Article
Modeling the Adaptive Immunity and Both Modes of Transmission in HIV Infection Khalid Hattaf 1,2, * 1 2
*
ID
and Noura Yousfi 2
Centre Régional des Métiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco;
[email protected] Correspondence:
[email protected]; Tel.: +212-664407825
Received: 15 February 2018; Accepted: 4 May 2018; Published: 8 May 2018
Abstract: Human immunodeficiency virus (HIV) is a retrovirus that causes HIV infection and over time acquired immunodeficiency syndrome (AIDS). It can be spread and transmitted through two fundamental modes, one by virus-to-cell infection, and the other by direct cell-to-cell transmission. In this paper, we propose a new mathematical model that incorporates both modes of transmission and takes into account the role of the adaptive immune response in HIV infection. We first show that the proposed model is mathematically and biologically well posed. Moreover, we prove that the dynamical behavior of the model is fully determined by five threshold parameters. Furthermore, numerical simulations are presented to confirm our theoretical results. Keywords: HIV infection; immunity; Lyapunov functional; global stability
1. Introduction Viruses are very small infectious agents that need to penetrate inside a cell of their host to replicate and multiply. Several viruses attack the human body, such as influenza virus, human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), Ebola virus, Zika virus, and so on. Viral infections caused by these viruses represent a major global health problem by causing the mortality of millions of people and the expenditure of enormous amounts of money in health care and disease control. In this study, we are interested in the viral infection caused by HIV. The World Health Organization (WHO) estimates that 36.7 million people were living with HIV at the end of 2016, and more than 1 million people died from HIV-related causes in 2016 [1]. In Morocco, the number of people living with HIV is estimated at 28,740, and 1097 people died from AIDS in 2014, while the cumulative number of HIV/AIDS cases reported since the beginning of the epidemic is 10,017 [2]. Many mathematical models have been developed to better understand the dynamics of HIV infection. One of the earliest of these models was presented by Perelson et al. [3] in 1996. This model is given by the following system: T˙ (t) I˙(t) V˙ (t)
= λ − dT (t) − β 1 T (t)V (t), = β 1 T (t)V (t) − aI (t), = kI (t) − µV (t),
(1)
where T (t), I (t), and V (t) are the concentrations of healthy CD4+ T cells, infected cells, and free virus at time t, respectively. Healthy cells are produced at rate λ, die at rate d, and become infected by free virus at rate β 1 . The parameter a is the death rate of infected cells. Free virus is produced by an infected cell at rate k and is removed at rate µ. Computation 2018, 6, 37; doi:10.3390/computation6020037
www.mdpi.com/journal/computation
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In the system given by Equation (1), the cell infection is instantaneous and is caused only by contact with free virus. In reality, there are two kinds of delays: one in cell infection, and the other in virus production. In addition, HIV can spread by two fundamental modes, one by virus-to-cell infection, and the other by direct cell-to-cell transmission. For these above reasons, Lai and Zou [4] improved the model of Perelson et al. [3] by incorporating the two modes of transmission and infinite distributed delay in cell infection. They obtained the following model: T˙ (t) I˙(t) V˙ (t)
= λ − dT (t) − β 1 T (t)V (t) − β 2 T (t) I (t), R∞ = 0 f 1 (τ )e−α1 τ [ β 1 T (t − τ )V (t − τ ) + β 2 T (t − τ ) I (t − τ )]dτ − aI (t), = kI (t) − µV (t),
(2)
where β 2 T (t) I (t) denotes the rate for a target cell to contact with an infected cell. It is assumed that the virus or infected cell contacts an uninfected target cell at time t − τ and the cell becomes infected at time t, where τ is a random variable taken from a probability distribution f 1 (τ ). The term e−α1 τ represents the probability of surviving from time t − τ to time t, where α1 is the death rate for infected but not yet virus-producing cells. The other parameters have the same biological meaning as in Equation (1). On the other hand, the adaptive immune responses of cytotoxic T lymphocytes (CTLs) and antibodies play an important role in the control of HIV infection. The first immune response exerted by CTL cells is called the cellular immunity. However, the second immune response mediated by antibodies is called the humoral immunity. In the literature, several authors are interested in modeling the role of these arms of immunity in viral infections. In 2016, Wang et al. [5] improved the model given by Equation (2) by considering the role of cellular immune response. In the same year, Elaiw et al. [6] improved the model given by Equation (2) by considering only the role of humoral immune response and infinite distributed delay in virus production. In 2017, Lin et al. [7] improved the models of Wang and Zou [8] and Murase et al. [9] by incorporating both modes of transmission, intracellular delay and humoral immunity. The aim of this work is to improve and generalize all the above models by proposing a new mathematical model that takes into account the role of the adaptive immune response in HIV infection and incorporates both modes of transmission. To this end, the next section deals with the presentation of our model and some properties of solutions, such as positivity and boundedness. In Section 2, we derive the threshold parameters of our model and discuss the existence of equilibria. The global stability of equilibria is investigated in Section 3. Some numerical simulations of our main results are presented in Section 4. The mathematical and biological conclusions are given in Section 5. 2. Presentation of the Model To model the role of the adaptive immune response in HIV infection with both virus-to-cell infection and cell-to-cell transmission, we propose the following model: T˙ (t) I˙(t)
= λ − dT (t) − β 1 T (t)V (t) − β 2 T (t) I (t), R∞ = 0 f 1 (τ )e−α1 τ [ β 1 T (t − τ )V (t − τ ) + β 2 T (t − τ ) I (t − τ )]dτ − aI (t) − pI (t) Z (t), R∞ V˙ (t) = k 0 f 2 (τ )e−α2 τ I (t − τ )dτ − µV (t) − qV (t)W (t), ˙ (t) = gV (t)W (t) − hW (t), W Z˙ (t) = cI (t) Z (t) − bZ (t),
(3)
where W (t) and Z (t) are the concentrations of antibodies and CTL cells at time t, respectively. Free HIV particles are neutralized by the antibodies at rate qV (t)W (t). However, the infected cells are killed by CTL cells at rate pI (t) Z (t). Antibodies develop in response to free virus at rate gV (t)W (t), and CTL cells expand in response to viral antigens derived from infected cells at rate cI (t) Z (t). The parameters h and b are, respectively, the death rates of antibodies and CTL cells. Further, we assume that the time necessary for the newly produced virions to become mature and infectious is a random
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variable with a probability distribution f 2 (τ ). The term e−α2 τ denotes the probability of surviving the immature virions during the delay period, where α12 is the average lifetime of an immature virus. R∞ Therefore, the integral 0 f 2 (τ )e−α2 τ I (t − τ )dτ describes the mature viral particles produced at time t. The other variables and parameters are defined as those in the systems given by Equations (1) and (2). In this section, we first investigate the nonnegativity and boundedness of solutions under the following nonnegative initial conditions:
= φ1 (θ ) ≥ 0, = φ4 (θ ) ≥ 0,
T (θ ) W (θ )
I (θ ) = φ2 (θ ) ≥ 0, Z (θ ) = φ5 (θ ) ≥ 0,
V (θ ) = φ3 (θ ) ≥ 0, θ ∈ (−∞, 0].
(4)
We define the Banach space for the fading memory type as follows: Cα
ϕ ∈ C ((−∞, 0], IR5+ ) : ϕ(θ )eαθ is uniformly continuous on (−∞, 0] and k ϕk = sup | ϕ(θ )|eαθ < ∞ ,
=
θ ≤0
where α is a positive constant and IR5+ = {( x1 , ..., x5 ) : xi ≥ 0, i = 1, ..., 5}. Theorem 1. For any initial condition φ = (φ1 , φ2 , φ3 , φ4 , φ5 ) ∈ Cα satisfying Equation (4), the model given by Equation (3) has a unique solution on [0, +∞). Furthermore, this solution is nonnegative and bounded for all t ≥ 0. Proof. By the fundamental theory of functional differential equations [10–12], the model given by Equation (3) with initial condition φ ∈ Cα has a unique local solution on [0, tmax ), where tmax is the maximal existence time for the solution of Equation (3). First, we prove that T (t) > 0 for all t ∈ [0, tmax ). In fact, supposing the contrary, we let t1 > 0 be the first time such that T (t1 ) = 0 and T˙ (t1 ) ≤ 0. By the first equation of the model given by Equation (3), we have T˙ (t1 ) = λ > 0, which is a contradiction. Thus, T (t) > 0 for all t ∈ [0, tmax ). By Equation (3), we have I (t)
= φ2 (0)e−
Rt
0 ( a + pZ ( s ))ds
+
Z t Z ∞ Rt − ξ ( a+ pZ (s))ds 0
e
0
f 1 ( τ ) e − α1 τ [ β 1 T ( ξ − τ ) V ( ξ − τ )
+ β 2 T (ξ − τ ) I (ξ − τ )]dτdξ, V (t)
= φ3 (0)e−
W (t)
= φ4 (0)e
Z (t)
= φ5 (0)e
Rt
0 ( µ + qW ( s ))ds
Rt
0 ( gV ( s )− h ) ds
Rt
0 ( cI ( s )− b ) ds
+k
Z t Z h2 Rt − ξ (µ+qW (s))ds 0
e
0
f 2 (τ )e−α2 τ I (ξ − τ )dτdξ,
,
,
which implies that I (t), V (t), W (t), and Z (t) are nonnegative for all t ∈ [0, tmax ). Next, we show the boundedness of each solution. From the first equation of the model given by Equation (3), we have T˙ (t) ≤ λ − dT (t), which implies that lim sup T (t) ≤ t→+∞
λ . d
Then T (t) is bounded. Let G1 (t) = I (t) +
p Z (t) + c
Z ∞ 0
f 1 (τ )e−α1 τ T (t − τ )dτ.
(5)
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Because T (t) is bounded and with respect to t. Hence, dG1 (t) dt
= λ
Z ∞ 0
R∞ 0
f 1 (τ )dτ = 1, the integral in G1 (t) is well defined and differentiable
f 1 (τ )e−α1 τ dτ − d
Z ∞ 0
f 1 (τ )e−α1 τ T (t − τ )dτ − aI (t) −
pb Z (t) c
≤ λη1 − δ1 G1 (t), where δ1 = min{ a, b, d} and ηi =
Z ∞ 0
f i (τ )e−αi τ dτ, i = 1, 2.
(6)
λη1 }, which implies that I (t) and Z (t) are bounded. It remains δ1 to prove that V (t) and W (t) are bounded. To this end, we consider Thus, G1 (t) ≤ M := max{ G1 (0),
q G2 (t) = V (t) + W (t); g then dG2 (t) dt
= k
Z ∞ 0
f 2 (τ )e−α2 τ I (t − τ )dτ − µV (t) −
hq W (t) g
≤ kMη2 − δ2 G2 (t), where δ2 = min{µ, h}. Similarly to the above, we deduce that V (t) and W (t) are also bounded. We have proved that all variables of Equation (3) are bounded, which implies that tmax = +∞ and that the solution exists globally. If in addition to Equation (4), we suppose that φi (0) > 0 for all i = 1, ..., 5; then we obtain the following remark. Remark 1. If φ = (φ1 , φ2 , φ3 , φ4 , φ5 ) ∈ Cα satisfies Equation (4) with φi (0) > 0, then each solution of Equation (3) with initial condition φ remains positive and bounded for all t ≥ 0. Next, we derive threshold numbers and identify biological equilibria for the model given by Equation (3). Clearly, Equation (3) always has an infection-free equilibrium of the form E0 ( T0 , 0, 0, 0, 0), λ where T0 = . Therefore, the basic reproduction number of Equation (3) can be defined as d R0 =
β 1 kλη1 η2 + β 2 λµη1 . daµ
As in [13], R0 can be rewritten as R0 = R01 + R02 , where R01 = number corresponding to the virus-to-cell infection mode, and R02
(7) β 1 kλη1 η2 is the basic reproduction daµ β λη = 2 1 is the basic reproduction da
number corresponding to the cell-to-cell transmission mode. When R0 > 1, Equation (3) has another infection equilibrium without immunity, E1 (T1 , I1 , V1 , 0, 0), where T1 =
λ dµ( R0 − 1) kdη2 ( R0 − 1) , I = and V1 = . dR0 1 β 1 kη2 + β 2 µ β 1 kη2 + β 2 µ
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If both humoral and cellular immune responses have not been established, we have gV1 − h ≤ 0 and cI1 − b ≤ 0. Thus, we define the reproduction number for humoral immunity: RW 1 =
gkdη2 ( R0 − 1) gV1 = h h( β 1 kη2 + β 2 µ)
(8)
and the reproduction number for cellular immunity: R1Z =
cI1 cdµ( R0 − 1) = . b b( β 1 kη2 + β 2 µ)
(9)
Z Hence, gV1 − h ≤ 0 and cI1 − b ≤ 0 are equivalent to RW 1 ≤ 1 and R1 ≤ 1, respectively. When RW > 1, Equation (3) has an infection equilibrium with only humoral immunity, 1 E2 ( T2 , I2 , V2 , W2 , 0), where
√ − B + B2 + 4AC h µ kη2 I2 aI2 , I2 = , V2 = , W2 = T2 = −1 , η1 ( β 1 V2 + β 2 I2 ) 2A g q µV2 with A = agβ 2 , B = a(dg + hβ 1 ) − λη1 β 2 g, and C = λhη1 β 1 . It is not hard to see that T2 , I2 and V2 are positive. It remains to check that W2 is positive. Because RW > 1 and 1 µahV2 µV2 W AµV2 + B − C = 2 2 1 − R1 , k2 η22 k η2 we easily deduce that W2 > 0. If cellular immunity has not been established, we have cI2 − b ≤ 0. For this, we define the reproduction number for cellular immunity in competition as R2Z =
cI2 , b
(10)
which implies that cI2 − b ≤ 0 is equivalent to R2Z ≤ 1. When R1Z > 1, Equation (3) has an infection equilibrium with only cellular immunity, E3 ( T3 , I3 , V3 , 0, Z3 ), where λcµ b kbη2 a T3 = , I3 = , V3 = , Z3 = dcµ + b(kβ 1 η2 + µβ 2 ) c cµ p
R0 1+
ab λcη1 R0
−1 .
We have that T3 , I3 , and V3 are positive. It suffices to check that Z3 is positive. Because R1Z > 1 and b(kβ 1 η2 + µβ 2 ) R0 Z − 1 = R − 1 , 1 ab ab 1 + λcη R0 dcµ(1 + λcη R0 ) 1
1
we deduce that Z3 > 0. If humoral immunity has not been established, we have gV3 − h ≤ 0. In this case, we define the reproduction number for humoral immunity in competition as RW 3 =
gV3 , h
(11)
which implies that gV3 − h ≤ 0 is equivalent to RW 3 ≤ 1. Z W When R2 > 1 and R3 > 1, Equation (3) has an infection equilibrium with both cellular and humoral immune responses, E4 ( T4 , I4 , V4 , W4 , Z4 ), where λgc b h µ a , I = , V4 = , W4 = ( RW − 1), Z4 = T4 = dgc + β 1 hc + β 2 bg 4 c g q 3 p
λcη1 ( β 1 hc + β 2 bg) −1 . ab(dgc + β 1 hc + β 2 bg)
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We have T4 > 0, I4 > 0, V4 > 0, and W4 > 0 (as RW 3 > 1). It remains to check that Z4 > 1. On the other b hand, R2Z > 1 is equivalent to C > 2 bA + cB . By a simple computation, we have c C−
b pb bA + cB = 2 dgc + β 1 hc + β 2 bg Z4 . 2 c c
Summarizing the above discussions, we obtain the following theorem. Theorem 2. λ (i) If R0 ≤ 1, then Equation (3) always has one infection-free equilibrium, E0 ( T0 , 0, 0, 0, 0), where T0 = . d (ii) If R0 > 1, then Equation (3) has an infection equilibrium without immunity, E1 ( T1 , I1 , V1 , 0, 0), where T1 =
λ dµ( R0 − 1) kdη2 ( R0 − 1) , I = and V1 = . dR0 1 β 1 kη2 + β 2 µ β 1 kη2 + β 2 µ
> 1, then Equation (3) has an infection equilibrium with only humoral immunity, (iii) If RW 1 E2 (T2 , I2 , V2 , W2 , 0), where √ − B + B2 + 4AC h µ kη2 I2 aI2 , I2 = , V2 = , W2 = −1 . T2 = η1 ( β 1 V2 + β 2 I2 ) 2A g q µV2 (iv) If R1Z > 1, then Equation (3) has an infection equilibrium with only cellular immunity, E3 (T3 , I3 , V3 , 0, Z3 ), where λcµ b kbη2 a T3 = , I3 = , V3 = , Z3 = dcµ + b(kβ 1 η2 + µβ 2 ) c cµ p
R0 1+
ab λcη1 R0
−1 .
(v) If R2Z > 1 and RW 3 > 1, then Equation (3) has an infection equilibrium with both cellular and humoral immune responses, E4 ( T4 , I4 , V4 , W4 , Z4 ), where T4 =
λgc b h µ , I4 = , V4 = , W4 = ( RW − 1) and dgc + β 1 hc + β 2 bg c g q 3 a Z4 = p
λcη1 ( β 1 hc + β 2 bg) −1 . ab(dgc + β 1 hc + β 2 bg)
It is important to note that RW 3 =
RW 1 gkbη2 µ W 1 = , W2 = ( R2Z RW . 3 − 1) and R3 > hµc q R1Z R2Z
(12)
3. Global Stability In this section, we investigate the global stability of the five equilibria of Equation (3) by constructing appropriate Lyapunov functionals. We first analyze the global stability of the infection-free equilibrium. Theorem 3. The infection-free equilibrium E0 is globally asymptotically stable when R0 ≤ 1.
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Proof. To study the global stability of E0 , we consider a Lyapunov functional defined as follows:
= T0 Φ(
L0 ( t )
β T qβ T p 1 T (t) Z (t) ) + I (t) + 1 0 V (t) + 1 0 W (t) + T0 η1 µ gµ cη1
t 1 ∞ [ β 1 T (s)V (s) + β 2 T (s) I (s)]dsdτ f 1 ( τ ) e − α1 τ η1 0 t−τ Z t Z ∞ kβ T + 1 0 I (s)dsdτ, f 2 ( τ ) e − α2 τ µ t−τ 0
+
Z
Z
where Φ( x ) = x − 1 − ln x for x > 0. It is not hard to see that Φ( x ) ≥ 0 for all x ∈ (0, +∞). Hence, the functional L0 is nonnegative. In order to simplify the presentation, we use the following notations: Ψ = Ψ(t) and Ψτ = Ψ(t − τ ) for any Ψ ∈ { T, I, V, W, Z }. Calculating the time derivative of L0 along the positive solution of Equation (3), we obtain dL0 | dt (3)
=
=
If follows from R0 ≤ 1 that
β T0 qβ T0 ˙ T0 ˙ 1 p ˙ Z 1− T + I˙ + 1 V˙ + 1 W + T η1 µ gµ cη1 Z 1 ∞ f (τ )e−α1 τ [ β 1 TV + β 2 TI − β 1 Tτ Vτ − β 2 Tτ Iτ ]dτ + η1 0 1 Z kβ 1 T0 ∞ + f 2 (τ )e−α2 τ ( I − Iτ )dτ µ 0 d a hqβ 1 T0 p − ( T − T0 )2 + ( R0 − 1) I − W− Z. T η1 gµ cη1 dL0 ≤ 0. It is straightforward to show that the largest invariant set in dt
dL {( T, I, V, W, Z )| 0 = 0} is { E0 }. By LaSalle’s invariance principle [14], the infection-free equilibrium dt E0 is globally asymptotically stable when R0 ≤ 1. When R0 > 1, Equation (3) has four infection steady states Ei , 1 ≤ i ≤ 4. The following theorem characterizes the global stability of these steady states. Theorem 4. Assume R0 > 1. Z (i) The infection equilibrium without immunity E1 is globally asymptotically stable if RW 1 ≤ 1 and R1 ≤ 1. W (ii) The infection equilibrium with only humoral immunity E2 is globally asymptotically stable if R1 > 1 and R2Z ≤ 1. (iii) The infection equilibrium with only cellular immunity E3 is globally asymptotically stable if R1Z > 1 and RW 3 ≤ 1. (iv) The infection equilibrium with both cellular and humoral immune responses E4 is globally asymptotically stable if R2Z > 1 and RW 3 > 1.
Proof. For (i), consider the following Lyapunov functional:
L1 ( t )
=
1 I (t) β 1 T1 V1 V (t) qβ T p + I1 Φ + VΦ + 1 1 W (t) + Z (t) η1 I1 kη2 I1 1 V1 gµ cη1 Z ∞ Z t β TV T ( s )V ( s ) dsdτ + 1 1 1 f 1 ( τ ) e − α1 τ Φ η1 T1 V1 0 t−τ Z ∞ Z t β T I T (s) I (s) + 2 1 1 f 1 ( τ ) e − α1 τ Φ dsdτ η1 T1 I1 0 t−τ Z ∞ Z t I (s) β TV f 2 ( τ ) e − α2 τ Φ dsdτ. + 1 1 1 η2 I1 0 t−τ T (t) T1 Φ T1
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Then dL1 | dt (3)
I V T1 ˙ 1 β TV qβ T ˙ p ˙ Z ) T + (1 − 1 ) I˙ + 1 1 1 (1 − 1 )V˙ + 1 1 W + T η1 I kη2 I1 V gµ cη1 Z ∞ TV Tτ Vτ β TV f 1 ( τ ) e − α1 τ Φ −Φ dτ + 1 1 1 η1 T V T1 V1 0 1 1 Z ∞ β T I Tτ Iτ TI −Φ + 2 1 1 f 1 ( τ ) e − α1 τ Φ dτ η1 T1 I1 T1 I1 0 Z ∞ β TV I Iτ + 1 1 1 f 2 ( τ ) e − α1 τ Φ −Φ dτ. η2 I1 I1 0
= (1 −
By λ = dT1 + β 1 T1 V1 + β 2 T1 I1 = dT1 + dL1 | dt (3)
a η1 I1
and kη2 I1 = µV1 , we obtain
d hqβ 1 T1 W pb Z ( R − 1) Z = − ( T − T1 )2 + ( R 1 − 1 )W + T gµ cη1 1 Z β 1 T1 V1 ∞ T1 Tτ Vτ I1 Tτ Vτ − α1 τ + f 1 (τ )e 3− − + ln dτ η1 T T1 V1 I TV 0 Z ∞ T β T I Tτ Iτ Tτ Iτ f 1 ( τ ) e − α1 τ 2 − 1 − + 2 1 1 + ln dτ η1 T T1 I TI 0 Z ∞ Iτ β TV V Iτ dτ. − 1 1 1 f 2 (τ )e−α2 τ 1 − ln η2 V I1 I 0
Thus, dL1 | dt (3)
d hqβ 1 T1 W pb Z = − ( T − T1 )2 + ( R 1 − 1 )W + ( R − 1) Z T gµ cη1 1 Z β 1 T1 V1 ∞ Tτ Vτ I1 T1 − α1 τ ) dτ − f 1 (τ )e Φ( ) + Φ( η1 T T1 V1 I 0 Z ∞ β T I T Tτ Iτ − 2 1 1 f 1 ( τ ) e − α1 τ Φ ( 1 ) + Φ ( ) dτ η1 T T1 I 0 Z ∞ β TV V Iτ − 1 1 1 f 2 (τ )e−α2 τ Φ( 1 )dτ. η2 V I1 0
dL1 | ≤ 0 with equality if and only if dt (3) T = T1 , I = I1 , V = V1 , W = 0, and Z = 0. It follows from LaSalle’s invariance principle that E1 is globally asymptotically stable. For (ii), consider the following Lyapunov functional: Z Because Φ( x ) ≥ 0 for x > 0, RW 1 ≤ 1, and R1 ≤ 1, we have
L2 ( t )
=
I (t) V (t) 1 β 1 T2 V2 + I2 Φ + V2 Φ η1 I2 kη2 I2 V2 qβ 1 T2 V2 W (t) p W2 Φ + Z (t) + gkη2 I2 W2 cη1 Z ∞ Z t β TV T ( s )V ( s ) + 1 2 2 f 1 ( τ ) e − α1 τ Φ dsdτ η1 T2 V2 0 t−τ Z Z t β 2 T2 I2 ∞ T (s) I (s) − α1 τ + f 1 (τ )e Φ dsdτ η1 T2 I2 0 t−τ Z ∞ Z t β TV I (s) + 1 2 2 f 2 ( τ ) e − α2 τ Φ dsdτ. η2 I2 0 t−τ T (t) T2 Φ T2
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Hence, dL2 | dt (3)
I T2 ˙ 1 β TV V ) T + (1 − 2 ) I˙ + 1 2 2 (1 − 2 )V˙ T η1 I kη2 I2 V W2 ˙ p ˙ qβ 1 T2 V2 Z (1 − )W + + gkη2 I2 W cη1 Z β 1 T1 V1 ∞ TV Tτ Vτ − α1 τ + f 1 (τ )e Φ −Φ dτ η1 T1 V1 T1 V1 0 Z ∞ Tτ Iτ β T I TI + 2 1 1 f 1 ( τ ) e − α1 τ Φ −Φ dτ η1 T1 I1 T1 I1 0 Z ∞ I Iτ β TV f 2 ( τ ) e − α1 τ Φ −Φ dτ. + 1 1 1 η2 I1 I1 0
= (1 −
By λ = dT2 + β 1 T2 V2 + β 2 T2 I2 = dT2 + dL2 | dt (3)
a η1 I2 ,
V2 =
h , and kη2 I2 = µV2 + qW2 V2 , we obtain g
d pb Z = − ( T − T2 )2 + ( R − 1) Z T cη1 2 Z β 1 T2 V2 ∞ Tτ Vτ I2 T2 Tτ Vτ − α1 τ + f 1 (τ )e 3− dτ − + ln η1 T T2 V2 I TV 0 Z ∞ β T I T2 Tτ Iτ Tτ Iτ + 2 2 2 f 1 ( τ ) e − α1 τ 2 − − + ln dτ η1 T T I TI 0 2 Z ∞ β TV Iτ V2 Iτ − 1 2 2 − ln f 2 ( τ ) e − α2 τ dτ. η2 V I2 I 0
Thus, dL2 | dt (3)
d pb Z = − ( T − T2 )2 + ( R − 1) Z T cη1 2 Z ∞ β TV T2 Tτ Vτ I2 − 1 2 2 f 1 ( τ ) e − α1 τ Φ ( ) + Φ ( ) dτ η1 T T2 V2 I 0 Z ∞ β 2 T2 I2 Tτ Iτ T2 − α1 τ − f 1 (τ )e Φ( ) + Φ( ) dτ η1 T T2 I 0 Z ∞ β TV V2 Iτ f 2 ( τ ) e − α2 τ Φ ( − 1 2 2 )dτ. η2 V I2 0
dL2 Because R2Z ≤ 1 and Φ( x ) ≥ 0 for x > 0, we have | ≤ 0 with equality if and only if T = T2 , dt (3) ˙ ˙ I = I2 , and V = V2 . Then I = 0 and V = 0, which leads to Z = 0 and W = W2 . Therefore, the dL2 = 0} is the singleton { E2 }, and the proof of (ii) largest compact invariant set in {( T, I, V, W, Z )| dt is completed.
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For (iii), consider the following Lyapunov functional: L3 ( t )
I (t) 1 β TV V (t) I3 Φ + 1 3 3 V3 Φ η1 I3 kη2 I3 V3 qβ T Z (t) p + 1 3 W (t) + Z2 Φ gµ cη1 Z3 Z t Z ∞ T ( s )V ( s ) β 1 T3 V3 − α1 τ Φ + f 1 (τ )e dsdτ η1 T3 V3 t−τ 0 Z ∞ Z t T (s) I (s) β T I f 1 ( τ ) e − α1 τ Φ dsdτ + 2 3 3 η1 T3 I3 0 t−τ Z t Z ∞ I (s) β TV Φ f 2 ( τ ) e − α2 τ + 1 3 3 dsdτ. η2 I3 t−τ 0
= T3 Φ
T (t) T3
+
From λ = dT3 + β 1 T3 V3 + β 2 T3 I3 = dT3 + dL3 | dt (3)
a η1 I2
+
p η1 I3 Z3 , I3
b = , and kη2 I3 = µV3 , we easily have c
d hqβ 1 T3 W = − ( T − T3 )2 + ( R 3 − 1 )W T gµ Z β 1 T3 V3 ∞ T3 Tτ Vτ I3 − α1 τ − f 1 (τ )e Φ( ) + Φ( ) dτ η1 T T3 V3 I 0 Z ∞ β T I Tτ Iτ T3 − 2 3 3 f 1 ( τ ) e − α1 τ Φ ( ) + Φ ( ) dτ η1 T T3 I 0 Z β 1 T3 V3 ∞ V I 3 τ − f 2 ( τ ) e − α2 τ Φ ( )dτ. η2 V I3 0
dL3 Consequently, | ≤ 0 with equality if and only if T = T3 , I = I3 , and V = V3 . It follows from dt (3) I˙ = 0 and V˙ = 0 that Z = Z3 and W = 0. By LaSalle’s invariance principle, we deduce that E3 is globally asymptotically stable. Finally, we show (iv) by considering the following Lyapunov functional: L4 ( t )
1 I (t) β TV V (t) I4 Φ + 1 4 4 V4 Φ η1 I4 kη2 I4 V4 qβ T V W (t) p Z (t) + 1 4 4 W4 Φ + Z Φ gkη2 I4 W4 cη1 4 Z4 Z ∞ Z t β 1 T4 V4 T ( s )V ( s ) − α1 τ + f 1 (τ )e Φ dsdτ η1 T4 V4 0 t−τ Z ∞ Z t T (s) I (s) β T I + 2 4 4 f 1 ( τ ) e − α1 τ Φ dsdτ η1 T4 I4 0 t−τ Z ∞ Z t β TV I (s) + 1 4 4 f 2 ( τ ) e − α2 τ Φ dsdτ. η2 I4 0 t−τ
= T4 Φ
T (t) T4
By λ = dT4 + β1 T4 V4 + β2 T4 I4 = dT4 +
+
a η1 I4
+
p η1 I4 Z4 , I4
=
b h V = , and kη2 I4 = (µ + qW4 )V4 , c 4 g
we obtain dL4 | dt (3)
=
Z ∞
Tτ Vτ I4 T4 Tτ Vτ f 1 (τ )e 3− − + ln dτ T T4 V4 I TV 0 Z ∞ β T I T Tτ Iτ Tτ Iτ + 2 4 4 f 1 ( τ ) e − α1 τ 2 − 4 − + ln dτ η1 T T4 I TI 0 Z ∞ V Iτ Iτ β TV − 1 4 4 f 2 (τ )e−α2 τ 4 − ln dτ. η2 V I4 I 0
d β TV − ( T − T4 )2 + 1 4 4 T η1
− α1 τ
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Hence, dL4 | dt (3)
Thus,
Z ∞ T d β TV Tτ Vτ I4 f 1 ( τ ) e − α1 τ Φ ( 4 ) + Φ ( = − ( T − T4 )2 − 1 4 4 ) dτ T η1 T T4 V4 I 0 Z ∞ T Tτ Iτ β T I f 1 ( τ ) e − α1 τ Φ ( 4 ) + Φ ( − 2 4 4 ) dτ η1 T T4 I 0 Z ∞ V Iτ β TV f 2 (τ )e−α2 τ Φ( 4 )dτ. − 1 4 4 η2 V I4 0
dL4 | ≤ 0 with equality holds if and only if T = T4 , I = I4 , and V = V4 . Let dt (3) Γ = {( T, I, V, W, Z )|
dL4 = 0}. dt
From the second and third equations of the model given by Equation (3), we have I˙ V˙
= η1 ( β 1 T4 V4 + β 2 T4 I4 ) − aI4 − pI4 Z = 0, = kη2 I4 − µV4 − qV4 W = 0,
which implies that Z = Z4 and W = W4 . Then the largest compact invariant set in Γ is the singleton { E4 }. Therefore, E4 is globally asymptotically stable. The conditions of the global stability of E2 and those of E3 given in (ii) and (iii) of Theorem 4 Z Z W do not hold simultaneously. In fact, supposing the contrary, then RW 1 > 1 ≥ R2 and R1 > 1 ≥ R3 . 1 Z Z Z Because RW 3 ≤ 1 and R2 > W , we have R2 > 1. This is a contradiction with R2 ≤ 1. R3 According to Equation (12) and Theorem 4, we have the following important result. Remark 2. Assume R0 > 1. Z 1. If max( RW 1 , R1 ) ≤ 1, then Equation (3) converges to E1 without immunity. W Z 2. If max( R1 , R1 ) > 1, two cases arise: Z W (i) When max( RW 1 , R1 ) = R1 , the humoral immunity is dominant, and Equation (3) converges to E2 if Z Z R2 ≤ 1 or to E4 if R2 > 1. Z Z (ii) When max( RW 1 , R1 ) = R1 , the cellular immunity is dominant, and Equation (3) converges to E3 without humoral immunity.
R2Z
From this important remark, we can define the over-domination of humoral immunity when Z W > 1 and RW 3 > 1 and the over-domination of cellular immunity when R2 > 1 and R3 < 1.
4. Numerical Simulations In this section, we present some numerical simulations in order to validate our theoretical results. For simplicity, we chose f 1 (τ ) = δ(τ − τ1 ) and f 2 (τ ) = δ(τ − τ2 ) with τ1 and τ2 to be the delays in cell infection and virus production, respectively, and δ(.) to be the Dirac function; then our model becomes T˙ (t) I˙(t)
= λ − dT (t) − β 1 T (t)V (t) − β 2 T (t) I (t), = [ β 1 T (t − τ1 )V (t − τ1 ) + β 2 T (t − τ1 ) I (t − τ1 )]e−α1 τ1 − aI (t) − pI (t) Z (t), V˙ (t) = kI (t − τ2 )e−α2 τ2 − µV (t) − qV (t)W (t), ˙ (t) = gV (t)W (t) − hW (t), W ˙ Z (t) = cI (t) Z (t) − bZ (t).
(13)
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This system improves the model presented in 2017 by Lin et al. [7], which considered only the humoral immunity and discrete delay in cell infection and ignored the time delay in virus production; that is, τ2 = 0. In addition, Equation (13) includes many special cases existing in the literature. For example, when β 2 = 0 and τ1 = τ2 = 0, we obtain the model presented by Wodarz in [15] and analyzed by Hattaf et al. in [16]. When β 2 = 0 and τ2 = 0, we obtain the model of Yan and Wang [17]. The algorithm for the numerical treatment of the delay differential system given by Equation (13) can be derived for the numerical method presented in [18,19]. Recently, this numerical method has been used for delayed partial differential equations [20]; it is called the “mixed” Euler method, as it is a mixture of both forward and backward Euler methods. In addition, it is shown that this mixed Euler method preserves the qualitative properties of the corresponding continuous system, such as positivity, boundedness, and global behaviors of solutions. Hence, we discretize the continuous system given by Equation (13) by this numerical method. Thus, we let ∆t be a time step size and assume that there exist two integers (m1 , m2 ) ∈ N I 2 with τ1 = m1 ∆t and τ2 = m2 ∆t. The grids points are tn = n∆t for n∈N I . By applying the mixed Euler method and using the approximations T (tn ) ≈ Tn , I (tn ) ≈ In , V (tn ) ≈ Vn , W (tn ) ≈ Wn , and Z (tn ) ≈ Zn , we obtain the following discrete model: Tn+1 In+1 Vn+1 Wn+1 Zn+1
= Tn + λ − dTn+1 − β 1 Tn+1 Vn − β 2 Tn+1 In ∆t, = In + [ β 1 Tn−m1 +1 Vn−m1 + β 2 Tn−m1 +1 In−m1 ]e−α1 τ1 − aIn+1 − pIn+1 Zn ∆t, = Vn + ke−α2 τ2 In−m2 +1 − µVn+1 − qVn+1 Wn ∆t, = Wn + gVn+1 Wn+1 − hWn+1 ∆t, = Zn + cIn+1 Zn+1 − bZn+1 ∆t,
where the discrete initial conditions are Ts
= φ1 (ts ), Is = φ2 (ts ), Vs = φ3 (ts ), Ws = φ4 (ts ), for s ∈ {−m, −m + 1, · · · , 0} and m = max(m1 , m2 ).
Zs = φ5 (ts ),
Z Z W The five threshold parameters R0 , RW 1 , R1 , R2 , and R3 for Equation (13) are given by − α τ − α τ 2 2 1 1 Equations (7)–(11) with η1 = e and η2 = e . In order to study the impact of cell-to-cell transmission and both arms of adaptive immunity on the HIV dynamics, we chose β 2 , g, and c as free parameters. The units of state variables T, I, and Z were given by cells µL−1 . Further, the units of V and W were given by virions µL−1 and molecules µL−1 , respectively. The other parameter values for the simulation are listed in Table 1.
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Table 1. List of parameters. Parameter λ d β1 a α1 µ k α2 p q h b τ1 τ2
Unit cells
µL−1
day−1
day−1 µL virion−1 day−1 day−1 day−1 day−1 virion cell−1 day−1 day−1 − 1 cell µL day−1 molecule−1 µL day−1 day−1 day−1 days days
Value
Range
10 0.0139 2.4 × 10−5 0.29 0.01 3 50 0.01 0.001 0.5 0.5 0.1 1.5 0.5
5.9770–24.1860 — 2.4 × 10−5 –4.8 × 10−3 0.2666–0.7073 — 2.06–3.81 27–7073 — 0.001–1 — — 0.05–0.15 0-2 —
Source [21] [22] [3,23] [3,24,25] [26] [3] [21] [26] [27–30] Assumed Assumed [28,29] [31] [32]
For the case in which β 2 = 10−6 , g = 10−5 , and c = 0.002, we obtained R0 = 0.9751. It follows from Theorem 2 that Equation (13) has one infection-free equilibrium E0 (719.4245, 0, 0, 0, 0). From Figure 1, we see that the concentration of uninfected CD4+ T cells increased and tended to the value T0 = 719.4245, while the concentrations of infected cells, free HIV particles, antibodies, and CTL cells decreased and tended to zero. This means that E0 is globally asymptotically stable and that the virus will be cleared. This confirms the result in Theorem 3. For the case in which β 2 = 1.5 × 10−4 , g = 10−5 , and c = 0.002, we obtained R0 = 1.3392, Z RW 1 = 0.0143, and R1 = 0.1721. It follows that case (i ) of Theorem 4 occurs, and the infection equilibrium without immunity E1 (537.9692, 8.5643, 142.0261, 0, 0) is globally asymptotically stable (see Figure 2). For the case in which β 2 = 3.4 × 10−4 , g = 10−3 , and c = 0.002, we obtained R0 = 1.8036, W R1 = 2.5099, and R2Z = 0.2000. From (ii ) of Theorem 4, the infection equilibrium with only humoral immunity E2 (507.5951, 10.0014, 99.9785, 3.9525, 0) is globally asymptotically stable (see Figure 3). For the case in which β 2 = 5.4 × 10−4 , g = 10−3 , and c = 0.02, we obtained R0 = 2.2923, Z R1 = 3.8301, and RW 3 = 0.8292. By (iii ) of Theorem 4, the infection equilibrium with only cellular immunity E3 (537.3806, 5.0015, 83.1539, 0, 211.6047) is globally asymptotically stable (see Figure 4). For the case in which β 2 = 5.4 × 10−4 , g = 10−2 , and c = 0.02, we obtained R0 = 2.2923, Z R2 = 1.8783, and RW 3 = 8.2918. Using (iv ) of Theorem 4, we deduce that the infection equilibrium with both arms of immunity E4 (594.0271, 4.9769, 10.0246, 43.3843, 53.3821) is globally asymptotically stable (see Figure 5).
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800
60
Infected cells
Uninfected cells
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700 600 500 400
0
200
400
Antibodies
Virus
400 200 0
200
400
400
600
400
600
Days
30 20 10 0
600
0
200
Days CTL cells
CTL cells
200
Days
200 150 100 50 0
0
40
600
0
20 0
600
Days
800
40
200 100
E0
0 40 20
0
200
400
600
Antibodies
Days
0
500
0
1000
Virus
700
60
Infected cells
Uninfected cells
Figure 1. Demonstration of the global stability of the infection-free equilibrium E0 for R0 = 0.9751 ≤ 1.
600 500 400
0
200
400
Antibodies
Virus
400 200 0
200
400
100 50 400
Days
400
600
20 10 0
200
200 100
E1
0 40 20
200
600
Days
150
0
400
Days
30
0
600
CTL cells
CTL cells
200
Days
200
0
0
40
600
0
20 0
600
Days
800
40
600
Antibodies
0
0
500
1000
Virus
Figure 2. Demonstration of the global stability of the infection equilibrium without immunity E1 for Z R0 = 1.3392 > 1, RW 1 = 0.0143 ≤ 1, and R1 = 0.1721 ≤ 1.
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700
60
Infected cells
Uninfected cells
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600 500 400
0
200
400
Antibodies
Virus
400 200 0
200
400
200
400
600
400
600
Days
30 20 10 0
600
0
200
Days CTL cells
CTL cells
0
Days
200 150 100 50 0
0 40
600
0
20
600
Days
800
40
200 100
E2
0 40 20
0
200
400
600
Antibodies
Days
0
500
0
1000
Virus
700
60
Infected cells
Uninfected cells
Figure 3. Demonstration of the global stability of the infection equilibrium with only humoral immunity Z E2 for R0 = 1.8036 > 1, RW 1 = 2.5099 > 1, and R2 = 0.2000 ≤ 1.
600 500 400
0
500
1000
1500
Antibodies
Virus
400 200 0
500
1500
1000
CTL cells 1000
Days
1500
2000
20 10 0
500
1000
2000
E3
1000 0 40 20
500
2000
Days
500
0
1500
Days
30
0
2000
1000
0
500
Days
1500
CTL cells
1000
0
40
600
0
20 0
2000
Days
800
40
1500
2000
Antibodies
0
0
500
1000
Virus
Figure 4. Demonstration of the global stability of the infection equilibrium with only cellular immunity E3 for R0 = 2.2923 > 1, R1Z = 3.8301 > 1, and RW 3 = 0.8292 ≤ 1.
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700
60
Infected cells
Uninfected cells
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600 500 400
0
200
200
0
200
400
200
400
600
CTL cells 400
Days
800
50 0
200
400
1000 500
E4
0 200 100
200
600
100
Days
500
0
800
150
0
800
600
Days
Days
1000
0
0
200
400
0
20 0
800
Antibodies
Virus
600
Days
600
CTL cells
400
40
600
800
Antibodies
0
0
200
400
600
Virus
Figure 5. Demonstration of the global stability of the infection equilibrium with both arms of immunity E4 for R0 = 2.2923 > 1, R2Z = 1.8783 > 1, and RW 3 = 8.2918 > 1.
5. Conclusions In this paper, we have modeled the role of the adaptive immunity in HIV infection by proposing a new mathematical model that takes into account the classical virus-to-cell infection, the direct cell-to-cell transmission, and the two kinds of delays during infection processes and virus production. By a rigorous mathematical analysis, we have proved that the global dynamics of the proposed model is fully determined by five threshold parameters, which are the basic reproduction number R0 , and Z the reproduction numbers for humoral immunity RW 1 , for cellular immunity R1 , for cellular immunity Z W in competition R2 , and for humoral immunity in competition R3 . More precisely, the infection-free equilibrium is globally asymptotically stable if R0 ≤ 1, which biologically means that the HIV is cleared and the infection dies out. When R0 > 1, our model has four infection equilibria, which are the following: (i ) the infection equilibrium without immunity is globally asymptotically stable if RW 1 ≤1 Z and R1 ≤ 1; (ii ) the infection equilibrium with only humoral immunity is globally asymptotically Z stable if RW 1 > 1 and R2 ≤ 1; (iii ) the infection equilibrium with only cellular immunity is globally asymptotically stable if R1Z > 1 and RW 3 ≤ 1; (iv ) the infection equilibrium with both cellular and humoral immune responses is globally asymptotically stable if R2Z > 1 and RW 3 > 1. Biologically, this implies that the HIV persists and the infection becomes chronic when the basic reproduction number is greater than 1. Additionally, the activation of one or both arms of immunity is unable to eliminate the virus in the human body. From Remark 2, we can deduce that the over-domination of cellular immunity leads to the absence of the humoral immunity, and the over-domination of the humoral immunity leads to the persistence of HIV infection with a weak response of both arms of immunity. This biological result may be an explanation for the dysfunction of the adaptive immunity in HIV-infected patients. Our model can be adapted for HBV infection, and the above result can also explain the dysfunction of the adaptive immune response in patients infected with HBV, which is still largely incomplete [33]. Furthermore, the model and the results presented in this study improve and generalize the models and the corresponding results in more recent papers with only cellular immunity [5], with only humoral immunity [6–9], and with both arms of immunity [15,17].
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Author Contributions: Both authors contributed equally to the writing of this paper. They both read and approved the final version of the manuscript. Acknowledgments: We would like to thank the editor and anonymous referees for their very helpful comments and suggestions that greatly improved the quality of this study. Conflicts of Interest: The authors declare no conflict of interest.
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