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In order to find out the effect of human (sexual) behavior change and immigration in spreading the HIV/AIDS, a deterministic model of HIV/AIDS with infective immigration is formulated. First, basic properties of the model, including non-negativity and boundedness of the solutions, existence of the endemic equilibrium and the basic reproduction number, R
_{0 }are analyzed. The geometrical approach is used to obtain the global asymptotic stability of endemic equilibrium. Then the basic model is extended to include several control efforts aimed at reducing infection and changing behavior. Pontryagin’s maximum principle is used to derive the optimality system and solve the system numerically. Our numerical findings are illustrated through simulations using MATLAB, which shows reliability of our model from the practical point of view.

Mathematical models used extensively to study the dynamics of epidemics both from the cellular level to the population level by many researchers [

To cope with these problems, Megan Coffee et al. [

The optimal control theory has been applied to quite a few HIV models (see [

The paper organized as follows: In section 2, we have developed our model and the non-negativity and the boundedness of the solutions are shown as a basic property of the system. Also, we have discussed existence of endemic equilibrium. We derive basic reproduction number

We divided the sexually active population

with initial conditions

There are a brief description of the model parameters:

Q_{0}: Total number of newly recruited individuals by birth (who come of age) and by immigration;

ε_{1}: The proportion of infected individuals in the recruited population;

ε_{2}: The proportion of susceptible individuals who changed their sexual habits in the recruited population;

δ: The proportion of unaware infected individuals and

β_{1}: The horizontal transmission rate for contact with the

β_{2}: The horizontal transmission rate for contact with the

d: The natural death rate of population;

θ: Proportion of susceptible individuals who changed their sexual habits;

κ_{1}: Proportion of unaware infected individuals who are screened;

κ_{2}: Progression rate of unaware infected individuals to the full-blown AIDS group;

κ_{3}: Progression rate of aware infected individuals to the full-blown AIDS group;

μ_{1}: Proportion of the

μ_{2}: Proportion of the

σ_{1}: Disease-induced death rate for full-blown AIDS individuals;

σ_{2}: Disease-induced death rate for the individuals who receive the treatment.

The model is formulated based on following assumptions:

・ Since our purpose in this model is to see what effect the human behavior (including movement, sexual habits) can play in the dynamics of HIV/AIDS disease, we avoid to consider detailed clinical stages of HIV/AIDS infection, instead we classed the population in two ways, uninfected and infected group. Uninfected group divided into two different compartments according to their behavior towards safe sex. Infected individuals divided four different compartments according to whether the infected individual aware of his/her HIV infection status, whether the infected individual received treatment and whether the infected individual has developed the last stage of the disease, the full blown AIDS.

・ Susceptible individuals are assumed to get infected by sexual contact with both aware and unaware infected individuals with different transmission rates. The assumption that aware infected individuals also take part in the transmission process is based on the fact that some aware infected individuals may practise low-efficiency safe sex measures (and a few of them may transmit the disease intentionally), and susceptible individuals may not be aware of the infected situation of his/her partner, which make them more vulnerable to the disease. So the new generated infected individuals by aware infective individuals are assumed to be not aware of his/her infection at first and go to the unaware infected individuals class.

・ The simplest conceptual framework based on homogeneous behavior gives us clear insights into how community based chemotherapy can influence epidemiological pattern and transmission success. Here the mixing of susceptibles with infectives is considered to be homogeneous and accordingly the incidence rate is assumed to be bilinear [

・ All new born are susceptible, i.e., in our model vertical transmission do not account for.

・ We assumed that individuals in the treatment class not only to receive the ART therapy, but also to be served with knowledge about the HIV/AIDS disease so that they were persuaded to avoid unsafe sexual behaviors. Full- blown AIDS individuals are assumed too ill to sexually active, So the suscep- tibles do not get infected through sexual contacts with individuals from these two groups.

・ Inclusion of compartment

The model system (1) describes human population and therefore it is necessary to prove that all the variables

is defined based on biological considerations and positively invariant with respect to the model system (1).

Hence the theorem

Theorem 2.1 Every solution of the system (1) with initial conditions (2) exists in the interval

Proof Since the right hand side of system (1) is completely continuous and locally Lipschitzian on C (space of continuous functions), the solution (

If

which is a contradiction meaning that

which implies that

This completes the proof.

Since the variables

with initial conditions

The system (3) does not exhibit a disease-free equilibrium due to direct inflow of population at a constant rate. However, there exists only one non-negative equilibrium point of the model (3), i.e., endemic equilibrium

By solving Equations (5)-(7), we get

where

and

Obviously

Theorem 2.2 The system (3) has a endemic equilibrium

Especially when

In this section we shall discuss the global stability of the endemic equilibrium

Theorem 3.1 The endemic equilibrium

globally asymptotically stable, if

Before start our proof, we first recall the following lemma by Li and Muldowney [

Lemma If the system

where

1) has a unique equilibrium

2) there exists a compact absorbing set

then the equilibrium

Here

the matrix

Proof of Theorem 3.1 The system is uniformly persistent in

The Jacobian matrix of system (3) is given by

where

The second additive compound matrix of

Consider the function

then

Therefore

and

Therefore,

where

Let

Let

where

where

Therefore,

So, we have

where

Furthermore, we obtain

then

Therefore

Then based on Theorem 3.5 of [

In this section we give following optimal control problem.

satisfying initial conditions given in (2). We consider following optimal control parameters

1)

2)

3)

4)

The objective functional [

the weight constants

the admissible control set is given as

Theorem 4.1 Given the objective function

1) The class of all initial conditions with the corresponding control functions in

2) The admissible control set

3) Each RHS of the system (14) is continuous and is bounded above by sum of bounded control and state and can be written as a linear function of the control variables with coefficients dependent on time and state variables;

4) The integrand

Proof 1) We refer to Theorem 3.1 proposed by Picard-Lindelof in [

2) The control set

3) We observe that the integrand

4)

where

The above establishes a bound on

Thus, we have a unique solution of the optimality system for small time intervals due to the opposite time orientations of the state equations and the adjoint equations [

Using Pontryagin’s Maximal principle [

where Hamiltonian is defined as

Hence solving for

We can now impose the bounds

or

In this section, first we present some numerical results of the system (1), when

Variable | ||||||
---|---|---|---|---|---|---|

Value | 15,932,420 | 9124 | 27,373 | 18,751 | 8058 | 358,751 |

Parameter | Value | Parameter | Value |
---|---|---|---|

20,000 population/year | 0.031/year | ||

0.0016/year | 0.032/year | ||

0.06/year | 0.07/year | ||

0.85/year | 0.05/year | ||

0.0000000000288/population/year | 0.45/year | ||

0.000000000000284/population/year | 0.63/year | ||

0.0024/year | 0.16/year | ||

0.001/year |

and _{0}, d, μ_{1},

Now we solved the optimality system numerically by using fourth-order iterative Runge-Kutta scheme and presented some graphical results which mostly are comparative figures of susceptibles

In Figures 2(a)-(c), we present some graphics when we apply all combination of control measures to the system. By comparison its easy to see that we have some positive results immediately after the time break. The decrease in number of susceptible

to practise safe sex measures and efficient medical testing is far more cost- efficient. From

In Figures 3(a)-(c), we consider implying two control measures

In this paper, we formulated a deterministic model for controlling HIV/AIDS disease. we proved that our system only has one endemic equilibrium and it’s

globally asymptotically stable if threshold-like conditions satisfied. And then we presented an optimal control problem. Our aim is to investigate combined role of the human behavior change and medical screening in the transmission of HIV/AIDS. We proved the existence and uniqueness of the optimal control and characterized the controls using Pontryagon’s Maximal Principal. In the end we solved the optimality system numerically, and results once again shows us that the optimal way of preventing further prevalence of HIV/AIDS is practicing safe sexual behaviors and conducting efficient, affordable testing as soon as possible for all people. The initial conditions we used in our simulations come from the statistical data, we have fixed some parameters and introduced some of the parameter values from similar works in this field to show our analytical findings.

This work was supported by the National Natural Science Foundation of China (Grants Nos. 11261056, 11271312).

Mastahun, M. and Abdurahman, X. (2017) Optimal Control of an HIV/AIDS Epidemic Model with Infective Immigration and Behavioral Change. Applied Mathematics, 8, 87-105. http://dx.doi.org/10.4236/am.2017.81008