Bifurcations and Structures of the Parameter Space of a Discrete-Time SIS Epidemic Model

Abstract

The dynamics of a discrete-time SIS epidemic model are reported in this paper. Three types of codimension one bifurcation, namely, transcritical, flip, Neimark–Sacker (N-S) bifurcations, and their intersection codimension two bifurcations including 1 : 2, 1 : 3, and 1 : 4 resonances are discussed. The necessary and sufficient conditions for detecting these types of bifurcation are derived using algebraic criterion methods. Numerical simulations are conducted not only to illustrate analytical results but also to exhibit complex behaviors which include period-doubling bifurcation in period $-2,-4,-8,-16$ orbits, invariant closed cycles, and attracting chaotic sets. Especially, here we investigate the parameter space of the discrete model. We also investigate the organization of typical periodic structures embedded in a quasiperiodic region. We identify period-adding, Farey sequence of periodic structures embedded in this quasiperiodic region.