MODELLING THE EFFECTS OF MULTI-INTERVENTION ... - NOMA

MODELLING THE EFFECTS OF MULTI-INTERVENTIONCAMPAIGNS FOR THE MALARIA EPIDEMICIN MALAWIPeter Mpasho Mwamusaku MwamtobeM.Sc. (Mathematical Modelling) DissertationUniversity of Dar es SalaamSeptember, 2010

MODELLING THE EFFECTS OF MULTI-INTERVENTIONCAMPAIGNS FOR THE MALARIA EPIDEMICIN MALAWIByPeter Mpasho Mwamusaku MwamtobeA dissertation submitted in partial fulfillment of the requirements for the degree ofMaster of Science (Mathematical Modelling) of the University of Dar es SalaamUniversity of Dar es SalaamSeptember, 2010

iCERTIFICATIONThe undersigned certify that they have read and hereby recommend for acceptanceby the University of Dar es Salaam the dissertation entitled: Modelling the Effects ofMulti-intervention Campaigns for the Malaria Epidemic in Malawi, in partialfulfillment of the requirements for the degree of Master of Science (MathematicalModelling) of the University of Dar es Salaam.__________________________________Dr. NYIMVUA SHABANI(Supervisor)Date: ..................................Dr. SENELANI DOROTHY HOVE-MUSEKWA(Supervisor)Date: ..................................Prof. JEAN MICHEL TCHUENCHE(Supervisor)Date: ...................................

iiDECLARATIONANDCOPYRIGHTI, Peter Mpasho Mwamusaku Mwamtobe, declare that this dissertation is myown original work and that it has not been presented and will not be presented toany other University for a similar or any other degree award.Signature:This dissertation is copyright material protected under the Berne Convention, theCopyright Act 1999 and other international and national enactments, in that behalf,on intellectual property. It may not be reproduced by any means, in full or inpart, except for short extracts in fair dealings, for research or private study, criticalscholarly review or discourse with an acknowledgement, without the written permissionof the Director of Postgraduate Studies, on behalf of both the author and theUniversity of Dar es Salaam.

iiiACKNOWLEDGEMENTSI would like to express my sincere gratitude to my supervisors, Dr. Senelani D. Hove-Musekwa (National University of Science and Technology, Zimbabwe), Prof. JeanMichel Tchuenche and Dr. N. Shabani (University of Dar es Salaam) for their constantsupport, guidance, continuous encouragement and constructive ideas throughoutmy research work. I have learned so much from them about epidemiologicalmodelling and its application.I extend my thanks to the coordinators of the Norad Progamme for Masters Studies(NOMA), University of Dar es Salaam for their sponsorship that enabled me toundertake this study. Special thanks go to Dr. W. C. Mahera, the local coordinatorof the programme, who made all efforts to provide me with a very conducive studyenvironment. I also express my sincere appreciation to all staff members in the Departmentof Mathematics for their support and encouragement for the whole periodof my study.I would like to thank University of Malawi (The Malawi Polytechnic), Malawi, forproviding family support for the whole period of my study in Dar es Salaam, Tanzania.It was a great opportunity for me to meet different expertise in the field ofcoursework, research and other close related fields.Warmest thanks to my fellow postgraduate students in the Department of Mathematics,for their contribution and encouragement during the whole period of mystudy.Last but not least; I would like to express my utmost thanks to my parents, wife,relatives and children for their love during the whole period of my study. Theircontinuous support and encouragement is highly appreciated. The limit is the skyper their say.

ivDEDICATIONTo God The Almighty,My Parents: Gilbert and Gloria Mwamtobe,My Wife: Maria-Angela Mwamlima Mwamtobe,andChildren: Gibson, Blessings, Lausha, Wantwa and Lwitiko.

vABSTRACTWe develop a basic deterministic malaria model with two latent periods in the nonconstanthost-vector populations, and formulate the model with intervention strategiesby adding the protected and treated classes in order to assess the potential impactof protection and treatment strategies on the transmission dynamics of malaria.The models are analysed qualitatively to determine criteria for control of a malariaepidemic, and are used to compute the basic reproduction and effective reproductionnumbers necessary for country-wide control of malaria. The equilibria of models aredetermined. In addition to having a disease-free equilibrium, which is locally asymptoticallystable when the R 0 < 1, the basic malaria model exhibits the phenomenonof backward bifurcation where a stable disease-free equilibrium coexists with a stableendemic equilibrium for a certain range of associated reproduction number less thanone. Furthermore, the malaria model with intervention strategies indicates that itexhibits a forward bifurcation, in which in the absence of a low-level unstable equilibriumwhentheeffectivereproductionnumberRe < 1,astableequilibriumbifurcatingfrom the disease-free equilibrium when R e > 1, arise naturally when the disease doesnot invade when R e = 1. Numerical results indicate the effect of the two controls(protection and treatment) in lowering exposed and infected members of each of thepopulations. The results also highlights the effects of some model parameters, theinfection rate and biting rate. Among the interesting dynamical behaviours of thebasic malaria model, numerical simulations show a backward bifurcation which givesa challenge to the designing of the effective control measures.

viTABLE OF CONTENTSPageCertification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Declaration and Copy Right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iiiiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vviList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiCHAPTER ONE: INTRODUCTION 11.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 General Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Specific Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Significance of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4CHAPTER TWO: LITERATURE REVIEW 5CHAPTER THREE: THE BASIC MALARIA MODEL 10

viiPage3.1 The Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Invariant Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Positivity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Existence and Stability of Steady-state Solutions . . . . . . . . . . . . . . . . . . . . . . . 183.4.1 The Existence of the Trivial Equilibrium Point . . . . . . . . . . . . . . . . . . . . . . . . 193.4.2 Disease-free Equilibrium Point E 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.3 The Reproduction Number R 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.4 Local Stability of the Disease-free Equilibrium E 0 . . . . . . . . . . . . . . . . . . . . . . 253.4.5 The Endemic Equilibrium Point (E 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4.6 Local Stability of the Endemic Equilibrium E 1 . . . . . . . . . . . . . . . . . . . . . . . 303.4.7 Existence of Backward Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.8 Determination of the Backward Bifurcation and Local Stability of theEndemic Equilibrium Point E 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38CHAPTER FOUR: THE MALARIA MODEL WITH INTERVENTIONSTRATEGIES 404.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Formulation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Mathematical Analysis of the Model with Intervention Strategies . . . . . . . . . 444.3.1 Disease-free Equilibrium E 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

viiiPage4.3.2 The Effective Reproduction Number, R e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3.3 Existence and Stability of Endemic Equilibrium Point E 3 . . . . . . . . . . . . . . . . 514.3.4 Existence and Uniqueness of Endemic Equilibrium E3 . . . . . . . . . . . . . . . . . . 564.3.5 Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3.6 Global Stability of the Disease-free Equilibrium E 02 . . . . . . . . . . . . . . . . . . . . 644.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67CHAPTER FIVE: NUMERICAL SIMULATIONS 695.1 Table of Parameter Values of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2 Dynamics of Human Population State Variables of the Basic Model . . . . . . . . 715.3 Prevalence in the Basic Malaria Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4 Dynamics between Infected and Susceptible Human Populations . . . . . . . . . . 725.5 Dynamics of the Human Population Variables of the Model with InterventionStrategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.6 Phase Diagrams of Prevalence, Human and Mosquito Populations . . . . . . . . . . 795.7 Simulation of Protected and Treated Human Populations . . . . . . . . . . . . . . . . 81CHAPTER SIX: CONCLUSION AND RECOMMENDATION 846.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.2 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

ixPage6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87REFERENCES 88APPENDICES 95Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

xLIST OF TABLESPage1 State variables of the basic malaria model . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Parameters of the basic malaria model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 State variables of the model with intervention strategies . . . . . . . . . . . . . . . . 424 Parameters of the model with intervention strategies . . . . . . . . . . . . . . . . . . . 435 Parameter values of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xiLIST OF FIGURESPage1 The basic malaria flow-chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 The curve illustrates the bifurcation diagram where the green continuouscurve corresponds to the stable disease-free equilibrium DFE, the bluecontinuous curve corresponds to the stable endemic equibrium EE, andthe dash line depicts instability. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 383 The malaria model with interventions flowchart. . . . . . . . . . . . . . . . . . . . . . . . 434 Bifurcation diagram for the model system (51) obtained from numericalsimulations, which show that the disease-free and endemic equilibria exchangestability when Re = 1 for arbitrary set of parameter values. Theblue continuous curves depicts stable equilibria and dashed red curves depictsunstable equilibria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 Illustrates the changes in the four state variables of the basic malaria modelshowing the dynamics, with time, of (a) susceptible human individuals, (b)exposed human individuals, (c) infected human individuals and (d) showsthe dynamics of recovered human individuals . . . . . . . . . . . . . . . . . . . . . . . . 716 Represents changes of prevalence with time. . . . . . . . . . . . . . . . . . . . . . . . . . . 727 Illustrates the dynamics of infected human population against susceptiblehuman population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

xiiPage8 Represents the phase plane of infected human population versus susceptiblehuman population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 Illustrates the changes in the susceptible and protected human individuals of themalaria model with intervention strategies for Re = 0.0850 and R0 = 0.0927. . 7510 Shows the phase diagrams of the exposed human individuals of the malariamodel with intervention strategies for R e = 0.0850 and R 0 = 0.0927 . . . . . . . . 7611 Represents the phase diagrams of the infected human individuals of themalaria model with intervention strategies for R e = 0.0850 and R 0 = 0.0927 . . 7612 Illustrates the treated human individuals of the malaria model with interventionstrategies for R e = 0.0850 and R 0 = 0.0927 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713 Shows the phase potrait of the treated human individuals with time of themalaria model with intervention strategies for R e = 0.0850 and R 0 = 0.0927 . . 7814 Illustrates the phase diagram of recovered human individuals with time of themalaria model with intervention strategies for R e = 0.0850 and R 0 = 0.0927 . . 7815 Illustrates the dynamics of protected human population and exposed mosquitopopulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916 Represents the phase plane, infected human individuals versus susceptiblehuman individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8017 Illustrates the changes of prevalence with time as treated rate, σ, varies . . . . . 8018 Shows the impact of varying the infection rate, ν . . . . . . . . . . . . . . . . . . . . . . . 81

xiii19 Shows the graphs of the protected human individuals and treated humanindividuals with time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8220 Illustrates the phase diagram of the protected and treated human populationwith time as we vary the protection rate g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8221 Shows the phase diagram of the protected and treated human population withtime as we vary treated rate, σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

1CHAPTER ONEINTRODUCTION1.1 General Introduction.Malaria is an infectious disease mainly found in tropical areas such as Sub-SaharanAfrica, Central and South America, the Indian subcontinent, South East Asia andthe Pacific islands which are called malaria regions. It is a life-threatening diseasecaused by parasites called Plasmodium that are transmitted to people through thebites of infected mosquitoes (WHO, 2009). The female Anopheles type of mosquitogets infected when it bites someone carrying the malaria parasite. There are fourdifferent types of Plasmodium parasites: Plasmodium falciparum is the only parasitewhich causes malignant malaria. It causes symptoms straight away and can bemild or severe. Secondly, Plasmodium vivax causes benign malaria with less severesymptoms. The vector can stay in the liver for up to three years and can lead to arelapse. Thirdly, Plasmodium malariae also causes benign malaria and is relativelyrare. Lastly, Plasmodium ovale also causes benign malaria and can stay in the bloodand liver for many years without causing symptoms. Plasmodium falciparum isresponsible for about three-quarters of reported malaria cases. Most of the othercases of malaria are caused by Plasmodium vivax with just a few caused by theother two species (Lalloo et al, 2007). It is possible to get infected with more thanone type of Plasmodium parasite. Each parasite causes a slightly different type ofillness. This study will be based mainly on the malignant malaria which is fatal inMalawi.The common first symptoms of malaria are similar to the flu. The patient may have:a headache, aching muscles, tummy ache, and weakness or lack of energy. A day orso later, the body temperature may rise (up to 40 degrees Celsius) and the patientmay have: a fever, shivers, mild chills, severe headache, vomiting, diarrhoea, andloss of appetite (Bupa, 2009). However, it takes at least six days for symptoms to

2appear. The time it takes symptoms to appear can vary with the type of parasitethat the mosquito was carrying.If the person gets infected with Plasmodium falciparum, malaria can progress tomore severe form called complicated malaria. The following symptoms may appear:low blood sugar levels, severe anaemia, jaundice, fluid on one ′ s lungs (pulmonaryoedema), acute respiratory distress syndrome, kidney failure, spontaneous bleeding,and state of shock (circulatory collapse), fits (convulsions), paralysis and coma. Severemalaria can affect the patients brain and central nervous system and can befatal (Bupa, 2009; WHO 2009).Malaria transmission rates can differ depending on local factors such as rainfallpatterns (mosquitoes breed in wet conditions), the proximity of mosquito breedingsites to people, and types of mosquito species in the area. Some regions have a fairlyconstant number of cases throughout the year-these countries are termed ”malariaendemic”. In other areas, there are malaria seasons usually coinciding with therainy season. Large and devastating epidemics can occur when the mosquito-borneparasite is introduced into areas where people have had little prior contact withthe infecting parasite and have little or no immunity to malaria, or when peoplewith low immunity move into areas where malaria is endemic. These epidemics canbe triggered by wet weather conditions and further aggravated by floods or masspopulation movements driven by conflict. Travelers from malaria-free regions, withlittle or no immunity, who go to areas with high disease prevalence and non-immunepregnant women, are very vulnerable and are at high risk of being infected withmalaria. The illness can result in high rates of miscarriage and causes over 10% ofmaternal deaths (soaring to a 50% death rate in cases of severe disease) annually;semi-immune pregnant women risk severe anaemia and impaired fetal growth evenif they show no signs of acute disease (WHO, 2009).Early treatment of malaria shortens its duration, prevent complications and avoidmajority of deaths. Because of its considerable drag on health in low-income coun-

3tries, malaria disease management is an essential part of global health development.Treatment aims to cure patients of the disease rather than to diminish the numberof parasites carried by an infected person. The best available treatment, particularlyfor Plasmodium falciparum malaria, is a combination of drugs known asartemisinin-basedcombinationtherapies(ACTs). However, thegrowingpotentialforparasite resistance to these medicines is undermining malaria control efforts. WHOrecommends: use of insecticide-treated nets for night-time prevention of mosquitobites; for pregnant women in highly endemic areas, preventive doses of sulfadoxinepyrimethamine(IPT/SP) to periodically clear the placenta of parasites are recommended;indoorresidualsprayingtokillmosquitoesthatrestonthewallsandceilingsof houses (Whitty, 2007).Beyond the human toll, malaria wreaks significant economic havoc in high-rate areas,decreasing Gross Domestic Product (GDP) by as much as 1.3% in countrieswith high levels of transmission. Over the long-term, these aggregated annual losseshave resulted in substantial differences in GDP between countries with and withoutmalaria (particularly in Africa). Malaria ′ s health costs include both personal andpublic expenditures on prevention and treatment. In some heavy-burden countries,the disease accounts for: up to 40% of public health expenditures, 30% to 50% ofinpatient hospital admissions, up to 60% of outpatient health clinic visits (WHO,2009). Malaria disproportionately affects poor people who cannot afford treatmentor have limited access to health care, and traps families and communities in a downspiral of poverty.1.2 Statement of the ProblemMalaria is by far the world’s most threatening tropical parasitic disease. The diseaseisendemicinMalawiandclaimssomanylives. Mathematicalmodelsofthedynamicsof this disease with special emphasis on Malawi are uncommon. Also, no previousmathematical study (in Malawi) has been conducted to establish the effects of multi-

4intervention campaigns on the malaria epidemic. Therefore, this study is intended toinvestigate the effects of multi-intervention campaigns on the transmission dynamicsof malaria in Malawi.1.3 Research Objectives1.3.1 General ObjectiveThe main purpose of this study is to understand the effects of multi-interventioncampaigns on the transmission dynamics of malaria in Malawi.1.3.2 Specific ObjectivesThe objectives of this study are to:(1). Formulate and analyze a mathematical model of malaria with control measuresin Malawi,(2). Investigate the transmission dynamics of malaria and assess the effects of controlmeasures in terms of the basic reproduction number, R 0 ,(3). Validate the model of malaria with data from Malawi.1.4 Significance of the StudyThe health as well as the socioeconomic impacts of emerging and re-emerging vectorbornediseases like malaria is significant. The disease is endemic and claims so manylives and consequently makes its study valuable. As a result of the study therewill be a model with special emphasis on Malawi on the disease. Since no previousmathematicalstudyhasbeendoneinMalawionthedisease, theresultsontheeffectsof the prevention and control measures will provide relevant guidance for decisionmakers on which intervention to focus on.

5CHAPTER TWOLITERATURE REVIEWTeklehaimanot et al., (2004) found that malaria was associated with rainfall andminimum temperature (with the strength of the association varying with altitude)in Ethiopia. Worall et al., (2007) used rainfall and maximum temperature at a lag offour months to successfully fit a biological transmission model to malaria case datain a district in Zimbabwe. Craig and colleagues linked inter-annual differences inmalaria to rainfall and temperature in South Africa. In Malawi, the main malariavector Anopheles culicifacies breeds primarily in river bed pools (WHO, 2009) whichoccur during dry periods, but also in other breeding sites such as seepage areas nextto irrigation tanks, hoof prints, and abandoned pits. Briet et al., (2008) explainedthat the extreme south west of Sri Lanka has always been virtually free of malaria.It is attributed to the wet climate in which rivers flow year round without pooling.This brings to our attention that some areas will affect our assumptions of the modelbecause the continuous flow of rivers reduces the availability of mosquitoes hencereducing the rate of mosquito-human contacts.Interventions to prevent or reduce the transmission of malaria are currently beingused, with a degree of success, in some parts of the world. Some of the methodsinclude: the situation of irrigated lands far from residential areas and cities, housespraying with residual insecticides and most recently the use of mosquito treated bednets. The methods operate by reducing the contact rates (and hence exposure toinfection) between the mosquitoes and humans. Other measures that employ the useof antimalarial drugs as a control measure may not be very effective when comparedwithcontrolmeasuresthatdirectlyaffectthedynamicsoftransmissionoftheparasite(that is based on the human mosquito interaction). This is because in endemic areas,drug coverage can only be effective if permanent prophylaxis is employed across anentire endemic human population. In most developed countries, where malaria hasbeen eradicated but the mosquito vector is still present, changes in world climate

6through global warming indicate that these malaria free zones risk being re-colonisedby malaria (Martens et al, 1999). Given these challenges be it in endemic areas orotherwise, predictive mathematical modelling and computer simulations remain ourgreatest hope (Carter, 2002; Ritchie and Montague, 1995).Ngwa (2004), formulated a variable humans and mosquitoes mathematical modelconsisting of susceptible-exposed-infectious-recovered-susceptible (SEIRS) patternfor humans and susceptible-exposed-infectious (SEI) pattern for mosquitoes. Theprimary objective was to study endemic malaria and the consequent disease relateddeathsin endemicregions. Theimportance ofincludingdemographic effectswith netpopulation growth was seen to enable the model to predict the number of fatalitiesthat may arise as a result of malaria. This type of prediction is not evident in theconstant population model and hence has been overlooked in previous models formalaria. However, no control measure was mentioned to contain the epidemic in anyregion.Malaria affects the health and wealth of nations and individuals alike. In Africatoday, malaria is understood to be both a disease of poverty and a cause of poverty(Greenwood and Mutabingwa, 2002; Sachs and Malaney, 2002). Malaria has significantmeasurable direct and indirect costs, and has been shown to be a majorconstraint to economic development (Sacks and Malaney, 2002). This means thegap in prosperity between countries with malaria and countries without malaria hasbecome wider every single year. Gallup and Sachs (2003) showed that where malariahas been eliminated, economic growth has increased substantially. Hence we need tofind cost effectiveness of the intervention strategies.The Global Malaria Control Strategy is a concerted effort meant to bring aboutchanges in the way malaria problem is addressed. As a result, this strategy stressesthe selective use of preventive measures wherever they can lead to sustainable results(WHO, 1993). The measures are aimed at halting the deteriorating effectsof the malaria situation, minimizing the wasteful use of resources and contributing

7appropriately to the development of health services, intersectoral cooperation andcommunity participation. The ultimate goal of malaria control will be to preventmortality and reduce morbidity and social and economic loss through the progressiveimprovementandstrengtheningoflocalandnationalcapacities(WHO,1993; FMoH,2000). Several interventions have been recommended to curb the rising burden ofthe disease in endemic regions. These interventions form the pillar of the global campaignfor effective malaria intervention, particularly in sub-Saharan Africa. In April25, 2000, African Heads of State and Government at the Abuja, Nigeria summit onRoll Back Malaria expressed their political will to vigorously pursue the interventions.The target set at the Summit was that by 2005 at least 60% of those at riskof malaria particularly pregnant women and children under five years of age willbenefit from the most suitable combination of personal and community protectivemeasures such as insecticide-treated mosquito nets and other interventions which areaccessible and affordable to prevent infection and suffering (FMoH, 2000).Chaves et al, (2008) suggested that the intervention using insecticide-treated bednets represents an excellent example of implementing an infectious disease controlprogramme. The results emphasize the need to implement infectious disease controlprogrammes focusing on the most vulnerable populations which is the basis of thisstudy. In addition, Morel et al, (2005) used a cost-utility analysis to examine thecosts and the effects of scaling-up seven interventions strategies against malaria andtheirpromisingcombinations. Theyusedefficacydatawhichcamefromtheliteratureand researchers calculations supported by expert opinion. The results showed thathigh coverage with artemisinin based combination treatments was found to be themost cost effective strategy for control of malaria in most countries in sub-SaharanAfrica. Since the researchers pointed out that, on the cost-effectiveness grounds, inmost areas in sub-Saharan Africa, greater coverage with highly effective combinationtreatmentshouldbethecornerstoneofmalariacontrol, thisstudywillalsodeterminethecost-effectivenessoftheselectedmalariacontrolinterventionsusingtheestimatedprimary data obtained in Malawi.

8Also Yang (2001), developed a mathematical model for malaria that incorporatesglobal warming and local socioeconomic conditions. The main objective was toapply sensitivity analysis to a mathematical model describing malaria transmissionrelating global warming and local socioeconomic conditions which represent the levelof malaria infection in a community. Their work was mainly based on the infectionand none of the interventions were tackled. In addition, Gomez-Elipe et al, (2003)studied a mathematical model involving malaria incidence based on monthly casereports and environmental factors. They predicted malaria incidence in an area ofunstable transmission by studying the association between environmental variablessuch as rainfall, temperature and vegetation density, and disease dynamics. Malariacontrol measures were not mentioned.Another project was done by Li (2008) who formulated a mathematical model formalaria transmission that includes incubation periods for both infected human hostsand mosquitoes. It was demonstrated that models having the same reproductivenumber but different number of progression stages can exhibit different transienttransmission dynamics. It was concluded that humans acquire partial immunity tomalaria after infection, although the mechanisms of immunity are not fully understood.The acquired immunity appears to depend on both the duration and theintensity of past exposure to infection. In addition, Nakul et al., 2006, presentedan ordinary differential equation mathematical model for the spread of malaria inhuman and mosquito populations. They assumed that both species follow logisticpopulation model, with immigration and disease-induced death of humans.The sophistication of the epidemiological modelling efforts has grown steadily. Acontainer-inhabitingmosquitosimulationmodelwasdevelopedbyFocksetal.,(1993).CompartmentalSEIR(susceptible-exposed-infected-recovered)differentialequationsmodels including asymptomatic immune humans were studied by Ngwa et al., (2004,2006). SEIR differential equations models with different levels of acquired immunityand the loss of immunity among human host population were formulated in Yang(2000) and the effects of social and economic conditions and temperature on the

9transmission were investigated by using numerical simulations in some of these studies.However, it seems that gradual partial immunity is induced by infections andhence multiple interventions have not been considered. Similarly, the prospects forthe success of malaria control depend on the reproductive number for malaria, R 0 .Smith et al., (2007) explained that the large number of R 0 estimates strongly supportsthe long-held notion that malaria control presents variable challenges across itstransmission spectrum. Therefore strategic planning malaria control should considerR 0 , the special scale of transmission, human population density and heterogeneousbiting.Most commonly used practices of combating vector-borne diseases focus on the reductionof vectors and raising the public ′ s awareness about prevention of host-vectorcontacts. A number of field and laboratory research have been conducted about vectorcontrol to find the most effective approaches to reduce vector population. Thisincludes practicing and monitoring the efficacy of larvaciding, adulticiding, sprayingpesticide (Peterson, 2005).Smith and Hove-Musekwa (2008) developed a mathematical model for both regularand non-fixed spraying, using impulsive differential equations in order to determinethe minimal effective spraying period, as well as the amount by which mosquitoesshould be reduced at each spraying events. The effects of climate change on theprevalence of mosquitoes were considered. The results showed that both regularand non-fixed spraying resulted in a significant reduction in the overall number ofmosquitoes, as well as the number of malaria cases in humans. However, only oneintervention was discussed.Comparative knowledge of the effectiveness and efficacy of different control strategiesis necessary to design useful and cost-effective malaria control programs. Mathematicalmodelling of malaria can play a unique role in comparing the effects of controlstrategies. The researcher, therefore, investigates the effect of such control measureson malaria dynamics and their costs.

10CHAPTER THREETHE BASIC MALARIA MODELIn this chapter, we consider the basic malaria model without any intervention strategiesin which we will explain the means of transmission of malaria in Malawi. Themodel will later be modified to consider the malaria model with intervention strategies.3.1 The Model FormulationWeformulateourbasicmalariamodelwiththepopulationunderstudybeingdividedinto compartments and with assumptions about the nature and time rate of transferfrom one compartment to another. We consider the total population sizes denotedby N h (t) and N v (t) for the human hosts and female mosquitoes, respectively. Wewill use the SEIRS framework to describe a disease with temporary immunity onrecovery from infection. SEIRS model indicates that the passage of individuals isfrom the susceptible class, S, to the exposed class, E, then to infective class, I, andfinally to the recovery class, R. S(t) represents the number of individuals not yetinfectedwiththemalariaparasiteattimet, orthosesusceptibletothedisease. Manydiseases like malaria have what is termed a latent or exposed phase, E(t), duringwhich an individual is said to be infected but not infectious. I(t) denotes the numberof individuals who have been infected with malaria and are capable of spreading thedisease to those in the susceptible category. This is done through infecting thesusceptible mosquitoes. The dynamic transmission of the malaria parasite betweenand amongst individuals in both species is driven by the mosquito biting habit ofthe humans. R(t) is the compartment for individuals who have recovered from thedisease. These humans can not transmit the infection to mosquitoes as we assumethat they have no plasmodium parasites in their bodies.The transfer rates between the subclasses are composed of several epidemiological

11parameters. Killeen et al., (2001) explained that susceptible human bitten by aninfectious anopheles mosquito may become infected with a finite probability thatdepends on the abundance of infectious mosquitoes and human hosts. The modelassumes horizontal standard incidence with homogenous mixing meaning that susceptibleindividuals get infected through contact with infected mosquitoes. Thesusceptible human population is increased by recruitment (birth and immigration)at a constant rate, Λ. All the recruited individuals are assumed to be naive whenthey join the community. Infected immigrants are not included because we assumethat most people who are sick will not travel. When an infectious female anophelesmosquito bites a susceptible human, there is some finite probability, β vh that theparasite (in the form of sporozoites) will be passed on to the human. The parasitethen moves to the liver where it develops into its next life stage. The infected personwill move to the exposed class. After a certain period of time, the parasite (in theform of merozoites) enters the blood stream, usually signaling the clinical onset ofmalaria. Then the exposed individuals become infectious and progress to infectedstate at a constant rate ν. We exclude the direct infectious-to-susceptible recovery byassuming that the individuals do not recover by natural immunity. This is a realisticsimplifying assumption because most people have some period of immunity beforebecoming susceptible again. After some time, individuals who have experienced infectionmay recover with natural immunity at a constant rate γ and move to therecovered class. The recovered individuals have some immunity to the disease anddo not get clinically ill. Since disease-induced immunity due to malaria is temporary,afractionω ofindividualsleavetherecoveredstatetothesusceptiblestate. Wemakethe simplifying assumptions that there is no immigration of the recovered humans.Humans leave the population through natural death, µ h and the infected humanshave an additional disease-related death rate constant δ h . The disease-induced rateis very small in comparison with the recovery rate.We divide the mosquito population into three classes: susceptible, X; exposed, Y;and infectious, Z. Female anopheles mosquitoes (male anopheles mosquito is not

12included in the model because only female mosquito bites animals for blood meals)enter the susceptible class through birth at a rate π. Susceptible mosquitoes becomeinfected by biting infectious humans at a rate α. The parasites (in the form ofgametocytes) enter the mosquito with probability β hv , when the mosquito bites aninfectious human, and the mosquito moves from the susceptible to the exposed class.After some period of time, dependent on the ambient temperature and humidity,the parasite develops into sporozoites and enters the mosquito’s salivary glands, andthe mosquito progresses at a rate θ, from the exposed class to the infectious class.We assume that the infective period of the vector ends with its death, and thereforethe vector does not recover from being infective (Aron, 1988). The mosquitoes leavethe population through natural death. Its per capita death rate is µ v and infectiousmosquitoes suffer an additional death at a rate δ v due to the presence of infectiousparasites in their bodies. The rate of infection of susceptible individual is λ h , andthe rate of infecting a susceptible mosquito is λ v .The model flow diagram is shown in Figure 1. The dash line from infected humanFigure 1: The basic malaria flowchartclass, I, to the susceptible mosquito population, X, shows that the infected humanindividuals infect the susceptible mosquito population whilst the dash line frominfected mosquito population, Z, to the susceptible human population, S, shows the

13transfer of plasmodium parasites from infected mosquito population to susceptibleindividuals.The state variables in Table 1 and parameters in Table 2 for the malaria basic modelsatisfy the equations (1). It is assumed that all state variables and parameters of thebasic model which monitors human and mosquito populations are positive ∀t ≥ 0,and will therefore be analysed in a suitable region. The parameters of the model areTable 1: State variables of the basic malaria modelSymbol DescriptionS(t)E(t)I(t)R(t)X(t)Y(t)Z(t)N h (t)N v (t)Number of susceptible humans at time tNumber of exposed humans at time tNumber of infectious humans at time tNumber of recovered humans at time tNumber of susceptible mosquitoes at time tNumber of exposed mosquitoes at time tNumber of infectious mosquitoes at time tTotal human population at time tTotal mosquito population at time tpresented in Table 3.2 below;

14Symbol DescriptionTable 2: Parameters of the basic malaria modelΛπRecruitment rate of humansBirth rate of mosquitoesµ h Per capita natural death rate for humansµ v Per capita natural death rate for mosquitoesνProgression rate of humans from the exposed state to theinfectious stateδ hδ vωθPer capita disease-induced death rate for humansPer capita disease-induced death rate for mosquitoPer capita rate of loss of immunityProgression rate of exposed mosquitoes to infectedmosquitoesγRecovery rate for humans from the infected state to therecovered state with natural immunityλ hForce of infection for susceptible humans to exposedindividualsλ vForce of infection for susceptible mosquitoes to exposedmosquitoesαβ vhbiting rate of mosquitoProbability that a bite results in transmission of infectionto the humanβ hvProbability that a bite results in transmission of theparasite from an infectious human to the susceptible mosquito

15From the above assumptions and the model flowchart together lead to the followingsystem of ordinary differential equations which describe the progress of the disease;⎫dS= Λ−λ h S +ωR−µ h SdtdE= λ h S −(ν +µ h )EdtdI= νE −(γ +µ h +δ h )Idt⎪⎬dR= γI −(ω +µ h )R(1)dtdX= π −λ v X −µ v XdtdY= λ v X −(θ+µ v )YdtdZ= θY −(µ v +δ v )Z⎪⎭dtwhere λ h = β vhαZN h, λ v = β hvαIN h.As in Tumwiine et al. (2007), the terms β vhαSZdenotes the rate at which theN hhuman hosts S get infected by infected mosquitoes Z and β hvαXIrefers to the rateat which the susceptible mosquitoes X are infected by the infected human hosts I.It indicates that the rate of infection of susceptible human S by infected mosquitoZ is dependent on the total number of humans N h available per vector.N h3.2 Invariant RegionThe total population sizes N h and N v can be determined by N h = S + E + I + Rand N v = X +Y +Z or from the differential equationsanddN hdtdN vdt= dSdt + dEdt + dIdt + dRdt= Λ−µ h N h −δ h I,= dX dt + dYdt + dZdt= π −µ v N v −δ v Z.Assuming the disease does not kill (δ h = 0), we have(2)(3)dN hdt= Λ−µ h N h −δ h I ≤ Λ−µ h N h . (4)

16Lemma 1 The model system (1) has solutions which are contained in the feasibleΩ = Ω h ×Ω v .Proof: Let (S,E,I,R,X,Y,Z) ∈ R 7 + be any solution of the system with nonnegativeinitial conditions.SincedN hdt≤ Λ−µ h N h , (5)and using Birkhoff and Rota (1989) Theorem on differential inequality, we have0 ≤ N h ≤ Λ µ h, henceΛ−µ h N h ≥ Ke −µ ht where K is constant. (6)Therefore, allfeasiblesolutionsofthehumanpopulationonlyofthemodelsystem(1)are in the regionΩ h ={(S,E,I,R) ∈ R 4 + : N h ≤ Λ µ h}. (7)Similarly, the feasible solutions of the mosquito population only are in the regionΩ v ={(X,Y,Z) ∈ R 3 + : N v ≤ π }. (8)µ vThus, the feasible set for model system (1) is given byΩ ={(S,E,I,R,X,Y,Z) ∈ R 7 + : S,E,I,R,X,Y,Z ≥ 0;N h ≤ Λ ;N v ≤ π }, (9)µ h µ vwhich is a positively invariant set under the flow induced by the model (1). Hencethe system (1) is epidemiologically meaningful and mathematically well-posed in thedomain Ω. Therefore, in this domain it is sufficient to consider the dynamics of theflow generated by the model (1). In addition, the usual existence, uniqueness andcontinuation of results hold for the system.3.3 Positivity of SolutionsLemma 2 Let the initial data be{(S(0),X(0)) > 0,(E(0),I(0),R(0),Y(0),Z(0)) ≥ 0} ∈ Ω.

17Then the solution set {S,E,I,R,X,Y,Z}(t) of the model system (1) is positive forall t > 0.Proof: The first equation of the model (1) givesdS= Λ−λ h S +ωR−µ h S ≥ −λ h S −µ h Sdt≥ −(λ h +µ h )S∫ ∫ 1S dS ≥ − (λ h +µ h )dtS(t) ≥ S(0)e −(∫ λ h dt+µ h t) ≥ 0.From the second equation of (1) we havedE= λ h S −(ν +µ h )E ≥ −(ν +µ h )E∫dt∫ 1E dE ≥ − (ν +µ h )dtE(t) ≥ E(0)e −(ν+µ h)t ≥ 0.We also get the following from third equation of (1)dI= νE −(γ +µ h +δ h )I ≥ −(γ +µ h +δ h )I∫dt∫ 1I dI ≥ − (γ +µ h +δ h )dtI(t) ≥ I(0)e −(γ+µ h+δ h )t ≥ 0.It follows also from fourth equation of (1) thatdR= γI −(ω +µ h )R ≥ −(ω +µ h )R∫dt∫ 1dt dR ≥ − (ω +µ h )dtR(t) ≥ R(0)e −(ω+µ h)t ≥ 0.Solving for X(t), we consider the fifth equation of (1) which givesdX= π −λ v X −µ v X ≥ −λ v X −µ v Xdt≥ −(λ v +µ v )X∫ ∫ 1X dX ≥ − (λ v +µ v )dtX(t) ≥ X(0)e −(∫ λ vdt+µ vt) ≥ 0.

18From the sixth equation of (1) we havedY= λ v X −(θ+µ v )Y ≥ −(θ+µ v )Y∫dt∫ 1Y dY ≥ − (θ+µ v )dtY(t) ≥ Y(0)e −(θ+µv)t ≥ 0.The seventh equation of (1) givesdZ= θY −(µ v +δ v )Z ≥ −(µ v +δ v )Z∫dt∫ 1Z dZ ≥ − (µ v +δ v )dtZ(t) ≥ Z(0)e −(µv+δv)t ≥ 0.Furthermore, we need to show that the region Ω is positively invariant so that itsuffices the dynamics of the above system. The right hand sides of equations (2)and (3) are both bounded by Λ−µ h N h and π −µ v N v , respectively, it follows thatdN hdt< 0 if N h (t) > Λ µ hand dN vdt< 0 if N v (t) > π µ v.Using a standard comparison theorem (Zhang, 1988), it has been shown above thatandN h (t) ≤ Λ µ h(1−e(−µ h t) ) +N h (0)e (−µ ht) ,N v (t) ≤ π µ v(1−e(−µ vt) ) +N v (0)e (−µvt) .In particular, if N h (0) < Λ µ hthen N h (t) ≤ Λ µ hand if N v (0) < π µ vthen N v (t) ≤ π µ v.Therefore Ω is positively invariant. If N v (0) > π µ vand N h (0) > Λ µ h, then either thesolution enters Ω in finite time, or N v (t) approaches π µ vand N h (t) approaches Λ µ hasymptotically, and the infected state variables E, I, Y, and Z approaches zero.3.4 Existence and Stability of Steady-state SolutionsThe E = (S ∗ ,E ∗ ,I ∗ ,R ∗ ,X ∗ ,Y ∗ ,Z ∗ ) is the steady-state of the system (1) which canbe calculated by setting the right hand side of the model (1) to zero, giving us the

following;19Λ−λ h S +ωR−µ h S = 0λ h S −νE −µ h E = 0νE −γI −(µ h +δ h )I = 0γI −ωR−µ h R = 0π −λ v X −µ v X = 0λ v X −θY −µ v Y = 0θY −(µ v +δ v )Z = 0⎫⎪⎬. (10)⎪⎭3.4.1 The Existence of the Trivial Equilibrium PointFor as long as the human recruitment term Λ and the mosquito recruitment term πare not zero, the population will not be extinct. This implies that there is no trivialequilibrium point, thus (S ∗ ,E ∗ ,I ∗ ,R ∗ ,X ∗ ,Y ∗ ,Z ∗ ) ≠ (0,0,0,0,0,0,0).3.4.2 Disease-free Equilibrium Point E 0Disease-free equilibrium points (DFE) are steady-state solutions where there is nodisease (malaria). We define the ”diseased” classes as the human or mosquito populationsthat are either exposed, or infected, that is; E,I,Y, and Z in the system (1).Hence, the DFE of the basic malaria model (1) is given by,E 0 = (S ∗ ,E ∗ ,I ∗ ,R ∗ ,X ∗ ,Y ∗ ,Z ∗ )( Λ= ,0,0,0, π ,0,0),µ h µ v(11)that represents the state in which there is no infection in the society and is knownas the disease-free equilibrium point (DFE).3.4.3 The Reproduction Number R 0We use the next generation operator approach as described by Diekmann, (1990)to define the basic reproductive number, R 0 , as the number of secondary infectionsthat one infectious individual would create over the duration of the infectious period,

20provided that everyone else is susceptible. It is an important parameter that playsa big role in the control of the malaria infection.R 0 = 1 is a threshold below which the generation of secondary cases is insufficient tomaintain the infection with human community. If R 0 < 1, each individual produces,on average, less than one new infected individual and hence the disease dies outwhile if R 0 > 1, each individual produces more than one new infected individual andhence the disease is able to invade the susceptible population. It is therefore a usefulquantity in the study of a disease as it sets the threshold for its establishment.The basic reproduction number can not be determined from the structure of themathematical model alone, but depends on the definition of infected and uninfectedcompartments. We define X s to be the set of all disease free states. That isX s = {x ≥ 0 | x i = 0, i = 1,...,m}.In order to compute R 0 , it is important to distinguish new infections from all otherchanges in the population. LetF i be the rate of appearance of new infections in compartment i,V i = V − i − V + i is the difference between the rate of transfer of individuals out ofcompartment i,(V − i), by all other means and the rate of transfer of individuals inthe compartment i,(V + i) by all other means.x 0 be the disease-free equilibrium pointIt is assumed that each function is continuously differentiable at least twice in eachvariable. The disease transmission model consists of nonnegative initial conditionstogether with the following system of equations:ẋ = f i (x) = F i (x)−V i (x), i = 1,...,n.[ ] [ ]∂Fi ∂ViLet F = (x 0 ) and V = (x 0 ) with 1 ≤ i,j ≤ m.∂x j ∂x jFurther, F is nonnegative, V is a nonsingular M-matrix in which both are the m×mmatrices, where m stands for the number of infected classes.Hence R 0 is the largest eigenvalue of FV −1 , where

21the (i,j) entry of F is the rate at which infected individuals in compartment j producenew infections in compartment i,the (j,k) entry of V −1 is the average length of time this individual spends in compartmentj during it’s lifetime, assuming that the population remains near the DFEand barring reinfection.Hence, the (i,k) entry of the product FV −1 is the expected number of new infectionsincompartmentiproducedbytheinfectedindividualoriginallyintroducedintocompartmentk. Following Diekmann et al., (2000), FV −1 is called the next generationmatrix for the model and we setR 0 = ρ(FV −1 ),where ρ(A) denotes the spectral radius of a matrix A.Rewriting the system (1) starting with the infected compartments for both populations;E,I,Y,Z, and then followed by unifected classes; S,R,X also from the twopopulations, givesdEdtdIdtdYdtdZdtdSdtdRdtdXdt= β vhαZSN h−(ν +µ h )E= νE −(γ +µ h +δ h )I= β hvαIX−(θ+µ v )YN h= θY −(µ v +δ v )Z= Λ− β vhαZS+ωR−µ h SN h= γI −(ω +µ h )R= π − β hvαIX−µ v XN h⎫⎪⎬. (12)⎪⎭The method of next generation matrix has been used to show the rate of appearanceof new infection in compartments; E and Y, from the system (12);⎡ ⎤β vh αZSN h0F =β hv αIX.⎢⎣ N h⎥⎦0

22The Jacobian matrix of F at the disease-free equilibrium point E 0 (11) whereN h ≤ Λ and N v ≤ π to form Jacobian matrix;µ h µ v⎡ ⎤0 0 0 β vh α0 0 0 0F =⎢0 β hvαπµ h. (13)0 0⎣ Λµ ⎥v ⎦0 0 0 0Calculating the transfer of individuals out of the compartments of the system (12)by all other means⎡ ⎤(ν +µ h )E(γ +µ h +δ h )I −νEV =.⎢ (θ +µ v )Y⎥⎣ ⎦(µ v +δ v )Z −θYThen Jacobian matrix of V⎡⎤ν +µ h 0 0 0−ν γ +µ h +δ h 0 0V =. (14)⎢ 0 0 θ+µ v 0⎥⎣⎦0 0 −θ µ v +δ vFurthermore, we have to find the inverse of the Jacobian matrix (14)⎡⎤1(µ h0 0 0+ν) ν 1V −1 (ν+µ=h )(γ+µ h +δ h ) (γ+µ h +δ h0 0) . (15)1⎢ 0 0(θ+µ⎣v)0⎥⎦θ 10 0(θ+µ v)(µ v+δ v) (µ v+δ v)We have to calculate the product of the equations (13) and (15) which gives⎡ ⎤0 0 a bFV −1 0 0 0 0=⎢c d 0 0⎥⎣ ⎦0 0 0 0(16)

23β vh αθwhere a =(θ +µ v )(µ v +δ v ) , b = β vhα(µ v +δ v ) , c = β hv απνµ hΛµ v (ν +µ h )(γ +µ h +δ h ) andβ hv απµ hd =Λµ v (γ +µ h +δ h ) .The eigenvalues of FV −1 are calculated from M =| FV −1 −Iλ |= 0M =∣−λ 0 a b0 −λ 0 00 0 −λ 0c d 0 −λ= 0.∣Lastly we calculate the eigenvalues from the matrix (16), we get⎡⎤000λ i =√(ν +µ h )(γ +µ h +δ h )(θ+µ v )(µ v +δ v )( β hvαπµ hµ )β vΛ vhανθ. (17)√ (ν +µ h )(γ +µ h +δ h )(θ +µ v )(µ v +δ v )⎢⎣ (ν +µ h )(γ +µ h +δ h )(θ +µ v )(µ v +δ v )( β hvαπµ h)β ⎥µ vΛ vhανθ⎦−(ν +µ h )(γ +µ h +δ h )(θ+µ v )(µ v +δ v )The above matrix can be simplified further and we get⎡⎤000λ i =√(ν +µ h )(γ +µ h +δ h )(θ+µ v )(µ v +δ v )(β hv β vh α 2 θν)( πµ hµ vΛ ). (18)√ (ν +µ h )(γ +µ h +δ h )(θ +µ v )(µ v +δ v )⎢⎣ (ν +µ h )(γ +µ h +δ h )(θ +µ v )(µ v +δ v )(β hv β vh α 2 θν)( πµ h) ⎥µ vΛ ⎦−(ν +µ h )(γ +µ h +δ h )(θ+µ v )(µ v +δ v )Thus, the reproduction number , R 0 , from (18) is the dominant eigenvalue of FV −1given byR 0 =√(ν +µ h )(γ +µ h +δ h )(θ+µ v )(µ v +δ v )β vh β hv α 2 νθπµ h(ν +µ h ) 2 (γ +µ h +δ h ) 2 (θ +µ v ) 2 (µ v +δ v ) 2 µ v Λ=√β vh β hv α 2 νθπµ h(ν +µ h )(γ +µ h +δ h )(θ+µ v )(µ v +δ v )µ v Λ(19)

24whereνis the probability of survival of individuals from latent (exposed) stage intoν +µ hthe infectious stage.θis the probability of survival of mosquitoes from the exposed stage into theθ +µ vinfectious stage of the mosquito population.β vh θαThe term describes the number of humans that one mosquito infects(through contact) during the lifetime it survives as infectious, when all humans(µ v +δ v )(θ+µ v )β hv ανare susceptibles. On the other hand, the term,describes the(ν +µ h )(γ +µ h +δ h )number of mosquitoes that are infected through contacts with one infectious human,while the human survives as infectious, assuming no infection among vectors.The threshold number, R 0 , is the product of R 0h defined as the number of humansthat one mosquito infects through its infectious lifetime, assuming all humans aresusceptible, and R 0v defined as the number of mosquitoes that one human infectsthrough the duration of the infectious period, assuming all mosquitoes are susceptibles.Our reproductive number includes the generation of infections of two populations,so is the square root.Therefore, manipulation of the R 0 , gives√R 0 =β vh ναµ hΛ(ν +µ h )(γ +µ h +δ h ) ·β hv θαπµ v (θ+µ v )(µ v +δ v )= √ R 0h ×R 0v ,whereR 0h =β vh ναµ hΛ(ν +µ h )(γ +µ h +δ h )(20)andR 0v =β hv θαπµ v (θ+µ v )(µ v +δ v ) . (21)β vh αµ his the number of latent infections produced by a typical infectiousΛ(γ +µ h +δ h )individual during the mean infectious period.

25β hv απis the number of latent infections produced by a typical infectiousµ v (µ v +δ v )mosquitoes during the mean infectious period.Notice that α appears twice in the expression since the mosquito biting rate controlstransmission from humans to mosquitoes and from mosquitoes to humans.Malaria infection exist in a community due to contact between the humans andmosquitoes. Whether the disease becomes persistent or dives out depends on magnitudeof the basic reproduction number, R 0 . Stability of the equilibruim points canbe analyzed using R 0 .3.4.4 Local Stability of the Disease-free Equilibrium E 0The local stability of the disease-free equilibrium can be discussed by examining thelinearized form of the system (1) at the steady state E 0 .Lemma 3 The disease-free equilibrium point E 0 for the system (1) is locally asymptoticallystable if R 0 < 1 and unstable if R 0 > 1.Proof: The Jacobian matrix of the model (1) with S = N h −(E+I +R) evaluatedat the disease-free equilibrium point is given by⎡⎤−(ν +µ h ) 0 0 0 0 β vh αν −(γ +δ h +µ h ) 0 0 0 00 γ −(ω +µ h ) 0 0 00 − β hvαπµ hΛν v0 −µ v 0 0β⎢ 0 hv απµ h⎣Λν v0 0 −(θ+µ v ) 0 ⎥⎦0 0 0 0 θ −(µ v +δ v )(22)The third and the fourth columns have diagonal entries. Therefore, the diagonalentries −(ω + µ h ) and −µ v are the two of the eigenvalues of the Jacobian. Thus,excluding these columns and the corresponding rows, we calculate the remainingeigenvalues. These eigenvalues are the solutions of the characteristic equation of the

26reduced matrix of dimension four which is given by(x+µ v +δ v )(x+ν +µ h )(x+γ +δ h +µ h )(x+θ+µ v )− β vhβ hv θα 2 πνµ hΛµ v= 0. (23)To simplify the notation, we let B 0 = µ v + δ v , B 1 = ν + µ h , B 2 = θ + µ v , B 3 =γ +δ h +µ h . This reduces (19) to R 2 0 = β vhβ hv α 2 θπνµ hΛµ v B 0 B 1 B 2 B 3and (23) tox 4 +A 3 x 3 +A 2 x 2 +A 1 x+A 0 = 0, (24)whereA 3 = B 1 +B 3 +2B 0 +θA 2 = (B 3 +B 1 )(2B 0 +θ)+B 0 B 2 +B 1 B 3A 1 = B 0 B 3 B 2 +B 1 B 3 (2B 0 +θ)+B 0 B 1 B 2A 0 = B 0 B 1 B 2 B 3 −να 2 θβ hv β vhπµ hΛµ v⎫⎪ ⎬⎪ ⎭. (25)The Routh-Hurwitz conditions (Murray, 1991), which usually have different formsare the sufficient and necessary conditions on the coefficients of the polynomial (24).These conditions ensure that all roots of the polynomial given by (24) have negativereal parts. For this polynomial, the Routh-Hurwitz conditions are A 2 > 0, A 3 >0, A 0 > 0, A 1 > 0, and H 1 = A 3 > 0,∣ ∣∣∣∣∣ AH 2 =3 1> 0,∣A 1 A 2∣ ∣∣∣∣∣∣∣∣ A 3 1 0H 3 =A 1 A 2 A 3> 0,∣ 0 A 0 A 1∣ ∣∣∣∣∣∣∣∣∣∣∣∣ A 3 1 0 0A 1 A 2 A 3 1H 4 => 0.0 A 0 A 1 A 2∣ 0 0 0 A 0ClearlyH 4 = A 0 H 3 .SinceB 0 > 0B 1 > 0,B 2 > 0,B 3 > 0,wehaveA i > 0,i = 1,2,3.Moreover, if R 0 < 0, it follows that A 0 > 0. Thus, it is enough to prove that

27H 2 > 0 and H 3 > 0. Clearly H 3 = A 1 (A 3 A 2 − A 1 ) − A 0 A 2 3 and H 2 = A 3 A 2 − A 1 .Using Maple, it is easy to see thatH 2 = A 3 A 2 −A 1= B3(B 2 0 +B 2 +B 1 )+B 2 B 3 (2B 0 +B 2 +2B 1 )+B0(B 2 3 +B 1 +B 2 )+B1(B 2 0 +B 2 +B 3 )+2B 0 B 1 (B 3 +B 2 )+B2(B 2 1 +B 0 )which is positive.⎫⎪⎬⎪⎭(26)Again, using Maple, we can also see thatH 3 = A 1 (A 3 A 2 −A 1 )−A 0 A 2 3⎫⎪ ⎬= (B 3 +B 0 )(B 0 +B 2 )(B 3 +B 2 )(B 1 +B 0 )(B 3 +B 1 )(B 1 +B 2 ) , (27)+να 2 πµ hθβ hv β ⎪ vh ⎭Λµ vwhich is clearly a positive quantity. Therefore, all of the eigenvalues of the Jacobianmatrix have negative real parts when R 0 < 1.However, R 0 > 1 implies that A 0 < 0, and since all of coefficients (A 1 , A 2 , and A 3 )of the polynomial (24) are positive, not all roots of this polynomial can have negativereal parts. This means, when R 0 > 1, the disease-free equilibrium point is unstable.NotethattheresultinLemma3islocal,thatis,wecouldonlyconcludethatsolutionswith fairly small initial size in the invariant set Ω are attracted to the disease-freeequilibrium point. It is possible to further reduce the dimension of the Jacobian inthe proof of Lemma (3) by using X = N v − (Y + Z) and S = N h − (E + I + R)without any technical difficulty.3.4.5 The Endemic Equilibrium Point (E 1 )Endemic equilibrium point E 1 is a steady-state solution where the disease persistsin the population. For the existence and uniqueness of endemic equilibriumE 1 = (S ∗ ,E ∗ ,I ∗ ,R ∗ ,X ∗ ,Y ∗ ,Z ∗ ),

28its coordinates should satisfy the conditionsS ∗ > 0,E ∗ > 0,I ∗ > 0,R ∗ > 0,X ∗ > 0,Y ∗ > 0,Z ∗ > 0. We need to solve the basicmalaria model (1) by equating to zero, at an arbitrary equilibriumE 1 = (S ∗ ,E ∗ ,I ∗ ,R ∗ ,X ∗ ,Y ∗ ,Z ∗ ).Solving the second equation of (1) for E ∗ we getwhere the sixth and seventh equations of (1) giveandE ∗ = β vhαZ ∗N h (ν +µ h ) S∗ (28)Y ∗ = β hvαI ∗N h (θ +µ h ) X∗ (29)Z ∗ =respectively. Substituting (29) into (30) for Y ∗ givesZ ∗ =But from fifth equation of (1), we know thatθ(µ v +δ v ) Y ∗ (30)θβ hv αI ∗N h (µ v +δ v )(θ+µ v ) X∗ . (31)X ∗ =πN h(β hv αI ∗ +µ v N h ) . (32)We continue by substituting the equation (32) into (31) to getZ ∗ =β hv αθπ(µ v +δ v )(θ+µ v )(β hv αI ∗ +µ v N h ) I∗ with equation (21) we get=R 0v µ v(β hv αI ∗ +µ v N h ) I∗ . (33)Futhermore, from the second equation of (1), we haveβ hv αZ ∗We can now substitute equation (33) into (34) to getN hS ∗ −(ν +µ h )E ∗ = 0. (34)β vh αR 0v µ v I ∗N h (β hv αI ∗ +µ v N h ) S∗ −(ν +µ h )E ∗ = 0. (35)

But from the third equation of (1) we getE ∗ = (γ +µ h +δ h )I ∗ which can be substituted into (34) to giveν29β vh αR 0v µ v I ∗ S ∗N h (β hv αI ∗ +µ v N h ) + (ν +µ h)(γ +µ h +δ h )I ∗ = 0 where (36)νI ∗ = 0 or β vh αR 0v µ v νS ∗ −N h (ν +µ h )(γ +µ h +δ h )(β hv αI ∗ +µ v N h ) = 0which means by algebraic manipulation with N h ≤ Λ µ hand R 2 0 = R 0h ×R 0v we have[] [ ]β vh ανµ hR 0v µ v S ∗ − β hv αI ∗ Λ+µ v = 0 with equation (20)Λ(ν +µ h )(γ +µ h +δ h )µ hR 0h R 0v µ v S ∗ −β hv αI ∗ − µ vΛ= 0µ hµ v R 2 0S ∗ −β hv αI ∗ − µ vΛµ h= 0 which givesS ∗ = β hvαµ h I ∗ +Λµ v. (37)µ h µ v R 2 0We can solve for I ∗ by considering equation (34), (37) and (33), with a lengthyalgebraic manipulation, to get the followingΛ−β vh αµ v R 0v I ∗N h (β hv αI ∗ +µ v N h ) S∗ + ωγI∗(ω +µ h ) −µ hS ∗ = 0N h Λ(ω +µ h )(β hv αI ∗ +µ v N h )−β vh αR ov µ v (ω +µ h )I ∗ +N h ωγI ∗ (β hv αI ∗ +µ v N h )−µ h N h (ω +µ h )(β hv αI ∗ +µ v N h ) = 0.Lastly we getA(I ∗ ) 2 +BI ∗ +C = 0, (38)[A = β hv αγωN h − β ]vhα 2 β hv R 0v (ω +µ h )R 2 0[B = β hv αΛN h (ω +µ h )− β ]vhαµ v ΛR 0v (ω +µ h )+ωγµµ h R 2 v Nh 2 −H 10C = [ µ v ΛNh(ω 2 +µ h )−µ h µ v Nh(ω 2 +µ h ) ] ,with H 1 = β hv αµ h N h (ω +µ h ), which gives√I ∗ B2 −4AC −B=2A(39)= Ψ. (40)

30We can now solve equation (37) using (40) to getS ∗ = β hvαµ h Ψ+µ v Λ, (41)µ h µ v R 2 0from the third and fourth equations of (1), and the substitution of (40) we getandE ∗ = (γ +µ h +δ h )ΨνR ∗ =(42)γΨ(ω +µ h ) . (43)In addition we are required to solve the X ∗ and Y ∗ for the susceptible and exposedmosquitoes in a malaria endemic area using (40) and N h ≤ Λ µ h, we getX ∗ =πΛ(β hv αµ h Ψ+µ v Λ) , (44)Y ∗ = (µ v +δ v )µ v µ h R 0v Ψ(β hv αµ h Ψ+µ v Λ) . (45)We can analyze from the quadratic (38) for the possibility of multiple equilibria. Itis important to note that the coefficient A is always positive and C is positive if R 0 isless than unity, and negative if R 0 is greater than unity. Hence, we have establishedthe following result:Remark 1 The basic malaria model (1) has(1). precisely one unique endemic equilibrium if B < 0, and C = 0 or B 2 −4AC = 0,(2). precisely one unique endemic equilibrium if C < 0 ⇔ R 0 > 1,(3). precisely two endemic equilibria if C > 0, B < 0 and B 2 −4AC > 0,(4). no endemic otherwise.3.4.6 Local Stability of the Endemic Equilibrium E 1The possible presence of two endemic equilibria shown in Remark 1,Case (3), aboveindicates the possibility of backward bifurcation in the model (1). This is exploredfurther below, using the Centre Manifold Theory (Carr, 1981).

313.4.7 Existence of Backward BifurcationWe intend to determine the stability of the endemic equilibrium and to investigatethe possibility of the existence of backward bifurcation due to existence of multipleequilibriumandreinfection. Asadiseaseinvadesitreducesthenumberofsusceptibleindividuals in the population, which tends to reduce its reproductive rate. For abackwardbifurcationtooccur, thismeansthatwhenR 0 < 1theendemicequilibriumpoint can exist as well as when R 0 > 1. Normally if R 0 < 1 the disease cannot invadea naive population, however in this case it could persist for R 0 < 1. When the modelexhibits backward bifurcation, reducing R 0 below unit is not sufficient to control theepidemic. Further when R 0 is slightly above 1, the disease would be expected topersist.In particular, when R 0 is precisely 1, each infection exactly replaces itself in thelinear approximation. Hence, whether the disease will invade when R 0 = 1 will bedetermined by whether the reproductive rate increases or decreases as the diseaseincreases along the center manifold (Dushoff, 1998). Hence, we would expect thedisease to be able to invade at R 0 = 1 in the case of a backward bifurcation withthe properties of unstable equilibrium bifurcating from the disease-free equilibriumwhen R 0 < 1, giving rise to multiple stable states. But not in the case of a forwardbifurcation, in which in the absence of a low-level unstable equilibrium when R 0 < 1and a stable equilibrium bifurcating from the disease-free equilibrium when R 0 > 1,arisenaturallywhenthediseasedoesnotinvadewhenR 0 = 1.Asimplecriterionforabackward bifurcation, then, is onein which thedisease can invade when R 0 = 1. Thisimplies that the disease-free equilibrium may not be globally asymptotically stableeven if R 0 < 1. There is a possibility of a backward bifurcation (bistability) wherea stable endemic state co-exist with the disease free equilibrium. The possibility ofthis phenomenon in model (1) is investigated below.

323.4.8 Determination of the Backward Bifurcation and Local Stability ofthe Endemic Equilibrium Point E 1The stability of the endemic equilibrium can be determined by computing the eigenvaluesof the Jacobian matrix and then evaluate it at the endemic equilibrium. Thisapproach is mathematically complicated for the system of equations (1). Insteadbifurcation analysis is perfomed at the disease free equilibrium by using Centre ManifoldTheory as presented in Chavez and Song, 2004.The system (1) is rewritten by introducing the dimensionless state variables of thebasic malaria model as follows; let x 1 = S, x 2 = E, x 3 = I, x 4 = R, x 5 = X,x 6 = Y, x 7 = Z.The system (1) can be written in vector form asdX idt= F(X i )where X i = (x 1 ,x 2 ,...,x 7 ) T , F = (f 1 ,f 2 ,...,f 7 ) T and (·) T denotes the matricestranspose.The system of equations (1) becomesdx 1dt = Λ− φ∗ αµ h x 7 x 1+ωx 4 −µ h x 1 = f 1Λdx 2dt = φ∗ αµ h x 7 x 1−(ν +µ h )x 2 = f 2Λdx 3dt = νx 2 −(γ +µ h +δ h )x 3 = f 3⎫⎪ ⎬dx 4dt = γx 3 −(ω +µ h )x 4 = f 4 , (46)dx 5dt = π − β hvαµ h x 3 x 5−µ v x 5 = f 5Λdx 6dt = β hvαµ h x 3 x 5−(θ+µ v )x 6 = f 6Λdx 7dt = θx ⎪6 −(µ v +δ v )x 7 = f 7 ⎭where N h = x 1 +x 2 +x 3 +x 4 and N v = x 5 +x 6 +x 7 , with φ ∗ = β vh from (19).Suppose that φ ∗ is a bifurcation parameter, the system (46) is linearized at diseasefree equilibrium point E 0 when φ = φ ∗ with R 0 = 1. Hence, solving for φ ∗ fromR 0 = 1 in (19) givesφ ∗ = (ν +µ h)(γ +µ h +δ h )(θ+µ v )(µ v +δ v )µ v Λβ hv α 2 νθπµ h.

33Then zero is a simple eigenvalue of the following Jacobian matrix, J bif with theapplication of the bifurcation parameters.⎡⎤−µ h 0 0 ω 0 0 −φα0 −(ν +µ h ) 0 0 0 0 φα0 ν ǫ 0 0 0 00 0 γ −(ω +µ h ) 0 0 0, (47)0 0 χ 0 −µ v 0 0⎢ 0 0 ̺ 0 0 −(θ+µ⎣v ) 0 ⎥⎦0 0 0 0 0 θ −(µ v +δ v )where ǫ = −(γ +µ h +δ h ), χ = − β hvαπµ h, ̺ = β hvαπµ h.Λµ v Λµ vA right eigenvector associated with the eigenvalue zero is w = (w 1 ,w 2 ,...,w 7 ).Solving gives the system−µ h w 1 +ωw 4 −φαw 7 = 0⎫−(ν +µ h )w 2 +φαw 7 = 0νw 2 −(γ +µ h +δ h )w 3 = 0γw 3 −(ω +µ h )w 4 = 0− β hvαπµ hw 3 −µ v w 5 = 0Λµ vβ hv απµ hw 3 −(θ+µ v )w 6 = 0Λµ v⎪⎭θw 6 −(µ v +δ v )w 7 = 0⎪⎬. (48)Solving the systems (48), gives the following right eigenvectorw 1 = ωw ⎫4 −φαw 7µ hw 2 = φαw 7ν +µ hνw 2w 3 =γ +µ h +δ hw 4 = γw ⎪⎬3. (49)ω +µ hw 5 = − β hvαπµ h w 3Λµ 2 vw 6 = β hvαπµ h w 3Λµ v (θ+µ v )w 7 = w 7 > 0⎪⎭

34and the left eigenvector satisfying v ·w = 1 is v = (v 1 ,v 2 ,...,v 7 ). To find these lefteigenvector associated with the eigenvalue 0, the matrix (47) should be transposedand gives matrix, J left⎡⎤−µ h 0 0 0 0 0 00 −(ν +µ h ) ν 0 0 0 00 0 ǫ γ χ ̺ 0ω 0 0 −(ω +µ h ) 0 0 0,0 0 0 0 −µ v 0 0⎢ 0 0 0 0 0 −(θ +µ⎣v ) θ ⎥⎦−φα φα 0 0 0 0 −(µ v +δ v )where ǫ = −(γ +µ h +δ h ), χ = − β hvαπµ h, ̺ = β hvαπµ h.Λµ v Λµ vFrom which the following system is calculated−µ h v 1 = 0−(ν +µ h )v 2 +νv 3 = 0−(γ +µ h +δ h )v 3 +γv 4 − β hvαπµ hΛµ vv 5 + β hvαπµ hΛµ vv 6 = 0ωv 1 −(ω +µ h )v 4 = 0−µ v v 5 = 0−(θ +µ v )v 6 +θv 7 = 0−φαv 1 +φαv 2 −(µ v +δ v )v 7 = 0.From which the left eigenvector is solved and resulting inv 1 = 0v 2 = v 2 > 0v 3 = (ν +µ h)v 2ν⎫⎪ ⎬v 4 = 0 . (50)v 5 = 0v 6 = θv 7θ +µ vv 7 = φαv 2 ⎪ ⎭µ v +δ vThe theorem in Chavez and Song, 2004, is reproduced below for convenience, andwill be useful to prove local stability of the endemic equilibrium point near R 0 = 1.

35Theorem 1 Consider the following general system of ordinary differential equationswith a parameter φ.dxdt = f(x,φ), f : Rn ×R → R and f ∈ C 2 (R n ×R),where 0 is an equilibrium point of the system (that is, f(0,φ) ≡ 0 for all φ) and( ) ∂fiA1. A = D x f(0,0) = (0,0) is the linearization matrix of the system around∂x ithe equilibrium 0 with φ evaluated at 0. Zero is a simple eigenvalue of A andother eigenvalues of A have negative real parts;A2. Matrix A has a nonnegative right eigenvector w and a left eigenvector v correspondingto the zero eigenvalue.Let f k be the k th component of f anda =b =n∑k,i,j=1n∑k,i=1v k w i w j∂ 2 f k∂x i ∂x j(0,0),v k w i∂ 2 f k∂x i ∂φ (0,0),then, the local dynamics of the system around 0 is totally determined by the sign ofa and b.(i). a > 0, b > 0. When φ < 0 with |φ|

36(iv). a < 0,b > 0. When φ changes from negative to positive, 0 changes its stabilityfrom stable to unstable. Correspondingly a negative unstable equilibriumbecomes positive and locally asymptotically stable.Particularly, if a > 0 and b > 0, then, a subcritical (or backward) bifurcation occursat φ = 0.Computation of a and bFor the system (46), the associated non-zero second order partial derivatives (atDFE) are given bya =3∑k,i,j=2b =v k w i w j∂ 2 f k∂x i ∂x j(0,0) +3∑k,i=27∑k,i,j=6v k w i∂ 2 f k∂x i ∂φ (0,0) + 7∑k,i=6v k w i w j∂ 2 f k∂x i ∂x j(0,0)v k w i∂ 2 f k∂x i ∂φ (0,0).Since v 1 = v 4 = v 5 = 0 for k = 1,4,5; then k = 2,3,6,7 should be considered.That is, the following functions will be used to find a and b from the system (46),f 2 = φαµ hx 7 x 1−(ν +µ h )x 2Λ= φαµ hx 7(N h −x 2 −x 3 )−(ν +µ h )x 2Λ= φαµ hx 7 N h− φαµ hx 7 x 2− φαµ hx 7 x 3,Λ Λ Λ−(ν +µ h )x 2f 6 = β hvαµ h x 3 x 5−(θ +µ v )x 6Λ= β hvαµ h x 3(N v −x 6 −x 7 )−(θ+µ v )x 6Λ= β hvαµ h x 3 N v− β hvαµ h x 3 x 6− β hvαµ h x 3 x 7.Λ Λ Λ−(θ+µ v )x 6Hence, the associated non-zero partial derivatives at the disease-free equilibrium aregiven by∂ 2 f 2= − φαµ h∂x 2 ∂x 7 Λ , ∂ 2 f 2= − φαµ h∂x 3 ∂x 7 Λ∂ 2 f 6= − β hvαµ h∂x 6 ∂x 3 Λ , ∂ 2 f 2= − β hvαµ h∂x 7 ∂x 3 Λ .

Therefore,where(a = v 2 w 2 w 7 − φαµ h( Λ+ v 6 w 7 w 3 − β )hvαµ hΛ37)+v 2 w 3 w 7(− φαµ hΛ= − αµ hΛ [v 2w 7 φ(w 2 +w 3 )+v 6 w 3 β hv (w 6 +w 7 )]= − αµ h[v2 w 2Λ7φ 2 αΓ+v 6 w 3 w 7 β hv ∆ ] < 0,∆ =( )+v 6 w 6 w 3 − β )hvαµ hΛ( ) γ +µh +δ h +νΓ =(γ +µ h +δ h )(ν +µ h )()β hv απµ h νφαΛµ v (θ +µ v )(γ +µ h +δ h )(ν +µ h ) +1 .Forthesignofb,itcanbeshownthattheassociatednon-vanishingpartialderivativesare∂f 2∂φ = αx 7,It follows from the above expressions that∂ 2 f 2∂x 7 ∂φ = α.b = v 2 w 7 α > 0.Thus, a < 0 and b > 0. So (by theorem 1, item (iv)), we have established thefollowing result (note that this result holds for R 0 > 1 but close to 1):Theorem 2 The unique endemic equilibrium guaranteed by Theorem 1 is locallyasymptotically stable for R 0 > 1 but near 1. In addition, by theorem 1, item (i), themalaria basic model undergoes the backward bifurcation when a > 0. This happensonly if v 2 < 0 and w 7 < 0. Otherwise it undergoes forward bifurcation.Numerical simulations are carried out, using an appropriate set of parameter values(satisfying the inequality a > 0 in Theorem 1) to illustrate the backward bifurcationphenomenon of model system (1), (see Figure 2). It should be emphasized thatthe parameter values used are chosen only to illustrate the backward bifurcationphenomenon of the basic model (1), and may not be realistic epidemiologically.

380.1The bifurcation diagram for basic modelForce of infection, λ h0.090.080.070.060.050.040.030.020.01Stable DFEStable EEUnstable EEUnstable DFE00.4 0.6 0.8 1 1.2 1.4Reproduction number, R 0Figure 2: The curve illustrates the bifurcation diagram where the blue continuous curve correspondsto the stable disease-free equilibrium DFE, the blue continuous curve corresponds to thestable endemic equibrium EE, and the dash line depicts instability.This clearly shows the co-existence of two stable equilibria when R 0 < 1, confirmingthat the model system (1) exhibits backward bifurcation for R 0 < 1. Finally, it isworth stating that the disease-free equilibrium of the basic malaria model (1) is notglobally asymptotically stable when the associated reproductive number, R 0 is lessthan unity, owing to the phenomenon of backward bifurcation. Consequently, thisstudy shows that the control of malaria spread in a population when R 0 < 1 willdepend on the initial sizes of the sub-populations of the basic malaria model (1.)3.5 SummaryThebasicdeterministicmodelofthetransmissiondynamicsofmalariawithavaryingtotal human population that incorporated recruitment of new individuals in thesusceptibleclassthroughrecruitmenthasbeenformulated. Analysisofthemodelhasshowed that there exists a domain where the model is epidemiologically meaningfuland mathematically well-posed. The model has been qualitatively analysed for theexistenceandstabilityofthedisease-freeequilibriumandendemicequilibriumpoints.Thereafter, the next generation method has been used to calculate the reproduction

39number, R 0 , as an important parameter that plays a big role in the control of themalaria infection. The stability of the equilibrium points has been analysed usingR 0 . We have proved that the disease-free equilibrium E 0 is locally asymptoticallystable if R 0 < 1, and when R 0 > 1 the endemic equilibrium E 1 appeared. The modelexhibited backward bifurcation where a stable disease-free equilibrium coexists witha stable endemic equilibrium for a certain range of associated reproduction numberless than one. This lead us to the addition of the protected and treated classes tothe basic model in order to achieve the disease-free region, hence the developmentof the malaria model with intervention strategies.

40CHAPTER FOURTHE MALARIA MODEL WITH INTERVENTIONSTRATEGIESInthischapterweextendthebasicmodeltoanalysetheeffectsofinterventionstrategiesin the malaria epidemic.4.1 IntroductionThe Goverment of Malawi has put in place several strategies through the NationalMalaria Control Programme to control malaria. The main strategic areas thathave been identified for scaling-up of malaria control activities, include malariacase management, intermittent preventive treatment (IPT) of pregnant women withsulfadoxine-pyrimethamine (SP), and malaria prevention with special emphasis onthe use of insecticide-treated mosquito nets (ITNs) and Indoor Residual Spraying(IRS), (NSO, 2008).A malaria model with preventive and control measures, temporary immunity, andvarying population sizes is studied as a follow up of the basic model (1). Epidemicmodels for varying population sizes are discussed in Mena-Lorca and Hethcote(1992). The epidemiological compartmental model for the spread of malaria withsusceptibles-exposed-infectious-recovered-susceptibles (SEIRS) pattern for humansand a susceptibles-exposed-infectious pattern for mosquitoes have been proposedin (Ngwa and Shu, (2000)) and (Chitnis et al, (2006)). This has been studied asthe basic model of this project in which we have excluded the direct infectious-tosusceptiblerecovery in humans that is considered in Ngwa and Shu, 2000. It hasbeen established that recoveries and temporary immunity keep the populations atoscillation patterns and eventually converge to a steady state. We extend our basicmodel to study the dynamics of malaria in which intervention strategies for control-

41ling the disease are incorporated and these include the protected and treated classesin the human population.Different sections will be established presenting the model formulation, explanationof the meaning of parameters and variables, and assumptions they satisfy. Theeffective reproduction number, R e , which determines the dynamical behaviour ofthe disease is computed. The existence and stability of the equilibrium points willbe established. We will analyze the stability of the steady states. Lastly we will givea brief discussion of results and make conclusions.4.2 Formulation of the Model.We have added two distinct epidemiological compartments of individuals who are inthe protected class denoted by P(t) and treated class T(t) to the human populationsystem (1). These classes have been included due to the use of insecticide treatedbed nets (ITN) and indoor residual spraying (IRS) as the preventive measures, andtreatement as a control measure. We still consider that the transfer rates betweenthe subclasses are composed of several epidemiological parameters.The fraction τ of the susceptible recruited individuals, noted in the basic model (1),aretakentobeunderpreventivecontrolandjointheprotectedclass. Thelikehoodofinfection is assumed to be reduced by a factor of ϑ. We can note that the protectionis effective if ϑ = 0 and ineffective if ϑ = 1. This is true just because this parameteris defined as the reduction of likelihood of infection by protection. For the protectionto be effective there should be no progression of individuals from the protected classto the exposed individuals. This happens when ϑ = 0. Having a protected class, wehave a proportion g (0 ≤ g ≤ 1) representing susceptible individuals who migrateto malaria free-areas, and thus become partially protected, but become exposed oncethey return to the malaria endemic areas by the force of infection λ hp . This happenswhen the prevention measures are relaxed or when the presumptive interventionsvanish. We assume that (g+τΛ) > 0 in order that there is a nonzero flow of humans

42into the protected class. The proportion g is due to the use of the protected mosquitobed nets and indoor residual spray.The exposed individuals as discussed in the system (1) progress to infected class ata constant rate ν. Individuals who have experienced infection may be treated at aconstant rate σ and they enter a treated class. After a successful treatment, theyrecovertemporarilyatacquiredimmunityrateε.Sincedisease-inducedimmunitydueto malaria is temporary, a fraction ω of individuals leave the recovered state to thesusceptible state while the complementary fraction (1 − ω) move to the protectedclass due to non compliance to treatment. They did not comply fully and theirrecovery is temporal.Therateofinfectionofsusceptibleindividualisλ h ,andtherateatwhichtheinfectedindividuals infect the susceptible mosquito is λ v , see Figure (3).The flow-diagram of the model is shown in Figure (3). The malaria model withintervention strategies has additional state variables in Table 3 and parameters inTable 4 which satisfy the system of equations (51). The above assumptions lead toTable 3: State variables of the model with intervention strategiesSymbol DescriptionP(t)T(t)Number of protected humans at time tNumber of treated humans at time tthe following deterministic system of nonlinear ordinary differential equations which

43Figure 3: The malaria model with interventions flowchartTable 4: Parameters of the model with intervention strategiesSymbol DescriptionτFraction of the susceptible recruited individuals whoare protectedϑgσReduction of likelihood of infection by protectionProgression rate of susceptible humans to protected classTreatment rate for humans from infected state totreated classεRecovery rate for humans from the treated state to therecovered state

44describe the progress of the disease with intervention strategies.⎫dS= (1−τ)Λ−λ h S +ωR−(g +µ h )SdtdP= τΛ+gS +(1−ω)R−λ hp P −µ h PdtdE= λ h S +λ hp P −(ν +µ h )EdtdI= νE −(γ +σ +µ h +δ h )Idt⎪⎬dT= σI −(ε+µ h )T, (51)dtdR= εT +γI −(1−ω)R−(ω +µ h )RdtdX= π −λ v X −µ v XdtdY= λ v X −(θ+µ v )YdtdZ= θY −(µ v +δ v )Z⎪⎭dtN hwhere λ h = β vhαZN h, λ v = β hvαIN h, and λ hp = β vhαϑZN h.The term β vhαϑZPdenotes the rate at which the protected individuals P, get infectedby infectious mosquitoes Z. It indicates that the rate of infection for theprotected individuals is reduced by a factor ϑ.4.3 Mathematical Analysis of the Model with Intervention StrategiesConsequently, adding the first six equations of the model (51), with the case thatthere is no disease-induced death, that is, δ h = δ v = 0), gives dN h= Λ − µ h N h ,dtso that N h (t) → Λ Λas t → ∞. Thus, is an upper bound of N h (t) providedµ h µ hthat N h (0) ≤ Λ . Further, if N h (0) > Λ Λ, then N h (t) will decrease to this level, .µ h µ h µ hSimilar calculation for the vector equations shows that N v → π as t → ∞.µ vHence, the following feasible region;Ω 1 ={(S,P,E,I,T,R,X,Y,Z) ∈ R 9 + : N h ≤ Λ = Nµh,N ∗ v ≤ π }= Nv∗h µ vis positive-invariant and attracting.

454.3.1 Disease-free Equilibrium E 2The disease-free equilibrium E 2 of the mathematical model with interventions (51)is given byE 2 = (S ∗ ,P ∗ ,E ∗ ,I ∗ ,T ∗ ,R ∗ ,X ∗ ,Y ∗ ,Z ∗ ).Setting the system (51) equal to zero, gives X ∗ = π , which is defined as theµ vasymptotic carrying capacity of the mosquito population. Solving for S ∗ and P ∗from the following equations(1−τ)Λ−(g +µ h )S ∗ = 0τΛ+gS ∗ −µ h P ∗ = 0(52)we getfrom first equation of (52), whereS ∗ = (1−τ)Λg +µ h= m 1 Λm 1 = 1−τg +µ h.Substituting in the second equation in (52), the result isτΛ+gm 1 Λ−µ h P ∗ = 0.Thus,whereHenceTherefore,P ∗ = (τ +gm 1)Λµ h= m 2 Λm 2 = τ +gm 1µ h= τµ h +gµ h (g +µ h ) .P ∗ = Λ(τµ h +g)µ h (g +µ h ) .E 2 = (S ∗ ,P ∗ ,E ∗ ,I ∗ ,T ∗ ,R ∗ ,X ∗ ,Y ∗ ,Z ∗ )( (1−τ)Λ= , Λ(τµ )h +g)g +µ h µ h (g +µ h ) ,0,0,0,0, π, 0,0 .µ v(53)

464.3.2 The Effective Reproduction Number, R eThe associated next generation matrices F 2 and V 2 can be found from F 2 and V 2respectively, whereF 2 =⎡ ⎤E ′I ′Y ′⎢Z ′⎥⎣ ⎦T ′=⎡⎢⎣β vh αZSN h+ β vhαϑZPN h0β hv αIXN h00⎤⎥⎦and⎡⎤(µ h +ν)E(γ +σ +µ h +δ h )I −νEV 2 =(θ+µ v )Y.⎢ (µ v +δ v )Z −θY ⎥⎣⎦(ε+µ h )T −σIThe Jacobian matrix of F 2 is given by⎡ ⎤0 0 0 m 00 0 0 0 0F 2 =0 n 0 0 0⎢0 0 0 0 0⎥⎣ ⎦0 0 0 0 0wherem = β vh α+ β vhαϑ(τµ h +g), and n = β hvαπ(g +µ h ).µ h (1−τ) µ v Λ(1−τ)The Jacobian matrix of V 2 is given by⎡ ⎤a 0 0 0 0−b c 0 0 0V 2 =0 0 e 0 0⎢ 0 0 −f q 0⎥⎣ ⎦0 −r 0 0 j(54)(55)

47wherea = (µ h +ν), b = ν, c = (γ +σ +µ h +δ h ), e = (θ +µ v ), f = θ q = (µ v +δ v ),r = σ, j = (ε+µ h ).Then the inverse matrix of V 2 is given by⎡ ⎤10 0 0 0ab 10 0 0ac c1V2 −1 =0 0 0 0e. (56)f 10 0 0⎢ eq q⎥⎣ rb r 1⎦0 0ajc jc jThe product of matrices (54) and (56) gives⎡ ⎤mf m0 0 0eq q0 0 0 0 0F 2 V2 −1=nb n0 0 0.ac c⎢ 0 0 0 0 0⎥⎣ ⎦0 0 0 0 0The eigenvalues areλ i =⎡ ⎤000√ . (57)aceqnbmf⎢ aceq⎣ √ ⎥aceqnbmf ⎦−aceqIt follows that the effective reproduction number for the model (51), denoted by R eis given byR e = ρ(F 2 V2 −1 )√ aceqnbmf=√ (aceq) 2nbmf=aceq .After lengthy algebraic manipulation, we get the effective reproductive number as√β vh β hv αR e =2 θπµ h ν(1+ϑµ h m 2 )(58)µ v Λ(µ h +ν)(γ +σ +µ h +δ h )(θ+µ v )(µ v +δ v )

48with m 2 > 0,whereR ev =β hv θαπµ v (θ+µ v )(µ v +δ v )is the contribution of mosquito population when it infects the humans, whileR eh =β vh αµ h ν(1+ϑµ h m 2 )Λ(µ h +ν)(γ +σ +µ h +δ h )is the human contribution when they infect the mosquitoes.Substituting m 2 in the R e , we get√β vh β hv αR e =2 θπµ h ν[g(1+ϑ)+µ h (1+τϑ)]µ v Λ(g +µ h )(µ h +ν)(γ +σ +µ h +δ h )(θ+µ v )(µ v +δ v ) . (59)The expression for the effective reproduction number, R e has a biological meaningthat is readily interpreted from terms under the square root. The termβ hv θπαµ v (µ v +θ)(µ v +δ v )representsthenumberofsecondaryhumaninfectionscausedbyoneinfectedmosquitovector. The second termβ vh αµ h ν[g(1+ϑ)+µ h (1+τϑ)]Λ(g +µ h )(µ h +ν)(γ +σ +µ h +δ h )represents the number of secondary mosquito infections caused by one infected humanshost. The square root represents the geometric mean R e for an average individualof both species combined. This effective reproduction number serves as aninvasion threshold both for predicting outbreaks and evaluating control strategiesthat would reduce the spread of the disease in the country through the reduction ofthe effective reproduction number and the parameters that enhance the spread ofthe disease due to the increase in the effective reproduction number.The main control measures that have been in place include use of insecticide treatedbed nets (ITN), indoor residual spraying (IRS) and treatment.In order to bring a population below the threshold is to reduce the number of susceptiblesby providing them protection from the disease. From the expression for

49the effective reproduction number R e , we note that the parameters τ, and ϑ playimportant roles in the spread of the disease. The following cases can be considered:The value of the basic reproduction number can be obtained from the value of theeffective reproduction number when control measures are effective (ϑ = 0) since thisis the reduction of likelihood of infection by protection in a sense that τ = 0. Thusthe basic reproduction number of the model (51) without control measures is givenby√β vh β hv αR 0 =2 νθπµ hµ v Λ(µ h +ν)(γ +µ h +δ h )(θ+µ v )(µ v +δ v ) .From the two reproduction numbers, we notice thatR e ≤ R 0for τ ≥ 0, due to reduction of likelihood of infection by protection. This implies thatcontrol intervention strategies have a positive impact on reduction of the spread ofmalaria.The Jacobian matrix, J E2 , of the model (51) evaluated at the disease-free equilibriumpoint (53) is given by⎡⎤−b 0 0 0 0 ω 0 0 −β vh αg −µ h 0 0 0 d 0 0 −ζ0 0 −a 0 0 0 0 0 f0 0 ν −c 0 0 0 0 00 0 0 σ −j 0 0 0 00 0 0 γ ε −h 0 0 00 0 0 −Υ 0 0 −µ v 0 0⎢ 0 0 0 Υ 0 0 0 −e 0⎥⎣⎦0 0 0 0 0 0 0 θ −q(60)whereζ = β vhαϑ(τµ h +g), Υ = β hvαπ(g +µ h ), a = ν +µ h , b = g +µ h ,µ h (1−τ) µ v Λ(1−τ)c = γ +σ +µ h +δ h , e = θ+µ v , q = µ v +δ v , j = ε+µ h d = 1−ω,f = β vh α+ β vhαϑ(τµ h +g), and h = 1+µ h .µ h (1−τ)

50We observe from second, first, sixth, seventh and fifth columns of matrix (60) hasthe following five distinct negative eigenvalues −µ h , −(g+µ h ), −(1+µ h ), −µ v and−(ε+µ h ) respectively. The remaining four eigenvalues are obtained from the 4×4block matrix given by⎡ ⎤−a 0 0 fν −c 0 0D =⎢ 0 Υ −e 0⎥⎣ ⎦0 0 θ −qwhose trace and determinant are given byTr(D) = −(ν +µ h )−(γ +σ +µ h +δ h )−(θ+δ v )−(µ v +δ v ) < 0,Det(D) = (ν +µ h )(γ +σ +µ h +δ h )(θ+µ v )(µ v +δ v )(1−R 2 e) > 0 if R e < 1,where√β vh β hv αR e =2 θπµ h ν[g(1+ϑ)+µ h (1+τϑ)]µ v Λ(g +µ h )(µ h +ν)(γ +σ +µ h +δ h )(θ+µ v )(µ v +δ v ) .Hence we establish the following resultsLemma 4 The disease-free equilibrium E 2 , of the malaria model with interventionstrategies (51), given by (53) is locally asymptotically stable if R e < 1, and unstableif R e > 1.The threshold quantity, R e , measures the average number of secondary cases generatedby a single infected individual in a susceptible human population, where afraction of the susceptible human population is under prevention and the infectedclass is under treatment.When the protection is not practised, the effective reproduction number R e withtreatment as the only intervention strategy being practised becomes√β vh β hv α 2 θπµ h νR et =µ v Λ(µ h +ν)(γ +σ +µ h +δ h )(θ+µ v )(µ v +δ v ) .

51Similarly, if the protection is the only intervention strategy being practised, theeffective reproduction number becomes√β vh β hv α 2 θπµ h ν[g(1+ϑ)+µ h (1+τϑ)]R ep =µ v Λ(g +µ h )(µ h +ν)(γ +µ h +δ h )(θ+µ v )(µ v +δ v ) .4.3.3 Existence and Stability of Endemic Equilibrium Point E 3We analyse the system (51) to obtain the endemic equilibrium point E 3 of the systemand its stability. We consider the equations for the proportions by first scaling thesubpopulations for N h and N v using the new variables.S P E IUsing fractions of population, we set: = s, = p, = e, = i,N h N h N h N hT R X Y Z= ξ, = r, = x, = y, = z, where dN h= Λ−µ h N h −δ h I,N h N h N h N h N h dtdN v= π−µ v −δ v Z, and N h = N v = 1 in the classes S,P,E,I,T,R,X,Y,Z in thedtpopulations and then differentiating with respect to time respectively. This is doneby differentiating the fractions with respect to time t and simplifying as follows:ds= 1 [ ] dSdt N h dt −sdN hdt= 1 N h[(1−τ)Λ−β vh αzsN h +ωrN h −(g +µ h )sN h ]dpdt− s [Λ−µ h N h −δ h iN h ]N h= (1−τ) Λ ] Λ−[+g −δ h i s−β vh αzs+ωrN h N h= 1 [ ] dPN h dt −pdN hdt= 1 N h[τΛ+gsN h +(1−ω)rN h −β vh αϑpN h −µ h pN h ]− p [Λ−µ h N h −δ h iN h ]N h= gs+τ Λ ] Λ+(1−ω)r−β vh αϑzp−[−δ h i pN h N hdedt= 1 [ ] dEN h dt −edN hdt= 1 N h[β vh αzsN h +β vh αϑzpN h −νeN h −µ h eN h ]− e [Λ−µ h N h −δ h iN h ]N h ] Λ= β vh αzs+β vh αϑzp−[+ν −δ h i eN h

didt52= 1 [ ] dIN h dt −idN hdt= 1 [νeN h −µ h iN h −γiN h −δ h iN h −σiN h ]− i [Λ−µ h N h −δ h iN h ]N h ] N hΛ= νe−[+σ +γ +δ h −δ h i iN hdξdtdrdt= 1 [ ] dTN h dt −tdN hdt= 1 [σiN h −µ h tN h −εtN h ]− t [Λ−µ h N h −δ h iN h ]N h ] N hΛ= σi−[+ε−δ h i tN h= 1 [ ] dRN h dt −rdN hdt= 1 N h[εtN h +γiN h −ωrN h −(1−ω)rN h −µ h rN h ]anddxdtdydtdzdt− r [Λ−µ h N h −δ h iN h ]N h ] Λ= εt+γi−[+1−δ h i rN h= 1 [ ] dXN v dt −xdN vdt= 1 N v[π −β hv αixN v −µ h xN v ]− x N v[π −µ v N v −δ v zN v ]= π N v(1−x)−β hv αix+δ v xz= 1 [ ] dYN v dt −ydN vdt= 1 [β hv αixN v −(θ+µ v )yN v ]− y [π −µ v N v −δ v zN v ]N v ] N vπ= β hv αix−[+θ −δ v z yN v= 1 [ ] dZN v dt −zdN vdt= 1 [θyN v −µ v zN v −δ v zN v ]− z [π −µ v N v −δ v zN v ]N v ] N vπ= θy −[+δ v −δ v z zN vdN hdtdN vdt==[ ΛN h−µ h −δ h i]N h[ πN v−µ v −δ v z]N v .

Thus we have the following reduced system of equationsds= (1−τ) Λ ] Λ−[+g −δ h i s−β vh αzs+ωrdt N h N hdp= gs+τ Λ ] Λ+(1−ω)r−β vh αϑzp−[−δ h i pdt N h ] N hdeΛ= β vh αzs+β vh αϑzp−[+ν −δ h i edtN h ]di Λ= νe−[+σ +γ +δ h −δ h i idt N h ]dξ Λ= σi−[+ε−δ h i tdt N h ]dr Λ= εt+γi−[+1−δ h i rdt N hdx= π (1−x)−β hv αix+δ v xzdt N v ]dy π= β hv αix−[+θ−δ v z ydt N v ]dz π= θy −[+δ v −δ v z zdt N v53⎫⎪⎬⎪⎭(61)together with total population sizes N h and N v satisfyingdN hdtdN vdt==[ ΛN h−µ h −δ h i]N h ,[ πN v−µ v −δ v z]N v .The system of proportions involves the total human population size N h in the proportionsfor human population and the total mosquito population size N v in theproportions for mosquito population. The system can now be reduced to a ninedimensionalsystem by eliminating s and x since s = 1−p−e−i−ξ −r andx = 1−y −z, respectively in the feasible region where the model makes biological

sense⎧⎛ ⎞peiξ⎪⎨Ω 2 =r∈ R 9 +N hy⎜ z⎟⎝ ⎠N v ⎪⎩∣54p ≥ 0,e ≥ 0,i ≥ 0,ξ ≥ 0,r ≥ 0,⎫⎪ ⎬p+e+i+ξ +r ≤ 1,N h ≤ Λ µ h,y ≥ 0,z ≥ 0,y +z ≤ 1,⎪N v ≤ π ⎭µ vthat can be shown to be positively invariant with respect to the system (62) whereR 9 + denotes the non-negative cone of R 9 including its lower dimensional faces. Thus,we have the following system of equationsdp= g(1−p−e−i−ξ −r)+τ Λ ⎫+(1−ω)r−β vh αϑzpdt ] N hΛ−[−δ h i pN h ]deΛ= β vh αz(1−p−e−i−ξ −r)+β vh αϑzp−[+ν −δ h i edt] N hdi Λ= νe−[+σ +γ +δ h −δ h i idt N h ]dξ Λ= σi−[+ε−δ h i t⎪⎬dt N h ]. (62)dr Λ= εt+γi−[+1−δ h i rdt N hdN h= Λ−µ h N h −δ h iN hdt]dyπ= β hv αi(1−y −z)−[+θ−δ v z ydt] N vdz π= θy −[+δ v −δ v z zdt N vdN v= π −µ v N v −δ v zN v .⎪⎭dtTo compute the steady states of the system (62), we set the derivatives with respectto the time in (62) equal to zero, and then on simplification the following algebraic

55expressions are obtained.g(1−p−e−i−ξ −r)+τ Λ ] ⎫ Λ+(1−ω)r = β vh αϑpz +[−δ h i pN h [ N hΛβ vh αz(1−p−e−i−ξ −r)+β vh αϑpz = +ν −δ h i]e[ N hΛνe = +σ +γ +δ h −δ h i]iN h [ Λσi = +ε−δ h i]t[ N h ⎪⎬ Λεt+γi = +1−δ h i]r .N hΛN h=β hv αi(1−y −z) =θy =µ h +δ h i[ π+θ −δ v z]y[ N vπ+δ v −δ v z]zN vπN v= µ v +δ v z.Hence the dimesionless proportions are calculated in terms of i as follows:p = d 2i 2 +d 1 i+d 3d 4 i+d 5e = c 1 it = c 2 ir = c 3 i⎪⎭(63)⎫⎪ ⎬(64)y = β hvαi[β hv θαi+(β hv αi+µ v +θ)(µ v +δ v )]−θβhv 2 α2 i 2(β hv αi+µ v +θ)[θβ hv αi+(β hv αi+µ v +θ)(µ v +δ v )]k 1 iz =⎪ ⎭k 6 i+k 5forc 1 = σ +γ +µ h +δ h, c 2 = σν µ h +ε , c σε3 =(µ h +ε)(µ h +1) ,d 1 = (g +τµ h )k 6 +(τδ h +(1−ω)c 3 −c 1 g −g −c 2 g −c 3 g)k 5 ,d 2 = τδ h +(1−ω)c 3 −c 1 g −g −c 2 g −c 3 g,d 3 = [g(1+ϑ)+µ h (1+τϑ)](µ v +δ v )(µ v +θ), d 4 = β vh αϑk 1 +(µ h +g)k 6 ,d 5 = (µ h +g)k 5 , k 1 = β hv θα, k 5 = (µ v +δ v )(µ v +θ), k 6 = β hv α(θ +µ v +δ v ),whereican beobtained bysubstitutingfor e intotheequation two ofthesystem (63)givesc 1 i = c 4i+c 4 pi−c 5 i 2c 7 i+c 8.

It is clear that either i = 0 (for the disease-free equilibrium point E 0 ) or56(endemic equilibrium point E 3 ) forc 4 = β vh β hv α 2 θ, c 5 = β vh αk 1 (1+c 2 +c 3 ),p = d 6i+d 7c 4(65)c 7 = β vh β hv α 2 θ+(µ h +ν)(θ+µ v +δ v )β hv α, c 8 = (µ h +ν)(µ v +δ v )(µ v +θ),d 6 = c 1 c 7 +c 5 and d 7 = c 1 c 8 −c 4 .In the next section, we seek to establish whether the unique endemic equilibriaexists. This is done by making more realistic assumption that the protective controlmeasures may not be totally effective, and thus 0 < ϑ < 1.4.3.4 Existence and Uniqueness of Endemic Equilibrium E 3The existence of the endemic equilibrium in Ω 2 , can be determined by assumingi ≠ 0, that is the equations for p in (64) and (65) will be used.Equating (64) and (65) we getAi 2 +Bi+C = 0 (66)whereA = d 4 d 6 −c 4 d[( 2)]σ +γ +µh +δ h= q 1 (β vh β hv α 2 θ +(µ h +ν)(θ+µ v +δ v )β hv α)ν(1−ω)σε− q 3[τδ h +(µ h +ε)(µ h +1) − σ +γ +µ ]h +δ hg −Hν> 0,B = d 4 d 7 +d 5 d 6 −c 4 d 1= q 1[( σ +γ +µh +δ hν> 0)](µ h +ν)(µ v +δ v )(µ v +θ)−β vh β hv α 2 θ+ (µ h +g)(µ v +δ v )(µ v +θ)q[ ( 2)]+ β vh β hv α 2 (µh +ε)(µ h +1)+σ(µ h +1)+σεθ(µ h +ε)(µ h +1)− β vh β hv θα 2 [(g +τµ h )(θ +µ v +δ v )β hv α+q 4 (µ v +δ v )(µ v +θ)]

57withH = g −σg(µ h +ε) − σεg(µ h +ε)(µ h +1) ,q 1 = β vh β hv α 2 θϑ+(µ h +g)(θ+µ v +δ v )β hv α,( ) σ +γ +µh +δ hq 2 = [β vh β hv α 2 θ +(µ h +ν)(θ+µ v +δ v )β hv α],νq 3 = β vh β hv α 2 θ,( )(1−ω)σε σ +γq 4 = τδ h +(µ h +ε)(µ h +1) − +µh +δ hg−g− σgν µ h +ε − σεg(µ h +ε)(µ h +1) ,C = d 5 d 7 −c 4 d 3= (µ h +g)(µ v +δ v ) 2 (µ v +θ) 2 ( σ +γ +µh +δ hν)(µ h +ν)− β vh β hv θα 2 [g(1+ϑ)+µ h (1+τϑ)](µ v +δ v )(µ v +θ)= (g +µ h )(µ v +δ v ) 2 (µ v +θ) 2 ( σ +γ +µh +δ hν)(µ h +ν)(1−R 2 e).For R e > 1, the existence of endemic equilibria is determined by the presence in(0,1] of positive real solutions of the quadratic expression (66).( ) σ +γC = (g +µ h )(µ v +δ v ) 2 (µ v +θ) 2 +µh +δ h(µ h +ν)(1−R 2νe) < 0.From the quadratic theorem, if x 1 ,x 2 are the roots of equation (66), then theirproductx 1 x 2 = C A .Since C < 0 and A > 0, then C . Hence, there exists exactly one positive endemicAequilibrium for i ∈ (0,1] whenever R e > 1. This gives the threshold for the endemicpersistence.Therefore, we have proved the existence and uniqueness of the endemic equilibriumE 3 for the system (61). This result is summarized in the following theorem:Theorem 3 If R e > 1, the system (61) has a unique endemic equilibrium E 3 .4.3.5 Bifurcation AnalysisFor models that exhibit forward bifurcation, the requirement R 0 < 1 (where R 0is the associated basic reproduction number) is necessary and sufficient for disease

58elimination (Hethcote, 2000). However, in the presence of a control measure (such asvaccination), the asymptotic dynamics of the disease transmission model is typicallygoverned by another threshold quantity, known as effective reproduction number,R e . Some studies have shown that whilst R e < 1 is necessary for the disease control,thisrequirementmaynotbesufficient. Thisisowingtothephenomenonofbackwardbifurcation, where a stable endemic equilibrium co-exists with a stable disease-freeequilibrium for R e . This phenomenon has been observed in numerous disease transmissionmodels, such as those for vaccination (Elbasha, 2006 and Kribs-zaleta etal., 2000) and vector-borne diseases (Garba et al., 2008).In a backward bifurcation setting, disease control is only feasible if R e is reduced furtherto values below another sub-threshold less than unity. Clearly, this phenomenonhas important public heath implications, since it has direct relevance on whether ornot the disease can be effectively controlled even when the classical epidemiologicalrequirement of having the associated reproduction number is satisfied. It is instructive,therefore, to explore whether or not the model (51) exhibits the phenomenonof backward bifurcation. This is done below.SupposeE 3 = (S ∗∗ ,P ∗∗ ,E ∗∗ ,I ∗∗ ,T ∗∗ ,R ∗∗ ,X ∗∗ ,Y ∗∗ ,Z ∗∗ ),represents any arbitrary endemic equilibrium of the model (51) (that is an equilibriumin which at least one of the infected components is non-zero). The existenceof backward bifurcation will be explored using the Centre Maniford Theory(Carr, 1981, and Castillo-Chavez and Song, 2004).To apply this theory, it is convenient to carry out the following change of variables.Let S = x 1 , P = x 2 , E = x 3 , I = x 4 , T = x 5 , R = x 6 , X = x 7 , Y = x 8 , Z = x 9so that N h = x 1 +x 2 +...+x 6 and N v = x 7 +x 8 +x 9 . By using vector notationX = (x 1 ,x 2 ,...,x 9 ) T , the model (51) can be re-written in the formdXdt = F(X),

59with F = (f 1 ,f 2 ,...,f 9 ) T , as follows:wheredx 1dtdx 2dtdx 3dtdx 4dtdx 5dtdx 6dtdx 7dtdx 8dtdx 9dt= (1−τ)Λ−λ h x 1 +ωx 6 −k 1 x 1 = f 1= τΛ+gx 1 +(1−w)x 6 −λ hp x 2 −µ h x 2 = f 2= λ h x 1 +λ hp x 2 −k 2 x 3 = f 3= νx 3 −k 3 x 4 = f 4⎫⎪ ⎬= σx 4 −k 4 x 5 = f 5 (67)= εx 5 +γx 4 −k 5 x 6 = f 6= π −λ v x 7 −µ v x 7 = f 7= λ v x 7 −k 6 x 8 = f 8= θx 8 −k 7 x 9 = f⎪9⎭k 1 = g +µ h , k 2 = ν +µ h , k 3 = σ +γ +µ h +δ h , k 4 = ε+µ h , k 5 = 1+µ h ,k 6 = θ +µ v , and k 7 = µ v +δ v .Consider the case when R e = 1. Suppose, further, that φ = φ ∗ = β vh is chosen as abifurcation parameter. Solving for φ from R e = 1, givesφ = φ ∗ = µ vΛ(g +µ h )(µ h +ν)(σ +µ h +δ h )(θ+µ v )(µ v +δ v ).β hv α 2 θπµ h ν[g(1+ϑ)+µ h (1+τϑ)]The Jacobian of the transformed system (67) at the DFE,(E 2 ), J(E 2 ), is given by⎡⎤−k 1 0 0 0 0 ω 0 0 −φαg −µ h 0 0 0 1−ω 0 0 −k 80 0 −k 2 0 0 0 0 0 k 90 0 ν −k 3 0 0 0 0 00 0 0 σ −k 4 0 0 0 0(68)0 0 0 γ ε −k 5 0 0 00 0 0 −Υ 0 0 −µ v 0 0⎢ 0 0 0 Υ 0 0 0 −k 6 0⎥⎣⎦0 0 0 0 0 0 0 θ −k 7whereΥ = β hvαπ(g +µ h ), k 8 = φαϑ(τµ h +g)µ v Λ(1−τ) µ h (1−τ)and k 9 = φ[αµ h(1−τ)+ϑα(τµ h +g)].µ h (1−τ)

60The right eigenvector of J(E 2 )| φ=φ ∗ is given by w = (w 1 ,w 2 ,...,w 9 ) T . Solving givesthe system−k 1 w 1 +ωw 6 −φαw 9 = 0gw 1 −µ h w 2 +(1−ω)w 6 −k 8 w 9 = 0−k 2 w 3 +k 9 w 9 = 0⎫Solving the system (69) givesw 1 = ωw 6 −φαw 9g +µ hνw 3 −k 3 w 4 = 0σw 4 −k 4 w 5 = 0εw 5 +γw 4 −k 5 w 6 = 0− β hvαπ(g +µ h )w 4 −µ v w 7 = 0µ v Λ(1−τ)β hv απ(g +µ h )w 4 −k 6 w 8 = 0µ v Λ(1−τ)θw 8 −k 7 w 9 = 0⎪⎭⎪⎬. (69)w 2 = gµ h(1−τ)w 1 +(1−ω)(1−τ)µ h w 6 −φαϑ(τµ h +g)w 9gµ h (1−τ)w 3 = φ[αµ h(1−τ)+αϑ(τµ h +g)]w 9µ h (1−τ)(ν +µ h )νw 3w 4 =σ +γ +µ h +δ hw 5 = σw 4ε+µ hw 6 = εw 5 +γw 41+µ hw 7 = − β hvαπ(g +µ h )w 4µ 2 vΛ(1−τ)w 8 = β hvαπ(g +µ h )w 4µ v Λ(1−τ)(θ +µ v )w 9 = w 9 > 0.Similarly, J(E 2 )| φ=φ ∗ has a left eigenvector v = (v 1 ,v 2 ,...,v 9 ) T . Transposing the

61matrix (68) gives⎡⎤−k 1 g 0 0 0 0 0 0 00 −µ h 0 0 0 0 0 0 00 0 −k 2 ν 0 0 0 0 00 0 0 −k 3 σ γ −Υ Υ 00 0 0 0 −k 4 ε 0 0 0.ω 1−ω 0 0 0 −k 5 0 0 00 0 0 0 0 0 −µ v 0 0⎢ 0 0 0 0 0 0 0 −k 6 θ⎥⎣⎦−φα −k 8 k 9 0 0 0 0 0 −k 7Hence we get the following system−k 1 v 1 +gv 2 = 0⎫−µ h v 2 = 0−k 2 v 3 +νv 4 = 0−k 3 v 4 +σv 5 +γv 6 − β hvαπ(g +µ h )v 7 + β hvαπ(g +µ h )v 8 = 0µ v Λ(1−τ) µ v Λ(1−τ)−k 4 v 5 +εv 6 = 0⎪⎬.ωv 1 +(1−ω)v 2 −k 5 v 6 = 0−µ v v 7 = 0−k 6 v 8 +θv 9 = 0−φαv 1 −k 8 v 2 +k 9 v 3 −k 7 v 9 = 0.⎪⎭Here we get the non-zero left eigenvector as followsv 3 = v 3 > 0v 4 = (ν +µ h)v 3νv 7 = 0v 8 = θv 9θ +µ vv 9 = φ[αµ h(1−τ)+αϑ(τµ h +g)].µ h (1−τ)(µ v +δ v )The theorem in Chavez and Song, 2004 is reproduced for convenience as theorem 2above.

62For the transformed system (67), the associated non-zero partial derivatives of theright-hand side functions, f i , i = 1, 2, ..., 9, are given bya 1 =4∑k,i,j=3b 1 =v k w i w j∂ 2 f k∂x i ∂x j(0,0) +4∑k,i=39∑k,i,j=8v k w i∂ 2 f k∂x i ∂φ (0,0) + 9∑k,i=8v k w i w j∂ 2 f k∂x i ∂x j(0,0)v k w i∂ 2 f k∂x i ∂φ (0,0)for k = 3,4,8,9, since for k = 1,2,5,6,7, we have v 1 = v 2 = v 5 = v 6 = v 7 = 0.We can find a and b from the following functions which have been developed fromthe system (67).f 3 = φα(g +µ h)Λ(1−τ) x 9x 1 + φαϑ(g +µ h)x 9 x 2 −k 3 x 3Λ(1−τ)= φα(g +µ h)x 9Λ(1−τ)−(ν +µ h )x 3 −k 3 x 3(N h −x 3 −x 4 )+ φαϑ(g +µ h)x 9(N h −x 3 −x 4 )Λ(1−τ)= φα(g +µ h)x 9 N hΛ(1−τ)+ φαϑ(g +µ h)x 9 N hΛ(1−τ)f 8 = β hvα(g +µ h )x 4 x 7Λ(1−τ)− φα(g +µ h)x 9 x 3Λ(1−τ)− φαϑ(g +µ h)x 9 x 3Λ(1−τ)−k 6 x 8− φα(g +µ h)x 9 x 4Λ(1−τ)− φαϑ(g +µ h)x 9 x 4Λ(1−τ)= β hvα(g +µ h )x 4(N h −x 8 −x 9 )−(θ +µ v )x 8 −k 6 x 8Λ(1−τ)= β hvα(g +µ h )x 4 N hΛ(1−τ)− β hvα(g +µ h )x 4 x 8Λ(1−τ)− β hvα(g +µ h )x 4 x 9Λ(1−τ)Furthermore, the associated non-zero partial derivatives are given by−k 3 x 3 ,−k 6 x 8 .∂ 2 f 3∂x 3 ∂x 9= − φα(g +µ h)Λ(1−τ) (1+ϑ)∂ 2 f 3= − φα(g +µ h)∂x 4 ∂x 9 Λ(1−τ) (1+ϑ)∂ 2 f 8= − β hvα(g +µ h )∂x 8 ∂x 4 Λ(1−τ)∂ 2 f 8= − β hvα(g +µ h ).∂x 9 ∂x 4 Λ(1−τ)

Therefore,where63a 1 = v 3 w 3 w 9[− φα(g +µ h)Λ(1−τ) (1+ϑ) ]+ v 8 w 8 w 4[− β hvα(g +µ h )Λ(1−τ)[+v 3 w 4 w 9 − φα(g +µ ]h)[ Λ(1−τ) ]+v (1+ϑ) 8 w 9 w 4 − β ]hvα(g +µ h )Λ(1−τ)= − α(g +µ h)Λ(1−τ) [v 3w 9 φ(1+ϑ)(w 3 +w 4 )+v 8 w 4 β hv (w 8 +w 9 )]= − α(g +µ h)Λ(1−τ) [v 3w 2 9φ 2 (1+ϑ)κ+v 8 w 4 w 9 β hv η] < 0,κ = (σ +γ +µ h +δ h )[αµ h (1−τ)+αϑ(τµ h +g)]+φν[αµ h (1−τ)+αϑ(τµ h +g)],µ h (1−τ)(ν +µ h )(σ +γ +µ h +δ h )η = νφβ hvαπ(g +µ h )[αµ h (1−τ)+αϑ(τµ h +g)]µ v µ h Λ(1−τ) 2 (ν +µ h )(θ+µ v )(σ +γ +µ h +δ h ) +1.It is shown that the associated non-vanishing partial derivatives for the sign of b 1 are∂f 3= φα+ φαϑ(τµ h +g)∂x 9 µ h (1−τ)∂ 2 f 3∂x 9 ∂φ = αµ h(1−τ)+αϑ(τµ h +g).µ h (1−τ)From the above equation we get( )µh (1−τ)+ϑ(τµ h +g)b 1 = v 3 w 9 α> 0.µ h (1−τ)Since the coefficient b 1 is always positive, the results are summarized by the followingtheorem belowTheorem 4 Since a 1 < 0 and b 1 > 0, then the endemic equilibrium of the malariamodel with intervention strategies guaranteed by the Theorem 1 is locally asymptoticallystable for R e > 1.The forward bifurcation is illustrated by simulating the malaria model with interventionsstrategies (51) using the following sets of parameters (chosen arbitrarilywith the objective of illustrating the backward bifurcation phenomenon): π =0.0089,β vh = 0.02341,δ h = 0.002,ω = 0.000136986,α = 0.123,ν = 0.07692andµ v =0.05 and other parameters are as shown in the Table (5). The bifurcation diagram

640.35The bifurcation diagram for intervention model0.3Force of infection, λ h0.250.20.150.10.05Stable DFEStable EEUnstable DFE00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Reproduction number, R eFigure 4: Bifurcation diagram for the model system (51) obtained from numerical simulations,which show that the disease-free and endemic equilibria exchange stability when R e = 1 for arbitraryset of parameter values. The blue continuous curves depicts stable equilibria and dashed redcurves depicts unstable equilibria.illustrated in Figure 4 shows forward bifurcation. The numerical simulations ofthe model system (51) using a program coded in MATLAB indicate that there istranscritical bifurcation at R e = 1, where stability changes from the disease-freeequilibrium to endemic equilibrium.The above result indicates the impossibility of backward bifurcation in the malariamodel with intervention strategies, since it has no endemic equilibria when R e < 1.This explains that the malaria model with intervention strategies (51) has a globallyasymptoticallystable disease-free equilibrium whenever R e ≤ 1, and has a uniqueendemic equilibrium point whenever R e > 1. The unique endemic equilibrium pointis locally asymptotically stable at least near R e = 1. A global stability result fordisease-free equilibrium, E 02 is given below.4.3.6 Global Stability of the Disease-free Equilibrium E 02Here, the global asymptotic stability property of the disease-free equilibrium E 02 ofthe model (51) will be explored for the case when δ h = δ v = 0. By considering the

65disease-free nation in the model (51), it follows that S = Nh ∗ −P −E −I −T −Rand X = Nv ∗ −Y −Z at the steady-state.Hence, the global stability of E 02 , can be established by considering the followinglimiting system of the malaria with intervention strategies (51).dPdtdEdtdIdtdTdtdRdtdYdtdZdt= τΛ+g(N ∗ h −P −E −I −T −R)+(1−ω)R−λ hpP −µ h P= λ h (N ∗ h −P −E −I −T −R)+λ hpP −(µ h +ν)E= νE −(γ +σ +µ h )I= σI −(ε+µ h )T= εT +γI −(1−ω)R−(ω +µ h )R= λ v (N ∗ v −Y −Z)−(θ+µ v )Y= θY −µ v Zin the positive-invariant region,⎫⎪⎬⎪⎭(70)where,Ω 2 = { (P,E,I,T,R,Y,Z) ∈ R 7 + : P +E +I +T +R ≤ N ∗ h, Y +Z ≤ N ∗ v},λ h = β vhαZ, λ hp = β vhθαZ, and λ v = β hvαI.N h N h N hThe associated threshold number for the limiting model (70), denoted by R ′ e is givenbyR ′ e = ρ(F 3 V2 −1 )√β vh β hv N ∗=vα 2 θ(A 1 +B 1 )(Nh ∗)2 µ v (θ+µ v )(µ v +δ v )(µ h +ν)(γ +σ +µ h +δ h )where, A 1 = νN ∗ h , B 1 = ϑP ∗ (ν +µ h ),(71)the matrix V 2 is as defined in (55). The matrix F 3 is given by⎡0 0 0 β vh α+ β vhαϑP ∗ ⎤0N ∗ h0 0 0 0 0F 3 =0 β hvαNv∗ 0 0 0.Nh∗ ⎢0 0 0 0 0⎥⎣⎦0 0 0 0 0Hence, the following results is established

66Lemma 5 The disease-free equilibrium E 2 , of the limiting system (70), given by (53)is locally asymptotically stable if R ′ e < 1 and unstable if R ′ e > 1.The feasible region can be defined asΩ 2 = {(P,E,I,T,R,Y,Z) ∈ Ω 1 : P ≤ P ∗ }.Hence, the following result is claimed.Theorem 5 The disease-free equilibrium of the model (70) given by (53), is globallyasymptotically stable in Ω 2 if R ′ e < 1.Proof: It can be seen from the first equation of the limiting system (70) thatdPdt= τΛ+g(Nh ∗ −P −E −I −T −R)+(1−ω)R−λ hpP −µ h P≤ τΛ+gNh ( ∗ −(g +µ h)P) τΛ+gN∗= (g +µ h ) h−Pg +µ h= (g +µ h )(P ∗ −P).Hence, P(t) ≤ P ∗ −(P ∗ −P 0 )e −(g+µ h)t .It follows that either P(t) approaches P ∗ asymptotically or P(t) ≤ P ∗ in finite timedP(note thatdt < 0 if P(t) > P∗ ). Therefore the set Ω 2 is positively-invariantand attracting. Further, the equations for the infected components of (70) can bewritten as⎡ ⎤ ⎡ ⎤E ′EI ′IY ′= (F 3 −V 2 −Q)Y⎢Z ′⎥ ⎢Z⎥⎣ ⎦ ⎣ ⎦T ′ T

67where the matrices F 3 and V 2 are as defined above, and the matrix Q is given by⎡β vh ϑαP λ h λ h 0(1− ∗ P ) ⎤0Nh∗ P ∗ 0 0 0 0 0Q =0 0 λ v λ v 0.⎢ 0 0 0 0 0⎥⎣⎦0 0 0 0 0Since P ≤ P ∗ (for all t ≥ 0) in Ω 2 , it follows that matrix Q is non-negative. Thus,⎡ ⎤ ⎡ ⎤E ′EI ′IY ′= (F 3 −V 2 )Y. (72)⎢Z ′⎥ ⎢Z⎥⎣ ⎦ ⎣ ⎦T ′ TIf R ′ e < 1, then ρ(F 3 V −12 < 1) (from the local stability result given in the Lemmaof (4)) which is equivalent to F 3 − V 2 (van den Driessche and Watmough, 2002).It follows that the linearised differential inequality system (72) is stable wheneverR ′ e < 1. Consequently, by comparison theorem (Lakshmikantham et al., 1989,) itfollows that (E,I,Y,Z,T,R) → (0,0,0,0,0,0). Substituting E = I = Y = Z =T = R = 0 into the first equation of the system (70) gives P(t) → P ∗ as t → ∞.Further, since Ω 2 is positively-invariant, it follows that disease-free equilibrium, E 2 ,is globally asymptotically stable in Ω 2 if R ′ e < 1.The epidemiological implication of the above result is that malaria will be eliminatedfrom the community if the prevention and treatment can lead to a situation whereR e is less than unity.4.4 SummaryWe have added two distinct epidemiological compartments of populations who arein the protected class (P(t)) and treated (T(t)) to the basic model due to the controlmeasures, prevention and treatment. We have maintained that the transfer

68rates between the subclasses are composed of several epidemiological parameters.Mathematical analysis of the model has shown that the existing domain is positiveinvariantandattracting.Themodelhasbeenqualitativelyanalysedfortheexistenceand stability of the disease-free equilibrium points. We have established the effectivereproduction number, R e , as a tool for effective disease management. If R e < 1, thedisease cannot persist in a country, hence R e is the useful indication of the effort requiredto eliminate an infection. It has been noted that R e ≤ R 0 which implied thatthe increasing preventive and control measures has a great effect on the reduction ofR e .Furthermore, themodelhasindicatedthatitexhibitsthephenomenonofforwardbifurcation, in which in the absence of a low-level unstable equilibrium when R e < 1and a stable equilibrium bifurcating from the disease-free equilibrium when R e > 1,arise naturally when the disease does not invade when R e = 1. This phenomenonhas important public health implications as it has direct relevance on whether ornot the disease can be effectively controlled even when the classical epidemiologicalrequirement of having associated reproduction number is satisfied. Therefore, thestability of the system can be estabished further by verifying some analytical results.

69CHAPTER FIVENUMERICAL SIMULATIONSWe carry out numerical simulations using a fourth order Rung-Kutta scheme inMatlab. The main aim is to verify some of the analytical results on the stabilityof the system (51). Some of the parameter values were obtained from the NationalStatistical Office (NSO) in Zomba, Malawi and others were obtained from literatureas well as assumptions. To control malaria, the Government of Malawi has put inplace several strategies through the National Malaria Control Programme.We simulate the basic malaria model in the absence of any intervention and themalaria model with intervention strategies, and find out the effects of varying eachintervention parameter. The figures are plotted using the parameter values in table(5) and the initial conditions: S 0 = 100, P 0 = 90, E 0 = 70, I 0 = 50, T 0 =35, R 0 = 30, X 0 = 1000, Y = 900, Z 0 = 700. These are estimated average valuesof the population which participated in the malaria survey which was conducted inthe country. The rates are given per day.

705.1 Table of Parameter Values of the ModelTable 5: Parameter values of the ModelSymbol Value Source Symbol Value SourceΛ 0.0015875 Niger, 2008 π 0.071 Niger, 20081µ h Blayneh, 2009 µ v 1/15 Blayneh, 2009(60×365)ν 1/17 Blayneh, 2009 δ h 0.02326 NSO, 2008δ v 0.07 Assumed ϑ 0.475 Assumed1ω Blayneh, 2009 θ 0.0017 Gumel, 2009(20×365)σ 0.01 NSO, 2008 α 0.86 Gumel, 2009β vh 0.86 Niger,2008 β hv 0.083 Miranda, 2009τ 0.11 Miranda, 2009 g 0.0833 NSO, 2008ε 0.00722 Gumel, 2009 γ 0.0035 Chitnis, 2008

715.2 Dynamics of Human Population State Variables of the Basic ModelThe simulation of basic model has been conducted to find out the dynamics ofthe disease in the population when there is no intervention to reduce or eradicatethe disease. In the absence of interventions strategies, the susceptible populations12000110002.2Susceptible Individuals100009000800070006000Exposed Individuals21.81.61.450001.2(a)40000 1 2 3 4 5Time in days(b)2.4 x 104 Time in days10 1 2 3 4 51.77000Infected Individuals1.8 x 104 Time in days80001.61.51.41.31.2Recovered Individuals600050004000300020001.11000(c)10 5 10 15 20 25(d)00 50 100 150Time in daysFigure 5: Illustrates the changes in the four state variables of the basic malaria model showingthe dynamics, with time, of (a) susceptible human individuals, (b) exposed human individuals , (c)infected human individuals and (d) shows the dynamics of recovered human individuals.decreases as shown by Figure (5)(a).This explains that the susceptible populationwill continue being exposed to the disease, as a result, the exposed population willincrease, see Figure (5)(b) with R 0 = 1.3034. This means that the plasmodium iscontinuously multiplying since there is no means of reducing or eradicating it. Theinfected population increases due to the increase in the exposure to the disease, seeFigure (5)(c). It supports the theorem that the disease is endemic when R 0 > 1. Therecovered population decreases as a result of availability of malaria in the society inwhich no any intervention is being practised. Figure (5)(d) shows that few recover

72with natural immunity.We also looked at the prevalence in the population. Prevalence is defined as the ratioof which the number of cases of a disease in a population and with the number ofindividuals in a population at a given time. The prevalence graph, Figure (6) withR 0 = 1.3034, increases with bigger gradient for a while and then drops asymptotically.This happens just because of the reduced number of susceptible individualswith time as most of the individuals in the society become affected with the disease,see Figure (5)(a) and Figure (5)(c).5.3 Prevalence in the Basic Malaria Model0.80.70.6Prevalence0.50.40.30.20.100 50 100 150Time in daysFigure 6: Represents changes of prevalence with time.5.4 Dynamics between Infected and Susceptible Human PopulationsThe relationship of the susceptible and infected human populations were considered.Thegraph(7)withR 0 = 0.1287showsthatataninitialstage, thesusceptiblehumanpopulation was free from the disease. The infection in the population is shown thatit increases with time when there is no intervention being practised. As most ofthe susceptible individuals were getting infected with time, the infected individuals

73180001600014000Infected Individuals1200010000800060004000200000 2 4 6 8 10Susceptible individualsx 10 4Figure 7: Illustrates the dynamics of infected human population against susceptible human population2 x 104 Susceptible individualsInfected Individuals1.81.61.41.210.80.6α 1= 60α = 40 2α 3= 80α 4= 200.40.200 2 4 6 8 10x 10 4Figure 8: Represents the phase plane of infected human population versus susceptible humanpopulation

74increases. This is evidenced further when the parameter of mosquito biting ratevaried, see Figure (8). It shows that the susceptible population is becoming lower asthe infectives increase.

755.5 Dyamics of the Human Populations Variables of the Model withIntervention StrategiesWesimulatedthemalariamodelwithinterventionstrategiestofindoutthedynamicsofthehumanstateofvariables. Thereafterwewillcomparetheprotectedandtreatedclasses to find out the impact of these interventions if they are combined.140120Susceptible IndividualsProtected Individuals100Population8060402000 50 100 150Time in daysFigure9: Illustratesthechangesinthesusceptibleandprotectedhumanindividuals ofthemalariamodel with intervention strategies for R e = 0.0850 and R 0 = 0.0927The figure (9) contains the the susceptible and protected populations graphs. Thereis a steady decrease in the susceptible population and an increase in the protectedpopulation. This shows that most recruited individuals join the protected class.Therefore the exposure of the recruited individuals to malaria is reduced. Thishighlights the effectiveness of preventive measures in controlling malaria disease.It has been shown in the Figure (10) that the exposed population decreases exponentiallywith time because of the preventive measure which is being practised.The trend remained the same despite variation in the biting rate of the mosquito α.Hence the protection as a measure of intervention reduces the contact rate betweenthe female anopheles mosquito and the host. This means that prevention is moreimportant in the beginning of disease outbreak.

76Exposed Individuals2500200015001000d = 0.29 1d 2= 0.029d 3= 0.37d 4= 0.9950000 50 100 150 200 250Time in daysFigure 10: Shows the phase diagrams of the exposed human individuals of the malaria modelwith intervention strategies for R e = 0.0850 and R 0 = 0.0927Infected Individuals2500200015001000ν 1= 0.05882ν 2= 0.06667ν 3= 0.05263ν 4= 0.150000 50 100 150 200 250 300 350 400Time in daysFigure 11: Represents the phase diagrams of the infected human individuals of the malaria modelwith intervention strategies for R e = 0.0850 and R 0 = 0.0927

77Furthermore, the results in the Figure (11) indicate that the infected populationdecreases with time due to the decrease in the exposed population. This shows thatspreading of the disease is being reduced due to the preventive measures which arebeing practised.180160140Treated Individuals120100806040200 50 100 150 200 250 300Time in daysFigure 12: Illustrates the treated human individuals of the malaria model with interventionstrategies for R e = 0.0850 and R 0 = 0.0927The graphs of the treated individuals in Figure (12) and Figure(13) increases whenthere are more infectives and decreased following the trend of the decrease in theinfected individuals. The graphs of treated human individuals have highlighted thattreatment measure is more needed when there are infectives in the society. As itcan be seen in the Figure (11) and Figure (13), if we have to use only one control,then, prevention is more effective than treatment to diminish the size of latent andinfectious hosts.The simulations of the recovered population in Figure (14), shows that there is anincrease in the recovered population because of the control measures which are beingpractised. Our conclusion from this is that these control measures have positiveimpact in the reduction or eradiction of the spreading of malaria disease.

78220200180σ 1= 0.01σ 2= 0.015σ 3= 0.02160Treated Individuals140120100806040200 50 100 150 200 250 300Time in daysFigure 13: Shows the phase potrait of the treated human individuals with time of the malariamodel with intervention strategies for R e = 0.0850 and R 0 = 0.0927Recovered Individuals109876543J 1= 0.00722J 2= 0.00952J 3= 0.00645210 20 40 60 80 100Time in daysFigure 14: Illustrates the phase diagram of recovered human individuals with time of the malariamodel with intervention strategies for R e = 0.0850 and R 0 = 0.0927

795.6 Phase Diagrams of Prevalence, Human and Mosquito Populations90008000Protected human IndividualsExposed mosquito population70006000Populations5000400030002000100000 100 200 300 400 500Time in daysFigure 15: Illustrates the dynamics of protected human population and exposed mosquito populationExposed mosquito population and protected human population graph, Figure (15),shows the decreasing survival probability of a mosquito as more humans are coveredby ITNs and IRS. These control measures reduce the availability of hosts, and killmosquitoes that are attempting to feed, in such way reducing malaria transmission.The Figure (16) shows the dynamics of proportion of infected individuals. There isan increase in the infection when the prevention was low, and with time the infectiondecreases with the increase in the protection, inspite of varying the infection rate ν.ITNs, as a preventive measures are effective and economical method to kill themosquitoes. The netting also acts as a protective barrier against bites, making it anideal prevention mechanism in poor areas.The prevalence shown in Figure (17), of the model with intervention strategies indicatesthat with time the disease is reduced asymptotically. When there is variationin the treatment parameter σ, as shown in Figure (17), we get the same result. In addition,in Figure (18), the variation of the infection rate ν showed that the prevalenceis reduced asymptotically to zero with time. This means the infection is reducing

80Infected Individuals1501401301201101009080ν 1= 0.05263ν 2= 0.05882ν = 0.066673ν 4= 0.076927060500 100 200 300 400 500Protected individualsFigure 16: Represents the phase plane, infected human individuals versus susceptible humanindividuals0.90.80.70.6σ 1=0.01σ 2=0.03σ 3=0.02σ 4=0.9Prevalence0.50.40.30.20.100 100 200 300 400 500 600 700 800Time in daysFigure 17: Illustrates the changes of prevalence with time as treated rate, σ, varies.

810.90.80.70.6ν 1= 0.05ν 2= 0.0667ν = 0.05883ν 4= 0.0769Prevalence0.50.40.30.20.100 100 200 300 400 500 600 700 800Time in daysFigure 18: Shows the impact of varying the infection rate ν.with time.5.7 Simulation of Protected and Treated Human PopulationsThe Figure (19) contains the plot of the protected and treated individuals. It hasbeen shown that at an early stage the treatment has more effect compared to prevention.But with time, the prevention measure plays a bigger part in reducing thespread of malaria disease. More evidence has been shown further with the variationin the prevention rate g as in Figure (20) and the treated rate σ, Figure (21). Thereis no change in the prevention graph. This explains that the effect of preventionremains constant provided it has positive impact to the reduction of the spreadingof the disease. The pattern for the treated graph remains the same even thoughthere are some differences in the gradient as we vary the treated rate. Treatment ishighly needed when there are more infected individuals,see Figure (11). In perspective,one could conclude from the controls in Figure (20) and Figure (21) that weshould give full prevention effort in the beginning of emergence of the disease whilegiving full treatment effort in the middle of time interval when control efforts arepractised. This means that prevention is more important in the beginning of the

8235003000Protected IndividualsTreated Individuals2500Populations20001500100050000 50 100 150 200 250 300 350 400Time in daysFigure 19: Shows the graphs of the protected human individuals and treated human individualswith timeProtected and Treated Individuals350030002500200015001000500g 1=0.0833g 2=0.78g 3=0.00613g 4=0.0549g 5=0.62100 50 100 150 200 250 300 350 400Time in daysFigure 20: Illustrates the phase diagram of the protected and treated human population withtime as we vary the protection rate g.

83Protected and Treated Individuals350030002500200015001000500σ 1=0.01σ 2=0.56σ =0.015 3σ =0.019 4σ 5=0.02500 50 100 150 200 250 300 350 400Time in daysFigure 21: Shows the phase diagram of the protected and treated human population with timeas we vary treated rate σ.disease outbreak. On the other hand, treatment is more important while the diseaseprevails.Hence , in such scenario the treatment programs must be complemented with otherinterventions (such as vector reduction strategies and personal protection) to havea realistic chance of effectively controlling the disease spread. Our conclusion fromthisisthatinterventionpracticesthatinvolvebothpreventionandtreatmentcontrolsyield a relatively better result. It shows that the combination of these interventionscan play a positive role in reducing or eradicating the disease in the country.Therefore, control efforts aimed at lowering the infectivity of infected individuals tothe mosquito vector will contribute greatly to the lowering of the malaria transmissionand this will eventually lower the prevalence of malaria and the incidence ofthe disease in the country. This can be achieved by prompt provision of effectiveprevention measures and antimalarial drug for treatment to reduce transmission andmorbidity.

84CHAPTER SIXCONCLUSION AND RECOMMENDATION6.1 ConclusionWe presented a basic deterministic model of the transmission dynamics of malaria.The model considered a varying total human population that incorporated recruitmentof new individuals into the susceptible class through either birth or immigration.Then protected and treated classes were added to the basic model to formulatethe malaria model with intervention strategies in order to assess the potential impactof protection and treatment strategies on the transmission dynamics of the disease.Our model incorporated features that are effective to control the transmission ofmalaria disease in Malawi. Analysis of the model showed that there exists a domainwhere the model is epidemiologically and mathematically well-posed. The prominentparameter in our model, the basic reproduction number, R 0 , as a modified controlintervention measure was computed. The model was then qualitatively analysed fortheexistenceandstabilityoftheirassociatedequilibria. Itwasprovedthatunderthecondition that R 0 < 1 the disease-free equilibrium E 0 is locally asymptotically stable,and when R 0 > 1 the endemic equilibrium E 1 , appeared. The model exhibits thephenomenon of backward bifurcation where a stable disease-free equilibrium coexistswith a stable endemic equilibrium for a certain range of associated reproductionnumber less than one.The malaria model with intervention strategies indicates that it exhibits the phenomenonof forward bifurcation, in which in the absence of a low-level unstableequilibrium when R e < 1 a stable equilibrium bifurcating from the disease-free equilibriumwhen R e > 1, arise naturally when the disease does not invade when R = 1.We established the effective reproduction number, R e , as an important tool for effectivedisease management. The threshold for effective reproduction number and thebasic reproduction number in the absence of the disease was compared. If R e < 1,

85the disease can not persist in a country, hence R e is the useful indication of the effortrequired to eliminate an infection. It was also noted that R e ≤ R 0 which impliedthat increasing preventive and control measures has a great effect on reduction of R e .Thus, malaria can be eradicated out of community by deployment of a combinationof strategies such as effective mass drug administration and vector control that areof significant in its fight.Numerical analysis of the model suggested that effective control or eradication ofmalaria can be achieved by the combination of protection and treatment measures.We showed that at the time when infection was high in the community, treatmentwas highly required compared to prevention. Further, we found that preventionmeasures were important to maintain the reduction or eradication of the diseasetransmission after lowering the infection. These results indicate the effect of the twocontrols (protection and treatment) in lowering exposed and infected members ofeach of the populations. This study concures with the Chavez (2008) suggestion thatthe intervention using insecticide-treated bed nets represents an excellent exampleof implementing an infectious disease control programme, and Smith et al, (2008) ′ sstudy, whichshowedthatbothregularandnon-fixedsprayingresultedinasignificantreduction in the overall number of mosquitoes, as well as the number of malariacase in humans. Hence the combination of these two findings, and treatment asan additional intervention strategies, showed great impact in the reduction of thespreading of the disease. Therefore, the combination of these interventions can playa bigger role in reducing or eradicating the transmission of the disease in the country.This study provides useful tools for assessing the effectivess of multi-interventionstrategies and analysing the potential impact of prevention with treatment.Among the interesting dynamical behaviours of the model, numerical simulationsshow a backward bifurcation in the basic model which gives a challenge to the designingof the effective control measures. We further expected to use a cost-utilityanalysis to examine the costs and effects of intervention strategies against malariaand their combinations. We failed to acquire the necessary data because of beau-

86cracy in the Ministry of Health and the time was too short to meet all the necessarypeople. There is a National Health Sciences Research Committee (NHSRC) to whichthe proposal is submitted for scientific and ethical review to accelerate implementationof quality health research in Malawi. The committee takes time to respond sincethey are expected to discuss and make decision to any healthy research proposal.6.2 RecommendationAstheresurgenceofmalariacontinuestotakeitstollonindividualsandcommunitiesin Malawi, the policy makers have to be informed about the research results. Thefollowing recommendations should be considered:(1). Since most of the reductions in transmission come from the protection of a fewhumans, it is far more important to improve the killing effects of insecticidemosquito treated bednets (ITNs) and indoor residual spray (IRS) around thosewho are mostly exposed to malaria; however, complete coverage and improvedkilling effects may be necessary to reach control goals.(2). Vector control interventions such as insecticide-treated nets (ITNs) and indoorresidual spraying (IRS) are proving effective to combat and prevent the diseasein Malawi. ITNs and IRS, with insecticidal and diversionary properties, wouldreduce the availability of hosts, and kill mosquitoes that are attempting to feedon human’s blood, and reducing malaria transmission.(3). Since the set of intervention strategies is appropriate for the transmissionregime, such as combination of prevention and treatment, and if it is implementedattheappropriatetargetedscaleinmany-endemicareasinthecountry,the malaria related millennium development goals can be achieved well beforean effective vaccine is available.(4). Because of the complications of measuring malaria at different transmissionlevels with different immunological status prevalent in different age and gender

87groups, and across different locations, some guidelines should be developed togive researchers and health professionals a more accurate foundation on whichto select their indicators.6.3 Future WorkWeareextendingthemodeltoincludetheeffectsoftheenvironmentonthespreadofmalaria. Someparameters,suchastheincubationperiodinmosquitoesandmosquitobirth rate depend on seasonal environmental factors such as rainfall, temperature,andhumidity. Wecanincludetheseeffectsbymodellingtheseparametersasperiodicfunctionsoftime. Thiswouldprovideamoreaccuratepictureofmalariatransmissionand prevalence than that obtained from models using parameters values that areaveragedoverseasons. Otherplannedimprovementtothemodelincludetheadditionof age, spatial variation on the implementation of insecticide mosquito treated nets(ITNs)andindoorresidualspraying(IRS)andcost-effectiveanalysisofinterventions.

88REFERENCESAnderson, R.M and May, R.M (1991), Infectious diseases of humans, Oxford, OxfordUniversity Press.Aron, J.L (1988), “Mathematical modelling of immunity to malaria, Nonlinearityin biology and medicine (Los Alamos, NM, 1987)”, Math.Biosci., Volume 90,pp 385396.Birkhoff, G and Rota, G.C (1982), Ordinary Differential Equations, Ginn.Blayneh Kbenesh, Cao Yanzhao, and Kwon Hee-Dae (2009), “Optimal Control ofVector-borne Diseases: Treatment and Prevention”, Discrete and ContinuousDynamical Systems Series B, Volume 11, No.3, pp 1-xx.Blayneh K.W and Jang S.R (2006), “A discrete SIS-model for a vector-transmitteddisease”, Applicable Analysis, Volume 85, No. 10, pp 1271-1284.Brauer F, and Castillo-Chavez C (2001), Mathematical model in population biologyand epidemiology, Applied Mathematics Series Text 40, Springer-Verlag, NewYork.Briet J.T.O, Vounatsou P, Gunawardena M.D, Galappaththy N.L.G and AmerasingheH.P (2008), “Models for short term malaria prediction in Sri Lanka”,BioMed Central, Volume 7, No. 76.Briet J.T.O, Vounatsou P, Gunawardena D, Galappaththy N.L.G and AmerasingheH.P (2008), “Temporal correlation between malaria and rainfall in Sri Lanka”,Biomed Central, Volume 7, No. 77.Bupa UK (2009), Malaria symptoms, causes and treatment of disease,http://hcd2.bupa.co.uk/fact sheets/html/malaria disease retrieved on Friday,19th June, 2009.Carr J (1981), Applications centre mainfold theory, Springer-Verg, New York.

89Carter R (2002), “Spatial simulation of malaria transmission and it control bymalaria transmission blocking vaccination”, International Journal for Parasitolog,Volume 32, pp 1617-1624.Chaves F.L, Kaneko A, Taleo G, Pascual M and Wilson L.M (2008), “Malariatransmission pattern resilience to variability is by insecticide-treated nets”,Biomed Central, Volume 7, No. 100.Chavez-Castillo C and Song B (2004), “Dynamical models of tuberculosis and theirapplications”, Math.Biosci.Eng., No. 2, pp 361-404.Chitnis N, Hyman J.M and Cushing J.M (2008), “Determining important parametersin the spread of malaria through the sensitivity analysis of a mathematicalmodel”, Math. Bio., Volume 70, pp 1272-1296.Craig M.H, Kleinschmidt I, Nawn J.B, Le Sueur D, Sharp B.L (2004), “Exploring30 years of malaria case data in KwaZulu-Natal, South Africa: part I. Theimpact of climatic factors”, Trop Med Int Health, Volume 9, pp 1247-1257.Diekmann O and Heesterbreek J.A.P, (2000), Mathematical Epidemiology of InfectiousDiseases: Model Building, Analysis and Interpretation, Wiley, NewYork.Diekmann O and Heesterbeek J.A.P (1990), “On the definition and computationof the basic reproduction ratio R 0 in the model of infectious disease in heterogeneouspopulations”, J. Math. Biol., Volume 2, No. 1, pp 265-382.Dietz K (2008), “The estimation of the basic reproduction number for infectiousdiseases”, Stat Methods Med Res., Volume 2, pp 23-41.Dushoff J, Huang W, and Castillo-Chavez C (1998), “Backwards bifurcations andcastastrophe in simple models of fatal diseases”, J. Math. Biol., Volume 36,pp 227-248.

90Focks D.A, Haile D.G, Daniel E and Mount G.A (1993), “Dynamics life table modelfor Aedes aegypti (L.) (Diptera: Culicidae)”, J. Med. Entomol., Volume 30,pp 1003-1017.Garba S.M, Gumel A.B and Abu Bakar M.R (2008), “Backward bifurcation indengue transmission dynamics”, Math.Biosci., Volume 215, pp 11-25.Gomez-Elipe A, Otero A, van Herp M and Aguirre-Jaime A (2007), “Forecastingmalaria incidence based on monthly case reports and environmental factors inKaruzi, Burundi, 1997-2003”, BioMed Central, Volume 6, pp 129.Greenwood B, and Mutabingwa T (2002), “Malaria in 2002”, Nature, Volume 415,pp 670-672.Hale J.K (1969), Ordinary Differential Equations, John Wiley, New York.Hethcote H.M (2000), “The mathematics of infectious diseases”, SIAM Rev., Volume42, No. 4, pp 599-953.Killeen G.F, Mckenzie F.E, Foy B.D, Bogh C and Beier J.C (2001), “The availabilityof potential hosts as a determinant of feeding behaviours and malariatransmission by African mosquito populations”, Trans. R. Soc. Trop. Med.Hyg., Volume 95, pp 469-474.Kribs-ZaletaC,andValesco-HernandezJ,(2000), “Asimplevaccinationmodelwithmultiple endemic state”, Math.Biosci., Volume 164, pp 183-201.Kwon H, Blayneh W.K and Cao Y (2007), “A model for vector-borne diseases: Optimaltreatment and prevention”, Discrete and Continuous Dynamical SystemsB, Volume 11, No. 3, pp 587611.Lalloo D.G, Shingadia D, Pasvol G, Chiodini P.L, Whitty C.J and Beeching N.J(2008), “UK malaria treatment guidelines”, J. Infect 2007, Volume 54, pp111-121.

91Li J (2008), “A malaria model with partial immunity in humans”, Math Biosciencesand Engineering, Volume 5, No. 4, pp 789-801.Martens P, Kovats R.S, Nijhof S, Vries P. de, Livermore M.T.J, Bradley D.J, CoxJ and McMichael A.J (1999), “Climate change and future populations at riskof malaria”, Global Environmental Change, Volume 9, pp S89-S107.Miranda I. Teboh-Ewungkem, Chandra N. Podder and Abba B. Gumel (2009),“Mathematical Study of the Role of Gametocytes and an Imperfect Vaccine onMalaria Transmission Dynamics”, Bulletin of Mathematical Biology DOI 10,Volume 1007, No. S11, pp 538-009-9437-3.Morel M.C, Laurer A.J and Evans B.D (2005), “Cost effectiveness analysis ofstrategies to combat malaria in developing countries”, BMJ, doi:10, Volume1136/bmj, No. 38639.702384.AE, pp 1-7.Murray J.D (1991), Mathematical Biology Volume 19. Springer-Verlag, New York.Nakul C, Hyman M.J and Cushing J.M (2006), “Bifurcation analysis of a mathematicalmodel for malaria transmission”, SIAM J. Appl. Math, Vol. 67, No.1, pp 24-45.Nakul C, Hyman M.J and Cushing M.J (2008), Determining important parametersin the spread of malaria through the sensitivity for mathematical biology, NewYork.National Statistical Office and UNICEF (2008), Malawi Multiple Indicator ClusterSurvey 2006, Final Report, Lilongwe, Malawi: National Statistical Office andUNICEF.Ngwa G.A and Shu W.S (2000), “A mathematical model for endemic malaria withvariable human and mosquito populations”, Math. Comp. Modeling, Volume32, pp 747-763.Ngwa G.A (2004), “Modeling the dynamics of endemic malaria in growing population”,Discrete and Continuous Dyn. Sys. Ser. B, Volume 4, pp 1173-1202.

92Ngwa G.A (2006), “On the population dynamics of the malaria vector”, Bull MathBiol, Volume 68, pp 2161-2189.Niger Ashrafi M and Gumel Abba B (2008), “Mathematical Analysis of the Roleof Repeated Exposure on Malaria Transmission Dynamics”, Differential Equationsand Dynamical Systems, Volume 16, No. 3, pp 251-287.Peterson L.J (2005), “An extension perspective on monitoring pesticide resistance”,Journal of Extension, Volume 43, No. 4, Article No. 4TOT6.Ritchie S.A and Montague C.L (1995), “Simulated populations of the black saltmarshmosquitoAedestaeniorhynchusinaFloradaMangroveForestEcologicalmodeling”, Volume 77, pp 123-141.Sachs J and Malaney P (2002), “The economic and social burden of malaria”,Nature, Volume 415, pp 680-685.Smith R.J and Hove-Musekwa D.S (2008), “Determining effective spraying periodsto control malaria via indoor residual spraying in sub-Saharan Africa”, ApplMath and Deci Sc., doi:10, Volume 1155, No. 745463, pp 1-19.Smith L.D, McKenzie F.E, Snow W.R and Hay I.S (2007), “Revisiting the basicreproduction number for malaria and its implications for malaria control”,PLoS Biology, Volume 5, pp 531-542.Teklehaimanot H.D, Lipsitch M and Teklehaimanot A (2004), “Weather-basedprediction of Plasmodium falciparum malaria in epidemic-prone regions ofEthiopia I. Patterns of lagged weather effects reflect biological mechanisms”,Malar J, Volume 3, No. 41.Thomson M.C and Connor S.J (2001), A framework for field research in Africa:Malaria early warning systems: concepts, indicators and partners. Geneva,Roll Back Malaria Cabinet Project.

93Tumwiine J, Mugisha J.Y.T and Luboobi L.S (2007), “On oscillatory pattern ofmalaria dynamics in a population with temporary immunity”, Computationaland Mathematical Methods in Medicine, Volume 8, No. 3, pp 191-203.United Nations, The Millennium Development Goals Report, accessed online atwww.un.org/millenniumgoals retrieved on Sunday, 3th May, 2009.Whitty C.J.M, Lalloo D and Ustianowski A (2007), “Malaria: an update on treatmentof adults in non-endemic countries”, BMJ., Volume 333, pp 214-245.WHO (2003), “Expert on Malaria”, WHO Tech Report Serie 892.12, Report: pp36.WHO (2001), Guidelines on the use of insecticidetreated mosquito nets for the preventionand control of malaria in Africa. CTD/MAL/AFRO/97.4.WHO (1993), “Implementation of the global malaria control strategy. Reports ofthe World Health Organization Study Group on The Implementation of theGlobal Plan of Action for malaria control 1993 20002”, WHO Technical ReportSeries, Number 839.WHO (2009), malaria, Fact Sheets, http://www.who.int/inf-fs/en/fact094.html retrievedon Wednesday, 17th June, 2009.WHO (1999), The WHO Report 1999-Making a difference, Geneva.Worrall E, Connor S.J and Thomson M.C (2007), “A model to simulate the impactof timing, coverage and transmission intensity on the effectiveness of indoorresidual spraying (IRS) for malaria control”, Trop Med Int Health, Volume 12,pp 75-88.Yang H.M and Ferreira M.U (2000), “Assessing the effects of global warming andlocalsocialeconomicconditionsonthemalariatransmission”, Re Saude Public,Volume 34, pp 214-222.

94Yang H.M (2000), “Malaria transmission model for different levels of acquired immunityand temperature-dependent parameters (vector)”, Re Saude Public,Volume 34, pp 223-231.

95APPENDICESAPPENDIX A:Matlab codes used for simulating the basic malaria model(i)The M-function files(a)function dydt = basicnew(t,y)dydt = zeros(size(y));c=0.08333; d=60; e=0.0057233; w=0.0018; u=0.02326;b=u*10^5;v=1/17; j=0.0001; j1=0.0035; k=0.03454; l=0.022; m=6;n=0.1429; r=0.1; q=0.07;S=y(1);E=y(2);I=y(3);R=y(4);X=y(5);Y=y(6);Z=y(7);N = S + E + I + R;a1=c*d/N; a2=e*d/N;%The malaria model without interventionsdydt(1) = b - a1*Z*S + w*R - u*S;dydt(2) = a1*Z*S - (v + u)*E;dydt(3) = v*E - (j + u + k)*I;dydt(4) = j*I - (w +u)*R;dydt(5) = m - a2*I*X - n*X;dydt(6) = a2*I*X - (r + n)*Y;

96dydt(7) = r*Y - (n + q)*Z;%The basic reproduction number for the basic malaria modelR0 =sqrt((c*e*d^2*v*r*m*u)/(n*b*(u+v)*(j+u+k)*(r+n)*(n+q)))\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\(ii).The executable file for finding the behavior ofsusceptible exposed infected and recovered humanpopulations by plotting the intended line graph.clear%Showing the behavior of susceptible individuals with nointerventionstspan = [0 5];y0 = [12000 10500 10000 8000 15000 11000 9000];[t,y] = ode45(@basicnew,tspan,y0);plot(t,y(:,1),'Linewidth',2,'Color','r')xlabel('Time in years')ylabel('Susceptible Individuals')legend('R_0 = 0.1277')////////////////////////////////////////////////////////////The executable file for showing the relationship between susceptibleand infected human populations.cleartspan = [0 250];y0 = [12000 10500 9000 8000 15000 11000 9000];[t,y] = ode45(@basicnew,tspan,y0);plot(y(:,1),y(:,3),'Linewidth',2,'Color','r')

97xlabel('Susceptible individuals')ylabel('Infected Individuals')legend('R_0 = 0.1277')//////////////////////////////////////////////////////////////The executable file for prevalence in the basic modelclear%Showing the dynamics of infected individualstspan = [0 90];y0 = [12000 900 850 600 10000 700 600];[t,y] = ode45(@basica1new,tspan,y0);N1=(y(:,1)+y(:,2)+y(:,3)+y(:,4));plot(t,(y(:,2)+y(:,3)+y(:,4))./N1,'r','Linewidth',2)xlabel('Time in days')ylabel('Prevalence')legend('R0 = 1.2766')

98APPENDIX B:Matlab codes used for simulating the malariamodel with intervention strategies.function dydt = pet1new(t,y)dydt = zeros(size(y));a=0.11; b=0.0015875; c=0.86; d=0.29; e=0.083; f=2; w=1/(20*365);u=1/(60*365); j1=0.0035;v=1/17; j=0.01; k=0.0003454; l=0.00722; m=0.071; n=1/15;r=0.0017; q=0.07; g=0.78;S=y(1);P=y(2);E=y(3);I=y(4);T=y(5);R=y(6);X=y(7);Y=y(8);Z=y(9);N = S + P + E + I + T + R;a1=c*d/N; a2=c*d*f/N; a3=e*d/N;%The malaria model with interventionsdydt(1) = (1 - a)*b - a1*Z*S + w*R - (g + u)*S;dydt(2) = a*b + g*S + (1 - w)*R - a2*Z*P - u*P;dydt(3) = a1*Z*S + a2*Z*P - (v + u)*E;dydt(4) = v*E - (j + j1 + u + k)*I;

99dydt(5) = j*I - (l + u)*T;dydt(6) = l*T + j1*I - (1 - w)*R - (w +u)*R;dydt(7) = m - a3*I*X - n*X;dydt(8) = a3*I*X - (r + n)*Y;dydt(9) = r*Y - (n + q)*Z;%The basic reproduction number for the basic malaria model%R0 = sqrt((c*e*d^2*v*r*m*u)/(n*b*(u+v)*(j1+u+k)*(r+n)*(n+q)))%The effective reproduction number of the malaria model withinterventions%RE =sqrt((c*e*d^2*v*r*m*u)*(g*(1+f)+u*(1+a*f))/(n*b*(g+u)*(u+v)*(j+j1+u+k)*(r+n)*(n+q)));%The force of infection for the susceptibles%a4 = c*d*Z/N%The force of infection for the protected class%a5 = c*d*f/N;

100The executable function used for malaria model withintervention strategies-The dynamics of protected and treated individuals.clear %CORRECTLY DONE 1 of 2tspan = [0 400];y0 = [1000 900 700 500 350 300 10000 9000 7000];[t,y] = ode45(@petnew,tspan,y0);%plot(t,y(:,1),t,y(:,2))plot(t,y(:,2),'r',t,y(:,5),'b','Linewidth',2)xlabel('Time in days')ylabel('Populations')legend('Protected Individuals','Treated Individuals');;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;clear %CORRECTLY DONE 2 of 2tspan = [0 400];y0 = [1000 900 700 500 350 300 10000 9000 7000];[t1,y1] = ode45(@pet1new,tspan,y0);[t,y] = ode45(@petnew,tspan,y0);[t2,y2] = ode45(@pet2new,tspan,y0);[t3,y3] = ode45(@pet3new,tspan,y0);[t4,y4] = ode45(@pet4new,tspan,y0);%plot(t,y(:,1),t,y(:,2))plot(t1,y1(:,2),'g',t,y(:,2),'r',t2,y2(:,2),'g:',t,y(:,5),'b',t3,y3(:,5),'r:',t4,y4(:,5),'y','Linewidth',2)xlabel('Time in days')ylabel('Protected and Treated Individuals')

101legend('g_1=0.78','g_2=0.0833','g_3=0.00613','\sigma_1=0.01','\sigma_2=0.015','\sigma_3=0.019')/////////////////////////////////////////////////////////////-Behavior of recovered individualsclear%Showing the behavor of the recovered individuals withchanges of recovered parameter.tspan = [0 400];%no backupy0 = [100 90 70 50 35 30 1000 900 700];[t,y] = ode45(@petnew,tspan,y0);[t1,y1] = ode45(@peta5new,tspan,y0);[t2,y2] = ode45(@peta6new,tspan,y0);plot(t,y(:,2),'b',t1,y1(:,2),'g',t2,y2(:,2),'r','Linewidth',2)xlabel('Time in days'ylabel('Protected Individuals')-Exposed human population behaviorcleartspan = [0 250];y0 = [1000 900 600 500 350 300 10000 9000 7000];[t,y] = ode45(@petnew,tspan,y0);plot(t,y(:,3),'Linewidth',2,'Color','r')xlabel('Time in days')ylabel('Exposed Individuals')

102-Behavior of infected human population.clear %CORRECTLY DONE 3tspan = [0 400];y0 = [1000 900 700 400 350 300 10000 9000 7000];[t,y] = ode45(@petnew,tspan,y0);[t1,y1] = ode45(@pet5new,tspan,y0);[t2,y2] = ode45(@pet6new,tspan,y0);[t3,y3] = ode45(@pet7new,tspan,y0);%plot(t,y(:,4),'Linewidth',2,'Color','r')plot(t,y(:,4),'g',t1,y1(:,4),'r',t2,y2(:,4),'g:',t3,y3(:,4),'b','Linewidth',2)xlabel('Time in days')ylabel('Infected Individuals')legend('\nu_1 = 0.05882','\nu_2 = 0.06667','\nu_3 = 0.05263','\nu_4 =0.1');;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;-For the behavior of recovered individuals.cleartspan = [0 100];y0 = [1000 900 70 50 30 10 10000 9000 7000];[t,y] = ode45(@petnew,tspan,y0);[t1,y1] = ode45(@peta5new,tspan,y0);[t2,y2] = ode45(@peta6new,tspan,y0);plot(t,y(:,6),'b','Linewidth',2)xlabel('Time in days')ylabel('Recovered Individuals')

103-For Prevalence in the malaria model with interventions.clear%Prevalence graphstspan = [0 400];y0 = [1000 900 700 500 350 300 10000 9000 7000];[t,y] = ode45(@petnew,tspan,y0);[t1,y1] = ode45(@petbnew,tspan,y0);[t2,y2] = ode45(@petb1new,tspan,y0);[t3,y3] = ode45(@petb2new,tspan,y0);%[t4,y4] = ode45(@pet4,tspan,y0);%N2 = (y(:,4)+y(:,5)+y(:,6))/N1;%N3 = (y1(:,4)+y1(:,5)+y1(:,6))/N1;%plot(t,y(:,1),t,y(:,2))N1=(y(:,1)+y(:,2)+y(:,3)+y(:,4)+y(:,5)+y(:,6));N2=(y1(:,1)+y1(:,2)+y1(:,3)+y1(:,4)+y1(:,5)+y1(:,6));N3=(y2(:,1)+y2(:,2)+y2(:,3)+y2(:,4)+y2(:,5)+y2(:,6));N4=(y3(:,1)+y3(:,2)+y3(:,3)+y3(:,4)+y3(:,5)+y3(:,6));hold onplot(t,(y(:,4)+y(:,5)+y(:,6))./N1,'r','Linewidth',2)plot(t1,(y1(:,4)+y1(:,5)+y1(:,6))./N2,'b','Linewidth',2)plot(t2,(y2(:,4)+y2(:,5)+y2(:,6))./N3,'g','Linewidth',2)plot(t3,(y3(:,4)+y3(:,5)+y3(:,6))./N4,'y','Linewidth',2)%plot(t1,y1(:,2),'g',t,y(:,2),'r',t2,y2(:,2),'g:',t,y(:,5),'b',t3,y3(:,5),'r:','Linewidth',2)hold offxlabel('Time in days')ylabel('Prevalence')legend('\sigma_1=0.01','\sigma_2=0.03','\sigma_3=0.02','\sigma_4 =0.9')

104;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;-The prevalence simulations.clear%Prevalence graphs for model with interventionstspan = [0 800];y0 = [1000 900 700 500 350 300 10000 9000 7000];[t,y] = ode45(@pettnew,tspan,y0);[t1,y1] = ode45(@petbbnew,tspan,y0);[t2,y2] = ode45(@petb11new,tspan,y0);[t3,y3] = ode45(@petb22new,tspan,y0);N1=(y(:,1)+y(:,2)+y(:,3)+y(:,4)+y(:,5)+y(:,6));N2=(y1(:,1)+y1(:,2)+y1(:,3)+y1(:,4)+y1(:,5)+y1(:,6));N3=(y2(:,1)+y2(:,2)+y2(:,3)+y2(:,4)+y2(:,5)+y2(:,6));N4=(y3(:,1)+y3(:,2)+y3(:,3)+y3(:,4)+y3(:,5)+y3(:,6));hold onplot(t,(y(:,4)+y(:,5)+y(:,6))./N1,'r','Linewidth',2)plot(t1,(y1(:,4)+y1(:,5)+y1(:,6))./N2,'b','Linewidth',2)plot(t2,(y2(:,4)+y2(:,5)+y2(:,6))./N3,'g','Linewidth',2)plot(t3,(y3(:,4)+y3(:,5)+y3(:,6))./N4,'y','Linewidth',2)hold offxlabel('Time in days')ylabel('Prevalence')legend('\sigma_1=0.01 \nu_1 = 0.05','\sigma_2=0.03 \nu_2 =0.0667','\sigma_3=0.02 \nu_3 = 0.0588','\sigma_4 =0.9 \nu = 0.0769')