Abstract
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.
Highlights
An important principle in theoretical biology is that of competitive exclusion: no two species can forever occupy the same ecological niche
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]
Summary
An important principle in theoretical biology is that of competitive exclusion: no two species can forever occupy the same ecological niche. 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. By calculation, it follows from M-matrix theory [27], if R1ƒ1 and R2ƒ1, the origin is locally asymptotically stable.
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