Original article Biomath 3 (2014), 1407211, 1–8 B f Volume ░, Number ░, 20░░ BIOMATH ISSN 1314-684X Editor–in–Chief: Roumen Anguelov B f BIOMATH h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / Biomath Forum Global Stability of an Epidemic Model with two Infected Stages and Mass-Action Incidence Mamadou Lamine Diouf1,2, Abderrahman Iggidr1, Mamadou Sy2 1 Inria, Université de Lorraine, CNRS. Institut Elie Cartan de Lorraine, UMR 7502. ISGMP Bat. A, Ile du Saulcy, 57045 Metz Cedex 01, France. 2 UMI-IRD-209 UMMISCO, and LANI Université Gaston Berger, Saint-Louis, Sénégal. e-mail: dioufabu@yahoo.fr, Abderrahman.Iggidr@inria.fr, mamadou.sy@ugb.edu.sn Received: 30 October 2013, accepted: 21 July 2014, published: 31 July 2014 Abstract—The goal of this paper is the estab- lishment of the global asymptotic stability of the model SI with two classes of infected stages and with varying total population size. The incidence used is the mass-action incidence given by (β1I1 + β2I2) S N . Existence and uniqueness of the endemic equilib- rium is established when the basic reproduction number is greater than one. A Lyapunov function is used to prove the stability of the disease free equilib- rium, and the Poincarré-Bendixson theorem allows to prove the stability of the endemic equilibrium when it exists. Keywords-Epidemic model, Global stability, Mass- action incidence I. INTRODUCTION Mathematical analysis became a major tool in the study of the evolution of epidemics. Indeed, more and more models were developed for the study of some epidemics. In order to model an epidemic disease, the population is divided into various classes. In some cases the population is divided into two senior classes: the class of the susceptible individuals, denoted by S, and the class of the infected individuals, denoted by I. Sometimes, the class of the infected can be split into several classes which allow to highlight the state of the disease. In our case, the infected are divided into two categories, denoted I1 and I2, with I1 the first stage of the disease and I2 the worsened case. If β1 and β2 are the per capita transmission rate of the infection in respectively the compart- ments I1 and I2, there are β1I1 + β2I2 infective contacts. If any contact with a susceptible gives a new infected, then there is (β1I1 + β2I2)P(S) new infected, where P(S) is the probability for an infected to meet a susceptible. The quantity (β1I1 + β2I2)P(S) is known in the literature as Citation: Mamadou Diouf, Abderrahman Iggidr, Mamadou Sy, Global Stability of an Epidemic Model with two Infected Stages and Mass-Action Incidence, Biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 Page 1 of 8 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.07.211 M Diouf et al., Global Stability of an Epidemic Model with two Infected Stages... the mass action incidence rate. One can notice that most of the classical models of disease use a bilinear mass action incidence (β1I1 + β2I2)S. For example, some of the most famous: the models of Kermack-Mckendrick (1927) and that of Lotka- Voltera (1926). The goal of our study is to analyze the global stability of the SI1I2 model. The system con- sidered can represent, for instance, the modeling of the HIV. For this model we suppose that an infected can have S N contact of susceptible, then P(S) is given by S N . Also the incidence is given by (β1I1 + β2I2) S N , where N represents the total population size: N = S + I1 + I2. The stability study of systems using this form of incidence is a very interesting subject to which some authors have already devoted some works. The work of C. Simon and J. Jacquez in [18] can be cited. Indeed, these authors addressed the problem for n classes of infected, using some elegant geometrical arguments, but they use a constant recruitment and also they suppose that transition rate from a class of infected to the next class and the rate of disease-induced death are equal. However, in our study, the recruitment is variable and the transition rate (denoted γ) from the first stage of infection I1 to the second stage I2 is different from the rate of disease-induced death (denoted d). This makes that for our system the explicit determination of the endemic equilibrium is very difficult if not impossible. So, the stability around possible endemic equilibrium is also more difficult to check than in the case of a constant recruitment. We can also cite more recent works. Particularly, the work of Melese and Gumel in [17], where for the proof of the endemic equilibrium stability, authors make a very strong assumption, which is very difficult to verify. We cite also and specially the work of M. Li, J. Graef, L. Wang and J. Karsai in [15], which deals with a similar system, but the authors used one contact rate. In the works made by C. C. McCluskey (2003) [16] and J. M. Hyman and J. Li (2005) [8], similar models have been considered, but the authors of [8] did not address the question of the global stability of the endemic equilibrium while in [16] the global stability of the endemic equilibrium was proved under the assumption that γ = d (i.e., the transition rate from I1 I2 is equal to the rate of disease- induced death) and β1 = β2. Besides, we mention the work of H. Guo and M. Y. Li (2006), where authors established the stability of the disease free equilibrium, but for the endemic equilibrium, they used bilinear incidence. We finish by mentioning the paper [10], where the authors considered similar systems but they used bilinear incidence. The paper is organized as follows. In Section II, we give the differential system governing the time evolution of the number of individuals in different classes is given, we derive the system governing the dynamics of the proportions and we compute the basic reproduction number R0. In Section III, we prove the existence and uniqueness of the endemic equilibrium when R0 is greater than one. The global asymptotic stability of the disease free equilibrium is studied in Section IV by using two Lyapunov functions. The local stability of the disease free equilibrium is given in Section V. We prove in Section VI that the system governing the proportions has no periodic orbit and that the endemic equilibrium is globally asymptotically stable. For the stability of the endemic equilibrium, the Poincaré-Bendixson theory is used. II. THE MODEL The SI models are well known in the dynamic of population. In this section, we present the SI model used in this paper. The population of size N is divided into subclasses of individuals who are susceptible, infected into the first stage of the disease and infected into the second stage, with sizes denoted by S, I1 and I2. The model we consider is given by the system   Ṡ = bN − (β1I1 + β2I2) S N −µS, İ1 = (β1I1 + β2I2) S N − (µ + γ)I1, İ2 = γI1 − (µ + d)I2. (1) Where N = S + I1 + I2 is the total population size; b and µ represent the per capita birth rate and the per capita natural death rate of the population, respectively. β1 and β2 are respectively the per capita transmission rate of the compartments I1 and I2. γ denotes the per capita rate of transfer of infected individuals from the infected stage 1 to stage 2, and d is the disease induced death rate. The total population size N satisfies the equation: Ṅ = (b−µ)N −dI2. Biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 Page 2 of 8 http://dx.doi.org/10.11145/j.biomath.2014.07.211 M Diouf et al., Global Stability of an Epidemic Model with two Infected Stages... The proportions s = S N , i1 = I1 N and i2 = I2 N satisfy the following differential system:  ṡ = b− bs− (β1i1 + β2i2)s + dsi2, i̇1 = (β1i1 + β2i2)s− (b + γ)i1 + di1i2, i̇2 = γi1 − (b + d)i2 + di22. (2) We determine the basic reproduction number, which represents the number of secondary cases produced by one infective host in an entirely susceptible population. We denote by Fj(s,i1, i2) the rate of appearance of new infections in compartment j, and by Vj(s,i1, i2) the rate of transfer of individuals in and out the com- partment j by all other means. The matrices F and V are given by: F =   0(β1i1 + β2i2)s 0   and V =   b− bs− (β1i1 + β2i2)s + dsi2−(b + γ)i1 + di1i2 γi1 − (b + d)i2 + di22   . The Jacobian matrices at the disease free equilibrium (1, 0, 0) are: DF =   0 0 00 β1 β2 0 0 0   and DV =   −b −β1 −β2 + d0 −(b + γ) 0 0 γ −(b + d)   . Let: F = ( β1 β2 0 0 ) and V = ( −(b + γ) 0 γ −(b + d) ) . It is well known [3] that the basic reproduction number is the spectral radius of the next generation matrix for the model, namely −FV −1. The basic reproduction number of system (2) is then R0 = β1 b + γ + β2γ (b + γ)(b + d) . III. THE EQUILIBRIUM POINTS The disease free equilibrium is given by DFE=(1,0,0). In the following, we show the existence and uniqueness of the endemic equilibrium for the system (2) assuming that b ≥ d. Recall that b and d represent the birth and the disease induced rate, respectively. Proposition III.1. If R0 > 1, the endemic equilibrium exists and is unique. Proof: At the equilibrium, the third equation of (2) gives: i∗1 = b + d γ i∗2 − d γ i∗22 . (3) Replacing i∗1 by its expression in the second equation of (2), we have after simplification by γi∗2: β1(b + d) −β1di∗2 + β2γ)s ∗ − (b + γ)(b + d) +d(b + d)i∗2 + d(b + d)i ∗ 2 −d 2 i∗22 = 0. (4) Also, in (4) we replace s∗ by its expression given by: s∗ = 1− i∗1 − i∗2 = 1− b + d γ i∗2 − d γ i∗22 − i∗2, then i∗2 is solution of the polynomial: P(i∗2) = a3(i ∗ 2) 3 + a2 i ∗2 2 + a1i ∗ 2 + a0 = 0, where a3 = −β1 d2 γ , a2 = 2β1d b + d γ + β1d + β2d−d2, a1 = −β1 (b+d)2 γ −β1(b + d) −β1d−β2(b + d) −β2γ + d(b + γ) + d(b + d), and = −R0 (b + d)(b + γ) ( 1 + b + d γ ) 4 −β1d + d(2b + d + γ) a0 = β1(b + d) + β2γ − (b + d)(b + γ) = (b + d)(b + γ) (R0 − 1). Using the fact that R0 > 1, it is easy to show that: a3 < 0, a2 > 0, a1 < 0, and a0 > 0. We have P(i∗2) = 0 ⇔ Q(i∗2) = R0, where Q is the polynomial given by Q(i∗2) = − a3 k (i∗2) 3 − a2 k i∗22 − a1 k i∗2 + 1, and k = (b + d)(b + γ). We have: Q(0) = 1 Biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 Page 3 of 8 http://dx.doi.org/10.11145/j.biomath.2014.07.211 M Diouf et al., Global Stability of an Epidemic Model with two Infected Stages... Q(1) = β1(b 2 + bγ + dγ) + γ(b(b−d + γ) + β2(b + γ)) k . Also Q(1) −R0 = b β1b + β2γ + bγ + γ 2 −dγ kγ , which is positive if and only if β1b + β2γ + bγ + γ 2 > dγ. (5) The relation (5) is satisfied thanks to the assumption b ≥ d. Thus, 1 = Q(0) < R0 < Q(1). Let us localize exactly the domain of i∗2. We have i∗1 + i ∗ 2 < 1, (6) and since, by relation (3), i∗1 = b + d γ i∗2 − d γ i∗22 , we deduce that i∗2 must verify the following inequality: R(i∗2) = −di ∗2 2 + (b + d + γ)i ∗ 2 −γ < 0. The discriminant of the polynomial R is ∆R = (b + d + γ)2 − 4dγ = b2 + 2b(d + γ) + (d − γ)2 > 0. The roots of R are r1 = (b + d + γ − √ ∆R)/2d and r2 = (b+d+γ+ √ ∆R)/2d. We have: r1 < γ/2d < r2, and with the assumption b ≥ d we have r2 > 1. i∗2 must satisfy o < i∗2 < min{r1, 1} ≤ min{γ/2d, 1}, that is i∗2 must belong to the interval I = (0, min{r1, 1}) ⊂ (0, min{γ/2d, 1}). On the other hand, we have Q(r1) −R0 = (1/(2k))[b(b + d + γ + √ ∆R)] > 0. Since Q(0) = 1 < R0, Q(r1) > R0, and Q(1) > R0, the graph of Q intersects the horizontal line y = R0 at least one time in I. Now let us show that there is exactly one intersection in I. The derivative of Q is: Q′(i∗2) = −(1/k)(3a3 i ∗2 2 + 2a2i ∗ 2 + a1). Note that by Descartes rules of signs there is no negative root. On the other hand, the discriminant of Q′ is ∆ = a22 − 3a3a1, we then have two cases: - If ∆ ≤ 0, Q′ is positive on R. - If ∆ > 0, we have two roots x1 and x2, and x1 + x2 = −(2a2/3a3). However: −2a2 3a3 = 4 3 b + d d + 2 3 { γ d + β2γ β1d − γ β1 } = 2 3 b+d d + 2 3 {b+d d + γ d + β2γ β1d − γ β1 } = 2 3 b + d d + 2 3β1d {β1(b + d) + β2γ +β1γ −dγ} = 2 3 b+d d + 2 3β1d {(b + d)(b + γ)R0 +β1γ −dγ}. Thus −2a2/3a3 = 2 3 b + d d + 2 3β1d {b(b + γ)R0 +bdR0 + β1γ + dγ(R0 − 1)}. We know that b(b + γ)R0 = β1b + β2bγ b + d . Since b ≥ d, we have −2a2/3a3 > 2, thus there is at least one root of Q′ larger than one. All these observations show that the graph of Q intersects the line y = R0 only once. i∗1 is deduced by i∗1 = b + d γ i∗2− d γ i∗22 , and s ∗ = 1− b + d + γ γ i∗2+ d γ i∗22 . Then, the endemic equilibrium exists and is unique. IV. GLOBAL STABILITY OF THE DFE Theorem IV.1. If R0 < 1, the DFE is globally asymptotically stable. Proof: To prove Theorem IV.1, we distinguish two cases, the first case corresponds to β2 ≥ d and the second is β2 < d. In both cases, we use Lyapunov functions. Case 1: β2 ≥ d.: We consider the following Lyapunov function: V = i1 + β2 b + d i2. The derivative of V is: V̇ = (β1i1 + β2i2)s− (b + γ)i1 + di1i2 + β2γ b + d i1 −β2i2 + β2 d b + d i22. Since β1i1s ≤ β1i1, we have V̇ ≤ β1i1 + β2i2s− (b + γ)i1 + di1i2 + β2γ b + d i1 −β2i2 + β2 d b + d i22 ≤ (b + γ)[ β1 b + γ + β2γ (b + γ)(b + d) − 1]i1 +β2i2(s− 1) + di1i2 + β2 d b + d i22, ≤ (b + γ)(R0 − 1)i1 + β2i2(s− 1) + di1i2 +β2 d b + d i22. We know that β2i2(s− 1) = −β2i2(i1 + i2), then V̇ ≤ (b + γ)(R0 − 1)i1 + (d−β2)i1i2 +( d b + d − 1)β2i22. Biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 Page 4 of 8 http://dx.doi.org/10.11145/j.biomath.2014.07.211 M Diouf et al., Global Stability of an Epidemic Model with two Infected Stages... Thus V̇ ≤ (b + γ)(R0 − 1)i1 + (d−β2)i1i2 − β2b b + d β2i 2 2 ≤ 0. It follows that V̇ is negative definite when R0 < 1. When R0 = 1, the time derivative of V V̇ is only nonpositive but in this case LaSalle invariance principle allows to prove the global asymptotic stability of the DFE. Case 2: β2 < d: We consider the following Lyapunov function defined on {0 < s ≤ 1, 0 ≤ i1 ≤ 1, 0 ≤ i2 ≤ 1}: V = s− ln s + i1 + ( b + γ γ − β1 γ )i2. We obtain V̇ = ṡ(1 − 1 s ) + i̇1 + ( b + γ γ − β1 γ )i̇2 = (b− bs)(1 − 1 s ) − (β1i1 + β2i2)s + (β1i1 +β2i2) + dsi2 −di2 + (β1i1 + β2i2)s −(b + γ)i1 + di1i2 + (b + γ)i1 − (b + γ)(b + d) γ i2 + d(b + γ) γ i22 −β1i1 +β1 b + d γ i2 −β1 d γ i22, we get : V̇ = − b s (1 −s)2 + β2i2 + di2(s + i1 − 1) − (b + γ)(b + d) γ i2 + bd γ i22 + di 2 2 +β1 b + d γ i2 −β1 d γ i22. We have the followings equalities: b s (1−s)2 = b s (i1 +i2) 2 and di2(s+i1−1) = −di22. Then V̇ becomes: V̇ = − b s (i1 + i2) 2 + β2i2 − (b + γ)(b + d) γ i2 + bd γ i22 + β1 b + d γ i2 −β1 d γ i22 = − b s (i1 + i2) 2 + (b + γ)(b + d) γ (R0 − 1)i2 + bd γ i22 −β1 d γ i22 = − b s (i1 + i2) 2 − (b + γ)(b + d) γ (1 −R0)i2 −β1 d γ i22 + bd γ i22. As 1/s ≥ 1 and i2 ≥ i22, we have V̇ ≤ −b(i1 + i2)2 − (b + γ)(b + d) γ (1 −R0)i22 −β1 d γ i22 + bd γ i22 = −bi21 − 2bi1i2 − bi 2 2 − (b + γ)(b + d) γ .(1 −R0)i22 −β1 d γ i22 + bd γ i22 = −bi21 − 2bi1i2 − i22 γ ( bγ + (b + γ)(b + d) .(1 −R0) + β1d− bd ) . Denote by D = bγ + (b+γ)(b+d)(1−R0) +β1d−bd, then V̇ ≤−bi21 − 2bi1 i2 −D i22 γ . Therefore, V̇ ≤ 0 if D ≥ 0. If bd < bγ + β1d holds then D ≥ 0. If not, we rewrite D in the following form: D = bγ + (b + γ)(b + d) −β1(b + d) −β2γ +β1d− bd = bγ + b2 + bd + bγ + dγ −β1(b + d) −β2γ + β1d− bd = b2 + 2bγ + dγ −β1b−β2γ = b(b−β1) + γ(2b + d−β2). The inequality bd ≥ bγ + β1d gives b > β1, and with the assumption β2 < d we get again D ≥ 0. We conclude that V̇ ≤ 0 if the assumption β2 < d holds. Once again LaSalle invariance principle allows to conclude. Conclusion: in both cases ( β2 ≥ d and β2 < d), we have proved that the disease free equilibrium is globally asymptotically stable. Biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 Page 5 of 8 http://dx.doi.org/10.11145/j.biomath.2014.07.211 M Diouf et al., Global Stability of an Epidemic Model with two Infected Stages... V. LOCAL STABILITY OF THE ENDEMIC EQUILIBRIUM With the assumption b ≥ d we have the following result: Theorem V.1. The endemic equilibrium is asymptoti- cally stable when it exists, i.e., when R0 > 1. Proof: Since s + i1 + i2 = 1, we can eliminate s in System (2). Therefore, we get the following system:   i̇1 = (β1i1 + β2i2)(1 − i1 − i2) − (b + γ)i1 +di1i2, i̇2 = γi1 − (b + d)i2 + di22. (7) The Jacobian of system (7) at the endemic equilibrium (EE = (i∗1, i ∗ 2)) is: J(EE) = ( β1 − 2β1i∗1 −β1i∗2 −β2i∗2 − (b + γ) + di∗2 γ β2 − 2β2i∗2 −β2i∗1 −β1i∗1 + di∗1 −b−d + 2di∗2 ) At the endemic equilibrium we have: β1 − 2β1i∗1 −β1i ∗ 2 −β2i ∗ 2 − (b + γ) + di ∗ 2 = −β2i∗2 1 − i∗2 i∗1 −β1i∗1. The determinant of J(EE) is given by: det(J(EE)) = β2(b + d)i ∗ 2 1−i∗2 i∗1 + β1(b + d)i ∗ 1 −2β2di∗22 1 − i∗2 i∗1 − 2β1di∗1i ∗ 2 −β2γ + 2β2γi∗2 + β2γi ∗ 1 +β1γi ∗ 1 −dγi ∗ 1 = β2(b + d)i ∗ 2 1 − i∗2 i∗1 + ( β1(b + d) +β2γ ) i∗1 + 2β2i ∗ 2 ( γ −di∗2 1−i∗2 i∗1 ) −2β1 di∗1 i ∗ 2 −β2γ + β1 γ i ∗ 1 −dγ i∗1. In the first term of the determinant, we replace (b+d)i∗2 by γi∗1 + di ∗2 2 and we get: det(J(EE)) = β2 ( γi∗1 + di ∗2 2 ) 1 − i∗2 i∗1 +(b + d)(b + γ)R0i∗1 +2β2 i∗2 i∗1 ( γi∗1 −di ∗ 2 + di ∗2 2 ) −2β1di∗1i∗2 −β2γ + β1γi∗1 −dγi∗1. We replace again γi∗1 − di∗2 + di∗22 by bi∗2 and by developing the first term of the determinant, we get: det(J(EE)) = β2γ −β2γi∗2 + β2di ∗2 2 1 − i∗2 i∗1 +(b + d)(b + γ)R0i∗1 + 2β2b i∗22 i∗1 −2β1di∗1i ∗ 2 −β2γ + β1γi ∗ 1 −dγi∗1 = β2i ∗ 2 ( −γ + di∗2 1 − i∗2 i∗1 + b i∗2 i∗1 ) +(b + d)(b + γ)R0i∗1 + β2b i∗22 i∗1 −2β1di∗1i ∗ 2 + β1γi ∗ 1 −dγi ∗ 1 = β2 i∗2 i∗1 ( −γi∗1 + (b + d)i ∗ 2 −di ∗2 2 ) +(b + d)(b + γ)R0i∗1 + β2b i∗22 i∗1 −2β1di∗1i ∗ 2 + β1γi ∗ 1 −dγi ∗ 1. Thus det(J(EE)) = b(b + d + γ)R0i∗1 + β2b i∗2 i∗1 +dγi∗1(R0 − 1) + β1i∗1(γ − 2di∗2). The determinant is positive because i∗2 ∈ (0, γ 2d ). Furthermore the trace is negative, because it is given by: trJ(EE) = −β2i∗2 1 − i∗2 i∗1 −β1i∗1 − b−d + 2di ∗ 2. Then the endemic equilibrium is asymptotically stable. VI. GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM Since s + i1 + i2 = 1, we can reduce system (2) to a planar system and hence we can use the Poincarré- Bendixson theorem to investigate the global attraction of the endemic equilibrium when R0 > 1. To this end, let us consider the following system:  ṡ = b(1 −s) − (β1i1 + β2(1 − i1 −s))s +ds(1 − i1 −s), i̇1 = (β1i1 + β2(1 − i1 −s))s− (b + γ)i1 +di1(1 − i1 −s), (8) defined on the set Ω = {0 ≤ s ≤ 1, 0 ≤ i1 ≤ 1, s + i1 ≤ 1}. We establish by the Dulac-Bendixson criterium that there is no periodic orbit for (8). Biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 Page 6 of 8 http://dx.doi.org/10.11145/j.biomath.2014.07.211 M Diouf et al., Global Stability of an Epidemic Model with two Infected Stages... Theorem VI.1. System (8) has no periodic orbit. Proof: Consider the function B(x,y) = 1 xy . We have: B ṡ(s,i1) = b si1 − b i1 −β1 − β2 i1 + β2 + β2s i1 + d i1 −d− ds i1 , thus ∂ ∂s B ṡ(s,i1) = − b s2i1 + β2 i1 − d i1 . And B i̇1(s,i1) = β1 + β2 i1 −β2 − β2s i1 − b + γ s + d s − di1 s −d, so ∂ ∂i1 B i̇1(s,i1) = − β2 i21 + β2s i21 − d s . It leads to ∂B ṡ(s,i1) ∂s + ∂Bi̇1(s,i1) ∂i1 = − b s2i1 + β2 i21 (s + i1 − 1) − d i1 − d s ∂B ṡ(s,i1) ∂s + ∂Bi̇1(s,i1) ∂i1 < 0 ∀s,i1 ∈ (0, 1]. By Dulac-Bendixson criterium, we conclude that there is no closed orbit for system (8). Thanks to Theorem VI.1 and the Poincaré-Bendixson theorem we have the following result: Theorem VI.2. If R0 > 1 the endemic equilibrium exists and is globally asymptotically stable in Ω − Γ, where Γ is the stable manifold of the disease free equilibrium. Proof: If R0 > 1, the Jacobian matrix of system of (8) at the point (1, 0) has a negative determinant. Therefore the DFE is unstable, but the eigenvalues of the Jacobian matrix at the DFE are equal to: λ1,2 = β1 − (b + γ) − (b + d) ± √ (β1 − (b + γ) − (b + d)) 2 − 4(b + γ)(b + d)(1 −R0). One of the two eigenvalues is negative, which gives that the disease free equilibrium has one dimensional stable manifold Γ. The ω−limit set of the system (8) on Ω−Γ is reduced to the endemic equilibrium point. Because of the local stability of the endemic equilibrium for R0 > 1, the endemic equilibrium is globally asymptotically stable. VII. CONCLUSION The model SI is one of the most important epi- demiological model. This paper gives a qualitative analysis of the stability of the model with a non-linear incidence. For this incidence, the system is analyzed by considering the differential system satisfied by the proportions, and the theory of Poincarré-Bendixson is used. It would be interesting to generalize the work to study the system with arbitrary n infected stages. 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Biomath 3 (2014), 1407211, http://dx.doi.org/10.11145/j.biomath.2014.07.211 Page 8 of 8 http://dx.doi.org/10.1093/imammb/dqi001 http://dx.doi.org/10.1016/S0025-5564(02)00149-9 http://dx.doi.org/10.11145/j.biomath.2014.07.211 Introduction The model The equilibrium points Global Stability of the DFE Local Stability of the endemic equilibrium Global stability of the endemic equilibrium Conclusion References