MODELLING THE EFFECTS OF TEMPERATURE ... - NOMA

iCERTIFICATIONThe undersigned certify that they have read and hereby recommend for acceptanceby the University of Dar es Salaam the dissertation entitled: Modelling the Effectsof Temperature Variations on the Dynamics of Malaria, in partialfulfillment of the requirements for the degree of Master of Science (MathematicalModelling) of the University of Dar es Salaam.—————————————Dr. Nyimvua Shaban(Supervisor)Date: ..................................—————————————Prof. Joseph T. Y. Mugisha(Supervisor)Date: ...................................

iiDECLARATIONANDCOPYRIGHTI, Stephano Walijena Sanga, declare that this dissertation is my own originalwork and has not been presented and will not be presented to any other universityfor 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 in part,except for short extracts in fair dealings, for research or private study, critical scholarlyreview or discourse with an acknowledgment, without the written permissionof the Directorate of Postgraduate Studies, on behalf of both the author and theUniversity of Dar es Salaam.

iiiACKNOWLEDGEMENTSIt is may sincere gratitude to thank the Almighty God for the wonderful opportunityof undertaking the M.Sc. Mathematical Modelling programme at theUniversity of Dar-es-Salaam. God’s power and love should be enriched to all initiatorsand facilitators of the programme for it should be kept on existence andconsolidated for the future benefit of our country and Africa at large as far the programmeobjectives are concerned.With prayer of thanks, strongly I would like to find some way of expressing myheart-felt gratitude to my Dissertation Supervisors Prof. Joseph T.Y. Mugisha(Makerere University, Uganda) and Dr. Nyimvua Shaban (University of Dar-es-Salaam, Tanzania), for all that they kindly and tirelessly took much of their time forsupport, guidance, encouragement and constructive ideas towards the accomplishmentof this work regardless of their many responsibilities in academic and socialaffairs in and out of their Universities. I have learnt a lot from them and God mayempower them with endless blessings in all.My sincere appreciation is to Prof.E.S. Massawe, the Head of Department ofMathematics, Dr. W.C. Mahera, the Coordinator of the NORAD Masters Programme(NOMA) and the academic staff of the University of Dar-es-Salaam formaintaining the programme based on facts enrichment of life application of Mathematicsacross multidimensional fields of studies prioritised to East and SouthernAfrica. This is together with accommodating our facilitators from other UniversitiesWorldwide devoted to support us, cooperatively worked as team in lectures andsupervisions at the University of Dar-es-Salaam particularly at the Department ofMathematics. I also thanks the supporting staff members who indirectly contributedto our learning empowerment through directives, material and learning environment.

xLIST OF FIGURES3.1 The model diagram for human-mosquito populations. . . . . . 143.2 The Forward or Transcritical bifurcation. . . . . . . . . . . . . 374.1 Illustrates changes in human state variables of the model. . 404.2 The dynamics of mosquito population with time under constanttemperature-dependent parameters. . . . . . . . . . . . 424.3 The dynamics of human and mosquito population’s epidemiologicalclasses plotted on the same axes. . . . . . . . . . . . . 444.4 Changes in prevalence of malaria in high and low temperatures 454.5 The impact of temperature-dependent parameter on the dynamicsof susceptible mosquitoes. . . . . . . . . . . . . . . . . . 464.6 The impact of temperature-dependent parameter on the dynamicsof infected mosquitoes. . . . . . . . . . . . . . . . . . . . 484.7 The impact of temperature-dependent parameter on dynamicsof exposed individuals. . . . . . . . . . . . . . . . . . . . . . . 484.8 The temperature in warm area (Dar-es-Salaam). . . . . . . . . 504.9 Deviations of temperature from 16 o C above which mosquitoesand malaria parasites life is supported in Dar-es-Salaam. . . 504.10 The temperature in cold area (Iringa). . . . . . . . . . . . . . . 534.11 Deviations of temperature from 16 o C above which mosquitoesand malaria parasites life is supported in Iringa . . . . . . . . 53

xiLIST OF TABLES3.1 State variables used in the Model formulation . . . . . . . . . 153.2 Parameters Used in the Model Formulation . . . . . . . . . . 164.1 Model Simulation Parameter Values given - per day . . . . . 397.1 The mean monthly maximum temperature in Dar-es-Salaam 847.2 The mean monthly minimum temperature in Dar-es-Salaam 857.3 The mean monthly maximum temperature in Iringa . . . . . 867.4 The mean monthly minimum temperature in Iringa . . . . . . 87

1CHAPTER ONEINTRODUCTION1.1 Background to the StudyThere is an outcry that malaria is spreading in almost all tropical areas being muchmore in Sub-Sahara Africa (Chang’a et al., 2010). There is a newly emergence ofmosquitoes and outbreaks of malaria every year in highland cold areas of Tanzaniaas compared to coastal warm areas where malaria existed originally. This tendencyis to be investigated to check whether it is supported by average annual temperaturevariations or not.Temperature is the degree of heat as an intrinsic quality of objects expressed ashotness or coldness relative to something else. It is related to global warming whichis described as the rising of global average land and sea surface temperatures andincreasing frequency of extreme weather conditions in many parts of the world. Ingeneral, climate constrains the range of malaria, while weather elements affect thetiming and intensity of malaria outbreaks (Dobson and Carper, 1993).Temperature varies inversely with height as one goes up from the sea level, this hasbeen associated with changes in ecology at high altitude to that of lower altitudes(Patz, 1996; Epstein, 2001). There is now considerable evidence that temperatureincreases by 0.6 ◦ C in every 100 metres upwards from the sea level since the midnineteenthcentury. This change has taken place since 1976. In this case 14 of thewarmest years on record have occurred since 1980, and predicted that an averageglobal temperatures would increase to between 1.4 ◦ C-5.8 ◦ C by the year 2100 (IntergovernmentalPanel on Climate Change (IPCC)).

2Temperature may alter the range and prevalence of malaria infections. Higher temperatures,along with changes in precipitation and humidity, can affect the biologyand ecology of disease vectors and intermediate hosts, the pathogens that they transmit,and consequently the risk of transmission (Githeko et al., 2000). Diseases carriedby mosquito vectors are particularly sensitive to meteorological conditions sincethese insects have difficult temperature thresholds for survival and are susceptibleto changes in average ambient temperature (Epstein, 2001).Malaria is one of the most prevalent and fatal human infectious disease worldwidecaused by the protozoa known as Plasmodium. It is transmitted to vertebrates byfemale genus Anopheles mosquitoes when they feed on blood. There are four speciesof the parasite, namely; Plasmodium falciparum, Plasmodium vivax, Plasmodiumovale and Plasmodium malaria which infect human beings (Encarta, 2008). Someclinical symptoms of malaria are fever, pain, chills and sweats may develop a fewdays after an infected mosquito bite. The gametocyte is a the malaria organism inthe stage in its life cycle during which it reproduces in the blood of a mosquito. Itis produced after some two weeks independent of the immediate surrounding temperature(Onwujekwe, 2009).Anopheles mosquitoes can only transmit Plasmodium falciparum malaria parasitesif the temperature remains above 16 ◦ C (Saker et al., 2004). Within their survivalrange, warmth accelerates the biting rate of mosquitoes and the maturation of parasiteswithin them (McArthur, 1972). Since insects have short lifespan, this increasesthe chances of their biting for blood meals from an infected person and uninfectedperson leading to transmission of the pathogen from one person to person. The lifecycle of the malaria parasite or other pathogen carried by the vector is thus accelerated.The precise effect on transmission requires continued study to determinewhether shorter, more intense, lower lifespan or increase transmission,are in balance.

3The influence of climate, particularly temperature change on the transmission ofPlasmodium falciparum malaria, continues to be a subject of considerable debate.Temperature influences anopheline mosquito feeding intervals, population density,and longevity, as well as the reproductive potential of the plasmodium parasite.This study gives an idea that there is a demand of exploration of the mode of existenceof mosquitoes depending on temperature particularly in cold high altitudes andlatitudes. Coluzzi (1992) supports this, that in other areas temperature may extendthe range of latitude and altitude at which malaria is transmitted. In Africa andSouth America, this has increased the distribution of Anopheles mosquitoes, allowingmalaria to spread to formerly unaffected areas, particularly cities. Thus temperaturevariations and mosquito life conditions are subject to investigation under this studyemploying the mathematical model to describe the overall transmission of malariaand its incidence in the community.1.2 Statement of the problemThe epidemiological impact of weather changes in communities is highly increasingand expanding faster regardless of all efforts to combat malaria, and hence malariatransmission dynamics persists worldwide. Malaria is re-emerging in areas wherecontrol efforts were once effective and emerging in areas thought free of the disease.This has been attributed to a number of factors and most importantly temperatureand global warming at large.There is a need to use mathematical models to explore the role and impact of climaticconditions more specifically the temperature variations in the dynamics of malaria.

41.3 Research Objectives1.3.1 General objectiveThe main objective of this study is develop understanding of malaria transmissiondynamics as a result of temperature changes using a Mathematical model.1.3.2 Specific ObjectivesThe specific objectives of this study were:(i). Modelling the impacts of temperature in the transmission dynamics of malaria.(ii). to Compare the modelled rates of malaria transmission in warm areas and incold areas.1.4 Significance of the StudyAn analysis of the mathematical malaria model to describe the understanding of thedisease characteristics focuses on the nation, community hospitals, health centersmanagement and educational sector.It is important to establish and have somebasic behavioural understanding of malaria dynamics that may admit planners to along run setting of predictive ways to investigate whether the disease may persist orbe led to extinction. There will be a need to undertake new entomological researchinto areas that have been invaded by malaria vectors and where transmission istaking place (Githeko, A.K., and Ndegwa W., (2001)).The significance of thisstudy are to:(i). Facilitate the understanding of the effects of temperature relative to globalwarming on malaria transmission dynamics,(ii). Help planners set up long term interventional strategies, educate the public onissues pertaining to environmental degradation,

5(iii). Disseminate effective malaria control, preventions and treatment information,educational campaigns through Media and Non-Government Organizations torestrain malaria outbreak,(iv). Analyse the determinants or causes of the disease persistence by attemptingto measure the effectiveness of different existing services and programs,(v). Predict malaria emergence for scientists to evaluate inoculation or isolationplans for significant effect on the mortality rate of malaria.

6CHAPTER TWOLITERATURE REVIEWIn this chapter we review what have been done by other researchers so far relating totemperature variations connected to malaria dynamics in areas of different annualaverage temperatures and the number of incidence of malaria.Malaria transmission, is a complex interaction of many factors, including not onlyvector and parasite densities and behaviour, but also public health control measures.More suitable climate conditions may facilitate malaria transmission, but evidencefor increased incidence has not been clearly linked to climate, nor have predictionsfor the spread of malaria under future climate change scenarios produced the rangechanges that some have incomplete information.The vast majority of people in sub-Saharan Africa live in regions of stable transmissionof malaria. Throughout their lives, people are regularly exposed to multiplebites from infective mosquitoes (Reiter, 2008). This has led to the continuous susceptibilityof large population and increase in the number of new malaria outbreaksin unexpected areas.Economic development, particularly in poor countries, has been constrained by theloss of children and young workers due to infectious diseases such as Malaria whichcauses half of all deaths among families (WHO, 2008). The spread of malaria hasbeen enhanced by modern transport systems that have increased human travel overlonger distances. This has enabled the movement of pathogens carried by infected humanand mosquitoes to uninfected areas further, faster and in greater numbers thanever before (Mugisha et al., 2009). This has led to an introduction of pathogens intoareas in which they had been absent, for example malaria is the most common dis-

7ease brought in the United States of America where uninfected Anopheles mosquitovectors still exist. It may be due to infected immigrants from malaria-endemic areaswho act as a source of malaria when they move to a malaria-free zone that hasuninfected mosquitoes.Castro et al. (2004), describe malaria cases in the range of 14 to 18 million newmalaria cases are annually reported in Tanzania and 100,000 to 125,000 deathsamong which 70,000 to 80,000 are children of age below five years The annual incidencerate is between 400 and 500 per 1,000 people, and this number doubles forchildren less than five years of age. This has high implications of loss of people invery large number in every year. Killeen et al. (2001) explained that the susceptiblehuman bitten by an infectious anopheles mosquito may become infected with a finiteprobability that depends on the abundance of infectious anopheles mosquitoes andhumans.The potential impact of temperature changes on human health repeatedly focuses onmalaria which depends on the existence of mosquitoes in particular area’s temperaturefavourable to their life. Reuter (2008) notes that in the coming decades, morecases will occur in regions where the disease is already present, and that transmissionwill extend to higher latitudes and altitudes. This supports the existed conditionsbefore 1980s southern and northern highlands of Tanzania where temperature werevery low and no mosquitoes and malaria were found.According to McMichael and Haines (1997), temperature variations are likely to undergoperturbation to the physical and biological systems to which human health isbiologically and culturally adjusted. Detecting the influence of the observed (andmuch larger predicted) changes in temperature on malaria transmission is not ob-

8servable and easy.Chitnis et al.(2007) performed a sensitivity analysis on a mathematical model ofmalaria transmission to determine the relative importance of model parameters tothe disease transmission and prevalence in areas of low and high transmission. Hefound that, both in areas of low and high transmission, the reproductive number R owas the same, but the equilibrium fraction of infectious humans were most sensitiveto mosquito biting rate in low transmission and most sensitive to human recoveryrate in high transmission. This suggests strategies that target mosquito biting rateand those that target human recovery rate can be successful in controlling malaria.Yang and Ferreira (2001) developed a model which was used to assess the effects ofglobal warming and local socioeconomic conditions on malaria transmission. Theseeffects were assessed analysing the equilibrium points calculated at different butfixed values of the parameters of the model. Regarding malaria transmission, it wasobserved that the effects of global warming posed a major challenge in the subsequentyears, and the effects of variation in local socioeconomic conditions are muchstronger than the effects of the increasing global temperatures. The major questionhere is how this can be experienced in the context of malaria transmission dynamicsin Tanzania.McMichael and Haines (1997) noted that natural environment was modified by localinfluences, such as local weather conditions, physical disasters and global forces, suchchanges in the great biophysical systems of the world alter the global environment.Over the past 50 years, huge increases in economic and industrial activity have ledto exceptional effects on air, land and water environments, and the resulting changeshave important and wide-ranging implications for human health, with different pop-

9ulations facing varying degrees of vulnerability to positive and negative impacts.Malaria is restricted to warm, moist climates that allow adequate time for developmentand transmission of the plasmodium parasite before the mosquito vector dies.Temperature under the global warming has prompted predictions that potentiallyfatal falciparum malaria will spread to currently non-endemic areas as their climateswarm up (Rogers and Randolph, 2000). These predictions have been supported bya combination of climate-forecasting and biological models of malaria distribution.Sutherst (2004) notes that temperature increase as a global change driver, has potentialeffects on vector, pathogen, and host environments. The expansion of warmclimatic zones, with longer growth seasons, less extreme low temperatures, and morefrequent extreme high temperatures fosters vector and pathogen development, andmore generations per year. It was found that shorter life spans of vectors at hightemperatures, reduced low-temperature mortality of vectors. Rainfall is too uncertainand regionally variable to estimate, but increased frequency of extreme rainfallevents altered patterns of breeding with more flushing of mosquito breeding withincreased flooding.Field studies and modelling suggest the scientific difficulties of obtaining empiricalproof that temperature is altering the transmission of malaria. However, availableevidence suggests that changes consistent with climate-change effects on Anophelinemosquitoes are found in parts of the US, and small outbreaks of locally transmittedmalaria have occurred during unseasonably hot weather spells (Zucker, 1996;Epstein, 1998). To the moment little is known about the impacts temperature variationsin the malaria transmission in both cold and warm areas of Tanzania.

10Temperature variations may decrease transmission of malaria in some locations byreducing rainfall or raising temperatures to levels too high for transmission. Morestudies are needed to investigate the possible effects of temperature on geographicdistribution of malaria in low and high lands, and vectorial capacity in malaria.These either incorporate average temperature effects on various components of thetransmission cycle (Martens, 1999), or simply make a statistical correlation betweenthe current distribution of malaria and the most important temperature variability( Hales et al., 2002; Rogers and Randolph, 2000).According to Riedel (2010), the potential number of adult mosquitoes produced byone female was less at the highland site than at the lowland site which gives explanationof low vector concentration in the East African highlands. Low vector density islikely to be one of the major factors contributing to low malaria transmission in thehighland, whereas the warm lowlands like Dar-es-Salaam remains a malaria endemicarea with high vector densities.The intensity and pattern of malaria are influenced because new local habitats encouragepathogens (Walsh et al., 1993).The shorter prolonged existence in thehighland may also reduce the chance a single female mosquito may transmit malariaduring its lifespan.The findings from the past studies carried out by Chitnis etal.(2007) who presents that in areas of low transmission R 0 and the disease prevalenceproportions of infectious humans are most sensitive to mosquitoes biting rate.This is sufficient to conclude that low ambient temperature is primarily responsiblefor the low abundance of malaria vectors in the East African highlands.There is a paucity of studies of this nature which have attempted to examine malariatransmission dynamics, especially in relation to temperature fluctuations in both

11warm and cold areas of Tanzania, hence the desideratum of this study.From the previous studies reviewed it was observed that the effects of temperaturein malaria dynamics involved other different factors generalized from climate andweather changes which includes different elements capable of affecting transmissionof malaria independently. These studies were carried out in several high and lowlands where most of them focused on climate and weather changes with little attentionto temperature as an independent element of weather capable of influencingthe dynamics of malaria in one or both warn or cold areas. Temperature variationsseem to be very significance on malaria dynamics.So, in this study we investigate temperature as an element of weather independententity which can affect the progress of malaria in the community. Annual temperaturevariations were done in model simulations and so was to validate the modelusing data from different areas behaving differently in temperature with respect tothe reviewed temperature point of 16 ◦ C favorable for population life of mosquitoes.Temperature records are from both cold and warm areas were used in graphicalsimulations.

12CHAPTER THREEMALARIA MODEL WITH TEMPERATURE CHANGES3.1 The Model formulationAn investigation of temperature behaviours that promote favourable conditions formalaria vector and parasites survivorship are compared between warm and cold areaswith reference to the minimum temperature of 16 o C favourable temperature formosquitoes and parasites life. We use temperature records from 1999-2009 in bothcold area (Iringa) and warm areas (Dar-es-Salaam) of Tanzania to other related areasin tropical regions worldwide.We formulate our malaria model with the population under study being divided intocompartments and with assumptions about the nature and time rate of transfer fromone compartment to another. The human total population N h (t) is divided into theepidemiological classes: Susceptible S h (t), Exposed E h (t), Infected I h (t) and Temporaryimmune R h (t). Thus, N h (t) = S h (t) + E h (t) + I h (t) + R h (t).The transfer rates between the subclasses are composed of several epidemiologicalparameters. We assume the horizontal standard incidence with homogeneous mixingto this model. The susceptible human population is increased by birth at a constantrate, b h . All the recruited individuals are assumed to be uninfected new borns whenthey join the community.The horizontal transmission of malaria is modelled using the standard incidenceterms λ mh (T ), at a particular time with temperature, T, that denotes the rate atwhich the human hosts S h (t) get infected by infected anopheles mosquitoes I m (t),where β mh (T ), the contact rate from mosquito to human in availability of temper-

15The state variables in Table 3.1 and the parameters in the Table 3.2 below, fromthe interaction between the human host and the malaria vector(mosquito) describedabove satisfy the system of Equations (3.1) which is shown just after the statedassumption below these tables.Table 3.1: State variables used in the Model formulationVariableDescription.S h (t)E h (t)I h (t)R h (t)N h (t)S m (t)E m (t)I m (t)N m (t)Number of susceptible human individuals at time tNumber of exposed human individuals at time tNumber of infected human individuals at time tNumber of recovered human individuals at time tTotal human population at time tNumber of susceptible mosquitoes at time tNumber of exposed mosquitoes at time tNumber of infected mosquitoes at time tTotal mosquito population at time t

16Table 3.2: Parameters Used in the Model FormulationParameterDefinition.b hb mRecruitment (birth) rate of humansPer capita birth rate of mosquitoesµ h Per capita natural death rate for humansµ m Per capita natural death rate for mosquitoesα hωε hPer capita disease-induced death rate for humansPer capita rate of loss of immunityProgression rate from exposed individualsto infected individualsε m (T )Progression rate from exposed mosquitoesto infected mosquitoes with temperature dependenceγ hRecovery rate for humans from the infectedstate to the recovered stateβ mh (T )Probability that a bite results in transmissionof infection to the human with temperature variationβ hm (T )Probability that a bite results in transmissionof infection to the mosquito with temperature variationa ′R 0Biting rate of mosquitoThe basic reproduction number is the mean number of secondarycases caused by an individual infected soon after disease outbreak.

17We assume that all state variables and parameters of the model which monitor humanand mosquito populations are positive for all t ≥ 0. Therefore, the model will beanalysed in a suitable region looking at the epidemiological impact of temperaturechanges on malaria incidence in communities.These assumptions lead to the following coupled system of ordinary differential equationswhich describe the progress of the disease:dS hdtdE hdtdI hdtdR hdtdS mdtdE mdtdI mdt= b h − λ mh S h + ωR h − µ h S h= λ mh S h − (ε h + µ h )E h= ε h E h − (α h + γ h + µ h )I h⎫⎪ ⎬= γ h I h − (ω + µ h )R h, (3.1)= b m (T ) − (λ hm + µ m )S m= λ hm (T )S m − (ε m (T ) + µ m )E m= ε m (T )E m − µ m I⎪m⎭where λ hm = β hm (T )a ′ I hand λ mh = β mh (T )a ′ I m.N hN hThis indicates that the rate of infection of susceptible human S h by infected mosquitoI m is dependent on the total number of humans N h available per vector.3.3 Invariant regionThe total population sizes N h and N m can be determined by N h = S h +E h +I h +R hand N m = S m + E m + I m or from the differential equationsdN hdt= dS hdt + dE h+ dI hdt dt + dR hdt(3.2)= b h − µ h N h − α h I hIt is noted that in the absence of the disease (α h = 0,) we havedN hdt= dS hdt + dE h+ dI hdt dt + dR hdt≤ b h − µ h N h .(3.3)

19Thus, the feasible set for the model system (3.1) is given by⎧⎛ ⎞S hE h⎪⎨I hΩ =R hS m⎜E m⎟⎝ ⎠I m⎪⎩∈ R 7 +∣S h ≥ 0,E h ≥ 0,I h ≥ 0,R h ≥ 0,⎫⎪ ⎬S m ≥ 0, ,E m ≥ 0,I m ≥ 0;N h ≤ b h;µ hN m ≤ b m ⎪ ⎭µ mwhich is a positively invariant set. Hence the model is mathematically well posed andbiologically meaningful (Gumel and Niger, 2008). In addition, the usual existence,uniqueness and continuation results hold for the system.3.4 Positivity of solutionsIn this section, we need to check whether or not the system’s solutions are positiveand show that the region is positively invariant for equations of the model system(3.1).Lemma 2 Let the initial conditions be{(S h (0), S m (0)) > 0, (E h (0), I h (0), R h (0), E m (0), I m (0)) ≥ 0} ∈ Ω.Then the solution set {S h , E h , I h , R h , S m , E m , I m }(t) of the model system (3.1) ispositive for all t > 0.

20Proof: From the first equation of system (3.1), we havedS h= b h − (µ h + λ mh )S h + ωR h ≥ −(µ h + λ mh )S hdtdS h≥ −(µ h + λ mh )S h∫dt∫1dS h ≥ − (µ h + λ mh )dtS hS h (t) ≥ S h (0)e −(∫ λ mh dt+µ h t) ≥ 0.Now considering the third equation of the model system (3.1), we get the followingdI h= ε h E h − (α h + γ h + µ h )I h ≥ −α h I h − γ h I h − µ h I h∫dt∫1dI h ≥ − (α h + γ h + µ h )dtI hI h (t) ≥ I h (0)e −(α h+γ h +µ h )t ≥ 0.Similarly, it can be shown that the remaining state variables of the model system(3.1) are also non-negative for all t > 0.Furthermore, we need to show that the region Ω is positively invariant. The righthand sides of equations (3.4) and (3.3) are both bounde above by b m −µ m N m and b h −µ h N h respectively. It follows that dN hdtN m (t) > b mµ m.< 0 if N h (t) > b hand dN mµ h dt< 0 ifWe use the Standard Comparison Theorem (Zhang, 1988), to show thatandN h (t) ≤ b hµ h(1 − e −µ ht ) + N h (0)e −µ ht ,N m (t) ≤ b mµ m(1 − e −µmt ) + N m (0)e −µmt .In particular, if N h (0) < b hµ hthen N h (t) ≤ b hµ hand if N m (0) < b mµ mthen N m (t) ≤ b mµ m.Therefore Ω is positively invariant. If N m (0) > b mµ mand N h > b hµ h, then either thesolution enters Ω in finite time, or N m (t) approaches b mand N h (t) approaches b hµ m µ hasymptotically, and the infected variables E h , I h , E m , I m approaches zero, as t → ∞.

213.5 Existence of the disease-free equilibrium (DFE)In this section, system (3.1) is analysed in order to obtain the equilibrium points ofthe system and their stability. Let E(S ∗ h , E∗ h , I∗ h , R∗ h , S∗ m, E ∗ m, I ∗ m) be the equilibriumpoints of the model system (3.1).Then, the equilibrium points are obtained bysetting the right hand sides of the model system (3.1) to zero, that is:dS hdt= dE hdt= dI hdt = dR hdt= dS mdt= dE mdt= dI mdt= 0. (3.7)The population will never be extinct as long as the human recruitment term b hand the mosquito birth term b m are not zero. This implies that there is no trivialequilibrium point. In the absence of infection, that is, E h = I h = E m = I m = 0, themodel system (3.1) has a steady state, E 0 called the disease-free equilibrium. Thus,from the equation (3.7), the disease-free equilibrium of the model is given byE 0 = (Sh ∗, E∗ h , I∗ h , R∗ h , S∗ m, Em, ∗ Im)∗(bh= , 0, 0, 0, b )m(T ), 0, 0 .µ h µ m(3.8)3.5.1 Local stability of the disease-free equilibrium pointTo assess both the stability of the disease-free equilibrium (DFE), and the endemicequilibrium points(EE), the computation of the basic reproductive number,R 0 , isnecessary because it is a useful constant in this study. This is an important parameterthat plays a big role in the control of the malaria infection.The method of next generation matrix operator described by Diekmann and Heesterbeek(1990) is used to define the basic reproductive number, R 0 , as the number of secondaryinfections that an infectious individual would create over the duration of theinfectious period, given that everyone else is susceptible to the disease. The modelsystem (3.1) is rewritten starting with the infected compartments E h , I h , E m , I m and

22followed by the uninfected classes. S h , R h , S m for both populations. We have,dE hdtdI hdtdE mdtdI mdtdS hdtdR hdtdS mdt= λ mh S h − (ε h + µ h )E h= ε h E h − (α h + γ h + µ h )I h= λ hm (T )S m − (ε m (T ) + µ m )E m⎫⎪ ⎬= ε m (T )E m − µ m I m. (3.9)= b h − λ mh S h + ωR h − µ h S h= γ h I h − (ω + µ h )R h= b m (T ) − (λ hm + µ m )S⎪m⎭The rate of appearance of new infection in compartments: E h and E m , from thesystem (3.1) is given by⎡F =⎢⎣β mh (T )a ′ I m S hN h0β hm (T )a ′ I h S mN hWe differentiate the matrix above, by partial derivatives with respect to the modelvariables using Jacobian matrix method at the disease-free equilibrium point E 0 ,0⎤.⎥⎦where N h ≤ b hµ hand N m ≤ b mµ mto get the Jacobian matrix,⎡⎤0 0 0 β mh a ′0 0 0 0F =⎢0 β hma ′ b m µ h. (3.10)0 0⎣b h µ m⎥⎦0 0 0 0Calculating the transfer of individuals out of the compartments of the system (3.9)by all other means, we have⎡⎤(ε h + µ h )E h(α h + γ h + µ h )I h − ε h E hV =.⎢ (ε m (T ) + µ m )E m ⎥⎣⎦µ m I m − ε m (T )E m

23Hence, the Jacobian matrix of V evaluated at E 0 is given by⎡⎤ε h + µ h 0 0 0−ε h α h + γ h + µ h 0 0V =. (3.11)⎢ 0 0 ε m (T ) + µ m 0 ⎥⎣⎦0 0 −ε m (T ) µ mWe find the inverse of the Jacobian matrix (3.11) as,⎡1(ε h +µ h0 0 0) ε h1V −1 (ε=h +µ h )(α h +γ h +µ h ) (α h +γ h +µ h0 0) . (3.12)1⎢ 0 0(ε⎣m(T )+µ m)0 ⎥⎦0 0The product of matrix (3.10) and matrix (3.12) gives,⎡ ⎤0 0 e bF V −1 0 0 0 0=⎢c d 0 0⎥⎣ ⎦0 0 0 0where e = β mha ′ ε m (T )(ε m (T ) + µ h )µ m, b = β mha ′µ m, c =ε m(T )(ε m(T )+µ m)µ m1µ m⎤β hm a ′ b m ε h µ hb h µ m (ε h + µ h )(α h + γ h + µ h )(3.13)β hm a ′ b m µ hand d =b h µ m (α h + γ h + µ h ) .Hence, we develop the matrix determinant M =| F V −1 − Iλ |= 0−λ 0 e b0 −λ 0 0M == 0.0 0 −λ 0∣ c d 0 −λ ∣We have the characteristic polynomial equation λ 4 − cλ 2 b = 0The calculated eigenvalues from the matrix M through the polynomial equationabove, are displayed in a row matrix below,

24λ i =⎡ √⎤(ε h + µ h )(α h + γ h + µ h )(ε m (T ) + µ m )µ m ( β hma ′ b mµ hµ mb h)β mh a ′ ε h ε m (T )0, 0, 0,,(ε h + µ h )(α h + γ h + µ h )(ε m (T ) + µ m )µ m √⎢⎣ (ε h + µ h )(α h + γ h + µ h )(ε m (T ) + µ m )µ m ( β hma ′ b mµ hµ mb h)β mh a ′ ⎥ε h ε m (T ) ⎦ (3.14)−(ε h + µ h )(α h + γ h + µ h )(ε m (T ) + µ m )µ mHence, the reproduction number, R 0 , from the matrix (3.14), which is the spectralradius ρ(F V −1 ), defined as the dominant eigenvalue of F V −1 is given by√(ε h + µ h )(α h + γ h + µ h )(ε m (T ) + µ m )µR 0 =2 mb h β mh β hm a ′2 ε h ε m (T )b m µ h. (3.15)((ε h + µ h )(α h + γ h + µ h )(ε m (T ) + µ m )µ 2 mb h ) 2Equation (3.15) can be simplified to give the basic reproduction number R 0 , as√β mh β hm aR 0 =′2 ε h ε m (T )b m µ h. (3.16)(ε h + µ h )(α h + γ h + µ h )(ε m (T ) + µ m )µ 2 mb hThe term β mhε m (T )a ′in Equation (3.15)describes the number of humans thatµ m (ε m (T ) + µ m )one mosquito infects during the lifetime it survives as infectious, when all humansβ hm a ′ ε hare susceptibles. In addition, the termdescribes the numberof mosquitoes that are infected through contacts with one infectious(ε h + µ h )(α h + γ h + µ h )human,while the human survives as infectious, assuming no infection among vectors. Theepidemiological implication of this is that the disease can be eliminated from thecommunity if the initial sizes of the infected sub-populations which consequentlydecrease mosquitoes biting rate leading to R 0 < 1.ε m (T )The termis the probability that the adult mosquitoes survive throughoutincubation period to the period when they become infectious, and(ε m (T ) + µ m )ε h(ε h + µ h ) isthe probability that human (the host) is exposed to the disease throughout incubationperiod to the period when they become infectious. The termβ mh (T )(µ h + α h + γ h )is the average number of times a human being is bitten by infectious mosquitoes.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,

25on 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 individualand hence the disease is able to invade the susceptible population. Therefore R 0 is auseful quantity in the study of a disease as it sets the threshold for its establishment.Notice that in this model, a ′ has double implications. It appears twice in the expressionsince the mosquito biting rate controls transmission from humans to mosquitoesand from mosquitoes to humans.The following theorem leads to results on local stability of the disease -free equilibrium(DFE) computed by the next generation method of (Van den Drieusche andWatwough, 2002) as in the basic reproduction number shown in equation (3.15)above where, the threshold quantity R 0 is the reproduction number for malaria. Biologicallyspeaking, the Theorem above implies that malaria can be eliminated fromthe community if R 0 < 1 but if R 0 > 1 the disease becomes endemic in the population.Thus, the local stability of the disease - free equilibrium result is establishedas,Theorem 1 The disease-free equilibrium (DFE) of the model system (3.9) is locallyasymptotically stable when R 0 < 1, and unstable when R 0 > 13.5.2 The endemic equilibrium pointIn the presence of malaria, that is, E h ≠ 0, I h ≠ 0, E m ≠ 0 and I m ≠ 0, the modelsystem (3.1) has an equilibrium point called the endemic equilibrium denoted by E 1and is given by,E 1 = (S ∗ h, E ∗ h, I ∗ h, R ∗ h, S ∗ m, E ∗ m, I ∗ m) ≠ 0,

26That is, E 1 is a steady-state endemic equilibrium point whereby the disease persistsin the population.Its coordinates should satisfy the following conditions for itsexistence and uniqueness of a particular point.0 < S ∗ h, 0 < E ∗ h, 0 < I ∗ h, 0 < R ∗ h, 0 < S ∗ m, 0 < E ∗ m, 0 < I ∗ mso, from the model system (3.1), we have the first order system of differential equationsexpressed asS ′ h = E′ h = I′ h = R′ h = S′ m = E ′ m = I ′ m = 0,from which the model system (3.1) turns into the homogeneous system of differentialequations (3.17). So we compute the system (3.17) below for endemic equilibriumpoints in terms of λ ∗ hm and λ∗ mh values.0 = b h − λ mh S h + ωR h − µ h S h0 = λ mh S h − (ε h + µ h )E h0 = ε h E h − (α h + γ h + µ h )I h⎫⎪ ⎬0 = γ h I h − (ω + µ h )R h. (3.17)0 = b m (T ) − (λ hm + µ m )S m0 = λ hm (T )S m − (ε m (T ) + µ m )E m⎪0 = ε m (T )E m − µ m I m ⎭The model system (3.1) has an equilibrium point E 1 = (Sh ∗, E∗ h , I∗ h , R∗ h , S∗ m, Em, ∗ Im)∗computed by using Maple programming language as shown in the (Appendix E). Thesolutions are presented in a simplified structure and refered to as Equation (3.18)below, given that;a = (µ h + ε h )(µ h + ω), b = µ h + α h + γ h ,c = b h (µ h + ω), d = ωγ h ,g = ε m (T ) + µ m , k = ε m (T )b m (T ),x = γ h cε h , q = ab − ε h d,y = q(µ h + ω), z = abµ h (µ h + ω),

27The endemic equilibrium of the malaria model system (3.1) in terms of the equilibriumvalue of the forces of infection, λ ∗ hm and λ∗ mhis given as equilibriam poitdenoted E 1 where,S ∗ h = b h(yλ ∗ mh + z) + ωxλ∗ mh(µ h + λ ∗ mh )(yλ∗ mh + z) ,E ∗ h = cε hλ ∗ mh (µ h + α h + γ h )λ ∗ mh ε h(ab − ε h d) + abµ h ε h,I ∗ h =cε h λ ∗ mhλ ∗ mh (ab − ε hd) + abµ h,R ∗ h =xλ∗ mhyλ ∗ mh + z ,⎫⎪ ⎬. (3.18)Sm ∗ b m (T )=λ ∗ hm (T ) + µ ,mEm ∗ = λ∗ hm (T )b m(T )g(λ ∗ hm (T ) + µ m) ,Im ∗ kλ ∗ hm=(T )g(λ ∗ hm (T ) + µ m) , ⎪ ⎭and λ ∗ mh =β mh (T )Im∗Sh ∗ + E∗ h + I∗ h + ,R∗ hSubstituting λ ∗ mh and (3.18) into the expression λ∗ hm = β hm (T )Ih∗Sh ∗ + E∗ h + I∗ h + , and afterR∗ hsome algebraic manipulations, the endemic equilibria of the malaria model (3.1)satisfy the polynomial given by,λ ∗ hm f(λ∗ hm ) = 0 whereFor simplification let,λ ∗ hmf(λ ∗ hm) = λ ∗ hm(Aλ ∗2hm + Bλ ∗ hm + C), (3.19)φ = (ε h ab − ε h d), ψ = (ab − ε h d), ϑ = (µ h + α h + γ h ),θ = (cε h z + µ h xε h ψ + abµ h ε h x + cε h µ h y + ωxε h ),Reflecting back to the the polynomial given by the Equation (3.19).

28we have,A = µ m g(Φ + (abµ h ε h y + yµ h Φ + zΦ + abµ 2 h ε hy + abµ h ε h z) + µ 2 mg(yΦµ 2 mg 2 +abµ h ε h y)) + (Φ(yµ h + zµ 2 mg 2 )) + abµ 2 mg 2 (ycε h ϑ + ycε h + xε h ψ) + abµ 2 mg(µ 2 mg 2 (ycε h ϑ + ycε h + xcε h ψ + θ)ψ + b h yε h ψ + zcε h + µ h cε h yϑ)µ 2 mg 2 )−β hm β mh cε h kψ,B = µ m g(yΦµ 2 mg 2 + µ 2 mg 2 (abµ h ε h y + yµ h Φ + zΦ) + µ 2 mg(abµ 2 h ε hy + abµ h ε h z))+abµ 2 h ε hz + µ 2 mg(µ 2 mg 2 yΦ + 2µ 2 mg 2 (abµ h ε h y + yµ h Φ + zΦ) + µ 2 mg(abµ 2 h ε hy + abµ h ε h z) + abµ 2 h ε hz − ((β hm β mh cε h kq + abµ 2 mg)(µ 2 mg 2 (ycε h ϑ+ycε h + xcε h ψ) + 2µ 2 h g2 θψ) + b h yε h ψ + zcε h + µ h cε h yϑ)+µ h g(abµ h ε h b h y + zb h ε h ψ + ωxabµ h ε h + zµ h cε h ϑ + zµ h cε h + µ 2 h abε hx)+abµ 2 mg(µ 2 mg 2 (ycε h ϑ + ycε h + xε h ψ + µ 2 g 2 θψ + b h yε h ψ + zcε h + µ h cε h yϑ)),C = µ 2 mg[(µ 2 mg 2 yΦ + µ 2 mg 2 (abµ h ε h yµ h Φ + zΦ) + µ 2 mg(abµ 2 h ε hy + abµ h ε h z)+abµ 2 h ε hz + abµ 2 mg(µ 2 mg(ycε h ϑ + ycε h + xε h ψ + 2µ 2 g 2 (cε h z + µ h xε h ψ+abµ h ε h x) + cε h yµ h β hm β mh cε h kq) + abµ 2 mg(µ 2 mg 2 (ycε h ϑ + ycε h + xε h ψ)+ωxε h ψ + b h yε h ψ + zcε h + µ h cε h yϑ))](1 − R0).2In this case, the value R 2 0 comes in C from the discriminant of the polynamial (3.19)such that B 2 − 4AC = 0 which is B 2 = 4AC, the critical value R 0 described inTheorem (2) and the Equation (3.20) below√R0 c = 1 − B24AHB 24AH = (1 − R2 0)B 2 = 4AH(1 − R0)2B 2= 4AC4AC = 4AH(1 − R 2 0)C = H(1 − R 2 0)

29and H is an Equation (3.21) which is descrbed below the Theorem (2).From Equation (3.19), it can be observed that the root λ ∗ hm= 0 corresponds tothe disease-free equilibrium. On the other hand, the relationship f(λ ∗ hm ) = 0 correspondsto the existence of multiple equilibria. It can be noted from equation (3.19)that the coefficient A is always positive, C is positive if R 0 < 1 and C is negativewhenever R 0 > 1. As in Mtisi et al. (2009), we make conclusions on the existenceof Endemic Equilibrium (EE) in the following theorem;Theorem 2 The malaria model (3.1) has(i) precisely one unique endemic equilibrium if C < 0 ⇔ R 0 > 1,(ii) precisely one unique endemic equilibrium if B < 0 , C = 0 or B 2 − 4AC = 0,(iii) precisely two endemic equilibria if C > 0, B < 0 and B 2 − 4AC > 0,(iv) No endemic equilibrium otherwise.Point (iii) suggests that backward bifurcation is a possibility in the model and thatthe disease may spread even though R 0 < 1. To find the backward bifurcation pointwhen R 0 < 1, the discriminant is set to zero, that is, B 2 − 4AC = 0 and solve forthe critical value of R 0 . After some manipulations and making R 0 the subject offormula, we getR c 0 =√1 − B24AH , (3.20)

30where,H = µ 2 mg[(µ 2 mg 2 yΦ + µ 2 mg 2 (abµ h ε h yµ h Φ + zΦ) + µ 2 mg(abµ 2 h ε hy + abµ h ε h z)+abµ 2 h ε hz + abµ 2 mg(µ 2 mg(ycε h ϑ + ycε h + xε h ψ + 2µ 2 mg 2 (cε h z + µ h xε h ψ+abµ h ε h x + cε h yµ h β hm β mh cε h kq + abµ 2 mg(µ 2 mg 2 (ycε h ϑ + ycε h + xε h ψ)+ωxε h ψ + b h yε h ψ + zcε h + µ h cε h yϑ].(3.21)Thus, from equation (3.20), it can be shown that backward bifurcation occurs forvalues of R 0 in the range R c 0 < R 0 < 1. The following theorem gives a condition forthe existence of the endemic equilibrium point, E 1 .Theorem 3 The endemic equilibrium point, E 1 exists whenever R 0 > 1.Proof 1 From the equation, f(λ ∗ hm ) = Aλ∗2 hm + Bλ∗ hm + C = 0, we obtainλ ∗ hm = −B ± √ (B 2 − 4AC),2Afrom which it is clear that the disease is endemic when,λ ∗ hm > 0 ⇒ B 2 − 4AC > B 2 ⇒ 4AH(1 − R 2 0) < 0 ⇒ R 0 > 1,where H is as in (3.21).Thus, the endemic equilibrium point E 1 exists whenever R 0 > 1.□3.6 Stability of the endemic equilibrium E 1It is possible that two endemic equilibria can be present in the model by consideringwhat has been shown in the Theorem 2 case (iii) that indicates the possibility ofbackward bifurcation in the model system (3.1).The Centre Manifold Theoremdeveloped by Castillo-Chavez and Song (2004) is used to explore further the presenceof backward bifurcation.

313.6.1 Existence of backward bifurcationThe existence and stability of endemic equilibrium is determined through the investigationof the possibility of the existence backward bifurcation due to existence ofmultiple equilibrium and reinfection. As a disease invades it reduces the number ofsusceptible individuals in the population, which tends to reduce its reproductive rate.Computation of eigenvalues of the Jacobian matrix can be used to determine thestability of the disease at an endemic equilibrium point. The bifurcation analysisis performed at the disease-free equilibrium by using Centre Manifold Theorem aspresented in Castillo-Chavez and Song (2004).The model system (3.1) is re-written by using state variables of the malaria modeland the center manifold approach on the system.Let x 1 = S h , x 2 = E h , x 3 = I h , x 4 = R h , x 5 = S m , x 6 = E m , x 7 = I m .Writing the system 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 model system (3.1) changes todx 1dtdx 2dtdx 3dtdx 4dtdx 5dtdx 6dtdx 7dt= f 1 = b h − φ∗ a ′ µ h x 7 x 1b h+ ωx 4 − µ h x 1= f 2 = φ∗ a ′ µ h x 7 x 1b h− (ε h + µ h )x 2= f 3 = ε h x 2 − (α h + γ h + µ h )x 3⎫⎪ ⎬= f 4 = γ h x 3 − (ω + µ h )x 4(3.22)= f 5 = b m (T ) − β hm(T )a ′ µ h x 3 x 5− µ m x 5b h= f 6 = β hm(T )a ′ µ h x 3 x 5− (ε m (T ) + µ m )x 6b h= f 7 = ε m (T )x 6 − µ m x⎪7⎭

32where N h = x 1 +x 2 +x 3 +x 4 and N m = x 5 + x 6 + x 7 , with φ ∗ = β mh from (3.15).Let φ ∗ be a bifurcation parameter, and Let φ be the value of the bifurcation parameterof the model system (3.22) when it is linearized at the disease-free equilibriumpoint E 0 such that φ = φ ∗ and R 0 = 1. Solving for the bifurcation parameter φ ∗from R 0 = 1, givesφ = φ ∗ = (ε h + µ h )(γ h + µ h + α h )(ε m + µ m )(µ m + α m )µ m b hβ hm a ′2 ε m b m µ h.A simple eigenvalue of the following Jacobian matrix, J w , is zero with the applicationof the bifurcation parameters.⎡⎤−µ h 0 0 ω 0 0 −φa ′0 −(ε h + µ h ) 0 0 0 0 φa ′0 ε h a 1 0 0 0 0J w =0 0 γ h −(ω + µ h ) 0 0 00 0 a 3 0 −µ m 0 0⎢ 0 0 a 4 0 0 −(ε m (T ) + µ m ) 0⎥⎣⎦0 0 0 0 0 ε m (T ) −µ m(3.23)where a 1 = −(α h + γ h + µ h ), a 3 = − β hm(T )a ′ b m µ h, a 4 = β hm(T )a ′ b m µ h.b h µ mb h µ mA right eigenvector associated with the eigenvalue zero is w = (w 1 , w 2 , . . . , w 7 ).Solving the system (3.23), such that J w w = 0 we get−µ h w 1 + ωw 4 − φa ′ w 7 = 0⎫−(ε h + µ h )w 2 + φa ′ w 7 = 0ε h w 2 − (α h + γ h + µ h )w 3 = 0γ h w 3 − (ω + µ h )w 4 = 0− β hm(T )a ′ b m µ hw 3 − µ m w 5b h µ m= 0β hm (T )a ′ b m µ hw 3 − (ε m (T ) + µ m )w 6b h µ m= 0ε m (T )w 6 − µ m w 7 = 0⎪⎬. (3.24)⎪⎭

33The following are the eigenvalues from the system (3.24)w 1 = ωw ⎫4 − φa ′ w 7µ hw 2 = φa′ w 7ε h + µ hε h w 2w 3 =α h + γ h + µ hw 4 = γ ⎪⎬hw 3. (3.25)ω + µ hw 5 = − β hm(T )a ′ b m µ h w 3b h µ 2 mw 6 = β hm(T )a ′ b m µ h w 3b h µ m (ε m (T ) + µ m )w 7 = w 7 > 0⎪⎭The left eigenvector satisfying v · w = 1 is v = (v 1 , v 2 , . . . , v 7 ). We transpose thematrix (3.23) to find the left eigenvalue zero, given by J v⎡⎤−µ h 0 0 0 0 0 00 −(ε h + µ h ) ε h 0 0 0 00 0 a 1 γ h a 3 a 4 0ω 0 0 −(ω + µ h ) 0 0 00 0 0 0 −µ m 0 0⎢ 0 0 0 0 0 −(ε m (T ) + µ m ) ε m (T )⎥⎣⎦−φa ′ φa ′ 0 0 0 0 −µ mwhere a 1 = −(α h + γ h + µ h ), a 3 = − β hm(T )a ′ b m µ h, a 4 = β hm(T )a ′ b m µ h.b h µ mb h µ mWe get the following system of equations.−µ h v 1 = 0−(ε h + µ h )v 2 + ε h v 3 = 0−(α h + γ h + µ h )v 3 + γ h v 4 − β hm(T )a ′ b m µ hv 5 + β hm(T )a ′ b m µ hv 6 = 0b h µ mb h µ mωv 1 − (ω + µ h )v 4 = 0−µ m v 5 = 0−(ε m (T ) + µ m )v 6 + ε m (T )v 7 = 0−φa ′ v 1 + φa ′ v 2 − µ m v 7 = 0.

34After solving we get the following left eigenvectorsv 1 = 0v 2 = v 2 > 0v 3 = (ε h + µ h )v 2ε hv 4 = 0⎫⎪ ⎬. (3.26)v 5 = 0v 6 = ε m(T )v 7ε m (T ) + µ mv 7 = φa′ v 2 ⎪ ⎭µ mThe Theorem below is reproduced for quick reference, and will be useful to provelocal stability of the endemic equilibrium point near R 0 = 1.Theorem 4 (Castillo-Chavez and Song, 2004) consider the following general systemof ordinary differential equations with 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) ≡ 0for 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=1∂ 2 f kv k w i w j (0, 0),∂x i ∂x j∂ 2 f kv k w i (0, 0),∂x i ∂φthen, the local dynamics of the system around 0 is totally determined by the sign ofa and b.

35(i). a > 0, b > 0. When φ < 0 with |φ|

36Furthermore, the associated non-zero partial derivatives at the disease-free equilibriumare solved as∂ 2 f 2= − φa′ µ h,∂x 2 ∂x 7 b h∂ 2 f 6∂x 6 ∂x 3= − β hm(T )a ′ µ hb h,∂ 2 f 2∂x 3 ∂x 7= − φa′ µ hb h∂ 2 f 2∂x 7 ∂x 3= − β hm(T )a ′ µ hb h.whereab 2 == v 2 w 2 w 7 (− φa′ µ h) + v 2 w 3 w 7 (− φa′ µ h)b hb h+ v 6 w 6 w 3 (− β hm(T )a ′ µ h) + v 6 w 7 w 3 (− β hm(T )a ′ µ h)b hb h= − a′ µ h[v 2 w 7 φ(w 2 + w 3 ) + v 6 w 3 β hm (T )(w 6 + w 7 )]b h= − a′ µ hb h[v2 w 2 7φ 2 a ′ b 1 + v 6 w 3 w 7 β hm (T )b 2]< 0,( )αh + γ h + µ h + ε hb 1 =(α h + γ h + µ h )(ε h + µ h )()β hm (T )a ′2 b m µ h ε hb h µ m (ε m (T ) + µ m )(α h + γ h + µ h )(ε h + µ h ) + 1 .For the sign of b, we can show that the associated non-vanishing partial derivatives∂f 2∂φ = a′ x 7 ,Hence from the above expression we get∂ 2 f 2∂x 7 ∂φ = a′ .b = v 2 w 7 a ′ > 0.Therefore, a < 0 and b > 0. So (by Theorem 4, item (iv)), as a result we haveestablished the following theorem:Theorem 5 The unique endemic equilibrium guaranteed by Theorem 2 is locallyasymptotically stable for R 0 > 1 but near 1. In addition, by Theorem 2 item (i) themalaria model undergoes backward bifurcation when a > 0, this happens only whenν 2 < 0 and ω 7 < 0, otherwise it undergoes forward bifurcation.The implication of Endemic Equilibrium point being locally asymptotically stablemeans that the disease can invade the population and persist if no purposive interventionalmeasures are required to reduce or if possible eradicate the disease.

37Figure 3.2: The Forward or Transcritical bifurcation.The curve below illustrates the bifurcation diagram where the thick blue continuouscurve R 0 < 1 corresponds to the stable disease-free equilibrium DFE, and the samecurve R 0 > 1 corresponds to the stable endemic equibrium EE. This clearly reinforcesthat the model system (3.1) exhibits forward bifurcation for R 0> 1 as seen inFigure 3.2 below.3.7 Forward BifurcationThe malaria model normally has backward bifurcation where multiple stable equilibriaco-exist when R 0 ≤ 1 (Mtisiet al. 2008), but this model is giving us theforward bifurcation as shown in Figure 3.2. It is generated by using the Matlab codsshown in Appendix A. This may be happening because of temperature dependentparameters incoperated in the model. According to Dushoff et al. (1998), we provedthe stability of the disease-free equilibrium point (locally asymptotically stable forR 0 < 1 and unstable for R 0 > 1) and the existence of the endemic equilibrium pointfor all R 0 > 1. From this figure simulations suggest that the endemic equilibrium isstable for R 0 > 1.

38However, for a biologically meaningful interpretation of the above result, we notethat when a < 0; making R 0 greater than 1 by a small change in parameters givesrise to a positive branch of equilibrium. But if we decrease R 0 to just less than 1there is no endemic steady state. As in Chinis et al. (2008), the bifurcation typeobtained as seen on the diagram above is often called a forward bifurcation.If we use directly the quantity R 0 to control the malaria, we must lower R 0 below 1to prevent it when a < 0. We recall that in most epidemic models investigated, thebifurcation tends to be forward at R 0 = 1. For small changes in the values of theseparameters, major changes in equilibrium behaviour can occur.3.8 SummaryIn this chapter we formulated malaria model to connect two sub-models humans(hosts) and mosquitoes (vectors). It has been found that the disease-free equilibriumE o is locally asymptotically stable if R o < 1 and has endemic equilibrium E 1 whenR o > 1. Normally malaria dynamics has a coexistence between DFE and EE insome points just R o < 1 and hence showing the existence of backward bifurcationfor the model. But the malaria model established in this study is incorporated withtemperature dependent parameters there is an emergence of forward bifurcationscalled transcritical bifurcation as described early.

39CHAPTER FOURNUMERICAL SIMULATIONSIn this section, we illustrate the analytical results of this work by carrying out numericalsimulations of the model. The simulations will be done using the parametervalues presented in Table 4.1 and the following initial conditions; S h = 100, E h =700, I h = 500, R h = 400, S m = 100, E m = 900 and I m = 700. These are average valueswhich have been calculated from the number of people who participated duringthe survey. The parameter values were obtained from TACAIDS (Tanzania) andliterature as shown in Table 4.1. The model is simulated using Matlab programminglanguage. Thus, we simulate the dynamics of each epidemiological class of themodel with time as well as the effect of temperature-dependent parameters on thedynamics of the infected classes of the model.Table 4.1: Model Simulation Parameter Values given - per day.Parameter Parameter Value Referenceb h 0.0015875 Gemperli, 2006b m (T ) 0.61 Assumedµ h 0.0004 TACAIDS, 2008µ m 0.05 Macdonald, 1957a ′ 0.29 Ishikawa, 2003ε m (T ) 0.75 Lakshmikantham, 2004ω 0.000017 Ishikawa, 2003γ h 0.00019 Coutinho, 2005α h 0.333 TACAIDS, 2008β hm 0.86 Niger, 2008β mh 0.083 Miranda, 2009ε h 1/17 Blayneh, 2009

40(a)(b)(c)(d)Figure 4.1: Illustrates changes in human state variables of the model.4.0.1 Simulation results of the human populationThe Figure 4.1 for human state variables has shown the dynamics of (a) susceptiblehuman population, (b) exposed human individuals, (c) infected human individualsand (d) recovered individuals, were excuted using matlab codes shown in Appendix BWe observe from Figure 4.1(a) that the number of susceptible individuals decreaseswith increasing time and eventually decays to zero. This is the case due to continuedexposure of humans to mosquitoes. Also, exposed individuals initially increase dueto higher numbers of susceptibles in the community and then decrease exponentiallyas a result of natural death as shown by Figure 4.1(b).Similarly, Figure 4.1(c)shows that the number of infected individuals decreases with time. This is caused

41by malaria-induced deaths, natural death and also the interventions that might bepracticed in the community.Such interventions might be education, drugs for reatment,used of preventive netsand repellents, and insectsides applications. In addition, natural immunity and possibleintervention strategies induce a decrease in the recovered human population asshown by Figure 4.1(d).

42(a)(b)(c)(d)Figure 4.2: The dynamics of mosquito population with time under constanttemperature-dependent parameters.4.0.2 Simulation results of the mosquito populationFigure 4.2 are Matlab graphs representing (a) susceptible mosquito population, (b)exposed mosquito population and (c) infected mosquitoes. (d) Model variables andsine wave changes of temperature. Graphical simulation of the above figures weredeveloped by using the Matlab Programming language codes shown in Appendix B.It is graphically seen in Figure 4.2 (a) that the number of susceptible mosquitoesdecreases with time and diminishes to zero. This is due to the fact that, mosquitoesget infected when they bite infected humans in addition to natural death. This alsoaffects the other classes. We observe in Figure 4.2(b) and Figure 4.2(c) respectivelythat, the number of exposed and infected mosquitoes decrease with time.

43It is also observed in Figure 4.2(d) that the sinusoidal fluctuation of the exposedmosquitoes has exchangeable amplitude with that of infected mosquitoes population(exposed decrease with increase of infected mosquitoes and vice vesa).Thegraph shows that initially, exposed individuals increased when infected individualsdecreased, but deviation is small as time goes. It reduces exponentially as sinusoidalfunction of temperature decreases. The sinusoidal waves represent temperature dependentexposed (E m ) and infected (I m ) mosquito variables which are in the sametrend with that of infected individuals I h . It implies that there is a relational impactof temperature in the spread of malaria disease.We used the sine function w1 = 0.00019+cos( 25πt1225πt)+sin( ) to describe temperature-12dependent parameters behaviour for Exposed and infected mosquitoes and infectedhuman I h in the same trend of the disease in Figure 4.2(d). It is showing seasonaloscillatorily decreasing with decrease in infected mosquitoes.Hoshen and Morse(2004) in their study, conclude that the interannual variation of malaria incidence inSouthern Africa, is determined by temperature variation. The variation of temperatureis of greatest importance. Thus in the case of a uniform heating of the region,we could expect the area to move from its present epidemic structure to seasonalendemicity.

44Figure 4.3: The dynamics of human and mosquito population’s epidemiologicalclasses plotted on the same axes.Figure 4.3 illustrates the behaviour of each of the classes of our model for both humanand mosquito populations as discussed in Section 4.0.1 and Section 4.0.2.4.0.3 The impact of temperature on malaria dynamicsIn the case of our model simulations, high temperatures shall mean all temperaturedependentparameters that fall in the range [0.5, 1) and low temperatures for anytemperature-dependent parameters in the range (0, 0.5).

45Figure 4.4: Changes in prevalence of malaria in high and low temperaturesThe red curve in Figure 4.4 is plotted with b m (T ) = 0.61 estimated in the [0.5, 1)interval, ε m (T ) = 0.75 and the blue curve with b m (T ) = 0.071, ε m (T ) = 0.0065.They show some pertains of relationship between the two coloured graphs.Thecodes in Appendix B are used to demonstrate the graphs. All values of parameterswere subjected into Matlab simulation as in shown in Appendix B.In Figure 4.4, we observe that the malaria prevalence is high in areas with hightemperatures. This is due to the fact that warm areas provide conducive environmentfor mosquito breeding and hence, the probability of transmission of the diseaseincreases. However, the prevalence changes as temperature decreases.This is alsosupported by Zhou,et al. (2003) who presents that temperature affects the developmentrates and survivorship of malaria parasites and mosquito vectors.

46Figure 4.5: The impact of temperature-dependent parameter on the dynamicsof susceptible mosquitoes.In the malaria model, the birth rate, b m (T ), of mosquitoes is affected by temperature,that is, the higher the temperature the higher the birth rate and vice-versa.Thus, from Figure 4.5 it can be noted that there are more susceptible mosquitoeswhenever the temperatures are high and low whenever the temperatures are low.In other words, areas that have higher temperatures are expected to have moremosquitoes than those with lower-temperature areas.The same phenomenon also applies for infected mosquitoes as confirmed by Figure4.6 which indicates that there are more infected mosquitoes in highly-temperaturedareas and less otherwise. This is in agreement with observations in Paaijmans et al.(2009) that temperature fluctuations can substantially alter the incubation periodof parasites and increase mosquitoes population and malaria transmissions rates inlowlands with higher temperatures. On the other hand, low temperatures in thehighlands of East Africa have shown low mosquito population hence low malariatransmission.

47Matlab codes in Appendix B were used to generate graphs as shown in,Figure 4.5Figure 4.6 and Figure 4.7 illustrating the impact of temperature-dependent parameteron dynamics of susceptible and infected mosquitoes as well as exposed individuals.The effect of temperature on the dynamics of susceptible and infected mosquitoesalso affects humans that are exposed to the disease. Figure 4.7 indicates that whentemperatures are high, the number of exposed humans increase to a certain level andthen gradually decreases until it reaches zero. However, low temperatures mean thatless exposed individuals will be available in the community. This is due to infectedmosquitoes that infect humans at an alarming rate in areas with high temperatures,since malaria epidemic is as a result of direct interaction between mosquitoes andhumans.4.1 Model validation using temperature recordThe model validation is a way to support the outcomes analysed in the dynamic behaviourof the model impacting its predictive ability. It encompass both quantitativeand qualitative elements possibly the most important step in the model sequence.Often the validation of a model seems to consist of measures of the variability inthe response accounted for the model under investigation. Validation can be conductedby comparing variations of elements distribution, or with observations aboutelements from the literature. In our case, a more convincing demonstration of theutility of temperature variations, to test predictions of mosquito population changeswith temperature using the collected data of temperature from warm and cold areasof Tanzania.According to Foley et al. (2008), numerical methods for model validation, such asstatistical ones, are also useful, but more finely if include graphical methods. Graphs

48Figure 4.6: The impact of temperature-dependent parameter on the dynamicsof infected mosquitoes..Figure 4.7: The impact of temperature-dependent parameter on dynamicsof exposed individuals.

49provide information on the adequacy of different aspects of the model because theyreadily illustrate a broad range of complex aspects of the relationship between themodel and the data. They often try to compress that information into a single descriptiveresult.We compared temperature variations in warm and cold areas of Tanzania usingmonthly recoded mean temperature from 1999 to 2009 as shown in Appendices Cand D. The records were obtained from Mufindi District(Iringa) at Mufindi Tea andCoffee Company Limited in Southern Highland cold area of Tanzania, and TanzaniaMeteorological Agency (TMA) weather forecasting department Dar-es-Salaam forcoastal lowland warm areas of Tanzania.Previous studies revealled that areas of high level of malaria prevalence are of lowlandsobviously with high temperature and areas of low temperature in some highlandswere said to be free from the disease. However, (Saker et al. 2004) presentsthat Anopheles mosquitoes can only transmit Plasmodium falciparum malaria parasitesif the temperature remains above 16 o C. Furthermore, within their survivalrange, warmths accelerate the biting rate of mosquitoes, and the maturation of parasiteswithin them (McArthur, 1972).It is well-known that temperature decreases with the increase in height from thesea level by 0.6 o C in every 100 meters. From the model simulations we saw thatthe higher the temperature the higher the prevalence of malaria, and the low thetemperature the low the diseases prevalence. This implies that areas of higher temperaturepromote mosquitoes and malaria parasites life and temperature below 16 o Chave little or else no support of such life.

50Figure 4.8: The temperature in warm area (Dar-es-Salaam).Areas with high temperature in Dar-es-Salaam from 1999 to 2009 were illustratedin Figure 4.8 using data shown in Table 7.1 of Appendix C. The two beams containline graphs which are annual temperature on monthly basis showing the maximumand minimum temperatures.Figure 4.9: Deviations of temperature from 16 o C above which mosquitoesand malaria parasites life is supported in Dar-es-Salaam.

51The temperature 16 o C in Figure 4.9 is ploted by using data in Table 7.2 of AppendixB, above it mosquitoes life is supported.It is noted that the minimum favorable temperature for mosquito survivorship is16 o C. Based on findings, the temperature we investigate shows deviation from themaximum and minimum level (monthly based) in each case and compare temperaturesin cold and worm areas as shown in Figures 4.9 and Figure 4.11 in Dar-es-Salaaam and Iringa respectively.We presented the maximum and minimum temperature in both cold and warm areasof Tanzania. The records represents the temperature variations in almost a decadefrom 1999 to 2009 as the data from the areas selected for other areas world alikeregions with data shown in Appendix C and D with temperature behavioural difference.The two beams contain line graphs which are annual temperature on monthlybasis; trucks being higher in warm areas (Dar es Salaam) and low in cold areas(Iringa) as Figure 4.8 and Figures 4.10 indicate.The maximum and minimum temperature in Dar es Salaam (warm area) are bothabove the minimum mosquito and parasites life supporting temperature 16 o C asshown in Figure 4.9.This may be an indicator of the possible existence of mosquitoesin warm areas that have implication of malaria susceptibility throughout the years.

524.2 SummarySimulations of results were carried out using the model parameter values obtainedfrom TACAIDS (Tanzania). We used a Matlab programming language and Microsoftexcel to simulate results of malaria dynamics based on its effect on individuals behaviourin each class as time goes. Investigation of increase and decrease of eachcompartment as a result of exposure and infection of both human and mosquitoesshow that temperature increase is associated with mosquito population increase.Temperature changes were found to be proportional to malaria incidence in particulararea depending on the intensity of temperature. We compared temperature inwarm and cold areas by using data in Appendix C and Appendix D for cold areas byinvestigating the maximum and minimum temperatures and their deviations from16 o C the mean of the minimum temperature required for mosquitoes and malariaparasites life support. It is suggested that malaria prevalence is higher in areas withhigher temperatures than areas with low temperatures.We are interested in the mosquitoes population change relative to temperaturechanges in malaria infected areas and those in risk of malaria invasion, where humans,pathogens, and vectors potentially overlap (Foley et al., 2008), depending onrealistic model of vector distribution an integral part of ecological role models forpotential malaria vectors development.

53Figure 4.10: The temperature in cold area (Iringa).Areas of warm temperature in Iringa from 1999 to 2009 were illustrated in Figure 4.10using data shown in Table 7.3 of Appendix D. The two beams contain line graphswhich are annual temperature on monthly basis showing the maximum and minimumtemperatures in cold areas. The two beams contain line graphs which are annualtemperature monthly based.Figure 4.11: Deviations of temperature from 16 o C above which mosquitoesand malaria parasites life is supported in Iringa.

54CHAPTER FIVECONCLUSION AND RECOMMENDATIONS5.1 DiscussionIn this dissertation we developed a model to study the effects of temperature changeson malaria dynamics. Under this study, we have presented and analysed a malariamodel with some temperature-dependent progression parameters. The study focuswas to investigate the effects of temperature variations on the transmission dynamicsof malaria in warm and cold areas in a community. In this study, we only consideredmuch the mosquito population as related to infection progress to both host andthe vector as it is affected by temperature in aspects like breeding conditions andmosquitoes population changes.Both qualitative and numerical analyses of the model were done. Qualitative analysisof the model involved the determination of the basic reproduction number, R 0 (bythe next generation operator approach) as well as the existence and stability of modelequilibria. It was indicated that the model has two equilibrium points namely; thedisease-free equilibrium which is locally asymptotically stable whenever R 0 < 1 andunstable otherwise giving rise to the existence of the endemic equilibrium for R 0 > 1.In numerical analysis, we observed that malaria prevalence rate is higher in areaswith high temperatures than in those with low temperatures.This is dueto the fact that areas with high temperatures provide conducive environment formosquito breeding hence, increasing the mosquito susceptible population. Minakawaet al (2006) suggest that malaria transmission is more susceptible to environmentalchanges in the highlands than in the lowlands because vector production is apparentlylimited by the cool highland climate.

55Production of malaria vectors is determined by their development time, survival rate,and fecundity,as well as habitat availability. As a result, of this, even the humanpopulation is affected as far as malaria dynamics are concerned. It was observedthat, as long as temperatures are high in a community, then that community willalways have individuals exposed to malaria and those that are sick from the disease.From the real field data findings, it is revealed that temperature deviation fromthe minimum temperature favorable for mosquito life and parasites development inDar-es-Salaam is still positive. This implies that the temperature in this warm areassupports life of mosquitoes throughout each year regardless of seasonal temperaturevariations above 16 o C. This situation indicates that all ranges of temperatures inwarm areas support mosquitoes life and hence malaria dynamics.Figure 4.10 shows the temperature of cold highland area in Iringa (Mufindi district)where the temperature is much low.The maximum temperature of this area isalmost the same as the minimum temperature in Dar es Salaam (warm area). Figure4.10 shows that the minimum temperature in the area is below 16 o C (indicated(0.0) deviation point) Figure 4.10 cold areas in highlands zone (Iringa) has the temperaturebelow 16 o C, which implies that in each year unfavorable temperatures formosquitoes survival emerge quite often.From the findings of this study, it is disclosed that there is a variation of temperaturewith a large interval from warm to cold areas. So malaria parasites and mosquitoespopulations grow faster with higher temperatures in warm areas; but decrease incold areas with low temperature. This implies that areas in higher temperatureshuman population is more exposed to malaria than those of lower temperatures in

56the community.However, it should be said here that, the same numerical analysis showed that althoughareas with high temperature have a high malaria prevalence, the diseasecan be eliminated at some point in the course of infection due to natural immunityof infected individuals, natural and malaria-induced deaths as well as possibleinterventions available in the community.5.2 ConclusionA deterministic malaria model has been formulated and analyzed using assumptionsmade.Analysis and determination of the model stability of the disease free andendemic equilibrium were carried out. The assigned parameters were used to investigatetemperature dependent variables behaviour in connection to temperaturevariation and mosquitoes vector population. Some temperature dependent parameterswere used to visualize graphical representations of temperature variations incold highlands and warm lowlands comparatively in support of life of mosquitoesand hence malaria parasites development.Basing on the results of this study, we conclude that, temperature has a substantialeffect on malaria dynamics. Areas with high temperatures are likely to be malariasusceptible to, unlike those areas with low temperatures. It is also supported byZhou et al. (2003) in their study on climate variability and malaria epidemics inthe East African highlands that temperature and rainfall affects the development ofrates of survivorship of malaria parasites and mosquito vectors.It is suggested that we suppress conducive environment for parasites and mosquitoessurvival as temperature significantly increase with this situation mostly in cold areas

57with temperature changes more over 16 ◦ C favoured by parasites and mosquito life.5.3 RecommendationsMalaria eradication remains a challenge to Malaria Control Programs in developingcountries, hence there is a need of immediate efforts to control the disease. Itis known that malaria epidemic has no permanent immunity as seen in the compartmentalmodel flow in Figure (3.1) that Removed individuals undergo temporaryimmunity that leads back to susceptible class and infected mosquitoes are infectedfor all life. We have to control mosquito’s population and minimize ways of directcontact between mosquitoes and human beings. From the study results of this work,we therefore recommend that;1. An introduction of vector control parameters in the model to analyse the causesor determinants of the disease, such parameters should represent insecticide tobe used in reduction of mosquitoes population in areas of high temperaturesand destruction of influential habitats of mosquitoes.2. Control of factors related with temperature changes is important. Simulationsshow that increase in temperature is associated with increase in mosquitoespopulation.The community should be communicated to monitor temperatureincrease factors made by human daily occupations leading to favourableconditions for mosquito and parasites development in areas of low and hightemperatures.3. Treatment parameters are to be introduced in the model to reduce the prevalenceof malaria and human death rate due to the disease caused by infectedmosquitoes interaction with infected and uninfected individuals.4. Relevant authorities should consider eliminating the birth rate of mosquitoessince it is directly related to mosquito breeding.Malaria warning systems

58should be developed to sensitise the community response to combat the diseasein infected areas and those regarded as being controlled or free from the disease.5. Intervention and control strategies of malaria should be introduced in places orareas of high malaria prevalence rates due to high temperatures. It is suggestedto suppress developmental stages of mosquitoes susceptible to the disease incold areas before the areas are invaded by the disease.5.4 Future WorkWe suggest the following areas for further study of the Malaria dynamics to becarried out for further investigation of barriers of control and develop new strategicmethods that may be appropriate to overcome malaria transmission.5.4.1 Social economic conditionThis should be modelled to check how human interactions in communities livingin different levels of malaria prevalence contribute to the disease progress. It focuson Transportation of goods, human traveling (infected and uninfected ones) to andfrom endemic to non-endemic areas. Temperature variation and rainfall, are to beinvestigated simultaneously with effective data obtained from weather stations ofareas under study of malaria dynamics.5.4.2 Global warmingThis can be studied to check whether or not that global warming significantly influencesexpansion of malaria prevalence with altitude and latitude involving experimentaldata collected in a variety of fields of studies using mathematical modelsvalidated by using information available at a particular area.

595.4.3 Water systems management and irrigation schemesIn urban drainage systems and rural water usage in either irrigation or swimmingand fishing, dames constructions including all factors of water lodging and storagesthat may alter and promote the biological effects on mosquitoes ecology are subjectto constraints to be studied.5.4.4 Temperature and rainfall effects on malaria dynamicsThe combination of the two weather elements is in suspect of having strong supportof mosquitoes life generation with high level of breeding stages development.This probably can lead to an increase in the rate of human-mosquito contact whicheventually lead to malaria intensity.5.4.5 Malaria intervention and experimental controlClinical and community trials can be used to investigate the effectiveness of newmethods used to control the disease or to improve the underlying conditions.

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67APPENDICESAPPENDIX AThe matlab codes for the forward bifurcation diagramTo find A, B, C for sue in quadratic equationFile name:quadsaved in Bifur Codesparameter valuesR0_value=0.4:0.01:1.6;Root_array=zeros(length(R0_value),2);b_h=0.0015875; bmT=0.61; mu_h=0.02; mu_m=0.05; ome=0.017;alp_h=0.333; bethm=0.86; betmh=0.0833;gam_h=0.03704; ...epsmT=0.75; a=0.29; r1=5; r2=0.5; b_m=0.61;eps_h=1/17;a=(mu_h+eps_h)*(mu_h+ome);c=b_h*(mu_h+ome);d=ome*gam_h;g=epsmT+mu_m;k=epsmT*bmT;x=gam_h*c*eps_h; ...v1=mu_h+alp_h+gam_h;b=v1;v=a*b-eps_h*d;q=v;y=v*(mu_h+ome);w=eps_h*(a*b-d);

68z=a*b*mu_h*(mu_h+ome);hold onfor i=1:1:length(R0_value);R0=R0_value(i);beta =((eps_h+mu_h)*v1*(epsmT+mu_m)*(mu_m^2*b_h)*R0^2)./(betmh*a*eps_h*epsmT*epsmT*b_m*mu_h);A=r1*r2*(mu_m*g*(w+a*b*mu_h*eps_h*y+y*mu_h*w+z*w+a*b*mu_h^2*eps_h...*y+a*b*mu_h*eps_h*z)+mu_m^2*g*(y*w*mu_m^2*g^2+(a*b*mu_h...*eps_h*y+w*(y*mu_h+z*mu_m^2*g^2))+a*b*mu_m^2*g^2*(y*c*eps_h...*(mu_h+alp_h+gam_h)+y*c*eps_h+x*eps_h*v))+a*b*mu_m^2*g*(mu_m^2...*g^2*(y*c*eps_h*v1+y*c...*eps_h+x*c*eps_h*v+(c*eps_h*z+mu_h*x*eps_h*v+a*b*mu_h*eps_h*x+c...*eps_h*y*mu_h+ome*x*eps_h*v+b_h*y*eps_h*v+z*c*eps_h+mu_h*c*eps_h...*y*v1)*mu_m^2*g^2)-bethm*betmh*c*eps_h*k*v));B=r1*(mu_h*g*(y*w*mu_m^2*g^2+mu_m^2*g^2*(a*b*mu_h*eps_h*y+y*mu_h...*w+z*w))+mu_m^2*g*(a*b*mu_h*y+a*b*mu_h*eps_h*z)+a*b*mu_h^2...*eps_h*z+mu_m^2*g*(mu_m^2*g^2*y*w-2*mu_m^2*g^2)*(a*b*mu_h...*eps_h*y+y*mu_h*w+z*w+mu_m^2*g*(a*b*mu_h^2*eps_h*y+a*b*mu_h...*eps_h*z))+a*b*mu_h^2*eps_h*z-((bethm*betmh*c*eps_h*k*q+a*b...*mu_m^2*g)*(mu_m^2*g^2*(y*c*eps_h*v1*y*c*eps_h...+x*c*eps*v)))+2*mu_h^2*g^2*(c*eps_h*z+mu_h*x*eps_h*v+a*b*mu_h...*eps_h*x+c*eps_h*mu_h*y+ome*x*eps_h*v+b_h*y*eps_h*v+z*c*eps_h...+mu_h*c*eps_h*y*v1)+mu_h*g*(a*b*mu_h*eps_h*b_h*y+z*b_h*eps_h*v...+ome*x*a*b*mu_h*eps_h+z*mu_h*c*eps_h*v1+z*mu_h*c*eps_h+mu_h^2...

69*a*b*eps_h*x)+a*b*mu_m^2*g*(mu_m^2*g^2*(y*c*eps_h*v1+y*c*eps_h...+x*eps_h*v+mu_m^2*g^2*(c*eps_h*z+mu_h*x*eps_h*v+a*b*mu_h*eps_h...*x+c*eps_h*y*mu_h+ome*x*eps_h)*v+b_h*y*eps_h*v+z*c*eps_h+mu_h...*c*eps_h*y*v1)));C=(mu_m^2*g*((mu_m^2*g^2*y*w+mu_m^2*g^2*(a*b*mu_h*eps_h*y*mu_h*w...+z*w))+ mu_m^2*g*(a*b*mu_h^2*eps_h*y +a*b*mu_h*eps_h*z)+ a*b...*mu_h^2*eps_h*z+a*b*mu_h^2*g*(mu_h^2*g*(y*c*eps_h*v1+y*c...*eps_h+x*eps_h*v+mu_m^2*g^2*(c*eps_h*z+mu_h*x*eps_h*v)+a*b...*mu_h*eps_h*x+c*eps_h*y*mu_h*bethm*betmh*c*eps_h*k*q+a*b...*mu_m^2*g*(mu_m^2*g^2*(y*c*eps_h*v1+y*c*eps_h+x*eps_h*v)+ome...*x*eps_h*v+b_h*y*eps_h*v+z*c*eps_h+mu_h*c*eps_h*y*v1)))))*(1-R0^2);p=[A,B,C];r =roots(p);len=length(r);for t=1:1:lenif (imag(r(t))~=0) || (real(r(t))

70Root_array(j,:) =sort(Root_array(j,:));endf1=R0_value_Cr;while (Root_array(f1,1)~=0)f1=f1+1;endR0_value_Cr2=f1;Zero_1st=R0_value(1,1:R0_value_Cr2-1);y_zero=zeros(1,length(Zero_1st));Unstable =R0_value(1,R0_value_Cr:length(R0_value));figure (1)plot(Zero_1st,y_zero,’b’,’LineWidth’,3)figure (2)plot(Unstable, Root_array(R0_value_Cr:length(R0_value),2),’r--’,...’LineWidth’,3)figure (3)plot(Unstable,Root_array(R0_value_Cr:length(R0_value),2),’b’,...’LineWidth’,3)ylabel(’Force of infection, \lambda_{hm}’,’FontSize’,12)xlabel(’Reproduction number, R_e’,’FontSize’,12)hold offfigure (4)plot(R0_value,Root_array(:,1),’r’,R0_value,Root_array(:,2),’b’,...R0_value,Root_array(:,2),’g’,’LineWidth’,2)

71The graph of human susceptible, exposed and infected populationscleary0=[100 700 50 30 1000 900 700];[t y0]=ode45(@sanga12,tspan,y0);plot(t,y0)Numerical simulation and graphical representation codes.Code Name: sangabfunction dy = sangab(t,y)dy = zeros(size(y));%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++b1=0.0015875;b2=0.071;r=0.083;m=0.86;u1=0.0000005;u2=0.05;e1=0.0588;e2=0.75;j1=0.333;w1=0.19;w=0.000017;a=0.29;%u1=0.02326; b1=u1*10^5; b2=6; r=0.0833; m=0.0057233; u2=0.1429;%e1=1/17; e2=0.1; j1=0.03454; w1=0.0035; w=0.0018; a=40;%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++S1=y(1);E1=y(2);I1=y(3);R1=y(4);S2=y(5);E2=y(6);I2=y(7);%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++N=S1+E1+I1+R1;g1=(r*I2*a)/N; g2=(m*I1*a)/N; B1=r; B2=m;dy(1)=b1-(u1+g1)*S1+w*R1;

72dy(2)=g1*S1-(u1+e1)*E1;dy(3)=e1*E1-(u1+j1+w1)*I1;dy(4)=w1*I1-(u1+w)*R1;dy(5)=b2-(u2+g2)*S2;dy(6)=g2*S2-(e2+u2)*E2;dy(7)=e2*E2-u2*I2;%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++R0=sqrt((B1*B2*a^2*e1*e2*b2*u1)/((e1+u1)*(j1+w1+u1)*(e2+u2)*u2^2*b1))%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

73APPENDIX BMain Matlab M-file used to produce different plots{\bf {Code Name: Sanga}}\begin{verbatim}function dy = sanga(t,y)dy = zeros(size(y));%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++%b1=0.0015875;b2=0.071;r=0.083;m=0.86;u1=0.00004;u2=0.05;%e1=0.0588;e2=0.75;j1=0.333;w1=0.00019;w=0.000017;a=0.29;%u1=0.02326; b1=u1*10^5; b2=6; r=0.0833; m=0.0057233;%u2=0.1429; e1=1/17;e2=0.1; j1=0.03454; w1=0.0035; w=0.0018;a=40;b1=0.0015875;b2=0.071;r=0.083;m=0.86;u1=0.00004;u2=0.05;e1=0.0588;e2=0.75+2*sin(t); j1=0.333;w1=0.00019;w=0.000017;a=0.29;%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++S1=y(1);E1=y(2);I1=y(3);R1=y(4);S2=y(5);E2=y(6);I2=y(7);%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++N=S1+E1+I1+R1;g1=(r*I2*a)/N; g2=(m*I1*a)/N; B1=r; B2=m;

74dy(1)=b1-(u1+g1)*S1+w*R1;dy(2)=g1*S1-(u1+e1)*E1;dy(3)=e1*E1-(u1+j1+w1)*I1;dy(4)=w1*I1-(u1+w)*R1;dy(5)=b2-(u2+g2)*S2;dy(6)=g2*S2-(e2+u2)*E2;dy(7)=e2*E2-u2*I2;%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++R0=sqrt((B1*B2*a^2*e1*e2*b2*u1)/((e1+u1)*(j1+w1+u1)*(e2+u2)*u2^2*b1))%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++Code Name: sanga1function dy = sanga1(t,y)dy = zeros(size(y));%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++b1=0.0015875;b2=0.071;r=0.083;m=0.86;u1=0.00004;u2=0.05;e1=0.0588;%e2=0.75;j1=0.333;w1=0.00019;w=0.000017;a=0.29;e2=0.0065%u1=0.02326; b1=u1*10^5; b2=6; r=0.0833; m=0.0057233; u2=0.1429;%e1=1/17;e2=0.1; j1=0.03454; w1=0.0035; w=0.0018; a=40;%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++S1=y(1);E1=y(2);I1=y(3);R1=y(4);S2=y(5);E2=y(6);

75I2=y(7);%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++N=S1+E1+I1+R1;g1=(r*I2*a)/N; g2=(m*I1*a)/N; B1=r; B2=m;dy(1)=b1-(u1+g1)*S1+w*R1;dy(2)=g1*S1-(u1+e1)*E1;dy(3)=e1*E1-(u1+j1+w1)*I1;dy(4)=w1*I1-(u1+w)*R1;dy(5)=b2-(u2+g2)*S2;dy(6)=g2*S2-(e2+u2)*E2;dy(7)=e2*E2-u2*I2;%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++R0=sqrt((B1*B2*a^2*e1*e2*b2*u1)/((e1+u1)*(j1+w1+u1)*(e2+u2)*u2^2*b1))%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++Code Name: sanga2function dy = sanga2(t,y)dy = zeros(size(y));%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++%b1=0.0015875;b2=0.071;r=0.083;m=0.86;u1=0.00004;u2=0.05;%e1=0.0588;e2=0.75;j1=0.333;w1=0.00019;w=0.000017;a=0.29;%u1=0.02326; b1=u1*10^5; b2=6; r=0.0833; m=0.0057233; u2=0.1429;%e1=1/17;e2=0.1; j1=0.03454; w1=0.0035; w=0.0018; a=40;b1=0.0015875;b2=0.071+0.01*cos(80*pi*t)+sin(80*pi*t);r=0.083;m=0.86;u1=0.00004;u2=0.05;e1=0.0588;e2=0.75;j1=0.333;w1=0.00019;w=0.000017;a=0.29;

76%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++S1=y(1);E1=y(2);I1=y(3);R1=y(4);S2=y(5);E2=y(6);I2=y(7);%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++N=S1+E1+I1+R1;g1=(r*I2*a)/N; g2=(m*I1*a)/N; B1=r; B2=m;dy(1)=b1-(u1+g1)*S1+w*R1;dy(2)=g1*S1-(u1+e1)*E1;dy(3)=e1*E1-(u1+j1+w1)*I1;dy(4)=w1*I1-(u1+w)*R1;dy(5)=b2-(u2+g2)*S2;dy(6)=g2*S2-(e2+u2)*E2;dy(7)=e2*E2-u2*I2;%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++R0=sqrt((B1*B2*a^2*e1*e2*b2*u1)/((e1+u1)*(j1+w1+u1)*(e2+u2)*u2^2*b1))%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++Code Name: sanga11function dy = sanga11(t,y)%dy = zeros(size(y));%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++%b1=0.0015875;b2=0.071;r=0.083;m=0.86;u1=0.00004;u2=0.05;

77e1=0.0588;%e2=0.75;%j1=0.333;w1=0.00019;w=0.000017;a=0.29; e2=0.0065;%u1=0.02326; b1=u1*10^5; b2=6; r=0.0833; m=0.0057233;%u2=0.1429; e1=1/17; e2=0.1; j1=0.03454; w1=0.0035; w=0.0018;%a=40;b1=0.0015875; b2=0.071;r=0.083;m=0.86;u1=0.00004+cos(25*pi*t/12)+sin(25*pi*t/12);u2=0.05;e1=0.0588;e2=0.75;j1=0.333;w1=0.00019+cos(25*pi*t/12)+sin(25*pi*t/12);w=0.000017;a=0.29;e2=0.0065+cos(25*pi*t/12)+sin(25*pi*t/12);%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++S1=y(1);E1=y(2);I1=y(3);R1=y(4);S2=y(5);E2=y(6);I2=y(7);%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++N=S1+E1+I1+R1;g1=(r*I2*a)/N+cos(25*pi*t/12)+sin(25*pi*t/12); g2=(m*I1*a)/N+cos(25*pi*t/1+sin(25*pi*t/12); B1=r; B2=m;dy(1)=b1-(u1+g1)*S1+w*R1;dy(2)=g1*S1-(u1+e1)*E1;dy(3)=e1*E1-(u1+j1+w1)*I1;dy(4)=w1*I1-(u1+w)*R1;dy(5)=b2-(u2+g2)*S2;dy(6)=g2*S2-(e2+u2)*E2;dy(7)=e2*E2-u2*I2;

78%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++R0=sqrt((B1*B2*a^2*e1*e2*b2*u1)/((e1+u1)*(j1+w1+u1)*(e2+u2)*u2^2*b1))%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++Code Name: sanga12function dy = sanga12(t,y)%dy = zeros(size(y));%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++%b1=0.0015875;b2=0.61;r=0.083;m=0.86;u1=0.00004;u2=0.05;%e1=0.0588;%e2=0.75;j1=0.333;w1=0.00019;w=0.000017;a=0.29;e2=0.75;%u1=0.02326; b1=u1*10^5; b2=6; r=0.0833; m=0.0057233; u2=0.1429;%e1=1/17;e2=0.1; j1=0.03454; w1=0.0035; w=0.0018; a=40;b1=0.0015875;b2=0.61+cos(2*pi*t/12)+sin(2*pi*t/12);r=0.083;m=0.86;u1=0.00004+cos(2*pi*t/12)+sin(2*pi*t/12);u2=0.05;e1=0.0588;%e2=0.75;j1=0.333;w1=0.00019;w=0.000017+4*cos(2*pi*t/12)+sin(2*pi*t/12);a=0.29;e2=0.75 + 4*cos(2*pi*t/12)+sin(2*pi*t/12);%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++S1=y(1);E1=y(2);I1=y(3);R1=y(4);S2=y(5);E2=y(6);I2=y(7);%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++N=S1+E1+I1+R1;g1=(r*I2*a)/N+4*cos(2*pi*t/12)+sin(2*pi*t/12); g2=(m*I1*a)/N;B1=r; B2=m;

79dy(1)=b1-(u1+g1)*S1+w*R1;dy(2)=g1*S1-(u1+e1)*E1;dy(3)=e1*E1-(u1+j1+w1)*I1;dy(4)=w1*I1-(u1+w)*R1;dy(5)=b2-(u2+g2)*S2;dy(6)=g2*S2-(e2+u2)*E2;dy(7)=e2*E2-u2*I2;%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++R0=sqrt((B1*B2*a^2*e1*e2*b2*u1)/((e1+u1)*(j1+w1+u1)*(e2+u2)*u2^2*b1))%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++Code Name: sangaabfunction dy = sangab(t,y)dy = zeros(size(y));%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++%b1=0.0015875;b2=0.071;r=0.083;m=0.86;u1=0.0000005;u2=0.05;%e1=0.0588;e2=0.75;j1=0.333;w1=0.19;w=0.000017;a=0.29;%u1=0.02326; b1=u1*10^5; b2=6; r=0.0833; m=0.0057233; u2=0.1429;%e1=1/17;e2=0.1; j1=0.03454; w1=0.0035; w=0.0018; a=40;b1=0.0015875;b2=0.071;r=0.083;m=0.86;u1=0.0000005;u2=0.05;e1=0.0588;e2=0.75+0.005*sin(t);j1=0.333;w1=0.19;w=0.000017;a=0.29;%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++S1=y(1);E1=y(2);I1=y(3);R1=y(4);

80S2=y(5);E2=y(6);I2=y(7);%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++N=S1+E1+I1+R1;g1=(r*I2*a)/N; g2=(m*I1*a)/N; B1=r; B2=m;dy(1)=b1-(u1+g1)*S1+w*R1;dy(2)=g1*S1-(u1+e1)*E1;dy(3)=e1*E1-(u1+j1+w1)*I1;dy(4)=w1*I1-(u1+w)*R1;dy(5)=b2-(u2+g2)*S2;dy(6)=g2*S2-(e2+u2)*E2;dy(7)=e2*E2-u2*I2;%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++R0=sqrt((B1*B2*a^2*e1*e2*b2*u1)/((e1+u1)*(j1+w1+u1)*(e2+u2)*u2^2*b1))%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++Code Name: sangaacleartspan=[0 150];y0=[100 70 50 30 1000 900 700];[t y]=ode45(@sanga,tspan,y0);%plot(t,y)cleartspan=[0 150];y0=[100 70 50 30 1000 900 700];[t y]=ode45(@sanga,tspan,y0);

81Code Name: proj2cleartspan = [0 150];y0=[100 70 50 40 1000 900 700];[t y]=ode45(@sanga,tspan,y0);[t1 y1]=ode45(@sanga1,tspan,y0);N1=(y(:,1)+y(:,2)+y(:,3)+y(:,4));N2=(y1(:,1)+y1(:,2)+y1(:,3)+y1(:,4));hold onplot(t,(y(:,2)+y(:,3)+y(:,4))./N1,’r’,’LineWidth’,3)plot(t1,(y1(:,2)+y1(:,3)+y1(:,4))./N2,’b’,’LineWidth’,3)hold offxlabel(’Time in days’,’FontSize’,11)ylabel(’Prevalence’,’FontSize’,11)legend(’High temparature’,’Low Temparature’)Code Name: projjcleartspan=[0 90];y0=[100 700 500 400 100 900 700];[t y]=ode45(@sangab,tspan,y0);plot(t,y(:,1),’LineWidth’,3,’Color’,’r’)plot(t,y(:,2),’LineWidth’,3,’Color’,’r’)plot(t,y(:,3),’LineWidth’,3,’Color’,’r’)plot(t,y(:,4),’LineWidth’,3,’Color’,’r’)plot(t,y(:,5),’LineWidth’,3,’Color’,’r’)plot(t,y(:,6),’LineWidth’,3,’Color’,’r’)plot(t,y(:,7),’LineWidth’,3,’Color’,’r’)

82plot(t,y,’LineWidth’,2)legend(’S_h’,’E_h’,’I_h’,’R_h’,’S_m’,’E_m’,’I_m’)xlabel(’Time in days’,’FontSize’,11)ylabel(’Susceptible Individuals’,’FontSize’,11)legend(’R_0=0.1099’)ylabel(’Exposed Individuals’,’FontSize’,11)ylabel(’Infected Individuals’,’FontSize’,11)ylabel(’Recovered Individuals’,’FontSize’,11)ylabel(’Susceptible Mosquitoes’,’FontSize’,11)ylabel(’Exposed Mosquitoes’,’FontSize’,11)ylabel(’Infected Mosquitoes’,’FontSize’,11)Code Name: projectcleartspan=[0 80];y0=[100 700 500 400 100 900 700];[t1 y1]=ode45(@sanga11,tspan,y0);[t2 y2]=ode45(@sanga12,tspan,y0);plot(t1,y1(:,6),t2,y2(:,6),’LineWidth’,3)plot(t1,y1(:,1),’LineWidth’,3,’Color’,’r’)plot(t,y(:,2),’LineWidth’,3,’Color’,’r’)plot(t,y(:,3),’LineWidth’,3,’Color’,’r’)plot(t,y(:,4),’LineWidth’,3,’Color’,’r’)plot(t,y(:,5),’LineWidth’,3,’Color’,’r’)plot(t,y(:,6),’LineWidth’,3,’Color’,’r’)plot(t,y(:,7),’LineWidth’,3,’Color’,’r’)plot(t,y,’LineWidth’,2)legend(’S_h’,’E_h’,’I_h’,’R_h’,’S_m’,’E_m’,’I_m’)xlabel(’Time in days’,’FontSize’,11)

83ylabel(’Susceptible Individuals’,’FontSize’,11)legend(’High temperature’,’Low temperature’)legend(’R_0=0.1099’)ylabel(’Exposed Individuals’,’FontSize’,11)ylabel(’Infected Individuals’,’FontSize’,11)ylabel(’Recovered Individuals’,’FontSize’,11)ylabel(’Susceptible Mosquitoes’,’FontSize’,11)ylabel(’Exposed Mosquitoes’,’FontSize’,11)ylabel(’Infected Mosquitoes’,’FontSize’,11)

84APPENDIX CTable 7.1: The mean monthly maximum temperature in Dar-es-SalaamYear Jan Feb Mar Apr May Jun Jly Aug Sept Oct Nov Dec1999 32.7 33.1 31.7 30.2 29.5 28.9 28.2 28.7 30.4 30.8 31.3 31.12000 32.2 33.0 31.8 30.9 30.2 29.3 28.8 29.4 30.6 31.6 31.9 31.52001 31.3 32.2 32.4 31.0 30.1 29.3 29.3 29.6 29.9 31.2 32.2 32.12002 31.8 32.5 31.3 29.7 30.8 29.9 29.8 28.9 29.5 31.2 31.5 32.22003 32.2 33.6 34.1 33.3 31.6 30.0 29.2 30.2 31.5 32.3 32.5 33.42004 29.6 31.1 32.1 30.2 31.2 29.8 29.4 30.0 30.2 30.9 31.6 31.12005 32.4 32.5 33.0 31.8 30.1 30.6 29.7 29.6 31.1 31.3 31.6 32.72006 33.3 33.6 31.5 30.8 30.0 28.6 29.6 30.0 30.7 31.7 31.1 31.52007 31.8 33.3 32.5 31.5 30.6 30.1 30.2 29.9 31.7 31.4 31.7 32.22008 32.6 31.7 32.9 29.5 30.5 29.5 29.5 30.0 30.9 32.0 31.5 32.42009 33.2 33.6 32.9 31.2 30.7 29.7 30.0 30.5 31.7 32.5 32.4 32.6Source of data: Tanzania Meteorological Agency (TMA) Dar-es-Salaam

85Table 7.2: The mean monthly minimum temperature in Dar-es-SalaamYear Jan Feb Mar Apr May Jun Jly Aug Sept Oct Nov Dec1999 24.5 24.0 23.4 22.5 21.4 19.6 18.9 18.4 19.0 19.1 21.5 22.22000 24.4 23.1 23.1 22.2 20.9 19.6 18.6 18.6 18.1 19.7 22.4 23.52001 24.1 23.3 23.6 23.0 22.1 20.1 18.3 18.5 18.3 20.3 21.8 24.62002 24.9 24.7 23.7 23.0 21.4 19.2 19.0 19.4 19.4 21.0 22.3 23.72003 24.6 24.9 24.7 23.1 22.4 21.2 19.7 18.3 19.5 20.8 22.9 25.02004 25.2 23.8 24.2 23.1 21.4 19.8 18.7 18.3 19.6 21.4 22.7 24.12005 24.2 25.0 24.6 23.6 21.9 20.3 19.3 18.2 19.0 20.0 22.2 24.52006 24.8 24.7 23.6 23.0 21.6 20.3 18.9 19.3 19.7 21.5 22.9 23.62007 25.2 24.7 23.9 23.1 22.6 20.0 19.2 19.5 19.8 20.7 22.7 24.32008 24.7 24.0 23.2 22.4 21.3 19.1 18.6 19.0 19.0 21.3 23.3 24.72009 25.2 24.2 23.8 23.5 21.8 19.4 19.7 19.2 19.3 21.9 23.8 24.8Source of data: Tanzania Meteorological Agency (TMA) Dar-es-Salaam

86APPENDIX DTable 7.3: The mean monthly maximum temperature in IringaYear Jan Feb Mar Apr May Jun Jly Aug Sept Oct Nov Dec1999 23 22 22 20 18 18 15 17 20 22 24 232000 23 23 22 21 19 17 17 20 20 23 23 232001 22 23 23 21 18 17 18 19 21 23 24 242002 21 23 21 19 19 17 19 21 18 23 24 232003 23 24 24 23 21 20 16 20 20 24 25 242004 24 24 23 22 19 18 18 18 21 22 23 232005 23 25 25 22 20 18 17 19 21 23 25 262006 24 25 23 20 20 17 17 19 21 23 25 262007 25 25 23 20 18 17 17 17 20 22 23 232008 23 23 23 20 21 21 16 16 22 22 24 242009 24 24 24 23 24 24 19 21 21 21 21 22Source: Mufindi Tea Coffee Company Limited (MTC) Co Ltd Itona Estate

87Table 7.4: The mean monthly minimum temperature in IringaYear Jan Feb Mar Apr May Jun Jly Aug Sept Oct Nov Dec1999 12 13 14 13 11 10 9 10 11 11 12 132000 13 13 14 13 11 8 9 10 10 11 13 142001 14 14 14 13 11 9 8 9 10 11 12 122002 13 13 13 13 11 9 9 12 10 11 12 132003 12 12 13 12 12 10 9 9 11 12 13 142004 14 14 14 13 11 10 9 9 11 12 13 142005 14 14 14 13 11 10 9 9 10 12 13 142006 14 14 14 13 13 9 9 10 10 18 13 142007 14 14 13 13 13 10 9 11 11 12 12 112008 12 12 13 11 11 11 9 11 13 13 15 132009 13 13 12 12 13 12 10 10 12 13 13 14Source: Mufindi Tea and Coffee Company Limited (MTC) Co Ltd ItonaEstate

88APPENDIX E