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MATHEMATICAL MODELLING OF HIV/AIDS DYNAMICSWITH TREATMENT AND VERTICAL TRANSMISSIONAbdalla Sheha WazirM.Sc. (Mathematical modelling) DissertationUniversity of Dar es salaamJuly, 2011

MATHEMATICAL MODELLING OF HIV/AIDS DYNAMICSWITH TREATMENT AND VERTICAL TRANSMISSIONByAbdalla Sheha WazirA Dissertation Submitted in Partial Fulfilment of the Requirements for theDegree of Master of Science (Mathematical Modelling) of the University ofDar es SalaamUniversity of Dar es salaamJuly, 2011

iCERTIFICATIONThe undersigned certify that they have read and hereby recommend for acceptanceby the University of Dar es salaam a dissertation entitled: Mathematical Modellingof HIV/AIDS Dynamics with Treatment and Vertical Transmission, in partialfulfilments of the requirements for the degree of Master of Science (Mathematicalmodelling) of the University of Dar es Salaam.………………………………Dr. E. S. Massawe(Supervisor)Date………………………………………………………..Prof. Oluwole D. Makinde(Supervisor)Date 14 th June 2011

iiDECLARATIONANDCOPYRIGHTI, Abdalla Sheha Wazir, declare that this dissertation is my own original work andthat it has not been presented and will not be presented to any other University for asimilar 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, criticalscholarly review or discourse with an acknowledgement, without the writtenpermission of the Directorate of Postgraduate Studies, on behalf of both the authorand the University of Dar es salaam.

iiiACKNOWLEDGEMENTSI have honoured to thank Almight God for giving me ability and effort to insurecompletion of the dissertation and taking me through my study successfully.I would like to express my sincere thanks and appreciation to my supervisors, Prof.O. D. Makinde and Dr. E. S. Massawe for their constructive ideas, guidance,continuous supporting, suggestions and comments through my research work. I wishto express my sincere appreciation to Dr. W. Mahera (Course coordinator) and allstuff members in the Department of Mathematics for their support andencouragement for the whole period of my study.I acknowledge NORAD for supported and co-organized M.Sc in MathematicalModelling Program at University of Dar es Salaam. I am also very grateful to theMinistry of Education and Vocational Training-Zanzibar for an incredible partiallyfinancial support for the whole period of my study at University of Dar es Salaam.My special thanks are directed to my lovely wife Zena H. Makame and my lovelyson Abdulswamad A. Sheha for their patience, tolerance, motivation and support incompleting the program.Moreover, my sincere thanks also to my classmates for their encouragement andsupport throughout our course work program. Finally I thank all who supported medirectly or indirectly toward the struggle of attaining this academic level.

vABSTRACTThis study proposes and analyzes a non-linear mathematical model for the dynamicsof HIV/AIDS with treatment and vertical transmission. The equilibrium points of themodel system are found and their stability is investigated.The model exhibits two equilibria namely, the disease-free and the endemicequilibrium. It is found that if the basic reproduction number R0 1, the disease-freeequilibrium is always locally asymptotically stable and in such a case the endemicequilibrium does not exist. If R0 1, a unique equilibrium exist which locallyasymptotically stable and becomes globally asymptotically stable under certainconditions showing that the disease becomes endemic due to vertical transmission.By using stability theory and computer simulation, it is shown that by usingtreatment measures (ARVs) and by controlling the rate of vertical transmission, thespread of the disease can reduced significantly and also the equilibrium values ofinfective, pre-AIDS and AIDS population can be maintained at desired levels.A numerical study of the model is also used to investigate the influence of certainkey parameters on the spread of the disease.

viTABLE OF CONTENTSCertification...............................................................................................................iDeclaration and Copyright ........................................................................................iiAcknowledgements................................................................................................. iiiDedication ...............................................................................................................ivAbstract ....................................................................................................................vTable of Contents.....................................................................................................viList of Figures .......................................................................................................viiiList of Tables............................................................................................................xCHAPTER ONE: INTRODUCTION ....................................................................11.1 General Introduction.....................................................................................11.2 Statement of the Problem..............................................................................51.3 General Objectives .......................................................................................61.4 Specific Objectives.......................................................................................61.5 Significant of the Study ................................................................................6CHAPTER TWO: LITERATURE REVIEW .......................................................82.1 Research Hypothesis ..................................................................................102.2 Methodology..............................................................................................11CHAPTER THREE: THE MODEL DESCRIPION AND ANALYSIS .............123.0 Historical Background................................................................................123.1 Model Formulation.....................................................................................153.2 Positivity of Solutions ................................................................................21

vii3.3 Stability Analysis of the Model...................................................................253.4 Equilibrium Points of the Model.................................................................253.5 Computation of the Basic Reproduction Number, R0.................................263.6 Local Stability of the Disease Free Equilibrium..........................................283.7 The Endemic Equilibrium and Local Stability ............................................313.8 Existence of Forward Bifurcation ...............................................................363.9 Global Stability of the Endemic Equilibrium ..............................................43CHAPTER FOUR: THE NUMERICAL SIMULATIONS OF THEMODEL...494.1 Model Simulation.......................................................................................49CHAPTER FIVE: DISCUSSION, CONCLUTION AND FUTUREWORK .....705.1 Discussion..................................................................................................705.2 Conclusions................................................................................................745.3 Future Work...............................................................................................75REFFERENCES...................................................................................................76

viiiLIST OF FIGURES3.1 The flow chart of the model. .............................................................................163.2 Forward Bifurcation of the model (3.3). ............................................................434.1(a). Endemic equilibrium of proportion of infectives with Susceptible................504.1(b). Endemic equilibrium of proportion of Pre-AIDS population........................504.1(c). Endemic equilibrium of proportion of treated class ......................................514.1(d). Endemic equilibrium of proportion of AIDS population ..............................514.2(a) Variation of population in different classes for 0 and 0 .....................534.2(b) Variation of population in different classes for 0.1 and 0.2. ............534.3(a) Variation of infected population for different values of . .............................544.3(b) Variation of AIDS population for different values of . ................................554.3(c) Variation of Treated class for different values of . .......................................554.4(a) Variation of Susceptible for different values of . .........................................564.4(b) Variation of Treated population for different values of . ............................574.4(c) Variation of AIDS population for different values of . ...............................574.5(a) Variation of Infective population for different values of . ...........................584.5(b) Variation of pre-AIDS population for different values of . .........................594.5(c) Variation of Treated class for different values of . ......................................594.5(d) Variation of AIDS population for different values of . ...............................604.6(a) Variation of pre-AIDS population for different values of 1.........................614.6(b) Variation of AIDS population for different values of 1...............................614.6(c) Variation of Treated population for different values of 2............................624.6(d) Variation of AIDS population for different values of 2...............................62

ix4.7(a) Variation of Susceptibles population for different values of 1.....................634.7(b) Variation of Infectives population for different values of 1.........................644.7(c) Variation of Susceptibles population for different values of 3.....................644.7(d) Variation of Infectives population for different values of 3.........................654.8(a) Variation of Susceptibles population for different values of c1......................664.8(b) Variation of Infectives population for different values of c1..........................664.8(c) Variation of Susceptibles population for different values of c3.....................674.8(d) Variation of Infectives population for different values of c3.........................674.9(a) Variation of AIDS patients for different values of k and . .......................684.9(a) Variation of AIDS patients for different values of k and . .......................684.10 Variation of AIDS populations for different values of ................................69

xLIST OF TABLES3.1 Definitions of Symbols used frequently. ........ …………………………………153.2 Definitions of Parameters used frequently………………………………………16

1CHAPTER ONEINTRODUCTION1.1 General IntroductionThe human immunodeficiency virus (HIV) infection which can lead to acquiredimmunodeficiency syndrome (AIDS), has become a hazard infectious disease in boththe developed and developing nations. It is a fatal disease, which breaks down thebody’s immune system, leaving the victim vulnerable to a host of life threateningopportunistic infections, neurological disorders or unusual malignancies. It hascaused mortality of millions of people and expenditure of enormous amount ofmoney in health care and disease control (Naresh et. al.,2006).AIDS is one of the deadliest epidemics in human history. It was first identified in1981 in New York and California (USA). Shortly after its detection in the UnitedStates, AIDS quickly developed into a worldwide epidemic, affecting virtually everynation. About 90 percent of all HIV-infected people live in the developing countries(http://data.unaids.org/pub/GlobalReport/2008). In 2007 it was estimated that twothirds (67%) of the global total of 32.9 million people with HIV live in this region,and three quarters (75%) of all AIDS deaths in 2007 occurred(http://data.unaids.org/pub/GlobalReport/2008). Consequently, it is estimated that bythe year 2020 the nine most severely hit Sub-Saharan countries may lose 13%–26%of their agricultural labour force to HIV/AIDS. The Growth Domestic Product incountries with HIV prevalence rates 10% or higher could contract AIDS by 18% by2020 (Amoako et. al., 2008). Women in Sub-Saharan Africa account for 58% of all

2HIV infections in the region, but a striking 78% of those young people (ages 15–24years) living with HIV are females (http://data.unaids.org/pub/GlobalReport/2008).Sub-Saharan African women and children are particularly at risk for HIV/AIDS. InJune 2000, UNAIDS reported that one in five women under the age of 25 areinfected with HIV in Africa. HIV/AIDS transmission in Africa is primarily throughheterosexual sex and vertical transmission (mother-to-child). Forty percent of allHIV/AIDS cases result from mother to child transmission. While fewer than 300infants in the U.S acquired HIV through vertical transmission in 1997. In sub-Saharan Africa over 2.5 million children under the age of 15 have died of AIDS,most of these children were exposed to HIV during labour or breastfeeding.HIV/AIDS is a serious threat to future development in Africa. The cost of one yearof basic healthcare for an individual living with HIV/AIDS is two to three timesmore than the average per-capita gross domestic product of some African countries.HIV prevention programs are less expensive than HIV care and should be a criticalcomponent of any HIV/AIDS intervention program (Adelman, 2001).In the United Republic of Tanzania there is a dynamic of HIV/AIDS epidemic,which is still growing. The first case of HIV/AIDS in Tanzania was reported in 1983.By 1985, an estimated number of people living with HIV/AIDS was 140,000 (1.3%prevalence) and by 1990 the estimated number was about 900,000 (7.2%prevalence). According to the surveillance report on HIV/AIDS and sexuallytransmitted infections issued July 2009 published by the National Aids ControlProgrammes(NACP), about 1.4 million Tanzanians (1,300,000 adults and 110,000children) are living with HIV infection, in a total population of 41 million. The

3social, economic, and environmental impact of the pandemic is sorely felt as anestimated 140,000 Tanzanians have perished, leaving behind as estimate 2.5 millionorphans and vulnerable children, representing approximately 10 to 12% of allTanzanian children. Since the National AIDS Control Programme was established in1985, the progress of the epidemic has been monitored through unlinked anonymoustesting of blood from pregnant women attending antennal clinics for the first time inselected sentinel sites.Treatment is the process of offering the HIV positive individual with a lifeprolonging drugs/medicines known as antiretroviral (ARV) medicines or ART. ARTdrugs are the main types of treatment for HIV/AIDS. It is not a cure but it canprolong the life of a person for many years. It consists of drugs that have to be takenevery day for the rest of person’s life. It keeps the amount of HIV in the body at lowlevel which retards any weakening of immune system and allows it to recover fromany damage that HIV might have caused already (AVERT, 2010). Treatment withanti-retroviral increases the life expectancy of people infected with HIV. Even afterHIV has progressed to diagnosable AIDS, the average survival time withantiretroviral therapy was estimated to be more than 5 years as of 2005 (Schneider et.al.,2005).Without antiretroviral therapy, someone who has AIDS typically dieswithin a year (Morgan et. al., 2002).WHO estimated that 1.2 million people started treatment in 2009 bringing the totalnumber of people receiving live saving HIV treatment in low and middle incomecountries to 5.2 million compared to 4 million at the end of 2008 (WHO, 2010). Bythe end of 2008, 2.9 millions adults and children in Sub-Saharan Africa were

4receiving ART while those in need were 6.7 million (UNAIDS/WHO 2009). InTanzania the number of HIV positive to be initiated on ART is expected to increasefrom 440,000 in 2008 to 600,000 in 2012 (ICAP Tanzania, 2004).Current treatment for HIV infection consists of Highly Active AntiretroviralTherapy, or HAART(http://www.hab.hrsa.gov/tools/HIVpocketguide05/PktGARTtables.htm). This hasbeen highly beneficial to many HIV-infected individuals since its introduction in1996, when the protease inhibitor-based HAART initially became available (Palellaet. al., 1998).The development of HAART as effective therapy for HIV infection has substantiallyreduced the death rate from this disease in those areas where these drugs are widelyavailable (Palella et. al., 1998). As the life expectancy of persons with HIV hasincreased in countries where HAART is widely used, the continuing spread of thedisease has caused the number of persons living with HIV to increase substantially.Vertical transmission is the transmission of the HIV virus from mother to the child.Vertical transmission from mother to child can occur in utero (during pregnancy),during labour and delivery, and through breastfeeding. In Sub-Saharan African, therisk of transmission through this means ranges between 10 percent and 30 percent.Without prophylaxis, between 20% and 25% of babies born to HIV-positive womenwill be infected with HIV. As a result, prevention of vertical transmission hasbecome a critical component of national and global responses to HIV. Women livingwith HIV have the right to enjoy the benefits of scientific progress and the healthservices that enable them to reduce the risk of transmitting HIV to their children.

5They also have the right, like all women, to become pregnant, to control the numberand spacing of their children, get married and form the family. In addition to theirHIV prevention impact, services to prevent vertical transmission are an obvious forrealising the integration of reproductive health and HIV/AIDS programmes, which isa public health goal in many countries and a recommendation of the World HealthOrganization (WHO) (Cset et. al., 2009).In the absence of treatment, the transmission rate up to birth between the mother andchild is around 25% (Coovadia, 2004). However, where combination antiretroivaldrug treatment and Cesarian section are available, this risk can be reduced to as lowas one percent. Postnatal mother-to-child transmission may be largely prevented bycomplete avoidance of breast feeding; however, this has significant associatedmorbidity. Exclusive breast feeding and the provision of extended antiretroviralprophylaxis to the infant are also efficacious in avoiding transmission(http://www.cochrane.org/reviews/en/ab006734.html).Naresh et. al.,(2006),carried a study on mathematical modelling of the spread ofHIV/AIDS epidemic with vertical transmission. However they did not includetreatment in their model. This study there fore aims at improving the model done by(Naresh et al.,2006) by considering treatment and vertical transmission in thedynamic of HIV/AIDS.1.2 Statement of the ProblemNaresh et. al.,(2006) developed a model for transmission of HIV into a population ofvarying size with vertical transmission, in which they assumed a varying size

6population recruiting non infected individuals. However, usually infected individualsare recruited or else immigrated into the population. In this research it is intended todevelop a mathematical model for transmission of HIV/AIDS into a population ofvarying size with vertical transmission in the presence of treatment and otherdemographic and epidemiological factors basing on the population into thecommunity. This will be an improved model from Naresh et al., (2006) as a result ofincluding treatment which wasn’t considered in their model1.3 General ObjectivesThe general objective of this study is to formulate and analyse a model of HIV/AIDSdynamics with treatment and vertical transmission.1.4 Specific ObjectivesThe specific objectives of this study are:i. To establish a mathematical model of HIV/AIDS showing the effect oftreatment and vertical transmission on the transmission of HIV/AIDS.ii. To determine the stability of the disease free equilibrium point.iii. To determine the stability of endemic equilibrium point.iv. To determine the impact of each embedded parameter on the HIV/AIDStransmission.1.5 Significant of the StudyThe outcomes of this study will help the Government and NGO’s to establishpolicies, programmes and optimal plan for control of the disease by taking into

7account the aspect of treatment and vertical transmission. It will help the society ingeneral to have an understanding on how the disease can be controlled throughtreatment of infectives with ARV. Also it will add more knowledge to the existingliterature on HIV/AIDS and help Researchers to do more research on this disease.Lastly this study will help educationalist to develop educational seminars, workshopsor training programmes to educate people on the impact of treatment and verticaltransmission in curbing down the HIV/AIDS epidemic.

9infections in many sub Sahara Africa countries. The use of ARVs for prevention isestablished in the prevention of mother-to-child programmes and through postexposureprophylaxis for sexual assault and needle-stick injuries. They alsosuggested that, providing antiretroviral therapy infected with HIV would stop theHIV epidemic. The researchers narrated that their model assumes little or notransmission would occur from people on antiretroviral treatment, based on findingsfrom Uganda, Spain and Taiwan. The model also assumes that instead of treatingonly those with low CD4 cell counts, all HIV-positive people would receivetreatment from the time of diagnosis to everyone in the world.Fred Brauer (1994), on Models for Diseases with Vertical Transmission andNonlinear Population Dynamics had a different approach regarding on the commonassumption to modelling communicable disease, most of what is known for verticaltransmission models assumes birth rates are proportional to population size, but hadconsidered models with nonlinear population dynamics and finite carrying capacityand analyze the stability of equilibrium in the special case in which the overall birthrate does not depend on infective population size.Busenberg and Cooke, (1993) discussed a variety of diseases that transmit bothhorizontally and vertically and gave a comprehensive survey of the formulation andthe mathematical analysis of compartmental models that also incorporate verticaltransmission.Naresh et. al., (2006) developed a model of transmission of HIV into population ofvarying size with vertical transmission and other demographical and epidemiolocalfactors. Their purpose was to formulate a model for AIDS epidemic that may be

10transmitted either directly or vertically in populations and to study its behaviourqualitatively and numerically. They found that an increase in the rate of verticaltransmission leads to increase the population of infective which in turn increases thepre-AIDS and AIDS population. Thus, they concluded that the vertical spread of thedisease should be controlled by the way of promoting effective treatment to keep theoverall infective population under control.Simwa et. al., (2003) in there paper dealt with a deterministic model for HIVepidemic with three stages of disease progression among infected patients. It isassumed that the patient once infected experiences disease progression up to fullblownAIDS. Using two systems of ordinary differential equations that are coupledthrough a delay in one of the systems, a compartmental model for the dynamics ofthe HIV/AIDS epidemic is constructed. The transmission of the disease is consideredto be only through heterosexual contact and vertically from an infected mother to herunborn child. Numerical integration of the equations is used for simulating the stagespecificepidemic curves, given the demographic and epidemiological parameters ofthe model. Through simulation it is noted that all the three stage-specific prevalencerate curves also satisfy this condition.2.1 Research HypothesisThe research hypotheses are:i. There exists a mathematical model of HIV/AIDS dynamics with treatment andvertical transmission.ii.The disease free equilibrium point can be made stable under certain condition.

11iii.iv.The endemic equilibrium point is stable under certain condition.The variations of the embedded parameters have positive correlation with modelresults.2.2 Methodology.It is proposed to formulate a mathematical model for the analysis of HIV/AIDSdynamics with treatment and vertical transmission of HIV/AIDS using differentialequations. The model to be established will be a modification of the modeldeveloped by Naresh et al., (2006). The basic reproduction number ( R ) which is thefundamental parameter governing the transmission or spreading of the disease will becomputed using the next generation operator method.oLocal and global stability of disease free equilibrium point (DFE) and local stabilityof an endemic equilibrium point (EEP) will be analysed.The model will besimulated to determine the behaviour of each embedded parameter for the impact oftreatment and vertical transmission on the transmission of HIV/AIDS.For numerical analysis and validation of the model, estimated and heuristic data willbe used. Simulation of the model will be done with mathematical computer package(MATLAB).

12CHAPTER THREETHE MODEL DESCRIPTION AND ANALYSIS3.0 Historical BackgroundEpidemiology is the study of the distribution and determinants of diseases bothinfectious and non infectious. Originally the term was used to refer only to the studyof epidemic infectious diseases but it is now applied more broadly to other diseasesas well (Johnson, 2004). Mathematical models have been used extensively inresearch into the epidemiology of HIV/AIDS, to help improve our understanding ofmajor contributing factors in a giving epidemic (Naresh at. el., 2006).The goal of any modelling exercise is to extract as much information as possiblefrom available data and provide an accurate representation of both the knowledgeand uncertainty about the epidemic. A number of different models of HIV and AIDShave been developed, ranging from simple extrapolations of past curves to complextransmission models. A major tradition in modelling HIV/AIDS epidemics has beento use back calculation, or back projection, techniques. These techniques producestatistical solutions to convolution equations that relate the number of AIDSdiagnoses over time to past trends in HIV infection, and to the distribution of theHIV incubation period (Salomon and Murray, 2001).In this study we have, therefore, developed a model for transmission of HIV into apopulation of varying size with vertical transmission and other demographic sandepidemiological factors. Our purpose is to formulate a model for AIDS epidemic thatmay be transmuted either horizontally or vertically in populations.

14Model ParametercDescriptionRate of recruitment into susceptible population.Average number of sexual partners per unit timeSexual contact rateRate of movement from infectious classFraction of newborns infected with HIV who dies immediatelyafter birthRate of newborns infected with HIVNatural mortality rateInduced death rate due to AIDS1Fraction of joining the pre-AIDS classRate of movements of pre-AIDS class individuals into AIDSclass2Fraction of joining treated classmkRate at which AIDS group get treatmentFraction of who get treatmentRate at which treated group becomes full blown AIDSTable 3.2 Definitions of Parameters used frequently

153.1 Model FormulationConsider a population of sizeNtat time t with constant inflow of susceptiblewith rate N . The population size tsusceptibles S t, infectives tpatientsN is divided into fives subclasses which areI (also assumed to be infectious), pre-AIDSP t, treated class T t and AIDS patients tA with natural mortality rate in all classes. is the disease induced death rate in the AIDS patients class and the rate at which AIDS patients get treatment.In the model, it is assumed that the susceptibles become HIV infected via sexualcontacts with infective which may also lead to the birth of infected children. It isassumed that a fraction of new born children are infected during birth and hence aredirectly recruited into the infective class with a rate 1 and others dieeffectively at birth 0 1.We do not consider direct recruitment of the infected persons but by verticaltransmission only. It is assumed that some of the infectives join pre-AIDS class,depending on the viral counts, with a rate 1 and then proceed with a rate todevelop full blown AIDS, it is also assumed that some of the infectives move to jointreated class with a rate 2 and then proceed with a rate k to develop full blownAIDS while others with serious infection directly join the AIDS class with a rate1 .1 2The interaction between susceptibles and infectives is assumed to be of standardmass action type.

17dA 11 2 I 1 m P kT A,dtwith the initial conditionsS S , I 0 I 0 ,P0 P 0 ,T 0 T 0 ,and A 000 A,To simplify the model, it is reasonable to assume that the AIDS patients and those inpre-AIDS class are isolated and sexually inactive and hence they are not capable ofproducing children i.e. P 1 1 A 0 and also they do not contribute toviral transmission horizontally i.e. 2and 4are taken negligible (Naresh et.al.,2006)..In view of the above assumptions, the system can now be written as follows:dSdtc11IS NNc33TSN SdIdtc11ISc33TS 1N N I IdPdt 1I P(3.2)dTdt 2 I mPAk TdAdt I mP kT A 111 2

18From the system (3.2), it is pointed out here that not all infected individuals take partin spreading the disease, as in the case of infected children, but they will all developtheir own AIDS. However, the model can be modified to include a delay to make theinfected children become adult to further spread the infection either vertically orhorizontally.The total population N can be calculated as:N S I P T A , thendNdtdS dI dP dT dA ,dt dt dt dt dtdN c11ISc33TS N Sdt N Nc ISNc TSN I 1 1 3 3 1 1 I P 2 I m P A k T I1 1 I m P kT A1 2Therefore,dN N NA 1 Idt

19S IThus, we normalize the model. Without loss of generality let s , i ,N Np PN,Th andNa ANTherefore, the normalized system can be calculated as followsS sNdSdtdNsdtdsNdtds 1dt NdS dtdN sdt dsdt c1 1is c33hs a 1 isI iNdIdtdNidtdiNdtdi 1dt NdI dtdN idt didt c11is c33hs 1 i a 1 iiP pNdPdtdNpdtdpNdt

20dp 1dt NdP dtdN pdt dpdt 1 i a 1 ipH hNdT dN dh h Ndt dt dtdh 1 dT dN hdt N dt dt dhdt 2 i m p a k a 1 ihA aNdAdtdNadtdaNdtda 1dt NdA dtdN adtdadt 1 1 11 2 i m p kh a ia

21Now, the system be written as followsdsdt c11is c33hs a 1 is,didt c11is c33hs 1 i a 1 ii,dp 1 i a 1 ip,dt (3.3)dh 2 i m p a k a 1 ih,dt da 1 1 2 i 1 m p kh a 1 ia,dt Such thats i p h a 1.3.2 Positivity of SolutionsFor the model (3.3) to be epidemiological meaningful and well posed, we need toprove that all state variables are non-negative t 0.Theorem 1 Let , , , , s i p h a s i p h a 5, , , , : 1, then the solutionss t i t p t h t a t of the system (3.1.3) are positive t 0.To prove the theorem, we will use differential equations of the system (3.3).Using the 1 st equation we have

22ds s(3.4)dtdss dtFinding the integrating factor, IF dt e etd setet dtIntegrating both sides we get 1s t ce tt 0, s t s o . s o 1cApplying initial conditions: When s t 1 s 0 1 e tWhen t , s t 1There fore, 0 s t 1Similarly, using the second equation of the system (3.3) we havedidt (3.5)didt 0

23 dtIF e e tddt ie 0Integrating both sides lead to t ce i tApplying initial conditions: at 0, 0 . 0t i t i i c 0 i t i e t , i t 0tTherefore,it 0.Also in the third equation of the system (3.3) we havedp p(3.6)dtdp dtpIntegrating both sides, we get 0 p t p e tApplying initial conditions; at t 0, pt p0

24When t , p t 0Similarly, using fourth equation of the system (3.3) we havedh k h(3.7)dtdh k dthIntegrating both sides, we get k t 0 h t h e Applying initial conditions; at t 0, ht h0When t , h t 0Finally, using fifth equation of the system (3.3) we haveda a(3.8)dtda dtaIntegrating both sides, we get 0 a t a e tApplying initial conditions; at t 0, at a0When t , a t 0

253.3 Stability Analysis of the Model.In this section, we present the results of stability analysis of the equilibrium points.3.4 Equilibrium Points of the ModelThe model (3.3) has two non negative equilibrium points which are; the disease freeequilibrium point E0, and endemic equilibrium point*Eds di dp dh daAt equilibrium point 0,dt dt dt dt dthence the following systems ofequations is solved for equilibrium points.0 c1 1is c33hs a 1 is 0 c11is c33hs 1 i a 1 ii0 1 i a 1 ip0 2 i m p a k a 1 ih 0 11 2 i 1 m p kh a 1 iaFor the disease free equilibrium point i p h a 0,when substituted above, thesystem of equations is reduced to s 0 , hence s 1.There fore, the disease free equilibrium E0is 1,0,0,0,0,

26The linear stability of E0can be established by its basic reproduction number. It isdetermined by using next generation method on the equation (3.3) in the form ofmatrices F andV .Let Fibe the rate of appearance of new infection in compartmentVibe the transfer of individuals out of compartment by another meansx0be the disease free equilibriumR0is the largest eigenvalue ofFi x Vi x 0 0 x xj j 1(3.9)3.5 Computation of the Basic Reproduction Number, R0The basic reproduction number R0is defined as the effective number of secondaryinfectious caused by typical infected individual during his interred periods ofinfectiousness (Diekman et al., 1990). It is obtaining by taking the largest (dominant)Eigen value (spectral radius) ofRFi x Vi x 0 00 x xj j 1Where;FFixx j0, and VVixx j0

27c110 c330 0 0 0 0F and 0 0 0 0 0 0 0 0V 1 011 m1120 m00 k k0 0 R0 F V1R0c11 2c3 32 2 2 2 1m 1m 1 2 k (3.10)Thus if R 1 0, the infection triggers an epidemic otherwise its prevalence is zero i.e.for R 01 . From the solution, it is noted that with an increase in R0, which can beviewed as a function of c , the number of sexual partners . the number of invectivesincreases which in turns increases the AIDS patient population. Thus in order to keepthe spread of the disease at minimum, the number of sexual partners should berestricted.

283.6 Local Stability of the EquilibriaTheorem 2.The disease free equilibrium of the system is locally asymptoticallystable if R 01 and unstable if R 1 0.Now to determine the local stability of E0, the following variational matrix iscomputed corresponding to equilibrium point E0dsdt c1 1is c33hs a 1 is D s, i, p, h,adidt c11is c33hs 1 i a 1 ii F s, i, p, h,adp 1 i a 1 ip G s, i, p, h,adt (3.1.1)dh 2 i m p a k a 1 i h L s, i, p, h,adt da 12 dt1 i 1 m p kh a 1 ia M s, i, p, h,aJacobean Matrix of the system (3.3) is given by

29J0 D E0 D E0 D E0 D E0 F E0 s i p h a F E0 F E0 F E0 F E0 F E0 s i p h aG E0 G E0 G E0 G E0 G E0 s i p h a L E0 L E0 L E0 L E0 L E0 s i p h aM E0 M E0 M E0 M E0 M E0 s i p h a V0 1c1 1 1 0 c3 3 c 2 m k m k 0 c11 1 0330 0 0 0 0 0 11 21The characteristic equation corresponding to V0is given byf4 3 2 a a a a 0, (3.1.2)1234Wherea 4 k c 1 1 1 a c k k k2 2 3 33 k c k k 3 2 1 1

302a3 k 2 k 2 2 2 k 3c k k k 3 22 2 2 21 12 2 c m k k3 3 2 2 2 1 1 2 2 2 k 3 2 3 2 2 2 k k 2 2 k 4 k k k k 2 2 3 2 22 2 2 3 2 3 4 3 3 .2 2 3 2 2 a k k k k 2 3 2 2c1 1k k k k c232 2 2 2 1m 1m13Thus by Routh-Hurwitz criteria, E0is locally asymptotically stable as it can be seenfora , a 0, a 0, a 0, a a a 0, and a a a a a a 0.102 3 4 1 2 32 22 2 3 3 1 4Thus, using a 40 , 2 2 k 3 k 2 k 2 k 2 3 2 2c1 1k k k k

31 c2 mm 0332 2 2 2 111This will lead to 2 2 k 3 k 2 k 2 k 2 2 3 2 2c1 1k k k k c232 2 2 2 1m 1m13Hence,2c1 c33 2 12 2 2 1m 1m 1 1 2 k k Therefore, R 1 0.This proofs the Theorem above i.e. The disease free equilibrium of the system islocally asymptotically stable if R 01 and unstable if R 1 0.3.7 The Endemic Equilibrium and Local StabilityThe endemic equilibrium point of the model (3.3) is given by E * s, i * , p * , h * , aTo obtain an endemic equilibriumthe model. .*E of this model, we set to zero each equation in

32Then by solving the system of equations in (3.3) we express each equilibrium pointin terms of i **and a at steady state, we get c1 1is c33hs a 1 is...............................................i c1 1is c33hs 1 i a 1 ii..............................ii 1 i a 1 ip............................................................iii 2 i m p a k a 1 ih.........................................iv1 i 1 m p kh a 1 ia............v1 2Add equation s* i and ii to get the value of1 1* * a 1 i *s ,we get i a i i* * * **Using equation iiito get value of p ,p**1i ,* a*1 iSolve equation iv and v to get the value of* *h and a we get;*ha** * * *1 1 2 i 1 m1 i k 2 i m 1i k

33Where: a 1 i* * k a 1 i* * a* 1 i* and** * *1 12 i 1 m 1i k 2i m1i * * 2 i m 1i kThe values of i * are in the form of polynomiali A i A i A i A i Ai A (3.1.3)* *5 *4 *3 *2 *5 4 3 2 1 00After some algebraic calculation, the expression can be reduced as:Ai Bi 0(3.1.4)*2 *Where, * *A k a a , 3 c c a 2c a 2c a c k c ka3 * * 2 * *1 1 1 1 1 1 1 1 1 1 1 1c a c k c a c k 2c a c * 2 * *1 1 1 1 1 1 1 1 1 1 3 3 2

34c m c m a c c m c *3 3 1 3 3 1 3 3 2 3 3 1 3 3c a c c k a 2c ka c c * 2 2 * *3 3 2 3 3 2 1 1 1 1 3 3 2 3 3 2c 2c a c a c a c a* * * *3 3 1 3 3 2 3 3 1 3 3 3 3 22 2 c c 3c a c k c k a 3c a2 * 2 * *1 1 1 1 1 1 1 1 1 1 1 12 2 2 c c a c a c a c c 2 * * * 2 21 1 1 1 1 1 1 1 1 1 1 1 c2 *33 2 a .B k a* 4 3 * 3 3 3 2 * 3 2 2 * 3 k 4 a k 4 a 3 a 2 2 3 4 k 3 k a 6 a 6 a 4 a a2 2 2 * 2 * 2 * * *3 4 3 4 a a k a k ka* 4 4 4 * * 3 2 *3 3 4 a 2 a a a c 3 k a 3 * 2 * * * 4 2 * 51 13 2 4 2 ka 2 a k 4 a k a a* * * * *2 2 2 3k a k 3 a 3 a k a c * 2 2 * * * 31 1

352 3 c c 3c a 3c a c k c a3 3 3 * 2 * 3 *1 1 1 1 1 1 1 1 1 1 1 13 2 2 2 a 2 k a 3 k a 3 a 3 a 3 a* 2 * * * * 2 2 *2 2 3 a k 3 a 3 a k a2 * 2 2 2 2 * * * 6 *2*23 2 *233 3 2 3 3 1 a a c a c m c c k c k c a c a c a2 * *3 3 1 1 1 1 1 1 1 1 3 32 2 c a c a c a c a c a2 * * * * 2 *3 3 2 3 3 1 1 1 1 1 1 1 2c k c m a c c m c a2 * 2 2 *1 1 3 3 1 3 3 2 3 3 1 1 1c k a 2c a c a 2c a 2c ka* 2 * * 2 * *1 1 1 1 3 3 2 3 3 2 1 1 2c k a c a 2 ka c c * * * 2 21 1 3 3 2 3 3 2 3 3 1c 2c a 2c a c k c c 2 2 * 2 2 * 2 2 23 3 1 1 1 1 1 1 1 1 1 14 2 3 4 4* 3 * 2 * * * c33 2 a 10 a 10 a 5 a k a2 3 2 3 6 k a 5 a 4 a 6 a k a3 2 * 4 * * 2 * *3 2 5 3 k 4 3 ka * 4 k a * 4 3 a * 4 3 k a * a3 3 3 a a k a 3 a 3 ka* * * 2 2 *

363.8 Existence of Forward BifurcationWe will analyse the local asymptotic stability of endemic equilibrium by using thetheorem of Centre manifold (Gumel et al., 2008). The linearizion matrix has at leastone Eigen value with zero real part. Also we shall describe the possibility of forwardbifurcation from the non hyperbolic equilibrium. To describe it, consider a generalsystem of ODEs with F . This theory is convenient by changing variables.Consider s x1 , i x2, p x3, h x4,a x5. By using vector notation1 2 3 4 5TX x , x , x , x , x in the model (3.3) can be written in the formdXdtF X , where by F f , f , f , f , f 1 2 3 4 5Tas follows:dxdt1 f c x x c x x x 1 x x1 1 1 2 1 3 3 4 1 5 2 1dxdt21 1 f c x x c x x x x x x2 1 1 2 1 3 3 4 1 2 5 2 2dxdt3 f3 1 x2 x5 1 x2 x3(3.1.5)dxdt4 f x m x x k x 1 x x4 2 2 3 5 5 2 4dxdt51 1 1 f x m x kx x x x5 1 2 2 3 4 5 2 5The Jacobian of the system (3.15) at the DFE is given by,

37J E0 1c1 1 1 0 c3 3 c 2 m k m k 0 c11 1 0330 0 0 0 0 0 11 21(3.1.6)WhereR02c11c332 mm 1 22 2 2 11 kSupposesolve for*1 is the chosen bifurcation parameter and if we consider R0 1*1 givesand* 2 2c1 k k k k c 3 3(3.1.7)Where m m 22 2 2 2 1 1 1The system of the transformed equation with*1 has a simple zero eigen values.Hence the Center Manifold theory can be used to analyze the dynamic ofdXdt , , , , 1 2 3 4 5T* F X F f f f f f near . It can be shown that the Jacobian1dXT 1 2 3 4 5atdtof F X F f , f , f , f , f 1* has right eigen vector associatedwith the zero eigen values by w w , w , w , w , w where1 2 3 4 5T

38 1 c33 c1 1 c33m 1 c33 2w w1 2c33w w13 2w c 1 1 14w2c33 c1 1 1 c33m 1 c33 2w w5 2c33w2 0,Where k 1 11 c The right eigenvector of J E 0 associated with the zero eigenvalues at*1 isgiven by u u u u u u , , , , , where1 2 3 4 5Tu1 0

39um 1 m 1 1 2 1 2 u2 4um 1m u3 4u u5 4u4 0,Theorem 3: (Castillo-Chavez and Song, 2004) Consider the following generalsystem of ordinary differential equations with a parameter such thatdxdtf ( x,),f:nn2 n and f c .Where 0 is an equilibrium point of the system (that is 0, 0f for all ) and f x x j 1. D f 0,0 i 0,0is the linearization matrix of the system aroundthe equilibrium 0 with evaluated at 0 .2. Zero is a simple eigenvalue of and all other eigenvalues of havenegative real parts.3. Matrix has a right eigenvectors and left eigenvector q correspondingto zero eigenvalues.

404. Let fkbe thethk component of f anda n2 fx xk ukij 0,0, (3.1.8)k, i, j1i jb n kukj 0,0. (3.1.9)k , j12 fxjThe local dynamics of the system around the equilibrium point is totally determinedby the signs of a and b particularly, if a 0 and b 0 then a backward bifurcationoccurs at 0 .1. a 0, b 0. When 0 with | |1,0 is locally asymptotically stable, and thereexist a positive unstable equilibrium; when 0 1,0is unstable and there exist anegative and locally asymptotically stable equilibrium;2. a 0, b 0 . When 0 with | |1,0 is unstable; when 0 1,0is locallyasymptotically stable, and there exist a positive unstable equilibrium;3. a 0, b 0. When 0 with | |1,0 is unstable and there exist a locallyasymptotically stable negative equilibrium; when 0 1,0is stable, and positiveunstable equilibrium appears;4. a 0, b 0. When changes from negative to positive, 0 changes its stabilityfrom stable to unstable. Correspondingly a negative unstable equilibrium becomespositive and locally asymptotically stable.

41Calculation of Bifurcation Parameters a and bFrom model (3.15), the associated none zero partial derivatives of F at the diseasefree equilibrium are given by, f f f f f f f f 0, 2 2 2 2 2 22 22 2 3 3 4 45 52 2x2 x2 x5 x2 x3 x3 x5 x2 x4 x4 x5 x2 x5 x5 f f f f 2 1 , , 1 , , 2 22 22 23 32 x2 x2 x5 x2 x3 x3 x5 f f f f 1 , , 1 , 2 ,x x x x x x x2 22 24 45 5 22 4 4 5 2 5 5It follows from above expressions that f f fa u w w u w w u w w u w w2 2 222 3 452 i j 3 i j 4 i j 5 i jxix jxix jxi x jxi xj (3.2.0)f2 u 2 w2 w5 2 1 w 2 u3 w3 w5 1 w2w31 2 21u4 w4w5 w2w4 u5 w5 w2w5 There fore a 0, ifq w3w5 2 3u11 w w 3Where q1 u w w 2 1 w u w w 1 w w u 2 w 1 w w2 22 2 5 2 4 4 5 2 4 5 5 2 5

42Hence, a 0, if1 c3 3m 1 c33 2 c33 11 c33 m 1mWhere c33m 1 c3 3 2 2 c33m 1 c33 2 c3 3 c33 c3321 m m 1 1 1 2 1 2 c331 c m c 33 1 33 2 c 3 3 c m c c333 3 1 3 3 2 2 1 For the sign of b it can be shown that the associated non vanishing partial derivativesof f is2 f2x1 2 c1So thatf 2221x(3.2.1)ib u wi u2w2c1

43m 1 m 1 1 2 1 2 c1 0,Therefore, b 0Hence, we have established the following result. Since b 0 and a 0 if theinequality satisfies, then there exists a forward bifurcation.Figure 3.2 Forward Bifurcation of the model (3.3).3.9 Global Stability of the Endemic EquilibriumTheorem 4. If the endemic equilibrium*E exist, then it is globally asymptoticallystable provided that R0 1.Consider the following positive definite definite function about*E

44* * ** * * * ** s * i * p V s , i , p , h , a s s slog i i i log p p p log s i p * ** h * a h h hlog a a a log h a (3.2.2)By direct calculation of the derivative of V along the solution of (3.3) can be writtenas dt s dt i dt p dt h dt a dt* * * * *dV s s ds i i di p p dp h h dh a a daBut from the model (3.3).dsdt c is c hs a 1 is1133didt1 i a c is c hs 11133 iidpdt i a 1 ip1dh 2i mpa k a 1 ihdtdadt i 1 mp kh a 111 2iaIt can now be written as*dV s s c1 1is c33hs a 1 isdt s

45* i i c1 1is c33hs 1 i a 1 iii * p p 1 i a 1 ipp * h h 2i m p a k a 1 ihh a a a*1 1 1 i m p kh a i a1 2Therefore,* * * * *substitute s s s , i i i , p p p , h h h and a a a , aftersome algebraic calculations we get2 2* *s s i i* *11 33 11 dV c i c h a i i c s adt s i2* p p h h2* * a i i a i i p * 2a a i i p h h i i c h i i mpa i i p h h*3 * * * ** * * * *3 3 1 2hh a a a a a a a i i i p m p kh iih a a a a a a* * * * * * ** * * *1 23i 3 i c h i i m p a a 3 a i2 *2 2 *23 3 1 2 i i p mp kh* * *1 2

46* *s s i i 2 2* * * c11i c3 3h a i i c1 1s as * * p p h h 2 2* * * * a i i k a i ip * 2a a s i p h a s i p h* * * **i i c3 3h 1i2ihih h i a a a am p a i i ih h i a a a a* * * *3 * * ** *31 2a a a p m p kh ii aa i i c h ia a a* * ** * * * * * *3 3 33 3 1 i m p a i i i p mp kh* * * * *2 1 2Hence, the sufficient condition for dVdtto be negative definite is that 0.Where, 2 2* *s s i i* *11 33 11 c i c h a i ic s asi2* p p h h2* * a i i a i i p * 2a a i i p h h i i c h i i mpa i i p h h*3 * * * ** * * * *3 3 1 2hh a a a a a a a i i i p m p kh iih a a a a a a* * * * * * ** * * *1 23

47i 3 i c h i i m p a a 3 a i2 *2 2 *23 3 1 2 i i p mp kh* * *1 2And * *s s i i 2 2* * * c11i c33h a i i c1 1s as i* * p p h h 2 2* * * * a i i k a i ip * 2a a s i p h a s i p h* * * **i i c3 3h 1i2ihh h i a a a am p a i i ih h i a a a a* * * *3 * * ** *31 2a a a p m p kh ii aa i i c h ia a a* * ** * * * * * *3 3 33 3 1 i m p a i i i p mp kh* * * * *2 1 2dVdVTherefore, 0, if and 0, ifdtdts s i i p p h h* * * * , , , anda* a . * * * * * dV Therefore, the largest compact invariant set in s , i , p , h , a : 0is thedt singleton E* ,where*E is endemic equilibrium of the normalized model system

48(3.3). By LaSalle’s invariant principle, then it implies that*E is globallyasymptotically stable in if . This completes the proof of theorem ( 4).

49CHAPTER FOURTHE NUMERICAL SIMULATIONS OF THE MODEL4.1 Model SimulationTo study the dynamical behaviour of the model (3.3) numerically, the system isintegrated by using Runge-Kutta method using the following set of parameter values: 0.3, 0.2, 0.1, 1 0.4, 3 0.05, 1 0.2, 2 0.01, k 0.08, 0.9,c1 3, c3 1,m 0.4 , 0.4, 0.6, 1With initial valuess 0 0.5, i0 0.3, p0 0.12, h0 0.07 , a 0 0.01.The endemic equilibrium values are computed as shown graphically in figure 4.1(a) -4.1(d). It is shown thatsusceptible population s * .* * *i , p , h and*a is plotted against the proportion of

50Figure 4.1(a). Endemic equilibrium of proportion of infectives with Susceptible.Figure 4.1(b). Endemic equilibrium of proportion of Pre-AIDS population

51Figure 4.1(c). Endemic equilibrium of proportion of treated classProrportion of Aids patients0.20.180.160.140.120.10.080.060.040.021234Equilibrium point00.45 0.5 0.55 0.6 0.65 0.7 0.75Proportion of SusceptiblesFigure 4.1(d). Endemic equilibrium of proportion of AIDS populationWe see from the figures that for any initial start, the solution curve tend toequilibrium E * . Hence, we infer that the system (3.3) may be globally stable about*this endemic equilibrium point E . The computed values are:*s 0.6028, i * 0.2433,*p 0.2284,*h 0.04236,*a 0.08929 .

52The results of numerical simulation are shown graphically in Figures 4.2. In figure4.2(a) the distribution of proportion of population with time is shown in differentclasses with neither new infected children into the population nor recruitments i.e. 0 and 0.It is seen that in the absence of vertical transmission into thecommunity, the proportion of susceptible population decreases continuously as thepopulation is closed which results in an increase in proportion of infective populationfirst and then it decreases as all infectives subsequently develop full blown AIDS andthe die out by natural or disease-induced death rate. The proportion of treated class isincrease continuously since the total population being constant in this case.Figure 4.2(b) shows the variation of proportion of population in all classes with bothrecruitment of susceptibles and fraction of new born children which are infected atbirth. It is found that susceptible first decrease with time after using ARVs theyprolong life time they increase and then reaches its equilibrium position. Since due tovertical transmission from infectives, susceptible population increase continuouslytherefore infection becomes more endemic, following that R0 1.

53Figure 4.2(a) Variation of population in different classes for 0 and 0.Figure 4.2(b) Variation of population in different classes for 0.1and 0.2.

54The role of vertical transmission in the model i.e. the rate of recruitment of infectedchildren directly into infective class is explicitly shown in figures 4.3(a) - 4.3(c). It isseen that in figure 4.3(a) as the rate of infected children born increase, theproportion infective population also increases. It may be noted here that the birth ofinfected children make the infective population increase. In figure 4.3(b) if weincrease the value of the proportion of AIDS population decreases with time thenit increases until reaches its equilibrium position. Thus, if the birth of infectedchildren is controlled by treatment, the overall infective population will remain undercontrol. This will help in reducing the AIDS population. In 4.3(c) it is found that asthe rate of infected children born increases, the treated population decreases.Figure 4.3(a) Variation of infected population for different values of .

55Figure 4.3(b) Variation of AIDS population for different values of .Figure 4.3(c) Variation of Treated class for different values of .Figure 4.4(a) – 4.4(c) shows the impact of recruitment rate for Susceptibles, Treatedclass and AIDS patients for different values of . It is seen that as recruitment rate

56increase the susceptible population increase compared rather than decrease the rate .Since the inflow of susceptible increase, but the treated population and AIDSpopulation are decreases with time till it reach equilibrium and then decreasecontinuously due to treatment in the dynamic of HIVFigure 4.4(a) Variation of Susceptible for different values of .

57Figure 4.4(b) Variation of Treated population for different values of .Figure 4.4(c) Variation of AIDS population for different values of .

58Figures 4.5(a) – 4.5(d) depict the variation of infectives, pre-AIDS, Treatment andAIDS population respectively with time for different values of movement rate ofindividuals from infective class to pre-AIDS, Treatment or AIDS depending uponviral counts. It is seen that with increase in the value of movement rate , theinfected population decreases which intern increase the Treated population. Also ifwe increases movement rate , the Pre-AIDS and AIDS population decreases withtime until it reach the equilibrium stabilise the model.Figure 4.5(a) Variation of Infective population for different values of .

59Figure 4.5(b) Variation of pre-AIDS population for different values of .Figure 4.5(c) Variation of Treated class for different values of .

60Figure 4.5(d) Variation of AIDS population for different values of .Figure 4.6(a) – 4.6(b) shows the Variation of population in the classes with fractionof movement rate from infectious class . It is seen that from the figures when 1increases, the pre-AIDS population also decreases continuously it might be oftreatment which is provided to the patients, but for the AIDS class this is different, itis found that as 1increases, the AIDS population decreases with time then reachesits equilibrium and start to increase. This is caused by ARVs it prolonging the lifespan, then the patient join full blown AIDS, therefore disease persist. Also it is foundfrom the graph that as 2increases Treated population initially increase, as timegone then it decreases, but in AIDS patients they decreases continuously up toequilibrium then they increase because disease still exist.

61Figure 4.6(a) Variation of pre-AIDS population for different values of 1.Figure 4.6(b) Variation of AIDS population for different values of 1.

62Figure 4.6(c) Variation of Treated population for different values of 2.Figure 4.6(d) Variation of AIDS population for different values of 2.Figure 4.7(a) – 4.7(d) predict the Variation of the contact rates of susceptibles withinfectives 1and susceptibles with treated class 3in the susceptibles andinfectives population. The figures shows that as the contact rate 1increase the

63susceptibles decrease, infectives class decrease and treatment increase. Also as thecontact rates of susceptible with treated population 3increase the susceptibleinitially increase with time and then it reach its equilibrium position, but this differ toinfectives as 3increases, the infectives decreases with time than reach itsequilibrium and increases.Figure 4.7(a) Variation of Susceptibles population for different values of 1

64Figure 4.7(b) Variation of Infectives population for different values of 1Figure 4.7(c) Variation of Susceptibles population for different values of 3

65Figure 4.7(d) Variation of Infectives population for different values of 3Figure 4.8(a) – 4.8(b) shows the variation of susceptibles and infectives populationfor different values of number of sexual partners of susceptibles with infectives c 1,and between susceptibles with treated population c 3. It is seen that as the number ofsexual partners of susceptibles with infectives c1increases with time, susceptiblesdecreases continuously and infectives increases continuously. Also it is seen that ifthe number of sexual partners of susceptibles with treated population c3increaseswith time, susceptibles first increase than decreases with time as shown in fig. 4.8(c)due to loss of immune. But for infectives it seen that as c3increases also infectivesincreases, thus in order to reduce the spread of the disease, the number of sexualpartners as well as unsafe sexual interaction with an infective is to be restricted.

66Figure 4.8(a) Variation of Susceptibles population for different values of c 1Figure 4.8(b) Variation of Infectives population for different values of c1

67Figure 4.8(c) Variation of Susceptibles population for different values of c3Figure 4.8(d) Variation of Infectives population for different values of c3Figure 4.9(a) and 4.9(b) shows the variation of treatment rates in treated class andAIDS population. It is found that if treatment rates is zero the disease increase with

68time whereas treatment population decreases. Also the graphs shows that as the ratesof treatment increases, the AIDS patients decreases due to treatment measures whichincreases the immune system and prolong life of AIDS population, therefore thisincreases the treated class.Figure 4.9(a) Variation of AIDS patients for different values of k and .Figure 4.9(a) Variation of AIDS patients for different values of k and .

69In figure 4.10 the effect of disease induced death rate is shown and it is found thatas increases, the population of AIDS patient’s decreases. It is found that the AIDSinduced death rate , can be controlled by provide antiretroviral drugs (ARVs) toAIDS patients or by advising AIDS patients to follow healthy regulation about theAIDS disease. From the figures, it can also be seen that the respective populationsare tending to the equilibrium level. This has also been observed for different initialvalues of the variables. Hence the endemic equilibrium*E is globallyasymptotically stable for the above set of parameter value.Figure 4.10 Variation of AIDS populations for different values of .

70CHAPTER FIVEDISCUSSION, CONCLUSION AND FUTURE WORK5.1 DiscussionIn this study, a non linear model has been proposed and analyzed to study thedynamic of HIV/AIDS model with treatment and vertical transmission. The diseasefree and endemic equilibria are obtained and their stability are investigated. Anumerical study of the model has been conducted to see the impact of certain keyparameters on the spread of HIV disease.It is observed that an increased ininfectives through vertical transmission would lead to increase the population ofinfectives which in turn increase the pre-AIDS, treated class and AIDS population.AIDS can be eradicated after some fixed years if we can reduce the value of R0toless than unity i.e. R0 1. In the case R0 1 all the infected classes which areinfectives, pre-AIDS and AIDS populations decrease with time. When this attained,immediately the treated class increases since the infective class start decreasing, andthen follows the pre-AIDS class and finally the AIDS can be eradicated.The following measures can be exercised in order to decrease the value of R0to beless than one so as to decrease the infections, which can lead on eradication of theHIV/AIDS disease.

71Use of Antiretroviral DrugsIn many cases, an HIV-infected individual may be invaded by a host of commondiseases ranging from very strong bacterial and viral infections to mild sicknesses.Since the infected person may be invaded by many infections simultaneously, it isnoted that a single drug may not be sufficient to clear such an attack. As a result, acouple of drugs recombinants that are particularly made to treat the HIV/AIDSpatients of combinations of attacks have been developed.Women who have reached the advanced stages of HIV disease require combinationsof antiretroviral drugs for their own health. This treatment, which must be takenevery day for the rest of women’s life, is also highly effective at preventing motherto-childtransmission (PMTCT).The most effective PMTCT therapy involves a combination of three antiretroviraldrugs taken during the later stages of pregnancy and during labour. This therapy isessentially identical to the treatment taken by HIV positive people for their ownhealth; except that it is taken only for a few months and the choice of drugs may beslightly different. Triple therapy is usually recommended to women in developedcountries, and is becoming more widespread in the rest of the world; this is reflectedby WHO 2009 Guidelines.Effective Prevention of Vertical transmissionEffective prevention of mother-to-child transmission (PMTCT) requires a three-foldstrategy (UNAIDS/WHO, 2005).

72Preventing HIV infection among prospective parent making HIV testing and otherprevention interventions available in services related to sexual health such asantenatal and postpartum care.Avoid unwanted pregnancies among HIV positive women-providing appropriatecounselling and support to women living with HIV to enable them to make informeddecisions about their reproductive lives.Preventing the transmission of HIV from HIV positive mothers to their infantsduring pregnancy, labour, delivery and breastfeeding.Integration of HIV care, treatment and support for women found to be positive andtheir families.Also it is needed to control the vertical transmission of HIV by suppressinghorizontal transmission; including condom use and other safer sexual contact. Fromnumerical simulation it is found that an increase in the number of sexual partnersfurther reduces the susceptible population by way of spreading the disease. Thus inorder to reduce the spread of the disease, the number of sexual partners should berestricted as well, and consequently the equilibrium values of infective, treated andAIDS population can be maintained at desired levels.Cesarean SectionsA cesarean section is an operation to deliver a baby through its mother’s abdominalwall. When a mother is HIV positive a cesarean section may be done to protect thebaby from direct contact with her blood and other bodily fluids. There are indicationsthat cesarean birth decreases the risk of mother-to-child transmission (Newell, 1998).

73If the mother is taking combination antiretroviral therapy then a cesarean section willoften not be recommended because the risk of HIV transmission will be very low.Cesarean delivery may be recommended if the mother has a high level of HIV in herblood, but the procedure is seldom available and/or safe in resource poor settings(Dabis, F., 2002).Safer Infant Feeding to HIV Positive MothersA number of studies have shown that the protective benefit of drugs is diminishedwhen babies continue to be exposed to HIV through breastfeeding (Lu L. etal.,2009). Mother with HIV are advised not to breastfeed whenever the use of breastmilk substitutes (formula) is acceptable, feasible, affordable sustainable and safe.For HIV positive women who choose to breastfeed, exclusive breastfeeding isrecommended for the first months of an infant’s life and should be discontinued oncean alternative form of feeding becomes feasible (UNAIDS, 2005). Mixed feeding isnot recommended because studies suggest it carries a higher risk than exclusivebreastfeeding. This is because mixed feeding damages the lining of the baby’sstomach and intestines and thus makes it easier for HIV in breast milk to infect thebaby. However, since ARVs have been more available, the risk from mixed feedingis reduced. Indirect evidence suggest that keeping the period of transmission fromexclusive breastfeeding to alternative feeding as short as possible may reduce the riskof transmission. Unfortunately, the best duration for this is not yet known and mayvary according to the infant’s age and/or the environment (UNAIDS 2005).

745.2 ConclusionsIn this research, a non-linear mathematical model is proposed and analyzed to studythe transmission of HIV/AIDS in a population of varying size with treatments andvertical transmission under the assumption that due to sexual interaction ofsusceptibles with infectives, the infected babies are born to increase the growth ofinfective population directly. It is assumed that people in pre-AIDS and AIDS classesare exposed and incapable of producing children. By analyzing the model, we havefound a threshold parameter R . It is noted that when 0R0 1 then the disease dies outand when R0 1 the disease becomes endemic. The model has two non-negativeequilibrium namely the disease-free equilibrium, * * * * * *equilibrium E 1,0,0,0,00and the endemicE s , i , p , h , a . It is found that the disease-free equilibrium E0islocally asymptotically stable if R0 1 and for R0 1 it is unstable and the infectionis maintained in the population. The endemic equilibrium E * , which exist only whenR0 1 , is always locally asymptotically stable. The equilibrium is also shown to beglobally asymptotically stable under certain conditions. It is found that an increase inthe rate of vertical transmission leads to increase the population of infectives whichin turn increases the pre-AIDS and AIDS population. Thus, the vertical spread of thedisease should be controlled by effective treatment or promoting the condom use tokeep the overall infective population under the control. This will help in reducing theAIDS population as well. Also by simulation it is shown that by controlling the rateof vertical transmission, the spread of the disease can be reduced significantlyconsequently the equilibrium values of infective, treatment and AIDS population canbe maintained at desired levels. It is also found that the increase in the number of

75sexual partners further reduces the total population by way of spreading the disease.Thus, in order to reduce the spread of the disease, the number of sexual partners aswell as unsafe sexual interaction with an infective is to be restricted. The effect of anincrease in disease-induced death rate is, however, to decrease the AIDS patientspopulation. From the analysis, it may also be speculated that if the HIV infection issuppressed at an early stage by effectively treating the infectives, the progression tothe AIDS can be slowed down and the life span of HIV infectives can be prolonged.5.3 Future WorkHIV/AIDS at present has no cure and no vaccine as yet. As the number of reportedcases of AIDS continues to swell, the magnitude and severity of the problembecomes increasingly evident and the need to develop effective strategies to preventand control HIV infection is becoming even more urgent. In this study it is pointedout here that not all infected individuals take part in spreading the disease, as in thecase of infected children, but they will all develop their own AIDS.From the base of this study, it is proposed that future work should consider that allinfected individuals to take part in spreading the disease, as in the case of infectedchildren and develop to AIDS.Also the model can be modified to include a delay to make the infected childrenbecome adult to further spread the infection either vertically or horizontally.

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