Mathematical Analysis of HIV/AIDS Prophylaxis Treatment Model

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1 Applied Mathematical Sciences, Vol. 12, 2018, no. 18, HIKARI Ltd, Mathematical Analysis of HIV/AIDS Prophylaxis Treatment Model F. K. Tireito, G. O. Lawi and C. O. Okaka Department of Mathematics Masinde Muliro University of Science and Technology, Kenya Corresponding author Copyright c 2018 F. K. Tireito, G. O. Lawi and C. O. Okaka. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Prophylaxis (PrEP is considered one of the promising interventions against HIV infection as trials on various groups and sites have reported its significant effectiveness. From the studies done, analysis of the data collected shows that PrEP has the potential to help prevent HIV infections and slow the HIV epidemic. In this paper, a deterministic model that incorporates PrEP is developed to assess the impact of the use PrEP on the transmission of HIV/AIDS. Stability analysis of the model showed that DFE is locally and globally asymptotically stable when R p < 1. The EE exists and is locally and globally asymptotically stable when R p > 1. PrEP effectiveness is achieved if taken effectively, hence lack of adherence to PrEP substantially increases the risk of HIV infection among the high risk individuals. Keywords: HIV/AIDS, PrEP, Sensitivity Analysis 1 Introduction Treatment of HIV/AIDS infection has improved dramatically in the past few decades but is still limited by the development of drug resistance and the inability of current therapies to completely eradicate the virus from an individual. The World Health Organization s (WHO Global Health Sector Strategy on

2 894 F. K. Tireito, G. O. Lawi and C. O. Okaka HIV embraces innovation in the HIV treatment, prescribing, for example, that people at substantial risk of HIV infection should be offered pre-exposure prophylaxis (PrEP as an additional prevention strategy [12]. Thus, a HIV/AIDS model that incorporates the use of PrEP and study its long term solutions is developed in this paper. 2 Model Formulation Pre-exposure prophylaxis (PrEP is a HIV prevention strategy whereby uninfected hosts take drugs that are similar to (or the same as ART in order to reduce their susceptibility to infection [12]. PrEP resembles anti-infection vaccines [2], however, PrEP differs from traditional anti-infection vaccines in an important way: if a host on PrEP becomes infected, the viruses they harbour will immediately be exposed to anti-retroviral drugs resulting in viral clearance. The dynamics of the model is described schematically in Figure 1; S(t I(t A(t d (1- P(t T(t Figure 1: A Flow Diagram of a HIV/AIDS Model Incorporating Prophylaxis Use The corresponding mathematical equations of the schematic diagram can be described by a system of ordinary differential equations; Ṡ(t = µs(t ωs(t βci(ts(t, (t

3 Use of prophylaxis in HIV/AIDS prevention 895 P (t = ωs(t µp (t I(t = βci(ts(t (t (1 ηβci(tp (t, (t (1 ηβci(tp (t (t (ν θ µi(t, T (t = δa(t θi(t (γ µt (t, A(t = γt (t νi(t (d µ δa(t, (1 where; recruitment rate of the susceptibles, µ is the natural mortality rate, β represents the contact rate with HIV, C is the average number of sexual partners, ω represents the PrEP uptake rate, η is the PrEP efficacy, ν represents the rate of progression of HIV infectives to Aids, θ is the ART uptake rate amongst the infected individuals, γ is the rate of progression to AIDS class due to resistance to treatment (drug resistance, δ is the rate at which Aids patients get treatment, and finally d is the disease induced death rate in the Aids patients class. Since the model in Equation (1 describes a human population, the model developed is positive and bounded for all t 0 with; (t = S(t P (t I(t T (t A(t. Thus all the state variables in Equation (1 will remain positive so that the solutions of the model with positive initial conditions will remain positive for all t > 0. Using the next generation matrix approach [4], the next generation matrix of model (1 R p = βc (ν θ µ ( µ ω(1 η µ ω. (2 3 Local Stability Analysis of the Disease Free Equilibrium The model developed in Equation (1 has a disease free equilibrium (DFE given by Σ 0 = (S(0, P (0, I(0, T (0, A(0 = ( Theorem 1. If R p < 1, then E 0 = Γ 0 and is locally asymptotically stable., µω ( µ ω, ω, 0, 0, 0 µ(µ ω (3 ω, 0, 0, 0 is the equilibrium in µ(µω

4 896 F. K. Tireito, G. O. Lawi and C. O. Okaka Proof. Evaluating the Jacobi matrix of Equation (1 at DFE we have µ ω 0 βc 0 0 ω µ (1 ηβc 0 0 J(E 0 = βcs(t (1 ηβcp (t 0 0 (ν θ µ 0 0 (t (t 0 0 θ (γ µ δ 0 0 ν γ (d µ δ (4 To investigate the stability of the DFE, we compute the eigenvalues of Equation (4. µ ω λ 0 βc 0 0 ω µ λ (1 ηβc (ν θ µ[r p 1] λ 0 0 = θ (γ µ λ δ 0 0 ν γ (d µ δ λ (5 Using the Routh Hurwitz criterion [6], the eigenvalues obtained by the matrix (5 are negative since the trace=γ 4µ d δ ω (ν θ µ[r p 1] < 0 and determinant= (ν θ µ[r p 1]µ(µ ω[γ(d µ µ(d µ δ] > 0 when R p < 1. Hence, the endemic equilibrium E is locally asymptotically stable whenever R p < 1. Given a small infective population, each infected individual in the entire period of infectivity will produce on average less than one infected individual when R p < 1. This shows that the disease is eliminated in the population when R p < 1. 4 Global Stability Analysis of the Disease Free Equilibrium In this section the Global asymptotic stability of the disease free equilibrium is analyzed using the theorem by Castillo Chavez et. al. [1]. Theorem 2. The fixed points E 0 = (X, 0 is Globally asymptotically stable provided R p < 1. Proof. Consider F (X, 0 = ( (µ ωs(t, ωs(t µp (t

5 Use of prophylaxis in HIV/AIDS prevention 897 where and A = Ĝ(X(t, Z(t = G(X, Z = AZ Ĝ(X, Z (ν θ µ 0 0 θ (γ µ δ ν γ (d µ δ Ĝ 1 (X(t, Z(t Ĝ 2 (X(t, Z(t = Ĝ 3 (X(t, Z(t ( βc 1 S(tηP (t 0 0 I(t It follows that Ĝ1(X(t, Z(t 0, Ĝ 2 (X(t, Z(t = Ĝ3(X(t, Z(t = 0 thus Ĝ(X(t, Z(t 0. Conditions M 1 and M 2 are satisfied and thus E 0 is Globally Asymptotically Stable for R p < 1. Global asymptotic stability shows that regardless of any starting solution, the solutions of the model will converge to DFE whenever R p < 1. 5 Local Stability Analysis of the Endemic Equilibrium The endemic equilibrium state is the state where the disease cannot be totally eradicated but remains in the population at manageable levels. HIV/AIDS is endemic or persistent in the population if S (t, I (t, T (t, A (t > 0 for all t > 0. To investigate the stability the endemic equilibria of model (1, linearization of model (1 at Endemic equilibria is done. The Jacobian of model (1 at E = (S, P, I, T, A is; J = µ ω βci 0 βcs ω 0 0 µ (1 ηβci (1 η βcp 0 0 (1 η βci a θ (γ µ δ 0 0 ν γ (d µ δ βci where a 1 = βcs (ν θ µ. For stability of system (6, we compute its eigenvalues which involves the solution of the system; µ ω βci λ 0 βcs 0 0 ω µ (1 ηβci λ (1 η βcp 0 0 βci (1 η βci a 1 λ 0 0 = θ (γ µ λ δ 0 0 ν γ (d µ δ λ (1 ηβcp (6

6 898 F. K. Tireito, G. O. Lawi and C. O. Okaka (7 The characteristic equation of (7 is given by; where P (λ = λ 5 Aλ 4 Bλ 3 Dλ 2 Eλ G = 0 (8 A = 4µ ω 2(1 η βci γ d δ βci B = (µ ω (2µ βci δ d (1 η βci β2 C 2 S I 2 µ(d µ δ (γ µ(µ ω βi [ (γ µ [ ( (d µ δ µ ω βci (νγ νµ βci ] (1 η 2 β2 C 2 I P 2 (R p 1(ν θ µ µ (1 η βci ] D = β2 C 2 S I [ 2 3µ (1 η βci ] d δ γ (µ ω ((1 βci ηω βci δθ (µ ω βci (µδ ωδ βδci (ν θ µ(r p 1 θω(1 η βci [ β 2 C 2 S I E = (µ 2 (1 η βci (µ ω βci ] (µ ν θ(r p 1 (d µ δ [ ( µ ω (µ βci (1 η βci (1 η 2 β2 C 2 P I ] 2 (γ µ [ βcs ω µ(d µ δ (1 ηβci (µ ω βci β2 CS I ] 2 [ β 2 C 2 S I G = (γ µ(d µ δ dγµ ωdγ βdγ (γ 2µ d δ [ β 2 ωcs 2 ] ( µ (1 η βci (1 η (µ ω βci [(ν θ µ(r p 1 µ (1 η βci ] ] Thus, the number of possible negative real roots of equation (9 depends on the signs of A, B, D, E&G. This can be analyzed using the Descartes Rules of Signs of the polynomial given by [13]: P (λ = Aλ 4 Bλ 3 Dλ 2 Eλ G (9 From Discartes Rule of Signs [13], the number of negative real zeros of P is either equal to the number of variations in sign of P ( λ or less than this by an even number. Thus, the possibilities of negative roots of Equation (9 is as summarized in Table 1.

7 Use of prophylaxis in HIV/AIDS prevention 899 Table 1: Roots of the Characteristic Equation Cases A B D E G R p > 1 o. of o. of Sign Changes - roots 1 R p > R p > R p > R p > R p > 1 4 4, R p > R p > R p > From Table 4.1, the maximum number of variations of signs in P ( λ is four, hence the characteristic polynomial (9 has four negative roots. Thus, P ( λ = λ 5 Aλ 4 Bλ 3 Dλ 2 Eλ G = 0 (10 has negative roots. Therefore, whenever 1 η > 0 and if cases 1-8 in Table 1 are satisfied, model (1 is locally asymptotically stable if R p > 1. Epidemiologically, given a small infective population, each infected individual in the entire period of infectivity will produce on average less than one infected individual when R p > 1. This shows that persistence of the infection occurs whenever R p > 1. 6 Global Stability Analysis of the Endemic Equilibrium Consider the following Lyapunov function; V = S S ln S P P ln P I I ln I T T ln T A A ln A (11 Differentiating V with respect to time gives V = (1 S Ṡ (1 P P (1 I I (1 T S P I T Replacing Ṡ, P, I, T, A from Equation (1 in (12, we obtain; V = (1 S S [ µs ωs βcis ] (1 I I [ βcis T (1 A Ȧ(12 A (1 ηβcip (ν θ µi]

8 900 F. K. Tireito, G. O. Lawi and C. O. Okaka (1 P P (1 A A [ωs µp (1 η βcip ] [γt νi (d µ δa] (1 T T [δa θi (γ µt ] At the boundary, thus we let = and using the following relations µ µ at the steady state, = µs ωs βµci S, γt νi = (d µ δa, µβcs βµcp = (ν θ µ, δa θi = (γ µt µp = ωs η βµci P and after simplifications, Equation (14 becomes [ V = (1 S µs ωs βµci S µs ωs βµcis ] S [βcisµ (1 I ηβcip µ ] (ν θ µi (1 P [ωs µp (1 η βcip µ ] I P (1 T [δa θi (γ µt ] (1 A [γt νi (d µ δa] T A ( βµci S ( = µs ωs 2 S S S ωs (1 P S (1 βµcis SS S P S (1 PP I (1 ηβµcp I δa ( 1 T T A A I θi (1 T γt ( 1 A A T T I I T νi (1 A A From the property that the geometric mean is less than or equal to the arithmetic mean, the inequality dv = 0 holds iff (S = S, P = P, I = I, T = dt T, A = A. By Lassalles s invariance [ principle [3], every solution ] of the system (1 with initial conditions in (S, P, I, T, A R 5 : approaches µ the endemic equilibrium, thus the endemic equilibrium E is Globally Asymptotically stable. Epidemiologically, this implies that any perturbation of the model by introduction of infectives into the population makes the model to converge to the endemic equilibrium E. 7 Conclusion Pre-exposure prophylaxis (PrEP is considered one of the promising interventions against HIV/AIDS infection as experiments on various groups and I I (14 (13

9 Use of prophylaxis in HIV/AIDS prevention 901 sites have reported its significant effectiveness [4]. Despite the advocacy of behaviour change and scale-up of ART theraphy, new incident infections continues to be a problem. Thus, the importance of combination of prevention approaches to the transmission of HOV/AIDS. From the results, if the reproduction number R p is less than unity then the DFE is locally and globally asymptotically stable, while if R p > 1, the endemic equilibrium is asymptotically stable. This shows that use of PrEP will lower the HIV/AIDS prevalence if used effectively. References [1] C. Castillo-Chavez, F. Zhilan and H. Wenzhan, On the computation of R 0 and its role in global stability, In Mathematical Approaches for Emerging and Reemerging Infectious Diseases, Institute for Mathematics and its Applications, Springer, Vol. 67, 2002, [2] A. Ducrot, P. Magal, Travelling wave solutions of an infection Age structured model with diffusion, Proc. Roy. Soc. Edinburg Sect. A: Mathematics, 139 (2009, [3] J. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, USA, [4] A.Y. Muhammad, Analysis of Spatial Dynamics and Time Delays in Epidemic Models, Diss, University of Sussex, [5] L. Rodgers, S. Chien, HIV/AIDS Epidemic in Kenya: An Overview, Journal of Experimental and Clinical Medicine, 4 (2012, [6] E. Routh, A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion, Macmillan Publishers, [7] J. Silva, M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete contin. Dyna. Syst., 35 (2015, no. 9, [8] UAIDS, Report on the global AIDS epidemic, (2016. [9] W. Wang, M. Wanbiao, L. Xiulan, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, onlinear Analysis: Real World Applications, 33 (2017,

10 902 F. K. Tireito, G. O. Lawi and C. O. Okaka [10] WHO Antiretroviral Theraphy for HIV Infection in Adults and Adolescents, Report, World Health Organization, [11] World Bank Data, Kenya, World Development Indicators. [12] WHO A Guidline on when to Start Antiretroviral Theraphy and on Pre- Exposure Prophylaxis for HIV, Report, World Health Organization, [13] W. Xiaoshen, A simple proof of Descartes s Rule of Signs, Amer. Math. Monthly, 111 (2004, Received: July 2, 2018; Published: August 2, 2018