Dynamics of a HIV Epidemic Model

Transcription

1 International Journal of Scientific & Engineering Research Volume 3 Issue 7 Jul- ISSN Dnamics of a HIV Epidemic Model Nguen Huu Khanh Abstract We stud a non-linear mathematical model for the transmission of HIV disease The model is represented b a sstem of differential equations depending on parameters We diide the population into three subclasses: uninfected cells infected cells and free irus particles A factor deciding the spread of irus is the basic reproduction number R We found that if R < then the disease goes etinct whereas if R > then the disease remains This phenomenon is eplained b a transcritical bifurcation A numerical inestigation for the model is carried b the software Mathematica and AUTO Inde Terms Bifurcation free disease equilibrium endemic disease equilibrium basic reproductie ratio transcritical bifurcation transmission irus INTRODUCTION H IV is primaril a seual transmitted disease its spread reflects the social pattern of human seual relationships Since it was discoered in 98 HIV has become one of the leading causes of death of people in countries HIV has killed an estimated 5 million indiiduals world wide [ 7 4] One of the most important concerns about an infectious disease is its abilit to inade a population The dnamics between irus infections and the immune sstem inole man different components and are multifactorial The understanding of the long-time behaiour of this disease will help us to find whether this epidemics will die out or sta in the population and to design strategies of fighting them Mathematical models and computer simulations hae become useful in analzing the spread and control of infectious diseases The build and test theories that are inoled with comple biological sstems related disease getting quantitatie conjectures determining dnamical parameters due to change and estimating parameters from data Various approaches for studing epidemiolog of HIV hae been deeloped in recent ears [ ] Man epidemiological models hae a free disease equilibrium (FDE) at which the population remains in the absence of disease These models usuall hae a threshold parameter known as the basic reproduction number R such that if R < then the FDE is locall asmptoticall stable and the disease cannot inade the population but if R > then the FDE is unstable and an epidemic is epected In this paper we stud a HIV endemic model as mentioned in [3] The model is gien b a sstem of three dimensional ordinar differential equations depending to parameters The population size is diided into three subclasses that are uninfected cells infected cells and free irus particles We assume that the enironment is homogeneous Nguen Huu Khanh is working at Department of Mathematics College of Natural Science Can Tho Uniersit Viet Nam nhkhanh@ctuedun IJSER Our emphasis lies on obtaining a mathematical understanding of the dnamics and bifurcation of the model We analze the stabilit of equilibria and ehibit phase portraits for competition dnamics We show that the factor that goern the dnamics of the model is the basic reproducctie number R Transcritical bifurcation is useful to eplain the echange of stabilit of equilibria Obtained results eplain the transmission of disease in the HIV model Numerical method is used to inestigate behaiour of the model THE STUCTURE OF THE MODEL The endemic model of HIV consists of three ariables: the population size of uninfected cells infected cells and free irus particles Uninfected cell Virus Infected cell () () () Fig Fig Transmission Flow diagram of diagram the HIV of endemic the irus model Free irus particles infect uninfected cells at a rate proportional to the product of their abundances The rate describes the impact of this process including the rate at which irus particles find uninfected cells the rate of irus entr and the rate and probabilit of successful infection Infected cells produce free irus at a rate Infected cells die at a rate and free irus particles are remoed from the sstem at a rate One can see that the aerage life-time of an infected cell is /

2 International Journal of Scientific & Engineering Research Volume 3 Issue 7 Jul- ISSN whereas the aerage life-time of a free irus particles is / The total amount of irus particles produced from one infected cell is / Uninfected cells are produced at a constant and die at a rate The aerage life-time of an uninfected cell is / In accordance with the preious assumptions the model is gien b the following differential equations d d d We assume that parameters and are positie The sstem () is a sstem of nonlinear differential equation To stud the dnamics of this sstem we consider the linearization at equilibria 3 ANALYSIS OF THE MODEL 3 Inariant set We establish the inariant set of the sstem () that is the first quadrant D {( ( t) ( t) ( t)): ( t) ( t) ( t ) } This means that the solution of the sstem is still in D for t > Hence for the rest of the paper we onl focus on sstem () restricted to D 3 Equilibria To find equilibria we set the right-hand side of the sstem () equal to zero There are two equilibria in the ( ) space: ) The free disease equilibrium E ( / ) ) The endemic disease equilibrium E ( z ) where () () The equilibrium E alwas eists while E onl eists for > 33 The basic reproductie ratio The dnamics of the HIV model is decided b the basic reproductie ratio R which is defined as the number of newl infected cells that arise from an one cell when almost all cells are uninfected We found that In the net section we will show that for R < the transmission is etinct whereas for R > the irus still remain We consider two cases to illustrate the characteristics of the basic reproductie ratio R Case R < Choosing = = = 7 = 5 = = 45 we hae R = 6349 < With the initial condition () = 9 () = 5 () = 35 one can see the infected cell component (t) and the free irus component (t) as t This means that the epidemics will die out (see Fig ) Case R > Choosing = = 5 = 55 = 85 = = 75 we hae R = > With the initial condition () = () = 5 () = 5 one can see the infected cell component (t) 75 and the free irus component (t) 435 as t This means that the epidemics still remain (see Fig 3) t Fig Time series of and for R < t Fig 3 Time series of and for R > R (3) As R < the sstem has an unique equilibrium E and it is stable For R > the sstem has two equilibria E and E where E is unstable and E is stable IJSER 4 STABILITY OF EQUILIBRIA The local stabilit for equilibria is determined b the Jacobian matri of the sstem () which is

3 International Journal of Scientific & Engineering Research Volume 3 Issue 7 Jul- 3 ISSN J Remark For R < if u(t) = ((t) (t) (t)) is a solution of the sstem () near to E ( / ) then u(t) E as t Therefore u(t) and (t) as t This means that the epidemics will die out 4 The free disease equilibrium E The free disease equilibrium E ( / ) alwas eists for > and > Theorem The free disease equilibrium E is locall asmptoticall stalble for R < and unstable for R > Proof The Jacobian matri at E is gien b J The eigenalues are obtained b soling the characteristic equation det(j λ I) = due to Mathematica We get the following eigenalues: 3 ( ) 4 ( ) 4 We see that < < When R < we hae < and 3 ( ) 4 This implies 3 < Therefore E is stable Similarl if R > then > It leads to 3 > and E is unstable The endemic disease equilibrium E For R > from () we can see the endemic disease equilibrium E ( z) has positie coordinates Therefore E is belonging to the inariant set D Theorem The endemic disease equilibrium E is onl eisted and locall asmptoticall stable for R > Proof The Jacobian matri ealued at E is gien b J where and are gien b () Soling the characteristic equation det(j λi) = b Mathematica we obtain the following eigenalues: < It is eas to see that For R > we hae > and This implies Therefore E is stable Fig 4 Phase trajectories of the sstem () near E on the ( ) plane The circle stands for E IJSER Fig 5 Phase trajectories of the sstem () on the ( ) plane The circle stands for E and the solid circle is for E

4 International Journal of Scientific & Engineering Research Volume 3 Issue 7 Jul- 4 ISSN Remark For R > if u(t) = ((t) (t) (t)) is a solution of the sstem () near to E ( z ) where z are gien b () then u(t) E as t Therefore u(t) and (t) > as t This shows that the transmission remains 5 BIFURCATION ANALYSIS In this section we stud the change of solution of the sstem () as parameters ar The change of dnamical feature of the sstem is called bifurcation In order to eplain bifurcation of the sstem we assume that the equilibrium E eists for R < although this is untrue The software package AUTO is useful tool to detect bifurcation points The free disease equilibrium E alwas eists As R > it is unstable When R < it is stable and this is correspondent to the transmission die out The disease irus equilibrium E onl eists for R > It is stable and this case corresponds to the epidemics still remain Transcritical bifurcation occurs as R = At this alue E loses its stabilit (the eigenalue 3 moes from negatie to positie) and change from stable to unstable For R > E go from unstable to stable Two equilibria echange their stabilit This bifurcation eplains the meaning of the basic reproductie ratio R that decide the epidemics die out or remain computed b AUTO The horizontal aes stands for the parameter and the ertical aes is for alues of The line contains solutions is the line of the free disease equilibria and the line consists of solutions and 6 is the line of the endemic disease equilibria The dashed line is unstable and the solid line is stable The transcritical bifurcation occurs at the solution 6 CONCLUSION A model for the infection of HIV was constructed and analzed We eamine the potential effects of the proimate determinants of HIV transmission dnamics alone and in combination There are two equilibrium points for this model first E occurs when all the cells are not infected and there is no contaminated process This point is called a disease-free equilibrium point Second E occurs when there are HIV-infected cells and contaminated process Seeral parameters affect on (t) (t) and (t) One of important parameters is the rate that relating to infection of irus on uninfected cells The basic reproductie ratio R is the factor that goerns the transmission of disease B goerning suitable parameters one can let R < and the disease will die out The theor of dnamical sstem shows that a transcritical bifurcation occurs for R = The software Mathematica and AUTO is used to describe the dnamics of the model This model reflects man stabilit properties of more complicated models ACKNOWLEDGMENT This publication was made possible through support proided b the IRD-DSF REFERENCES Fig 6 Bifurcation duagram of the HIV model in the ( ) plane B using the software package AUTO [3] one can detect the transcritical bifurcation of the equilibria E and E Fiing the following parameters = = 7 γ = = 5 = = 45 and let aries We note that is important parameter that cause the infection of irus on cells Continue from the free disease equilibrium E with = 7486 = and = AUTO detects a transcitical bifurcation occurs at alue = 3499 that correspond to R = At this alue two equilibria echange their stabilit Figure 6 shows the bifurcation diagram of the HIV model [] F Bararama L S Luboobi and J Y T Mugisha "Periodicit of the HIV/AIDS Epidemic in a Mathematical Model" American Journal of Infectious Disease pp [] D S Callwa R M Nowak "Virus Phenotpe Switching and Disease Progression in HIV- intection" Proc R Soc Lond B Biol Sci 56 pp [3] J M Coffin "HIV Population Dnamics in Vio: Implications for Genetic Variation" Pathogennesis and Therap Science 67 pp [4] E J Doedel RC Paffenroth AR Champnes TF Fairgriee YA Kuznetso B Sandstede and X Wang AUTO : Continouation and Bifurcation Software for Ordinar Diffeential Equations sourceforgenet/projects/auto/ [5] G Guckenheimer and P Holmes Nonlinera Oscillations Dnamical Sstems and Bifurcations of Vector Fields NewYork Springer-Verlag 983 [6] Z Ma J Zhou and J Wu Modeling and Dnamics of Infectious Diseases World Scientific Publishing 9 [7] M A Nowak R M Ma Virus Dnamics Mathematical Principles of Immunolog Oford Uniersit Press [8] N Nuraini R Irawan and E Soewono "Mathematical Modeling of HIV/AIDS Transmission among Injecting Drug Users with Metha- IJSER

5 International Journal of Scientific & Engineering Research Volume 3 Issue 7 Jul- 5 ISSN done Therap" Asian Transactions on Basic & Applied Sciences Vol Issue 5 pp -5 [9] E S Rosenberg M Altfeld S H Poon M N Philip "Immune Control of HIV- after earltreatment of Acute Infection" Nature 47pp [] S Kocacs "Global Dnamics of a HIV transmission model" Acta Uni Sapientiae Mathematica pp [] J Liszziewicz and F Lori "Structured Treatment interuption in HIV/AIDS therap" Microbes Infect 4 pp 7-4 [] S A Sheikh F Musali and M Alsolami "Stabilit Analsis of an HIV/AIDS Epidemic Model with Screening" International Mathematical Forum Vol 6 66 pp [3] D Wodarz and M A Nowak "Mathematical Models of HIV Pathogenesis and Treatment" BioEssas 4 pp [4] D Wodarz and M A Nowak "Specific Therap Regimes coulldlead to Long-Term Control of HIV" Proc Natl Acad Sci USA 96 pp IJSER