Epidemiological Model and Public Health Sensitization in Mali

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 R0 < 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.


Introduction
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.[1]- [11], etc.).Some of these models showed that a change in risky behaviour was necessary to prevent the spread of HIV even in the presence of a treatment (see for example [12]- [16]).Thus, it is instructive to study models that focus on non-pharmaceutical interventions, such as the use of public health sensitization campaign.
The models developed in [15] [17]- [19] study the impact of public health sensitization campaign on the spread of HIV-AIDS.In this paper we propose and study a mathematical model to estimate the impact of public health sensitization campaign on the spread of HIV-AIDS in Mali.We divide for it the population into two classes: "class with high-risk behavioral or class without public health sensitization" and "class with low risk behavioral or class with public health sensitization".Every class consists of susceptible individuals and infected individuals.The class of the individuals at high-risk behavioral is split into susceptibles individuals ( ) N t and can be expressed as the following sum: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .

N t S t S t I t I t I t I t I t I t
Our model is given by the following system of ODEs with constant coefficients: ) ( ) ( ) )  and .
By adding of (2) to (9), we obtain: where the parameters of the model are defined in Table 1.
Our mathematical model is an extension of the models developed in [15] [17]- [19].In our model, we suppose: 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 h S and that the rate of mortality induced by the HIV is neglected;

Existence of Solutions
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: 1.
Now we can enounce the following result: Theorem 1.For any initial condition in Ω , the system has a unique solution globally defined and which stays in Ω for any time 0 t ≥ .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
, , Step 2: We show that Ω is positively invariant.A.
Adding all the equations of (15), we obtain: By integrating (40) between 0 and t, we have: will be verified for all 0 t > .This second stage shows that the solutions are limited for everything 0 t ≥ .We can conclude that the solutions of the model exist globally in Ω .□

Global Stability of the Disease Free Equilibrium
We have the following theorem.Theorem 5.For the system (15)-( 22), if 0 1 R < then the disease free equilibrium is globally asymptotically stable.
M and 6 M are positives.Consequently the function V is posi- tive, 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 ( ) .

Numerical Simulations
Before closing this section, we verify numerically the theoretical results obtained in subsections 2, 2 and 2 for an initial condition 0.96104 For numerical simulations, the system (15) ( 22) is discretized with a Runge-Kutta's method (ODE45).We collect a set of values of biological parameters for the model corresponding to the data on the spread of the HIV-SIDA in two cases: First case: the disease goes extinct in the population (see Figure 2).Second case: the disease persists in the population (see Figure 3).These parameters are obtained in the literature and are summarized in the Table 2.

Model without Public Health Sensitization
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 15)-( 22) reduces to: For this sub-model by using the same reasoning in the theorem 1, we demonstrate that for any initial condition in h Ω , the system has a unique solution globally defined and stays in h Ω for any time 0 t ≥ where , , , 0,1 0 1 .
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: 0 0 0 0 the next-generation matrix is defined by: ( ) ( ) The basic reproduction ratio for the sub-model (61)-( 64) is given by the formula (66): ( ) ( ) Theorem 7. The disease free equilibrium dfe x of the sub-model (61)-( 64) is locally-asymptotically stable if

Global Stability of the Disease Free Equilibrium
Theorem 8.For the system (61)-(64), if 0 1 h R < then the disease free equilibrium dfe x is globally asymptoti- cally stable.
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: 1 .
We can also write .
then V  is negative along the trajectories.This ends the proof of the theorem.□

Existence and Uniqueness of an Endemic Equilibrium
It is found that an unique endemic equilibrium of (61)-(64) for 0 1 h R > .Thus, we solve the system: From (72), we have: From (73), we have: From ( 74) et (76), we have: .

Numerical Simulations
Before closing this section, we verify numerically the theoretical results obtained in this section for an initial condition 0.97075 For numerical simulations, the system (61)-( 64) is discretized with a Runge-Kutta's method (ODE45).We collect a set of values of biological parameters for the sub-model ( 61)-( 64) corresponding to the data on the spread of the HIV-SIDA in two cases: First case: the disease goes extinct in the population (see Figure 4).Second case: the disease persists in the population (see Figure 5).These parameters are obtained in the literature and are summarized in the Table 3.

Evaluation of Impact of Public Health Sensitization
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.,  4 and Table 5, simulations of the model ( 15)-( 22) show that in Mali the proportion of infected individuals would reach approximately 0.0686 (let 499550 cas) in 9 years from 2001 (Figure 6(b)).These projections of the model are compatible with the EDSM III projections over the year 2010 which predicted that by the year 2010 in Mali, if measures are not taken to control the epidemic of the HIV-AIDS, about 50000 people could be infected by the virus (see Figure 6).
We resume in Table 4 and Table 5, data of Mali concerning the spread of the HIV-AIDS.

hS
, indi- viduals who are in stage 1 of the infection ( ) 1 h I , individuals who are in stage 2 of the infection ( ) of the individuals at low-risk behavioral is split into susceptibles individuals ( ) f S , individuals who are in stage 1 of the infection ( ) total population (Figure 1) at time t is denoted by ( )

Figure 1 .
Figure 1.Behavioral representation of the HIV-AIDS model.

2 . 3 .
Local Stability of Disease Free EquilibriumBy 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.

Figure 2 .
Figure 2. Dynamics of the system (15)-(22) in case where the disease goes extinct in the population.With the parameters of the Table 2 (first case), we have R 0 = 0.5826.Figure 2(a) shows the evolution of susceptibles individuals, whereas Figures 2(b)-(d)show the evolution of infected individuals.The system converges towards the desease free equilibruim (0.226, 0, 0, 0, 0.774, 0, 0, 0).The simulation was realized with the MATLAB logiciel.
equilibrium dfe x for the system (67)-(70) corresponds to the point ( ) 0, 0, 0, 0 .Now, let us consider the following function: = ) and using the data in Table

Figure 4 .
Figure 4. Dynamics of the system (61)-(64) in case where the disease goes extinct in the population.With the parameters of the Table 3 (first case), we have R h0 = 0.3077.Figure 4(a) shows the evolution of susceptibles individuals, whereas Figures 4(b)-(d)show the evolution of infected individuals.The system converges towards the desease free equilibruim (1, 0, 0, 0).The simulation was realized with the MATLAB logiciel.

Figure 5 .
Figure 5. Dynamics of the system (61)-(64) in case where the disease persists in the populationn.With the parameters of the Table3(second case), we have R h0 = 1.6.Figure5(a)shows the evolution of susceptibles individuals, whereas Figures5(b)-(d)show the evolution of infected individuals.The system converges towards the endemic equilibrium (0.6298, 0.038, 0.0195, 0.0111).The simulation was realized with the MATLAB logiciel.

Figure 5 (
Figure 5. Dynamics of the system (61)-(64) in case where the disease persists in the populationn.With the parameters of the Table3(second case), we have R h0 = 1.6.Figure5(a)shows the evolution of susceptibles individuals, whereas Figures5(b)-(d)show the evolution of infected individuals.The system converges towards the endemic equilibrium (0.6298, 0.038, 0.0195, 0.0111).The simulation was realized with the MATLAB logiciel.

Figure 6 .
Figure 6.Dynamics of the system (15)-(22) in the nose of the cases, R h0 = 1.6.Figure 6(a) shows the evolution of susceptibles individuals, whereas Figure 6(b) shows the evolution of infected individuals.We use the parameters of the Table4.The simulation was realized with the MATLAB logiciel.

Figure 6 (
Figure 6.Dynamics of the system (15)-(22) in the nose of the cases, R h0 = 1.6.Figure 6(a) shows the evolution of susceptibles individuals, whereas Figure 6(b) shows the evolution of infected individuals.We use the parameters of the Table4.The simulation was realized with the MATLAB logiciel.

Figure 7 .
Figure 7. Dynamics of the system (15) in case the population of Mali is submitted to Public sensitization compaign on the spread of HIV-AIDS, R 0 = 0.5109.Figure 7(a) shows the evolution of susceptibles individuals, whereas Figures 7(b)-(d) show the evolution of infected individuals.We use the parameters of theTable 4. The simulation was realized with the MATLAB logiciel.

Figure 7 (
Figure 7. Dynamics of the system (15) in case the population of Mali is submitted to Public sensitization compaign on the spread of HIV-AIDS, R 0 = 0.5109.Figure 7(a) shows the evolution of susceptibles individuals, whereas Figures 7(b)-(d) show the evolution of infected individuals.We use the parameters of theTable 4. The simulation was realized with the MATLAB logiciel.

1: Local existence of the solutions.
The local existence of the solutions ensues directly from the regularity of the function

Table 5 .
Mali demographic data of 2001 used as initial conditions.

Table 4 .
The simulation was realized with the MATLAB logiciel.