Competitive Exclusion and Coexistence of Pathogens in a Homosexually-Transmitted Disease Model

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population has been studied completely by Jiang and Chai in (J Math Biol 56:373–390, 2008). In this paper, we give a analysis for a SIS STD with two competing strains, where populations are divided into three differential groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the boundary equilibria that characterizes competitive exclusion of the two competing strains; we also investigate the existence and stability of the positive coexistence equilibrium, which characterizes the possibility of coexistence of the two strains. We obtain sufficient and necessary conditions for the existence and global stability about these equilibria under some assumptions. We verify that there is a strong connection between the stability of the boundary equilibria and the existence of the coexistence equilibrium, that is, there exists a unique coexistence equilibrium if and only if the boundary equilibria both exist and have the same stability, the coexistence equilibrium is globally stable or unstable if and only if the two boundary equilibria are both unstable or both stable.


Introduction
An important principle in theoretical biology is that of competitive exclusion: no two species can forever occupy the same ecological niche. Classifications on the meaning of competitive exclusion and niche have been central to theoretical ecology [1][2][3][4]. On the other hand, biologists and mathematical modelers have long been concerned with the evolutionary interactions that result from changing host and pathogen populations. Continuous advances in biology and behavior have brought to the forefront of research the importance of their role in disease dynamics [5][6][7][8][9][10][11][12][13][14][15][16][17]. Sexually transmitted diseases, such as gonorrhea have incredibly high incidences throughout the world, providing the necessary environment and opportunities for the evolution of new strains(see [18] and the references therein). The coexistence of gonorrhea strains has become an increasingly serious problem. Understanding the mechanisms that lead to coexistence or competitive exclusion is critical to the development of disease management strategies, as well as to our understanding of STD dynamics.
In previous papers [18,19], they have shown that coexistence of multiple strains is not possible in a heterosexually-active homogenous population where individuals have the same mean behavior by investigating SIS STD models and establishing that such populations are unable to support multiple strains. However, using simple heterosexual mixing models, Castillo-Chaves et al. [20,21] have shown that heterogeneity(behavioral or genetically or a combination of both) of one sex population(the female population) is enough to maintain heterogeneity and to lead possible coexistence of multiple strains. Chai [22] and Qiu [23] has given the completely classification for this model. Li et al. [24] have determined what is the minimum level of heterogeneity required to support multiple strains to coexist. They formulated and analyzed a one-sex, SIS STD model with two competing strains under the same assumptions. Furthermore, in [25], we have presented a thorough classification of dynamics for this model in terms of the first and the second so called reproductive numbers, and discussed the biological meaning of our results in the finally. This paper focus on the dynamics of sexually transmitted pathogens in a homosexually active population, where populations are divided into three groups based on their susceptibility to infection(colonization) by two distinct pathogenic strains of an STD. It is assumed that a host cannot be invaded simultaneously by both disease agents(that is, there is no superinfection) and that when symptoms appear-a function of pathogen, strain, virulence, and an individual's degree of susceptibility-then individuals are treated and/or recover.

Methods
Let S k , k~1,2,3, denote the susceptibles with sexual activity r k , which is the number of contacts per individual in group k per unit of time, and use I k and J k to denote the infectives with sexual activity k and infected by strain 1 and strain 2, respectively. The dynamics of the disease transmission then is described by the following equations: where are the rates of incidence with T k~Sk zI k zJ k being the population size of group k, m k S 0 k are the constant input flows entering the sexually active sub-populations, 1 m k are the average sexual life spans for people in group k, b I and b J are the transmission probabilities per contact with individuals infected by strains 1 and 2, respectively, and c I k and c J k are the rates of recovery for classes I k and J k , respectively. It is assumed that people with different sexual activity having different rates of recovery as highly sexually-active individuals may have health examinations more frequently.
The limiting system of (1) is Then a 11 a 22~a12 a 21 , a 11 a 33~a13 a 31 , a 22 a 33~a23 a 32 , With these notations, the system (2) can be rewritten into the following compact form: Note that p i is the total population of group i,i~1,2,3.
Throughout this paper will consider only the dynamics of (3) in and x~(x 1 ,x 2 ,x 3 ),y~(y 1 ,y 2 ,y 3 ) [ R 3 . Let Q t (x,y) denote the solution flow generated by (3). It is not difficult to see that the flow is positively invariant in V.

Discussion
Next, we consider the necessary thresholds and the stability of the infection-free state, established the principle of competitive exclusion and coexistence for SIS models with heterogeneous mixing.

Thresholds
The linearization about the infection-free equilibrium of (3) is _ x x :~Px, _ y y :~Qy: where Now we define the reproductive numbers Hence, by calculation, it follows from M-matrix theory [27], if R 1 ƒ1 and R 2 ƒ1, then the origin is locally asymptotically stable. If R 1 w1 or R 2 w1, the infection-free equilibrium is unstable. As in [24], it can be shown that the locally stable infection-free equilibrium and the locally stable boundary equilibrium associated with model (3), which will be studied in the following section, are globally stable. We only state the results as follows and omit the details. The interested reader is referred to [24].
In summary, we state the threshold conditions for the disease as follows.
Theorem 1. Let the reproductive number R 1 and R 2 be defined in (5). Then, if R 1 ƒ1 and R 2 ƒ1, the infection-free equilibrium is globally asymptotically stable so that the epidemic goes extinct regardless of the initial levels of infection. If R 1 w1 or R 2 w1, then the infection-free equilibrium is unstable and the epidemic spreads in the population.  respectively. Following Smith [28], both (3) I and (3) J are strongly concave. From [28] it follows that the origin is globally asymptotically stable, or there is exists and equilibrium E x~( x x 1 , x x 2 , x x 3 ,0,0,0) with x x 1 w0, x x 2 w0, x x 3 w0 such that it is globally asymptotically stable in H I \fOg. Moreover, E x is also linearly stable, that is,

The computation of boundary equilibria
, then the origin is globally asymptotically stable in H I , otherwise, R 1 w1, E x exists. Next, we discuss the computation for where Then u,w,h 1 satisfy the equations where h a1 (h 1 )~a 11 By (7), we have Now, we assume that a 12~a22 and b 12~b22 , (9) is equivalent to Let Solving u in (10),we get that which implies that h 1 must be the positive root of Since g a (h 1 ) is a quadratic function in h 1 with g a (0)v0 and the coefficient of second order positive, there exists a unique real number h Ã a1 w0 such that In addition Similarly, the origin is globally asymptotically stable in H J if R 2 ƒ1. Otherwise, if R 2 w1, then (3) J has an equilibrium E y~( 0,0,0, y y 1 , y y 2 , y y 3 ) with y y 1 w0, y y 2 w0, y y 3 w0 such that it is globally asymptotically stable in H J \fOg. Moreover, E y is also linearly stable, that is, is stable. The positive components y y 1 , y y 2 , y y 3 can be calculated by y y 1~v , y y 2~b and h Ã b1 is the unique positive root for We have the following inequalities All above computation results will be very useful in the classification for various dynamical behavior. Before finishing this section, we present a result for (3) that is easily obtained by the theory of monotone dynamical systems. Theorem 2. (i) The infection-free equilibrium x~y~0 is globally asymptotically stable if and only if the reproductive numbers R 1 ƒ1,R 2 ƒ1.
(ii) If R 1 w1,R 2 ƒ1, then E x is globally asymptotically stable in V\H J . (iii) If R 1 ƒ1,R 2 w1, then E y is globally asymptotically stable in V\H I .

The stability of boundary equilibria
First, in the case that either R 1 ƒ1 or R 2 ƒ1, Theorem 2 tell us that the global behavior for (3) is clear. So it suffices to consider the case both R 1 w1 and R 2 w1. Let From now on, we discuss the stability of the boundary equilibrium E x .
The Jacobian matrix J(E x ) of (3) at E x takes the form where A 11 is a stable matrix in the above section and It follows from [27]  From the first equation of (3) I and (6) we get that and by (7), we have It deduces from (16) and (17) that , E x is stable; , E x is unstable.
In a quite similar way, we can discuss the stability for the boundary equilibrium E y , its stability is completely determined by the determinant of the matrix B B 11 :~{ The computation shows that where h b1 is given in (12) and (13).
we get the following stability results from (18): Theorem 4. The stability for E y is confirmed by using (18) as follows: (I) D 1 v0,D 2 v0,D 3 v0, E y is stable; , E y is stable; , E y is unstable.
Remark 1. In Theorem 3 and Theorem 4, we only give the results in this case D 1 w0,D 2 w0,D 3 v0. The other cases can be considered analogously by changing the relevant parameters.
Let s(J(E x )) and s(J(E y )) denote the largest real part of its eigenvalues respectively, which is an eigenvalue for J(E x )) and J(E y ) respectively by Perron-Frobenius theory [27].
Proof Suppose that D 1 w0,D 2 w0,D 3 v0. We have : Then The discussion in the above has shown that s(J(E x ))ƒ0M is equivalent to : ð20Þ (19) and (20) deduce that By Theorem 4, s(J(E y ))w0.
The other cases can be considered analogously. Remark 3. Suppose D 1 w0,D 2 w0,D 3 v0 and s(J(E x ))ƒ0. If there is no positive equilibrium in V, then E x is globally asymptotically stable in V H J . Similar result holds for E y .

The existence of endemic equilibrium
It follow from Theorem 2 that one of the necessary conditions for existence of positive equilibrium is that R 1 w1 and R 2 w1: Now, let we assume (x 1 ,x 2 ,x 3 ,y 1 ,y 2 ,y 3 ) is a positive equilibrium for (3), and set Then q i w0 for i~1,2,3 and (x 1 ,x 2 ,x 3 ,y 1 ,y 2 ,y 3 ) satisfies Thus x 2~q 2 a 22 q 1 a 12 x 1 , x 3~1 {q 1 a 11 {q 2 a 22 q 1 a 13 x 1 , Substituting (22) into (21) yields q 1 a 11 zq 2 a 22 zq 3 a 33~1 , hence q 1 (a 11 {b 11 )zq 2 (a 22 {b 22 )zq 3 (a 33 {b 33 )~0, ð24Þ which implies by q 1 w0,q 2 w0,q 3 w0 that either In order to study the existence of positive equilibrium, we only need to consider the case (I) and (II). Suppose first the former holds. Without loss of generality, we assume that Substituting (25) and (23) into (22), we conclude that such a positive equilibrium must have the form where w~q 2 q 1 : Substituting (26) into (21), we obtain the equations for u,v,w in the form

#~0
: By calculation, we have Notice that (27) is reduced to the system By (28), we have Notice that From (29), we obtain Then, (30) has a unique positive solution if and only if Moreover, we have the result as follows: Theorem 5. If a 12~a22 ,b 12~b22 , and D 1 w0,D 2 w0,D 3 v0. System (3) has a unique positive solution if and only if the following conditions is satisfied: It follows from (4) and Smith [26] that (3) is type-K monotone system, hence Q t (x,y) tends to an equilibrium as t??: Then we can give stability conditions for the positive coexistence equilibrium as follows.
Theorem 6. The positive coexistence equilibrium is stable if and is unstable if : It remains to consider the case D 1~D2~D3~0 . In this case, it is easy to verify a ij~bij for i,j~1,2,3: Thus (3) I and (3) J are the same. Let E x~( x x 1 , x x 2 , x x 3 ,0,0,0). Then E y~( 0,0,0, x x 1 , x x 2 , x x 3 ). Set L~fE(m)jE(m)~mE x z(1{m)E y : 0ƒmƒ1g: Then a straight proof by using a ij~bij shows that all points in segment L are nontrivial equilibria for (3).
The proof refer to the proof of Theorem 4.2 in [25].

Results
In this article, we have given the stability analysis of the nontrivial boundary equilibria and the positive coexistence equilibrium. Our results can be summarized as the following: System (3) (and hence (1))has a unique positive coexistence equilibrium if and only if the two nontrivial boundary equilibria have the same stability. (Both are stable or unstable.) The positive coexistence equilibrium is stable if the boundary equilibria are both unstable. In this case the positive coexistence is a globally attractor. The positive coexistence equilibrium is unstable if and only if the boundary equilibria are both stable. The sufficient and necessary conditions for both boundary equilibria to be stable (unstable) and hence for the positive coexistence equilibrium to be unstable (stable) are given by (H1), (H2) in Theorem 5. Furthermore, if there is no coexistence equilibrium, then the locally stable boundary equilibrium, if it exist, is also globally stable.
In the paper [25], we have given the biological meanings for our results. The biological meanings for the results in this paper which can be given in the same way. The interested reader is referred to [25].