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]–[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]–[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, denote the susceptibles with sexual activity , which is the number of contacts per individual in group per unit of time, and use and to denote the infectives with sexual activity and infected by strain 1 and strain 2, respectively. The dynamics of the disease transmission then is described by the following equations:(1)whereare the rates of incidence with being the population size of group , are the constant input flows entering the sexually active sub-populations, are the average sexual life spans for people in group , and are the transmission probabilities per contact with individuals infected by strains 1 and 2, respectively, and and are the rates of recovery for classes and , 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(2)where

SetThenWith these notations, the system (2) can be rewritten into the following compact form:(3)

Note that is the total population of group . Throughout this paper will consider only the dynamics of (3) in , whereand . Let denote the solution flow generated by (3). It is not difficult to see that the flow is positively invariant in .

For two vectors , define the vector order as follows:and also define the type-K order in in the sense thatThe Jacobian-matrix at each point has the form(4)It follows from Smith [26] that the flow is type-K monotone in the sense that

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.

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].

Acknowledgments

The authors greatly appreciate an anonymous referee for his valuable comments which improve the paper very much.