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International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 638021, 8 pages
http://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

Academic Editor: Thomas Witelski

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 SIR 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)𝐵 the birth density in the population; (ii)𝐽 the density of the immature individuals; (iii)𝑀 the density of the mature individuals; (iv)𝑁 the density of the population; (v)𝐷 the density of the dead individuals; (vi)𝑑 the probability of mature individuals to die of HIV; (vii)𝑚 the probability of immature individuals becoming mature individuals; (viii)𝑛 the probability of mature individuals to die of other causes.

Then a simple model with compartments and a single population with stage structure reads:638021.xy.001(1.1)SIRFor describing the disease transmission, a dynamics between the compartments due to the disease has to be specified. A traditional 𝑆 model is introduced. Each member of the population is considered to belong to one of the three classes: susceptible individuals (denoted by 𝐼), infected individuals (denoted by 𝑅) and removed individuals (denoted by 𝑆). Each individual begins in the class 𝐼, only to move to the class 𝑅 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 𝐽 and are unable to be infected one again. The disease is fueled by supply of susceptible issued from the compartment 𝑁(𝑡). The size of the population is denoted by 𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡)+𝑅(𝑡)+𝐽(𝑡).(1.3) and can be expressed as the following sum:SIR The 𝑑𝑆𝑑𝑡=𝑟1𝑚(1𝜏)𝐽(𝑡)𝐹𝑖(𝐼,𝑡)𝑆+𝑟2𝑆,𝑑𝐼𝑑𝑡=𝐹𝑖(𝐼,𝑡)𝑆𝑟3(𝜎+𝛼)𝐼,𝑑𝑅𝑑𝑡=𝑟3𝜎𝐼𝜇𝑅,(1.4) model reads𝐹𝑖(𝐼,𝑡) where 𝐹𝑖(𝐼,𝑡)=Ω(𝑡)𝐼 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 Ω(𝑡) in which Ω(𝑡)=Ω0(1+Ω1sin(𝜔𝑡)) is the transmission rate; it is either constant, or a periodic modulation about a constant value, for example, 𝑚; 𝑟1 is the rate of immature individuals becoming mature individuals; 𝑟2 is the survival rate of the immature individuals; 𝑟3 is the survival rate of the mature individuals; 𝜎 is the survival rate of the infected mature individuals and 𝛼 is the rate of the survivors subjected to the antiretroviral treatment. 𝜏 is the rate of death due to the disease; 𝜇 is the fraction of infected immature from their mother; and 𝐶 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 2 of 𝐶, we prove that system (1.4) is controllable with three of its parameters. Whatever the initial conditions are, system (1.4) reaches the subset 𝐶 and remains in 𝐼. 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 𝑅 and 𝑆(𝑡)=𝑁(𝑡)𝐼(𝑡)𝑅(𝑡)𝐽(𝑡). According to (1.3) the susceptible compartment is expressed as 𝑑𝐼𝑑𝑡=𝐹𝑖[](𝐼,𝑡)𝑁(𝑡)𝐽(𝑡)𝑅(𝑡)𝐼(𝑡)𝑟3(𝜎+𝛼)𝐼,𝑑𝑅𝑑𝑡=𝑟3𝜎𝐼𝜇𝑅.(1.5), thus (1.4) is reduced to𝜇>0 Since 𝑡=𝜇𝑡, a new timescale 𝑑𝐼𝑑𝑡𝑡=Ω𝐼(𝑁𝐽𝑅𝐼)𝑟3(𝜎+𝛼)𝐼,𝑑𝑅𝑑𝑡=𝑟3𝜎𝐼𝑅.(1.6) is introduced. System (1.5) becomes:Δ=𝑁(𝑡)𝐽(𝑡)𝑅(𝑡) We assume that 𝛾=𝑟3𝜎 is constant. Defining 𝑑𝐼𝑑𝑡=Ω(𝑡)𝐼(Δ𝐼)𝛾𝐼𝛾𝛼𝜎𝐼,𝑑𝑅𝑑𝑡=𝛾𝐼𝑅.(1.7), and omitting the prime notations, system (1.6) becomes:Ω

Theorem 1.1. Assumes that 𝐶1(+;) is a (𝐼,𝑅)2+ function with bounded primitive. For every initial condition (𝐼(),𝑅())+2+, the solution 𝐾 of (1.7) belongs to 𝐾 where 2+. is a compact subset of 𝜃=𝛾(1+𝛼/𝜎)

Proof. Set 𝐼𝐼(𝑡)=0𝑒𝑡0(Ω(𝜏)Δ𝜃)𝑑𝜏1+𝑡0𝐼0𝑒𝑡0(Ω(𝜏)Δ𝜃)𝑑𝜏.𝑑𝑠(1.8), by integrating the first equation of (1.7) we have 𝑀 Let Ω be a bound from below of a primitive of 𝐼0𝐼(𝑡)<0𝑒𝑀Δ𝑡𝑒𝜃𝑡1+𝑡0𝐼0𝑒𝑡0(Ω(𝜏)Δ𝜃)𝑑𝜏=𝑑𝑠𝐼.(1.9), the we have 𝑑𝑅/𝑑𝑡=𝛾𝐼𝑅 From 𝑅(𝑡)=𝑅0𝑒𝑡+𝛾𝑒𝑡𝑡0𝑒𝑠𝐼(𝑠)𝑑𝑠𝑅0𝑒𝑡+𝛾𝑒𝑡𝑒𝑡1𝐼<𝑅0𝑒𝑡+𝛾𝐼=𝑅.(1.10) we deduce (𝐼,𝑅)
So the Poincaré-Bendixson's theorem [3] claims either the solution 𝑡 of system (1.7) tends to a critical point when the time 𝐶 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 (𝐼,𝑅) of the plane 0<𝑥1, for any given initial condition? For 𝐶 fixed, we define the convex domain 𝐶𝑇 of the plane and its associated truncated cylinder 𝑥𝐶=1,𝑥22+;𝑥1𝑥13;and4𝑥1𝑥2,𝐶𝑇=𝑡,𝑥1,𝑥23+;0𝑡𝑇;𝑥1𝑥13;and4𝑥1𝑥2.(2.1) by:𝐶𝑇

Definition 2.1 (contingent and exterior contingent cone). The contingent cone to 𝑥 at 𝑇𝐶𝑇(𝑥)𝑣3 is constituted by vectors lim0+𝑑inf𝐶𝑇𝑥+𝑣,𝐶𝑇=0,(2.2) verifying 𝑑𝐶𝑇 where 𝐶𝑇 denotes the distance to the subset 𝑇𝐶𝑇(𝑥). The exterior contingent cone 𝑣3 is constituted by vectors lim0+𝑑inf𝐶𝑇𝑥+𝑣,𝐶𝑇𝑑𝐶𝑇(𝑥)0.(2.3) verifying 𝑥

When a point 𝐶𝑇 belongs to the boundary of 3.4.1 the definition of exterior contingent cone is equivalent to the definition of the contingent cone. We have the following result [6, Theorem 𝐶𝑇 page 102].

Lemma 2.2. The exterior contingent cone to 𝑥 at point 𝑣3 is constituted by vectors 𝑥𝑃𝐶𝑇𝑥,𝑣0;(2.4) satisfying: (,) where 𝑃𝐶𝑇 denotes the Euclidean inner product, and 𝐶𝑇 stands for the orthogonal projection on 𝐹𝑡,𝑥1,𝑥2=1Ω(𝑡)𝑥1Δ𝑥1𝑥1𝛾𝛼1+𝜎𝛾𝑥1𝑥2,(2.5).

Before stating the result of controllability, we give some technicalities. Setting 𝑋{(𝑡,𝑥1,𝑥2),0<𝑡<𝑇 we have the following.

Lemma 2.3. Let 0<𝑥1; 0<𝑥2}𝐶𝑐𝑇; 𝑋𝑃𝐶𝑇𝑋 be fixed. Then 𝐶𝑇 is the outward normal to 0𝑠1 whenever it exists, and for 𝑋𝑃𝐶𝑇014𝑋=3𝑠.(2.6) is given by 𝐹(𝑋) Furthermore, a sufficient condition for the vector 𝑇𝐶𝑇 to belong to the exterior contingent cone 𝑥1Ω(𝑡)Δ𝑥1𝛼𝛾1+𝜎+10.(2.7) read as follows: 𝑇𝐶𝑇

Proof. From the definition of the exterior contingent cone (Figure 1) []𝑠0,1,Ω(𝑡)𝑥21+𝑥1𝛼Ω(𝑡)Δ𝛾1+𝜎4𝑠34𝑠3𝑥2.(2.8) we have: 𝑠 A sufficient condition independent of 𝑥2(3/4)𝑥1 for condition (2.8) to be satisfied is obtained when 0𝑥1 with Ω(𝑡)𝑥21+𝑥1𝛼Ω(𝑡)Δ𝛾1+𝜎𝑥1.(2.9) and read as follows: 0max0𝑡Ω(𝑡)=Ω

638021.fig.001
Figure 1: Exterior contingent cone.

Theorem 2.4. Let 𝛼, and let parameters 0<𝑥1<Δ, 𝜎, (𝐼0,𝑅0)2 be fixed. Whatever 𝑟3 are, choose 𝛾=𝑟3 in such a way that 𝜎30<𝛾4;ΩΔ𝑥1𝛼𝛾1+𝜎+10.(2.10) verifies: 0𝑇𝑟 Then, there exists 𝑡𝑇𝑟 such that for all time (𝐼(𝑡),𝑅(𝑡)), the solution 𝐶 of problem (1.7) belongs to the subset 𝑌=(𝑡,𝐼(𝑡),𝑅(𝑡)).

Proof. Set 𝑑𝑌(𝑡)𝑑𝑡=𝐹(𝑌(𝑡));0<𝑡,𝑌(0)=0,𝐼0,𝑅0.(2.11), then problem (1.7) is expressed as the following autonomous system: 𝐺(𝑡,𝐼) Define the function 𝛼𝐺(𝑡,𝐼)=Ω(𝑡)(Δ𝐼)𝛾1+𝜎+1.(2.12) by 𝐺 Function 𝐼 is a decreasing function with respect to 𝐺(𝑡,𝑥1)0 for all time. Thus if 𝐼>𝑥1, it will be negative for all 𝐹2(𝑡,𝐼,𝑅). Condition (2.10) implies that 𝐹3(𝑡,𝐼,𝑅) is negative and 0<𝑡 is positive for all 𝑥1<𝐼; 0𝑅; (𝐼(𝑡),𝑅(𝑡)). Theorem 1.1 asserts the existence of 𝑡 for all time 𝐶. A simple continuity argument implies that the subset 𝑇𝑟 defined in (2.1) is reached for a time (𝐼0,𝑅0) by the trajectory starting at the point 𝑇>𝑇𝑟. Fix 𝐹(𝑌)𝑇𝐶𝑇(𝑌), Lemma 2.2 claims that condition (2.10) is sufficient for 𝑌 when 𝐶𝑇 belongs to the boundary of (𝑇𝑟,𝐼(𝑡𝑟),𝑅(𝑇𝑟)). Nagumo's theorem applies for equation (2.11) with initial conditions (𝐼(𝑡),𝑅(𝑡))𝐶, and we get 𝑇𝑟𝑡 for 1 [7, Theorem 𝑟3, 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 𝜎 of the infected mature individuals recovered with the rate Δ=6066573.

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: 𝛾=0.25; 𝛼=0.01;; 𝑥1=1.5105𝐼. The following graphs represent the phase portrait of system (1.7). When the time elapses, the values of the function 𝑥 are along the 𝑅-axis and the values of the function 𝑦 are along the (𝐼0,𝑅0)=(1.59105,5.8104)-axis. There is no limit cycle, and the last point of the simulation is represented with the green point. The initial conditions are Ω=4.3108 (denoted by the red point). In Figure 2 we have considered the case (a) with 𝜎=0.5; (𝐼(𝑡),𝑅(𝑡)). The sufficient condition (2.10) is not satisfied, nevertheless, it can be checked that after a long time the computed solution 𝑥1 has a first component less than or equal to Ω=4.6108. In case (b), we have 𝜎=0.8; 𝐶𝑇. The sufficient condition (2.10) is not satisfied. Here the trajectory is outside the cone Ω(Δ𝑥1)𝛾(1+𝛼/𝜎)1.

fig2
Figure 2: (a) Ω(Δ𝑥1)𝛾(1+𝛼/𝜎)+1>0, (b) 𝐶𝑇.

The cone 𝑟3, roughly speaking, characterizes the treatment effort. The sufficient condition (2.10) is basically governed by two parameters: the transmission rate, and the survival rate 𝜎 of the infected mature individuals recovered at a rate 𝑟3. In the following examples, keeping the same values for parameters as in case (a) except for 𝑟3=0.85. For 𝐶𝑇, The sufficient condition (2.10) is not satisfied, and we have in Figure 3(a) the trajectory outside 𝑟3=2. For (0,0), the sufficient condition (2.10) is satisfied, and the trajectory is concentrated in a neighborhood of the disease-free equilibrium (3/4)𝑥1𝑥2 see Figure 3(b).

fig3
Figure 3: (a) 𝑟3=0.85 is not satisfied with (3/4)𝑥1𝑥2; (b) 𝑟3=2. is satisfied with 𝑟3

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 2% 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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