A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and Cure of Infected Cells in Eclipse Stage

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1 Applied Mathematical Sciences, Vol. 1, 216, no. 43, HIKARI Ltd, A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and Cure of Infected Cells in Eclipse Stage Mehdi Maziane 1 Department of Mathematics and Computer Science, Faculty of Sciences Ben M sik Hassan II University, P.O Box 7955 Sidi Othman, Casablanca, Morocco El Mehdi Lotfi Department of Mathematics and Computer Science, Faculty of Sciences Ben M sik Hassan II University, P.O Box 7955 Sidi Othman, Casablanca, Morocco Marouane Mahrouf Department of Mathematics and Computer Science, Faculty of Sciences Ben M sik Hassan II University, P.O Box 7955 Sidi Othman, Casablanca, Morocco Khalid Hattaf Centre Régional des Métiers de l Education et de la Formation (CRMEF) 234 Derb Ghalef, Casablanca, Morocco Noura Yousfi Department of Mathematics and Computer Science, Faculty of Sciences Ben M sik Hassan II University, P.O Box 7955 Sidi Othman, Casablanca, Morocco Copyright c 216 Mehdi Maziane et al. 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. 1 Corresponding author

2 2122 Mehdi Maziane et al. Abstract In this paper, we propose a delayed human immunodeficiency virus (HIV) infection model with cure of infected cells in the eclipse phase described by ordinary differential equations (ODEs). The disease transmission process is modeled by a specific nonlinear function that generalizes many incidence rates existing in the literature. The delay represents the time needed for a newly productive infected cell to produce new viruses. In addition, the global stability of the infection-free equilibrium and chronic infection equilibrium is established by using the direct Lyapunov method. Moreover, the models and results presented in many previous studies are extended and generalized. Finally, we check our theoretical results with numerical simulations. Keyword: HIV, nonlinear incidence rate, delay, global stability. 1 Introduction The purpose of this work is to investigate the dynamical behavior of the following HIV infection model governed by the ODE model, which given by dt = λ T (t) f ( T (t), V (t) ) V (t) + ρe(t), de = f( T (t), V (t) ) V (t) (µ E + ρ + γ)e(t), di = γe(t) µ II(t), dv = ke mτ I(t τ) µ V V (t). The first equation of (1) describes the dynamics of the concentration of the healthy CD4 + T cells (T ). λ is the recruitment rate and is the death rate of uninfected cells. During infection, the concentration of healthy cells decreases proportionally to the product f(t, V )V. The second equation represents the dynamics of the concentration of the infected cells in eclipse stage (E), i.e. infected cells that are not yet producing virus. These cells die at a rate µ E, revert to the uninfected stage at a rate ρ and become productive cells at a rate γ. The third equation describes the dynamics of the concentration of productive infected cells (I), which die at a rate µ I. The last equation represents the concentration of viruses which are produced by infected cells at a rate k and die at a rate µ V. The delay τ represents the time needed for a productive infected cell to produce virions and e mτ is the probability of surviving from time t τ to time t. The disease transmission process is modeled by Hattaf s incidence rate βt f(t, V ) =, where α 1+α 1 T +α 2 V +α 3 T V 1, α 2, α 3 are constants and β is the (1)

3 Dynamics of HIV infection model 2123 infection rate. Recently, this incidence rate was used in [1, 4, 7, 8]. In [2], Buonomo and Vargas-De-Leon proved the global stability of (1) in the case when α 1 = α 2 = α 3 =. In 214, Hu et al. [5] replaced the bilinear incidence rate βt V by a saturated infection rate, i.e., the case that α 1 = α 2 =, and they investigated the the global stability of equilibria. The paper is organized as follows. The equilibria and the basic reproduction number of (1) is presented in section 2. In section 3, we establish the global stability of the disease-free equilibrium and the chronic infection equilibrium. In section 4, we give some numerical simulations in order to illustrate our theoretical results. Finally, the conclusion of our paper is the section 5. 2 Basic results Let C = C([ τ, ], IR 4 ) be the Banach space of continuous functions mapping from [ τ, ] to IR 4 equipped with the sup-norm. Using the fundamental theory of differential equations [3], we can easily show that there exists a unique solution (T (t), E(t), I(t), V (t)) for system (1) with initial conditions (T, E, I, V ) C. For biological reasons, we assume that the initial conditions satisfy T (s), E (s), I (s), V (s), for s [ τ, ]. (2) Proposition 2.1. The solution of system (1) satisfying condition (2) remains non-negative and bounded for all t. Proof. It is easy to show the positivity of the solution of system (1) with initial conditions satisfying (2). Now, we show the boundedness of solution. We define G(t) = T (t) + E(t) + I(t), then dg = λ T (t) µ E E(t) µ I I(t) (3) λ µg(t), (4) where µ = min(, µ E, µ I ). Hence G(t) max { G(), λ µ}. Therefore T (t), E(t) and I(t) are bounded. On the other hand, we have dv = ke mτ I(t τ) µ V V (t) (5) ke mτ I µ V V (t). (6) Then V (t) max { V (), ke mτ µ V I }.

4 2124 Mehdi Maziane et al. We deduce that V (t) is bounded. This completes the proof. Next, we discuss the existence of equilibria for system (1). For this, we need to define the basic reproduction number of disease which is given by R = e mτ f( λ, )kγ µ I µ V (ρ + µ E + γ) = e mτ λβkγ µ I µ V (λα 1 + )(ρ + µ E + γ). (7) It is easy to see that system (1) has an infection-free equilibrium of the form Q ( λ,,, ), which is the unique steady state when R 1, corresponding to the extinction of free virus. Theorem 2.2. (i) If R 1, then the system (1) has a unique infection-free equilibrium Q ( λ,,, ). (ii) If R > 1, the infection-free equilibrium is still present and the system (1) has a chronic infection equilibrium Q 1 (T 1, E 1, I 1, V 1 ) with T 1 (, λ ), E 1 >, I 1 > and V 1 >. Proof. The equilibria of system (1) satisfy the following system λ T f(t, V )V + ρe =, (8) f(t, V )V (µ E + ρ + γ)e =, (9) γe µ I I =, (1) e mτ ki µ V V =. (11) From (8)-(11), we get E = λ T µ E + γ, I = γ(λ T ) µ I (µ E + γ), V = e mτ kγ(λ T ) µ I µ V (µ E + γ) and f ( T, e mτ kγ(λ T ) µ I µ V (µ E + γ) ) e mτ (ρ + µ E + γ)µ V µ I =. kγ We have E = λ T µ E + γ, which implies that T λ. Then, there is no positive equilibrium when T > λ. Now, we consider the following function defined on the interval [, λ ] by g(t ) = f ( T, e mτ kγ(λ T ) µ I µ V (µ E + γ) ) e mτ (ρ + µ E + γ)µ V µ I. kγ Clearly, g() = emτ (ρ + µ E + γ)µ V µ I <, g( λ µ kγ T ) = emτ (ρ + µ E + γ)µ V µ I (R kγ 1) and g (T ) = f T e mτ kγ f µ I µ V (µ E + γ) V >. Hence, if R > 1, system (1) λ admits a chronic infection equilibrium Q 1 (T 1, E 1, I 1, V 1 ), with T 1 (, ), E 1 >, I 1 > and V 1 >.

5 Dynamics of HIV infection model Global stability of equilibria Now, we establish the global stability of our system (1). Theorem 3.1. The infection-free equilibrium Q is globally asymptotically stable if R 1. Proof. To discuss the global stability of Q for (1), we propose the following Lyapunov functional T f(t, ) W (T, E, I, V ) = T T T f(s, ) ds + ρ(t T + E) 2 2(1 + α 1 T )( + µ E + γ)t + ρ + µ E + γ I + E + emτ µ I (ρ + µ E + γ) V γ kγ + µ I(ρ + µ E + γ) t I(S)dS, γ where T = λ. To simplify, we will use the notation: u(t) = u and u(t τ) = u τ. It is not difficult to show that the functional W is non-negative. By calculating the time derivation of W along the positive solution of system (1), we get t τ dw = ( 1 f(t ) dt, ) dt ρ(t T + E)( f(t, ) + + de ) + ρ + µ E + γ (1 + α 1 T )( + µ E + γ)t γ + de + emτ µ I (ρ + µ E + γ) dv kγ + µ I(ρ + µ E + γ) γ di I µ I(ρ + µ E + γ) I τ. γ Using λ = T, we obtain dw = ( 1 T + ρ ) (T T ) 2 ρ(µ E + γ)e 2 ( + µ E + γ)t 1 + α 1 T (1 + α 1 T )( + µ E + γ)t ρ(t T ) 2 E + emτ µ I µ V (ρ + µ E + γ) (R 1)V (1 + α 1 T )T T kγ (α 2 + α 3 T )V 1 + α 1 T + α 2 V + α 3 T V f(t, )V. Since R 1, we have dw. Further, we can easily verify that the largest compact invariant set in {(T, E, I, V ) dw = } is just the singleton Q. From LaSalle invariance principle [6], we deduce that Q is globally asymptotically stable.

6 2126 Mehdi Maziane et al. Theorem 3.2. The chronic infection equilibrium Q 1 is globally asymptotically stable if 1 < R 1+ [ µ I µ V (µ E + γ) + e mτ α 2 λkγ](µ E + ρ + γ) + e mτ ρα 3 kγλ 2. ρµ I µ V (µ E + ρ + γ)( + α 1 λ) (12) Proof. To study the global stability of Q 1 for (1), we propose the following Lyapunov functional T f(t 1, V 1 ) W 1 (T, E, I, V ) = T T 1 T 1 f(s, V 1 ) ds ρ(1 + α 2 V 1 )(T T 1 + E E 1 ) 2 + 2(1 + α 1 T 1 + α 2 V 1 + α 3 T 1 V 1 )( + µ E + γ)t 1 + f(t 1, V 1 )V 1 γe 1 + µ If(T 1, V 1 )V 1 γe 1 I 1 Φ ( I ) + E1 Φ ( E ) e mτ µ I f(t 1, V 1 )V 1 + I 1 E 1 kγe 1 τ Φ ( I θ I 1 ) dθ, where Φ(x) = x 1 ln(x). Note that Φ has a global minimum at 1 and Φ(1) =. Calculating the time derivative of W 1 along the positive solutions of the system (1) and applying λ = µt 1 + f(t 1, V 1 )V 1 ρe 1, we get dw 1 = ( 1 f(t ) dt 1, V 1 ) dt ρ(1 + α 2 V 1 )(T T 1 + E E 1 )( f(t, V 1 ) + + de ) (1 + α 1 T 1 + α 2 V 1 + α 3 T 1 V 1 )( + µ E + γ)t 1 V 1 Φ ( V V 1 ) + f(t 1, V 1 )V 1 (1 I 1 γe 1 I )di + (1 E 1 E )de + emτ µ I f(t 1, V 1 )V 1 (1 V 1 kγe 1 V )dv + µ If(T 1, V 1 )V 1 γe 1 τ d Φ( I θ I 1 ) dθ [ (1 + α 2 V 1 )(T T 1 ) 2 = ( T 1 ρe 1 ) + T T 1 (1 + α 1 T 1 + α 2 V 1 + α 3 T 1 V 1 ) ρ(e E 1 ) 2 (1 + α 2 V 1 )(µ E + γ) T 1 (1 + α 1 T 1 + α 2 V 1 + α 3 T 1 V 1 )( + µ E + γ) f(t 1, V 1 )V 1 (1 + α 1 T )(α 2 + α 3 T )(V V 1 ) 2 V 1 (1 + α 1 T + α 2 V 1 + α 3 T V 1 )(1 + α 1 T + α 2 V + α 3 T V ) [ f(t 1, V 1 )V 1 φ ( f(t 1, V 1 )) (I 1 E ) ( f(t, V ) V E 1 ) + φ + φ f(t, V 1 ) E 1 I f(t 1, V 1 ) V 1 E ) ]. +φ ( V 1 I τ ) (f(t, V 1 ) + φ V I 1 f(t, V ) ] ρ T + µ E + γ + ρe

7 Dynamics of HIV infection model 2127 If R > 1 and ρe 1 T 1, then dw 1, for all T, E, I, V >. It is not hard to show that the condition ρe 1 T 1 is equivalent to R 1 + [ µ I µ V (µ E + γ) + e mτ α 2 λkγ](µ E + ρ + γ) + e mτ ρα 3 kγλ 2. ρµ I µ V (µ E + ρ + γ)( + α 1 λ) In addition, the largest compact invariant set in {(T, E, I, V ) dw 1 = } is just the singleton Q 1. By LaSalle invariance principle [6], we conclude that Q 1 is globally asymptotically stable. Remark 3.1. We have [ µ I µ V (µ E + γ) + e mτ α 2 λkγ](µ E + ρ + γ) + e mτ ρα 3 kγλ 2 lim ρ ρµ I µ V (µ E + ρ + γ)( + α 1 λ) =, [ µ I µ V (µ E + γ) + e mτ α 2 λkγ](µ E + ρ + γ) + e mτ ρα 3 kγλ 2 lim =. γ ρµ I µ V (µ E + ρ + γ)( + α 1 λ) Suppose that R > 1, then, according to Theorem 3.2, the chronic infection equilibrium Q 1 is globally asymptotically stable if ρ sufficiently small or γ sufficiently large. 4 Numerical simulations In this section, we present some numerical simulations in order to illustrate our theoretical results. Based on the biological parameter estimations presented in [8], we consider system (1) with the following parameter values Λ = 1 cells µl 1 day 1, =.2 day 1, β = µl virion 1 day 1, α 1 =.2, α 2 =.1, α 3 =.1, ρ =.8 day 1, γ =.1 day 1, µ I =.1 day 1, µ E =.5 day 1, k = 1 virion cell 1 day 1, µ V = 3 day 1, m = 2.5 day 1 and τ = 1.2 day. By calculation, we have R = Hence, system (1) has an infection-free equilibrium Q = (5,,, ). By using Theorem 3.1, we see that Q is globally asymptotically stable and the solution of (1) converges to Q (see Fig. 1). In this case, the virus is cleared and the infection dies out. Now, we choose τ =.8 day and we keep the other parameter values. By calculation, we have R = and [ µ I µ V (µ E + γ) + e mτ α 2 λkγ](µ E + ρ + γ) + e mτ ρα 3 kγλ 2 ρµ I µ V (µ E + ρ + γ)( + α 1 λ) =.431. Hence, the condition (12) is satisfied. By applying Theorem 3.2, the unique chronic infection equilibrium Q 1 ( , 7.225, 7.246, ) is globally asymptotically stable, which means that the virus persists in the host and the infection becomes chronic (see Fig. 2). In the figure 3, we keep all the parameter values and we observe the behavior of the HIV population with and without delay.

8 2128 Mehdi Maziane et al. 55 Approximation T 8 Approximation E T(t) 4 35 E(t) Approximation I 2 Approximation V 4 15 I(t) V(t) Figure 1: Shows the global stability of the infection-free equilibrium Q. 5 Approximation T 8 Approximation E T(t) 35 E(t) Approximation I 2 Approximation V I(t) 2 V(t) Figure 2: Shows the global stability of the chronic infection equilibrium Q 1. 5 Conclusion In this paper, we have studied a delayed HIV infection model with cure rate of infected cells in eclipse stage and specific nonlinear incidence rate that in-

9 Dynamics of HIV infection model Approximation T Without dealy With delay 8 6 Approximation E Without dealy With delay T(t) 3 E(t) Approximation I Without dealy With delay 1.5 x Approximation V 14 2 Without dealy With delay I(t) 3 V(t) Figure 3: Shows the dynamics of HIV infiction with and without time delay. cludes the traditional bilinear, the saturated, the Beddington-DeAngelis and the Crowley-Martin incidence rates. The global stability of the model is investigated by constructing suitable Lyapunov functionals. More precisely, the global stability of the disease-free equilibrium Q is characterized by R 1, which means that the virus is cleared and the infection dies out. When R > 1, the chronic infection equilibrium Q 1 is globally asymptotically stable provided that condition (12) is satisfied. However, if the cure rate ρ is sufficiently small or γ is sufficiently large, then the global asymptotic stability of the chronic infection equilibrium could be relaxed to R > 1. In this case, the virus persists in the host. These results show that R is the main threshold parameter to determine if the virus will persist in the population of cells or will die out. On the other hand, the basic reproduction number R is a decreasing function of the delay. Hence, the delay prevents the virus by reducing the value of R to a level lower than one. Moreover, ignoring the delay in a virus dynamics model will overestimate R. References [1] J. Adnani, K. Hattaf, N. Yousfi, Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate, International Journal of Stochastic Analysis, 213 (213), Article ID ,

10 213 Mehdi Maziane et al. [2] B. Buonomo and C. Vargas-De-Leon, Global stability for an HIV-1 infection model including an eclipse stage of infected cells, Journal of Mathematical Analysis and Applications, 385 (212), [3] J.K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Science, Vol. 99, Springer-Verlag, New York, [4] K. Hattaf, N. Yousfi, A. Tridane, Stability analysis of a virus dynamics model with general incidence rate and two delays, Applied Mathematics and Computation, 221 (213) [5] Z. Hu, W. Pang, F. Liao and W. Ma, Analysis of a CD4 + T cell viral infection model with a class of saturated infection rate, Discrete and Continuous Dynamical Systems-Series B, 19 (214), no. 3, [6] J.P. La Salle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, [7] E.M. Lotfi, M. Maziane, K. Hattaf, N. Yousfi, Partial Differential Equations of an Epidemic Model with Spatial Diffusion, International Journal of Partial Differential Equations, 214 (214), Article ID , [8] M. Maziane, E.M. Lotfi, K. Hattaf, N. Yousfi, Dynamics of a Class of HIV Infection Models with Cure of Infected Cells in Eclipse Stage, Acta Biotheoretica, 63 (215), Received: April 12, 216; Published: June 24, 216