key: cord-0752181-yyj0iemn
authors: Elaiw, A. M.; AlShamrani, N. H.
title: Modeling and analysis of a within-host HIV/HTLV-I co-infection
date: 2021-03-29
journal: Bol Soc Mat Mex
DOI: 10.1007/s40590-021-00330-6
sha: f787d1037ca92721443f81aeae0ac83e93c1f6c3
doc_id: 752181
cord_uid: yyj0iemn

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that attack the [Formula: see text] T cells and impair their functions. Both HIV and HTLV-I can be transmitted between individuals through direct contact with certain body fluids from infected individuals. Therefore, a person can be co-infected with both viruses. HIV causes acquired immunodeficiency syndrome (AIDS), while HTLV-I is the causative agent for adult T-cell leukemia (ATL) and HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP). Several mathematical models have been developed in the literature to describe the within-host dynamics of HIV and HTLV-I mono-infections. However, modeling a within-host dynamics of HIV/HTLV-I co-infection has not been involved. The present paper is concerned with the formulation and investigation of a new HIV/HTLV-I co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible [Formula: see text] T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell transmission. On the other side, HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact through the virological synapse, and (ii) vertical transmission through the mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We define a set of threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and Lyapunov–LaSalle asymptotic stability theorem. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

Mathematics Subject Classification 34D20 Á 34D23 Á 37N25 Á 92B05

Nowadays, humans are vulnerable to infection with many different viruses such as human immunodeficiency virus (HIV), human T-lymphotropic virus type I (HTLV-I), hepatitis B virus (HBV), hepatitis C virus (HCV), dengue virus and lastly coronavirus. These viruses cause many fatal diseases. HIV is a retrovirus that infects the susceptible CD4 þ T cell and destroys its functions. Acquired immunodeficiency syndrome (AIDS) is the advanced stage of HIV infection. An individual can be infected with HIV by direct contact with certain body fluids (blood, semen (cum), pre-seminal fluid (pre-cum), vaginal fluids, rectal fluids and breast milk) from an HIV-infected individual. Till now, the available antiviral treatments can significantly suppress HIV replication but they can not eliminate the HIV from the body. According to global health observatory (GHO 2018) data of HIV/AIDS published by WHO [1] that says, globally, about 37.9 million HIV-infected people in 2018, 1.7 million newly HIV-infected and 770,000 HIV-related death in the same year. During the last decades, mathematical modeling of within-host HIV infection has witnessed a significant development. Nowak and Bangham [2] have introduced an initial HIV infection model to describe the interaction between three compartments, susceptible CD4 þ T cells (S), active HIV-infected cells (I) and free HIV particles (V). Silent viral reservoirs remain one of the major hurdles for eradicating the HIV by current antiviral therapy [3] . Silent HIV-infected cells include HIV virions but do not produce them until they become activated. Mathematical modeling of HIV dynamics with silent infected cells can help in predicting the effect of antiviral drug efficacy on HIV progression [4] . Rong and Perelson [5] have included the silent infected cells in the initial HIV model presented in [2] as:

where S ¼ SðtÞ; L ¼ LðtÞ, I ¼ IðtÞand V ¼ VðtÞ are the concentrations of susceptible CD4 þ T cells, silent HIV-infected cells, active HIV-infected cells and free HIV particles at time t, respectively. The susceptible CD4 þ T cells are produced at specific constant rate q. The HIV virions can replicate using virus-to-cell transmission. The term g 1 SV refers to the rate at which new infectious appears by virusto-cell contact between free HIV particles and susceptible CD4 þ T cells. Silent HIVinfected cells are transmitted to be active at rate kL. The free HIV particles are generated at rate bI. The natural death rates of the susceptible CD4 þ T cells, silent HIV-infected cells, active HIV-infected cells and free HIV particles are given by aS, cL; aI and eV, respectively. A fraction b 2 ð0; 1Þ of new HIV-infected cells will be active, and the remaining part 1 À b will be silent. Over past decades, mathematical modeling and analysis of HIV mono-infection with both silent and active HIV-infected cells have witnessed a significant development (see e.g., [6] [7] [8] [9] [10] [11] [12] [13] and the review article [3] ). HTLV-I can lead to two diseases, adult T-cell leukemia (ATL) and HTLV-Iassociated myelopathy/tropical spastic paraparesis (HAM/TSP). HTLV-1 can be transmitted to human from sexual contact, needle sharing, contaminated blood products and breastfeeding [14] . HTLV-I is a global epidemic that infects about 10-25 million persons [15] . The infection is endemic in the Caribbean, southern Japan, the Middle East, South America, parts of Africa, Melanesia and Papua New Guinea [16] . HTLV-I is a provirus that targets the susceptible CD4 þ T cells. HTLV-I is a single-stranded RNA retrovirus that reverse transcribe its RNA genome into a proviral DNA copy which in turn reaches the host chromatin and integrates into the DNA of the host genome, at which point the virus is referred to as a provirus. Later, cell infected with this virus enters a silent period and it is not capable to produce DNA and infects susceptible cells. Although, silent HTLV-infected cells can survive for a long-lasting time, however, they may be suddenly activated by antigen and become able to infect susceptible cells. During the primary infection stage of HTLV-I, the proviral load can reach high level, approximately 30-50% [17] . Unlike in the case of HIV infection, however, only a small percentage of infected individuals develop the disease and 2-5% percent of HTLV-I carriers develop symptoms of ATL and another 0.25-3% develop HAM/TSP [18] . Many researchers have been concerned to study mathematical modeling and analysis of HTLV-I mono-infection in several works [19] [20] [21] .

There are some differences between HIV and HTLV-I. HIV can break free from a CD4 þ T cell and infect other susceptible CD4 þ T cell, while cell-free HTLV-I does not trigger infection. HTLV-I has two modes of transmission, the first is the horizontal transmission via direct cell-to-cell contact through the virological synapse [22] , and the second is the vertical transmission through the mitotic division of Tax-expressing HTLV-infected cells [23] . Tax-expressing HTLV-infected cells proliferate faster than susceptible CD4 þ T cells and silent HTLV-infected cells. This leads to an increase of proviral load. Therefore, vertical mitotic transmission plays an important role in the persistence of HTLV-I infection [23] . Li and Lim [24] have formulated an HTLV-I dynamics model that takes into account both horizontal and vertical routs of transmission as:

where S ¼ SðtÞ; E ¼ EðtÞ and Y ¼ YðtÞ are the concentrations of susceptible CD4 þ T cells, silent HTLV-infected cells and Tax-expressing HTLV-infected cells, at time t, respectively. The rate at which new infectious appears by cell-to-cell contact between Tax-expressing HTLV-infected cells and susceptible CD4 þ T cells is assumed to be g 2 SY. The fraction u 2 ð0; 1Þ is the probability of new HTLV infections via horizontal transmission could enter a silent period. The other route of transmission for HTLV is the vertical caused by selective expansion of Tax-expressing CD4 þ T cells that are driven into proliferation by HTLV Tax gene at a rate r à Y 1 À SþE K À Á , where K is the CD4 þ T cells carrying capacity. The term jr à Y 1 À SþE K À Á accounts for the HTLV-infected cells that being silent and, therefore, escaping from the immune system, where j 2 ð0; 1Þ. The natural death rates of the silent HTLV-infected cells and Tax-expressing HTLV-infected cells are represented by xE and d à Y, respectively. The term wE accounts for the rate of silent HTLVinfected cells that become Tax-expressing HTLV-infected cells. Asquith and Bangham [25] have been reported that, even in the presence of rapid selective mitotic division, target cell populations are less than the total CD4 þ T cells carrying capacity i.e. S þ E\K. Therefore, Lim and Maini [15] have replaced the logistic term r à Y 1 À SþE K À Á by an exponential growth term r à Y. Cytotoxic T lymphocytes (CTLs) are recognized as the significant component of the human immune response against viral infections. CTLs inhibit viral replication and kill the cells which are infected by viruses. In fact, CTLs and antibodies are necessary and universal to control HIV infection for years [26] . The incorporation of the immune response in the HIV dynamics models gives us a better understanding of within-host HIV dynamics. During recent years, great efforts have been made to formulate and analyze the within-host HIV mono-infection models under the influence of CTL immune response (see e.g. [2] and [27] ). In [28, 29] , silent HIVinfected cells have been included in the HIV dynamics models with CTL immune response. In the case of HTLV-I infection, it has been reported in [25] and [30] that the CTLs play an effective role in controlling such infection. CTLs can recognize and kill the Tax-expressing HTLV-infected cells, moreover, they can reduce the proviral load. In the literature, several mathematical models have been proposed to describe the dynamics of HTLV-I under the effect of CTL immune response (see e.g. [16] and [31] [32] [33] [34] ). HTLV-I dynamics models with the mitotic division of Taxexpressing HTLV-infected cells and CTL immune response have been developed in [15, [35] [36] [37] . Li and Zhou [36] have assumed that Tax-expressing HTLV-infected cells proliferate at rate r à Y, with 1 À j ð Þr à Y staying in the Tax-expressing HTLVinfected cells compartment, while jr à Y being silent and, therefore, escaping from the immune system.

Simultaneous infection by HIV and HTLV-I and the etiology of their pathogenic and disease outcomes have become a global health matter over the past 10 years. The importance of studying HIV/HTLV-I co-infection comes from the fact that both viruses share the same ways of transmission in a population as mentioned above. This means that co-infection with both viruses can occurred in the areas where both viruses are endemic [38] . Although CD4 þ T cells are the major targets of both HIV and HTLV-I, however, these viruses present a different biological behavior that causes diverse impacts on host immunity and ultimately leads to numerous clinical diseases [39] . It has been reported that the HTLV-I co-infection rate among HIV infected patients as increase as 100 to 500 times in comparison with the general population [40] . In seroepidemiologic studies, it has been recorded that HIVinfected patients are more exposure to be co-infected with HTLV-I, and vice versa compared to the general population [41] . HIV/HTLV-I co-infection is usually found in individuals of specific ethnic or who belonged to geographic origins where these viruses are simultaneously endemic [42] . As an example, the co-infection rates in individuals living in Bahia have reached 16% of HIV-infected patients [43] . The prevalence of dual infection with HIV and HTLV-I has become more widely in several geographical regions throughout the world such as South America, Europe, the Caribbean, Bahia (Brazil), Mozambique (Africa), and Japan [41, 43] . HIV and HTLV-I dual infection appears to have an overlap on the course of associated clinical outcomes with both viruses [41] . Several reports have concluded that HIV/ HTLV-I co-infected patients were found to have an increase of CD4 þ T cells count in comparison with HIV mono-infected patients, although there is no evident to result in a better immune response [39, 44] . Indeed, simultaneously infected patients by both viruses with CD4 þ T counts greater than 200 cells/mm 3 are more exposure to have other opportunistic infections as compared with HIV mono-infected patients who have similar CD4 þ T counts [44] . Studies have reported that higher mortality and shortened survival rates were accompany with co-infected individuals more than mono-infected individuals [45] . Considering the natural history of HIV, many researchers have noted that co-infection with HIV and HTLV-I can accelerate the clinical progression to AIDS. On the other hand, HIV can adjust HTLV-I expression in co-infected individuals which leads them to a higher risk of developing HTLV-I related diseases such as ATL and TSP/HAM [40, 41, 45] .

Although a great number of mathematical HIV and HTLV-I mono-infections models has been developed and analyzed, however, to the best of our knowledge there is no mathematical model for HIV/HTLV-I co-infection. Therefore, our aim in the present paper is to formulate a new HIV/HTLV-I co-infection model. The HIV can spread via virus-to-cell transmission, while HTLV-I has two routes of transmission, (i) horizontal transmission via direct cell-to-cell contact through the virological synapse, and (ii) vertical transmission through the mitotic division of Tax-expressing HTLV-infected cells. We first show that the model is well-posed by establishing that the solutions of the model are nonnegative and bounded. We calculate all equilibria and derive a set of threshold parameters which govern the existence and stability of the equilibria of the model. We study the global stability of equilibria by constructing suitable Lyapunov functions and utilizing Lyapunov-LaSalle asymptotic stability theorem. We conduct some numerical simulations to illustrate the theoretical results.

We remark that, our proposed HIV/HTLV-I co-infection model can be developed and extended to incorporate different biological phenomena such as intracellular time delay [46] [47] [48] [49] [50] , reaction-diffusion [51, 52] and stochastic interactions [53] .

We set up an ordinary differential equation (ODE) model that describes the change of concentrations of eight compartments with respect to time t; susceptible (uninfected) CD4 þ T cells S(t), silent HIV-infected cells L(t), active HIV-infected cells I(t), silent HTLV-infected cells E(t), Tax-expressing HTLV-infected cells Y(t), free HIV particles V(t), HIV-specific CTLs C I ðtÞ and HTLV-specific CTLs C Y ðtÞ. The dynamics of HIV/ HTLV-I co-infection is schematically shown in the transfer diagram given in Fig. 1 .

Our proposed model is given by the following system of ODEs:

where ðS; L; I; E; Y; V; C I ; C Y Þ ¼ ðSðtÞ; LðtÞ; IðtÞ; EðtÞ; YðtÞ; VðtÞ; C I ðtÞ; C Y ðtÞÞ. The term l 1 C I I is the killing rate of active HIV-infected cells due to their specific immunity. The term l 2 C Y Y is the killing rate of Tax-expressing HTLV-infected cells due to their specific immunity. The proliferation and death rates for both effective HIV-specific CTLs and HTLV-specific CTLs are given by r 1 C I I, r 2 C Y Y, p 1 C I and p 2 C Y , respectively. All remaining parameters have the same biological meaning as explained in the previous section. Table 1 summarizes all parameters and their definitions.

In [15] , it is assumed that r à 	 à ¼ min a; x; d à f g , which corresponds to experimental evidence indicating that the proliferation rate of HTLV-infected cells is generally lower than the rate of removal due to natural death. Since r à \d à and 0\j\1, then 1 À j ð Þr à \d à and

Let d ¼ d à À 1 À j ð Þr à and r ¼ jr à . Then, system (3) will take the following form of ODEs: 

Let X j [ 0, j ¼ 1; . . .; 5 and define H ¼ ðS; L; I; E; Y; V; C I ; C Y Þ 2 R 8 ! 0 : 0 SðtÞ; LðtÞ; IðtÞ X 1 ; È 0 EðtÞ; YðtÞ X 2 ; 0 VðtÞ X 3 ; 0 C I ðtÞ X 4 ; 0 C Y ðtÞ X 5 É :

The compact set H is positively invariant for system (4) .

This insures that, ðSðtÞ; LðtÞ; IðtÞ; EðtÞ; YðtÞ; VðtÞ; C I ðtÞ; C Y ðtÞÞ 2 R 8 ! 0 for all t ! 0 when ðSð0Þ; Lð0Þ; Ið0Þ; Eð0Þ; Yð0Þ; Vð0Þ; C I ð0Þ; C Y ð0ÞÞ 2 R 8 ! 0 . To show the boundedness of all state variables, we let

where / ¼ minfa; c; a 2 ; x; d à À r à ; e; p 1 ; p 2 g. Hence, 0 WðtÞ X 1 if Wð0Þ X 1 for t ! 0; where X 1 ¼ q / : Since S, L, I, E, Y, V, C I , and C Y are all nonnegative then 0 SðtÞ; LðtÞ; IðtÞ X 1 , 0 EðtÞ; YðtÞ X 2 , 0 VðtÞ X 3 , 0 C I ðtÞ

In this section, we derive eight threshold parameters which guarantee the existence of the equilibria of the model. Let ðS; L; I; E; Y; V; C I ; C Y Þ be any equilibrium of system (4) satisfying the following equations:

0 ¼ bg 1 SV þ kL À aI À l 1 C I I; ð7Þ

The straightforward calculation finds that system (4) admits eight equilibria.

(1) Infection-free equilibrium, Ð 0 ¼ ðS 0 ; 0; 0; 0; 0; 0; 0; 0Þ, where S 0 ¼ q=a. This case describes the situation of healthy state where both HIV and HTLV-I are absent.

(2) Chronic HIV mono-infection equilibrium with inactive immune response,

Therefore, Ð 1 exists when

At the equilibrium Ð 1 the chronic HIV mono-infection persists while the immune response is unstimulated. The basic HIV mono-infection reproductive ratio for system (4) is defined as:

The parameter R 1 determines whether or not a chronic HIV infection can be established. In terms of R 1 , we can write

(3) Chronic HTLV mono-infection equilibrium with inactive immune response,

Therefore, Ð 2 exists when

At the equilibrium Ð 2 the chronic HTLV mono-infection persists while the immune response is unstimulated. The basic HTLV mono-infection reproductive ratio for system (4) is defined as:

The parameter R 2 decides whether or not a chronic HTLV infection can be established. In terms of R 2 , we can write

(4) Chronic HIV mono-infection equilibrium with only active HIV-specific CTL,

We note that Ð 3 exists when

The HIV-specific CTLmediated immunity reproductive ratio in case of HIV mono-infection is stated as:

The parameter R 3 determines whether or not the HIVspecific CTL-mediated immune response is stimulated in the absent of HTLV infection.

(5) Chronic HTLV mono-infection equilibrium with only active HTLV-specific CTL,

We note that Ð 4 exists when

The HTLVspecific CTL-mediated immunity reproductive ratio in case of HTLV mono-infection is stated as:

The parameter R 4 determines whether or not the HTLV-specific CTL-mediated immune response is stimulated in the absent of HIV infection. (6) Chronic HIV/HTLV co-infection equilibrium with only active HIV-specific CTL,

We note that Ð 5 exists when R 1 =R 2 [ 1 and

The HTLV infection reproductive ratio in the presence of HIV infection is stated as:

The parameter R 5 determines whether or not HIV-infected patients could be coinfected with HTLV. Thus,

Þ.

(7) Chronic HIV/HTLV co-infection equilibrium with only active HTLV-specific CTL,

We note that Ð 6 exists when

The HIV infection reproductive ratio in the presence of HTLV infection is stated as:

Thus,

The parameter R 6 determines whether or not HTLVinfected patients could be co-infected with HIV. (8) Chronic HIV/HTLV co-infection equilibrium with active HIV-specific CTL and HTLV-specific CTL,

It is obvious that Ð 7 exists when

Clearly, Ð 7 exists when R 7 [ 1 and R 8 [ 1 and we can write

The parameter R 7 refers to the competed HIVspecific CTL-mediated immunity reproductive ratio in case of HIV/HTLV co-infection. On the other hand, the parameter R 8 refers to the competed HTLV-specific CTL-mediated immunity reproductive ratio in case of HIV/HTLV co-infection. The eight threshold parameters are given as follows:

According to the above discussion, we sum up the existence conditions for all equilibria in Table 2 .

In this section, we prove the global asymptotic stability of all equilibria by constructing Lyapunov function and applying Lyapunov-LaSalle asymptotic stability theorem [54] [55] [56] . We will use the arithmetic-geometric mean inequality

Let a function U j ðS; L; I; E; Y; V; C I ; C Y Þ and ! 0 j be the largest invariant subset of

We define a function F ðtÞ ¼ t À 1 À ln t:

Proof Constructing a Lyapunov function candidate U 0 ðS; L; I; E; Y; V; C I ; C Y Þ:

It is seen that, 

0Þ Chronic HIV mono-infection equilibrium with only active HIV-specific CTL

Chronic HIV/HTLV co-infection equilibrium with only active HIV-specific CTL

Chronic HIV/HTLV co-infection equilibrium with only active HTLV-specific CTL 

The fifth and sixth equations of system (4) imply

In addition, from the third equation of system (4) we have

which yields LðtÞ ¼ 0 for all t. Therefore, and applying Lyapunov-LaSalle asymptotic stability theorem [54] [55] [56] 

Proof Define a function U 1 ðS; L; I; E; Y; V; C I ; C Y Þ as:

Calculating dU 1 dt as:

Using the equilibrium conditions for Ð 1 , we get

Then, we obtain

Therefore Eq. (16) becomes

Since R 2 =R 1 1 and R 3 1; then using inequalities (13)- (14) we get dU 1 dt 0 for all S; L; I; Y; V; C I ; C Y [ 0 with equality holding when S ¼ S 1 , L ¼ L 1 , I ¼ I 1 , V ¼ V 1 and Y ¼ C I ¼ C Y ¼ 0: The trajectories of system (4) converge to ! 0 1 which includes elements with Y ¼ 0 and then _ Y ¼ 0. The fifth equation of system (4) implies

which yields EðtÞ ¼ 0 for all t. Hence, and Ð 1 is G.A.S using Lyapunov-LaSalle asymptotic stability theorem.

h Theorem 3 If R 2 [ 1, R 1 =R 2 1 and R 4 1, then Ð 2 is G.A.S.

Proof Define U 2 ðS; L; I; E; Y; V; C I ; C Y Þ as:

We calculate dU 2 dt as:

Using the equilibrium conditions for Ð 2 :

we obtain

Thus, if R 1 =R 2 1 and R 4 1, then using inequality (15) we get dU 2 dt 0 for all S; L; I; E; Y; V; Proof Define a function U 3 ðS; L; I; E; Y; V; C I ; C Y Þ as:

We calculate dU 3 dt as:

Using the equilibrium conditions for Ð 3 :

Hence, if R 5 1, then using inequalities (13)-(14) we get dU 3 dt 0 for all S; L; I; E; Y; V; C I ; C Y [ 0. Moreover, dU 3 

We note that the solutions of system (4) tend to ! 0 3 which includes elements with SðtÞ ¼ S 3 , LðtÞ ¼ L 3 ; IðtÞ ¼ I 3 ; VðtÞ ¼ V 3 ; YðtÞ ¼ 0, then _ IðtÞ ¼ 0; _ YðtÞ ¼ 0 and from the third and fifth equations of system (4) we have

which give C I ðtÞ ¼ C I 3 and EðtÞ ¼ 0 for all t. Therefore, . Applying Lyapunov-LaSalle asymptotic stability theorem we get Ð 3 is G.A.S. h Theorem 5 If R 4 [ 1 and R 6 1, then Ð 4 is G.A.S.

Proof Define U 4 ðS; L; I; E; Y; V; C I ; C Y Þ as:

Calculating dU 4 dt as:

Using the equilibrium conditions for Ð 4 :

We obtain

Hence, if R 6 1, then using inequality (15) we get dU 4 dt 0 for all S; L; I; E; Y; V; C I ; C Y [ 0. In addition, we have dU 4 

The fifth and sixth equations of system (4) imply

which give C Y ðtÞ ¼ C Y 4 and IðtÞ ¼ 0 for all t. Moreover, we have _ I ¼ 0, then from the third equation of system (4) we get 0 ¼ _ I ¼ kL;

which yields LðtÞ ¼ 0 for all t. Therefore, . By applying Lyapunov-LaSalle asymptotic stability theorem we get that Ð 4 is G.A.S.

h

Proof Define U 5 ðS; L; I; E; Y; V; C I ; C Y Þ as:

Calculating dU 5 dt as:

Using the equilibrium conditions for Ð 5 :

We obtain

Then, Eq. (20) will be reduced to the form

Hence, if R 8 1, then using inequalities (13)-(15) we get dU 5 dt 0 for all S; L; I; E; Y; V;

which yields C I ðtÞ ¼ C I 5 for all t. Therefore, . By applying Lyapunov-LaSalle asymptotic stability theorem we get Ð 5 is G.A.S. h Theorem 7 If R 6 [ 1, R 7 1 and R 2 =R 1 [ 1, then Ð 6 is G.A.S.

Proof Define U 6 ðS; L; I; E; Y; V; C I ; C Y Þ as:

Calculating dU 6 dt as:

Using the equilibrium conditions for Ð 6 :

We obtain

Then, Eq. (21) will be reduced to the form

Hence, if R 7 1, then using inequalities (13)-(15) we get dU 6 dt 0 for all S; L; I; E; Y; V; C I ; C Y [ 0, where dU 6 dt ¼ 0 occurs at S ¼ S 6 , L ¼ L 6 ; I ¼ I 6 ; E ¼ E 6 ; Y ¼ Y 6 ; V ¼ V 6 and C I ¼ 0: The solutions of system (4) converge to ! 0 6 which contains elements with E ¼ E 6 ; Y ¼ Y 6 , and then _ Y ¼ 0. The fifth equation of system (4) implies

which yields C Y ðtÞ ¼ C Y 6 for all t. Therefore, and then by applying Lyapunov-LaSalle asymptotic stability theorem we get that Ð 6 is G.A.S.

h

Proof Define U 7 ðS; L; I; E; Y; V; C I ; C Y Þ as:

Calculating dU 7 dt as:

For solving system (3) numerically we fix the values of some parameters taken from literature as mentioned in Table 4 . To verify the stability of the eight equilibria given in Theorems 1-8, we vary some parameters that affect the values of the threshold parameters which in turn control the existence and stability of the equilibria. We confirm that we have assumed some values of the model's parameters just to conduct the numerical simulations. In fact, it is challenging to collect real data from HIV/HTLV-I co-infected patients. However, if one has real data then the model's parameters can be estimated and the validity of the model can be established.

In this subsection, we illustrate our global stability results given in Theorems 1-8.

To do so, we show that the solution of the system starting from any initial point (at any disease stage) in the feasible set H will converge to only one of the eight equilibria of the system. Therefore, we choose three different initial conditions for the system (3) as follows:

Initial-1 : ðSð0Þ; Lð0Þ; Ið0Þ; Eð0Þ; Yð0Þ; Vð0Þ; C I ð0Þ; C Y ð0ÞÞ ¼ ð600; 1:5; 1:5; 30; 0:3; 5; 1; 3Þ, Initial-2: ðSð0Þ; Lð0Þ; Ið0Þ; Eð0Þ; Yð0Þ; Vð0Þ; C I ð0Þ; C Y ð0ÞÞ ¼ ð500; 1; 1; 20; 0:2; 2; 2; 2Þ;

Initial-3: ðSð0Þ; Lð0Þ; Ið0Þ; Eð0Þ; Yð0Þ; Vð0Þ; C I ð0Þ; C Y ð0ÞÞ ¼ ð300; 0:5; 0:5; 10; 0:1; 1:5; 3; 1Þ.

Choosing selected values of g 1 , g 2 , r 1 and r 2 under the above initial conditions leads to the following scenarios:

Scenario 1 (Stability of Ð 0 ): g 1 ¼ 0:0001; g 2 ¼ 0:001 and r 1 ¼ r 2 ¼ 0:2. For this set of parameters, we have R 1 ¼ 0:49\1 and R 2 ¼ 0:23\1. Figure 2 displays that the trajectories initiating with Initial-1, Initial-2 and Initial-3 reach the Table 3 Global stability conditions of the equilibria of model (4) Equilibrium point Global stability conditions Ð 0 ¼ ðS 0 ; 0; 0; 0; 0; 0; 0; 0Þ R 1 1 and R 2 1

equilibrium Ð 0 ¼ ð1000; 0; 0; 0; 0; 0; 0; 0Þ. This shows that Ð 0 is G.A.S according to Theorem 1. In this situation both HIV and HTLV-I will be died out. Scenario 2 (Stability of Ð 1 ): g 1 ¼ g 2 ¼ 0:0005; r 1 ¼ 0:003 and r 2 ¼ 0:2. With such choice we get R 2 ¼ 0:12\1\2:43 ¼ R 1 , R 3 ¼ 0:47\1 and hence R 2 =R 1 ¼ 0:05\1. Therefore, the conditions in Table 2 are verified. In fact, the equilibrium point Ð 1 exists with Ð 1 ¼ ð411:22; 8:03; 11:45; 0; 0; 28:64; 0; 0Þ. Figure 3 displays that the trajectories initiating with Initial-1, Initial-2 and Initial-3 tend to Ð 1 . Therefore, the numerical results support Theorem 2. This case corresponds to a chronic HIV mono-infection but with unstimulated CTL-mediated immune response.

Scenario 3 (Stability of Ð 2 ): g 1 ¼ 0:0001; g 2 ¼ 0:01, r 1 ¼ 0:001 and r 2 ¼ 0:05. Then, we calculate R 1 ¼ 0:49\1\2:33 ¼ R 2 , R 4 ¼ 0:78\1 and then R 1 =R 2 ¼ 0:21\1. Hence, the conditions in Table 2 are satisfied. The numerical results show that Ð 2 ¼ ð421:67; 0; 0; 89:2; 1:37; 0; 0; 0Þ exists. Figure 4 illustrates that the trajectories initiating with Initial-1, Initial-2 and Initial-3 tend to Ð 2 . Thus, the numerical results consistent with Theorem 3. This situation leads to a persistent HTLV-I mono-infection with unstimulated CTL-mediated immune response.

Scenario 4 (Stability of Ð 3 ): g 1 ¼ 0:001; g 2 ¼ 0:005 and r 1 ¼ r 2 ¼ 0:01. Then, we calculate R 3 ¼ 1:39 [ 1 and R 5 ¼ 0:33\1. Table 2 and Fig. 5 show that the trajectories initiating with Initial-1, Initial-2 and Initial-3 tend to Ð 3 ¼ ð285:71; 9:74; 10; 0; 0; 25; 0:97; 0Þ. Therefore, Ð 3 is G.A.S and this agrees with Theorem 4. Hence, a chronic HIV mono-infection with HIV-specific CTLmediated immune response is attained.

Scenario 5 (Stability of Ð 4 ): g 1 ¼ 0:0007; g 2 ¼ 0:1, r 1 ¼ 0:05 and r 2 ¼ 0:3. Then, we calculate R 4 ¼ 5:37 [ 1 and R 6 ¼ 0:79\1. According to Table 2 , Ð 4 exists with Ð 4 ¼ ð230:77; 0; 0; 118:4; 0:33; 0; 0; 4:35Þ. In Fig. 6 , we show that the trajectories initiating with Initial-1, Initial-2 and Initial-3 tend to Ð 4 and then it is G.A.S which agrees with Theorem 5. Hence, a chronic HTLV-I mono-infection with HTLV-specific CTL-mediated immune response is attained. Scenario 6 (Stability of Ð 5 ): g 1 ¼ 0:001; g 2 ¼ 0:01, r 1 ¼ 0:05 and r 2 ¼ 0:08. Then, we calculate R 5 ¼ 1:55 [ 1, R 8 ¼ 0:85\1 and R 1 =R 2 ¼ 2:09 [ 1. Table 2 equilibrium Ð 7 exists. Figure 9 illustrates that the trajectories initiating with Initial-1, Initial-2 and Initial-3 tend to Ð 7 ¼ ð476:19; 1:95; 2; 58:64; 0:2; 5; 0:97; 3:42Þ. The numerical results displayed in Fig. 9 show that Ð 7 is G.A.S based on Theorem 8. In this case, a chronic co-infection with HIV and HTLV-I is attained where both HIV- To further confirmation, we calculate the Jacobian matrix J ¼ JðS; L; I; E; Y; V; C I ; C Y Þ of system (3) as in the following form:

Then, we calculate the eigenvalues k i ; i ¼ 1; 2; :::; 8 of the matrix J at each equilibrium. The examined steady will be locally stable if all its eigenvalues satisfy the following condition:

Reðk i Þ; i ¼ 1; 2; :::; 8:

We use the parameters g 1 , g 2 , r 1 and r 2 the same as given above to compute all positive equilibria and the corresponding eigenvalues. From the scenarios 1-8, we present in Table 5 the positive equilibria, the real parts of the eigenvalues and whether the equilibrium is locally stable or unstable. The numerical results are consistent with the global stability results. For each of the above mentioned scenarios, only one equilibrium is asymptotically stable, while the others are unstable.

In this subsection, we study the influence of HTLV-I infection on HIV monoinfection dynamics, and how affect the HIV infection on the dynamics of HTLV-I mono-infection as well.

To investigate the effect of HTLV-I infection on HIV mono-infection dynamics, we make a comparison between model (3) and the following HIV mono-infection model:

We fix parameters g 1 ¼ 0:0006, r 1 ¼ 0:05, and r 2 ¼ 0:5 and consider the following initial condition: Initial-4: ðSð0Þ; Lð0Þ; Ið0Þ; Eð0Þ; Yð0Þ; Vð0Þ; C I ð0Þ; C Y ð0ÞÞ ¼ 600; 2:4; 1:8; 60; ð 0:2; 4:5; 1:8; 3:5Þ.

We choose two values of the parameter g 2 as g 2 ¼ 0:04 (HIV/HTLV-I coinfection), and g 2 ¼ 0:0 (HIV mono-infection). It can be seen from Fig. 10 

To investigate the effect of HIV infection on HTLV-I mono-infection dynamics, we make a comparison between model (3) and the following HTLV-I monoinfection model:

We fix parameters g 2 ¼ 0:01; r 1 ¼ 0:05, and r 2 ¼ 0:5 and consider the following initial condition: Initial-5: ðSð0Þ; Lð0Þ; Ið0Þ; Eð0Þ; Yð0Þ; Vð0Þ; C I ð0Þ; C Y ð0ÞÞ ¼ ð700; 4; 2; 21; 0:198; 5; 4:5; 0:6Þ.

We choose two values of the parameter g 1 as g 1 ¼ 0.001 (HIV/HTLV-I coinfection), and g 1 ¼ 0:0 (HTLV-I mono-infection). It can be seen from Fig. 11 that when the HTLV-I mono-infected individual is co-infected with HIV then the concentrations of susceptible CD4 þ T cells, silent HTLV-infected cells and HTLVspecific CTLs are decreased. Although, the concentration of Tax-expressing HTLVinfected cells tend to the same value in both HTLV-I mono-infection and HIV/ HTLV-I co-infection.

This paper investigates the global behavior of solutions of system that was used to study HIV/HTLV-I co-infection dynamics. We incorporated the effect of HIVspecific CTLs and HTLV-specific CTLs into the model. The HIV can be transmitted to the susceptible CD4 þ T cells by virus-to-cell transmission, while HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact, and (ii) vertical transmission through mitotic division of Tax-expressing HTLV-infected cells. We studied the basic properties of the model by showing that the solutions are nonnegative and bounded. We derived eight threshold parameters that governed the existence and stability of the eight equilibria of the model. We constructed suitable Lyapunov functions and utilized Lyapunov-LaSalle asymptotic stability theorem to establish the global asymptotic stability of all equilibria. We conducted numerical simulations to support and clarify our theoretical results. We studied the effect of HIV infection on HTLV-I mono-infection dynamics and vice versa. The model analysis suggested that co-infected individuals with both viruses will have smaller number of healthy CD4 þ T cells in comparison with HIV or HTLV-I monoinfected individuals. It was reported in [62] that no treatments exist for acute or chronic HTLV-I infection. However, antiviral treatments of HIV infection is currently used to suppress viral replication. For example, reverse transcriptase inhibitors (RTIs) can prevent the establishment of productive infection of a cell. Model (4) under the effect of RTIs can be written as: Table 5 Local stability of positive equilibria Ð _ S ¼ q À aS À ð1 À 'Þg 1 SV À g 2 SY;

_ L ¼ ð1 À 'Þ 1 À b ð Þg 1 SV À k þ c ð ÞL; _ I ¼ ð1 À 'Þbg 1 SV þ kL À aI À l 1 C I I;

where ' 2 ½0; 1 is the effectiveness of RTIs. We noted that the threshold parameters R 2 and R 4 does not depend on the effectiveness of RTIs ', while the remaining threshold parameters depend on ' as follows:

R 7 ¼ ð1 À 'Þg 1 br 1 r 2 q bc þ k ð Þ a c þ k ð Þ ð1 À 'Þp 1 g 1 br 2 þ p 2 g 2 er 1 þ aer 1 r 2 ð Þ ;

R 8 ¼ wug 2 er 1 r 2 q ðd À rÞw þ dxÞ ½ ð 1 À 'Þp 1 g 1 br 2 þ p 2 g 2 er 1 þ aer 1 r 2 ð Þ :

Therefore to clear the HIV from the body by RTIs we have three cases: (i) R 1 1 and R 2 1: Let ' be chosen such that

:

It follows that when ' ! ' min 1 and R 2 1, the system will converge to Ð 0 where both HIV and HTLV-I will be cleared from the body.

(ii) R 2 [ 1, R 4 1 and R 1 =R 2 1: Let ' be chosen such that

Thus, if R 2 [ 1, R 4 1 and ' ! ' min 2 , then the system will converge to Ð 2 where the HIV is cleared while the HTLV-I will be chronic with inactive HTLV-specific CTL immune response.

(iii) R 4 [ 1 and R 6 1. Let ' is chosen such that R 6 1 as:

WHO: Global Health Observatory (GHO) data

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Using the equilibrium conditions for Ð 7 :We obtainThen using inequalities (13)-(15) we get dU 7 dt 0 for all S; L; I; E; Y; V;The third and fifth equations of system (4) implywhich ensure that C I ðtÞ ¼ C I 7 and C Y ðtÞ ¼ C Y 7 for all t. Thus and by applying Lyapunov-LaSalle asymptotic stability theorem we get that Ð 7 is G.A.S. hIn Table 3 , we summarize the global stability results given in Theorems 1-8.

In this section, we illustrate the results of Fig. 11 The influence of HIV infection rate (g 1 6 ¼ 0) on HTLV mono-infection dynamics (23) will cause a chronic HIV/HTLV-I co-infection R 6 ¼ ð1 À 'Þqbg 1 r 2 bc þ k ð Þ ae c þ k ð Þ p 2 g 2 þ ar 2 ð Þ 1;:Therefore, if R 4 [ 1 and ' ! ' min 3 , then the system will converge to Ð 4 where the HIV is cleared while the HTLV-I will be chronic with active HTLV-specific CTL immune response.