Modeling the Adaptive Immunity and Both Modes of Transmission in HIV Infection

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.


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: 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.
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: 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 −α 1 τ represents the probability of surviving from time t − τ to time t, where α 1 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.

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: 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 variable with a probability distribution f 2 (τ).The term e −α 2 τ denotes the probability of surviving the immature virions during the delay period, where 1 α 2 is the average lifetime of an immature virus.Therefore, the integral ∞ 0 f 2 (τ)e −α 2 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: We define the Banach space for the fading memory type as follows: where α is a positive constant and IR 5 + = {(x 1 , ..., x 5 ) : x i ≥ 0, i = 1, ..., 5}.
Proof.By the fundamental theory of functional differential equations [10][11][12], the model given by Equation (3) with initial condition φ ∈ C α has a unique local solution on [0, t max ), where t max is the maximal existence time for the solution of Equation ( 3).First, we prove that T(t) > 0 for all t ∈ [0, t max ).In fact, supposing the contrary, we let t 1 > 0 be the first time such that T(t 1 ) = 0 and Ṫ(t 1 ) ≤ 0. By the first equation of the model given by Equation (3), we have Ṫ(t 1 ) = λ > 0, which is a contradiction.Thus, T(t) > 0 for all t ∈ [0, t max ).By Equation (3), we have Then T(t) is bounded.Let Because T(t) is bounded and is well defined and differentiable with respect to t.Hence, where δ 1 = min{a, b, d} and Thus, G 1 (t) ≤ M := max{G 1 (0), λη 1 δ 1 }, which implies that I(t) and Z(t) are bounded.It remains to prove that V(t) and W(t) are bounded.To this end, we consider where δ 2 = 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 t max = +∞ and that the solution exists globally.
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 E 0 (T 0 , 0, 0, 0, 0), where T 0 = λ d .Therefore, the basic reproduction number of Equation ( 3) can be defined as As in [13], R 0 can be rewritten as is the basic reproduction number corresponding to the virus-to-cell infection mode, and R 02 = β 2 λη 1 da is the basic reproduction number corresponding to the cell-to-cell transmission mode.When R 0 > 1, Equation (3) has another infection equilibrium without immunity, E 1 (T 1 , I 1 , V 1 , 0, 0), where If both humoral and cellular immune responses have not been established, we have gV 1 − h ≤ 0 and cI 1 − b ≤ 0. Thus, we define the reproduction number for humoral immunity: and the reproduction number for cellular immunity: Hence, 3) has an infection equilibrium with only humoral immunity, E 2 (T 2 , I 2 , V 2 , W 2 , 0), where we easily deduce that W 2 > 0. If cellular immunity has not been established, we have For this, we define the reproduction number for cellular immunity in competition as 3) has an infection equilibrium with only cellular immunity, E 3 (T 3 , I 3 , V 3 , 0, Z 3 ), where We have that T 3 , I 3 , and V 3 are positive.It suffices to check that Z 3 is positive.
we deduce that Z 3 > 0. If humoral immunity has not been established, we have gV 3 − h ≤ 0. In this case, we define the reproduction number for humoral immunity in competition as which implies that 3) has an infection equilibrium with both cellular and humoral immune responses, E 4 (T 4 , I 4 , V 4 , W 4 , Z 4 ), where We have T 4 > 0, I 4 > 0, V 4 > 0, and Summarizing the above discussions, we obtain the following theorem.

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 E 0 is globally asymptotically stable when R 0 ≤ 1.

Proof.
To study the global stability of E 0 , we consider a Lyapunov functional defined as follows: 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 L 0 is nonnegative.
In order to simplify the presentation, we use the following notations: Ψ = Ψ(t) and Ψ τ = Ψ(t − τ) for any Ψ ∈ {T, I, V, W, Z}.Calculating the time derivative of L 0 along the positive solution of Equation ( 3), we obtain It is straightforward to show that the largest invariant set in {(T, I, V, W, Z)| dL 0 dt = 0} is {E 0 }.By LaSalle's invariance principle [14], the infection-free equilibrium E 0 is globally asymptotically stable when R 0 ≤ 1.
When R 0 > 1, Equation (3) has four infection steady states E i , 1 ≤ i ≤ 4. The following theorem characterizes the global stability of these steady states.Theorem 4. Assume R 0 > 1.

(i) The infection equilibrium without immunity E
Proof.For (i), consider the following Lyapunov functional: Then I 1 and kη 2 I 1 = µV 1 , we obtain Thus, ≤ 0 with equality if and only if T = T 1 , I = I 1 , V = V 1 , W = 0, and Z = 0.It follows from LaSalle's invariance principle that E 1 is globally asymptotically stable.
For (ii), consider the following Lyapunov functional: Hence, Thus, Because R Z 2 ≤ 1 and Φ(x) ≥ 0 for x > 0, we have ) ≤ 0 with equality if and only if T = T 2 , I = I 2 , and V = V 2 .Then İ = 0 and V = 0, which leads to Z = 0 and W = W 2 .Therefore, the largest compact invariant set in {(T, I, V, W, Z)| dL 2 dt = 0} is the singleton {E 2 }, and the proof of (ii) is completed.
Finally, we show (iv) by considering the following Lyapunov functional: we obtain Hence, Thus, ≤ 0 with equality holds if and only if T = T 4 , I = I 4 , and From the second and third equations of the model given by Equation ( 3), we have which implies that Z = Z 4 and W = W 4 .Then the largest compact invariant set in Γ is the singleton {E 4 }.Therefore, E 4 is globally asymptotically stable.
The conditions of the global stability of E 2 and those of E 3 given in (ii) and (iii) of Theorem 4 do not hold simultaneously.In fact, supposing the contrary, then This is a contradiction with R Z 2 ≤ 1.According to Equation (12) and Theorem 4, we have the following important result.
, the humoral immunity is dominant, and Equation (3 , the cellular immunity is dominant, and Equation (3) converges to E 3 without humoral immunity.
From this important remark, we can define the over-domination of humoral immunity when R Z 2 > 1 and R W 3 > 1 and the over-domination of cellular immunity when R Z 2 > 1 and R W 3 < 1.

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 − τ 1 ) and f 2 (τ) = δ(τ − τ 2 ) with τ 1 and τ 2 to be the delays in cell infection and virus production, respectively, and δ(.) to be the Dirac function; then our model becomes 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, τ 2 = 0.In addition, Equation ( 13) includes many special cases existing in the literature.For example, when β 2 = 0 and τ 1 = τ 2 = 0, we obtain the model presented by Wodarz in [15] and analyzed by Hattaf et al. in [16].When β 2 = 0 and τ 2 = 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 (m 1 , m 2 ) ∈ I N 2 with τ 1 = m 1 ∆t and τ 2 = m 2 ∆t.The grids points are t n = n∆t for n ∈ I N.By applying the mixed Euler method and using the approximations and Z(t n ) ≈ Z n , we obtain the following discrete model: where the discrete initial conditions are The five threshold parameters R 0 , R W 1 , R Z 1 , R Z 2 , and R W 3 for Equation ( 13) are given by Equations ( 7)- (11) with η 1 = e −α 1 τ 1 and η 2 = e −α 2 τ 2 .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.For the case in which β 2 = 10 −6 , g = 10 −5 , and c = 0.002, we obtained R 0 = 0.9751.It follows from Theorem 2 that Equation ( 13) has one infection-free equilibrium E 0 (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 T 0 = 719.4245,while the concentrations of infected cells, free HIV particles, antibodies, and CTL cells decreased and tended to zero.This means that E 0 is globally asymptotically stable and that the virus will be cleared.This confirms the result in Theorem 3.
1 is globally asymptotically stable if R W 1 ≤ 1 and R Z 1 ≤ 1. (ii)The infection equilibrium with only humoral immunity E 2 is globally asymptotically stable if R W The infection equilibrium with both cellular and humoral immune responses E 4 is globally asymptotically stable

Table 1 .
List of parameters.