Survival and extinction on \(\mathbb{Z}\) for isolation and SIRS models

Characterize survival and extinction for the contact process with isolation and the SIRS model on \(\mathbb{Z}\), including the parameter regimes in which each process survives or dies out, in order to determine whether the one-dimensional strong-survival results needed for the higher-dimensional theory hold.

Background

The paper explains that its higher-dimensional results for the contact process with isolation and the SIRS model do not include the analogue of the one-dimensional survival result used for the contact process with avoidance. Existing work gives only partial information, such as weak survival in selected regimes or strong survival under restricted conditions. Thus, the one-dimensional phase behavior of both models remains unresolved.

References

However, fully characterizing the survival and extinction of the contact process with isolation and the SIRS model on $\mathbb{Z}$ remains an open problem.

— Epidemics with avoidance and isolation on $\mathbb{Z}^d$  (2609.03359 - Heeszel et al., 3 Sep 2026) in Section 1, Section “Models with Vertex Isolation”