Georgia Institute of Technology College of Sciences

We study the SIRS process on sparse random graphs with power--law degree distributions.
A large physics literature reports numerical evidence for a positive epidemic threshold for SIRS with waning immunity on scale--free networks, suggesting a transition between short--lived and exponentially long--lived regimes.
In contrast, for the SIS/contact process on power--law graphs with exponent $\tau>3$, it is rigorously known that the critical value is $\lambda_c=0$ and that survival is exponentially long for every $\lambda>0$.
We show that, in a survival--time sense, the true threshold for SIRS on power--law random graphs with $\tau>3$ is also zero. Joint work with Debankur Mukherjee and Souvik Dhara.