Stability and Dynamic of HIV-1 Mathematical Model with Logistic Target Cell Growth, Treatment Rate, Cure Rate and Cell-to-cell Spread

. One way of HIV infection spreading is through the cell division of infected cells by mitosis expressed in mathematical models as a logistic process. Cell-to-cell transmission is another factor in the spread and speed of disease. In this work, we present a ﬁve-dimensional Ordinary Diﬀerential Equation model (ODE) with the logistic form for proliferation of uninfected cells, cell-to-cell and virus-to-cell transmission rate, two types of cellular and humoral immune responses, the cure rate for return-ing infected cells to non-infectious cells, and two treatment rates, one for reducing infectious cells and the other for blocking free viruses. We discuss the positivity and boundedness of solutions, free-equilibrium points, steady-state equilibrium points, and stability by the Routh Hurwitz criterion. The rate of reproduction is analyzed, and the useful parameters for increasing or decreasing it are identiﬁed. Numerical simulations are performed to investigate the dynamic behavior of model responses to treatment eﬀects on disease.


Introduction
AIDS, acquired immune deficiency syndrome, was first reported in the United States in 1981 by the Centers for Disease Control and Prevention. AIDS is a syndrome caused by a retrovirus, Human Immunodeficiency Virus. The disease involves severe weakness of the immune system that, by arrival, opportunistic infections in the body, eventually resulting in death. AIDS is recognized as a global epidemic disease [14].
HIV is a lentivirus from the family of retroviruses. There are two types of HIV viruses: HIV-1 and HIV-2. Most of the registered cases are HIV-1. HIV is a virus that attacks components of the immune system such as lymphocyte cells with the CD + 4 receptor, macrophages, and dendritic cells [3]. It should be noted that the interval between the entry of the virus into the body and the time of diagnosis by the laboratory (sufficient amount of antibody secreted in the blood) is the window period. It usually lasts about 2 to 12 weeks, but maybe longer if you have an autoimmune disease or hepatitis B. If the The first HIV models at the late 1980 are expressed immediately after the discovery of the HIV, by Merrill [26] in 1987, Anderson and May in 1989 [4]. In 1989, Perelson proposed a model explaining many of the HIV infection's immunological consequences. In this model, the cell proliferation rate of CD + 4 T is logically expressed. Note that the logistic law is based on the fact that the growth of T cells (or any normal cell) stops as their population approaches T max . The term logistical growth here refers to the growth of T cells' population in the vicinity of a viral infection. One assumption of logistic growth of T cells is that if T (0) < T max , for all t then T (t) < T max . Thus, when the T cell population reaches its maximum, the presence of HIV will infect these cells [22,28].
In 1993, Perelson et al. studied a model dynamics that showed the effects of AZT antiviral drugs that inhibited the Reverse Transcription enzyme to prevent virus replication [30]. In 1996, Nowak and Bangam presented a simple model for antiviral immune response, virus replication, and virus diversity. They proposed three models. In the first model, a simple interaction between virus replication and host cells was demonstrated. Since the immune responses reduce viral load, the second model included immune responses against infected cells and, finally, a model in which the virus can reproduce and evade the immune responses that are the result of an HIV mutation [27]. In 1997, Kirschner 4 Najmeh Akbari, Rasoul Asheghi and Maryam Nasirian of [29] state that when a virus enters a CD + 4 T, chemotherapy can affect it, and this can change some infected cells into uninfected ones. Moreover, the current combination of antiretroviral therapies, including the Reverse Transcription and Protease Inhibitors, can restore some infected CD + 4 T cells to uninfected ones, by losing all cccDNA from their nuclei [5]. In 2016, Kaminski developed a method for curing infected cells by gene therapy or loss of all cccDNA from their nucleus [20]. With the above hypotheses, in the papers [2,7,41], the reversibility process of a part of infected cells to uninfected cells is considered with one parameter.
The mathematical model seems more realistic when all the factors contributing to the spread of the disease are expressed in it. However, since the dynamic study of a mathematical model with many variables and parameters is difficult and practically impossible, in most of the above models, for a simpler study of its dynamics, some existing factors are assumed to be constant and not included in the model. The innovation of this model is that with adding several factors from different papers such as cure rate, the proliferation rate of healthy cells with logistic function, the infection rate of healthy cells non-linearly, the spread of infection through cell-to-cell and virus-to-cell process, and cellular and humoral immunity, dynamics are examined.
The paper is organized as follows. In Section 2, we formulate a model with five state variables. In Section 3, we study the non-negative and boundedness of solutions.
In Section 4, we analyze infection-free equilibrium points. In Section 5, we investigate steady-state equilibrium points. In Section 6, we examine the role of parameters in the rate of reproduction. In Section 7, we consider the numerical simulation of the model (2.1). In the last section, we conclude.

Model formulation
The population of CD + 4 T cells is stimulated by antigens and then divided. Therefore, when their community reaches its maximum, their reproduction stops. Hence, their reproduction rate can be expressed as a logistic function. Also, to make a realistic model, the saturation function was used instead of the bilinear infection rate. On the other hand, because cell-to-cell transmission of the virus plays an essential role in reducing antibodies effect on viral Inhibitors, it has been considered along with virus-to-cell transmission. Also, due to antiviral drugs effectiveness in HIV chemotherapy's success, their effect is examined in the model. Then, both cellular and humoral immune systems have been used together for the effectiveness of the principle of Foutz vaccine design. Besides, Kaminsky gene therapy has been used to cure infected cells. A cure rate is stated as a method for averting of infected cells to uninfected cells by gene therapy or loss of all cccDNA from their nucleus.

Parameters Description r
The rate at which the CD + 4 T-cells are reproduced. m Maximum value of CD + 4 T-cells in the absence of the virus. β 1 The rate at which the virus infects the CD + 4 T-cells. β 2 The rate at which the CD + 4 T-cells are infected by the infected cells. α The rate at which the viruses are saturated. ρ The cure rate of the infected CD + 4 T-cells that reverted to the uninfected T-cells. d The death rate of the CD + 4 T-cells. δ The death rate of the infected CD + 4 T-cells. ρ 1 The rate at which the T-cells kill the infected CD + 4 T-cells. η The treatment rate at which measure the efficacy of Reverse Transcriptase Inhibitor. ε The treatment rate at which measure the efficacy of protease Inhibitor.
n Number of viruses produced by the infected CD + 4 T-cells. µ The death rate of the viruses.  The death rate of the T-cells.
The rate at which the presence of the virus activates the B-cells.

b 2
The death rate of the B-cells. Now, we extend a mathematical model for HIV infection with two treatment rates, cure rate, the transmission of infection by the virus-to-cell and cell-to-cell, logistic growth for CD + 4 T-cell uninfected, the saturation function for the infection rate, and both types of cellular and humoral immune systems. We use five state variables in the model. Population of uninfected CD + 4 T-cells (x), Population of infected CD + 4 T-cells (y), Population of infectious HIV virions (v), Population of T-cells (z), Population of B-cells (w). Also, two parameters η, ε ∈ [0, 1] have been introduced as treatment rates of Reverse Transcriptase Inhibitors (RTIs) and Protease Inhibitors (PIs) respectively. Reverse Transcriptase Inhibitor prevents the transcriptase process in cells infected by the virus HIV, and Protease Inhibitor blocks the protease enzyme, thereby preventing the production of infectious and mature viruses. The proposed model is illustrated below.
(2.1) 6 Najmeh Akbari, Rasoul Asheghi and Maryam Nasirian All parameters in model (2.1) are positive and assumed to be independent of time. They are described in

Non-negative and boundedness of solutions
For biological reasons, we suppose that the initial values of the variables of the system (2.1) are non-negative. Then, we prove that all of its solutions are also non-negative and bounded.
Proof. We know that Γ(t) is the solution of system (2.1) with the initial values The above computations show that the solutions of system (2.1) so long as y(x) > 0 with non-negative initial values remain in the first quadrant for all t ≥ 0, and hence, they are non-negative. Now we prove the boundedness property. To this end, we set T = x + y with x, y ≥ 0, and from the sum of the first two equations of (2.1), we geṫ It is clear that when x + y ≥ m, we haveṪ < 0, and so, T is decreasing, and hence it is bounded. This implies that the functions x and y are bounded on [0, ∞). From the third Noting that the function y is bounded on [0, ∞), we can assume that |y(s)| ≤ K 1 for some From the fourth equation,ż = c 1 yz − b 1 z, and by using the first and second equations of (2.1), we haveż Therefore, Now we consider the following cases: (c) If (r − d + b 1 ) ≥ 0 and (b 1 − δ) ≤ 0, then by using that x(s) is bounded on [0, ∞), we take |x(s)| ≤ K 2 and we get By integrating, we obtain that Now we consider the following: then by the boundedness of v, we assume |v| ≤ K 3 and we get Putting all together with the above calculations, we conclude that all the solutions of system (2.1) are bounded.

Infection-free equilibrium points
In this section, we study the conditions of existence and stability of the infection-free equilibrium points of the system (2.1), and we compute the basic reproduction number associated with them. According to [6], the basic reproduction number indicates the number of secondary infections by an infectious cell during proliferation. In the HIV model, the basic reproduction number plays an important role in spreading the reproduction of infection among healthy CD + 4 T-cells. System (2.1) has two infection-free equilibrium points E 00 = (0, 0, 0, 0, 0) and E 01 = m(r−d) r , 0, 0, 0, 0 . At E 01 , the uninfected CD + 4 cells reach to the maximal level. In this case, the disease cannot raid.

Local stability of the infection-free equilibrium points
We suppose R 0 is the basic reproduction number of system (2.1) for the infection-free equilibrium point E 00 that is given by R 0 = r d .
Proof. The Jacobian matrix of system (2.1) at E 00 = (0, 0, 0, 0, 0) is given by It is obvious that, the eigenvalues of the matrix J E 00 are Now it is clear that for R 0 < 1, all the eigenvalues of the Jacobian matrix J E 00 are negative, and hence, the free-equilibrium point E 00 is locally asymptotically stable, and it is unstable for R 0 > 1.
By a simple calculation, we can show that the basic reproduction number of system (2.1) for the infection-free equilibrium point E 01 is given by The threshold number R 1 denotes the number of infectious cells produced by the proliferation of an infectious cell [9].
Theorem 4.2. The disease-free equilibrium point E 01 is asymptotically stable when 0 < R 1 < 1 and it is unstable when R 1 > 1.
Proof. To simplify the calculations, we set A = µ(δ+ρ) δ1 and B = µβ 2 δ 1 . Then, the Jacobian matrix of system (2.1) at the disease-free equilibrium point E 01 is given by The characteristic polynomial of J E 01 is equal to Thus, the eigenvalues of the Jacobian matrix J E 01 are The above gives a 1 > 0. Therefore, the eigenvalues λ 1 , λ 2 , λ 3 , λ 4 and λ 5 are negative, and this shows that E 01 is locally asymptotically stable.

Steady-state equilibrium points
In this section, we discuss the conditions of existence and stability of the infection steadystate equilibrium points. All the steady-state equilibrium points exist when R 1 > 1. It shows the presence and spread of the disease. We can show that for R 1 > 1, system (2.1) has four equilibrium points. It has an equilibrium point and v 1 is the positive real root of the equation In addition, by setting A 2 = δ 1 µ (η 1 +Bα 2 ) and α 2 = 1+αv 2 , system (2.1) has an equilibrium point E 2 = (x 2 , y 2 , v 2 , z 2 , 0), where and z 2 is the unique positive solution of the equation and w 3 is the positive real root of the equation By letting ρ 2 = N µ, and α 4 = 1 + αv 4 , we see that the fourth equilibrium point of the system (2.1) is and z 4 is the positive real root of the quadratic equation Proof. It is clear that if R 1 < 1, then all the coefficients defined in (5.1) of the polynomial ψ 3 v 3 + ψ 2 v 2 + ψ 1 v + ψ 0 are positive and it has no positive roots. If R 1 = 1, then ψ 0 = 0 and the only non-negative root of the polynomial To show the existence of E 1 = (x 1 , y 1 , v 1 , 0, 0), we prove that the equation with ψ 0 < 0 has at least a positive real root. Since then ψ 0 < 0 when R 1 > 1. Now we use the linearization method to determine the stability of the equilibrium point E 1 . The Jacobian matrix evaluated at the equilibrium point E 1 is given by The characteristic equation of the Jacobian matrix J E 1 is given by Therefore, the eigenvalues are given by For v 1 < min b 1 δ 1 c 1 µ , b 2 c 2 , we have λ 1 , λ 2 < 0. The other eigenvalues of J E 1 have negative real parts. So the equilibrium point E 1 is locally asymptotically stable when v 1 < Theorem 5.2. The following holds: , then the equilibrium point E 2 exists and it is locally asymptotically stable for v 2 < b 2 c 2 .
Proof. According to Descartes's rule, there exists at least one positive real root z = z 2 for the equation provided T 2 > 0 and T 0 < 0, with the coefficients defined in (5.2). We use the linearization method to study the local stability of the equilibrium point E 2 . The Jacobian matrix calculated at the equilibrium point E 2 is given by The characteristic equation of the matrix J E 2 is given by . Now, we apply the Routh-Hurwitz criterion for the fourth-order polynomial λ 4 +n 1 λ 3 +n 2 λ 2 +n 3 λ+n 4 to study the remaining eigenvalues. On this basis, we compute That due to lengthy calculations, we will not write them here. Therefore, all the roots of the characteristic polynomial λ 4 + n 1 λ 3 + n 2 λ 2 + n 3 λ + n 4 have negative real parts, and hence, E 2 is locally asymptotically stable.
then the equilibrium point E 3 exists and it is locally asymptotically stable when Proof. If R 0 ≤ 1, then Q i > 0 for i = 0, 1, 2, 3. Hence, we do not have any positive real root for w. We assume R 0 > 1 and Q 0 < 0, then the equation has at least a positive real root and we get the equilibrium point The Jacobian matrix evaluated at E 3 is given by The characteristic polynomial of the matrix J E 3 is as follows: where One of the eigenvalues of the Jacobian matrix J E 3 is λ 1 = c 1 y 3 −b 2 , which is negative when y 3 < b 1 c 1 . Now, to investigate the other eigenvalues of J E 3 , we apply the Routh-Hourwitz criterion to the fourth-order polynomial λ 4 + e 1 λ 3 + e 2 λ 2 + e 3 λ + e 4 , and we get ∆ 1 = e 1 > 0, ∆ 2 = e 1 e 2 − e 3 > 0, ∆ 3 = ∆ 2 e 3 − (e 1 e 4 )e 1 > 0.
Thus, all the eigenvalues of the matrix J E 3 have negative real parts. Hence, the equilibrium point E 3 is locally asymptotically stable.
For the fourth endemic equilibrium point E 4 , we have the following result.
Theorem 5.4. The following statements are satisfied: (i) If R 0 ≤ 1, then the equilibrium point E 4 does not exist.
(ii) If R 0 > 1 and then the endemic equilibrium point E 4 exists and it is locally asymptotically stable.
Proof. If R 0 ≤ 1, then Ψ i > 0 for i = 0, 1, 2. Hence we do not have any positive value for z 4 . We assume R 0 > 1 and Ψ 0 < 0. Then, the equation Ψ 2 z 2 + Ψ 1 z + Ψ 0 = 0, with the coefficients defined in (5.3), has exactly one positive root z = z So we get the unique equilibrium point E 4 = (x 4 , y 4 , v 4 , z 4 , w 4 ). To analyze its stability, we compute the Jacobian matrix at the point E 4 , which is given by The characteristic polynomial of the matrix J E 4 is obtained as follows: By using the Routh-Hurwitz criterion for the fifth-order polynomials, we get As ∆ 3 q 4 + q 1 q 4 q 5 > ∆ 2 q 2 q 5 + q 2 5 , we deduce that ∆ 4 > 0. This implies that all the eigenvalues of the Jacobian matrix J E 4 have negative real parts. Therefore E 4 is locally asymptotically stable.

Analysis of the basic reproductive number
In this section, we analyze the basic production rate, and we examine the effect of different parameters for the viral load related to the basic reproduction number, R 1 , mentioned in the equation (4.1). The basic production number indicates the prevalence of infection in the population, and it is important ideas in epidemic theory for mathematicians [17]. The different ways exist for behavior Sensitivity Analysis. In this section, we apply the normalized forward index. The normalized forward index is defined as follows: where s states existence parameters in (4.1). By replacing that with partial derivation (6.1) respect to s, we get The index (6.2) obtained from the parameters existing in the system (2.1) indicates the sensitivity of the system parameters. It is positive for the parameters r, m, n, β 1 and β 2 , which means that increasing them raises the reproduction rate number R 1 and as a result the disease spreads. While this index is negative for the parameters d, ρ, η, ε and µ and it shows that increasing these parameters decreases the reproduction rate number R 1 . When the value of R 1 reaches less than one, the disease disappear. When (1 − η)(1 − ε)nρβ 1 − β 2 µ > 0, the index for δ is positive and the index is negative for (1 − η)(1 − ε)nρβ 1 − β 2 µ < 0.

Stability of equilibrium points
In this section, we examine the theoretical results of model (2.1) by numerical simulations. First, we illustrate the stability of equilibrium points for the different values of R 0 and R 1 . In the following, we examine the effect of cell-to-cell transmission on the spread of infection for different values β 2 . Then, the sensitivity analysis is used to identify important factors in increasing or decreasing the rate of reproduction. The parameters of ρ 2 , c 2 and b 2 may vary with different types of antibodies [35]. Initial values in Table 7.2 are expressed according to clinical data of HIV-infected patients during the symptomatic phase [16].
Since several biological studies indicate that some patients with HIV are treated after three to six months, our chosen interval will change between 100 days to 150 days [24]. In Figure 7.1, by using from Column 1 of Table 7.1, we have R 0 = 0.41 < 1, then E 0 = (0, 0, 0, 0, 0) or disease-free equilibrium point is stable. It shows that the treatment reduces the virus load and the disease will disappear.       , v(t), z(t) and w(t) of system (2.1) with treatment and without treatment for Column 1 of Table 7.1 at time t for R 0 = 0.41 < 1 at free-equilibrium point of E 0 = (0, 0, 0, 0, 0).
In Figure 7.2, from Column 2 of Table 7.1, we obtain R 1 = 0.05122764706 < 1 and R 0 = 100 > 1, then E 01 = (990, 0, 0, 0, 0) is asymptotically stable. This leads to the saturation of the population of uninfected cells and eventually eliminate the disease.  Table 7.1 with R 0 = 100 > 1 and R 1 = 0.05122764706 < 1 at E 01 = (990, 0, 0, 0, 0). In Figure 7.3, by replacing values from Column 3 of Table 7.1 in system (2.1), we obtain R 1 = 2.026297059 > 1 that despite the condition ψ 0 = −0.0002085467695 < 0, we have E 1 = (14841.86236, 6.496271350, 1039.403416, 0, 0) is asymptotically stable. In other words, by increasing the proliferation rate of the uninfected cells population and decreasing the level of treatment relative to Column 2, the values of infected cells and the virus in Table 7.1 are decreased such that they converge to y 1 and v 1 . Still, the levels of cellular and humoral immune cells converge to zero.  Table 7.1, we obtain R 1 = 4.703470589 > 1 and since T 0 = −1.464981642 < 0, the equilibrium E 2 = (98984.77489, 9.523809524, 1047.619048, 685.1145481, 0) exists and it is asymptotically stable. This means that treatment reduces the level of proliferation of viral and infectious cells and prevents excessive cellular immunity.  Figure 7.5, we choose parameters from Column 5 of Table 7.1 into system (2.1) and derive R 1 = 2.351735294 > 1 and Q 0 = −0.00005940827650 < 0. Then, it can be concluded that E 3 = (49497.49885, 1.996874149, 100, 0, 3589.684691) is an equilibrium point and it is asymptotically stable. By observing Figure 7.5, stimulation of the humoral immune system and timely treatment can significantly reduce virus replication and the number of infected cells.  We find R 1 = 20.39982353 > 1 and Ψ 0 = −0.001109190902 < 0 for parameters chosen from Column 6 of Table 7.1; see Figure 7.6. Therefore, E 4 = (296974, 9.523809524, 1000, 2997.258841, 1571.428572) is the asymptotically stable equilibrium point for the differential system. In this case, trajectories associated with different initial values typically converge to this equilibrium. Here, the virus load is controlled by the treatment, yet the virus remains in the body.
We simulate the stability of E 4 by the Routh Hurwitz method. The Jacobian matrix at the point E 4 is given by The characteristic polynomial of matrix J E 4 is obtained as follows: P J E 4 (λ) = λ 5 + 9.918934470λ 4 + 24.17243443λ 3 + 20.00924654λ 2 + 6.567388432λ + 0.186498605.
The Routh Hurwitz table is derived as There is no change of sign in the first Column of the Routh Hurwitz table. Hence, P J E 4 (λ) has no positive root. We derive the Routh Hurwitz table for computing the number of negative roots as There are five sign changes in the first Column of table RL. Thereby, P J E 4 (λ) has five negative roots. Figure 7.7 depicts the impact of the accident rate β 2 on the infection cell-to-cell transmissions (i.e., through cell-to-cell transmission) for different values of β 2 = 0, 0.00000047, and 0.0000047. By using values from Column 3 of Table 7.1, we compute the reproduction rate numbers in Table 7.3. These illustrate the role of essential parameters in the system for the increase or decrease of the infection.

Global dynamics and convergence dynamics
For reproduction rate number R 1 > 1, model (2.1) has four equilibrium points, where only E 4 is an internal (positive) equilibrium point (see Theorem 5.4). Numerical simulation shows that model (2.1) with parameter values taken from Column 3 of Table 7    If we use the parameter values given in Column 5 of Table 7.1, then model (2.1) admits three equilibrium points E 3 = (49497.49885, 1.996874149, 100, 0, 3589.684692), E 2 = (49487. 6, 9.5, 1048, 87.5001, 0) and E 1 = (17534. 8,8.847, 1415.5, 0, 0). In this case, all so-lutions starting from outside the invariant subspace {w = 0} converge to E 3 ; the solutions with initial values z = w = 0 converge to E 1 , and the other solutions with w = 0 converge to E 2 . See Figure 7.10. Here, we have three equilibria. When immune of cellular is inactive and humoral immune is active, the cellular dynamics of the system converge to E 3 . Since the cellular immune of E 3 is zero, the cellular immune in the body remains inactive. By simulating the model with the parameter values from Column 6 of Table 7.1, we see that there are four equilibrium points E 1 = (171905, 166.8, 26683.9, 0, 0), E 2 = (296970, 9.5, 1524, 3696.2, 0), E 3 = (296892, 86.55, 1000, 0, 38545), and E 4 = (296974, 9.523809524, 1000, 2997.258841, 1571.428572). By choosing different initial values, we observe that the solutions with any initial point with positive coordinates converge to E 4 . But the solutions with z = w = 0 converge to E 1 , the solutions with w = 0 and z = 0 converge to E 2 , and the solutions with z = 0 and w = 0 converge to E 3 . See Figure 7.11. The above simulation suggests that E 4 is globally asymptotically stable. Among the equilibria, E 4 stands to attract the cellular dynamics of the system for when immune of cellular and humoral are both active.

Conclusion
In this paper, we first formulate and then analyse a definitive in-vivo mathematical model for the dynamics of the human immunodeficiency virus (HIV). This model investigates the effect of cellular and humoral immune response on the CD + 4 uninfected cells, the CD + 4 infected cells, and the virus. Since infected CD + 4 T-cells affect the rate of proliferation of uninfected CD + 4 T-cells, the rate of the spread of uninfected CD + 4 T-cells in the model was considered as a logistic function. Also, in this model, the effect of cell-to-cell and virus-to-cell transmission in the spread of the disease have considered. In this model, the rate of cure and its impact on improving infected CD + 4 T-cells to uninfected CD + 4 T-cells have impacted. Then, compatibility with biological hypotheses, the boundedness, and the non-negativity of solutions was proved. We also showed that this model has six equilibrium points, including two free infection points and four endemic points. Then, the equilibrium points asymptotic stability was investigated by the Routh Hurwitz method.
The treatment parameters were considered as two parameters (the effectiveness of Reverse Transcriptase and Protease Inhibitors) time-independent in the model (2.1) and were examined their dynamics. Since global dynamic is biologically relevant, global dynamic analysis for various parameter values and the infected equilibrium points of model (2.1) is numerically examined. In all cases, the treatment is not sufficient to eliminate the disease and the virus remains in the body once the body is infected. However, the treatment appears to beneficial to control the amount of virus in the body and rate of infections.
Entry of any amount of virus into the body and the production of infected cells cause initial increases in cellular and humoral immune cells. However, they are finally stabilized and the treatment balances the number of viruses, healthy and infected cells. In there words, the virus load, the number of infected and healthy cells, cellular and humoral immune systems are either reduced or increased according to their relative values to the globally asymptotic equilibrium point E 4 .
Finally, by simulating the model (2.1), using Mathematica and Mapple software, the convergence of equilibrium points for the appropriate parameters, the effect of the cell-tocell transfer on disease spread, sensitivity analysis of reproduction for some parameters, and its effect on decreasing or increasing this amount was shown.