Dynamics of HIV-1/HTLV-I Co-Infection Model with Humoral Immunity and Cellular Infection

Human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses which infect the same target, CD4+ T cells. This type of cell is considered the main component of the immune system. Since both viruses have the same means of transmission between individuals, HIV-1-infected patients are more exposed to the chance of co-infection with HTLV-I, and vice versa, compared to the general population. The mathematical modeling and analysis of within-host HIV-1/HTLV-I co-infection dynamics can be considered a robust tool to support biological and medical research. In this study, we have formulated and analyzed an HIV-1/HTLV-I co-infection model with humoral immunity, taking into account both latent HIV-1-infected cells and HTLV-I-infected cells. The model considers two modes of HIV-1 dissemination, virus-to-cell (V-T-C) and cell-to-cell (C-T-C). We prove the nonnegativity and boundedness of the solutions of the model. We find all steady states of the model and establish their existence conditions. We utilize Lyapunov functions and LaSalle’s invariance principle to investigate the global stability of all the steady states of the model. Numerical simulations were performed to illustrate the corresponding theoretical results. The effects of humoral immunity and C-T-C transmission on the HIV-1/HTLV-I co-infection dynamics are discussed. We have shown that humoral immunity does not play the role of clearing an HIV-1 infection but it can control HIV-1 infection. Furthermore, we note that the omission of C-T-C transmission from the HIV-1/HTLV-I co-infection model leads to an under-evaluation of the basic HIV-1 mono-infection reproductive ratio.


Introduction
Human immunodeficiency virus type 1 (HIV-1) is a retrovirus that attacks and infects healthy CD4 + T cells; the crucial components of the human immune system. HIV-1 causes a fatal infectious disease called acquired immunodeficiency syndrome (AIDS). The World Health Organization (WHO) reported that there were about 36.7 million people living with HIV-1 at the end of 2016, and 1.8 million people become newly infected globally in 2016 [1]. In vivo, HIV-1 has two modes of dissemination, virus-to-cell (V-T-C) and cell-to-cell (C-T-C). In V-T-C dissemination, HIV-1 particles emitted from HIV-1-infected CD4 + T cells search for new healthy CD4 + T cells to infect. In the C-T-C mode of dissemination, HIV-1 can be transferred from HIV-1-infected CD4 + T cells to healthy CD4 + T cells via direct contact through the formation of virological synapses. Many studies have shown that HIV-1 propagation in the case of direct C-T-C dissemination is more efficient and potent than in The modeling of HIV-1 infection with V-T-C and C-T-C modes of dissemination has attracted the attention of many researchers who have included additional biological mechanisms in model (1), such as: • Time delay models: In reality, biological transitions such as infection interactions are not instantaneous but take time. In virology, intracellular delay accounts for the time of initial infection until the production of new virions. Lai and Zou [18] studied an HIV-1 infection model with C-T-C dissemination and two types of distributed time delays. Adak and Bairagi [19] investigated an HIV-1 infection model with C-T-C dissemination and discrete delays. They assumed logistic growth for healthy CD4 + T cells and a saturated incidence rate for V-T-C infection in the form ψ 1 H n P a n + H n , where n ≥ 1 and a > 0. • Latent infected cell models: The impact of latent infected cells and antiretroviral therapy on the dynamics of HIV-1 with C-T-C dissemination was studied in [20]. Wang et al. [21] included latent infected cells and intracellular delays into their model of HIV-1 dynamics with C-T-C dissemination. In [20,21], both the local and global stability of steady states were investigated. • CTL immune response models: Guo and Qiu [22] included the CTL immune response, latent infected cells and antiretroviral therapy in their model (1). Wang et al. [23] investigated the global stability of HIV-1 dynamics with C-T-C dissemination, a CTL immune response and a distributed delay. The model presented in [23] was generalized by Yan et al. [24], considering (i) two distributed delays and (ii) general functions for the V-T-C and C-T-C infection rates and the production/stimulation and removal of cells and HIV-1 particles. Elaiw and AlShamrani [25] investigated HIV-1 dynamics more generally in relation to the CTL immune response in cases where C-T-C dissemination is caused by both latent and active infected cells. • Diffusion models: Ren et al. [26] addressed the effects of C-T-C dissemination and the mobility of viruses and cells on HIV-1 dynamics. Gao and Wang [27] investigated a reaction-diffusion HIV-1 dynamics model with delay and C-T-C dissemination. In [28], a diffusive viral infection model was developed, assuming that each latent and active infected cell collaborated in C-T-C infection. Sun and Wang [29] presented a diffusive HIV-1 infection model with C-T-C dissemination, assuming that the V-T-C infection rate could be expressed by a general function F(H, P). • Age-structured models: Wang et al. [30] analyzed an age-structured HIV-1 infection model with C-T-C dissemination.

HIV-1 Mono-Infection Model with C-T-C Dissemination and Humoral Immunity
The HIV-1 infection model with C-T-C dissemination and a humoral immune response can be formulated as: where B = B(t) denotes the concentration of HIV-1-specific antibodies at time t. The proliferation rate for HIV-1-specific antibodies is given by ηBP. The HIV-1 particles are neutralized by HIV-1-specific antibodies at a rate of πBP. The death rate of HIV-1-specific antibodies is represented by λB. Lin et al. [31,32] extended model (2) by considering an intracellular discrete-time delay. The effect of B-cell impairment on HIV-1 infection with C-T-C and a distributed time delay was investigated by Elaiw and Alshehaiween [33]. Guo et al. [34] extended model (2) by incorporating two intracellular discrete delays and both CTL and humoral immune responses.

HIV-1/HTLV-I Co-Infection Models
In recent works, Elaiw and AlShamrani [52][53][54] studied HIV-1/HTLV-I co-infection models with a CTL immune response. Alshaikh et al. [55] investigated HIV-1/HTLV-I co-infection models with humoral immunity by assuming that the healthy CD4 + T cells were infected by HIV-1 only via V-T-C dissemination. Our aim in this paper was to develop an HIV-1/HTLV-I co-infection model with humoral immunity and both modes of HIV-1 infection, V-T-C and C-T-C taking into account both latent HIV-1-infected CD4 + T cells and latent HTLV-I-infected CD4 + T cells. We prove the nonnegativity and boundedness of the solutions of the models. We utilize the Lyapunov method to investigate the global stability of all steady states of the models. We illustrate the theoretical results with numerical simulations.

HIV-1/HTLV-I Co-Infection Model with Latent Infected Cells
In this section, we present the following system of ordinary differential equations (ODEs), which describe the interactions between seven compartments: with initial conditions (5) where I L and Y L represent, respectively, latent HIV-1-infected CD4 + T cells and latent HTLV-I-infected CD4 + T cells. The terms εI L and ρY L represent the activation rates of latent HIV-1-infected and latent HTLV-I-infected CD4 + T cells, respectively. The fraction coefficient δ ∈ (0, 1) is the probability that new HIV-1-infected CD4 + T cells could be active and the remaining fraction 1 − δ will be latent. The natural death rates of latent HIV-1-infected CD4 + T cells and latent HTLV-I-infected CD4 + T cells are demonstrated by θ I L and Y L , respectively. Table 1 summarizes the biological meanings of all variables and parameters.

Symbol Biological Description
Populations H Healthy CD4 + T cells Next, we will determine a bounded domain for the concentrations of the model's compartments to ensure that our model is biologically acceptable. Particularly, the concentrations should not become negative or unbounded.

Properties of Solutions
Lemma 1. All Solutions (H(t), I L (t), I A (t), Y L (t), Y A (t), P(t), B(t)) of system (4) with initial conditions (5) are nonnegative and ultimately bounded.
Proof. According to (4), we have It follows from Proposition B.7 of [56] that I L (t), I A (t), Y L (t), Y A (t), P(t), B(t) ≥ 0 for all t ≥ 0 whenever the initial conditions (5) are satisfied. Next, we aim to show the ultimate boundedness of the solutions. Form the first equation of system (4) we have dH dt ≤ ξ − αH and this implies that lim Let us define a function Ψ as: Then It follows that where Ω 3 = τΩ 2 , Ω 4 = 2κΩ 2 γ and Ω 5 = 2ηκΩ 2 πγ .
It can be verified that the compact set is positively invariant for system (4).

Steady States and Threshold Parameters
In this section, we find all steady states of the model and establish their existence in terms of four threshold parameters. To calculate the steady states of model (4), we solve We find that system (4) has five steady states: (i) The infection-free steady state,∆ 0 = (H 0 , 0, 0, 0, 0, 0, 0), whereH 0 = ξ/α. This steady state describes the case of a healthy state where both HIV-1 and HTLV-I are cleared out from the body.
(iii) The infected HTLV-I mono-infection steady state, and¯ 2 is the basic HTLV-I mono-infection reproductive ratio for system (4) and is defined as: (iv) The infected HIV-1 mono-infection steady state with efficacious humoral immunity, andĪ A 3 satisfies the quadratic equation Sinceκ 1 > 0 andκ 3 < 0, thenκ 2 2 − 4κ 1κ3 > 0 and Equation (8) has a positive root The HIV-1-specific humoral immunity reproductive ratio in the case of HIV-1 mono-infection is given as: The infected HIV-1/HTLV-I co-infection steady state with efficacious humoral immunity, We note that ∆ 4 exists when¯ 1 2 > 1,¯ * 4 > 1 and¯ 4 > 1. The competed HTLV-I reproductive ratio in the case of HIV-1/HTLV-I co-infection is stated as: According to the above discussion, we sum up the existence conditions for all steady states in Table 2.
We will use the following geometric-arithmetic mean inequality: Define functionΓ j (H, I L , I A , Y L , Y A , P, B) and letΠ j be the largest invariant subset of Theorem 1. (a) Assume that¯ 1 ≤ 1 and¯ 2 ≤ 1; then∆ 0 is globally asymptotically stable (GAS).
Proof. (a) Construct a functionΓ 0 (H, I L , I A , Y L , Y A , P, B) as: We calculate dΓ 0 dt as: Furthermore, we have dI A (t) dt = 0 and, from the third equation of system (4), we obtain which indicates that I L (t) = 0 for all t. Therefore,Π 0 = ∆ 0 and, applying the Lyapunov-LaSalle asymptotic stability theorem [60][61][62], we can observe that∆ 0 is GAS. To prove (b), we need to find the characteristic equation atthe steady state. We calculate the Jacobian matrix J = J(H, I L , I A , Y L , Y A , P, B) of system (4) in the following form: Then, the characteristic equation at the steady state∆ 0 is given by where ∆ is the eigenvalue and Clearly, if¯ 2 > 1, then Equation (11) has a positive root and hence∆ 0 is unstable.
Proof. Consider a functionΓ 2 (H, I L , I A , Y L , Y A , P, B) as: We calculate dΓ 2 dt as: Utilizing the steady state conditions for∆ 2 : we obtain If¯ 1 2 ≤ 1, then, applying inequality (9), we obtain dΓ 2 dt ≤ 0 in Θ with dΓ 2 dt = 0 when H =H 2 , Y L =Ȳ L 2 , Y A =Ȳ A 2 and I A = B = 0. The solutions of system (4) tend to the invariant set Π 2 which contains elements with H(t) =H 2 , Y A (t) =Ȳ A 2 , I A (t) = 0, then dH(t) dt = 0. The first equation of system (4) leads to Furthermore, we have dI A (t) dt = 0 and the third equation of system (4) yields Therefore,Π 2 = ∆ 2 and∆ 2 is GAS using the Lyapunov-LaSalle asymptotic stability theorem.
Proof. Define a functionΓ 3 (H, I L , I A , Y L , Y A , P, B) as: We calculate dΓ 3 dt as: We collect the terms of Equation (15) as: The steady state conditions for∆ 3 give: Then, we obtain In this case, It follows that This happens only whenH 3 ≤ ϕ(ρ+ ) ρτψ 3 =H 4 . Clearly, dΓ 3 dt ≤ 0 in Θ with dΓ 3 dt = 0 when H =H 3 , I L =Ī L 3 , I A =Ī A 3 , P =P 3 and Y A = 0. The solutions of system (4) tend to the invariant setΠ 3 which consists of elements satisfyingΠ 3 , we have I A (t) =Ī A 3 , P(t) =P 3 , Y A (t) = 0 and then dY A (t) dt = 0, dP(t) dt = 0. The fifth and sixth equations of system (4) give Therefore,Π 3 = ∆ 3 and, from the Lyapunov-LaSalle asymptotic stability theorem, we observe that∆ 3 is GAS.
Proof. DefineΓ 4 (H, I L , I A , Y L , Y A , P, B) as: Calculating dΓ 4 dt as: Equation (16) can be simplified as: The steady state conditions for∆ 4 yield Then, we obtain

Effect of Humoral Immunity on the HIV-1/HTLV-I Co-Infection Dynamics
In this subsection, we study the impact of HIV-1-specific antibodies on the HIV-1/HTLV-I co-infection dynamics. We note that the stability of the infection-free steady state∆ 0 depends on the parameters¯ 1 and¯ 2 . These parameters do not depend on the proliferation of the HIV-1-specific antibodies η. Therefore, HIV-1-specific antibodies do not play the role of clearing the HIV-1 infection, but they have an important role in controlling and suppressing HTLV-I infection. To observe the effect of HIV-1-specific antibodies on the solutions of the model, we fixed the parameters ψ 1 = ψ 2 = 0.0005, ψ 3 = 0.003 and varied the parameter η. We chose the following initial conditions: IS-4: (H, I L , I A , Y L , Y A , P, B)(0) = (500, 3, 1.5, 0.8, 3, 2, 3). We can see from Figure 6 that when η is increased, the concentrations of HIV-1 particles and latent/active HIV-1-infected CD4 + T cells are decreased, whereas the concentrations of latent/active HTLV-I-infected CD4 + T cells are increased. Therefore, HIV-1-specific antibodies can control HIV-1 infection, but they may enhance the progression of HTLV-I.

Conclusions and Discussion
In this article, we have studied a within-host HIV-1/HTLV-I co-infection model with humoral immunity with both V-T-C and C-T-C modes of transmission. We have presented some preliminary results regarding the positivity and boundedness of the models' solutions. By constructing suitable Lyapunov functions and using LaSalle's invariance principle, we have identified four threshold parameters for the global stability of steady states. More precisely, it has been shown that, if¯ 1 ≤ 1 and¯ 2 ≤ 1, then the infection-free steady state∆ 0 is GAS; if¯ 1 > 1,¯ 2 1 ≤ 1 and¯ 3 ≤ 1, then the infected HIV-1 mono-infection steady state with inefficacious humoral immunity∆ 1 is GAS; if¯ 2 > 1 and¯ 1 2 ≤ 1, then the infected HTLV-I mono-infection steady state∆ 2 is GAS; If¯ 3 > 1 and¯ 4 ≤ 1, then the infected HIV-1 mono-infection steady state with efficacious humoral immunity∆ 3 is GAS; and if¯ 1 2 > 1,¯ * 4 > 1 and¯ 4 > 1, then the infected HIV-1/HTLV-I co-infection steady state with efficacious humoral immunity∆ 4 is GAS. Numerical simulations have been provided to show the strength and credibility of our theoretical results.
Here 1 denotes the basic HIV-1 mono-infection reproductive ratio for system (17) that corresponds to V-T-C only. Let us consider 2 ≤ 1. We note that the incorporation of C-T-C transmission into the dynamics causes an increase in the parameter¯ 1 , sincē

Conflicts of Interest:
The authors declare no conflict of interest.