Stability analysis of a non-linear HIV/AIDS epidemic model with vaccination and antiretroviral therapy

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1 Int. J. Adv. Appl. Math. and Mech. ( (7 (IN: 7-9 IJAAMM Journal homepage: International Journal of Advances in Applied Mathematics and Mechanics tability analysis of a non-linear HIV/AID epidemic model with vaccination and antiretroviral therapy Research Article Debashis Biswas a, amares Pal b, a Department of Mathematics, Calcutta Girls College, Kolkata-, India b Department of Mathematics, University of Kalyani, Kalyani- 7, India Received August 7; accepted (in revised version November 7 Abstract: MC: In this paper, we like to propose and analyze a non-linear HIV/AID epidemic model with vaccination and antiretroviral therapy. For our convenient study we have divided the total populations into five classes such as susceptible class, unaware HIV infected class, aware HIV infected class, pre AID class and AID class respectively. For our present purpose we have taken only the disease spread through horizontal transmission into consideration. In this paper we have tried to develop a non-linear HIV/AID mathematical model to study the transmission dynamics of HIV at four compartments of the populations with vaccination and antiretroviral therapy and to prove the positivity and boundedness of its solutions. In this paper we have added a treatment procedure i.e. antiretroviral therapy and tried to find out its effect. We have also analyzed the stability behaviour of the system. Finally we have found that vaccination and antiretroviral therapy is an effective way to control the disease transmission. The mathematical model solved numerically by using an iterative numerical recipe which supports the theoretical or analytical results. 9B Keywords: HIV/AID Epidemic model Antiretroviral therapy Basic reproduction number tability Vaccination Numerical results 7 The Author(s. This is an open access article under the CC BY-NC-ND license ( Introduction HIV/AID is one of the most enormous diseases humankind has already ever faced all over the World. The number of people living with HIV is around 8 million in 99 and it increases to million at the end of (UAID, []. The first AID (Acquired Immunodeficiency yndrome case was documented by Centers of Disease Control (CDC of America in 98 []. AID is very dangerous disease that suppresses the normal function of the immune system of human body and such types of virus is called Human Immunodeficiency Virus or HIV. uch types of diseases now spread Worldwide [, ]. The first known case of HIV in India was diagnosed amongst female sex workers in Chennai in 986. In the next year 987 more or less than new cases were reported. The first National AID Control Programme was launched in 99, and National AID Control Organisation (NACO was constituted to implement the programme in India through HIV/AID Prevention and Control ocieties []. Mathematical models play a crucial part to study the disease transmission and shows the path to treat the HIV/AID affected patient. Many a Mathematician have already introduced various mathematical model about such disease [6 ]. From the initial models of May and Anderson [ ] various refinements have been added into modelling Corresponding author. Tel.: address(es: deba.biswas@gmail.com (Debashis Biswas, samaresp@yahoo.co.in (amares Pal.

2 tability analysis of a non-linear HIV/AID epidemic model with vaccination and antiretroviral therapy frameworks and specific issues have been expected by researchers [ 6]. Their preliminary endeavour is how to prevent AID from spreading [7]. There is so many difficulty to treating HIV infected patient. Numerous health organization has taken initiative to arrange programme to slow down the spreading of this virus [7]. Mainly, AID is spread by sexual transmission and vertical transmission from mother to child at birth and through breast milk [8]. Vaccination and antiretroviral therapy are also reduces the transmission rate of HIV [9]. Treatment (vaccination and antiretroviral therapy procedure can not eliminate HIV virus completely from the body but it can provide some resistance or slow down the processes of disease transmission. Although the treatment procedure is very slow and currently there is no effective vaccine. Antiretroviral therapy is very expensive and routine access of ART medicine is not available in all countries of the World [, ]. If we can avail the treatment processes properly then the transmission rate HIV/AID will be diminished. In this paper we have developed a nonlinear HIV/AID Mathematical model to study the transmission dynamics of HIV disease at four compartments of the populations with vaccination and antiretroviral therapy. The Mathematical model is solved numerically by using an iterative numerical recipe, which support the theoretical results.. The mathematical model Here we consider a non-linear HIV/AID epidemic model with Vaccination and antiretroviral therapy. We divided the total populations into five subclasses that is; the susceptible class (t, the HIV infected un-aware class U (t, the HIV infected aware class W (t, the pre AID class P(t, the AID class A(t. Our assumption is that the disease spread only by horizontal transmission i.e. sexual contact, blood transfusion. The disease is spread by unprotected sexual contact at optimum level. Here we may assume that aware HIV infected class does not spread the disease by unprotected sexual contact due to their proper awareness about the disease. Our mathematical model has been built up on the basis of this assumptions. d dt = R βu s (µ + σ, + α U + α U du dt = βu (µ + γ + ηu, + α U + α U dw = ηu (µ + γ W, ( dt dp dt = γu + γ W (µ + εp, d A = εp (µ + αa. dt Where ( >, U ( >, W ( >, P( > and A( >, t. ince the state variable A of the system ( does not appear in the first four equations. Therefore A class do not transmit infection and we consider the following subsystem: d dt = R βu s (µ + σ, + α U + α U du dt = βu (µ + γ + ηu, + α U + α U ( dw = ηu (µ + γ W, dt dp dt = γu + γ W (µ + εp, Where, R s is the constant recruitment rate of susceptible class, β is the per capita contact rate between susceptible class and un-aware HIV infected class, α, α are two positive constants, γ is the conversion rate of un-aware HIV infected class to AID class, γ is the conversion rate of aware HIV infected class to pre AID class, σ is the successful vaccination rate of the susceptible individuals, ε is the rate by which pre AID infective individuals develop AID, µ is the natural mortality rate of adult class, α is the AID related death rate. η is the antiretroviral therapy rate of an un-aware HIV infected class. The system ( has to be analysed with the following initial conditions, ( >, U ( >, W ( > and P( >.. Basic properties of the model.. Positivity of the olutions ince our proposed mathematical model indicates human population we must show that all the state variables remain non-negative s.

3 Debashis Biswas, amares Pal / Int. J. Adv. Appl. Math. and Mech. ( (7 Lemma.. All the solutions, U, W and P for the system ( is non-negative t with the initials condition ( >, U ( >,W ( > and P( > in the region Γ. Proof. Taking the first equation of the system ( in consideration, we get d dt =R βu s + α U + α U (µ + σ = R βu s + (µ + σ. + α U + α U d dt > βu + (µ + σ. + α U + α U (t > (e β U (ξ +(µ+δ +α U (ξ+α U (ξ dξ Considering the second equation of the system (, we get du dt = βu (µ + γ + ηu > (µ + γ + ηu. + α U + α U U (t > U (e (µ+γ+ηt. imilarly we can prove that W (t W ( e (µ+γ t and P(t P( e (µ+εt Therefore all solutions of the system ( are non-negative t... Invariant Region and Boundedness Lemma.. All the non-negative solutions of system ( are contained in the feasible region Γ = (, U, W, P R+ : +U +W + P R s (µ + σ is bounded. Proof. We have from the system ( d dt + du dt + dw + dp dt dt = R s µ( +U +W + P σ εp R s µ( +U +W + P. R s (µ+σ. Hence lim t sup ( +U +W + P Therefore all the non-negative solutions of the system ( ultimately lies in the region Γ = (, U, W, P R+ : +U +W + P R. s (µ + σ Hence the vector field points into the interior of Γ t is positively invariant.. Equilibrium points and basic reproduction Number The system ( possesses the following equilibria namely, (i The disease free equilibrium E = ( (µ+σ,,,, R s (ii The endemic equilibrium E = (,U,W,P with = W = R s ( + α U + α U (µ + σ( + α U + α U + βu, ηu (γ + µ, P = (µγ + γγ + ηγ U (µ + ε(γ + µ and U is the root of the quadratic equation a U +b U + c =, where a = α (µ + σ(µ + γ + η >, b = α (µ + σ(µ + γ + η + β(µ + γ + η >, c = (µ + σ(µ + γ + η( R.

4 tability analysis of a non-linear HIV/AID epidemic model with vaccination and antiretroviral therapy If R >, then c <. Therefore the above quadratic equation has a positive root if R >, where R is a basic reproduction number given as follows Hence R = R s β (µ + σ(µ + γ + η. U = b + a where, = α (µ + σ(µ + γ + η + β(µ + γ + η α (µ + σ(µ + γ + η(µ + σ(µ + γ + η( R. If R <, then there is no positive equilibrium. The local stability of E and E is governed by the basic reproduction number R which we may find by using the next generation matrix. The non-negative matrix, f of the infection terms and the non singular matrix v of the transition terms are and f = We get, f = Jacobian of f at disease free equilibrium = v = Jacobian of v at disease free equilibrium = Therefore R = ρ f v. pectral of the matrix R = βr s (µ+σ(µ+γ+η. βu +α U+α U (µ + γ + ηu v = ηu + (µ + γ W γu γ W + (µ + εp βr s (µ+σ and (µ + γ + η η (µ + γ γ γ (µ + ε.. tability Analysis.. Local tability In this current section we have found the local stability of the disease free and the endemic equilibrium point. We have presented the local stability of the disease free equilibrium point E = Rs (µ+σ,,,, in the following theorem. Theorem.. R The disease free equilibrium E = ( s (µ+σ,,,, is locally asymptotically stable if, R <, and unstable while R >. Proof. The variational matrix around the disease free equilibrium point E is given by (µ + σ J = βr s (µ+σ βr s (µ+σ (µ + γ + η η (µ + γ γ γ (µ + ε The characteristics equation associated with the Jacobian matrix J is given by (µ + ε λ (µ + γ λ (µ + σ λ βr s (µ + γ + η λ = (µ + σ

5 Debashis Biswas, amares Pal / Int. J. Adv. Appl. Math. and Mech. ( (7 The eigenvalues associated to J are λ = (µ + σ <, λ = (µ + γ <, λ = (µ + ε <, and λ = βr s (µ + σ (µ + γ + η = (R (µ + γ + η <. It is clear the all the eigenvalues of the Jacobian matrix J is negative when R <. o the disease free equilibrium is locally asymptotically stable. Further when R >, then there exists one positive and three negative eigenvalues, so an saddle unstable equilibrium exists... Global stability of disease free equilibrium Lemma.. If R <, then the disease free equilibrium point E of the model ( is globally asymptotically stable in Γ and unstable if R >. Proof. Define Lyapunov function: L = ηu + (µ + γ + ηw L = ηu + (µ + γ + ηw L βu = η (µ + γ + ηu + (µ + γ + η ηu (µ + γ + α U + α U W, L R η(µ + γ + η + α U + α U U ince = R s (µ + σ If U =, L = but if U and R <, L < therefore, the disease free equilibrium is globally asymptotically stable []... Local stability of an endemic equilibrium Lemma.. The endemic equilibrium point E is locally asymptotically stable, when R >. Proof. The Jacobian matrix J associate with the endemic equilibrium point E is given by, βu (µ + σ β β U α +α U +α U (+α U +α U J βu = β β U α (µ + γ + η +α U +α U (+α U +α U η (µ + γ γ γ (µ + ε The characteristics equation related to the Jacobian matrix J, (µ + ε λ λ (µ + γ [ βu + α U (µ + σ λ + α U βu + + α U + α U β β U α ( + α U + α U (µ + γ + η λ The two eigenvalues associated to the J are λ = (µ + ε, λ = (µ + γ and the other two eigenvalues of the quadratic equation p λ + q λ + r = β β U α ( + α U + α U ] =.

6 6 tability analysis of a non-linear HIV/AID epidemic model with vaccination and antiretroviral therapy where p = >, βu q = + α U + α U + (µ + +σ + (µ + γ + η β ( + α U + α U + β U α ( + α U + α U β U α = ( + α U + α U + βu + α U + α U + (µ + +σ + (µ + γ + η(α U + α U + α U + α U > r = βu (µ + γ + η + α U + α U + (µ + σβ U α ( + α U + α U β (µ + σ ( + α U + α U + (µ + +σ(µ + γ + η = βu (µ + γ + η + α U + α U + (µ + σβ U α ( + α U + α U + (µ + γ + η(µ + σ(α U + α U + α U + α U > It follows by Routh-Hurwitz Criteria that the roots of the above quadratic equation all have negative real parts. Therefore the endemic equilibrium point E is locally asymptotically stable when R >... Global stability of the endemic equilibrium Lemma.. If R >, then the endemic equilibrium E of the model ( is globally asymptotically stable in Γ. Proof. We define Lyapunov function V as follows [, ]. ( V = log + (U U U log UU + ( µ + ε η + ( (P P P log PP µγ + γγ + ηγ ( µ + γ + η (W W W log WW dv ( dt = d ( dt + U ( du µ + γ + η ( U dt + W ( dw µ + ε η + ( ( P dp η W dt µγ + γγ + ηγ P dt = ( ( µ + σ ( β U + ( + α U + α U βu ( + α U + α U + ( U βu U + α U + α U ( µ + γ + η ( µ + γ + η U + ( W ηu ( µ + γ W η W ( µ + ε η + ( ( P ( µγ + γγ + ηγ W ( µ + ε P µγ + γγ + ηγ P η = ( (µ ( + σ β U + ( + α U + α U β U ( + α U + α U + β U ( + α U + α U βu ( + α U + α U + ( µ + γ + η ( ( µ + U γ µ + γ + η W ( µ + γ + η U W ( ( µ + γ µ + γ + η W η W + η + (µ + γw η( µ + ε P ( µ + ε W P µγ + γγ + ηγ P + η ( P µ + ε µγ + γγ + ηγ = ( (µ ( + σ + ( µ + γ + η U β U ( + α U + α U + β U ( + α U + α U ( µ + γ + η U + ( µ + γ + η U ( µ + γ + η U + ( µ + γ + η U ( µ + γ + η U W W ( µ + ε W + ( µ + ε W ( µ + ε W P P + ( µ + ε W = ( (µ ( + σ + ( µ + γ + η U ( µ + γ + η U + ( µ + γ + η U ( µ + γ + η U + ( µ + γ + η U ( µ + γ + η U ( µ + γ + η U W W + ( µ + γ + η U η + ( µ + ε W ( µ + ε W P P = ( (µ ( + σ ( µ + γ + η ( U + UW U W ( µ + γ + η U ( ( µ + ε ( W P W W P

7 Debashis Biswas, amares Pal / Int. J. Adv. Appl. Math. and Mech. ( (7 7 Here we have used the properties i.e. the arithmetic mean is always greater than or equal to the geometric mean, therefore ( + UW U W, ( and ( W P dv W P. Then dt = only when =, U = U, W = W and P = P. Hence the maximum invariant set of the system ( on the set (, U, W, P : dv dt = is the singleton and by using Laalle invariant principle [], the endemic equilibrium is globally asymptotically stable if R >. 6. Numerical imulations In this part of our paper, we have presented some numerical simulations of the system ( to explain our analytical results.. The results of numerical simulation are exhibited in Fig. to Fig.. Fig. shows that the disease free equilibrium point is locally asymptotically stable when R =.97 < for the set of parameters value which is taken from Table. While R =. >, then Fig. shows that the disease free equilibrium point is unstable for the set of parameters value R s =., η =., µ =., γ =., σ =., β =. and the other parameters value are taken from Table. Also we say that when R =.868 >, then Fig. shows that the endemic equilibrium is locally asymptotically stable for the set of parameters value R s =.7, η =.6, µ =., γ =.9, σ =., β =., ε =., α =., γ =. and α =.. Now we choose the set of parameters value η =.9, µ =., γ =., σ =., β =.9, ɛ =., γ =. and α =. and the other parameters value are taken from Table with the initials value ( =, U ( =, W ( =, its phase portrait is given in Fig., which shows that the system ( has a stable limit cycle. Now for the parameters value R s =.7, η =.6, µ =., γ =.9, σ =.,.,.9, β =., ε =., α =., γ =. and α =. then Fig., Fig. 6 and Fig. 7 shows that if we increase the vaccination rate then U, W and P classes are also decreases. We have Fig. 8, Fig. which illustrate that if we increase the antiretroviral therapy then U, and P class are decrease but W class is increase which is given in Fig. 9 for the set of parameters value R s =.7, η =.,.,.8, µ =., γ =.9, σ =., β =., ε =., α =., γ =. and α =.. Table. A hypothetical set of parameter values. Parameter Definition Default value R s Is the constant recruitment rate of susceptible class.. α A positive constants.. α A Positive constants.. µ Is the natural mortality rate of adult class.. σ Is the successful vaccination rate of the susceptible individuals.. γ Is the conversion rate of un-aware HIV infected class to AID class.. γ Is the conversion rate of aware HIV infected class to pre AID class.. β Is the per capita contact rate between susceptible class and un-aware HIV infected class.. η Is the antiretroviral therapy rate of an un-aware HIV infected class.. ε Is the rate by which pre AID infective individuals develop AID.. Population U population W Population P population Population U population W Population P population Fig.. The equilibrium point E is asymptotically stable while R <. Fig.. The equilibrium point E is unstable while R >.

8 8 tability analysis of a non-linear HIV/AID epidemic model with vaccination and antiretroviral therapy Population U population W Population P population W... Fig.. The interior equilibrium point E (,U,W is stable while R >. E* U Fig.. The equilibrium point E is locally asymptotically stable while R >. U population σ=.= σ=.= σ=.9= W Population.. σ=.= σ=.= σ=.9= Fig.. This figure shows that when σ increases then U class decreases. Fig. 6. This figure shows that when σ increases then W class decreases. P population σ=.= σ=.= σ=.9= U population η=.= η=.= η=.8= Fig. 7. This figure shows that when σ increases then P class decreases. Fig. 8. This figure shows that when η increases then U class decreases. W Population η=.8 η=. η= P population η=.= η=.= η=.8= Fig. 9. This figure shows that when η increases then W class increases. Fig.. This figure shows that when η increases then P class decreases.

9 Debashis Biswas, amares Pal / Int. J. Adv. Appl. Math. and Mech. ( ( Conclusion In this paper, we have proposed and analyzed a non-linear HIV/AID epidemic model with vaccination and antiretroviral therapy and studied its stability analysis. By analyzing the mathematical model, first we have found a basic reproduction number R because it plays an important role. It is noted that when R < then disease dies out and R when R > then disease become endemic. The model has two non-negative equilibria namely, E = ( s (µ+σ,,,, the disease free equilibrium and E = (,U,W,P, the endemic equilibrium. It is noted that the equilibrium state E corresponding to disappearance of disease is locally as well as globally asymptotically stable if R < and unstable if R >. The endemic equilibrium E, which exists only when R > is locally asymptotically stable by using Jacobian matrix and globally asymptotically stable which proof by using Laalle invariant principle. If there is no vaccination then R reduces to R = βr s µ(µ+γ+η and if there is no antiretroviral therapy then R reduces to R = βr s µ(µ+γ. Therefore we say that R < R < R which ensure that the disease spread among the population rapidly in a very short. Therefore vaccination and antiretroviral therapy must decrease the disease transmission. We also say that if we increase the vaccination rate then U, W and P class are decreases. How ever if we increase the antiretroviral therapy rate then U, P class are decrease and W class is also increase. Also if we restrict the migration rate into susceptible community, then the spread of the disease can be kept under control. References [] UNAID Report on the Global AID Epidemic, HIV estimates with uncertainty bounds, 99 -, ( Last Accessed:, st January. [] Centers for Disease Control, Pneumocystis Pneumonia-Los Angeles, Morbidity and Mortality Weekly Report, (98 -. [] Centers for Disease Control, : Update on acquired immune deficiency syndrome (AID-United tates, Morbidity and Mortality Weekly Report ( [] J. Coffin, A. Hasse, J. A. Levy, L., Montagnier,., Oroszlan : Human immunodeficiency viruses, cience,, ( [] Annual report : published by Department of AID Control, Ministry of Health and Family Welfare, Government of India,. [6] L. M. Cai, X. Z. Li : tability analysis of an HIV/AID Epidemic Model with Treatment, Journal of Computational and Applied Mathematics 9(9 -. [7] L. M. Cai,. L. Guo : Analysis of an Extended HIV/AID Epidemic Model with Treatment, Applied Mathematics and Computation 6, ( [8] H. -F. Huo, L. -X., Feng : Global tability for an HIV/AID Epidemic Model with Different Latent tages and Treatment, Applied Mathematical Modeling 7 ( [9] A. M. ELaiw, Global Properties of a Class of HIV Models, Nnolinear Analysis: Real World Applications ( -6. [] D. M. Xiao,. G., Ruan : Global Analysis of an Epidemic Model with Non-Monotone Incidence Rate, Mathematical Biosciences 8 ( [] R. M. Anderson, The role of mathematical models in the study of HIV transmission and the epidemiology of AID, J. AID ( []. Busenberg, K. Cooke, H. Ying-Hen : A model for HIV in Asia, Math. Biosci. 8 ( [] O. Diekmann, P.J. A. Heesterbeek, J. A.J. Metz, On the definition and the basic reproduction ratio R in models for infectious diseases in heterogeneous populations, J. Math. Biol. 8 ( [] K. Dietz, On the transmission dynamics of HIV, Math. Biosci. 9 ( [] Y. -H. Hsieh, C. H. Chen, : Modeling the social dynamics of a sex industry: Its implications for spread of HIV/AID, Bull. Math. Biol. 66 ( -66. [6] National AID Control Organization Country cenario AID, Published by NACO, Ministry of Health, Government of India, NewDelhi,. [7] Ram Naresh, Agraj Tripathi, andip Omar, : Modelling the spread of AID epidemic with vertical transmission, Applied Mathematics and Computation, 78, ( [8] R. O. imwa, G.P., Pokhariyal, : A dynamical model for stage-specific HIV incidences with application to ub- aharan Africa, Applied Mathematics and Computation, Elsevier 6 (. [9]. Issa, E.. Massawe, O. D. Makinde, Modelling the effect of screening on the spread of HIV infection in a Homogeneous population with infective immigrants, cientific Research and Essays (RE ( 97-. [] J.., Montaner, R. Hogg, E. Wood, T. Kerr, M. Tyndall, The case for expanding access to highly active antiretroviral therapy to curb the growth of the HIV epidemic, Lancet 68 (9 (6-6.

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