Multi stability and global bifurcations in epidemic model with distributed delay SIRnS-model

Abstract

Many diseases, such as influenza and the common cold, cause recurrent epidemics. The classical SIRS model fails to obtain recurrent epidemics as it predicts a globally stable endemic fixed point. This endemic fixed point is, however, linearly unstable for most parameters, if one assumes that the time spent in the recovered state is deterministic rather than exponentially distributed. In that case all trajectories converge to a stable epidemic limit cycle. It has been shown that a similar region of instability exists for systems with intermediate immune time distributions. Furthermore, it has been suggested that a bistable region could exist. Here, we first characterize this bistable region using a combination of direct simulation and bifurcation theory. We find that it has a bound where the stable epidemic limit cycle annihilates with an unstable limit cycle in a non-local bifurcation. Secondly, we extend the bifurcation-analysis to narrower immune time distributions than previous studies. Here, we find new levels of complexity in the bifurcation diagram, including the possibility for at least two different epidemic limit cycles at the same disease parameters. Overall our study highlights that a given disease may have multiple epidemic signatures, dependent on how it is introduced.