Stability analysis of within‐host SARS‐CoV‐2/HIV coinfection model

The world has been suffering from the coronavirus disease 2019 (COVID‐19) since late 2019. COVID‐19 is caused by a virus called the severe acute respiratory syndrome coronavirus 2 (SARS‐CoV‐2). The human immunodeficiency virus (HIV) coinfection with SARS‐CoV‐2 has been reported in many patients around the world. This has raised the alarm for the importance of understanding the dynamics of coinfection and its impact on the lives of patients. As in other pandemics, mathematical modeling is one of the important tools that can help medical and experimental studies of COVID‐19. In this paper, we develop a within‐host SARS‐CoV‐2/HIV coinfection model. The model consists of six ordinary differential equations. It depicts the interactions between uninfected epithelial cells, infected epithelial cells, free SARS‐CoV‐2 particles, uninfected CD4+ T cells, infected CD4+ T cells, and free HIV particles. We confirm that the solutions of the developed model are biologically acceptable by proving their nonnegativity and boundedness. We compute all possible steady states and derive their positivity conditions. We choose suitable Lyapunov functions to prove the global asymptotic stability of all steady states. We run some numerical simulations to enhance the global stability results. Based on our model, weak CD4+ T cell immune response or low CD4+ T cell counts in SARS‐CoV‐2/HIV coinfected patient increase the concentrations of infected epithelial cells and SARS‐CoV‐2 viral load. This causes the coinfected patient to suffer from severe SARS‐CoV‐2 infection. This result agrees with many studies which showed that HIV patients are at greater risk of suffering from severe COVID‐19 when infected. More studies are needed to understand the nature of SARS‐CoV‐2/HIV coinfection and the role of different immune responses during infection.

to get an estimation of the parameters of their model. 36 Nath et al 37 mathematically analyzed the model of Li et al 36 by establishing the nonnegativity and boundedness of solutions and accomplishing stability analysis. Du and Yuan 38 studied a similar model with adding the effect of antiviral treatments and making a comparison between SARS-CoV-2 and influenza viruses. Almocera et al 39 40 developed a model that describes the occurrence of SARS-CoV-2 in cancer patients and the role of immune responses in this group of patients. Hattaf and Yousfi 41 investigated a within-host model that depicts the interactions between host epithelial cells, SARS-CoV-2 virus, and cytotoxic T lymphocytes (CTLs) with virus-to-cell and cell-to-cell transmission modes.

established a model that studies the interactions between SARS-CoV-2 and T cells with performing bifurcation and stability analyses. Elaiw and Al Agha
Some coinfection models between SARS-CoV-2 and other viruses have been established. For example, Pinky and Dobrovolny 42 used a within-host model to depict the effect of coinfection between SARS-CoV-2 and influenza virus. Ahmed et al. 43 developed a fractional epidemiological model to assess the pandemic situation in many countries affected by HIV and COVID-19 such as South Africa and Brazil. Pinky and Dobrovolny 44 formulated an epidemiological model to study the coinfection between SARS-CoV-2 and other viruses like influenza, respiratory syncytial virus (RSV), and rhinovirus. However and to the best of our knowledge, no within-host SARS-CoV-2/HIV coinfection models with a mathematical analysis have been established yet. Coinfection models are needed to help understand the dynamics of SARS-CoV-2/HIV coinfection, to support the medical and experimental studies and to find more effective ways to treat HIV patients who get SARS-CoV-2 infection. In this paper, we develop a within-host model of SARS-CoV-2/HIV coinfection. The construction of this model follows similar principals to those used in constructing models of HIV and other viruses. For the developed model, we (i) prove that all model's solutions are nonnegative as negative values are not accepted in biological models, (ii) confirm that all solutions have positive upper bounds, (iii) compute all model's steady states which have only positive components, (iv) find the basic reproduction numbers and other threshold numbers, (v) choose appropriate Lyapunov functions and use them to prove the global stability of the different steady states, (vi) run some numerical simulations to enhance the global stability results, (vii) discuss the impact of low CD4 + T cell counts in SARS-CoV-2/HIV coinfected patients, and (viii) suggest some modeling ideas for future works.
The paper is arranged as follows. Section 2 explains the meanings of the different variables and parameters that form the model. Section 3 shows the basic properties of the model which include nonnegativity and boundedness of solutions. Furthermore, it presents all possible steady states of the developed model with the derivation of threshold numbers. Section 4 demonstrates the global stability of steady states. Section 5 presents some numerical simulations. In the last section, we summarize and discuss the results. Also, we suggest some potential future works.

SARS-COV-2/HIV COINFECTION MODEL
This section describes the model intended to be studied in this paper. The proposed model takes the form where X(t), Y (t), V(t), S(t), W(t), and H(t) denote the concentrations of uninfected epithelial cells, infected epithelial cells, free SARS-CoV-2 particles, uninfected CD4 + T cells, infected CD4 + T cells, and free HIV particles at time t. Epithelial cells are produced from a source at a constant rate , die at rate X(t), and get infected by SARS-CoV-2 at rate X(t)V(t). Infected epithelial cells proliferate at rate X(t)V(t) and die at rate kY (t). CD4 + T cells help CD8 + T cells or CTLs to kill infected epithelial cells at rate Y (t)S(t). Here, the target cells of HIV are responsible for the immune response against SARS-CoV-2. SARS-CoV-2 particles are produced from infected cells at rate aY (t) and die at rate V(t). Uninfected CD4 + T cells are produced at a constant rate , get stimulated by infected epithelial cells at rate uY (t)S(t), die at rate S(t), and get infected at rate S(t)H(t). Infected CD4 + T cells proliferate at rate S(t)H(t) and die at a natural death rate W(t). HIV particles are produced by infected cells at rate W(t) and die at rate H(t). The meanings of the different parameters are summarized in Table 1. The formulation of the model is based on Nowak and Bangham's model that was used widely to  19 Their model was also used to model SARS-CoV-2 mono-infection. Here, we used the same principals to model SARS-CoV-2/HIV coinfection and connect the two infections together.

BASIC PROPERTIES
This section proves the nonnegativity and boundedness of solutions of model (1). Additionally, it computes all possible steady states with the associated threshold numbers.
Then, we obtain .

Steady states
In this subsection, we find all possible steady states of model (1) and deduce four threshold parameters that cover the existence of these steady states.
To compute the steady states of the model, we solve the following system of algebraic equations Accordingly, we find that model (1) has four steady states: where R 1 = ∕ . Here, R 1 is the basic reproduction number for HIV infection. It determines the establishment of HIV infection in the body. We see that and V 2 satisfies the following equation: Now, we show that there exists a positive root for Equation (3). We define a function G(V) as We have As a result, we deduce that Δ V exists when R 2 > 1. Note that R 2 can be rewritten as , . where Single HIV-infection steady state SARS-CoV-2/HIV coinfection steady state R 3 > 1 and R 4 > 1 Therefore, Δ VH exists when R 3 > 1 and R 4 > 1. Here, R 3 and R 4 are threshold numbers that determine the occurrence of SARS-CoV-2/HIV coinfection. Note that we can rewrite R 3 as All steady states of model (1) and their existence conditions are summarized in Table 2. For simplicity, we will use the following contractions in the next sections:

GLOBAL PROPERTIES
In this section, we show the global asymptotic stability of all steady states by constructing Lyapunov functions following the method presented in other works. [48][49][50] We define F( ) = − 1 − ln . We will use the arithmetic-geometric mean inequality Theorem 1. If R 1 ≤ 1 and R 2 ≤ 1, then the steady state Δ 0 is globally asymptotically stable (G.A.S).
By using the steady state conditions for Δ H , we get Then, we obtain Since R 4 ≤ 1 and by using the inequality in (4) for = 1, we find that Differentiating 2 , we obtain By using the steady state conditions for Δ V we obtain ) . If .
Differentiating 3 , we obtain By using the steady state conditions for Δ VH ) .
Since 1 < R 3 ≤ 1 + a ∕ (u + a ) and using inequalities (4) and (5) The global stability conditions of all steady states are summarized in Table 3.

NUMERICAL SIMULATIONS
This section presents some numerical simulations to assist the results gained in the previous sections. In addition, it shows the impact of low number of CD4 + T cells on SARS-CoV-2/HIV coinfection. To achieve this goal, we consider three sets of initial conditions as follows:  The selection of these values is optional. Furthermore, it is divided into three sets to ensure that the global stability of the steady states is not affected by the choice of initial conditions. We use the MATLAB solver ode45 to solve system (1). According to the global stability of the steady states Δ 0 , Δ H , Δ V , and Δ VH in Theorems 1-4, we split the simulations into four cases. In these cases, we vary the values of , , , and of model (1). The values of all other parameters are fixed and listed in Table 1. The four cases are given as follows: (i) We take = 0.9, = 1, = 5.36, and = 0.0001. The thresholds in this case are given by R 1 = 0.5 < 1 and R 2 = 9.03×10 −4 < 1. In harmony with Theorem 1, the steady state Δ 0 = (22.41, 0, 0, 1000, 0, 0) is G.A.S (see Figure 1). This is an optimal case when the person does not suffer from neither SARS-CoV-2 infection nor HIV infection. (ii) We choose = 0.55, = 1, = 5.36, and = 0.0016. This provides us with R 1 = 8 > 1 and R 4 = 0.0044 < 1. According to Theorem 2, the steady state Δ H = (22.41, 0, 0, 125, 17.5,43.75) is G.A.S (see Figure 2). This point represents the case when a person has HIV infection with low CD4 + T cell counts, while SARS-CoV-2 infection is not detected. (iii) We select = 2.9, = 0.02, = 0.03, and = 0.0001. This gives R 2 = 25.8534 > 1 and R 3 = 0.5054 < 1. In this case, the solutions globally converge to the steady state Δ V = (0.876, 0.0011, 0.0085, 1010.71, 0, 0). This result agrees with Theorem 3 and is displayed in Figure 3. This case represents a person with SARS-CoV-2 infection, but he does not have HIV disease. (iv) We consider = 2.9, = 0.02, = 0.1, and = 0.0016. This implies that R 3 = 8.0743 > 1, R 3 < 1 + a u +a = 8.9885, and R 4 = 59.76 > 1. In harmony with Theorem 4, the steady state Δ VH = (0.375, 0.0084, 0.02, 125, 17.71,44.28) is G.A.S (see Figure 4). In this case, SARS-CoV-2/HIV coinfection occurs, where an HIV patient gets infected with COVID-19. CD4 + T cells, which are the main target of HIV, are recruited to clear SARS-CoV-2 infection from the body. However, if the patient has low CD4 + T cell counts, the clearance of SARS-CoV-2 may not be achieved. This can lead to severe infection and death. For further confirmation of the asymptotic stability of Δ VH , we calculate the Jacobian matrix of model (1) at the steady state Δ = (X, Y , V, S, W, H) as Next, we calculate the eigenvalues  ( = 1, 2, … , 6) of the Jacobian matrix at all possible steady states. For asymptotic stability of Δ VH , we have to prove that Re( ) < 0, for all = 1, 2, … , 6, and all other steady states posses eigenvalues with positive real parts. The computations are organized in Table 4.

The effect of killing rate of CD4 + T cells
To see the effect of changing the value of on the stability of system (1), we pick up the same values considered in case (iv) ( = 2.9, = 0.1, and = 0.0016) with increasing the value of from = 0.02 to = Furthermore, to observe the impact of increasing or decreasing the value of on the concentrations of infected epithelial cells and SARS-CoV-2, we consider case (iv) with different values of (see Figure 5). We see that reducing the killing rate constant of CD4 + T cells increases the concentrations of infected epithelial cells and SARS-CoV-2 particles, while increasing the killing rate does the opposite effect. The above results indicate that high killing rate of CD4 + T cells is required to clear SARS-CoV-2 from the body of HIV patient. On the other hand, low killing rate or low CD4 + T cell counts allows SARS-CoV-2 to replicate and infect more healthy epithelial cells. As a result, SARS-CoV-2/HIV coinfecetd patient may suffer from severe SARS-CoV-2 infection and die.

Sensitivity analysis
Sensitivity analysis allows us to measure a relative change in a variable when a parameter changes. We perform sensitivity analyses for R 1 and R 2 due to their pivotal role in determining the stability of the uninfected steady state Δ 0 . Following Chitnis et al., 54 the normalized forward sensitivity index of a variable v with respect to a parameter p is defined as where v is a differentiable function.

Sensitivity analysis of R 1
The normalized forward sensitivity index of R 1 with respect to a parameter p is given by We compute the sensitivity indices of R 1 to each parameter that it includes using the parameter values given in Table 1.
The results are presented in Table 5. It is easy to see that the sensitivity indices of R 1 do not depend on any parameter values. For example, the sensitivity index of R 1 with respect to is Thus, it is helpful to interpret the sign of the sensitivity indices of R 1 . Based on the results in Table 5, the recruitment rate of unifected CD4 + T cells, , the infection rate of CD4 + T cells, , and the production rate of HIV, , are the parameters that contribute to the growth of HIV in the body. On the other hand, the death rate of uninfected CD4 + T cells, , the death rate of infected CD4 + T cells, , and the death rate of HIV particles, , are the parameters that have important role in fighting the growth of HIV in the body.

Sensitivity analysis of R 2
The normalized forward sensitivity index of R 2 with respect to a parameter p is given by Similar to R 1 , we compute the sensitivity indices of R 2 to each parameter that it includes using the values in Table 1. The results are shown in Table 6. We can see that the least sensitive parameter is the death rate of infected epithelial cells, k.
In general, it can be observed from Table 6 that when one of the parameters with a positive sensitivity index (a, , , ) is increased while the other parameters remain constant, this increases the value of R 2 . This implies that these parameters contribute to the growth of SARS-CoV-2. On the other hand, the parameters with negative sensitivity indices contribute to the reduction in the load of SARS-CoV-2.
Parameter Sensitivity index  of a patient having SARS-CoV-2 infection, but he is HIV-free. (iv) The SARS-CoV-2/HIV coinfection steady state Δ VH exists and is G.A.S if R 3 > 1, R 4 > 1, and R 3 < 1 + a ∕ (u + a ). This represents the case of coinfected patient.
We confirmed the compatibility of numerical simulations with theoretical results. We found that the killing rate of CD4 + T cells affects the global stability of steady states. Increasing the value of causes the SARS-CoV-2/HIV coinfection steady state to lose its stability, while the single HIV infection becomes G.A.S. This represents the situation when the coinfecetd patient becomes SARS-CoV-2-free, which can be obtained under effective treatments. Additionally, based on our model, decreasing the value of increases the concentrations of infected epithelial cells and SARS-CoV-2 particles. This means that low CD4 + T cell counts can increase the severity of SARS-CoV-2 infection in HIV patients with low immunity. These results agree with many studies which showed that HIV patients with low CD4 + T cell counts or not receiving antiretroviral therapy are at increased risk of getting SARS-CoV-2 infection and death. [14][15][16] However, some studies suggested that weak immunity and low CD4 + T cell counts in HIV patients can protect them from developing severe COVID-19 when infected. 2,14 Therefore, more studies are needed to fill this gap in SARS-CoV-2/HIV coinfection research and understand the full impact of coinfection on patients' lives. 14 Comparing with the existing models, the model studied and analyzed in this research is the first model that takes into account the effect of SARS-CoV-2/HIV coinfection. The data on SARS-CoV-2/HIV coinfection are still very limited. 2,15 As data become available, the model can be fitted with real data to find a better approximation of model's parameters. Consequently, the model can be used to support experimental and medical studies in (i) determining the value of parameters required to clear SARS-CoV-2 infection from HIV-infected patients, (ii) testing the effect of low CD4 + T cell counts on the severity of COVID-19 in HIV patients, (iii) checking the importance of model (1) on the coinfection between HIV and SARS-CoV-2, and (iv) understanding the dynamics of coinfection, which can have an important role in developing new treatments for this group of patients. The principal limitation of this paper is that we did not utilize real data to estimate the parameters' values in model (1). We took the values from well-studied HIV models or SARS-CoV-2 models. This is because the scarcity of data on SARS-CoV-2/HIV coinfection as we mentioned before. Accordingly, the results can be verified when more data are available.