GLOBAL DYNAMICS OF HIV INFECTION WITH TWO DISEASE TRANSMISSION ROUTES-A MATHEMATICAL MODEL

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we have studied the global dynamics of HIV model with two transmission paths: direct transmission through cells-to-cells contact and indirect transmission through virus. We have derived a four dimensional mathematical model including uninfected CD+T 4 cells, infected CD4 T cells, virus and the CTL immune response cells. The nonnegativity and boundedness property of the solutions the proposed mathematical system have been analysed, and the basic reproduction ratio R0 has been derived with the help of next generation matrix method. We also discussed the local and global stability with respect to the basic reproduction ratio of both disease-free and interior equilibrium points under certain conditions. Through numerical simulations, we have validated the all analytical findings. We have established that the disease-free equilibrium is globally stable for R0 < 1 and endemic equilibrium is globally stable for R0 > 1 whenever exists. It is also observed that cells-to-cells transmission rate is more effective compare to virus-to-cells infection rate.


INTRODUCTION
Human Immunodeficiency Virus (HIV) is treated as the most serious infectious disease worldwide and Acquired Immunodeficiency Syndrome (AIDS) is the last stage of this deadly infection process. Till date, 35 million people (approx.) have died from AIDS related illness and 30-40 million individual are living with HIV [1,2]. Mainly, HIV targets the human immune system and as a consequence immune system breaks down and can't work properly. As a result, HIV infected people can easily infected by the other infectious disease (influenza, pneumonia, tuberculosis etc.). When CD + 4 T cells count is less than 200 mm −3 , then HIV infected patient is treated as an AIDS patient [3,4].
CD + 4 T cells plays various important role in human immune system.It also acts as the main defender against the deadly RNA-virus. Viruses can spread by infecting CD +

T cells in two ways namely 'virus-to-cells' HIV infection as well as 'cells-to-cells' transmission. The pathway
of Virus-to-cells HIV infection is considered as a multistage process [5,6,7,8]. Firstly, the envelope protein (gp120) on the surface of HIV binds with CD + 4 receptor and two co-receptors (CXCR4 and CCR5) of healthyCD + 4 T cells. Then, virus injects the genetic material in to the healthy T cells by fusion process. This genetic material transforms viral RNA genome to DNA copy by reverse transcriptase enzyme. Then by another viral enzyme this DNA copy integrates into the viral DNA and at last by protease enzyme it transforms to infected provirus, thus the cells becomes infected. On the other hand, by the cells-to-cells transmission process virus is spread in our body through virological synapses which are a predominant mode of viral transmission [9,10,11]. These Virological synapses are formed for the interaction between CD + 4 and HIV envelope glycoprotein. When the donor and target cells interact with each other, a large number of infectious particles are accumulated and released at the places of 'cells-tocells' contacts [12]. It is well known that this cells-to-cells transmission process is significantly more efficient and faster viral replication mechanism. Also, during both process of infection due to huge replication of infected T cells and virus, immune system of our body produces an another type of T cells i.e. Cytotoxic T-lymphocyte cells (CTL). This immune cells has a significant role in suppressing HIV replication in acute infection [13,14].
Since few years, several mathematical models have been proposed to capture the HIV viral dynamics in theoretical perceptive. In some of models, authors have considered that infection occurs only from free HIV virus to CD + 4 T cells i.e. by "virus-to-cells transmission" [15,16,17,27]. Recently, many eminent researchers have described that there is an another viral transmission mechanism for the case of HIV -the "cells-to-cells transmission" [20,21].
Lai et al. have also proposed a mathematical model considering both mode of infection transmission viz. virus-to-cells and cells-to-cells [22]. Roy et al. [25] have described a mathematical model by considering the qualitative behavior of CTL response in a HIV model. Yet the global dynamics of a HIV model including both mode of infection transmission has not been explored.
In this paper, we develop a mathematical model by incorporating both mode of transmission with the immune response of CTL, which attacks infected cells and play a critical role for antiviral defense. We also consider that the virus can be proliferated by external viral sources other than infected T cells. Furthermore, we have analyzed the global stability of our formulated model by considering a suitable Lyapunov function. By using this global dynamical behavior, we also try to find out the answers of the following questions: (a) which transmission process is most effective? (b) how the CTLs response can prevent this effective transmission process?
This article is arranged as follows. Firstly, we formulate the model with initial condition in section 2. We have found the equilibrium points of the system and determine the basic reproductive ratio in Section 3. In Section 5 and Section 6, we discuss the local and global stability depending on the value of basic reproductive ratio respectively. Numerical simulations are presented in section 7 with discussion. Finally, section 8 concludes the paper with important findings.

MODEL FORMULATION
We construct a mathematical model of HIV disease dynamics considering both virus-to-cells infection and cells-to-cells transmission. Here CD4 + T cells population is partitioned into uninfected CD4 + T cells(x) and infected CD4 + T cells(y), with x(t) and y(t) representing their concentration respectively at a time t. We also consider virus population (v) and CTL population (z), with v(t), z(t) representing their concentration at a time t respectively.
The uninfected CD4 + T cells are produced from bone marrow and mature in thymus at a constant rate. Here λ be the constant production rate of this uninfected immune cells. We assume that the uninfected CD4 + T cells become infected by direct cells-to-cells transmission at a rate β 1 and by free virus at a rate β 2 . Here d 1 and d 2 are the per capita mortality rate of uninfected CD4 + T cells and infected CD4 + T cells respectively. Infected T cells are assumed to produce on average N mature viruses during its lifetime i.e we assume that Nd 2 is the growth rate of virus by infected CD4 + T cells. We also consider that av b+v is the proliferation rate of virus from other infected cells like macrophages. It should be noted here that the growth rate of external viral source other than T cells is a and half saturation constant of external viral source is b. Here d 3 represents the natural removing rate of virus.
We also consider CTL immune response to defend the virus replication and α is the proliferation rate due to immense growth of infected CD4 + T cells. Here we assume d 4 as removing rate of CTL response. The following equation demontrate the CTL population dynamics.
Here we also consider that the CTL response has a negative impact on infected CD4 + T cells and assume that β 3 is the apoptosis rate of infected cells due to CTL response. Based on the considerations, growth rate of the uninfected CD4 + T can rewrite as follows Assembling together the above three system of equation (1, 2, 3), we can rewrite the compact proposed mathematical model > 0 are the initial condition for above system and for this given initial condition, the solution (x(t), y(t), v(t), z(t)) of the system (4) is positively invariant and uniformly bounded in a region Π for t > 0 where

EQUILIBRIA AND STABILITY ANALYSIS
We will analyse the existence and stability of equilibria of the system (4) using the basic reproductive ratio which is determined below.
3.1. Basic reproductive ratio, R 0 . Here, in case of both mode of transmission the infectious compartments of the Jacobian matrix of the system at E 0 can be written as Now we define two matrices F and V as Thus the basic reproduction number R 0 can be defined as the spectral radius of the next generation operator FV −1 i.e the largest eigen value of the matrix FV −1 , where Therefore, where, , .

3.2.
Existence of equilibria. The system has four equilibria namely the infection-free equi- Since Θ 1 > 0 and Θ 3 < 0 then the above equation has a unique positive root. From the above expression of x * , y * , z * , it can be shown that the endemic equilibrium point

LOCAL STABILITY ANALYSIS
Now the Jacobian of the system at any point E(x, y, v, z) is given by Now at the infection-free equilibrium E 0 , the characteristic equation becomes If A 1 and A 2 both positive then by "Routh-Hurwitz Criterion", all the eigenvalues have negative real parts and consequently the infection-free equilibrium point E 0 is locally asymptotically stable. Now A 1 > 0 and A 2 > 0 implies R 0 < 1. This concludes that the infection-free equilibrium point is locally asymptotically stable if R 0 < 1.
Thus we have the following proposition, Moreover, the system (4) has a unique endemic equilibria for R 0 > 1.
At the endemic equilibrium E * , the characteristic equation becomes Then by "Routh-Hurwitz Criterion" at the endemic equilibrium point the system is locally asymptotically stable for R 0 > 1..
From the above discussion, we have the following theorem. Theorem 1. The endemic equilibrium E * is locally asymptotically stable if R 0 > 1.

GLOAL STABILITY ANALYSIS
Theorem 2. The infection-free equilibrium point E 0 is globally asymptotically stable if R 0 < 1.

Proof. We define a Lyapunov function as
Here V (t) > 0 for all positive values of x(t), y(t), v(t), z(t) and V (t) = 0 at infection-free equilibrium point E 0 . Now calculating time derivative of V (t), we geṫ

Proof. Let us consider a Lyapunov function as
Here W (t) > 0 for all positive values of x(t), y(t), v(t), z(t) and W (t) = 0 at endemic equilibrium point E * . Now calculating time derivative of W (t), we geṫ Therefore using A.M. ≥ G.M., we conclude that the 2nd, 3rd, 4th term of the last equation is less than zero. HenceẆ ≤ 0 if R 0 > 1. Moreover at E * ,Ẇ = 0. Using the Lyapunov-LaSalle invariance principle, we conclude that E * is global asymptotically stable for R 0 > 1.

NUMERICAL SIMULATIONS
In this section, on the basis of analytical findings, we carry out the numerical results of our system. We check the numerical results considering parameter values from different articles given in Table 1. The dynamics of cells are plotted with respect to time to investigate the qualitative behavior of considered cells between 100 days. Numerical simulations are done using MATLAB. In this section, we have tried to focus numerical view on dynamical cells interaction globally of the system that is considered in our proposed model.  Figure 1 shows the contour plot of the basic reproductive ration R 0 as a function of β 1 (the rate at which the uninfected CD4 + T cells become infected by cells-to-cells transmission) and   In Figure 3, we demonstrate the stability property of endemic steady state. When R 0 > 1, the model variables goes to a stable steady state E * after approximately 100 days. Due to both type of transmissions, the infected CD + 4 T cells increases rapidly for first 5-6 days approximately, but due to CTL immune response after 5-6 days infected cells can't proliferate swiftly. As a result, infected cells population decreases to a positive steady state after 30 days (approximately).
Here, uninfected cells decreases and virus population increases as time increases. CTL population responses after 5-6 days due to rapid growth of infection and as a result CTL increases smoothly till infected cells goes to stable state. Then after 30 days, CTL immune responses decreases to a certain level as infected cells grows. According to Theorem 3, the steady state E * is globally asymptotically stable for the set of parameters used for this figure.  Considering the value of β 1 as 0.001, a dramatically behaviour is observed as the infected T cells increases not much rapidly and goes to stable condition at 1300 mm −3 after 100 days approximately. Figure 4(B) depict that for different values of β 2 the infected T cells shows different qualitative nature but after 100 days all trajectories converge to a single stable region at density level 1550 mm −3 .
Therefore Figure 4 reveals that for variation of β 2 the trajectories of infected T cells reaches to almost same density but for β 1 the trajectories goes to different density after 100 days. So finally from this figure we conclude that β 1 is more efficient compare to β 2 in disease progression.

DISCUSSION AND CONCLUSIONS
In this research, we have studied the global dynamics of HIV infection considering for both cells-to-cells and virus-to-cells infection. Accordingly, a mathematical model has been formulated and analyzed analytically and numerically. The local stability criterion for infection-free equilibrium and endemic equilibrium has also been studied depending on the basic reproductive ratio, R 0 . The global stability of the infection-free equilibrium and the endemic equilibrium of system has been established by considering a suitable Lyapunov function.
We have observed that cells-to-cells transmission rate (β 1 ) is more affective compare to virusto-cells infection rate (β 2 ) i.e. cells-to-cells transmission have a great impact on spread of HIV infection. In presence of CTL, uninfected cells reaches to stable state at density level 375 mm −3 whereas without CTL it was 200 mm −3 after 100 days approximately. Therefore, the theoretical and the numerical results are in good agreement. Furthermore, our numerical studies also revel that when cells-to-cells transmission rate is high then CTL response is also quick and prominent.
In a nutshell, our model based results suggest that the system immunity represented by CTL can control viral replication and reduce the infection under appropriate conditions.
The present study can be extended in many ways. Effect of time delay can be observed incorporating a time delay in immune response [27,18]. Another important event is to see the effect of optimal therapy for controlling the disease in cost-effective and with minimum side effects [28,29]. How we can enhance the CTL response when cells-to-cells transmission rate is high, that will be the great challenge of our future work.