The complete classification for dynamics of an SIS STD model in terms of four parameters

In this paper, a nine-dimensional sexually transmitted disease model proposed by Castillo-Chavez et al. [C. Castillo-Chavez, Wenzhang Huang and Jia Li, The effects of female’s susceptibility on coexistence of multiple pathogen strains of sexually transmitted diseases, J. Math. Biol. 35 (1996) 503–522] is studied. The model involves two competing strains 1 and 2 in a two-sex heterosexually active population that includes a single group of males and two different groups of females. The first and second reproduction numbers RiRi, ℛiℛi are defined for strain i,i=1,2i,i=1,2 respectively. By applying the theory of type-KK monotone dynamical systems, a complete classification for the dynamics of this model is presented in terms of the first and second reproduction numbers Ri,ℛi,i=1,2Ri,ℛi,i=1,2. The classification not only is different to the complete classification given by Castillo-Chavez et al. [C. Castillo-Chavez, Wenzhang Huang and Jia Li, Competitive exclusion and coexistence of multiple strains in an SIS STD model, SIAM J. Appl. Math. 59 (1999) 1790–1811], but also can be easily explained in biology. Our results also show that the dynamics of the model is completely determined by the ubiety of two functions.