Effect of humoral immunity on HIV 1 dynamics with virus to target and infected to target infections Effect of humoral immunity on HIV 1 dynamics with virus to target and infected to target infections[.]
Effect of humoral immunity on HIV-1 dynamics with virus-to-target and infected-totarget infections A M Elaiw, A A Raezah, and A S Alofi Citation: AIP Advances 6, 085204 (2016); doi: 10.1063/1.4960987 View online: http://dx.doi.org/10.1063/1.4960987 View Table of Contents: http://aip.scitation.org/toc/adv/6/8 Published by the American Institute of Physics AIP ADVANCES 6, 085204 (2016) Effect of humoral immunity on HIV-1 dynamics with virus-to-target and infected-to-target infections A M Elaiw,a A A Raezah, and A S Alofi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O Box 80203, Jeddah 21589, Saudi Arabia (Received 26 May 2016; accepted 29 July 2016; published online August 2016) We consider an HIV-1 dynamics model by incorporating (i) two routes of infection via, respectively, binding of a virus to a receptor on the surface of a target cell to start genetic reactions (virus-to-target infection), and the direct transmission from infected cells to uninfected cells through the concept of virological synapse in vivo (infected-to-target infection); (ii) two types of distributed-time delays to describe the time between the virus or infected cell contacts an uninfected CD4+ T cell and the emission of new active viruses; (iii) humoral immune response, where the HIV-1 particles are attacked by the antibodies that are produced from the B lymphocytes The existence and stability of all steady states are completely established by two bifurcation parameters, R (the basic reproduction number) and R (the viral reproduction number at the chronic-infection steady state without humoral immune response) By constructing Lyapunov functionals and using LaSalle’s invariance principle, we have proven that, if R ≤ 1, then the infection-free steady state is globally asymptotically stable, if R ≤ < R 0, then the chronic-infection steady state without humoral immune response is globally asymptotically stable, and if R > 1, then the chronic-infection steady state with humoral immune response is globally asymptotically stable We have performed numerical simulations to confirm our theoretical results C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4960987] I INTRODUCTION Mathematical modeling and analysis of human immunodeficiency virus type-1 have attracted the interest of several researchers during the last decades This virus attacks the immune system and causes Acquired Immunodeficiency Syndrome (AIDS) Mathematical models are helpful to predict the dynamics of HIV-1, understand the interactions of all the components related to the infection of virus and improve diagnosis and treatment strategies.1 Cytotoxic T Lymphocyte (CTL) cells and B cells play central roles in controlling the HIV-1 infection The CTL cells attack the infected cells, while the B cells produce antibodies to attack the HIV-1 particles In the virus dynamics literature, several mathematical models have incorporated CTL immune response2–5 and humoral immune response.6–11 Intracellular time delay discrete or distributed has also been considered in the mathematical models of virus dynamics in several works (see e.g., Refs 13 and 12–18 All of the above mentioned studies assume that the uninfected CD4+ T cells becomes infected due to HIV-1 contacts However, the uninfected CD4+ T cells can become infected due to direct contact with infected cells (see Refs 19–23) The HIV-1 infection model with both infected-to-target and virus-to-target infections includes three state variables, T, T ∗ and V representing the concentrations of the uninfected CD4+ T cells, infected cells, and free HIV-1 particles, respectively.23 ˙ T(t) = ρ − dT(t) − β1T(t)V (t) − β2T(t)T ∗(t), (1) a Email: a_m_elaiw@yahoo.com (A Elaiw) 2158-3226/2016/6(8)/085204/16 6, 085204-1 © Author(s) 2016 085204-2 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) ∞ f 1(s)e−δ1s [ β1T(t − s)V (t − s) + β2T(t − s)T ∗(t − s)] ds − δ1T ∗(t), T˙ ∗(t) = (2) ∞ V˙ (t) = b f 2(s)e−δ2sT ∗(t − s)ds − cV (t), (3) where, ρ and d represent the birth rate and death rate constants of the uninfected CD4+ T cells, respectively The uninfected CD4+ T cells become infected at rate β1T(t)V (t) + β2T(t)T ∗(t), where β1 and β2 are the virus-target and infected-target incidence rate constants, respectively The infected cells and free HIV-1 particles die at rates δ1T ∗ and cV , respectively An infected cell produces an average number b of HIV-1 particles Functions f 1(s) and f 2(s) are two probability distributions where s is a random variable It is assumed that, if the virus or infected cell contact an uninfected CD4+ T cell at time t − s; the cell becomes infected cell at time t The term e−δ1s represents survival rate of the cell during this delay period, where δ1 > is the death or clearance rates of contacted cells before becoming infected.21 In addition, a cell that is infected at time t − s starts to generate new infectious HIV-1 particles at time t, and e−δ2s denotes the survival rate of the infected cell during this delay period, δ2 > is the death or clearance rates of the infected cells before becoming productively infected It is observed that, no HIV infection model with this type of infected-to-target infection has considered the effect of immune response Humoral immunity has been incorporated into virus dynamics models in several works,6–11 however, in these papers, only virus-to-cell transmission has been considered Therefore, reasonable mathematical models for HIV-1 with virus-to-target and infected-to-target infections should take humoral immunity into consideration Model (1)-(3) has two steady states, infection-free steady state and chronic-infection steady state Moreover, the dynamics is governed by only one threshold parameter R (the basic reproduction number) which is defined as the average total number of newly infected cells that arise from any one infected cell in the beginning of the infection.1 In the present paper, a mathematical model for HIV-1 with infected-to-target and virus-to-target infections and humoral immunity is formulated The model has two distributed time delays The model has three steady states, an infection-free steady state, a chronic-infection steady state without humoral immune response, and a chronic-infection steady state with humoral immune response The dynamics are governed by two threshold parameters, the basic reproduction number R and the humoral immunity number R We use Lyapunov functionals and LaSalle’s invariance principle to prove the global stability of all the steady states of the model II MATHEMATICAL MODEL We propose a mathematical model for HIV-1 with virus-to-target and infected-to-target infections and humoral immune response: ˙ T(t) = ρ − dT(t) − β1T(t)V (t) − β2T(t)T ∗(t), ∞ ∗ ˙ T (t) = f 1(s)e−δ1s ( β1T(t − s)V (t − s) + β2T(t − s)T ∗(t − s)) ds − µT ∗(t), ∞ V˙ (t) = b f 2(s)e−δ2sT ∗(t − s)ds − cV (t) − pV (t)W (t), (4) (5) (6) W˙ (t) = kV (t)W (t) − qW (t), (7) where, µ represents the death rate constant of the infected cells W (t) is the concentration of B cells (B lymphocytes) at time t The HIV-1 particles are attacked by the B cells at rate pV (t)W (t) The terms kV (t)W (t) and qW (t) represent the proliferation and death rates of the B cells, respectively All the parameters are positive Let us assume that the probability distribution function f i (s) satisfy f i (s) > 0, i = 1, and ∞ ∞ f i (s)ds = 1, 0 f i (u)eℓu du < ∞,i = 1, 2, 085204-3 Elaiw, Raezah, and Alofi where ℓ > Denote η i = fading memory type ∞ AIP Advances 6, 085204 (2016) f i (s)e−δ i s ds, i = 1, Thus, < η i ≤ Define the Banach space of Cα = {φ ∈ C((−∞, 0], R) : φ(θ)eαϑ is uniformly continuous for ϑ ∈ (−∞, 0] and ∥φ∥ < ∞}, where α is a positive constant and ∥φ∥ = sup |φ(ϑ)| eαϑ Let ϑ ≤0 Cα+ = {φ ∈ Cα : φ(ϑ) ≥ for ϑ ∈ (−∞, 0]} We consider the following initial conditions: T(ϑ) = ϕ1(ϑ),T ∗(ϑ) = ϕ2(ϑ),V (ϑ) = ϕ3(ϑ),W (ϑ) = ϕ4(ϑ) for ϑ ∈ (−∞, 0], ϕi ∈ Cα+, i = 1, , (8) System (4)-(7) with initial conditions (8) has a unique solution.24 A Basic properties Lemma Let (T(t),T ∗(t),V (t),W (t)) be the solution of model (4)-(7) satisfying (8) Then T(t) > 0, T ∗(t) ≥ 0, V (t) ≥ and W (t) ≥ and ultimately bounded Proof From Eq (4), we have T˙ |T =0=ρ > 0, therefore T(t) > for t ∈ (0, ϖ1) where (0, ϖ1) is the maximal interval of existence of solution of system (4)-(7) with (8) Moreover, from Eqs (5)(7), we get T ∗(t) = ϕ2(0)e−µt + t ∞ f 1(s)e−δ s β1T(η − s)V (η − s) + β2T(η − s)T ∗(η − s) dsdη ≥ 0, 0 t t ∞ t − [c+pW (ζ)]dζ − [c+pW (ζ)]dζ V (t) = ϕ3(0)e +b e η f 2(s)e−δ sT ∗(η − s)dsdη ≥ 0, W (t) = ϕ4(0)e −qt+k t e−µ[t−η] V (η)dη ≥ ∞ ρ From Eq (4) we get lim sup T(t) ≤ Let us define F1(t) = f 1(s)e−δ1sT(t − s)ds + T ∗(t) Then t→ ∞ d ∞ F˙1(t) = f 1(s)e−δ1s (ρ − dT(t − s) − β1T(t − s)V (t − s) − β2T(t − s)T ∗(t − s)) ds ∞ + f 1(s)e−δ1s ( β1T(t − s)V (t − s) + β2T(t − s)T ∗(t − s)) ds − µT ∗(t) ∞ = ρη − d f 1(s)e−δ1sT(t − s)ds − µT ∗(t) ( ∞ ) −δ s ∗ ≤ ρ − σ1 f 1(s)e T(t − s)ds + T (t) = ρ − σ1F1(t), where, σ1 = min{d, µ} It follows that, lim supt→ ∞ F1(t) ≤ σρ Since T(t) > and T ∗(t) ≥ 0, then lim supt→ ∞ T ∗(t) ≤ σρ Moreover, let us consider F2(t) = V (t) + kp W (t), then ∞ pq ˙ F2(t) = b f 2(s)e−δ2sT ∗(t − s)ds − cV (t) − W (t) k ( ) ρ p bρ ≤ bη − σ2 V (t) + W (t) ≤ − σ2F2(t) σ1 k σ1 where, σ2 = min{c, q} Hence, lim supt→ ∞ F2(t) ≤ plies that lim supt→ ∞ V (t) ≤ ultimately bounded bρ σ 1σ bρ σ 1σ and lim supt→ ∞ W (t) ≤ The non-negativity of the solution imkbρ pσ 1σ Then, T(t),T ∗(t),V (t) and W (t) are The existence of the steady state of the model (4)-(7) will be shown in the next lemma 085204-4 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) Lemma There exist two bifurcation parameters R and R such that (i) if R ≤ and R ≤ 1, then there exists only positive steady state S0, (ii) if R ≤ and < R 0, then there exist two positive steady states S0 and S1, and (iii) if R > and R > 1, then there exist three positive steady states S0, S1 and S2 The Proof The steady states of system (4)-(7) satisfy the following equations: = ρ − dT − β1TV − β2TT ∗, = η ( β1TV + β2TT ∗) − µT ∗, = η 2bT ∗ − cV − pVW, = kVW − qW (9) (10) (11) (12) Solving Eqs (9)-(12) we find that the system has three steady states, infection-free steady state S0 = (T0, 0, 0, 0), where T0 = dρ , chronic-infection steady state without humoral immune response S1(T1,T1∗,V1, 0) and chronic-infection steady state with humoral immune response S2(T2,T2∗,V2,W2), where bη 2T1∗ T0 dc T1 = , T1∗ = (R − 1), V1 = , R0 b β 1η + c β c µT2∗ −B + (B2 + 4AC q c ∗ , T2 = T2 = , V2 = , W2 = (R − 1) , (13) ∗ 2A k p η β1V2 + β2T2 where A = β2 µk, B = µ(dk + β1q) − ρη β2 k, C = ρη β1q, (14) and T0 β1bη 1η T0 β2η + C R 01 + R 02, µc µ bη 2T2∗ R1 = cV2 R0 = (15) In fact, R 01 measures the average number of secondary infected generation caused by an existing free virus, while R 02 measures the average number of secondary infected generation caused by an infected cell.22 Therefore, R 01 and R 02 are the basic reproduction number corresponding to virus-to-target and infected-to-target infections, respectively, and hence R represent the basic infection reproduction numbers for system (4)-(7) which determines whether or not a chronic-infection can be established.21 The parameter R represents the viral reproduction number at the chronic-infection steady state without humoral immune response We define the basic reproduction number for the humoral immune response R H um which comes from the limiting (linearized) W -dynamics near W = as: R H um = V1 V2 Lemma (i) if R < 1, then R H um < 1, (ii) if R > 1, then R H um > 1, (iii) if R = 1, then R H um = cV2 Proof (i) Let R < 1, then from Eq (15) we have T2∗ < bη , and then using Eq (13) we get √ cV2 −B + B2 + 4AC < , 2A bη which implies that ( 2AcV2 +B bη )2 − (B2 + 4AC) > (16) 085204-5 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) Using Eq (14), we get β2 µ2cqk ( β2c + β1bη 2) V2 (1 − R H um ) > 0, b2η 22 (17) then R H um < Similary, one can proof (ii) and (iii) B Global properties In this section, the global stability of system (4)-(7) will be established by using Lyapunov method and LaSalle’s invariance principle Through the paper we will use the function g(x) = x − − ln x and the notation (T,T ∗,V,W ) = (T(t),T ∗(t),V (t),W (t)) Theorem If R ≤ 1, then S0 is globally asymptotically stable (GAS) Proof Constructing a Lyapunov functional L 0(T,T ∗,V,W ) as the following ( ) T β1T0 p β1T0 ∗ L 0(T,T ,V,W ) = T0g + T∗ + V+ W T0 η1 c kc ∞ s ( β1T(t − ϑ)V (t − ϑ) + β2T(t − ϑ)T ∗(t − ϑ)) dϑds + f 1(s)e−δ1s η1 0 s b β1T0 ∞ −δ s + f 2(s)e T ∗(t − ϑ)dϑds c 0 Calculating the derivative of L along the solutions of system (4)-(7), we obtain ( ) dL T0 (ρ − dT − β1TV − β2TT ∗) = 1− dt T ∞ + f 1(s)e−δ1s ( β1T(t − ϑ)V (t − ϑ) + β2T(t − s)T ∗(t − s)) ds − µT ∗ η1 ∞ β1T0 p β1T0 [kVW − qW ] b + f 2(s)e−δ2sT ∗(t − s)ds − cV − pVW + c kc ∞ + f 1(s)e−δ1s ( β1TV + β2TT ∗ − β1T(t − s)V (t − s) − β2T(t − s)T ∗(t − s)) ds η1 b β1T0 ∞ + f 2(s)e−δ2s [T ∗ − T ∗(t − s)] ds c µ b β1T0η ∗ p β1T0 (T − T0)2 = −d + β2T0T ∗ − T ∗ + T − qW T η1 c kc ( ) (T − T0)2 µ β2T0η b β1T0η 2η ∗ p β1T0 + −1+ T − qW = −d T η1 µ µc kc = −d (T − T0)2 µ p β1T0 (R − 1) T ∗ − + qW T η1 kc dL dt ≤ if R ≤ 1, for dL dt = if and only if T ˙∗ Therefore, all T,T ∗,W > Let D0 = (T,T ∗,V,W ) : (18) dL dt = Clearly from Eq (18), = T0, T ∗ = and W = For each element of D0 we have T = T0 ∗ and T = then T = It follows from Eq (5) that ∞ ∗ ˙ f 1(s)e−δ1s β1T0V (t − s)ds 0=T = dL dt It yields V = Hence = if and only if T = T0, T ∗ = 0,V = and W = Applying LaSalle’s invariance principle we get that S0 is globally asymptotically stable when R ≤ Theorem If R ≤ < R 0, then S1 is GAS 085204-6 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) Proof We define the functional L as: ( ) ( ) ( ) β1T1V1 V T ∗ T∗ p β1T1 ∗ L 1(T,T ,V,W ) = T1g V1g + T1 g ∗ + + W T1 η1 T1 bη 2T1∗ V1 kc ) ∞ s ( β1T1V1 T(t − ϑ)V (t − ϑ) + f 1(s)e−δ1s g dϑds η1 T1V1 ) 0 s ( ∗ ∞ β2T1T1 T(t − ϑ)T ∗(t − ϑ) dϑds + f 1(s)e−δ1s g η1 T1T1∗ 0 ) s ( ∗ β1T1V1 ∞ T (t − ϑ) + dϑds f 2(s)e−δ2s g η2 T1∗ 0 Calculating dL dt along the trajectories of (4)-(7) we get ) dL T1 (ρ − dT − β1TV − β2TT ∗) = 1− dt T ( ) ( ∞ ) T∗ + − 1∗ f 1(s)e−δ1s ( β1T(t − s)V (t − s) + β2T(t − s)T ∗(t − s)) ds − µT ∗ η1 T )( ∞ ( ) β1T1V1 V1 p β1T1 −δ s ∗ (kVW − qW ) + b − f (s)e T (t − s)ds − cV − pVW + bη 2T1∗ V kc )) ( ( β1T1V1 ∞ T(t − s)V (t − s) T(t − s)V (t − s) −δ s TV + ds f 1(s)e − + ln η1 T1V1 T1V1 TV ( ) ) ( ∗ β2T1T1∗ ∞ T(t − s)T ∗(t − s) T(t − s)T ∗(t − s) −δ s TT + − + ln ds f 1(s)e η1 T1T ∗ T1T1∗ TT ∗ ( ∗ )) ( ∗ ∗ ∞ T (t − s) T (t − s) β1T1V1 T + ln ds (19) + f 2(s)e−δ2s ∗ − ∗ η2 T T T∗ 1 ( Collecting terms of Eq (19) and applying the steady state conditions for S1: µ cµ ρ − dT1 = β1T1V1 + β2T1T1∗ = T1∗ = V1, η1 bη 1η we get ( ) d T1 dL = − (T − T1) + − β1T1V1 + β2T1T1∗ dt T T T1∗ ∞ f 1(s)e−δ1s [ β1T(t − s)V (t − s) + β2T(t − s)T ∗(t − s)] ds + β1T1V1 − η1 T ∗ V1T ∗(t − s) β1T1V1 ∞ β1 pT1 ( q) f 2(s)e−δ2s + β2T1T1∗ − ds + V − W η2 VT1∗ c k ( ) β1T1V1 ∞ T(t − s)V (t − s) + f 1(s)e−δ1s ln ds η1 TV ( ) β2T1T1∗ ∞ T(t − s)T ∗(t − s) f 1(s)e−δ1s ln ds + η1 TT ∗ ( ∗ ) 0∞ β1T1V1 T (t − s) + f 2(s)e−δ2s ln ds η2 T∗ Consider the following equalities ( ) ( ) ( ) ( ) T(t − s)V (t − s)Ti∗ T(t − s)V (t − s) Ti ViT ∗ ln = ln + ln + ln , TV TiViT ∗ T VTi∗ ) ) ( ( ( ) T(t − s)T ∗(t − s) Ti T(t − s)T ∗(t − s) ln = ln + ln , ∗ ∗ TT TiT T ( ∗ ) ( ) ( ) VTi∗ T (t − s) ViT ∗(t − s) ln = ln + ln ,i = 1, T∗ VTi∗ ViT ∗ (20) 085204-7 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) Using Eqs (20) with i = we get ( ) T1 dL d T1 ∗ = − [T − T1] − β1T1V1 + β2T1T1 − − ln dt T T T ( ) ∗ T(t − s)V (t − s)T1∗ T(t − s)V (t − s)T β1T1V1 ∞ −δ s − − ln f 1(s)e ds − η1 T1V1T ∗ T1V1T ∗ ( ) β2T1T1∗ ∞ T(t − s)T ∗(t − s) T(t − s)T ∗(t − s) − − ln ds − f 1(s)e−δ1s η1 T1T ∗ T1T ∗ ( ) β1T1V1 ∞ V1T ∗(t − s) β1 pT1 ( q) V1T ∗(t − s) − − − ln ds + V − W f 2(s)e−δ2s η2 VT1∗ VT1∗ c k ) ( ( ) T(t − s)V (t − s)T1∗ T1 d β1T1V1 ∞ = − [T − T1]2 − β1T1V1 + β2T1T1∗ g − ds f 1(s)e−δ1s g T T η1 T1V1T ∗ ) ) ( ( β2T1T1∗ ∞ β1T1V1 ∞ T(t − s)T ∗(t − s) V1T ∗(t − s) −δ s ds − ds − f 1(s)e−δ1s g f (s)e g η1 T1T ∗ η2 VT1∗ 0 β1 pT1 ( q) + V1 − W (21) c k dL ∗ From Lemma 3, we have V1 ≤ qk Then dL dt ≤ and dt = occurs at S1 Let D1 = {(T,T ,V,W ) : dL dt = 0} It is clear that D1 = {S1 } Using LaSalle’s invariance principle we obtain that S1 is globally asymptotically stable when R ≤ and < R Theorem If R > 1, then S2 is GAS Proof Define the functional L as: ( ) ( ∗) ( ) ( ) T β1T2V2 p β1T2V2 T V W L 2(T,T ∗,V,W ) = T2g + T2∗g ∗ + + V g W g 2 T2 η1 T2 bη 2T2∗ V2 kbη 2T2∗ W2 ) s ( β1T2V2 ∞ T(t − ϑ)V (t − ϑ) dϑds + f 1(s)e−δ1s g η1 T2V2 0 ) s ( β2T2T2∗ ∞ T(t − ϑ)T ∗(t − ϑ) dϑds + f 1(s)e−δ1s g η1 T2T2∗ 0 ) s ( ∗ β1T2V2 ∞ T (t − ϑ) + f 2(s)e−δ2s g dϑds η2 T2∗ 0 Then dL dt along the trajectories of (4)-(7) is given by ( ) dL T2 (ρ − dT − β1TV − β2TT ∗) = 1− dt T ) ∞ ( T∗ + − 2∗ f 1(s)e−δ1s ( β1T(t − s)V (t − s) + β2T(t − s)T ∗(t − s)) ds η1 T ( ) ( )( ∞ ) T2∗ ∗ β1T2V2 µ V2 −δ s ∗ − 1− ∗ T + 1− b f 2(s)e T (t − s)ds − cV − pVW η1 T bη 2T2∗ V ( ) p β1T2V2 W2 (kVW − qW ) + 1− kbη 2T2∗ W ( ( )) β1T2V2 ∞ T(t − s)V (t − s) T(t − s)V (t − s) −δ s TV + − + ln f 1(s)e ds η1 T2V2 T2V2 TV ( ( ) ) ∗ β2T2T2∗ ∞ T(t − s)T ∗(t − s) T(t − s)T ∗(t − s) −δ s TT + f 1(s)e − + ln ds η1 T2T2∗ T2T2∗ TT ∗ ( ∗ ( ∗ )) β1T2V2 ∞ T T ∗(t − s) T (t − s) + f 2(s)e−δ2s ∗ − + ln ds η2 T2 T2∗ T∗ (22) 085204-8 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) Collecting terms of Eq (22) and applying the steady state conditions for S2: ρ − dT2 = β1T2V2 + β2T2T2∗ = µ ∗ ∗ cV2 pV2W2 q T ,T = + , V2 = , η 2 bη bη k we get ) ( d dL T2 = − (T − T2) + − β1T2V2 + β2T2T2∗ dt T T ∞ T∗ − 2∗ f 1(s)e−δ1s [ β1T(t − s)V (t − s) + β2T(t − s)T ∗(t − s)] ds + β1T2V2 + β2T2T2∗ η 1T ( ) β1T2V2 V2 ∞ β1T2V2 ∞ T(t − s)V (t − s) −δ s ∗ −δ s − f (s)e T (t − s)ds + f (s)e ln ds η 2T2∗ V η1 TV ) ) ( ( ∗ β2T2T2∗ ∞ β1T2V2 ∞ T(t − s)T ∗(t − s) T (t − s) −δ s ds + ds f 1(s)e−δ1s ln f (s)e ln + η1 TT ∗ η2 T∗ 0 Using Eq (20) with i = we get ( ( )) ( ( )) dL d T2 T2 T2 ∗ T2 = − (T − T2) − β1T2V2 − − ln − − ln − β2T2T2 dt T T T T T ( ) ∗ T(t − s)V (t − s)T2∗ T(t − s)V (t − s)T β1T2V2 ∞ −δ s − − ln ds − f 1(s)e η1 T2V2T ∗ T2V2T ∗ ( ) β2T2T2∗ ∞ T(t − s)T ∗(t − s) T(t − s)T ∗(t − s) − − − ln ds f 1(s)e−δ1s η1 T2T ∗ T2T ∗ ) ( 0∞ β1T2V2 V2T ∗(t − s) V2T ∗(t − s) − ds f 2(s)e−δ2s − − ln ∗ η2 VT2 VT2∗ Then ( ) ( ) dL d T2 T2 ∗ = − (T − T2) − β1T2V2g − β2T2T2 g dt T T T ) ( ∞ T(t − s)V (t − s)T2∗ β1T2V2 −δ s ds − f 1(s)e g η1 T2V2T ∗ ) ( β2T2T2∗ ∞ T(t − s)T ∗(t − s) − ds f 1(s)e−δ1s g η1 T2T ∗ ( ) β1T2V2 ∞ V2T ∗(t − s) − f 2(s)e−δ2s g ds η2 VT2∗ (23) dL 2 Since R > 1, then T2,T2∗,V2,W2 > From Eq (23) we have dL dt ≤ and dt = occurs at S2 Let D2 = (T,T ∗,V,W ) : dL dt = It is clear that D2 = {S2 } Applying LaSalle’s invariance principle we prove that S2 is globally asymptotically stable III NUMERICAL RESULTS In order to illustrate our theoretical results, we perform numerical simulations for the model (4)-(7) with particular distribution functions f 1(s) and f 2(s) as: f 1(s) = δ(s − s1), f 2(s) = δ(s − s2), where s1 and s2 are positive constants and δ(.) is the dirac delta function Then, we can see that, ∞ f i (s)ds = 1, η i = ∞ δ(s − s i )e−δ i s ds = e−δ i s i , i = 1, 2, ∞ δ(s − s i )e−δs φ(t − s)ds = e−δs i φ(t − s i ), i = 1, 085204-9 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) TABLE I The values of the parameters of the HIV-1 model with discrete time delays given by Eqs (24)-(27) Parameter Value Parameter Value ρ d β1 β2 µ p q 10 0.01 varied 0.0001 0.5 0.1 0.1 s1 s2 c b k δ1 δ2 varied varied 10 varied 0.9 0.1 for any function φ With such choice, model (4)-(7) leads to: ˙ T(t) = ρ − dT(t) − β1T(t)V (t) − β2T(t)T ∗(t), T˙ ∗(t) = [ β1T(t − s1)V (t − s1) + β2T(t − s1)T ∗(t − s1)] e−δ1s1 − µT ∗(t), V˙ (t) = be−δ2s2T ∗(t − s2) − cV (t) − pV (t)W (t), (24) ˙ = kV (t)W (t) − qW (t) Z(t) (27) (25) (26) For system (24)-(27) we define the parameters R and R as: e−δ1s1 ρ(b β1e−δ2s2 + β2c) , cµd − µdk + µ β1q − ρ β2 ke−δ1s1 + (µdk + µ β1q − ρ β2 ke−δ1s1)2 + β1 β2 µk ρqe−δ1s1 + R = be−δ2s2 * 2cq β µ , R0 = We note that, the derived R and R depend on time delay parameters s1 and s2 To conduct numerical results for system (24)-(27), we use the data given in Table I We mention that, the values in Table I not come from a real HIV-1 propagation model, but they are just provided ad hoc A Effect of the parameters β1 and k on the HIV-1 dynamics In order to show the global dynamics of the system we consider three different initial conditions: IC1: ϕ1(ϑ) = 600, ϕ2(ϑ) = 1, ϕ3(ϑ) = 1, ϕ4(ϑ) = 10, IC2: ϕ1(ϑ) = 200, ϕ2(ϑ) = 0.5, ϕ3(ϑ) = 3, ϕ4(ϑ) = 5, IC3: ϕ1(ϑ) = 90, ϕ2(ϑ) = 4, ϕ3(ϑ) = 9, ϕ4(ϑ) = 12, where, ϑ ∈ [− max{s1, s2}, 0] FIG The concentration of uninfected CD4+ T cells with initial conditions IC1-IC3 in case of R ≤ 1, (S0 is GAS) 085204-10 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) In this case we choose s1 = 0.7, s2 = 0.9 and study the following subcases for the initial conditions IC1-IC3: (i) R ≤ We choose, β1 = 0.0001, k = 0.008 With these data we get R = 0.4310 < and R = 0.3146 < It follows from Lemma that, the system has one steady state S0 Figures 1-4 show that, the concentration of uninfected CD4+ T cells increase and reach the value ρ/d = 1000 FIG The concentration of infected CD4+ T cells with initial conditions IC1-IC3 in case of R ≤ 1, (S0 is GAS) FIG The concentration of free HIV-1 with initial conditions IC1-IC3 in case of R ≤ 1, (S0 is GAS) FIG The concentration of B lymphocytes with initial conditions IC1-IC3 in case of R ≤ 1, (S0 is GAS) 085204-11 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) for large t Moreover, the concentrations of infected cells, free HIV-1 and B cells decrease and tend to zero for all the three initial conditions IC1-IC3 It means that, S0 is GAS and the virus will be removed This result support the result of Theorem (ii) R ≤ < R and We choose β1 = 0.001, k = 0.001 and then R = 3.3515 and R = 0.2953 Lemma and Theorem state that S1 exists and is GAS It is clear from Figures 5-8 that, both numerical and theoretical results are consistent The solutions of the system converges to FIG The concentration of uninfected CD4+ T cells with initial conditions IC1-IC3 in case of R ≤ < R 0, (S1 is GAS) FIG The concentration of infected CD4+ T cells with initial conditions IC1-IC3 in case of R ≤ < R 0, (S1 is GAS) FIG The concentration of free HIV-1 with initial conditions IC1-IC3 in case of R ≤ < R 0, (S1 is GAS) 085204-12 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) FIG The concentration of B lymphocytes with initial conditions IC1-IC3 in case of R ≤ < R 0, (S1 is GAS) FIG The concentration of uninfected CD4+ T cellss with initial conditions IC1-IC3 in case of R > 1, (S2 is GAS) FIG 10 The concentration of infected CD4+ T cells with initial conditions IC1-IC3 in case of R > 1, (S2 is GAS) the steady state S1(298.3709, 7.4736, 22.7680, 0), for all the three initial conditions IC1-IC3 This implies that the infection becomes chronic with no sustained humoral immune response (iii) R > 1: In this case, we choose β1 = 0.001, k = 0.01 and then R = 3.3515 and R = 1.6657 > According to Lemma and Theorem 3, the system has three steady states S0, S1 and S2 and S2 is GAS Figures 9-12 show that, the solutions of the system approach the steady state S2(486.6946, 5.4676, 10, 19.9705) for large t and for all the initial conditions IC1-IC3 This support 085204-13 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) FIG 11 The concentration of free HIV-1 with initial conditions IC1-IC3 in case of R > 1, (S2 is GAS) FIG 12 The concentration of B lymphocytes with initial conditions IC1-IC3 in case of R > 1, (S2 is GAS) TABLE II The values of steady states, R and R for the HIV-1 model with discrete delays given by (24)-(27) with different values of the delay parameter S Delay parameter S = 0.0 S = 0.4 S = 0.9 S = 1.2205 S = 1.4 S = 1.7 S = 1.93287 S = 2.3 S = 2.6 Steady states R0 R1 E = (475.06234, 10.4988, 10, 74.9876) S2 = (482.5792, 7.21984, 10, 39.3675) S2 = (488.8840, 4.5475, 10, 11.5608) S1 = (491.6680, 3.3894, 10, 0) S1 = (587.98951, 2.3374, 6.7734, 0) S1 = (792.8973, 0.8969, 2.5223, 0) S0 = (1000, 0, 0, 0) S0 = (1000, 0, 0, 0) S0 = (1000, 0, 0, 0) 6.8667 4.6083 2.7994 2.0339 1.7007 1.2612 0.6936 0.5144 3.4996 2.3123 1.3854 0.8337 0.6155 0.4867 0.3363 0.2488 the result of Theorems This indicates that the infection becomes chronic with persistent humoral immune response B Effect of time delay on the HIV-1 dynamics In this case, we consider the initial condition IC2 We take the values β1 = 0.001, k = 0.01 Let us consider the case S = s1 = s2 The values of R 0, R and the steady states of system (24)-(27) with different values of S are presented in Table II 085204-14 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) Table II show that as S is increased, the values of R and R are decreased Moreover, we have the following cases: (i) if ≤ S < 1.220500210, then S2 exists and it is GAS, (ii) if 1.220500210 ≤ S < 1.932870191, then S1 exists and it is GAS, (iii) if S ≥ 1.932870191, then S0 is GAS FIG 13 The concentration of uninfected CD4+ T cells with different values of the delay parameter S FIG 14 The concentration of infected CD4+ T cells with different values of the delay parameter S FIG 15 The concentration of free HIV-1 with different values of the delay parameter S 085204-15 Elaiw, Raezah, and Alofi AIP Advances 6, 085204 (2016) FIG 16 The concentration of B lymphocytes with different values of the delay parameter S From Figures 13-16 and Table II we can see that increasing the delay times can reduce the value of R and prevent the invasion of viruses This also show that the numerical results are compatible with the results of Theorems 1-3 IV CONCLUSION In this paper, we have studied an HIV-1 virus dynamics model with two types of infections, virus-to-target and infected-to-target We have incorporated two types of distributed-time delays and humoral immunity into the model We have shown that solutions of the system are positive and ultimately bounded which ensure the model is well-posed We have shown that this model admits three steady states: infection-free steady state, chronic-infection steady state without humoral immune response, and chronic-infection steady state with humoral immune response We have determined two bifurcation parameters, the basic infection reproduction number R and the viral reproduction number at the chronic-infection steady state without humoral immune response R We have established the existence and the global stability of all steady states of the model The global asymptotic stability of the model has been proven using Lyapunov method and LaSalle’s invariance principle More 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(2 016 ) FIG 11 The concentration of free HIV- 1 with initial conditions IC1-IC3 in case of R > 1, (S2 is GAS) FIG 12 The concentration of B lymphocytes with initial conditions IC1-IC3 in case of. .. = − (T − T1) + − β1T1V1 + β2T1T1∗ dt T T T1∗ ∞ f 1( s)e−δ1s [ β1T(t − s)V (t − s) + β2T(t − s)T ∗(t − s)] ds + β1T1V1 − ? ?1 T ∗ V1T ∗(t − s) β1T1V1 ∞ ? ?1 pT1 ( q) f 2(s)e−δ2s + β2T1T1∗ − ds +