Research Article Global Stability of Humoral Immunity HIV Infection Models with Chronically Infected Cells and Discrete Delays

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1 Discrete Dynamics in Nature and Society olume 215, Article ID 37968, 25 pages Research Article Global Stability of Humoral Immunity HI Infection Models with Chronically Infected Cells and Discrete Delays A. M. Elaiw and N. A. Alghamdi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 823, Jeddah 21589, Saudi Arabia Correspondence should be addressed to A. M. Elaiw; a m elaiw@yahoo.com Received 13 May 215; Revised 7 August 215; Accepted 25 August 215 Academic Editor: Zizhen Zhang Copyright 215 A. M. Elaiw and N. A. Alghamdi. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the global stability of three HI infection models with humoral immune response. We consider two types of infected cells: the first type is the short-lived infected cells and the second one is the long-lived chronically infected cells. In the three HI infection models, we modeled the incidence rate by bilinear, saturation, and general forms. he models take into account two types of discrete-time delays to describe the time between the virus entering into an uninfected CD4 + cell and the emission of new active viruses. he existence and stability of all equilibria are completely established by two bifurcation parameters, R and R 1. he global asymptotic stability of the steady states has been proven using Lyapunov method. In case of the general incidence rate, we have presented a set of sufficient conditions which guarantee the global stability of model. We have presented an example and performed numerical simulations to confirm our theoretical results. 1. Introduction During last decades, many researchers have developed and analyzed several mathematical models which describe human immunodeficiency virus (HI dynamics (see, e.g., [1 12]. HI mainly targets the CD4 + cells,leadingto Acquired Immunodeficiency Syndrome (AIDS. Most of the HI mathematical models presented in the literature consider only one type of infected cells called short-lived infected cells. However, it was shown that there is another source for the virus which is called long-lived chronically infected cells. his type of cells generates smaller number of viruses than the short-lived infected cells, but it lives longer [4]. he basic HI dynamics model with long-lived chronically infected cells presented in [4] is given by =π (1 ε k d, = (1 ξ(1 ε k b, C =ξ(1 ε k ac, =N b +N C ac c. (1 Here,,C,and are the concentrations of the uninfected CD4 + cells, short-lived infected CD4 + cells, long-lived chronically infected CD4 + cells,andfreevirus particles, respectively. π represents birth rate constant of the uninfected CD4 + cells.k is the infection rate constant. Parameters d, b, a, andc are the death rate constants of uninfected CD4 + cells, short-lived infected CD4 + cells, long-lived chronically infected CD4 + cells, and free viruses, respectively. he fractions ξ and (1 ξ with <ξ<1are the probabilities that, upon infection, an uninfected CD4 + cell will become either long-lived chronically infected or shortlived infected. N and N C denote the average numbers of free virus particles produced in the lifetime of the short-lived infected and long-lived chronically infected cells, respectively. Model (1 incorporates reverse transcriptase inhibitor drugs with drug efficacy ε and ε<1. In model (1, the immune response has not been modeled. he immune response plays an important role in controlling the diseases. In reality, the immune response needs indispensable components to do its job such as antibodies, cytokines, natural killer cells, and cells. he antibody immune response is a part of the adaptive system in which

2 2 Discrete Dynamics in Nature and Society the body responds to pathogens by primarily using the antibodies which are generated by the B cells, while the other part is the Cytotoxic Lymphocytes (CL immune responsewheretheclattacksandkillstheinfectedcells [3]. In malaria disease, the humoral immune response is more effective than the CL immune response. In the virus dynamics literature, several models have considered theeffectofclimmuneresponse[3,13]orthehumoral immune response [14 16]. Obaid and Elaiw [15] proposed the following model which takes into consideration the humoral immune response: =π (1 ε k d, = (1 ξ(1 ε k b, C =ξ(1 ε k ac, =N b +N C ac rz c, Z=gZ μz. Here, the variable Z represents the concentration of B cells. he HI are attacked by the B cells at rate rz. heterms gz and μz represent the proliferation and death rates of the B cells, respectively. In model (2, it is assumed that once the HI contacts the CD4 + cell, it becomes infected producing new viruses. Actually, there exists an intracellular time delay between the time the HI contacts an uninfected CD4 + cell and the time it becomes actively infected CD4 + cell[17]. In the literature, several papers have proposed various HI models with time delays [17 21]. Our aim in this work is to propose three HI dynamics models with two types of infected cells, two types of intracellular delays, and humoral immunity. Bilinear and saturated incidenceshavebeenproposedinthefirstandsecondmodel, respectively, while a general nonlinear incidence rate is proposed in the third model. For each model, we derive two bifurcation parameters, R and R 1,andestablishtheglobal stability using Lyapunov functional. 2. Model with Bilinear Incidence We propose the following HI infection model with humoral immunity, two types of infected cells and two types of intracellular delays: (t =π (1 ε k (t (t d(t, (3 (t =e m 1τ 1 (1 ξ(1 ε k (t τ 1 (t τ 1 (4 b (t, C (t =e m 1τ 1 ξ (1 ε k (t τ 1 (t τ 1 (5 ac (t, (t =e m 2τ 2 N b (t τ 2 N C ac (t τ 3 (6 r(t Z (t c(t, Z (t =g(t Z (t μz(t. (7 (2 Parameter τ 1 represents the time between HI contact with an uninfected CD4 + cell and the cell becoming infected but not yet producer cell. he factor e m 1τ 1 represents the loss of CD4 + cellsduringtheinterval[t τ 1,t]. he parameters τ 2 and τ 3 represent the time necessary for producing new infectious viruses from the short-lived infected and longlived chronically infected cells, respectively. he factors e m 2τ 2 and e m 3τ 3 represent the loss of short-lived and long-lived chronically infected cells during the intervals [t τ 2,t] and [t τ 3,t],respectively.Here,m 1,m 2,andm 3 are positive constants. he initial conditions for system (2 are given by (u =φ 1 (u, (u =φ 2 (u, C (u =φ 3 (u, (u =φ 4 (u, Z (u =φ 5 (u, φ j (u, φ j ( >, u [ τ,, j=1,2,...,5, j=1,2,...,5, where (φ 1 (u, φ 2 (u,...,φ 5 (u C([ τ., R 5 +,wherec is the Banach space of continuous functions and τ = max{τ 1,τ 2,τ 3 }. We note that system (3 (7 with initial conditions (8 has a unique solution satisfying t [22] Positive Invariance Proposition 1. he solution ((t, (t, C (t, (t, Z(t of (3 (7 with initial conditions (8 is nonnegative for t and ultimately bounded. Proof. Assume that (t on [, ρ] for some constant ρ and (t =, t [,ρ].from(3,weget(t =π> and hence (t >,forsomet (t,t +,andsufficiently small >. his leads to a contradiction; therefore, (t >, for all t.from(4,(5,and(6,wehave (t =φ 2 ( e bt +e m 1τ 1 (1 ξ(1 ε t k e b(t η (η τ 1 (η τ 1 dη, C (t =φ 3 ( e at +e m 1τ 1 ξ (1 ε t k e a(t η (η τ 1 (η τ 1 dη, (t =φ 4 ( e t (c+rz(ςdς +e m t 2τ 2 N b e t η (c+rz(ςdς (η τ 2 dη +e m t 3τ 3 N C a e t η (c+rz(ςdς C (η τ 3 dη. (8 (9

3 Discrete Dynamics in Nature and Society 3 his confirms that (t, C (t, and(t, for all t [,τ].byarecursiveargument,wegetthat (t, C (t,and(t,forallt. Moreover, from (7, we obtain Z (t =φ 5 ( e t (μ g(ςdς. (1 Clearly, Z(t, t.now,weletg 1 (t = e m 1τ 1 (t τ 1 + (t + C (t, G 2 (t = (t + (r/gz(t, s 1 = min{d, a, b}, s 2 = min{c, μ}, L 1 = π/s 1, L 2 = (N b+n C al 1 /s 2,and L 3 =gl 2 /r;then, G 1 (t =e m 1τ 1 [π d (t τ 1 ] b (t ac (t πe m 1τ 1 s 1 (e m 1τ 1 (t τ 1 + (t +C (t =πe m 1τ 1 s 1 G 1 (t <π s 1 G 1 (t. (11 Hence, lim sup t G 1 (t L 1.Since(t >, (t, and C (t, then limsup t (t L 1 and lim sup t C (t L 1.Also, G 2 (t =e m 2τ 2 N b (t τ 2 N C ac (t τ 3 c(t rμ Z (t g (e m 2τ 2 N b N C a L 1 s 2 ( (t + r Z (t g <(N b+n C a L 1 s 2 G 2 (t. (12 Hence, lim sup t G 2 (t L 2.Since(t and Z(t are nonnegative, then lim sup t (t L 2 and lim sup t Z(t L 3. herefore, (t, (t, C (t, (t,andz(t are ultimately bounded Steady States. System (3 (7 always admits an uninfected steady state S = (π/d,,,,. Let =π/d.now, we define the basic reproduction number for system (3 (7 as and infected steady state with humoral immune response S 2 = ( 2, 2,C 2, 2,Z 2 : 1 = R, 1 =e m 1τ 1 (1 ξ π (R, b R C 1 =e m 1τ 1 ξπ (R, a R 1 = 2 = d (1 ε k (R, πg gd + (1 ε kμ, 2 =e m 1τ 1 (1 ξ(1 ε kπμ b(dg+(1 ε kμ, C 2 =e m 1τ 1 2 = μ g, ξ (1 ε kπμ a(dg+(1 ε kμ, Z 2 = c r ( dgr dg + (1 ε kμ. (14 We note that 1, 1, C 1,and 1 are positive when R >1and where 2, 2, C 2,and 2 >and Z 2 >when dgr /(dg + (1 εkμ > 1. Now, we define humoral immune response reproduction number R 1 as Clearly, R lemma. R 1 = R 1+(1 ε kμ/dg. (15 > R 1.Fromabove,wecanstatethefollowing Lemma 2. For system (3 (7, one has the following: (i If R 1, then the system has only one positive steady state S. (ii If R 1 1<R, then the system has two positive steady states S and S 1. (iii If R 1 >1, then the system has three positive steady states S, S 1,andS 2. R = (1 ε k [e (m 1τ 1 +m 2 τ 2 (1 ξ N +e (m 1τ 1 +m 3 τ 3 ξn C ]. c (13 he system has two other steady states, infected steady state without humoral immune response S 1 = ( 1, 1,C 1, 1, 2.3. Global Stability Analysis. We establish the global stability of all the steady states of system (3 (7 employing the method of Lyapunov function. Let us define H (θ =θ ln θ. (16 heorem 3. For system (3 (7, if R 1,thenS is GAS.

4 4 Discrete Dynamics in Nature and Society Proof. Define b +η 2 (e m 1τ 1 ξ (1 ε k (t τ 1 (t τ 1 U = H( +η 1 +η 2 C +η 3 +η 4 Z ac +η 3 (e m 2τ 2 N b (t τ 2 N C ac (t τ 3 rz c+η 4 (gz τ 1 +η 5 (t θ (t θ dθ τ 2 +η 6 τ 3 (t θ dθ + η 7 C (t θ dθ, (17 μz+η 5 η 5 (t τ 1 (t τ 1 +η 6 η 6 (t τ 2 +η 7 C η 7 C (t τ 3. (2 where η i, i = 1,...,7, are positive constants satisfying the following equations: hen, e m 1τ 1 (1 ε k((1 ξ η 1 +ξη 2 =η 5, η 6 bη 1 =, η 7 aη 2 =, η 6 η 3 e m 2τ 2 N b=, (18 du =(1 (π d + (1 ε k η 3 c η 4 μz = d ( 2 +cη 3 (R η 4 μz. (21 he solution of (18 is given by η 1 = η 2 = η 3 = η 4 = η 7 η 3 e m 3τ 3 N C a=, gη 4 rη 3 =, η 5 = (1 ε k. N e m 1τ 1 +m 3 τ 3 e m 3τ 3 (1 ξ N +e m 2τ 2ξNC, N C e m 1τ 1 +m 2 τ 2 e m 3τ 3 (1 ξ N +e m 2τ 2ξNC, e m 1τ 1 +m 2 τ 2 +m 3 τ 3 e m 3τ 3 (1 ξ N +e m 2τ 2ξNC, re m 1τ 1 +m 2 τ 2 +m 3 τ 3 g[e m 3τ 3 (1 ξ N +e m 2τ 2ξNC ], η 5 = (1 ε k, η 6 = η 7 = bn e m 1τ 1 +m 3 τ 3 e m 3τ 3 (1 ξ N +e m 2τ 2ξNC, an C e m 1τ 1 +m 2 τ 2 e m 3τ 3 (1 ξ N +e m 2τ 2ξNC. (19 herefore, if R 1,thendU /, forall,, Z >. he solutions of system (3 (7 limit M, thelargestinvariant subset of {du / = } [22]. We note that du / = if and only if (t =, (t =,andz(t =. For each element of M,wehave(t = and Z(t =,; then, (t = and = (t =e m 2τ 2 N b (t τ 2 N C ac (t τ 3. (22 Since (t, C (t, then (t = C (t =. Hence, du / = if and only if (t =, (t =, C (t =, (t =, andz(t =. It follows from LaSalle s invariance principle (LIP that S is GAS when R 1. heorem 4. For system (3 (7, assume that R 1 1<R ; then, S 1 is GAS. Proof. Define U 1 = 1 H( 1 +η 1 1 H( 1 +η 3 1 H( 1 +η 4 Z +η 2 C 1 H(C C1 he values of η i, i=1,...,7,willbeusehroughthepaper. Calculating the derivative of U along the solutions of system (3 (7 and applying π= d,weobtain du =(1 (π (1 ε k d +η 1 (e m 1τ 1 (1 ξ(1 ε k (t τ 1 (t τ 1 τ 1 (t θ (t θ +η H( dθ 1 1 +η 6 1 τ 2 +η 7 C 1 τ 3 H( (t θ 1 dθ H( C (t θ C1 dθ. (23

5 Discrete Dynamics in Nature and Society 5 hen, / is given by Applying π=d 1 +(1 εk 1 1,weget = (1 1 (π (1 ε k d +η 1 (1 =(1 1 (d 1 d+(1 ε 1 (e m 1τ 1 (1 ξ(1 ε k (t τ 1 (t τ 1 k 1 1 (1 1 e m 1τ 1 η 1 (1 ξ(1 ε b +η 2 (1 C 1 C (e m 1τ 1 ξ (1 ε k (t τ 1 (t τ 1 k(t τ 1 (t τ 1 1 +η 1b 1 e m 1τ 1 η 2 ξ (1 ε k (t τ 1 (t τ 1 C 1 C ac +η 3 (1 1 (e m 2τ 2 N b (t τ 2 N C ac (t τ 3 rz c+η 4 (gz μz+η 5 [ (t τ 1 (t τ 1 (24 +η 2 ac 1 bη (t τ aη 2 C (t τ 3 1 +cη 3 1 +rη 3 1 Z μη 4 Z e m 1τ 1 η 1 (1 ξ(1 ε k ( ln ( (t τ 1(t τ 1 ] + η 6 [ ln ( (t τ 1(t τ 1 (t τ ln (t τ 2 ( ] + η 7 [C C (t τ 3 +C 1 ln (C (t τ 3 C ]. Equation(24canbesimplifiedas =(1 1 (π d e m 1τ 1 η 2 ξ (1 ε k ln ( (t τ 1(t τ 1 +η 1 b 1 ln ( (t τ 2 +η 2 ac 1 ln ( C (t τ 3 C. Using the steady state conditions for S 1, (1 ξ(1 ε k 1 1 =e m 1τ 1 b 1, e m 1τ 1 η 1 (1 ξ(1 ε k (t τ 1 (t τ 1 1 ξ (1 ε k 1 1 =e m 1τ 1 ac 1, (27 +η 1 b 1 c 1 =e m 2τ 2 N b 1 +e m 3τ 3 N C ac 1, e m 1τ 1 η 2 ξ (1 ε k (t τ 1 (t τ 1 C 1 C we get (1 εk 1 1 =η 1 b 1 +η 2aC 1 and +η 2 ac 1 bη (t τ (25 = d ( 1 2 +η 1 b 1 (1 1 aη 2 C (t τ 3 1 +cη 3 1 +rη 3 1 Z μη 4 Z +η 2 ac 1 (1 1 +η ln ( (t τ 1(t τ 1 η 1 b 1 (t τ 1 (t τ η 1 b 1 +η 1 b 1 ln ( (t τ 2 η 2 ac 1 (t τ 1 (t τ 1 C C +η 2 ac 1 +η 2 ac 1 ln (C (t τ 3 C. η 1 b 1 1 (t τ 2 1

6 6 Discrete Dynamics in Nature and Society η 2 ac 1 C (t τ 3 1 C 1 +rη 3 1 Z μη 4 Z +η 1 b 1 +η 2aC 1 +η 1 b 1 ln ((t τ 1(t τ 1 +η 2 ac 1 ln ((t τ 1(t τ 1 +η 1 b 1 ln ( (t τ 2 +η 2 ac 1 ln (C (t τ 3 C. Consider the following equalities: ln ( (t τ 1(t τ 1 = ln ( j +ln ((t τ 1(t τ 1 j j j + ln ( j j, ln ( (t τ 1(t τ 1 = ln ( j +ln ((t τ 1(t τ 1 Cj j j C + ln ( jc, C j ln ( (t τ 2 = ln ( (t τ 2 j j +ln ( j j, ln ( C (t τ 3 C = ln ( C (t τ 3 j Cj +ln ( C j j C, Using(29incaseofj=1,weobtain = d ( 1 2 η 2 ac 1 [ 1 ln ( 1 ] j = 1, 2. η 1 b 1 [ 1 ln ( 1 ] (28 (29 η 1 b 1 [(t τ 1(t τ ln ( (t τ 1(t τ ] η 2 ac 1 [(t τ 1(t τ 1 C C ln ( (t τ 1(t τ 1 C C ] η 1 b 1 [ (t τ ln ( (t τ ] η 2 ac (t τ [C C1 ln ( C (t τ 3 1 C1 ] + rη 3 ( 1 μ g Z = d ( 1 2 η 1 b 1 [H ( 1 +H( (t τ 1(t τ H( (t τ ] η 2 ac 1 [H ( 1 +H( (t τ 1(t τ 1 C C +H( C (t τ 3 1 dg+μ(1 ε k C1 ] + rη 3 ( g (1 ε k [R 1 ]Z. (3 We have if R >1,then 1, 1,C 1, 1 >.SinceR 1 1,then for all,,c,,z >,wehave /.Wenotethat / = at S 1.hen,fromLIP,S 1 is GAS. heorem 5. For system (3 (7, assume that R 1 >1;then,S 2 is GAS. Proof. We consider U 2 = 2 H( 2 +η 1 2 H( 2 +η 3 2 H( 2 +η 4 Z 2 H( Z Z 2 +η 2 C 2 H(C C2 τ 1 (t θ (t θ +η H( dθ 2 2

7 Discrete Dynamics in Nature and Society 7 +η 6 2 τ 2 +η 7 C 2 τ 3 H( (t θ 2 dθ H( C (t θ C2 dθ. (31 Function U 2 along the trajectories of system (3 (7 satisfies du 2 =(1 2 (π (1 ε k d +η 1 (1 2 (e m 1τ 1 (1 ξ(1 ε k (t τ 1 (t τ 1 b +η 2 (1 C 2 C (e m 1τ 1 ξ (1 ε k (t τ 1 (t τ 1 ac +η 3 (1 2 (e m 2τ 2 N b (t τ 2 N C ac (t τ 3 rz c+η 4 (1 Z 2 Z (gz μz+η 5 [ (t τ 1 (t τ ln ( (t τ 1(t τ 1 ] + η 6 [ (t τ ln (t τ 2 ( ] + η 7 [C C (t τ 3 +C 2 ln (C (t τ 3 C ]. Using the steady state conditions for S 2, π=d 2 + (1 ε k 2 2, (1 ξ(1 ε k 2 2 =e m 1τ 1 b 2, ξ (1 ε k 2 2 =e m 1τ 1 ac 2, c 2 =e m 2τ 2 N b 2 +e m 3τ 3 N C ac 2 r 2 Z 2, we get (1 εk 2 2 =η 1 b 2 +η 2aC 2 and du 2 = d ( 2 2 +η 2 ac 2 (1 2 η 1 b 2 +η 1 b 2 (1 2 (t τ 1 (t τ η 1 b 2 (32 (33 η 2 ac 2 η 1 b 2 (t τ 1 (t τ 1 C C +η 2 ac 2 2 (t τ 2 2 +η 1 b 2 +η 2aC 2 η 2 ac 2 +η 1 b 2 ln ((t τ 1(t τ 1 +η 2 ac 2 ln ((t τ 1(t τ 1 +η 1 b 2 ln ( (t τ 2 +η 2 ac 2 ln (C (t τ 3 C. Using(29incaseofj=2,weobtain du 2 = d ( 2 2 η 1 b 2 [H ( 2 +H( (t τ 1(t τ C (t τ 3 C 2 +H( (t τ ] η 2 ac 2 [H ( 2 +H( (t τ 1(t τ 1 C C +H( C (t τ 3 2 C2 ]. (34 (35 Since R 1 >1,then 2, 2,C 2, 2,andZ 2 >.Itisobserved that du 2 / = if and only if = 2, = 2, C =C 2,and = 2. herefore, if = 2,then=and (6 becomes =e m 2τ 2 N b 2 +e m 3τ 3 N C ac 2 c 2 r 2 Z, (36 which gives Z=Z 2.Hence,dU 2 / isequaltozeroats 2.he global stability of S 2 follows from LIP. 3. HI Dynamics Model with Saturated Incidence We present an HI infection model with saturated incidence: (1 ε k (t (t (t =π d(t, 1+β(t (t =e m 1τ 1 (1 ξ(1 ε k (t τ 1 (t τ 1 1+β(t τ 1

8 8 Discrete Dynamics in Nature and Society b (t, C (t =e m 1τ 1 ξ (1 ε k (t τ 1 (t τ 1 1+β(t τ 1 ac (t, (t =e m 2τ 2 N b (t τ 2 N C ac (t τ 3 r(t Z (t c(t, Z (t =g(t Z (t μz(t, (37 where β>is the saturation incidence rate constant. Similar to the previous section, one can show that the solutions of the model are nonnegative and bounded Steady States. System (37 always admits an uninfected steady state S = (,,,,,where = π/d.now,we define the basic reproduction number for system (37 as R = (1 ε k [e (m 1τ 1 +m 2 τ 2 (1 ξ N +e (m 1τ 1 +m 3 τ 3 ξn C ]. c (38 he system has two other steady states S 1 ( 1, 1,C 1, 1, and S 2 ( 2, 2,C 2, 2,Z 2,where 1 = βπ [e (m 1τ 1 +m 2 τ 2 (1 ξ N +e (m 1τ 1 +m 3 τ 3 ξn C ]+c ((1 ε k+dβ[e (m 1τ 1 +m 2 τ 2 (1 ξ N +e (m 1τ 1 +m 3 τ 3 ξn C ], 1 =e m 1τ 1 C 1 =e m 1τ 1 1 = (1 ξ cd b((1 ε k+dβ[e (m 1τ 1 +m 2 τ 2 (1 ξ N +e (m 1τ 1 +m 3 τ 3 ξn C ] (R, ξcd a((1 ε k+dβ[e (m 1τ 1 +m 2 τ 2 (1 ξ N +e (m 1τ 1 +m 3 τ 3 ξn C ] (R, d (1 ε k+dβ (R, 2 = π(g+βμ gd + (1 ε kμ + dβμ, (39 2 =e m 1τ 1 C 2 =e m 1τ 1 (1 ξ(1 ε kπμ b(gd+(1 ε kμ + dβμ, ξ (1 ε kπμ a(gd+(1 ε kμ + dβμ, 2 = μ g, Z 2 = c r ( dgr dg + (1 ε kμ + dβμ. We note that 1, 1, C 1,and 1 arepositivewhenr >1.And 2, 2,C 2, 2 >and Z 2 >when dgr /(dg+(1 εkμ > 1. Now, we define another threshold parameter R 1 as R 1 = R 1+((1 ε kμ + dβμ /dg. (4 Clearly, R >R 1.Fromabove,wehavethefollowingresult. Lemma 6. Forsystem(37,onehasthefollowing: (i If R 1, then the system has only one positive steady state S. (ii If R 1 1<R, then the system has two positive steady states S and S 1. (iii If R 1 >1, then the system has three positive steady states S,S 1,andS Global Stability Analysis. In this subsection, we investigate the global stability of system (37 by constructing suitable Lyapunov functionals and applying LaSalle invariance principle. heorem 7. For system (37, if R 1,thenS is GAS.

9 Discrete Dynamics in Nature and Society 9 Proof. Define Proof. Consider U = H( +η 1 +η 2 C +η 3 +η 4 Z τ 1 +η 5 τ 2 +η 6 (t θ (t θ dθ 1+β(t θ τ 3 (t θ dθ + η 7 C (t θ dθ. (41 Calculating the derivative of U along the solutions of system (37, we obtain U 1 = 1 H( 1 +η 1 1 H( 1 +η 3 1 H( 1 +η 4 Z+η τ 1 +η 2 C 1 H(C C1 H( (t θ (t θ(1 + β 1 dθ 1 1 (1 + β (t θ +η 6 1 τ 2 +η 7 C 1 τ 3 H( (t θ 1 dθ H( C (t θ C1 dθ. (44 du =(1 (1 ε k (π 1+β d +η 1 (e m 1τ 1 (1 ξ(1 ε k (t τ 1 (t τ 1 1+β(t τ 1 hen, / is given by =(1 1 (1 ε k (π d (1 + β +η 1 (1 b +η 2 (e m 1τ 1 ξ (1 ε k (t τ 1 (t τ 1 1+β(t τ 1 ac +η 3 (e m 2τ 2 N b (t τ 2 N C ac (t τ 3 rz c+η 4 (gz (42 1 (e m 1τ 1 (1 ξ(1 ε k (t τ 1 (t τ 1 1+β(t τ 1 b +η 2 (1 C 1 C (e m 1τ 1 ξ (1 ε k (t τ 1 (t τ 1 1+β(t τ 1 μz+η 5 1+β η (t τ 1 (t τ β(t τ 1 ac +η 3 (1 1 (e m 2τ 2 N b (t τ 2 (45 +η 6 η 6 (t τ 2 +η 7 C η 7 C (t τ 3. N C ac (t τ 3 rz c+η 4 (gz Collecting terms of (42, we obtain μz+η 5 [ 1+β (t τ 1(t τ 1 (1 + β (t τ 1 du =(1 (π d + (1 ε k 1+β η 3c η 4 μz = [d ( 2 +η 3 cβr 2 1+β +η 4μZ] ( ln ( (t τ 1(t τ 1 (1+β ] (1 + β (t τ 1 +η 6 [ (t τ ln ( (t τ 2 ] +η 7 [C C (t τ 3 +C 1 ln (C (t τ 3 C ]. +cη 3 (R. Simplifying (45, we get It follows that S is GAS when R 1. heorem 8. For system (37, assume that R 1 1<R ;then, S 1 is GAS. =(1 1 (π d + (1 ε k 1 1+β em 1τ 1 η 1 (1 ξ(1 ε k (t τ 1(t τ β(t τ 1

10 1 Discrete Dynamics in Nature and Society +η 1 b 1 e m 1τ 1 η 2 ξ (1 ε k (t τ 1(t τ 1 C 1 1+β(t τ 1 C +η 2aC 1 bη 1 (t τ 2 1 aη 2 C (t τ 3 1 cη 3 +cη 3 1 +rη 3 1 Z μη 4 Z+η (1 + β 1 ln ( (t τ 1(t τ 1 (1+β (1 + β (t τ 1 +η 1 b 1 ln ( (t τ 2 +η 2 ac 1 ln ( C (t τ 3 C. Applying π=d 1 +(1 εk 1 1 /(1 + β 1,weget =(1 1 (d 1 d + (1 ε k 1 1 (1 1 +(1 ε k 1 1+β (46 Using the steady state conditions for S 1, (1 ξ(1 ε k 1 1 =e m 1τ 1 b 1, ξ (1 ε k 1 1 =e m 1τ 1 ac 1, c 1 =e m 2τ 2 N b 1 N C ac 1, we obtain (1 εk 1 1 /(1 + β 1 =η 1 b 1 +η 2aC 1 and = d ( (1 ε k 1 1 (1 + β 1 (1 + β 1 +η 1 b 1 (1 1 +η 2aC 1 (1 1 η 1 b (t τ 1 (t τ 1 1 (1 + β (1 + β (t τ 1 +η 1 b 1 η 2 ac (t τ 1 (t τ 1 C 1 (1 + β C (1 + β (t τ 1 (48 e m 1τ 1 η 1 (1 ξ(1 ε k (t τ 1(t τ β(t τ 1 +η 1 b 1 e m 1τ 1 η 2 ξ (1 ε k (t τ 1(t τ 1 C 1 1+β(t τ 1 C +η 2aC 1 +η 2 ac 1 η 1b 1 η 2 ac 1 (1 ε k 1 1 C (t τ 3 1 C 1 1 (t τ 2 1 +η 1 b 1 +η 2aC 1 1 +rη 3 1 Z μη 4 Z (49 bη 1 (t τ 2 1 aη 2 C (t τ 3 1 cη 3 +cη 3 1 +rη 3 1 Z μη 4 Z e m 1τ 1 η 1 (1 ξ(1 ε k ln ( (t τ 1(t τ 1 (1+β (1 + β (t τ e m 1τ 1 η 2 ξ (1 ε k ln ( (t τ 1(t τ 1 (1+β (1 + β (t τ 1 +η 1 b 1 ln ( (t τ 2 +η 2 ac 1 ln ( C (t τ 3 C. (47 +η 1 b 1 ln ((t τ 1(t τ 1 (1+β (1 + β (t τ 1 +η 2 ac 1 ln ((t τ 1(t τ 1 (1+β (1 + β (t τ 1 +η 1 b 1 ln ( (t τ 2 +η 2 ac 1 ln (C (t τ 3 C. Consider the following equalities: ln ( (t τ 1(t τ 1 (1+β (1 + β (t τ 1 = ln ( j

11 Discrete Dynamics in Nature and Society 11 + ln ( (t τ 1(t τ 1 ( j j j (1 + β (t τ 1 + ln ( j j +ln ( 1+β, 1+β j ln ( (t τ 1(t τ 1 (1+β (1 + β (t τ 1 = ln ( j + ln ( (t τ 1(t τ 1 (1+β j C j j j (1 + β (t τ 1 C + ln ( jc C j ln ( (t τ 2 +ln ( 1+β 1+β j, = ln ( (t τ 2 j j +ln ( j j, ln ( C (t τ 3 C = ln ( C (t τ 3 j Cj +ln ( C j j C, Using (5, when j=1,weget = d ( (1 ε k β ] η 1 b 1 [ 1 j=1,2. (5 [ (1 + β 1 (1 + β 1 ln ( 1 ] η 2aC 1 [ 1 ln ( 1 ] η 1 b 1 [(t τ 1(t τ 1 ( (1 + β (t τ 1 ln ( (t τ 1(t τ 1 ( (1 + β (t τ 1 ] η 1 b 1 [ (t τ ln ( (t τ ] η 2 ac (t τ [C C1 ln ( C (t τ 3 1 C1 ] η 1 b 1 [ 1+β ln ( 1+β ] η 2 ac 1 [ 1+β ln ( 1+β ]+rη 3 ( 1 μ g Z = d ( 1 2 (1 ε k 1 1 β( [ 1 2 ] (1 + β 1 (1+β 1 η 1 b 1 [H ( 1 +H( (t τ 1(t τ 1 ( (1 + β (t τ 1 +H( (t τ H( 1+β ] η 2 ac 1 [H ( 1 +H( (t τ 1(t τ 1 ( C (1 + β (t τ 1 C +H( C (t τ 3 1 C1 +H( 1+β ] +rη 3 ( dg + μ (1 ε k+dβμ [R g (1 ε k+dβg 1 ]Z. (51 We have if R 1 1<R,then /, whereequality occurs at S 1. LIP implies global stability of S 1. heorem 9. For system (37, assume that R 1 >1;then,S 2 is GAS. Proof. We consider U 2 = 2 H( 2 +η 1 2 H( 2 +η 2 C 2 H(C C2 η 2 ac 1 [(t τ 1(t τ 1 ( C (1 + β (t τ 1 C ln ( (t τ 1(t τ 1 ( C (1 + β (t τ 1 C ] +η 3 2 H( 2 +η 4 Z 2 H( Z Z 2 +η τ 1 H( (t θ (t θ(1 + β 2 dθ 2 2 (1 + β (t θ

12 12 Discrete Dynamics in Nature and Society +η 6 2 τ 2 +η 7 C 2 τ 3 H( (t θ 2 dθ H( C (t θ C2 dθ. Calculate du 2 / along the trajectories of system (37 as du 2 =(1 2 (1 ε k (π d 1+β +η 1 (1 2 ( e m 1τ 1 (1 ξ(1 ε k (t τ 1 (t τ 1 1+β(t τ 1 b +η 2 (1 C 2 C ( e m 1τ 1 ξ (1 ε k (t τ 1 (t τ 1 1+β(t τ 1 (52 we get (1 εk 2 2 /(1 + β 2 =η 1 b 2 +η 2aC 2,and du 2 = d ( (1 ε k 2 2 (1 + β 2 (1 + β 2 +η 1 b 2 (1 2 +η 2aC 2 (1 2 η 1 b (t τ 1 (t τ 1 2 (1 + β (1 + β (t τ 1 +η 1 b 2 η 2 ac (t τ 1 (t τ 1 C 2 (1 + β C (1 + β (t τ 1 +η 2 ac 2 η 1b 2 η 2 ac 2 (1 ε k 2 2 C (t τ 3 2 C 2 2 (t τ η 1 b 2 +η 2aC 2 (55 ac +η 3 (1 2 (e m 2τ 2 N b (t τ 2 N C ac (t τ 3 rz c+η 4 (1 Z 2 Z (gz μz+η 5 [ 1+β (t τ 1(t τ 1 1+β(t τ ln ( (t τ 1(t τ 1 (1+β ] (1 + β (t τ 1 +η 6 [ (t τ ln ( (t τ 2 ] +η 7 [C C (t τ 3 +C 2 ln (C (t τ 3 C ]. Using the steady state conditions for S 2, (53 +η 1 b 2 ln ((t τ 1(t τ 1 (1+β (1 + β (t τ 1 +η 2 ac 2 ln ( (t τ 1 (t τ 1 (1+β (1 + β (t τ 1 +η 1 b 2 ln ( (t τ 2 +η 2 ac 2 ln (C (t τ 3 C. Using (5, when j=2,weget du 2 = d ( (1 ε k β ] η 1 b 2 [ 2 [ (1 + β 2 (1 + β 2 ln ( 2 ] η 2aC 2 [ 2 ln ( 2 ] π=d 2 + (1 ε k 2 2, η 1 b 2 [(t τ 1(t τ 1 (1+β (1 + β (t τ 1 (1 ξ(1 ε k 2 2 =e m 1τ 1 b 2, ξ (1 ε k 2 2 =e m 1τ 1 ac 2, (54 ln ( (t τ 1(t τ 1 (1+β (1 + β (t τ 1 ] η 2 ac 2 [(t τ 1(t τ 1 (1+βC C (1 + β (t τ 1 c 2 =e m 2τ 2 N b 2 +e m 3τ 3 N C ac 2 r 2 Z 2, ln ( (t τ 1(t τ 1 (1+βC C (1 + β (t τ 1 ]

13 Discrete Dynamics in Nature and Society 13 η 1 b 2 [ (t τ ln ( (t τ ] η 2 ac (t τ [C C2 ln ( C (t τ 3 2 C2 ] η 1 b 2 [ 1+β ln ( 1+β ] η 2 ac 2 [ 1+β ln ( 1+β ] = d ( 2 2 (1 ε k 2 2 β( [ 2 2 ] (1 + β 2 (1+β 2 η 1 b 2 [H ( 2 +H( (t τ 1(t τ 1 (1+β (1 + β (t τ 1 +H( (t τ H( 1+β ] η 2 ac 2 [H ( 2 +H( (t τ 1(t τ 1 (1+βC C (1 + β (t τ 1 +H( C (t τ 3 2 C2 +H( 1+β ]. (56 Similar to the proof of heorem 5, one can show that S 2 is GAS. 4. Model with General Incidence In the following model, we assume the incidence rate is given by a general function of the concentration of CD4 + cells and viruses: (t =π Φ( (t,(t d, (57 (t =e m 1τ 1 (1 ξ Φ((t τ 1,(t τ 1 (58 b (t, C (t =e m 1τ 1 ξφ ( (t τ 1,(t τ 1 ac (t, (59 (t =e m 2τ 2 N b (t τ 2 N C ac (t τ 3 r(t Z (t c(t, (6 Z (t =g(t Z (t μz(t. (61 he incidence rate is given by Φ((t, (t which is assumed to be continuously differentiable; moreover, it satisfies the following conditions. Condition C1. (i Consider Φ(, > and Φ(, = Φ(, =, for all, (,. (ii Consider Φ(, / >, Φ(, / >, and Φ(, / >, for all, (,. Condition C2. (i Consider Φ(, ( Φ(, /,forall, (,. (ii Consider (d/d( Φ(, / >. he nonnegativity and boundedness of the solutions of model (57 (61 can be shown as given in Section Steady States Lemma 1. Suppose that Conditions C1 and C2 are satisfied; then, there exist two bifurcation parameters R and R 1 with R >R 1 >.Moreover, (i if R 1, then the system has only one positive steady state S, (ii if R 1 1<R, then the system has two positive steady states S and S 1, (iii if R 1 >1, then the system has three positive steady states S, S 1,andS 2. Proof. Let π Φ(, d=, (62 e m 1τ 1 (1 ξ Φ (, b =, (63 e m 1τ 1 ξφ (, ac =, (64 e m 2τ 2 N b N C ac rz c=, (65 gz μz =. (66 Equation (66 admits two solutions, Z = and = μ/g. Substituting Z=into (63 and (64, we obtain and C as = e m 1τ 1 (1 ξ Φ (,, b C = e m 1τ 1 ξφ (,. a Substituting (67 into (65, we get [e (m 1τ 1 +m 2 τ 2 (1 ξ N +e (m 1τ 1 +m 3 τ 3 ξn C ]Φ(, c=. (67 (68

14 14 Discrete Dynamics in Nature and Society Using Condition C1, we have = is a solution of (68. herefore, = C = and = which leads to S = (,,,,.If =, then, from (62 and (68, we obtain = γφ (, c = γ (π d c = c dγ, (69 where γ=[e (m 1τ 1 +m 2 τ 2 (1 ξn +e (m 1τ 1 +m 3 τ 3 ξn C ].hen, (68 becomes γφ ( c, c=. (7 dγ Let us define a function Ψ 1 as Ψ 1 ( =γφ( c, c =. (71 dγ Condition C1 implies that Ψ 1 ( =, andwhen= = πγ/c >,thenψ 1 ( = c <.Wehave Ψ c Φ ( 1 ( =γ[, dγ + Φ (, ] c. (72 From Condition C1, we have Φ(,/ =;then, Ψ 1 ( =γ Φ (, c=c( γ c Φ (,. (73 herefore, if Ψ 1 ( >,thatis,(γ/c( Φ(, / > 1,then there exists 1 (, such that Ψ 1 ( 1 =.From(62,we define a function Ψ 2 as Ψ 2 ( =π d Φ(, 1 =. (74 Using Condition C1, we have Ψ 2 ( = π > and Ψ 2 ( = Φ(, 1 <.SinceΨ 2 is a strictly decreasing function, then there exists a unique 1 (, such that Ψ 2 ( 1 =. Itfollowsthat 1 = e m 1τ 1 (1 ξφ( 1, 1 /b > and C 1 =e m 1τ 1 ξφ( 1, 1 /a >. It means that an infected steady state without humoral immune response S 1 ( 1, 1,C 1, 1, exists when (γ/c( Φ(, / > 1. Now,wedefinethe parameter R as R = γ Φ (,. (75 c he other possibility of (66 is 2 =μ/g.insert 2 in (62 and define Ψ 3 as Ψ 3 ( =π d Φ(, 2 =. (76 Applying Condition C1, then Ψ 3 is a strictly decreasing function. Moreover, Ψ 3 ( = π > and Ψ 3 ( = Φ(, 2 <. hus, there exists a unique 2 (, such that Ψ 3 ( 2 =. It follows from (63 (65 that 2 = e m 1τ 1 (1 ξ Φ( 2, 2, b C 2 = e m 1τ 1 ξφ ( 2, 2, a Z 2 = c r [γφ ( 2, 2 c 2 ]. (77 hus, 2 > and C 2 > ;moreover,z 2 > when γφ( 2, 2 /c 2 >1. Now, we define parameter R 1 as R 1 = γφ ( 2, 2 c 2. (78 Hence, Z 2 can be rewritten as Z 2 = (c/r(r 1.Itfollows that S 2 ( 2, 2,C 2, 2,Z 2 exists when R 1 >1. Conditions C1 and C2 imply that R 1 = γφ ( 2, 2 c 2 =R. γ c 4.2. Global Stability Analysis Φ ( 2, < γ c Φ (, (79 heorem 11. Let Conditions C1 and C2 be satisfied and R 1; then, S for system (57 (61 is GAS. Proof. Define Φ( U = lim, + Φ(ρ, dρ+ η 1 +η 2 C hen, du τ 1 +η 3 +η 4 Z+ Φ ( (t θ,(t θ dθ τ 2 +η 6 τ 3 (t θ dθ + η 7 C (t θ dθ. Φ( =(1 lim, (π d Φ(, + Φ (, +η 1 (e m 1τ 1 (1 ξ Φ((t τ 1,(t τ 1 b +η 2 (e m 1τ 1 ξφ ( (t τ 1,(t τ 1 ac +η 3 (e m 2τ 2 N b (t τ 2 N C ac (t τ 3 rz c+η 4 (gz μz+φ(, Φ ( (t τ 1,(t τ 1 +η 6 η 6 (t τ 2 +η 7 C η 7 C (t τ 3. Equation(81canbesimplifiedas du =(1 lim + Φ(, Φ (, (π d +Φ(, lim + Φ(, Φ (, cη 3 η 4 μz (8 (81

15 Discrete Dynamics in Nature and Society 15 =π(1 Φ (,/ Φ (, / (1 +Φ(, Φ (,/ Φ (, / cη 3 η 4 μz π(1 Φ (,/ Φ (, / (1 + Φ (, cη 3 η 4 μz =π(1 Φ (,/ Φ (, / (1 +cη 3 ( 1 Φ (, η cη 3 4 μz =π(1 Φ (,/ Φ (, / (1 and using inequalities (85 and (86, we obtain sgn( 1 2 =sgn( 2 1, and this leads to contradiction. herefore, sgn( 1 2 =sgn( 2 1.FromtheconditionsofS 1,we have γφ( 1, 1 /c 1 =1;then, R 1 = γ c ( 1 2 (Φ ( 2, 2 Φ( 1, (Φ ( 1, 2 1 Φ( 1, 1 2. (9 Using inequalities (85 and (88, we obtain sgn(r 1 = sgn( 1 2. heorem 13. Assume that R 1 1<R and Conditions C1 C3 are satisfied; then, S 1 for system (57 (61 is GAS. Proof. Let us define +cη 3 (R η 4 μz. (82 Based on Condition C2, the first term of (82 is less than or equal to zero. herefore, if R 1,thendU /, forall,, Z >. Similar to the previous sections, one can show that S is GAS. Condition C3. Consider Φ (, ( Φ(, i (1 Φ(, i i Φ (,,, (,, i = 1,2. (83 Lemma 12. Suppose that R >1andConditions C1 C3 hold. hen, 1, 2, 1, 2,C 1,C 2, 1, 2 exist satisfying sgn (R 1 =sgn ( 2 1 =sgn ( 1 2. (84 Proof. From Condition C1, for 1, 2, 1, 2 >,wehave (Φ ( 2, 2 Φ( 1, 2 ( 2 1 >. (85 (Φ ( 1, 2 Φ( 1, 1 ( 2 1 >, (86 Applying Condition C3 when i=1,= 1,and= 2,we get (Φ ( 1, 2 1 Φ( 1, 1 2 (87 (Φ( 1, 2 Φ( 1, 1. It follows from inequality (86 that (Φ ( 1, 2 1 Φ( 1, 1 2 ( 1 2 >. (88 We assume that sgn( 2 1 =sgn( 2 1.hen,wehave (π d 2 (π d 1 =Φ( 2, 2 Φ( 1, 1 =Φ( 2, 2 Φ( 1, 2 +Φ( 1, 2 Φ( 1, 1, (89 U 1 Φ( = 1 1, 1 1 Φ(ρ, 1 dρ+ η 1 1 H( hen, +η 2 C 1 H(C C1 +η 3 1 H( +η 4 Z 1 τ 1 Φ ( (t θ,(t θ +Φ( 1, 1 H( dθ Φ( 1, 1 +η 6 1 τ 2 +η 7 C 1 τ 3 H( (t θ 1 dθ H( C (t θ C1 dθ. 1 =(1 Φ( 1, 1 (π d Φ(, Φ(, 1 +η 1 (1 1 (e m 1τ 1 (1 ξ Φ((t τ 1,(t τ 1 b +η 2 (1 C 1 C (e m 1τ 1 ξφ ( (t τ 1,(t τ 1 ac +η 3 (1 1 (e m 2τ 2 N b (t τ 2 N C ac (t τ 3 rz c+η 4 (gz μz+φ(, Φ((t τ 1,(t τ 1 (91

16 16 Discrete Dynamics in Nature and Society +Φ( 1, 1 ln Φ((t τ 1,(t τ 1 Φ (, +η 6 [ +η 2 ac 1 η 1b (t τ 2 1 η 2 a (t τ ln (t τ 2 ( ] + η 7 [C C (t τ 3 +C 1 ln (C (t τ 3 C ]. Collecting terms of (92, we obtain =(1 Φ( 1, 1 Φ(, 1 (π d +Φ(, Φ( 1, 1 Φ(, 1 e m 1τ 1 η 1 (1 ξ Φ((t τ 1,(t τ 1 1 +η 1 b 1 (92 C (t τ 3 1 cη 3 +cη 3 1 +η 3 r 1 Z η 4 μz + Φ ( 1, 1 e m 1τ 1 η 1 (1 ξ ln ( Φ((t τ 1,(t τ 1 Φ (, +Φ( 1, 1 e m 1τ 1 η 2 ξ ln ( Φ((t τ 1,(t τ 1 +η Φ (, 1 b 1 ln ( (t τ 2 +η 2 ac 1 ln ( C (t τ 3 C. Using the steady state conditions for S 1, (94 e m 1τ 1 η 2 ξφ ( (t τ 1,(t τ 1 C 1 C +η 2 ac 1 bη (t τ (93 (1 ξ Φ( 1, 1 =e m 1τ 1 b 1, ξφ ( 1, 1 =e m 1τ 1 ac 1, c 1 =e m 2τ 2 N b 1 +e m 3τ 3 N C ac 1, (95 C (t τ aη η 4 μz cη 3 +cη 3 1 +η 3 r 1 Z then we have Φ( 1, 1 =η 1 b 1 +η 2aC 1 and +Φ( 1, 1 ln ( Φ((t τ 1,(t τ 1 Φ (, +η 1 b 1 ln ( (t τ 2 +η 2 ac 1 ln (C (t τ 3 C. Applying π=d 1 +Φ( 1, 1,weget =d 1 (1 Φ( 1, 1 Φ(, 1 (1 1 +Φ( 1, 1 Φ (, Φ(, 1 +η 1 b 1 (1 Φ( 1, 1 Φ(, 1 +η 2 ac 1 (1 Φ( 1, 1 Φ(, 1 =(1 Φ( 1, 1 Φ(, 1 (d 1 d+φ( 1, 1 (1 Φ( 1, 1 Φ(, 1 +Φ(, Φ( 1, 1 Φ(, 1 e m 1τ 1 η 1 (1 ξ Φ((t τ 1,(t τ 1 1 +η 1b 1 e m 1τ 1 η 2 ξφ ( (t τ 1,(t τ 1 C 1 C η 1 b 1 +η 1 b 1 η 2 ac 1 +η 2 ac 1 η 1b 1 η 2 ac 1 Φ((t τ 1,(t τ 1 1 Φ( 1, 1 Φ((t τ 1,(t τ 1 C 1 Φ( 1, 1 C C (t τ 3 1 C 1 (t τ η 1 b 1 +η 2aC 1

17 Discrete Dynamics in Nature and Society 17 Φ( 1, 1 1 +η 3 r( 1 μ g Z +η 1 b 1 ln (Φ((t τ 1,(t τ 1 Φ (, +η 2 ac 1 ln (Φ((t τ 1,(t τ 1 Φ (, +η 1 b 1 ln ( (t τ 2 +η 2 ac 1 ln (C (t τ 3 C. Using the following equalities, ln ( Φ((t τ 1,(t τ 1 Φ (, = ln ( Φ((t τ 1,(t τ 1 j Φ( j, j + ln ( Φ( j, j Φ(, j +ln (Φ(, j Φ (, j + ln ( j j, ln ( Φ((t τ 1,(t τ 1 Φ (, = ln ( Φ((t τ 1,(t τ 1 C j Φ( j, j C + ln ( Φ( j, j Φ(, j ln (Φ(, j Φ (, j + ln ( jc, C j ln ( (t τ 2 = ln ( (t τ 2 j j +ln ( j j, ln ( C (t τ 3 C = ln ( C (t τ 3 j Cj +ln ( C j j C, j = 1, 2, (96 (97 in case of j=1,weobtain =d 1 (1 Φ( 1, 1 Φ(, 1 (1 1 +Φ( 1, 1 Φ (, [ Φ(, Φ(, 1 Φ (, 1 ] η 1 b 1 [Φ( 1, 1 Φ(, 1 ln ( Φ( 1, 1 Φ(, 1 ] η 2aC 1 [Φ( 1, 1 Φ(, 1 ln ( Φ( 1, 1 Φ(, 1 ] η 1 b 1 [Φ((t τ 1,(t τ 1 1 Φ( 1, 1 ln ( Φ((t τ 1,(t τ 1 1 Φ( 1, 1 ] η 2 ac 1 [Φ((t τ 1,(t τ 1 C 1 Φ( 1, 1 C ln ( Φ((t τ 1,(t τ 1 C 1 Φ( 1, 1 C ] η 1 b 1 [ (t τ ln ( (t τ ] η 2 ac (t τ [C C1 ln ( C (t τ 3 1 C1 ] η 1 b 1 [Φ(, 1 Φ (, 1 ln ( Φ(, 1 Φ (, 1 ] η 2 ac 1 [Φ(, 1 Φ (, 1 ln ( Φ(, 1 Φ (, 1 ] +rη 3 ( 1 μ g Z=d 1 (1 Φ( 1, 1 Φ(, 1 (1 Φ (, +Φ( 1, 1 ( 1 Φ(, 1 (1 1 Φ(, 1 Φ (, η 1b 1 [H (Φ( 1, 1 Φ(, 1 +H( Φ((t τ 1,(t τ 1 1 Φ( 1, 1 +H( (t τ H( Φ(, 1 ] Φ (, 1

18 18 Discrete Dynamics in Nature and Society η 2 ac 1 [H (Φ( 1, 1 Φ(, 1 +H( Φ((t τ 1,(t τ 1 C 1 Φ( 1, 1 C +H( C (t τ 3 1 C1 +H( Φ(, 1 ] Φ (, 1 +rη 3 ( 1 μ g Z. (98 Since R 1 1, then Conditions C1 and C3 and Lemma 12 imply that /, forall,,c,,z >,wherethe equality occurs at S 1. LIP implies that S 1 is GAS. heorem 14. Let R 1 >1and Conditions C1 C3 be satisfied; then, S 2 forsystem(57 (61isGAS. Proof. We consider a Lyapunov functional: Φ( U 2 = 2 2, 2 2 Φ(ρ, 2 dρ+ η 1 2 H( +η 2 C 2 H(C C2 +η 3 2 H( 2 2 Z 2 (gz μz+φ(, Z Φ ( (t τ 1,(t τ 1 + Φ ( 2, 2 ln ( Φ((t τ 1,(t τ 1 +η Φ (, 6 [ (t τ ln (t τ 2 ( ] + η 7 [C C (t τ 3 +C 2 ln (C (t τ 3 C ]. Using the steady state conditions for S 2, π=d 2 +Φ( 2, 2, (1 ξ Φ( 2, 2 =e m 1τ 1 b 2, ξφ ( 2, 2 =e m 1τ 1 ac 2, c 2 =e m 2τ 2 N b 2 +e m 3τ 3 N C ac 2 r 2 Z 2, we get Φ( 2, 2 =η 1 b 2 +η 2aC 2,and (1 (11 +η 4 Z 2 H( Z Z 2 +Φ( 2, 2 τ 1 Φ ( (t θ,(t θ H( dθ Φ( 2, 2 (99 du 2 =d 2 (1 Φ( 2, 2 Φ(, 2 (1 2 +Φ( 2, 2 Φ (, Φ(, 2 +η 6 2 τ 2 H( (t θ 2 dθ +η 1 b 2 (1 Φ( 2, 2 Φ(, 2 +η 7 C 2 τ 3 H( C (t θ C2 dθ. +η 2 ac 2 (1 Φ( 2, 2 Φ(, 2 hen, du 2 =(1 Φ( 2, 2 (π d Φ(, Φ(, 2 η 1 b 2 +η 1 b 2 Φ((t τ 1,(t τ 1 2 Φ( 2, 2 +η 1 (1 2 (e m 1τ 1 (1 ξ Φ((t τ 1,(t τ 1 η 2 ac 2 Φ((t τ 1,(t τ 1 C 2 Φ( 2, 2 C b +η 2 (1 C 2 C (e m 1τ 1 ξφ ( (t τ 1,(t τ 1 +η 2 ac 2 η 1b 2 η 2 ac 2 C (t τ 3 2 C 2 (t τ η 1 b 2 +η 2aC 2 ac +η 3 (1 2 (e m 2τ 2 N b (t τ 2 N C ac (t τ 3 c rz+η 4 (1 Φ( 2, 2 2 +η 1 b 2 ln (Φ((t τ 1,(t τ 1 Φ (,

19 Discrete Dynamics in Nature and Society 19 +η 2 ac 2 ln (Φ((t τ 1,(t τ 1 Φ (, +η 1 b 2 ln ( (t τ 2 +η 2 ac 2 ln (C (t τ 3 C. Using (97, when j=2,weget du 2 =d 2 (1 Φ( 2, 2 Φ(, 2 (1 2 Φ (, +Φ( 2, 2 [ Φ(, 2 + Φ(, 2 ] 2 Φ (, 2 η 1 b 2 [Φ( 2, 2 Φ(, 2 ln (Φ( 2, 2 Φ(, 2 ] η 2 ac 2 [Φ( 2, 2 Φ(, 2 ln (Φ( 2, 2 Φ(, 2 ] (12 η 1 b 2 [H (Φ( 2, 2 Φ(, 2 +H( Φ((t τ 1,(t τ 1 2 Φ( 2, 2 +H( (t τ H( Φ(, 2 ] Φ (, 2 η 2 ac 2 [H (Φ( 1, 1 Φ(, 1 +H( Φ((t τ 1,(t τ 1 C 2 Φ( 2, 2 C +H( C (t τ 3 2 C2 +H( Φ(, 2 ]. Φ (, 2 It is easy to show that S 2 is GAS. 5. Numerical Simulations (13 In this section, we show an example of the general model (5- (57 where Conditions C1 C3 can be satisfied: η 1 b 2 [Φ((t τ 1,(t τ 1 2 Φ( 2, 2 ln ( Φ((t τ 1,(t τ 1 2 Φ( 2, 2 ] (1 ε k (t (t (t =π d(t 1+ω(t +β(t, (t =e m 1τ 1 (1 ξ (1 ε k (t τ 1(t τ ω (t τ 1 +β(t τ 1 η 2 ac 2 [Φ((t τ 1,(t τ 1 C 2 Φ( 2, 2 C ln ( Φ((t τ 1,(t τ 1 C 2 Φ( 2, 2 C ] η 1 b 2 [ (t τ ln ( (t τ ] η 2 ac (t τ [C C2 ln ( C (t τ 3 2 C2 ] η 1 b 2 [Φ(, 2 Φ (, 2 ln ( Φ(, 2 Φ (, 2 ] η 2 ac 2 [Φ(, 2 Φ (, 2 ln ( Φ(, 2 Φ (, 2 ] =d 2 (1 Φ( 2, 2 Φ(, 2 (1 2 +Φ( 2, 2 Φ (, ( Φ(, 2 (1 Φ(, 2 2 Φ (, b (t, C (t =e m 1τ 1 ξ (1 ε k (t τ 1(t τ 1 1+ω(t τ 1 +β(t τ 1 ac (t, (t =e m 2τ 2 N b (t τ 2 N C ac (t τ 3 r(t Z (t c(t, Z (t =g(t Z (t μz(t, (14 where ω, β. In this example, the incidence rate is given by We have the following: Φ (, >, Φ (, = Φ (, =Φ(, =, (1 ε k 1+ω+β. (15 >, >,

20 2 Discrete Dynamics in Nature and Society Φ (, Φ (, Φ (, Φ (, = = = = = (1 ε k (1 + β (1 + ω + β 2 >, (1 ε k (1+ω (1 + ω + β 2 >, (1 ε k 1+ω >, >, (1 ε k (1 ε k 1+ω+β 1+ω Φ (,, >, >, Φ (, ( Φ(, i (1 Φ(, i i Φ (, = β (1+ω ( i 2 i (1 + ω + β i (1+ω+β, >, >, >, >,,>, i=1,2. (16 hen, Conditions C1 C3 are satisfied. he parameters R and R 1 will be R = (1 ε k [e (m 1τ 1 +m 2 τ 2 (1 ξ N +e (m 1τ 1 +m 3 τ 3 ξn C ], c(1+ω R 1 = (1 ε k 2 [e (m 1τ 1 +m 2 τ 2 (1 ξ N +e (m 1τ 1 +m 3 τ 3 ξn C ], c(1+ω 2 +β 2 where 2 = 1 2ω (ω (1+ζ 2 + [(1 + ζ 2 ω ] 2 +4ω (1 + β 2, 2 = μ g (1 ε k, ζ = β+. d (17 (18 he values of some parameters of model (14 are listed in able 1. he remaining parameters of the model will be chosenvaried.withoutlossofgenerality,weletτ e =τ 1 =τ 2 = τ 3.WehaveusedMALABforthenumericalcomputations. Now, we study the following cases. Parameter π d b able 1: he parameters of model (14. alue 1 cells mm 3 day.1 day.3 day ω.5 cells mm 3 β.5 virus mm 3 a N.25 day 1 virus cells ξ.5 N C 5 virus cells r.1 cells mm 3 day μ m 1 m 2 m 3 c.5 day 1day 1day 1day 3day Case 1 (effect of the parameters k and g on the stability of the system. Now, we confirm the results of heorems he evolution of the dynamics of model (14 was observed over a time interval [, 1]. We have chosen the initial conditions as follows: ( = 6, ( = 1, C ( = 5, ( = 1, and Z( = 8. We use three sets for the parameters k and g to get the following three subcases. We fix the values τ e =.5 and ε=: (i R 1. he values of k and g are chosen as k =.2 virus mm 3 day and g =.5 virus mm 3 day. he values of the two bifurcation parameters are computed as R =.367 < 1 and R 1 =.3114 < 1. hismeansthats is GAS. One can see from Figures 1 5 that the numerical results confirm the results of heorem 11. We observe that the states of the system eventually approach the steady state S (1,,,,. Inthiscase,theHIwillbe removed from the blood. (ii R 1 1 < R. We take the values k =.1 virus mm 3 day and g =.5 virus mm 3 day.inthiscase,r 1 =.8291 < 1 < R = According to heorem 13, S 1 is GAS. From Figures 1 5, we can see that there is a consistency between the numerical and theoretical results of heorem 13. Moreover, the states of the system converge to the steady S 1 (12.22, 8.89, 1.67, 8.91,. (iii R 1 >1.Wechoosek =.1 virus mm 3 day and g =.2 virus mm 3 day.hen,wecompute R = > 1 and R 1 = > 1.FromFigures 1 5, we can see that the states of the system approach the infected steady state with humoral immune response S 2 (538.56, 4.66, 5.6, 25., his

21 Discrete Dynamics in Nature and Society Uninfected cells (cells/mm Long-lived infected cells (cells/mm ime (days ime (days R 1 R 1 1<R R 1 >1 R 1 R 1 1<R R 1 >1 Figure 1: he concentration of uninfected CD4 + cells for model (14. Figure 3: he concentration of long-lived chronically infected cells for model (14. Short-lived infected cells (cells/mm ime (days R 1 R 1 1<R R 1 >1 Figure 2: he concentration of short-lived infected cells for model (14. supports the results of heorem 14 that the infected steady state without humoral immune response S 2 is GAS. Case 2 (effect of the parameter ε on the stability of the system. For this case, we take τ e =.5 day, k =.1 virus mm 3 day,andg =.1 virus mm 3 day.in Figures 6 1, we show the effect of the drug efficacy ε on the stability of the steady states of the system. We can see from Figures 6 1 that as the drug efficacy ε is increased, the concentration of uninfected CD4 + cellsisincreased, Free virus (virus/mm ime (days R 1 R 1 1<R R 1 >1 Figure 4: he concentration of free virus particles for model (14. while the concentrations of free virus particles, B cells, shortlived infected cells, and long-lived chronically infected cells are decreased. From these figures and able 2, we can see that the values of R and R 1 are decreased as ε is increased. Using the values of the parameters given in able 1, we obtain the following: (i If ε <.373,thenS 2 exists and it is GAS. (ii If.373 ε <.4455,thenS 1 exists and it is GAS. (iii If.4455 ε<1,thens is GAS. It means that the numerical results and the results of heorems are compatible. In this case, the treatment

22 22 Discrete Dynamics in Nature and Society B cells (cells/mm Short-lived infected cells (cells/mm ime (days ime (days R 1 R 1 1<R R 1 >1 Figure 5: he concentration of B cells for model (14. ε =. ε =.2 ε =.4 ε =.6 Figure 7: he concentration of short-lived infected cells for model (14. 1 Uninfected cells (cells/mm ime (days ε =. ε =.2 ε =.4 ε =.6 Figure 6: he concentration of uninfected CD4 + cells for model (14. Long-lived cells (cells/mm ime (days ε =. ε =.2 ε =.4 ε =.6 able 2: he values of equilibria, R,andR 1 for model (14 with different values of ε. Equilibria R R 1 ε =. E 2 (231.88, 7.77, 9.32, 5, ε =.2 E 2 (337.44, 6.7, 8.4, 5, ε =.373 E 1 (456.34, 5.5, 6.6, 5, ε =.4 E 1 (572.42, 4.32, 5.19, 39.32, ε =.4455 E (1,,,, ε =.6 E (1,,,, ε =.8 E (1,,,, with sufficient drug efficacy can succeed to clear the HI from the plasma. Figure 8: he concentration of long-lived chronically infected cells for model (14. Case 3 (effect of the parameter τ e on the stability of the system. We fix the parameters ε =.4, k =.1 virus mm 3 day,andg =.1 virus mm m 3 day.in Figures 11 15, we show the effect of the time delay parameter τ e on the stability of the steady states of the system. We can see from Figures that the time delay parameter τ e plays a similar role as the drug efficacy parameter ε. Fromthese figures and able 3, we can see that the values of R and R 1 are decreased when τ e is increased. Using the values of the parameters given in able 1, we obtain the following: (i If τ e <.488,thenS 2 exists and it is GAS.

23 Discrete Dynamics in Nature and Society Free virus (virus/mm Uninfected cells (cells/mm ime (days ime (days ε =. ε =.2 ε =.4 ε =.6 τ e =.1 τ e =.3 τ e =.5 τ e =.7 Figure 9: he concentration of free virus particles for model ( Figure 11: he concentration of uninfected CD4 + cells for model ( B cells (cells/mm ime (days Short-lived infected cells (cells/mm ε =. ε =.2 ε =.4 ε =.6 Figure 1: he concentration of B cells for model (14. able 3: he values of equilibria, R,andR 1 for model (14 with different values of τ e. Delay Equilibria R R 1 τ e =. E 2 (476.81, 8.72, 1.46, 5., τ e =.1 E 2 (476.81, 7.89, 9.47, 5, τ e =.3 E 2 (476.81, 6.46, 7.75, 5, τ e =.488 E 1 (476.81, 5.39, 6.47, 5, τ e =.5 E 1 (572.42, 4.32, 5.18, 39.32, τ e =.5394 E (1,,,, τ e =.7 E (1,,,, τ e = 1. E (1,,,, τ e = 2. E (1,,,, (ii If.488 τ e <.5394,thenS 1 exists and it is GAS. (iii If.5394 τ e,thens is GAS ime (days τ e =.1 τ e =.3 τ e =.5 τ e =.7 Figure 12: he concentration of short-lived infected cells for model (14. Figures show that the numerical results are also compatible with the results of heorems Conclusions In this paper, we have proposed three HI infection models with humoral immune response and two types of infected cells. he models incorporate two types of discrete delays. he incidence rate of infection is given by bilinear, saturated functional response and general nonlinear function in the

24 24 Discrete Dynamics in Nature and Society 25 6 Long-lived infected cells (cells/mm B cells (cells/mm ime (days ime (days τ e =.1 τ e =.3 τ e =.5 τ e =.7 τ e =.1 τ e =.3 τ e =.5 τ e =.7 Figure 13: he concentration of long-lived chronically infected cells for model (14. Figure 15: he concentration of B cells for model (14. Acknowledgments Free virus (virus/mm ime (days τ e =.1 τ e =.3 τ e =.5 τ e =.7 Figure 14: he concentration of free virus particles for model (14. first, second, and third model, respectively. We have derived two bifurcation parameters, R and R 1. he global stability of all steady states of the models has been established using Lyapunovmethod.Wehavepresentedanexampleforthe general incidence rate and performed numerical simulations to support our theoretical results. Conflict of Interests he authors declare that there is no conflict of interests regarding the publication of this paper. hispaperwasfundedbythedeanshipofscientificresearch (DSR, King Abdulaziz University, Jeddah. he authors, therefore, acknowledge with thanks DSR technical and financial support. he authors are also grateful to Professor Zizhen Zhang for constructive suggestions and valuable comments, which improve the quality of the paper. References [1] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, vol. 272, no. 5258, pp , [2] P.W.Nelson,J.D.Murray,andA.S.Perelson, AmodelofHI-1 pathogenesis that includes an intracellular delay, Mathematical Biosciences,vol.163,no.2,pp ,2. [3]M.A.NowakandR.M.May,irus Dynamics: Mathematical Principles of Immunology and irology, Oxford University Press, Oxford, UK, 2. [4]D.S.CallawayandA.S.Perelson, HI-1infectionandlow steady state viral loads, Bulletin of Mathematical Biology, vol. 64,no.1,pp.29 64,22. [5]A.M.ElaiwandS.A.Azoz, Globalpropertiesofaclassof HI infection models with Beddington-DeAngelis functional response, Mathematical Methods in the Applied Sciences,vol.36, pp , 213. [6] A. M. Elaiw, Global properties of a class of HI models, Nonlinear Analysis. Real World Applications, vol.11,no.4,pp , 21. [7] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynamics,vol.69,no. 1-2, pp , 212. [8] A. M. Elaiw and A. S. Alsheri, Global dynamics of HI infection of CD4 + cells and macrophages, Discrete Dynamics in Nature and Society,vol.213,ArticleID264759,8pages,213.

25 Discrete Dynamics in Nature and Society 25 [9] P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HI-1 infection, Mathematical Biosciences,vol.179,no.1,pp.73 94,22. [1]Y.Wang,L.Liu,X.Zhang,andY.Wu, Positivesolutionsof an abstract fractional semipositone differential system model for bioprocesses of HI infection, Applied Mathematics and Computation,vol.258,pp ,215. [11] P. Yu, J. Huang, and J. Jiang, Dynamics of an HI-1 infection model with cell mediated immunity, Communications in Nonlinear Science and Numerical Simulation,vol.19,no.1,pp , 214. [12]X.Chen,L.Huang,andP.Yu, Dynamicbehaviorsofaclass of HI compartmental models, Communications in Nonlinear Science and Numerical Simulation, vol.23,no.1 3,pp , 215. [13] X.Y.Zhou,X.Y.Shi,Z.H.Zhang,andX.Y.Song, Dynamical behavior of a virus dynamics model with CL immune response, Applied Mathematics and Computation, vol. 213, no. 2, pp , 29. [14]A.Murase,.Sasaki,and.Kajiwara, Stabilityanalysisof pathogen-immune interaction dynamics, Journal of Mathematical Biology,vol.51,no.3,pp ,25. [15] M.A.ObaidandA.M.Elaiw, Stabilityofvirusinfectionmodels with antibodies and chronically infected cells, Abstract and Applied Analysis,vol.214,ArticleID65371,12pages,214. [16]. Wang, Z. Hu, and F. Liao, Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, Journal of Mathematical Analysis and Applications, vol.411,no.1,pp.63 74,214. [17] N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay, JournalofheoreticalBiology, vol. 226,no.1,pp.95 19,24. [18] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bulletin of Mathematical Biology, vol. 72, no. 6, pp , 21. [19] M. Y. Li and H. Shu, Impact of intracellular delays and targetcell dynamics on in vivo viral infections, SIAM Journal on Applied Mathematics,vol.7,no.7,pp ,21. [2] D. Li and W. Ma, Asymptotic properties of a HI-1 infection model with time delay, Journal of Mathematical Analysis and Applications,vol.335,no.1,pp ,27. [21] Z. Yuan and X. Zou, Global threshold dynamics in an HI virus model with nonlinear infection rate and distributed invasion and production delays, Mathematical Biosciences and Engineering, vol. 1, no. 2, pp , 213. [22] J.K.HaleandS..Lunel,Introduction o Functional Differential Equations, Springer, New York, NY, USA, 1993.

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