GLOBAL STABILITY OF HIV-1 AND HIV-2 MODEL WITH DRUG RESISTANCE COMPARTMENT

In this paper we propose a new mathematical model describing the dynamics of the human immunodeficiency virus 1 and 2 with drug resistant compartment. We proved the positivity and boundedness of solutions with non-negative initial conditions. The dynamical system admits four equilibrium states: free equilibrium disease, one endemic equilibrium of each strain and one of the two strains. Two basic reproduction numbers are calculated. The global stability of the four equilibrium points is proved by using suitable Lyapunov functions. Numerical simulations were carried out to illustrate our results and a parameter sensitivity analysis completed this work.


INTRODUCTION
Human Immunodeficiency Virus infection remains a major public health problem of global proportions, resulting in nearly 33 million deaths to date [1]. However, with improved access to effective prevention, diagnosis, treatment and care, including for opportunistic infections, HIV infection has become a chronic condition that can be managed with the assurance of a long and healthy life, and is classified into two types: HIV-1 and HIV-2. HIV-1 was first discovered and is more prevalent worldwide, while HIV-2 is less pathogenic and is largely housed in West Africa. So when we generally say HIV, we refer to HIV-1. The main differences between HIV-1 and HIV-2 infections lie in the mechanism of retroviral pathogenesis, which is not entirely clear yet, but they have the same symptoms. The advanced stage of HIV infection is the Acquired Immunodeficiency and HIV-2 infection. Therefore, it is recommended to ensure that the differentiation between HIV-1 and HIV-2 is correctly carried out at the time of HIV diagnosis. This is essential in order to use the appropriate and specific virological monitoring tests and to choose an appropriate. Currently World Health Organization (WHO) [1] is developing a new five-tear global action plan for 2017-2021 to support a coordinated emergence of HIV drug resistance, and to strengthen country efforts to achieve the global HIV targets treatment [2].
Mathematical modelling is an important tool for describing and understanding the dynamics of numerous infectious diseases, which allows monitoring. The classical mathematical model for infectious diseases is the compartment model, first proposed by Kermack and McKendric in the year of 1927, in which, individuals are divided into multiple compartments dependent on their epidemiological status [3]. Since then, several mathematical models have been proposed for HIV/ AIDS transmission dynamics to find out the mechanism of HIV transmission and to determine the effective measures in preventing and controlling the spread of HIV/AIDS [4], [5], [6], [7], [8].
In [9] the author's derived HIV therapeutic strategies by formulating and analyzing an optimal control problem using two types of dynamics treatments while [10] investigated the fundamental role of chemotherapy treatment in controlling the virus reproduction in an HIV patient. Moreover, [11] presented the impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives on the transmission dynamics of the disease in a homogeneous population with constant immigration of susceptible incorporating use of condom, screening of unaware infectives and treatment of the infected. And the authors in [12] proposed a new epidemiological model for HIV/AIDS transmission including PrEP and study a control problem to determine the PrEP strategy that satisfies the mixed state control constraint and minimizes the number of individuals with pre-AIDS HIV infection balanced against the costs associated with PrEP. Most recently, Gurmu et all. [13] studied the role of passive immunity and drug therapy in reducing the replication and transmission of the disease for a mathematical model of HIV/AIDS transmission dynamics with drug resistance compartment.
In this work, we continue the investigation of this last kind of problems by taking into account twostrain HIV-1 and HIV-2 model with drug resistance compartment. In our paper we will establish the global stability of all our two-strain HIV model equilibria.
The rest of the paper is organized as follows. In the next section, we introduce the HIV-1 and HIV-2 model with drug resistance compartment, and we show the positivity and boundedness of the solutions of our model with positive initial conditions. In section 3, we calculate the basic reproduction number, and we study the global stability of the equilibria. Section 4 presents numerical simulations to assess the dynamics of a HIV-1 and HIV-2 transmission with drug resistance compartment.
The sensitivity analysis of the basic reproduction number with respect to the parameters of our model is given in Section 5. Finally, a brief conclusion sums up the paper. The dynamics of the model described by the following nonlinear system of differential equations : with : The corresponding flow chart and description of the parameters for the model (2.1) are given in figure 1 and table 1, respectively.   Proof. We have According to Lemma 2 in [14], the positivity of all solutions initiating in R 6 + under positive initial conditions is guaranteed.

Theorem 2. The biologically region
is bounded, and all the solutions of the model (2.1) starting in Ω confined within the region. This completes the proof.

MODEL ANALYSIS
3.1. Basic Reproduction Number R 0 . The basic reproduction number, denoted R 0 , is one of the most important concepts about the dynamics of epidemic models, which represents the expected number of secondary cases caused by a typical infected individual in a completely susceptible population [15].
Mathematically the basic reproduction number is defined as a spectral radius of the next generation matrix FV −1 [16]: R 0 = ρ(FV −1 ), where F is the non-negative matrix of the new infection terms, and V is the matrix of the transition infections associated.
In our model, the infected compartments are I 1 , I 2 and A, then the matrices F and V are : The dominant eigenvalues of FV −1 are : Consequently, the basic reproduction number of model (2.1) is :

3.2.
Model steady states. The model (2.1) admits four equilibrium points, whose one diseasefree equilibrium (DFE) and three endemic equilibrium as follows : The HIV-1 only endemic equilibrium exist when R 1 0 > 1, and is given by • The HIV-2 only endemic equilibrium exists when R 2 0 > 1, and is given by • The interior endemic equilibrium point for the model (2.1) is given by where : 3.3. Global stability. In this section, we will prove the global stability of the equilibrium points.
Theorem 3. If R 0 ≤ 1, the disease-free equilibrium point E f is globally asymptotically stable.
Theorem 5. The HIV-2 only endemic equilibrium E S 2 is globally asymptotically stable if R 1 0 1 < Proof. Consider the following Lyapunov function : Differentiating V 2 with respect to time gives : Substituting the expressions for the derivatives inV 2 , it follows from 2.1 thaṫ We have Λ = β 2 S * 2 I * 2,2 + µS * 2 from the first equation of 2.1 at steady-state E S 2 , therefore,V 2 can be written aṡ then,V 2 can be simplified to : This impliesV 2 < 0, by the relation between geometric and arithmetic means. The equalityV 2 = 0 holds if and only if (S, I 1 , I 2 , D R , A, R) take the equilibrium values (S * 2 , I * 2,2 , 0, D * R,2 , A * 2 , R * 2 ). Therefore, by LaSalle's Invariance Principle [18], the endemic equilibrium E S 2 is globally asymptotically stable.
Theorem 6. The interior endemic equilibrium E S t is globally asymptotically stable if R 0 > 1.
Proof. Consider the following Lyapunov function : t − S * t ln S) + (I 1 − I * 1,t − I * 1,t ln I 1 ) + (I 2 − I * 2,t − I * 2,t ln I 2 ) Differentiating V t with respect to time gives : Substituting the expressions for the derivatives inV t , it follows from 2.1 thaṫ We have Λ = β 1 S * t I * 1,t + β 1 S * t I * 2,t + µS * t from the first equation of 2.1 at steady-state E S t , therefore, V t can be written aṡ which can then be simplified tȯ This impliesV t < 0, by the relation between geometric and arithmetic means. The equalityV t = 0 holds if and only if (S, I 1 , I 2 , D R , A, R) take the equilibrium values (S * t , I * t,1 , I * t,2 , D * R,t , A * t , R * t ). Therefore, by LaSalle's Invariance Principle [18], the interior endemic equilibrium E S t is globally asymptotically stable.

NUMERICAL SIMULATIONS
In this section, we give some numerical simulations of the model 2.1 to verify the validity of our theoretical results. The parameter values for each numerical simulation are displayed in table 2.
From figure 2, we show that all curves are decreases to zero, unless the susceptible individuals, this is because both the basic reproduction numbers are less than one (R 1 0 = 0.47 and R 2 0 = 0.37). This shows that the disease persists and thus agrees with Theorem 3 which says that the disease-free equilibrium is globally asymptotically stable.
Next, from Figure 3, we show that HIV-1 infected individuals persists while the HIV-2 infected individuals dies out, this is because the basic reproduction number for I 2 is less than unity while the other is great than 1 (R 1 0 = 2.39 and R 2 0 = 0.37). Thus result agrees with Theorem 4. From Figure 4, we observe the persistence of the HIV-1 infected individuals and HIV-2 infected individuals, we can also remark that the persistence of I 1 is higher than the one of I 2 , this is because ( R 1 0 = 7, 09 > 1 and R 2 0 = 6.67 > 1). Thus result agrees the global stability of the interior endemic equilibrium. Table 2. Model parameters and their interpretations.

SENSITIVITY OF THE BASIC REPRODUCTION NUMBER
The sensitivity analysis of the basic reproduction numbers aims to determine the influence of some parameters on dynamic of the model 2.1, using the normalized forward sensitivity index follows.
Definition 7. [19]; [20] The normalized forward sensitivity index of R 0 that depends differentiability on a parameter p is defined by We examine the sensitivity index firstly for the basic reproduction number R 1 0 with respect to Λ and β 1 .
Proposition 8. The normalized forward sensitivity index of R 1 0 with respect to Λ and β 1 is 1 : Proof. It is a simple application of definition 7.
The sensitivity index of R 1 0 with respect to θ 1 , ω 1 and µ is given, respectively, by : 111. Secondly, we compute the sensitivity index for the basic reproduction number R 2 0 with respect to Λ, β 2 , θ 2 , ω 2 and µ.  Table 3. Sensitivity index of R 1 0 and R 2 0 for parameter values given in Table 2 The parameters that they have a positive sensitivity indices means that they have a great impact on persistence of the disease in the population if their values are increasing, will lead to an increase in the basic reproduction number. Furthermore, the parameters that they have a negative sensitivity indices means that they have an influence to minimizing the burden of the disease in the population as their values increasing while the others are left constant, will lead to a decreases in the basic reproduction number, which leads to minimizing the rate of infection in the population [21].

CONCLUSION
In this work, we have formulated a mathematical model for the transmission dynamics of HIV-1 and HIV-2 with drug resistance compartment. Moreover, existence, positivity and boundedness are verified. We have computed the basic reproduction numbers, then, we found two basic reproduction numbers, we have proved the global stability of both the disease-free and endemic equilibrium by using Lyapunov's direct method and LaSalle's invariance principle. Furthermore, we have introduced a numerical simulations illustrate and extend the obtained theoretical results. Sensitivity analysis of the model is analyzed to establish which parameter has high effect on the transmission of the disease.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.