Global Stability of HIV Infection Model with Adaptive Immune Response and Distributed Time Delay

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1 pplie Mathematical Sciences, Vol. 8, 24, no. 4, HIKRI Lt, Global Stability of HIV Infection Moel with aptive Immune Response an Distribute Time Delay Brahim EL Boukari Department of Mathematics an Computer Science, Faculty of Sciences Ben M sik Hassan II University, P.O Box 7955 Sii Othman, Casablanca, Morocco Khali Hattaf Department of Mathematics an Computer Science, Faculty of Sciences Ben M sik, Hassan II University, P.O Box 7955 Sii Othman, Casablanca, Morocco & Centre Régional es Métiers e l Eucation et e la Formation, Derb Ghalef, Casablanca, Morocco Noura Yousfi Department of Mathematics an Computer Science, Faculty of Sciences Ben M sik, Hassan II University, P.O Box 7955 Sii Othman, Casablanca, Morocco Copyright c B. EL Boukari, K. hattaf an N. Yousfi. This is an open access article istribute uner the Creative Commons ttribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. bstract In this paper, we propose a mathematical moel with five nonlinear ifferential equations escribing the interactions between HIV, CD4 + T cells, an the aaptive immune response represente by cytotoxic T lymphocytes (CTL) cells an the antiboies. In aition, the lag between the time when the virus enters a cell an when the cell becomes infecte is moele by a istribute time elay. The global stability of the steay states o the moel is etermine by using the irect Lyapunov functional metho.

2 654 B. EL Boukari, K. Hattaf an N. Yousfi Mathematics Subject Classification: 34D23, 34K2, 92B5, 92D3 Keywors: aptive immune response, istribute time elay, global stability Introuction Human immunoeficiency virus (HIV) is a lentivirus that causes acquire immunoeficiency synrome (IDS). Nowaays, HIV infection is an important an serious worlwie cause of mortality. Recently, many mathematical moels (see [8,, 5, 4, 6, 2,, ]) have been evelope in orer to unerstan the ynamics of HIV infection. In this paper, we consier the following HIV infection moel with aaptive immune response an istribute time elay governe by nonlinear ifferential equations: ẋ(t) ẏ(t) v(t) ẇ(t) ż(t) = λ x(t) βx(t)v(t), = β f(τ)e mτ τ ay(t) py(t)(t), = ky(t) μv(t) qv(t)w(t), = gv(t)w(t) hw(t), = cy(t)(t) b(t), () where x(t), y(t), v(t), w(t) an (t) enote the concentration of uninfecte cells, infecte cells, free virus particles, antiboy response an CTL response at time t, respectively. The uninfecte cells are prouce at a constant λ, ie a rate x an become infecte by free virus at a rate βxv. Infecte cells are lost at a rate ay an are kille by the CTL response at a rate py. Free viruses are prouce by infecte cells at a rate ky, cleare at a rate uv an are neutralie by antiboies at a rate qvw. ntiboies evelop in response to free virus at a rate gvw an ecay at a rate hw. CTL cells expan in response to viral antigen erive from infecte cells at a rate cy an ecay in the absence of antigenic stimulation at a rate b. Here, we assume that the uninfecte cells are contacte by the virus particles at time t τ becomes infecte cells at time t, where τ is a ranom variable with a probability istribution f(τ) over the interval [,h] an h is limit superior of this elay. This probability istribution is assume to be a positive an integrable function on [,h], an satisfying f(τ)τ =. The term e mτ is the probability of surviving from time t τ to time t, where m is the eath rate for infecte but not yet virus-proucing cells The novelty of our stuy is that it incorporates both humoral an cellular responses an investigates the impact of the istribute elay on the ynamical behavior of HIV infection. The rest of the paper is organie as follows. Section 2 eals with positivity an bouneness of solutions. The equilibria of the moel an five threshol parameters are given in Section 3. In Section 4, we iscuss the global stability of the equilibria. The paper ens with a conclusion in Section 5.

3 HIV infection moel with immunity an istribute elay Positivity an boueness of solutions In this section, we establish the positivity an bouneness of solutions of system (). Let C = C([ h, ], R 5 ) be the Banach space of continuous functions mapping the interval [ h, ] into R 5 with the topology of uniform convergence. By the funamental theory of functional ifferential equations [3], it is easy to show that there exists a unique solution (x(t),y(t),v(t),w(t),(t)) of system () with initial ata (x,y,v,w, ) C. For biological reasons, we assume that the initial ata for system () satisfy: x (s), y (s), v (s), w (s), (s), s [ h, ] (2) Proposition 2. ll solutions of system () subject to conition (2) remains nonnegative an boune for all t [, + ). Proof. From (), we have x(t) = x()e t t (+βv(s))s + λ y(t) = y()e t t (a+p(s))s + v(t) = v()e t (μ+qw(s))s + k βe t ξ e (+βv(s))s t ξ (3) ξ t (a+p(s))s f(τ)e mτ v(ξ τ)x(ξ τ)τξ (4) ξ y(ξ)e (μ+qw(s))s t ξ (5) w(t) = w()e t (6) (t) = ()e t (7) From (3), (6) an (7), we have x(t), w(t) an (t) for all t [, + ). Let t [,h], it is easy to show, from (4)an (5), that y(t) an v(t) for all t [,h]. This metho can be repeate to euce non-negativity of y, an v on the interval [h, 2h] an then on successive intervals [nh, (n+)h], n 2, to inclue all positive times. Next, we show the bouneness of solutions. Let T (t) =x(t)+y(t)+ a aq v(t)+ 2k We have 2kg w(t)+p c (t)+β T (t) =λ x(t) a 2 y(t) aμ 2k v(t) aqh 2gk w(t) pb c (t) mβ t f(τ) e m(t s) x(s)v(s)sτ. t τ t f(τ) e m(t s) x(s)v(s)sτ. t τ Hence, T (t) λ δt(t) with δ = min{, a,μ,h,b,m}. SoT (t) max{ λ,t()}. an T 2 δ is boune. Therefore, all variables of () are boune. This completes the proof.

4 656 B. EL Boukari, K. Hattaf an N. Yousfi 3 Equilibria an threshol parameters In orer to establish the global ynamics of system (), we erive five threshol parameters, that are: R = λkβ aμ, (8) D w = λkg aμh, D = λc ab, (9) H w = R + D w, H = R + D, () where = f(τ)e mτ τ. The above threshol parameters R, D w, D, H w an H represent, respectively, the basic reprouction number, the basic efense number by antiboy response, the basic efense number by CTL response, the half harmonic mean of R an D w, an the half harmonic mean of R an D. ( λ It is no ifficult to show that the system () always a isease-free equilibrium E ),,,,, an four enemic equilibrium points: E 4 ( λ s in [9], we remark that Remark 3. E ( λ, μ R kβ (R ), ) β (R ),,, E 2 ( λ E 3 ( λ H, b R c, kb μc,, a ) p (H ), H w R, hμ kg Hw, h g, μ q (Hw ), ), H w R, b c, h g, μ q (Dw D ), a p ( D H w D w ), ). If R <, E oes not exists an E = E when R =. 2. If H <, E 2 oes not exists an E 2 = E when H =. 3. If H w <, E 3 oes not exists an E 3 = E when H w =. 4. If D H w D w < or Dw <, E D 2 oes not exists. In aition, E 4 = E 3 if D H w D w =, an E 4 = E 2 if Dw =. D

5 HIV infection moel with immunity an istribute elay Global Stability ( In this section, we stuy the global stability of five equilibria E i x i,y i,v i,w i, i ), i {,, 2, 3, 4}. Let the function F (s) =s ln(s), it is clear that F (s) for any s> an F has the global minimum F () =. For each component s i of E i, we efine on R + the following function { Fi (s) = s i F ( s s i ), if s i F i (s) = s, if s i =. Therefore, we have the following main result. Theorem 4. (i) If R, then E is globally asymptotically stable. (ii) R >, H an H w, then E is globally asymptotically stable. (iii) H > an Dw D, then E 2 is globally asymptotically stable. (iv) If H w > an Hw D D w, then E 3 is globally asymptotically stable. (v) If Dw D > an Hw D D w >, then E 4 is globally asymptotically stable. Proof. For each E i, we consier the following Lyapunov functional L i = F i (x)+ F i(y) + β τ ) f(τ)e mτ F i (x(t θ)v(t θ) θτ q + m i F i (v)+m i g F i(w)+ p c F i(), where m i = a k when i {,, 3}, an m i = βx iv i ky i when i {2, 4}. First, we prove (i). Fori =, we get L = ( x x )[λ x(t) βx(t)v(t)] + β f(τ)e mτ τ a y(t) p y(t)(t)+ β f(τ)e mτ [x(t)v(t) ]τ + a aq p [ky(t) μv(t) qv(t)w(t)] + [gv(t)w(t) hw(t)] + [cy(t)(t) b(t)]. k kg c Since x = λ, we have that L = x (x x ) 2 +(βx aμ aqh )v k kg w bp c. = x (x x ) 2 aμ k ( R )v aqh kg w bp c.

6 658 B. EL Boukari, K. Hattaf an N. Yousfi If R then L. L = x = x,w =, =. Using (), we have v = an y =. By LaSalle s invariance principle, we have E is globally asymptotically stable. For i we have L i = ( x i )[λ x(t) βx(t)v(t)] x + ( y i y )[ β f(τ)e mτ τ a y(t) p y(t)(t)] + β f(τ)e mτ [x(t)v(t) +x i v i ln( )]τ xv + m i ( v i v )[ky(t) μv(t) qv(t)w(t)] + m q i g ( w i )[gv(t)w(t) hw(t)] w + p c ( i )[cy(t)(t) b(t)]. Using the fact that λ = x i + βx i v i, we have L i = x (x x i) 2 x i + βx i v i βx i v i x β + βx iv i f(τ)e mτ ln( f(τ)e mτ y i y τ v i y )τ m i ky i xv y i v + m i[μv i + hqw i g ] + [ay i + pb c i]+[ a + m ik p i ]y +[βx i m i μ qm i w i ]v + qm i [v i h g ]w + p [y i b c ]. Using the relation: ln( x(t τ)v(t τ) xv L i )=ln( x(t τ)v(t τ)y i x i v i )+ln( x i )+ln( v iy y x vy i ), we obtain = x (x x i) 2 x i + βx i v i βx i v i x + βx iv i ln( x i x ) βx iv h ( i f(τ)e mτ y i x i v i y ln(y ) i ) τ + x i v i y [ay i + pb c i] v i y + m i ky i y i v + m i[μv i + hqw i g ]+βx iv i ln( v iy ) vy i + [ a + m ik p i ]y +[βx i m i μ qm i w i ]v + qm i [v i h g ]w + p [y i b c ]. For i =, we have m = a, [ay k + pb c ]=m [μv + hqw ]=m g ky = βx v, an [ a + m k p ]=[βx m μ qm w ] =, then L = x (x x ) 2 βx v F ( x x ) βx v h ( ) f(τ)e mτ y F τ y x v βx v F ( v y vy ) aq k + hβ (g )( H w )w p gβ + bkβ (μc )( H kβc ).

7 HIV infection moel with immunity an istribute elay 659 Since F (s), s > an H, Hw then L. If L =. Then x = x, an v y = vy. n from first equation of () we have v = v, an thereafter y = y. n from the other equation of () we have w = an =. By LaSalle s invariance principle [7], we have E is globally asymptotically stable. If i = 2, we get m 2 = βx 2v 2 ky 2, [ay 2 + pb c 2]=m 2 [μv 2 + hqw 2 ]=βx g 2 v 2, an [ a + m 2k p 2 ]=[βx 2 m 2 μ qm 2 w 2 ]= p [y 2 b ] =, then c L 2 = x (x x 2) 2 βx 2 v 2 F ( x 2 x ) βx 2v 2 βx 2 v 2 F ( v 2y ) qβx 2v 2 h [ Dw ]w. vy 2 gky 2 D Since F (s), s > an Dw D then L 2. f(τ)e mτ F ( y 2 y )τ x 2 v 2 If L 2 =. Then x = x 2 an v 2 y = vy 2. n from first equation of () we have v = v 2 an from the other equation of () we have y = y 2, w = an = 2. By LaSalle s invariance principle, E 2 is globally asymptotically stable. If i = 3, we have m 3 = a k, [ay 3 + pb p 3 ]=[βx 3 m 3 μ qm 3 w 3 ]=qm 3 [v 3 h g L 3 = x (x x 3) 2 βx 3 v 3 F ( x 3 x ) βx 3v 3 βx 3 v 3 F ( v 3y ) pb vy 3 c [ Hw D ]. D w c 3]=m 3 [μv 3 + hqw 3 ]=βx g 3 v 3, an [ a + m 3k ] =, then f(τ)e mτ F ( y 3 y )τ x 3 v 3 Since F (s), s > an Hw D then L D w 3. If L 3 =. Then x = x 3 an v 3 y = vy 3. s above, we have y = y 3, v = v 3, w = w 3 an =. By LaSalle s invariance principle, E 3 is globally asymptotically stable. If i = 4, we have m 4 = βx 4v 4 ky 4, [ay 4 + pb c 4]=m 4 [μv 4 + hqw 4 m 4 k p 4 ]=[βx 4 m 4 μ qm 4 w 4 ]=qm 4 [v 4 h g ]= p [y 4 b c L 4 = x (x x 4) 2 βx 4 v 4 F ( x 4 x ) βx 4v 4 βx 4 v 4 F ( v 4y ). vy 4 f(τ)e mτ F ( y 4 y ]=βx g 4 v 4, an [ a + ] =, then )τ x 4 v 4 Since F (s), s >, then L 4. If L 4 =. Then x = x 4 an vy = vy 4. s above, we have y = y 4, v = v 4, w = w 4 an = 4. By LaSalle s invariance principle, E 4 is globally asymptotically stable. This completes the proof of the theorem. 5 Conclusion In this work, we have propose a mathematical moel with istruste time elay an both humoral an cellular immune responses. The global stability of equilibria is completely

8 66 B. EL Boukari, K. Hattaf an N. Yousfi etermine by five threshols parameters. More precisely, we have prove that when R, the isease-free equilibrium E is globally asymptotically stable. When R >, four enemic equilibria appear that are: the immune-free equilibrium E which is globally asymptotically stable if H an Hw ; the enemic equilibrium with CTL response E 2 which is globally asymptotically stable if H > an Dw ; the enemic equilibrium D with antiboy response E 3 which is globally asymptotically stable if H w > an Hw D D w ; the enemic equilibrium with both CTL an antiboy responses E 4 that is globally asymptotically stable if Dw > an Hw D D >. D w From our main result, we conclue that the time istribute elay plays an important role to control the concentration of viral loa by reucing the basic number R (see (8)). In aition, this R is inepenent of the immune parameters that are b, c, p, g, h an q, which means that the aaptive immune response is unable to eliminate the virus, but it can reuce the concentration of viral loa an increase the concentration of healthy CD4 + cells, which can see by comparing the components of the viral loa an those of the concentration of CD4 + before an after the activation of both CTL an antiboy immune response. References []. M. Elaiw,. lhejelan, an M.. lghami, Global Dynamics of Virus Infection Moel with ntiboy Immune Response an Distribute Delays, Discrete Dynamics in Nature an Society 23 (23), 9 pages. [2] B. EL Boukari, K. Hattaf, N. Yousfi, Moeling the Therapy of HIV Infection with CTL Response an Cure Rate, International Journal of Ecological Economics an Statistics, 28 (23) 7. [3] J. Hale an S. M. Veruyn Lunel, Introuction to Functional Differential Equations, Springer-Verlag, New York, (993). [4] K. Hattaf, N. Yousfi, Dynamics of HIV Infection Moel with Therapy an Cure Rate, International Journal of Tomography an Statistics, 6 () (2) [5] K. Hattaf an N. Yousfi, Optimal Control of a Delaye HIV Infection Moel with Immune Response Using an Efficient Numerical Metho, ISRN Biomathematics, (22), oi:.542/22/ pages. [6] K. Hattaf, N. Yousfi, Two optimal treatments of HIV infection moel, Worl Journal of Moelling an Simulation, 8, (22) [7] J.P. LaSalle, The stability of ynamical systems, Regional Conference Series in pplie Mathematics, SIM, Philaelphia, 976

9 HIV infection moel with immunity an istribute elay 66 [8] M. Nowak, C. Bangham, Population ynamics of immune responses to persistent viruses, Science 272 (996) [9] N. Yousfi, K. Hattaf, M. Rachik, nalysis of a HCV moel with CTL an antiboy responses, ppl. Math. Sci. 3 (57) (29) [] Z. Yuan, X. Zou, Global threshol ynamics in an HIV virus moel with nonlinear infection rate an istribute invasion an prouction elays, Mathematical Biosciences, (23), [] H. Zhu an X. Zou, Dynamics of a HIV- infection moel with cell-meiate immune response an intracellular elay, Discrete Contin. Dyn. Syst. Ser. B, 2 (2) (29) Receive: December 8, 23