Stability analysis of SARS-CoV-2 / HTLV-I coinfection dynamics model

: Although some patients with coronavirus disease 2019 (COVID-19) develop only mild symptoms, fatal complications have been observed among those with underlying diseases. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the causative of COVID-19. Human T-cell lymphotropic virus type-I (HTLV-I) infection can weaken the immune system even in asymptomatic carriers. The objective of the present study is to formulate a new mathematical model to describe the co-dynamics of SARS-CoV-2 and HTLV-I in a host. We ﬁrst investigate the properties of the model’s solutions, and then we calculate all equilibria and study their global stability. The global asymptotic stability is examined by constructing Lyapunov functions. The analytical ﬁndings are supported via numerical simulation. Comparison between the solutions of the SARS-CoV-2 mono-infection model and SARS-CoV-2 / HTLV-I coinfection model is given. Our proposed model suggest that the presence of HTLV-I suppresses the immune response, enhances the SARS-CoV-2 infection and, consequently, may increase the risk of COVID-19. Our developed coinfection model can contribute to understanding the SARS-CoV-2 and HTLV-I co-dynamics and help to select suitable treatment strategies for COVID-19 patients who are infected with HTLV-I.


Introduction
In November 2019, a dangerous type of virus named severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) appeared first in Wuhan, China.This virus infects the human body and causes coronavirus disease 2019 (COVID- 19), which can lead to death.According to the update provided by the World Health Organization (WHO) on December 4, 2022 [1], over 641 million confirmed cases and over 6.6 million deaths have been reported globally.SARS-CoV-2 is transmitted to people when they are exposed to respiratory fluids carrying infectious viral particles.The implementation of preventive measures such as hand washing, using face masks, physical and social distancing, disinfection of surfaces and getting the COVID-19 vaccine can reduce SARS-CoV-2 transmission.Ten vaccines for COVID-19 have been approved by the WHO for emergency use.These include Novavax, Bharat Biotech, Serum Institute of India (Novavax formulation), Sinopharm, Pfizer/BioNTech, Sinovac, Janssen (Johnson & Johnson), Oxford/AstraZeneca, Serum Institute of India (Oxford/AstraZeneca formulation) and that presented in [2].
SARS-CoV-2 is a single-stranded positive-sense RNA virus that infects epithelial cells.SARS-CoV-2 can lead to acute respiratory distress syndrome, which has high mortality rates, particularly in patients with other viral infections [3].It was discovered in [4] that, 94.2% of individuals with COVID-19 were also coinfected with several other microorganisms, such as fungi, bacteria and viruses.
Important viral copathogens include the respiratory syncytial virus, rhinovirus/enterovirus, influenza A and B viruses (IAV and IBV), metapneumovirus, parainfluenza virus, human immunodeficiency virus (HIV), cytomegalovirus, dengue virus, hepatitis B virus, Epstein-Barr virus and other coronaviruses, among which the rhinovirus/enterovirus and IAV are the most common copathogens [5].Disease progression and outcome in SARS-CoV-2 infection are highly dependent on the host immune response, particularly in the elderly in whom immunosenescence may predispose them to increased risk of coinfection [6].Immunosenescence renders vaccination less effective and increases the susceptibility to viral infections [7].
Human T-cell lymphotropic virus type-I (HTLV-I) is a single-stranded RNA virus that infects essential human system immune cells, CD4 + T cells.CD4 + T cells are considered "helper" cells because they do not neutralize infections, but rather trigger the body's response to infections [8].They are considered essential in the activation and growth of cytotoxic T lymphocytes (CTLs).The role of CTLs is to destroy cells infected with microorganisms, such as bacteria or viruses.HTLV-I can cause immune dysfunction even in asymptomatic carriers [3].HTLV-I can lead to two diseases: adult T-cell leukemia (ATL) and HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP) [9].Although HTLV-I can cause fatal diseases (ATL and HAM/TSP), most of the infected persons remain asymptomatic throughout their lives [3].An estimation by the WHO stated that about 5 to 10 million individuals are infected with HTLV-I worldwide [10].The primary method of HTLV-I transmission is through bodily fluids including semen, blood and breast milk [11].In [3,12], two cases of COVID-19 patients with HTLV-I infection have been reported.These reports highlighted the need for the accumulation of similar cases to illustrate the risk factors for severe illness, the best-in-class antiviral agent, the way to manage and prevent secondary infection and the optimal treatment strategy for patients with SARS-CoV-2-HTLV-I coinfection.
Over the years, mathematical models have demonstrated their ability to provide useful insight to gain a further understanding of virus dynamics within the host.These models may assist in the development of viral therapies, as well as in the selection of appropriate therapeutic approaches.Stability analysis of the model's equilibria may help researchers to establish the conditions that ensure the persistence or termination of this infection.Mathematical models of SARS-CoV-2 mono-infection within a host have recently been developed in several works.
Stability analysis for models describing the within-host dynamics of SARS-CoV-2 infection was studied in [19][20][21]36,39,41].Hattaf and Yousfi [19] studied a within-host SARS-CoV-2 infection model with cell-to-cell transmission and CTL immune response.The model included both lytic and nonlytic immune responses.The Lyapunov method was used to prove the global stability of the three equilibria of the model.A SARS-CoV-2 infection model with both CTL and antibody immunities was developed and analyzed in [21].Mathematical analysis of the model presented in [14] was studied in [41].Both local and global stability analyses of the model's equilibria were established.Almocera et al. [20] studied the stability of the two-dimensional SARS-CoV-2 dynamics model with an immune response presented in [13].Elaiw et al. [25] studied the global stability of a delayed SARS-CoV-2 dynamics model with logistic growth of the uninfected epithelial cells and antibody immunity.In very recent works, the Lyapunov method was used to establish the global stability of coinfection models, including SARS-CoV-2/HIV-1 [35,36], SARS-CoV-2/IAV [39] and SARS-CoV-2/malaria [37,38].
During the last decades, modeling and analysis of HTLV-I mono-infection have attracted the interest of several researchers.Stilianakis and Seydel [42] constructed an HTLV-I model within a host as follows: ) respectively denotes the concentrations of healthy (or uninfected) CD4 + T cells, latently HTLV-I-infected CD4 + T cells, actively HTLV-I-infected CD4 + T cells and ATL cells.Some biological factors have been considered in the HTLV-I mathematical models by incorporating (i) CTL immunity [9,[43][44][45], (ii) the mitotic transmission of actively infected cells [46][47][48][49][50], (iii) intracellular time delay [51,52] or immune response delay [43,53] and (iv) reaction and diffusion [54].Elaiw et al. [55] developed and analyzed a general HTLV-I with CTL immunity, mitosis and time delay.HIV-1 and HTLV-I have similar ways of transmission between individuals.Therefore, we presented and analyzed some models for within-host HIV-1/HTLV-I coinfection [56,57].
To the best of our knowledge, mathematical modeling of within-host SARS-CoV-2-HTLV-I coinfection has not been studied before.The objective of this work is to formulate a new model for within-host SARS-CoV-2-HTLV-I coinfection.We study the properties of the model's solutions, calculate all equilibrium points, investigate the global stability of equilibria and conduct some numerical simulations.
The SARS-CoV-2/HTLV-I coinfection model presented in this paper can be helpful to describe the co-dynamics of several human viruses.In addition, the model may be used to predict new treatment regimens and strategies for patients who are coinfected with different viruses or multi-variants of a virus [58].

Mathematical SARS-CoV-2 and HTLV-I coinfection model
The dynamics of SARS-CoV-2-HTLV-I coinfection is schematically shown in the transfer diagram given in Figure 1.Now, we propose a new ordinary differential equation model for SARS-CoV-2-HTLV-I coinfection within a host as follows: (2.7)All parameters of the model described by (2.1)-(2.7)are positive.Since the CD4 + T cells help CTLs to kill the actively SARS-CoV-2-infected epithelial cells, we assume implicitly that the actively SARS-CoV-2-infected epithelial cells are killed at a rate µYU and the CD4 + T cells are proliferated at a rate θYU.We assume that actively HTLV-I-infected cells proliferate at a rate ε * A, with a part ωε * A turning into latent, where ω ∈ (0, 1).All parameters of the model are positive.In [49,56], it was proposed that (2.14) We have If HTLV-I does not exist and we neglect the regeneration of the uninfected epithelial cells, the death of the uninfected epithelial cells and the death of the latently SARS-CoV-2-infected epithelial cells, then the model described by (2.8)-(2.14)will lead to the model described by (1.1)- (1.4).Moreover, in the absence of SARS-CoV-2, then the model described by (2.8)-(2.14)leads to the HTLV-I mono-infection models (without considering the HTLV-I-specific CTL ) presented in [49,50].

Properties of solutions
Let M i > 0, i = 1, 2, 3 be defined as Additionally, define the following compact set: Lemma 3.1.The compact set Ω is positively invariant for the model described by (2.8)-(2.14).
Proof.We have that To investigate the boundedness of the model's solutions, we define t) and A(t) are all bounded.

Equilibrium points
To calculate the equilibrium points of the system given by (2.8)-(2.14),we solve the following system: We find that the system admits four equilibrium points.
(iv) HTLV-I and SARS-CoV-2 coinfection equilibrium point , where It follows that, since ξ A − ε > 0, X 3 and U 3 are always positive, while Therefore, we can rewrite the components of EP 3 as .
Thus, EP 3 exists when R 3 > 1 and R 4 > 1.At this point, R 3 and R 4 are threshold numbers that determine the occurrence of HTLV-I/SARS-CoV-2 coinfection.Now, we summarize the above results in the following lemma.
Lemma 4.1.There exist four threshold numbers R i , i = 1, 2, 3, 4, such that then, in addition to EP 0 , there is an HTLV-I mono-infection equilibrium point, then, in addition to EP 0 , there is an HTLV-I and SARS-CoV-2 coinfection equilibrium point,

Global stability analysis
In this section, we discuss the global stability of four equilibrium points, EP i , i = 0, 1, 2, 3. We will utilize the following arithmetic-mean-geometric-mean inequality: Let ∆ j be the largest invariant subset of To prove the results given in the next Theorems 5.1-5.4,we follow the works of [60,61] to build suitable Lyapunov functions and apply LaSalle's invariance principle [62].
Proof.Define Φ 2 as follows: Utilizing the equilibrium point conditions for EP 2 : we obtain We have Collecting terms, we get Hence, if R 3 ≤ 1, then EP 3 does not exist since A 3 ≤ 0 and L 3 ≤ 0. This implies that α ≤ 0, and, by using the inequality of (5.1), we obtain Thus, dΦ 2 dt ≤ 0 for all X, N, Y, V, U, L, A > 0 and The solutions of the system converge to ∆ 2 , which comprises elements with A = 0.It follows that Ȧ = 0, and Eq (2.14) becomes Therefore, ∆ 2 = {EP 2 } .LaSalle's invariance principle implies that EP 2 is globally asymptotically stable [62].
Let us define a parameter R as follows: .
Based on the above findings, we summarize the existence and global stability conditions for all equilibrium points in Table 1.
Table 1.Conditions of existence and global stability of the system's equilibria.

Equilibrium point
Existence conditions Global stability conditions

Numerical simulations
In this section, we present some numerical results for the model described by (2.1)-(2.7) to illustrate the stability of equilibrium points.We used the ODE45 solver in MATLAB to solve the system numerically.
Case 3. (R 2 > 1 and R 3 ≤ 1): We choose ρ = 3, µ = 0.01, ξ V = 0.04 and π = 0.0001.Then, we calculate R 2 = 20.7589> 1 and R 3 = 0.2979 < 1. Lemma 4.1 and Theorem 5.3 state that the SARS-CoV-2 mono-infection equilibrium point EP 2 = (0.529, 0.025, 0.011, 0.066, 916.87, 0, 0) exists and is globally asymptotically stable.Figure 4 displays the numerical solutions of the system converge to EP 2 for all three initials IC1-IC3.The results support the theoretical results presented in Theorem 5.3.In this situation, the patient becomes infected by SARS-CoV-2, while the HTLV-I infection is removed.Figure 5 shows that the solutions of the system converge to EP 3 for all initials IC1-IC3.The results support the theoretical results presented in Theorem 5.4.In this case, a SARS-CoV-2 and HTLV-I coinfection happens, where an HTLV-I-patient gets contaminated with COVID-19.CD4 + T cells are animated to dispense with SARS-CoV-2 disease from the body.In any case, assuming that the patient has low CD4 + T cell counts, the freedom of SARS-CoV-2 may not be accomplished.This can prompt extreme contamination and passing.Now, we check the local stability of the model's equilibria.The Jacobian matrix J = J (X, N, Y, V, U, L, A) can be calculated as follows: .
At each equilibrium point, we compute the eigenvalues λ j , j = 1, 2, ..., 7 of J.If Re(λ j ) < 0, j = 1, 2, ..., 7, then the equilibrium point is locally stable.We select the parameters ρ, µ, ξ V and π as given in Cases 1-4; then, we compute all nonnegative equilibria and the accompanying eigenvalues.Table 3 outlines the nonnegative equilibria, the real parts of the eigenvalues and whether or not the equilibrium point is stable.We found that the local stability agrees with the global one.

Comparison results
In this subsection, we present a comparison between the HTLV-I single infection and the coinfection with HTLV-I and SARS-CoV-2.We compare the solutions of the model described by (2.1)-(2.7)and the following SARS-CoV-2 mono-infection model: We consider ρ = 3, µ = 0.01, ξ V = 0.6 and π = 0.0015 with the initial condition IC3.We observe from Figure 6 that the presence of HTLV-I reduces the concentrations of uninfected epithelial cells and uninfected CD4 + T cells, while it increases the concentrations of latently SARS-CoV-2-infected epithelial cells, actively SARS-CoV-2-infected epithelial cells and free SARS-CoV-2 particles.This means that HTLV-I suppresses the immune response and enhances the SARS-CoV-2 infection.

Discussion
SARS-CoV-2 and HTLV-I coinfection cases were reported in [3] and [12].Therefore, it is important to understand the within-host dynamics of this coinfection.In this paper, we developed and examined a within-host SARS-CoV-2 and HTLV-I coinfection model.We studied the basic and global properties of the model.We found that the system has four equilibria, and we proved the following: (I) The uninfected equilibrium point EP 0 always exists.It is globally asymptotically stable when R 1 ≤ 1 and R 2 ≤ 1.This result suggests that, when R 1 ≤ 1 and R 2 ≤ 1, both SARS-CoV-2 and HTLV-I infections are predicted to die out regardless of the initial conditions.From a control viewpoint, setting R 1 ≤ 1 and R 2 ≤ 1 will be a good strategy.The parameter R 2 may be reduced by multiplying the parameters by ρ or η by (1 − 1 ) or (1 − 2 ), respectively.Here, 1 ∈ [0, 1] and 2 ∈ [0, 1] represent the effectiveness of antiviral drugs for blocking the infection and production of SARS-CoV-2 particles, respectively [18].Since there is no treatment for HTLV-I infection, R 1 ≤ 1 is rarely achieved.
(II) The HTLV-I mono-infection equilibrium point EP 1 exists if R 1 > 1.It is globally asymptotically stable when R 1 > 1 and R 4 ≤ 1.This result establishes that, when R 1 > 1 and R 4 ≤ 1, the HTLV-I mono-infection is always established, regardless of the initial conditions.
(III) The SARS-CoV-2 mono-infection equilibrium point EP 2 exists if R 2 > 1.It is globally asymptotically stable when R 2 > 1 and R 3 ≤ 1.This result shows that, when R 2 > 1 and R 3 ≤ 1, the SARS-CoV-2 mono-infection is always established, regardless of the initial conditions.
(IV) The HTLV-I and SARS-CoV-2 coinfection equilibrium point EP 3 exists if R 3 > 1 and R 4 > 1.It is globally asymptotically stable when R 4 > 1 and 1 < R 3 ≤ 1 + R.This result shows that, when R 4 > 1 and 1 < R 3 ≤ 1 + R, the HTLV-I and SARS-CoV-2 coinfection is always established, regardless of the initial conditions.
We discussed the impact of HTLV-I infection on the SARS-CoV-2 mono-infection dynamics.We found that the presence of HTLV-I suppresses the immune response and enhances the SARS-CoV-2 infection.This result agrees with [3], where it was reported that HTLV-I can cause immune dysfunction even in asymptomatic carriers.Therefore, HTLV-I may increase the risk of COVID-19.
The main limitation of the present research work is that we did not use real data to estimate the values of the model's parameters.The reasons are as follows: (i) The real measurements from HTLV-I/COVID-19 coinfection patients are still very limited; (ii) Comparing our results with a small number of real studies may not be very precise; (iii) Collecting real data from patients with HTLV-I and COVID-19 coinfection is not an easy task.Thus, the theoretical results obtained in the present paper need to be tested against empirical findings when real data become available.

Conclusions
Mathematical models are frequently used to understand the complex behavior of biological systems.In this paper, we constructed a new SARS-CoV-2/HTLV-I coinfection model within a host.The model considers the interactions between uninfected epithelial cells, latently SARS-CoV-2-infected epithelial cells, actively SARS-CoV-2-infected epithelial cells, free SARS-CoV-2 particles, uninfected CD4 + T cells, latently HTLV-I-infected CD4 + T cells and actively HTLV-I-infected CD4 + T cells.We examined the nonnegativity and boundedness of the solutions.We calculated the equilibrium points of the model and established the conditions of their existence and stability.We established the global stability of

Figure 1 .
Figure 1.Schematic diagram of the SARS-CoV-2 and HTLV-I coinfection dynamics in vivo.