Download Sexuality-Based Model for HIV/AIDS with the Effect of Immigration

Sexuality-Based Model for HIV/AIDS with the Effect of Immigration and
Emigration
Pratibha Rani1, V.P. Saxena2, D.S. Hooda1
1
Department of Mathematics,
Jaypee University of Engineering & Technology, Guna (M.P.)
Email: [email protected].
Email: [email protected]
2
Department of Mathematics,
Sagar Institute of Research & Technology, Bhopal (M.P.)
Email: [email protected]
Abstract
In this paper, a nonlinear mathematical model for heterosexual transmission of HIV/AIDS
epidemic with the effect of immigration and emigration is proposed. Here, the total adult
population is divided into three different classes: male, female and female sex workers. The
results of the model are analyzed by using stability theory of differential equations and also, the
numerical simulation is implemented. This paper has been influenced by the importance of
female sex workers in the -transmission of the HIV/AIDS.
Keywords: HIV/AIDS, Immigration, Emigration, Female Sex Workers, Stability.
1. Introduction
One of the most important public-health problem in developing countries is the Acquired
Immune Deficiency Syndrome (AIDS) epidemic, which is caused by Human Immunodeficiency
Virus (HIV). The first case of AIDS was recognized in 1981 and every year the number of
infected people rises rapidly. As per the estimates of WHO and UNAIDS in 2013, approximately
35 million people were living with HIV globally. Of these, around 17.2 million were men, 16
million were women and 3.2 million were less than 15 years old. In the same year, around 2.1
million people were becoming newly infected and 1.5 million died of AIDS. HIV prevalence in
India varies geographically [9, 10, 11, 20]. The four states in India with the highest numbers of
people living with HIV are Andhra Pradesh, Karnataka, Maharashtra and Tamilnadu. HIV is
primarily transmitted through unprotected sexual intercourse or blood transfusion. It is limited to
those people who are capable to transmit HIV either by profession, such as sex workers, age
group, social class, heredity such as mother to child transmissions, lifestyles which include drug
users and misfortune those who acquired it by blood transfer [17]. As the infection progresses,
the virus begins to reproduce itself and affects specific cells of the immune system, called CD4
cells. Within a few weeks, people are experiencing flu-like symptoms such as fever, headache,
upset stomach and muscle aches. This stage of infection is called acute infection or primary HIV
infection. It may last over 8-10 years depending on many factors and then the immune system
has become seriously damaged. Over the time, HIV can destroy many of these CD4 cells and
then the body can’t fight against disease. Thus, HIV infection contributes to AIDS. The life
expectancy of an AIDS patient is 1 to 3 years [12]. In recent years, a sexually transmitted HIV
infection poses an alarming threat in many countries. It ceased to increased death rates in various
risk groups throughout the world. Currently, there are no known vaccines to protect HIV
infection, but it is possible to protect yourself and others from infection through public
awareness efforts. Therefore, it is relevant and important to investigate models for HIV/AIDS
disease with different demographic structures in order to judge the influences of the disease
along the population dynamics [2, 3, 7, 8].
Recently, because of high infection rates and large numbers of sexual partners, a sex worker has
been considered as a core group for the transmission of HIV. Sex workers can increase the peril
of infection of HIV and other sexually transmitted infections (STIs) by engaging in dangerous
sexual behaviors. Female sex workers (FSW) form heterogeneous groups who have exchanged
sex for money, goods and services. These groups are a critical effort for public health [9]. In
some of the recent studies, it has been noted that HIV prevalence among female sex workers
varies widely and it is more than 20 times higher than the HIV prevalence in the general
population [4, 15]. Not only those sex workers, who are engrossed in sexual services, are at
higher risk of HIV infection, but also who are unaware about their HIV status, can put in danger
their own wellness and increases their danger of transmitting HIV infection. A mathematical
model for HIV/AIDS disease is needed in order to obtain a numerical description of these trends
and to make forecasts. In recent years, many studies have been made to model the transmission
of infectious diseases and different issues like immigration, emigration, antiretroviral drugs, the
role of sex workers etc., have been identified [2, 3]. May and Anderson [1] studied an HIV
transmission dynamics model that represents the progression from HIV status of AIDS, where
the population is split into categories of progressive infectious stages. They have been added
various refinements into modeling frameworks [17]. A simple deterministic model has been
proposed by Naresh and Omar [16], to study the transmission dynamics of HIV/AIDS in a
population with variable size structure. Daabu and Seidu [6] developed a model to study the
impact of migration on the spread of HIV/AIDS in South Africa using observed data. Bhunu et.
al [5] presented a mathematical model to assess the link between prostitution and HIV
transmission. Final results from this study suggest that effectively controlling HIV/AIDS calls
for strategies that come up to both prostitution and HIV/AIDS transmission. In India, Eighty-five
per cent of HIV transmission occurs through heterosexual contact with sex workers and their
clients [20, 21]. A theoretical framework for the transmission of HIV/AIDS epidemic in India
has been presented by Srinivasa Rao [18]. Kaur et. al [13] introduced a nonlinear mathematical
model for studying the transmission dynamics of HIV/AIDS epidemic with emphasis on the role
of female sex workers. The surroundings and context in which sex workers live and exercise do
not affirm them to inhibit the risk factors [10, 11]. Because of these causes, sex workers have
been seen a key population and in our opinion the interventions need to be planned and carried
out to lead and empower this specific population if the epidemic is to be controlled [13, 14].
Thus, in the present analysis, the attempts are made to explore the role of female sex-workers,
their clients in the transmission of HIV infection and also study the effect of immigration and
emigration of the individuals in the population.
In this paper, a nonlinear mathematical model is proposed for studying the transmission
dynamics of HIV/AIDS in section 2. In section 3, analysis of the proposed model is discussed.
The stability of the model is explained in section 4 and the numerical result for fixed values of
parameters is discussed in section 5.
2. Mathematical Model
The differential equations for the transmission dynamics of HIV/AIDS are given as follows:
dSm
 1  Sm  1 Sm I f   2 Sm I fs   Sm 1 ,
dt
dI m
 1 Sm I f   2 Sm I fs  (b1   ) I m  I m ,
dt
dAm
 b1 I m  (   d ) Am ,
dt
dS f
dt
  2   S f  3 S f I m   S f  2 ,
dI f
dt
 3 S f I m  (b2   ) I f  I f ,
dAf
dt
dS fs
dt
 b2 I f  (   d ) Af ,
 3   4 S fs I m   S fs 3 ,
dI fs
dt
  4 S fs I m  (b3   ) I fs ,
dAfs
dt
 b3 I fs  (   d ) Afs .
(1)
Here, the total population
N (t )  Sm (t )  I m (t )  Am (t )  S f (t )  I f (t )  Afs (t )  S m (t )  S fs (t )  I fs (t )  Afs (t )
and the parameters used in this model is defined as follows in Table 1:
Table 1. Interpretation of variables and parameters
Symbol
Description
Sm (t ), I m (t ), Am (t )
Population of susceptible, infective and
AIDS-infected male individuals at time ' t '
S f (t ), I f (t ), Af (t )
Population of susceptible, infective and
AIDS-infected female individuals at time
't '
S fs (t ), I fs (t ), A fs (t )
Population of susceptible, infected and
AIDS-infected female sex workers at time
't '
1 ,  2 , 3
Immigration rate of susceptible male,
female and female sex-workers;
1 ,2 ,3
Migration rate of susceptible male, female
and female sex-workers;
1
Interaction rate of susceptible male and
infected female individuals;
2
Interaction rate of susceptible male and
infected female sex workers;
3
Interaction rate of susceptible female and
infected male individuals;
4
Interaction rate of susceptible female sexworkers and infected male individuals;

Birth rate of susceptibles and infectives in
male and female populations;

Natural death rate;
Progression rates from HIV infective male,
female and female sex-workers.
b1 , b2 , b3
The flow diagram of a given model (1) is given as follows:
1
 Sm
Sm
 Sf
Sf
 Sm
( 1 I f   2 I fs ) S m
 Im
2
2
1
Sfs
 S fs
 4 S fs I m
Ifs
If
 If
 I fs
 If
 Im
b1 I m
 S fs
 Sf
3 I m S f
Im
b3 I fs
b2 I f
Af
Am
3
3
Afs
(   d ) Af
(   d ) Am
Fig. 1
(   d ) A fs
3. Analysis of the model
3.1 Positivity and boundedness of solutions
Theorem 3.1: For the time t  0, all the solutions of the system (1) are eventually confined in
the compact subset
{(Sm , S f , S fs , I m , I f , I fs , Am , Af , Afs )
9

: N  (Sm (t )  S f (t )  S fs (t )
 I m (t )  I f (t )  I fs (t )  Am (t )  A f (t )  Afs (t )) 
(   )
}.
(   )
Proof: Since, we have
N (t )  Sm (t )  S f (t )  S fs (t )  I m (t )  I f (t )  I fs (t )  Am (t )  A f (t )  A fs (t ).
Differentiating ' N (t ) ' with respect to ' t ', we get
dN dSm dS f dS fs dI m dI f dI fs dAm dAf dAfs









.
dt
dt
dt
dt
dt
dt
dt
dt
dt
dt
By using (1), it becomes
dN
 (1   2   3 )   ( Sm  S f  I m  I fs )   ( S m  S f  S fs
dt
I m  I f  I fs  Am  Af  Afs )  (1  2  3 )  d ( Am  Af  Afs ),
it can also be written as
dN
 (   ) N    , where   1   2   3 ,   1  2  3 .
dt
(2)
Then the solution of (2) becomes
Ne
(   )t
(  ) e(   ) t

 c.
(   )
At t  0, (3) implies
N (t ) 
(  )
(   )   t
 [ N0 
]e .
(   )
(   )
(3)
lim Sup N 
As t ,
t
(  )
.
(   )
This shows that the solutions of the model (1) are bounded in the interval [0, ), i.e., confined in
the region  .
3.2 Disease-Free Equilibrium and the Basic Reproduction Number
At the disease-free equilibrium state, I m  I f  I fs  Am  Af  Afs  0. Substituting these values in
the above system of equations, the disease free equilibrium of the model (1) is given by
E0  (Sm0 , S 0f , S 0fs , I m0 , I 0f , I 0fs , Am0 , A0f , A0fs )
(
1  1  2  2  3  3
,
,
, 0, 0, 0, 0, 0, 0).
 
 

The linear stability of E0 is established by the basic reproduction number R0 and it is obtained
by taking the largest Eigenvalue of the next generation matrix. The basic reproduction number
R0 is the effective number of secondary infections caused by a typical infected individual during
his entire period of infectiousness. Now, the next generation matrix is calculated as follows:
let F be the rate of appearance of new infection corresponding to (1) and it is given as
0
1 Sm0

0
0
 3 S f

 S0
0
F   4 fs
0
0

0
0
0
0

where S m0 
1 Sm0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0

0
,
0

0
0 
(1  1 ) 0 ( 2  2 ) 0 ( 3  3 )
, Sf 
, S fs 
.
(   )
(   )

Let V be the remaining transfer terms corresponding to the system (1) and it is given as follows:
0
0
0
0
0 
(b1    )

0
(b2    )
0
0
0
0 


0
0
(b3   )
0
0
0 
V 
.
b1
0
0
(  d )
0
0 


0
b2
0
0
(  d )
0 


0
0
b3
0
0
(   d ) 

Now,

0


 3 ( 2 2 )
 (   ) (b    )
1

1

 4 (  3  3 )
FV 

  (b1    )

0

0


0

1 (1 1 )
(   ) (b2    )
1 (1 1 )

0 0 0
(b3   )
0
0
0 0
0
0
0 0
0
0
0
0
0
0
0 0
0 0
0 0


0

.
0

0

0
0 
The corresponding characteristic equation is given by
det( F V 1   I )  0,
it can also be written as

1 (1 1 )
(   ) ( b2    )
 2 (1 1 )
(   ) ( b3   )
0
0
0

0
0
0
0
0

0
0
0
0
0
0
0
0
0

0
0
0

0
0
0

3 ( 2 2 )
(   ) ( b1    )
 4 (  3  3 )
 ( b1    )
0
0
0
Solving the above determinant (4), we get
 0.
(4)
  2


1 3 (1  1 ) ( 2  2 )
    


2
(    ) (b1     ) (b2     ) 

3

 
  0.
  2  4 (1  1 ) ( 3  3 ) 



 (    ) (b1     ) (b3   ) 

It can also be written as


4  2
  



  1 3 (1 1 ) ( 2 2 ) (b3   )
  (   )   (  ) (  )(b    )

2 4
1
1
3
3
2

2

(



)
(
b




)
(
b




)
(
b3   )
1
2





  0.


Therefore, the Eigenvalues of the characteristic equation (5) are   0, 0, 0, 0 and
  1 3 (1 1 ) ( 2 2 ) (b3   )  (   )
   (  ) (  )(b    )
3
3
2
 2   2 4 21 1
  (   ) (b1    ) (b2    ) (b3   )




.



Thus, the basic reproduction number R0 is given by
(1 1 ){  1 3 ( 2 2 ) (b3   )
R0   
 (   )  2  4 (3 3 )(b2    )}
.
 (   )2 (b1    ) (b2    ) (b3   )
3.3 Existence of Endemic Equilibrium
The endemic equilibrium point E1  (Sm* , S *f , S *fs , I m* , I *f , I *fs , Am* , A*f , A*fs ) is given by
(1 1 )  (b1    ) I m* *
( 2 2 )
S 
, Sf 
,
(   )
(    3 I m* )
*
m
S *fs 
(3 3 )
( 2 2 ) 3 I m*
*
,
I

,
f
(    4 I m* )
(    3 I m* ) (b2    )
I *fs 
(3 3 )  4 I m*
b1 I m*
*
,
A

,
m
(  4 I m*   ) (b3   )
(  d )
(5)
A*f 
( 2 2 ) 3 b2 I m*
(3 3 )  4 b3 I m*
*
,
A

.
fs
(   d ) ( 3 I m*    ) (b2    )
(   d ) (  4 I m*   ) (b3   )
Each of the variables are positive if I m*  0. Here I m* is given by the following quadratic
equation:
D1 I m* 2  D2 I m*  D3  0,
where
(6)
D1  [1 3 4 (2 2 ) (b1    ) (b3   )
 2 3 4 (3 3 ) (b1    ) (b2    )
 3 4 (b1    ) (b2    ) (b3   ) (  )]  0,
D2   1 3 4 (1 1 ) (2 2 ) (b3   )  1 3  (2 2 )
(b1    ) (b3   )  2 3 4 (1 1 ) (3 3 ) (b2    )
 (  ) 2 4 (3 3 ) (b1    ) (b2    )   (  )
(b1    ) (b2    ) (b3   ){ 3  (  ) 4 },
It can also be written as
D2   (   ) 2  4 (b1    ) (b2    ) (b3   )
(1 1 ){( 2 2 ) 1 3 (b3   )




 ( 3 3 )  2 3 (b2    )}


 1

(   ) 2 (b1    ) (b2    ) (b3   ) 




  3 (b1    ) (b3   )(( 2 2 ) 1   (  )(b2    ))
 (3 3 ) (  ) 2 4 (b1    ) (b2    ),
and
D3   (   ) 2 (b1    ) (b2    ) (b3   ) (1  R02 ).
For R0  1, we have D1  0, D3  0 and the first bracketed term of D2 is positive as the
transmission rate of infection  4 in female sex workers is always greater than or equal to the
transmission rate of infection  3 of female individuals. Hence, Descartes rule of signs suggests
that there is a zero positive root of the quadratic equation (6). For R0  1, we have
D1  0, D3  0 and D2 may be positive or negative. Therefore, (6) gives a unique positive root
corresponding to D2 , say I m* . Thus the existence of one positive real root I m*
implies the
positivity of the endemic equilibrium for R0  1.
4. Stability Analysis
Theorem 4.1: The disease-free equilibrium E0 of the model (1) is locally asymptotically stable
when R0  1 and unstable otherwise.
Proof: To prove this, suppose that
Sm (t )  Sm0  A1 e t , S f (t )  S 0f  A2 e t , S fs (t )  S 0fs  A3 e t ,
I m (t )  I m0  B1 e t , I f (t )  I 0f  B2 e t , I fs (t )  I 0fs  B3 e t ,
Am (t )  Am0  C1 e t , Af (t )  A0f  C2 e t , Afs (t )  A0fs  C3 e t .
Then, the system of equations (1) becomes
(     ) A1 e t  1
(1 1 )
(1 1 )
B2 e t   2
B3 e t  0,
(   )
(   )
(b1      ) B1 e t  1
(1 1 )
(   )
B2 e t   2 1 1 B3 e t  0,
(   )
(   )
 b1 B1 e t  (   d   ) C1 e t  0,
(     ) A2 e t  3
 3
( 2 2 )
B1 e t  0,
(   )
( 2 2 )
B1 e t  (b2      ) B2 e t  0,
(   )
b2 B2 e t  (   d   ) C2 e t  0,
(    ) A3 e t   4
4
(3 3 )

(3 3 )

B1 e t  0,
B1 e t  (b3     ) B3 e t  0,
b3 B3 e t  (   d   ) C3 e t  0.
(7)
To study the stability criterion of DFE E0 , the Jacobian matrix M 1 has been calculated as
follows:
 (     )


0



0


0

M1  
0



0



0

0


0

0
0
(     )
0
0
(   )
0
0
0
0
0
0
0
0
0
0
b1
0
0
0
0
1 (1 1 )
(   )
 2 (1 1 )
(   )
0
0
0
0
1 (1 1 )
 (  )
 2 1 1
(   )
(   )
(b2      )
0
0
(b3     )

0
b2
0
0
0
b3
0
3 ( 2 2 )
(   )
 4 (  3  3 )

(b1      )
 3 ( 2 2 )
(   )
  4 (  3  3 )




0
0
0


0
0
0


0
0
0

.

0
0
0

0
0
0


(  d   )
0
0

0
(  d   )
0

0
0
(   d   ) 
0
0
0
Therefore,
the
first
six
Eigenvalues
of
the
Jacobian
determinant
are
  , (   d ),  (   d ), (   d ),  (   ),  (   )
and the rest of the Eigenvalues are
calculated by the following cubic equation:
a0  3  a1  2  a2   a3  0,
where
a0 1 0, a1  x  y  z,
(8)
   (  ) (  )   (  ) (3 3 )
a2  x y  y z  z x   1 3 1 1 2 2 2  2 4 1 1
(  )
 (  )


,

and a3  x y z (1  R02 ). Also, x  (b1    ), y  (b2    ), z  (b3   ).
Clearly, for R0 1, we have a0 , a1 , a3  0 and a1 a2  a3 a0 is given as follows:
1 3 (1 1 ) ( 2 2 )

 x ( y  z) 
(   ) 2
a1 a2  a3 a0  ( x  y ) 

  (    ) (  3  3 )
 2 4 1 1

 (   )








  (  ) (3 3 ) 
 z  x ( y  z)  2 4 1 1

 (  )



  (  ) (3 3 ) 
 y  z ( y  z)  2 4 1 1
.
 (  )


Hence, by using Routh-Hurwitz criterion for a third-order polynomial, since a0 , a1 , a3  0 and
a1 a2  a3 a0  0 for R0 1, therefore, the roots of the cubic equation (7) have negative real parts.
Thus the disease free equilibrium E0 is locally asymptotically stable for R0  1.
Theorem 6.2: The endemic equilibrium E1 of the model (1) is globally asymptotically stable
when R0  1.
Proof: To establish the global stability of the endemic equilibrium E1 , we construct the
following Lyapunov function:
V ( Sm* , S *f , S *fs , I m* , I *f , I *fs )  c1 ( S m  S m*  S m* log
 S *f log
Sf
S
*
f
)  c3 ( S fs  S *fs  S *fs log
 c5 ( I f  I *f  I *f log
The time-derivative of V is given by
Sf s
S
*
fs
Sm
)  c2 ( S f  S *f
S m*
)  c4 ( I m  I m*  I m* log
If
If s
If
I *f s
)  c6 ( I fs  I *fs  I *fs log
*
).
Im
)
I m*
 S f  S *f
 S  S *  d Sm
dV
 c1  m m 
 c2 
 Sf
dt
 S m  dt

 S f s  S *f s
 c3 
 Sf s

 I f  I *f
 c5 
 If

 d Sf

 dt
 d Sf s
 I m  I m*  d I m

c


4
 I m  dt
 dt
 I f s  I *f s
 d If
 c6 

 Ifs
dt


 d Ifs
.

dt

(9)
Now, by using (1) in (9), we get
 Sm  Sm* 
dV
 c1 
 (1 1  Sm   Sm  1 Sm I f  2 Sm I fs )
dt
 Sm 
 S f  S *f
 c2 
 Sf


 S fs  S *fs
 ( 2 2  S f   S f  3 S f I m )  c3 

 S fs



 I  I* 
(3 3   S f  4 S fs I m )  c4  m m  ( 1 Sm I f  2 Sm I fs
 Im 
 I f  I *f
 I m  b1 I m   I m )  c5 
 If

 I fs  I *fs
 c6 
 I fs


 ( 3 S f I m  I f  b2 I f   I f )


 (  4 S fs I m  b3 I fs   I fs ).

It can also be written as
 S  S* 
dV
 c1  m m  ((   ) ( Sm  Sm* )  1 Sm* I *f   2 Sm* I *fs
dt
 Sm 
 S f  S *f
 1 S m I f   2 S m I fs )  c2 
 Sf

 S fs  S *fs
 3 S f I m )  c2 
 S fs


*
* *
 ((    ) ( S f  S f )   3 S f I m


*
*
*
 (  ( S fs  S fs )   4 S fs I m   4 S fs I m )

(10)

I *f
I *fs
 I m  I m*  
*
*
 c4 
  1 Sm I f   2 Sm I fs  I m  1 Sm *   2 Sm *
Im
Im
 I m  

 I f  I *f
 c5 
 If

 I fs  I *fs
 c6 
 I fs


  3 S f I m  I f



 


* 

* I
 3 S f *m  
I f  

* 


* Im
   4 S fs I m  I fs   4 S fs *   .

I fs  


Thus, we have
( S f  S *f )2
( Sm  Sm* )2
dV
  c1 (   )
 c2 (   )
dt
Sm
Sf
 c3 
( S fs  S *fs )2
S fs
 f ( x1 , x2 , x3 , x4 , x5 , x6 ),
where f ( x1 , x2 , x3 , x4 , x5 , x6 )  (c1  c4 ) 1 a x1 x5  (c1  c4 ) 2 d x1 x6
 (c2  c5 ) 3 b x2 x4  (c3  c4 ) 4 c x3 x4  (c1 1 a  c5 3 d ) x5
 (c1 2 d  c6 4 c) x6  (c2 3 b  c3 4 c  c4 1 a  c4 2 d ) x4


1
1
 (c1 1 a  c1 2 d ) 1    c2 3 b 1  
x1 
x2 



 x x 
1
 c3 4 c 1    c4 1 a 1 1 5   c4 2 d
x3 
x4 


 x3 x4 
 x2 x4
 x1 x6 
  c5 3 b 1
1 
  c6  4 c 1
x4 
x6 
x5



Also,
Sf
S fs
If
I fs
Sm
I
 x1 , *  x2 , *  x3 , m*  x4 , *  x5 , *  x6 ,
*
Sm
Sf
S fs
Im
If
I fs
Sm* I *f  a, S *f I m*  b, S *fs I m*  c, Sm* I *fs  d .

.

(12)
To determine c1 , c2 , c3 , c4 , c5 , c6 , we set the coefficients of x1 x5 , x1 x6 , x2 x4 , x3 x4 , x5 , x6 , x4
equal to zero and solving these resulting equations, we have c1  c4 , c2  c5 and c3  c6 . Hence,
by using these results in (10) and taking c1  c4  1, we get
( S f  S *f )2
( Sm  Sm* )2
dV
  c1 (   )
 c2 (   )
dt
Sm
Sf
 c3 
( S fs  S *fs )2
S fs

1 1 x x x x 
 1 a  4    1 5  2 4 
x1 x2
x4
x5 


1 1 x x x x 
  2 d  4    1 6  3 4   0.
x1 x3
x4
x6 

dV
dV
 0 in the region  and the inequality
 0 holds only if
dt
dt
x1  x2  x3  x4  x5  x6 1, for which S m  S m* , S f  S *f , S fs  S *fs , I m  I m* , I f  I *f , I fs  I *fs .
Thus, it implies
Thus, by LaSalle’s invariance principle, the unique endemic equilibrium E1 of the model (1) is
globally asymptotically stable for R0  1.
5. Numerical Analysis
In this section, the system (1) is presented graphically when all the parameters are constant and
are the following values:
1  80; 2  60; 3  50; 1  0.00005; 2  0.0002; 3  0.0001; 4  0.0003; b1  0.107261;
b2  0.0924; b3  0.25;   0.0743; d  0.123;  0.03.
Fig. 2 Population versus time
Conclusion
In the present communication, a nonlinear mathematical model is developed to examine the
transmission of HIV/AIDS with the influence of role of female sex workers. Also, study the
effect of immigration and emigration of susceptible male, female and female sex workers. By
analyzing the model, we find that the disease free equilibrium is globally asymptotically stable
for the basic reproduction number is less than unity and the endemic equilibrium is globally
stable for the basic reproduction number is greater than unity.
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