`International Journal of Mathematics and Mathematical SciencesVolume 2010, Article ID 638021, 8 pageshttp://dx.doi.org/10.1155/2010/638021`
Research Article

## Dynamics and Thresholds of a Simple Epidemiological Model: Example of HIV/AIDS in Mali

1Département de Mathématiques et D'Informatique, Faculté des Sciences et Techniques, B.P.E 3206, Bamako, Mali
2Université de Lyon, INSA, ICJ UMR CNRS 5208, 69100 Lyon, France

Received 23 March 2010; Revised 13 August 2010; Accepted 23 September 2010

Copyright © 2010 Ouaténi Diallo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The dynamics of many epidemiological models for infectious diseases that spread in the sexually active population presents a crucial period: the period of the influx or recruitment of susceptible. In this paper, we assume that the recruitment of susceptible is done among the juvenile group. We propose a dynamical system to modelize the disease spread, and we study the dynamical behavior of this system. Then, the controllability of the system is studied. We prove that the survival rate allows to control the dynamic of the system. Numerical simulations are given to illustrate the results.

#### 1. Introduction

In recent years several authors have described interesting dynamical behavior of epidemiological models in which the population can be portioned into two age structured classes: immature individuals and mature ones (see, e.g., [1, 2]). The HIV disease belongs to the class of diseases which spread essentially among sexually active individuals. Thus, it is meaningful to consider stage structure in epidemiological models. The population is initially divided into two compartments: those, who are mature individuals or adults and those who are in youthful age or immature individuals. All population groups are subject to the risk of dying from AIDS.

We denote by (i) the birth density in the population; (ii) the density of the immature individuals; (iii) the density of the mature individuals; (iv) the density of the population; (v) the density of the dead individuals; (vi) the probability of mature individuals to die of HIV; (vii) the probability of immature individuals becoming mature individuals; (viii) the probability of mature individuals to die of other causes.

Then a simple model with compartments and a single population with stage structure reads:(1.1)For describing the disease transmission, a dynamics between the compartments due to the disease has to be specified. A traditional model is introduced. Each member of the population is considered to belong to one of the three classes: susceptible individuals (denoted by ), infected individuals (denoted by ) and removed individuals (denoted by ). Each individual begins in the class , only to move to the class after coming into contact with an infected person. Infected individuals eventually recover from the disease due to a medical treatment and then move to the class and are unable to be infected one again. The disease is fueled by supply of susceptible issued from the compartment . The size of the population is denoted by and can be expressed as the following sum: The model reads where is the incidence function which may vary periodically because a part of the infected population represented by the truck drivers, for example, moves regularly. It is usual to take in which is the transmission rate; it is either constant, or a periodic modulation about a constant value, for example, ; is the rate of immature individuals becoming mature individuals; is the survival rate of the immature individuals; is the survival rate of the mature individuals; is the survival rate of the infected mature individuals and is the rate of the survivors subjected to the antiretroviral treatment. is the rate of death due to the disease; is the fraction of infected immature from their mother; and is the rate of death due to other causes.

The aim of this work is to provide simple conditions for the parameters of the SIR model (1.4) that makes possible to control the infected individuals. By using the notion of the exterior contingent cone to a convex subset of , we prove that system (1.4) is controllable with three of its parameters. Whatever the initial conditions are, system (1.4) reaches the subset and remains in . The paper is organized as follows: the introduction ends with an existence and uniqueness result. In Section 2 the controllability of system (1.4) is studied and several numerical results are presented in connection with available data concerning Mali.

The dynamic behavior of (1.4) is determined by the variation of and . According to (1.3) the susceptible compartment is expressed as , thus (1.4) is reduced to Since , a new timescale is introduced. System (1.5) becomes: We assume that is constant. Defining , and omitting the prime notations, system (1.6) becomes:

Theorem 1.1. Assumes that is a function with bounded primitive. For every initial condition , the solution of (1.7) belongs to where is a compact subset of

Proof. Set , by integrating the first equation of (1.7) we have Let be a bound from below of a primitive of , the we have From we deduce
So the Poincaré-Bendixson's theorem [3] claims either the solution of system (1.7) tends to a critical point when the time goes to infinity, or it is a periodic solution.
A complete bifurcation analysis is beyond the objectives of this paper. For a precise study of the orbits the reader is referred to [4] or [5], for example.

#### 2. Controllbility of the Model with Its Coefficients

The question we address in this section reads: does there exist parameters which allow system (1.7) to evolve toward a fixed region of the plane , for any given initial condition? For fixed, we define the convex domain of the plane and its associated truncated cylinder by:

Definition 2.1 (contingent and exterior contingent cone). The contingent cone to at is constituted by vectors verifying where denotes the distance to the subset . The exterior contingent cone is constituted by vectors verifying

When a point belongs to the boundary of the definition of exterior contingent cone is equivalent to the definition of the contingent cone. We have the following result [6, Theorem page 102].

Lemma 2.2. The exterior contingent cone to at point is constituted by vectors satisfying: where denotes the Euclidean inner product, and stands for the orthogonal projection on .

Before stating the result of controllability, we give some technicalities. Setting we have the following.

Lemma 2.3. Let ; ; be fixed. Then is the outward normal to whenever it exists, and for is given by Furthermore, a sufficient condition for the vector to belong to the exterior contingent cone read as follows:

Proof. From the definition of the exterior contingent cone (Figure 1) we have: A sufficient condition independent of for condition (2.8) to be satisfied is obtained when with and read as follows:

Figure 1: Exterior contingent cone.

Theorem 2.4. Let , and let parameters , , be fixed. Whatever are, choose in such a way that verifies: Then, there exists such that for all time , the solution of problem (1.7) belongs to the subset .

Proof. Set , then problem (1.7) is expressed as the following autonomous system: Define the function by Function is a decreasing function with respect to for all time. Thus if , it will be negative for all . Condition (2.10) implies that is negative and is positive for all ; ; . Theorem 1.1 asserts the existence of for all time . A simple continuity argument implies that the subset defined in (2.1) is reached for a time by the trajectory starting at the point . Fix , Lemma 2.2 claims that condition (2.10) is sufficient for when belongs to the boundary of . Nagumo's theorem applies for equation (2.11) with initial conditions , and we get for [7, Theorem , page 27].

As consequence of Theorem 2.4 the SIR models allow to improve the efficiency of medical policies. The sufficient condition (2.10) characterizes the treatment effort through the survival rate of the infected mature individuals recovered with the rate .

Let us end this section with numerical examples. The system (1.7) is discretized with a Runge-Kutta's method (RK4). By using available data from Mali in system (1.7) we have the following values for parameters: ; ; . The following graphs represent the phase portrait of system (1.7). When the time elapses, the values of the function are along the -axis and the values of the function are along the -axis. There is no limit cycle, and the last point of the simulation is represented with the green point. The initial conditions are (denoted by the red point). In Figure 2 we have considered the case (a) with ; . The sufficient condition (2.10) is not satisfied, nevertheless, it can be checked that after a long time the computed solution has a first component less than or equal to . In case (b), we have ; . The sufficient condition (2.10) is not satisfied. Here the trajectory is outside the cone .

Figure 2: (a) , (b) .

The cone , roughly speaking, characterizes the treatment effort. The sufficient condition (2.10) is basically governed by two parameters: the transmission rate, and the survival rate of the infected mature individuals recovered at a rate . In the following examples, keeping the same values for parameters as in case (a) except for . For , The sufficient condition (2.10) is not satisfied, and we have in Figure 3(a) the trajectory outside . For , the sufficient condition (2.10) is satisfied, and the trajectory is concentrated in a neighborhood of the disease-free equilibrium see Figure 3(b).

Figure 3: (a) is not satisfied with ; (b) is satisfied with

#### 3. Conclusion

In this paper, it is shown by using the exterior contingent cone and a viability theorem, simple convex subsets are reachable with a SIR model by adjusting some coefficients. Thus, it will be possible to predict with a certain accuracy the evolution level of the disease by adjusting one or another of parameters. In our example, it is important to see that, if the survival rate attains , the disease almost goes back at a level disease-free equilibrium. The controllability of the dynamical system has been established by using the exterior contingent cone technique. The mathematical model could be improved by introducing new compartments for describing, for example, the transmission of the disease between mothers and children. If the exterior contingent cone, for ordinary differential system of higher dimension, can be defined in the same way as before, it has to be calculable that which is an open question in general.

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