re-lationship between a population of repli ating parasites and a replenishing populationof

red blood ells. Thestudy exploresthe hypothesis thatsomemalaria-parasitespe iesthat

infe t humans su h as Plasmodium malariae and Plasmodium vivax have preferen e for

parti ularage lassesofred blood ells. Ourstudy onsiderstheinfe tionofredblood ells

by Plasmodium fal iparum whi h has dierent infe tion hara teristi s [34, 55℄. Wewant

to investigate whether Plasmodium fal iparum too has a tenden y for age sele tion like

the other malaria spe ies, noting that the dieren e in infe tion hara teristi s between

Plasmodiumfal iparumandthe spe ies onsideredbyM Queen andM Kenzie [55℄o urs

primarily inthe range of parameters used [34℄.

Areview byEngwerda andGood[23℄ onsideredtheintera tion betweenmalariaparasites

and the host immune system and revealed the potential for designing and implementing

new va ineand drugprograms through understanding of ell-immune, ell-parasite

inter-a tion. This study [23℄ in luded the adaptiveresponse whi his not part of this study but

providesinsight intothe ell-parasite dynami s whi h has guided our study.

A review by Mideo et al. [58℄ re ommended the use of mathemati almodels as a tool for

rening knowledge of within-host pro esses and has suggested why under ertain

ir um-stan es mathemati al models may be better than experimentation. Generally, however,

Mideo et al. [58℄re ommends the use of both approa hes sin e together these approa hes

possessthe potentialforinformingthedesignofinterventionand healthpoli yfor

Chapter 2. Literature review 13

stages asshown in (FIG.2.2). This ompli ated repli ation y le has impli ations

regard-ingeradi ationofmalariawhi histooexpensiveandprobablyunrealisti forpoorresour e

sub-Sahara Afri an ountries. A treatment or va ination strategy suggested by

mathe-mati almodels may ontribute to utting ost of intervention programs.

Thisstudy willattempttoquantitativelypredi tthe patternof pathogenesisasafun tion

someunderlyingwithin-hostregulatoryfa ts. However,asthebiologyofmalaria

pathogen-esis isvery omplexandinvolvesmanyfa tors, thisstudy willin ludeonlythosebiologi al

pro ess that we believe an plausibly explain the dynami of malariapathogenesis.

M Kenzie and Bossert [34, 53, 55℄ modelled malaria pathogenesis using a system of

ou-pleddierentialequationsinvolvingonlyuninfe ted RBCs,infe tedRBCsand merozoites.

Thesestudiesomittedthe detailedbiologysu hashowthe merozoitesare repli ated

asex-ually and on entrated onthe basi infe tion dynami s that lead to lini al malaria. The

aim in those studies was to understand pathogenesis and not how the immune system

responds to infe tion. However, the study by Hoshen et al. [34℄ gives useful insights and

on lusionsthathaveguidedourstudy. Thequestiononeaskiswhetherasimpliedmodel

su hasHoshenet al. [34℄ an yieldreliableinformationtoguideplanningandpoli y. F

ur-thermore, one wonders whether these simple models an reliably estimate the severity of

the disease?

Somelargermodels [61℄havein luded morebiologi alpro ess su hasinnateandadaptive

immuneresponses. However, su hstudieshavela ked lini allydeterminedparameter

val-ues to alibrate and validate their models. The obje tive behind su h large models is to

determine parasite repli ation me hanisms but in the absen e of biologi ally determined

laws it has proved di ult to justify results from su h models. It is known that when

merozoites infe t red blood ells, to initiate the asexual repli ation of the merozoite

pop-ulation[55, 61℄, not allinfe ted red blood ells ontribute tothe populationof merozoites

sin esomeofthemareapoptosedbythenaturalkiller ells. Adjustmenttoree tthisfa t

has been done inan adho manner[40℄. The dieren es ina ounting for the repli ation

laws for malaria, TB and HIV [25, 55℄ are indi ative of the absen e of lear biologi al

Chapter 2. Literature review 14

denition of the parasite repli ation lawwhi hwe admit has no biologi albasis.

These omplexmathemati almodelsonmalariapathogenesishave onsideredthe

develop-mentof erebralmalariain hildrenandadulttravelerslivinginnonendemi malariaareas

and have on luded that severe malariais animmune-mediateddisease [4℄. This study [4℄

onsidered the role of innate and adaptive immune responses in terms of (i) prote tion

from lini almalariaand(ii)theirpotentialrole inimmunopathologyandthesubsequen e

development of lini al immunity. Another study [3℄ has determined the potential

on-tribution of innate immune responses to the early pro-inammatory ytokine response to

Plasmodium fal iparummalaria. The study examined the kineti s and ellular sour esof

interferon-gamma produ tion in response to infe tion of red blood ells. The study

on- ludes that earlyinterferon-gammaresponse ouldredu e red blood ells infe tion.

Thereis onsiderable ontroversyoverwhi hfa torsdrivethemalariapathogenesisshortly

after infe tion [1, 30, 31, 55℄. Some of the assumptions made to explain the dieren es

in the initial dynami s of malaria strains in lude virulen e evolution [2, 14, 27, 49℄, (ii)

red blood ellage spe i infe tion strategy [55℄ and (iii) innate orearly spe i immune

responses to regulate the initialdynami s of infe tion and anaemia [21, 29℄. Antia et al.

[2℄,however, havegiventworeasonswhyitisdi ulttoas ertainthe ontributionofthese

fa tors to the dynami s of a ute infe tion namely (i)limited data on the dynami s of the

parasiteandlossofRBCsfollowinginfe tionofhumanswithhumanmalariaparasitesand

(ii)the dynami sof the infe tion ould involve many intera ting populations.

Several studieshave investigated the potentialforava ineasthe best strategyto ombat

malaria targeting the liver stage for a va ine [23, 33, 59℄. A ording to [23℄, this stage

poses many obsta les to anti-infe tion va ines and drugs. These in lude: (i) the liver

stage malaria parasiteshavedistin t metabolismwhi h helps them toevade anti-malarial

drugs (ii)Plasmodiummalaria parasites an lie dormant inthe liverand relapseto blood

infe tionaftermonths oreven years. A ordingtoMorrowandMoss[63℄,liverstage

para-sites annotbetargeted by anyli enseddrugex eptprimaquinewhi hisfataltopregnant

women and diabeti individuals.

OurhypothesisinthisstudyfollowsfromtheobservationmadebyM QueenandM Kenzie

Chapter 2. Literature review 15

stru tured model, we have manipulated the natural death term in the infe ted red blood

ell population to a hieve our goal. It is suggested in [39, 104℄ that CD

8

ells may not

fun tion optimally in individuals suering from hroni illnesses, we have investigate this

s enarioanda ordinglyhavemadeare ommendationregardingthe treatmentofmalaria

for individualssuering from hroni infe tions.

Despite alargearea inresear h onmalariapathogenesis(within-hostme hanismsthrough

whi h plasmodium parasite auses disease) [5, 23, 55, 58℄ many questions remain

unan-swered. Issues inpathogenesisneedto beexploredtodevelop better treatment [58,60℄. It

isknownthatmostofthedrugsa t bestagainstrepli atingpathogensin ombinationwith

ee tive immunologi alresponses [5℄. There is a need also to better understand ee tor

Chapter 3

Mathemati al tools

3.1 The denition and omputation of

R

0

Denition 3.1.1 The basi reprodu tion number (

R

0

), sometimes alled the basi repro-du tive rate or the basi reprodu tive ratio of an infe tion is the average number of

se -ondary ases aused by an infe ted individual introdu ed in a ompletely sus eptible

popu-lation [20℄.

For the ase of a single infe ted ompartment,

R

0

is simply the produ t of the infe tion

rate and the mean duration of the infe tion [93℄. But for models with several infe ted

ompartments, this denition is not su ient. That is a more sophisti ated te hnique is

required asreviewed below. Inthis thesis, weonly give abriefoverview ofthe al ulation

of the basi reprodu tion number

R

0

, using the next generation method dis ussed by

Diekmann [20℄ and van den Driess he et. al [93℄. Let

x = (x

1

, . . . , x

n

)

t

with

x

i

≥ 0

, be

the numberof individuals inea h ompartment

(i = 1, . . . , n).

Wesort the ompartments

so that the rst

m

ompartments orrespond to infe ted individuals and then, dene the

parasitefree equilibria as

x

0

= {x ≥ 0 | x

i

= 0, i = 1, . . . , m}

.

For the omputation of

R

0

, it is important to distinguish new infe tions from all other

hanges in the population. As illustrated in gure (3.1), we let:

F

i

(x)

be the rate of

appearan e of new infe tions in ompartment

i

,

V

+

i

be the rate of transfer of individuals into ompartment

i

and

V

−

i

(x)

be the rate of transfer of individuals out of ompartment

Chapter 3. A mathemati altools 17 PSfrag repla ements

F

i

(x)

V

+

i

(x)

V

i

−

(x)

FIG. 3.1. Representation of

F

and

V

F

i

(x) , V

i

+

(x) ,

and

V

−

i

(x)

are dierentiable ontinuous fun tions. In general, we an express a system of dierentialequations of the form

˙x = f

i

(x)

in the formthat a ounts

for inowand outow(3.1) as:

˙x (t) = F

i

(x) − V

i

(x) ,

and

V

i

(x)

= V

−

i

(x) − V

i

+

(x),

for

i = 1, ..., n.

Let the fun tions

F

i

(x) , V

+

i

(x) ,

and

V

−

i

(x)

satisfy the following onditions.

(A1) : If

x ≥ 0,

then

F

i

, V

+

i

, V

i

−

≥ 0.

(A2) : If

x

i

= 0,

then

V

−

i

= 0

. In parti ular, if

x ∈ x

0

, then

V

−

i

= 0,

for

i = 1, ...., m.

(A3) : If

i > m, F

i

= 0.

(A4) : If

x ∈ x

0

,

then

F

i

(x) = 0,

and

V

+

i

(x) = 0.

(A5) : If

f (x) = 0

,then theeigenvalues of

Df (x

0

)

havenegativereal parts and

x

0

isthe

parasitefree equilibrium.

The onditions listed above allow us to partition the matrix

DF (x)

as shown in Remark

1.

Remark 1

DF (x

0

)

is the derivative

 ∂f

i

∂x

j



evaluated at DFE. The onditions

(A

1

− A

5

)

allow us to partition the matrix

DF(x

0

)

and

DV(x

0

)

as;

DF(x

0

) =



F 0

0 0



and

DV(x

0

) =



V

0

J

3

J

4



,

where

F

and

V

are

m × m

matri esdenedas

F =

 ∂F

i

∂x

j

(x

0

)



,

and

V =

 ∂V

i

∂x

j

(x

0

)



.

Sin e

F

is non-negative,

V

is a non-singular matrix and all eigenvalues of

J

4

have positive real parts.

Chapter 3. A mathemati altools 18

If

f (x)

satises

(A

1

− A

5

)

, thenthe reprodu tiveratio isdened as

R

0

= η F V

−1

, where

η

is the spe tral radius [93℄.

When

R

0

< 1,

the infe tion dies out, ex ept for a model whi h exhibits a ba kward

bifur ation. For

R

0

> 1,

the infe tion spreads in a population. Thus

R

0

is a threshold

parameter su h that when

R

0

< 1,

the population remains healthy be ause the disease

failstoestablishitself. The stabilityofthe disease freeequilibriumpoint(maintainingthe

onditionsforabsen e ofdiseaseinapopulation)isdis ussedbyDiekmann([20℄),and van

den Driess he [93℄. The stability of parasite free equilibrium point is stated in theorem

3.1.2.

Theorem 3.1.2 The parasite free equilibrium point is lo ally asymptoti ally stable for

R

0

< 1

and unstable for

R

0

> 1.

The proof of this theorem is given inseveral studies [16, 43, 44℄.

3.2 The Routh-Hurwitz riterion

Let

a

n

S

n

+ a

n−1

S

n−1

+ · · · + a

1

S

0

+ a

0

= 0,

(3.1)

be a hara teristi equation ofa given Ja obianmatrix. The Routh-Hurwitztable[36, 62℄

forthe hara teristi equation(3.1)ofdegree

n

anbedeterminedasillustratedinTABLE

3.1.

TABLE. 3.1. The Routh-Hurwitz tableshowing the hara teristi equation

S

n

_a

n

a

n−2

a

n−4

a

n−6

S

n−1

_a

n−1

a

n−3

a

n−5

a

n−7

S

n−2

_b

1

=

a

n−1

a

n−2

− a

n

a

n−3

a

n−1

b

2

=

a

n−1

a

n−4

− a

n

a

n−5

a

n−1

b

3

=

a

n−1

a

n−6

− a

n

a

n−7

a

n−1

S

n−3

b

1

a

n−3

− b

2

a

n−1

b

1

b

1

a

n−5

− b

3

a

n−1

b

1

. . . . . . . . . . . . . . .

S

0

Chapter 3. A mathemati altools 19

This table an be used as follows:

ˆ If there are sign hanges in the rst olumn,then the eigenvalues have positivereal

parts.

ˆ The number of sign hanges in the rst olumn is equal to the number of positive

real roots of the hara teristi equation.

ˆ If there exists a mixture of positive and negative signs, then the given system is

unstable.

ˆ Iftherearenosign hangesintherst olumn,thenalleigenvaluesareeitherpositive

or negative. If there are negativesigns only, then the eigenvalues have negative real

parts and the system is stable.

3.3 Sensitivity analysis

The parameter values and assumptions of any model are subje t to hanges and errors.

Sensitivity analysis isate hnique for establishingthe signi an eof aparameter and how

it impa ts the dynami s of the model. An independent variable will impa t a parti ular

dependent variable if the variable isa dierentiablefun tion of that parameter [12, 74℄.

Sensitivity analysis is avery useful tool for hara terizing the un ertainty asso iatedwith

a parameter with regard to model on lusions. Its importan eis part and par el of good

modelling pra ti e and requires a modeller to provide an evaluation of onden e in the

model results. Furthermore, it validates the relevan e of the inputs by determining the

output of the model[74, 99℄.

Un ertainty analysis may be used to asses the variability in the out ome variable that

is due to the un ertainty in estimating the input values. Sensitivity analysis an extend

un ertaintyanalysisbyidentifyingimportantparametersthatyieldreliablepredi tions[7℄.

Chapter 3. A mathemati altools 20

of

K − 1

parameters and then vary only the value of the

K

th

parameter over a spe ied

range. This sensitivity analysis design has the advantage that it is simple and qui k, but

suers from major disadvantages. That is only one parameter may be varied at a time,

onlyasmallregionofa

K−

dimensionparameterspa e anbeexploredandvaluesof

K −1

parameters have to be estimated [7℄.

Latin Hyper ubeSampling(LHS) isthe typeof stratied Monte Carlosamplingand may

beviewedasanextensionoftheLatinSquaresampling. InLHS,theun ertaintyestimation

forea hinputparameterismodelledbytreatingea hinputparameterasarandomvariable.

It is an extremely e ient samplingdesign be ause is used only on e in the analysis. An

inputve tor isgenerated forea h omputer simulationof thedeterministi modeland the

model is then run

N

times [7, 74℄.

A distribution fun tion for ea h of the out ome variable an be dire tly derived be ause

of the probabilitysele tionte hnique. LHSenablesthe results ofa deterministi modelto

be interpreted within a statisti al framework. The distribution may be hara terized by

simpledes riptivestatisti s. Sensitivity analysis maybethenbeperformed by al ulating

thepartialrank orrelation oe ients(PRCC)forea hinputparameterandea hout ome

variable [7,75℄.

The LHS/PRCC te hnique involvesseven steps [7℄:

(a) Dening the probability distribution fun tion for parameters and state variables. A

mathemati almodel ontains a ertainnumberof parameters and state variables,the

estimated values forall oronly asubset of these willbeun ertain.

(b) Cal ulatingthenumberofsimulations(N).TheLHSdesigninvolvessamplingwithout

repla ement. Therefore if only

k

draws are made (where

k

equals the number of

un ertain variables), the

k

th

draw would be predetermined. Hen e the lower limit to

the value of

N

(where N equals the number ofsimulations)should be atleast

(k + 1).

( ) Dividing the range of ea h of the

K

parameters into

N

equal probable intervals. The

rangeofea hparameterisdividedinto

N

non-overlapingequiprobableintervals(where

Chapter 3. A mathemati altools 21

(d) Creating the LHS table. The LHS design involves random samplingwithout

repla e-ment and every equiprobable intervalof ea h input variableis sampledon e.

(e) Sampling the values of the input parameters and performingthe

N

simulations. The

LHS table isused togenerate for example

100

by

30

input matrix.

(f) Analysing modelout omes of un ertainty analysis. The results of the simulationruns

ofthemodel onsistof

N

observationsofea hout omevariable. Distributionfun tions

ofea hout omevariable anbedire tlyderivedand hara terizedbysimpledes riptive

statisti s.

(g) Analysing the model out ome with respe t to the

N

observations of ea h out ome

variable whi h may be used to assess the sensitivity of the out ome variables and to

Chapter 4

A within host model of blood stage

malaria

4.1 Introdu tion

Malaria parasites are transmitted from a mosquito to a human host. Upon entering the

human host, extra ellular malaria sporozoites must rst take up residen e in the liver

before initiatingred blood ellinfe tion. In the liver, the sporozoites undergo spe ta ular

phenotypi hangesprior tomultipli ation[37, 69℄. Thesporozoites matureintos hizonts

whi h rapture and release merozoites [11℄, whi h are fully ompetent to infe t red blood

ells and instigate the pathologyasso iated with malaria[83℄.

Several studies onthe innate immune response to malariainfe tion have been formulated

[13, 65, 84, 89℄. A study by Su et. al [84℄ on the syn hronization of parasite repli ation

in dierent red blood ells, onsiders an age-stru tural human malaria infe tion of red

blood ells. The numeri al simulation results in that study showed that syn hronization

with regularperiodi os illationo urswhen the repli ationrates in rease. A morere ent

study by Niger and Gumel [65℄ has investigated the innate immune response to malaria

infe tion and the ee t of imperfe t va ines assuming a s enario of parasite life y les.

The simulation results [65℄ showed that a va ine e a y of at least

87%

is ne essary to

eliminateInfe ted Red BloodCells (IRBCs )invivo.

Chapter 4. A withinhost modelof bloodstage malaria 23

stage1: A shortstagewhenmerozoitesare released fromthe livertoinitiatethered blood

ell infe tion. It is estimated that a primary infusion of

10

4

to

10

5

parasites are released

into the blood[55℄. (ii)stage 2: The red bloodstage during whi hasexual multipli ation

of merozoites and infe tion of red blood ells by the merozoites o urs on urrently. Our

modeldoesnot in ludethe earlystagesof the exo-erythro yti y le whi hisknownto be

a reservoir for the erythro yti stage [59℄. Despite this, the results of this model an still

ontribute towards understanding the repli ative dynami s of the parasite and hen e the

development of lini almalaria.

In-host mathemati almodels are important and ne essary to enhan e our understanding

ofthe dynami sofMalaria pathogenesis[58℄. Su hmodels analsobeused togiveinsight

into the ee tiveness of drug treatment and other intervention strategies [65℄. In this

Chapter, we investigate the dynami s of the malaria parasite during the red blood y le.

Our model in ludesthe red blood ells, extra ellular parasites, intra ellular parasitesand

ee tor ells. Thismodeldiersfromthemodelsintheearlierstudies[5,34,55,65℄inthat

we introdu e a lass of intra ellular parasitesin the pathogenesispro ess. Webelieveit is

important to in lude this pro ess in the dynami s so that intervention strategies an be

targetedatdierentstagesoftherepli ationpro essasisthe ase inHIV/AIDStreatment

[38℄. Our study addresses the following questions: (i)what repli ative hara teristi soer

the parasite opportunities to evade the host immune system? (ii) A signi ant number

of individuals in sub-Sahara Afri a are o-infe ted with viral (for example HIV, simian

immunode ien yvirus(SIV)),ba terialandparasiti infe tionsotherthanmalaria[38,42,

88,104℄,itisimportanttoinvestigatehowsu hindividualsrespond toamalariainfe tion.

Studies by Kalia et el [39℄ and Zhang et el [104℄ have shown that in hosts suering from

hroni infe tions, CD

8

T- ellfun tions are ompromisedand are dysfun tional. To give

insightintowhat happens when CD

8

T- ell responseshave been alteredand impaired,we

shall onsider a hypotheti al situation where a host is infe ted with the malaria parasite

whi h would lear if CD

8

T- ell responses were normal. We shall then alter the CD

8

T- ell response parameters

m

and

k

tp

at time

t < t

c

,

where,

t

c

is the time of learan e of

the malaria infe tion, and investigate the prognosis of the malaria disease by nding the

riti alee tor ellkillingrate whi hmustbemaintained(orex eeded) toensurethatthe

Chapter 4. A withinhost modelof bloodstage malaria 24

prompttreatment are the prin iplete hni al omponents of the globalstrategy to ontrol

malaria. This strategy is highly dependent on the drug e a y. Ee tive anti-malarial

drugs not onlyredu e mortalityand morbidityof malariabut alsoredu e the risk ofdrug

resistan e of the parasites toward availableanti-malarialdrugs [90℄.

Our model in ludes the repli ation of the parasite within the infe ted red blood ells.

Whilethe pro ess of entry into the red blood ells by the merozoites has been extensively

studied, the repli ationpro ess ofthe parasitewithin aninfe ted red blood ellisnot well

understood. In this study, we have assumed a repli ation law for intra ellular parasite

similar to that of the tuber ulosis ba teria in ma rophages [25℄. This assumption may

not be an a urate representation of the malaria parasite repli ation law, however, we

have utilized the available data [18℄ to ensure that parasite repli ation in a red blood

ell produ es between

8

to

32

merozoites. Using this information, we have determined

the rate of bursting,

k

b

,

for infe ted red blood ells taking the ell's arrying apa ity

to be

32

. Furthermore, we have assumed that the release of the merozoites is through

burstingof the infe ted red blood ell asis the ase for ba teriain ma rophages [25℄. We

have made this assumption be ause very little is known about the a tual parasite release

me hanism involved. We believe, however, that this study will stimulate experimental

biologists into investigating the reprodu tive law for intra ellular malaria parasites and

the release me hanismof parasites frominfe ted red blood ells.

We also extend the model to investigate the e a y level required to lear the parasites.

We introdu e treatment with a drug of onstant e a y

ǫ

1

targeting the infe tion terms

[71℄.

4.2 Methodology

4.2.1 Model formulation

The system of equations des ribing the dynami s of in-host malaria is given below: The

model represents the human blood stage of the malaria disease alled the erythro yti

Chapter 4. A withinhost modelof bloodstage malaria 25

SUSCEPTIBLE

LATENT

ACTIVE

INTRACELLULAR

EXTRACELLULAR

RBCs

RBCs

PARASITES

EFFECTOR

RBCs

Interaction,

Surce of

RBCs

Growth of

Growth of

Intracellular parasite

Movement

effector cells

Source of

effector cells

PARASITES

Source extracellular

parasites

PSfragrepla ements

µ

rl

αkRP

e

(1 − α)kRP

e

γR

l

µ

ra

µ

r

k

∗

_n

∗

_RP

e

n

1

µ

ra

P

i

k

tp

NEP

e

k

∗

_n

∗

_RP

e

k

11

NR

a

G

µ

pe

k

b

R

a

G

mER

a

FIG. 4.1. A diagrammati representation of withinhost malaria model.

TABLE. 4.1. The table with the variables,des riptions and units.

Variables Des riptions Units

R

Sus eptiblered blood ell ell/ml

R

l

Latent red blood ell ell/ml

R

a

A tivated red blood ell ell/ml

P

i

Intra ellular parasites ell/ml

P

e

Extra ellularparasites ell/ml

E

Ee tor ell ell/ml

Red blood ells (RBCs)

˙

Chapter 4. A withinhost modelof bloodstage malaria 26

TABLE. 4.2. The tablethat shows parameters and their des riptions.

Parameter Des riptions Units

S

r

Constant sour esof red blood ell

cell/ml.day

µ

r

Natural death rates of sus eptible RBCs

day

−1

k

Infe tion rate

ml/cell.day

α

Proportion of RBCs

dimensionless

γ

Rate of a tivation

day

−1

µ

rl

Natural death rate of latentRBCs

day

−1

m

Rate of killing a tivated RBCs by ee tor ell

ml/cell.day

k

b

Rate of bursting

day

−1

µ

ra

Natural death rates of a tivated RBCs

day

−1

N

Number of parasites that ll the RBCs

dimensionless

k

pi

Rate of growth of intra ellularparasites

day

−1

k

11

Rate of loss due toburst of a tivated ell

day

−1

n

1

A threshold numberof intra ellular parasites released

as aresults of the natural death of ana tivated RBC

dimensionless

k

tp

Rate of loss of extra ellular parasitesthat are killed

by ee tor ells

ml/cell.day

n

∗

_k

∗

Threshold numberas aresults of gain due to infe tion

of sus eptible RBC by extra ellular parasites

ml/cell.day

µ

pe

Natural death rate of extra ellular parasites

day

−1

ω

e

Growth rate of ee tor ells

day

−1

r

e

Carrying apa ity of ee tor ell

cell/ml

S

pe

Sour e of extra ellular parasites

cell/ml.day

During thehumanbloodstage, thesporozoitesinje tedintothehumanhostby thefemale

anopheles mosquito enter the bloodstream and infe t sus eptible red blood ells (RBCs).

The dynami s of the sus eptible red blood ell (RBC) populationare given in (4.1). The

terms in this equation have the following meaning: The rst term represents a onstant

naturalsour eforthe redblood ellpopulation. These ondtermrepresentsnaturaldeath

of the sus eptible red blood ells at a onstant rate

µ

r

and the third term represents

infe tion of RBCs by extra ellular parasites(merozoites) at a onstant rate

k

. The newly

infe ted red blood ells may be ome latently infe ted with the malaria parasite, a state

whi h inhibits parasite repli ation, or the RBCs may be ome a tively infe ted, meaning

that parasiterepli ation persists inthem. The rate of hange for the latentlyinfe ted red

blood ellpopulation,

R

l

, is given by equation(4.2)

˙

Chapter 4. A withinhost modelof bloodstage malaria 27

Thetermsinthis equationhavethefollowingmeaning: Thersttermrepresentsa

propor-tion of infe ted RBCs that have be ome latentlyinfe ted, and the se ond term represents

lossesduetoa tivationata onstantrate

γ

andduetonaturaldeathata onstantrate

µ

rl

.

Thea tivelyinfe tedred blood ellpopulationevolvesa ording tothe followingequation

(4.3) below:

˙

R

a

= (1 − α)kRP

e

+ γR

l

− mER

a

− k

b

R

a



P

i

2

P

i

2

+ (NR

a

)

2



− µ

ra

R

a

.

(4.3)

The rst term in equation (4.3) represents the proportion of sus eptible red blood ells

thatbe omea tivelyinfe ted,these ondtermrepresentsgainduetoa tivationoflatently

infe ted red blood ells,

R

l

,

the third term represents the removal of a tivated infe ted

RBCs due to killing by ee tor ells. When the merozoites infe t red blood ells, they

start to repli ate within the infe ted red blood ells. This pro ess an go on until the

number of parasiteswithin the infe ted red blood ell rea hes

32

[55℄ ausing it to burst.

The fourth term measures an ee tive number of infe ted red blood ells that burst to

release intra ellular parasite. The burstingrate in this modelis assumed to bedependent

onthedensitiesofintra ellularparasitesandinfe tedredblood ells[25,28℄. Thisbursting

lawhasbeenusedforpathogenssu hasTB[25℄. Tothebestofourknowledgethishasnot

been used for malaria and is not supported by any literature. Not all infe ted red blood

ells bursttoreleaseparasite. There isanee tivenumberofinfe ted redblood ellsthat

burst torelease parasiteintothe bloodstream. The fa tor;

P

2

i

P

2

i

+ (NR

a

)

2

,

measures the proportion of infe ted red blood ells that burst to release parasite. We

have hosen this ratio so that the repli ation pro ess has an upper bound.

Thislaw hasrevealed orre t repli ative dynami s for the TB ba teria[25℄ and

isadopted inthis study. Thefthterma ountsfornaturaldeathofinfe tedred blood

ells at a onstant rate

µ

ra

.

Parasites

Chapter 4. A withinhost modelof bloodstage malaria 28

TBba teriainma rophages[25℄. Intheabsen e ofexperimentallyor lini allydetermined

growth lawfor the intra ellular malaria parasite, we assume a growth law of intra ellular

parasitesinside an infe tedred blood ell of the form.

˙

P

i

= k

pi

P

i



1 −

P

i

2

P

i

2

+ (NR

a

)

2



+ k

∗

n

∗

RP

e

− k

11

NR

a



P

i

2

P

i

2

+ (NR

a

)

2



−n

1

µ

ra

P

i

.

(4.4)

Inequation(4.4),thersttermrepresentsthegrowthofintra ellularparasites. These ond

term represents gain of

P

i

due to infe tion of sus eptible RBCs by extra ellular parasites

(merozoites),

P

e

,

at a threshold

n

∗

_k

∗

, the third term represents an ee tive number of

intra ellularparasiteslostduetoburstingofa tivatedRBCsandtheforthtermrepresents

lossof intra ellular parasites due tonaturaldeath of aninfe ted red blood ell,

R

a

,

where

n

1

denotes a threshold number of intra ellular parasites released. Upon bursting of an a tivelyinfe tedredblood ell,itreleasesthemerozoitesintothebloodstreamto ontinue

the parasite y le.

k

pi

P

i



1 −

P

2

i

P

2

i

+ (NR

a

)

2



= k

pi

P

i



(NR

a

)

2

P

2

i

+ (NR

a

)

2



.

This term has the following hara teristi s;

lim

R

a

→0

k

pi

P

i



(NR

a

)

2

P

2

i

+ (NR

a

)

2



= 0.

There isno growth of intra ellular parasites.

lim

R

a

→∞

k

pi

P

i







1



P

i

N R

a



2

+ 1







= k

pi

P

i

.

In this ase the intra ellular in reases exponentially. As

P

i

→ ∞

the number of bursting

infe ted red blood ells de reases and the loss of infe ted red blood ells may be

exponential. The author is not aware of the repli ation law for malaria parasite hen e,

the repli ation law hosen here is similar to that for infe tion of ma rophages by TB

ba teria [25℄. This is at best an approximation whi h needs further investigation. The