_{1}

^{*}

In this paper we propose a mathematical model to evaluate the impact of public health sensitization campaign on the spread of HIV-AIDS in Mali. We analyse rigorously this model to get insight into its dynamical features and to obtain associated epidemiological thresholds. If R
_{0} < 1, we show that the disease-free equilibrium of the model is globally asymptotically stable when the public health sensitization program is 100% effective. The impact of public health sensitization strategies is assessed numerically by simulating the model with a reasonable set of parameter values (mostly chosen from the literature) and initial demographic data from Mali.

AIDS is the most deadly disease caused by a human immunodeficiency virus (HIV). The virus destroys all the immune system and leaves individuals susceptible to any other infections. It multiplies inside lymphocytes and finally destroys them. When the lymphocytes are reduced to a certain numbers, the immune system stops functioning correctly. Therefore, the individual can catch any kind of disease that might kill him easily because of the failure of the immune system. However, there exist drugs that can slow down the evolution of the virus. HIV is usually transmitted in three different ways: sexual contacts, blood transfusion, and exchange between mother and child during pregnancy, childbirth and breastfeeding.

Many mathematical models are used to study the impact of preventive control strategies on the spread of HIV-AIDS in given populations (cf. [

The models developed in [

viduals who are in stage 1 of the infection

individuals who are in stage AIDS

stage 2 of the infection

time t is denoted by

Our model is given by the following system of ODEs with constant coefficients:

with

By adding of (2) to (9), we obtain:

Our mathematical model is an extension of the models developed in [

H1: that he mode of transmission of the virus is the horizontal transmission;

H2: that every individual is susceptible at high risk before his recruitment in the compartment

H3: that

To show that the model is mathematically and biologically possible, we begin by rewriting it in terms of proportions. So, we introduce the following scalings:

Consequently:

By using what precedes, the rates of infection (10) become:

If we introduce the following parameters:

In the new variables, (2)-(9) reduces to:

Parameters | Biological description |
---|---|

Recruitment rate, natural mortality rate. | |

Sensitization rates. | |

Transfer rates. | |

Reduction probability of the susceptible contamination | |

Transmission rates. |

We suppose that the initial conditions belong in

Now we can enounce the following result:

Theorem 1. For any initial condition in

Before giving the proof of this theorem, we give at first a technical result which we shall use after.

Lemma 1. Let a(t) and y(t) be n X n matrices of bounded measurable functions on

with

Proof. Indeed, this follows from the integrated form of the differential Equation (24),

We rewrite the system (15)-(22) in the form

Now we can give the proof of the theorem 1.

Proof. Step 1: Local existence of the solutions.

The local existence of the solutions ensues directly from the regularity of the function

Step 2: We show that

A.

By Lemma 1,

We next show that

Proceding by contradiction:

We suppose that

Then,

By Lemma 1,

Now we consider the Equations (16) and (20) in time

which is a contradiction, Consequently

B. The following inequalities hold:

Adding all the equations of (15), we obtain:

By integrating (40) between 0 and t, we have:

So if the initial condition verifies

then the relation

will be verified for all

This second stage shows that the solutions are limited for everything

There will be absence of desease in the population if the proportions

Theorem 2. The model of HIV-SIDA (2)-(9) or (15)-(22) possesses a unique desease free equilibruim in

and

Proof. Let be

Let be

where

By using the method of Van den Drissche and Watmough, we denote by F the rate of appearance of new infections in compartments of the infectious, and by Vs the rate of transfer of individuals in and out the compartments of the infectious by all other means. Then

The next-generation matrix is defined by:

where

Proposition 3. The basic reproduction ratio for HIV-SIDA model (15)-(22) is explicitly given by the formula (52) where

Theorem 4. The disease free equilibrium

We have the following theorem.

Theorem 5. For the system (15)-(22), if

Proof. We begin by rewriting the system (15)-(22) in the form:

where

Now, let us consider the following function:

If

We can also write

Algebraic manipulations give

or

If

Before closing this section, we verify numerically the theoretical results obtained in subsections 2, 2 and 2 for an initial condition

First case: the disease goes extinct in the population (see

Second case: the disease persists in the population (see

These parameters are obtained in the literature and are summarized in the

In this section all sensitization-related parameters and variables are fixed to zero in order to understand the dynamic behavior of the population without public health sensitization campaign.

So, we pose

Parameters | First case | Second case |
---|---|---|

0.0146; 0.014 | 0.0146; 0.014 | |

0.05; 0.07 | 0.005; 0.007 | |

0.02; 0.02 | 0.02; 0.02 | |

0.02; 0.02 | 0.02; 0.02 | |

0.01; 0.01 | 0.005; 0.005 | |

0.01 | 0.005 | |

0.015; 0.0075 | 0.028; 0.0075 |

where

with

The desease free equilibruim

By using the method of Van den Drissche and Watmough, we denote by F the rate of appearance of new infections in compartments of the infectious, and by Vs the rate of transfer of individuals in and out the compartments of the infectious by all other means. Then:

the next-generation matrix is defined by:

Proposition 6. The basic reproduction ratio for the sub-model (61)-(64) is given by the formula (66):

Theorem 7. The disease free equilibrium

Theorem 8. For the system (61)-(64), if

Proof. We begin by rewriting the system (61)-(64) in the form:

where

The disease free equilibrium

Now, let us consider the following function:

The function V is positive, and it nulle at the disease free equilibrium. The derivative of this Lyapunov function V along the trajectories of the ordinary differentiel system is:

We can also write

Algebraic manipulations give

or

If

It is found that an unique endemic equilibrium of (61)-(64) for

From (72), we have:

From (73), we have:

From (74) et (76), we have:

From (75), (76) and (77), we have:

(76), (77) and (78) in (71) give:

(79) dans (76) give:

(79) in (77) give

If

Before closing this section, we verify numerically the theoretical results obtained in this section for an initial condition

First case: the disease goes extinct in the population (see

Second case: the disease persists in the population (see

These parameters are obtained in the literature and are summarized in the

Before using the model (15)-(22) to evaluate the impact of public health sensitization in combatting HIV-AIDS spread in a population, it is instructive to evaluate the behaviour of the model under the worst case scenario (i.e., the case where no public health sensitization is provided in the population). By setting all sensitization related parameters to zero (i.e.,

We resume in

Parameters | First case | Second case |
---|---|---|

0.01625 | 0.01625 | |

0.014 | 0.014 | |

0.02 | 0.02 | |

0.015 | 0.015 | |

0.005 | 0.026 |

Parameters | Values |
---|---|

0.05, 0.7 | |

0.04, 0.02, 0.04, 0.02 | |

0.05, 0.05, 0.05 | |

0.0026 0.00718 |

Demographic data | Values |
---|---|

5812498, 0.014 | |

170000, 0.0146 | |

0.96104, 0.00971 | |

0.01872, 0.00468 | |

0.00374, 0.00094 | |

0.00094, 0.00023 |

We evaluate now the behavior of the model (15)-(22) by considering the impact of Public Health sensitization compaign on the spread of HIV-AIDS in Mali. Using the data in

MahamadouAlassane, (2015) Epidemiological Model and Public Health Sensitization in Mali. Applied Mathematics,06,1696-1711. doi: 10.4236/am.2015.610151