Stability Analysis of a Delayed HIV / AIDS Epidemic Model with Treatment and Vertical Transmission

A delayed HIV/AIDS epidemic model with treatment and vertical transmission is investigated. The model allows some infected individuals to move from the symptomatic phase to the asymptomatic phase; next generation of infected individuals may be infected and it will take them some time to get maturity and infect others. Mathematical analysis shows that the global dynamics of the spread of the HIV/AIDS are completely determined by the basic reproduction number R0 for our model. If R0 < 1 then disease free equilibrium is globally asymptotically stable, whereas the unique infected equilibrium is globally asymptotically stable if R0 > 1.


Introduction
Mathematical models play an important role in the study of the transmission dynamics of HIV/AIDS, and in some sense, delay models give better compatibility with reality, as they capture the dynamics from the time of infection to the infectiousness.Some HIV/AIDS models are introduced in [1]- [5].In recent years, a few studies of vertical transmission have been conducted to describe the effects of various epidemiological and demographical factors [6]- [8], and some models considered vertical transmission with time delay [9] [10].Some specific HIV models with imperfect vaccine were introduced in [11]- [13].

S K c I bJ S S t I c I bJ S k I
In model (1), it is assumed that some individuals with the symptomatic phases J can be transformed into asymptomatic individuals I after treatment and they get the result that when 0 1 R < the disease free equilibrium is globally asymptotically stable and if 0 1 R > the endemic equilibrium is globally asymptotically stable.In [9], Ram Naresh et Here, the authors assume that a fraction of newborns, who sustain treatment, join the infective class while others, who do not sustain treatment, join the AIDS class after getting sexual maturity.The infectives through vertical transmission at any time t are given by . The authors proved the local and global stability of disease free equilibrium and endemic equilibrium under some conditions.Inspired by these works, we consider an HIV/AIDS model with vertical transmission and with time delay.
The organization of the paper is as follows.In the next section we present the model with delay.Section 3 presents the basic properties of the model.In Section 4, we analyze local and global stability of equilibrium points.In the last section, we present a brief conclusion.

Mathematical Model
We propose an HIV/AIDS model which incorporates time delay during which a newly born infected child attains sexual maturity and becomes infectious.In this model, the sexually mature population is divided into four subclasses: the susceptibles (S), the asymptomatic infectives (I), the symptomatic infectives (J) and full-blown AIDS group (A).The number of total population is denoted by ( ) N t , for any time t.We assume that the susceptibles become HIV infected via sexual contacts with infectives.It is also assumed that all newborns are infected at birth ( ) . It is reasonable to assume that full-blown AIDS patients are sexually inactive and symptomatic stage patients feel uncomfortable (some may know they are AIDS) and the possibility of producing children is small, so can be taken negligible.We also assume that a fraction of infected newborns, who sustain treatment, joins the asymptomatic infective class while others, who do not sustain treatment, joins AIDS class after getting sexual maturity.The infectives through vertical transmission at any time t is given by ( ) because those who are infected at time ( ) becomes infectoius (asymptomatic stage infectious) at time t, if they do not develop to AIDS patient by that time.The fraction of infectives which became AIDS patient during the period of getting sexual maturity, if they survive to the maturity, joins to the AIDS class.However, for the model to be biologically reasonable, it may be more realistic to assume that not all those infected will survive after τ time units, and this claim support the introduction of the survival term e µτ − .Thus, in our model the term ( )e pI t µτ γ τ − − also represents the introduction of infectives through vertical transmission.If the birth rate of newborns γ equals to zero, then our model will back to the model (1).
With the above considerations and assumptions, the spread of the disease is assumed to be governed by the following model: where K µ is the recruitment rate of the population, µ is the death rate.c is the average number of contacts of an individual per unit of time.β and bβ are the probability of disease transmission per contact by an in- fective in the first stage and in the second stage, respectively.1 k and 2 k are transfer rate from the asympto- matic phase I to the symptomatic phase J and from the symptomatic phase to the AIDS cases, respectively.α is transformation rate from the symptomatic phase J to asymptomatic phase I. d is the disease-related death rate of the AIDS cases.γ is the birth rate of infected newborns, p is the fraction of infected newborns joining the asymptomatic infective class after getting sexual maturity and remaining part ( ) joins the AIDS class after getting sexual maturity ( ) It is also assumed that all the parameters of the model are non-negative.Based on it's biological meaning, we always assume that 2 1 k k ≥ .

Basic Properties
For model ( 2), let the initial condition be ( ) ( ) This implies that if µ γ > all solutions of model (3) starting in 3 R + are bounded and eventually enter the attracting set Ω .
It is reasonable to assume that the general death rate µ is greater than the birth rate of infected newborns γ , that is µ γ > .In some models, death rate equal to birth rate.However, in this model, γ is smaller than birth rate.Below we assume µ γ > .Since the variable A of model (3) does not appear in the first three equation, in the subsequent analysis, we only consider the submodel:  ( ) ( )( ) ( ) By straightforward computation, when 0 1 R > model (4) has the unique positive equilibrium ( ) , , E S I J , where .

Stability Analysis
First we will study the local and global stability of disease free equilibrium 0 E .The variational matrix of model ( 4) is given by e .
R < , the disease free equilibrium 0 E is locally asymptotically stable.Proof.The Jacobian matrix corresponding to model (4) about 0 E as follows, ( ) ( ) ( ) where  τ > , we assume that ( ) is the root of characteristic Equation ( 6), then ω satisfies ( )( ) Through simple computation, we can found that all the coefficients of this equation is positive, so Equation ( 7) have no solution, it implies that Equation ( 6) have not the root like i λ ω = .Hence all roots of (6) have negative real part.
We are now in a position to investigate the global stability of the disease-free equilibrium 0 E .Theorem 4.2.If 0 1 R < , then the infection free equilibrium 0 E is globally asymptotically stable.Proof.Consider the following Lyapunov functional.

∫
Calculating the derivative of L along with the solution of model ( 4), we have ≤ when 0 1 R < , the equality 0 L′ = holds if and only if 0, 0 E is globally asymptotically stable by the LaSalle invariance principle [14].Now, when 0 1 R > we will study the local and global stability of * E .Theorem 4.3.If 0 1 R > , the infected equilibrium * E is locally asymptotically stable.Proof.For this purpose, we obtain the Jacobian matrix corresponding to model (4) about * E as follows, ( ) m w c S v q Obviously, 0 1 2 0, 0, 0 l l l > > > and 1 2 0 0 l l l − > .This implies that when 0 1 R > and 0 τ = , * E is locally asymptotically stable by the Hurwitz criterion.Now we study the stability behavior of * E in the case 0 τ > .We assume that ( ) is the root of characteristic equation, then η satisfies ( )( Separating the real and imaginary parts, we have ( ) ( ) Eliminating τ by squaring and adding ( 9) and (10), we get the equation determining for η as, when 2 1 k k ≥ , through simple computation we can see that 0 0 d > , 1 0 d > , 2 0 d > , in this circumstance (11)  has not positive root.So all roots of (8) has negative real part.
Next, we consider the global stability of * E when 0 1 R > .Theorem 4.4.If 0 1 R > , then the infected equilibrium * E is globally asymptotically stable.
Proof.Firstly, we define a function, ( ) . Take the Lyapunov functional as follows.
( ) Next calculating the derivative of V along with the solution of model ( 4), we have

Conclusion
In this paper, we have considered an HIV/AIDS model with treatment, vertical transmission and time delay.Under the assumption that asymptomatic infectives (J) have the symptoms of AIDS, AIDS patients (A) are isolated; hence their probability of producing children is small; and it is neglected.From the local stability of disease free equilibrium, we calculated the basic reproduction number 0 R .Further we get the results that when 0 1 R < the disease free equilibrium is globally asymptotic stable, and when 0 1 R > the endemic equilibrium is globally asymptotic stable.
Cai and X. Li studied local and global stability of the equilibria of a SIJA model with treatment:
I is the unit matrix.
root of this equation is λ µ = − .So we consider the following equation.
+ > .Hence the roots of this equation have negative real part by the Hurwitz criterion.If 0 Eliminating τ by squaring and adding above the two equation, we get that we consider the following variables substitutions by letting, Since the arithmetic mean is greater than or equal to the geometric mean and function g is a positive function, V′ = is the singleton { } * E .Thus, the endemic equilibrium * E is globally asymptotically stable if 0 1 R > by LaSalle Invariance Principle [14].
al. considered the following SIA model with vertical transmission: