Characterization of intermediate survival regimes

Characterize the behavior of the contact process with avoidance on \(\mathbb{Z}^d\) for infection rates in the intervals \(\lambda(\alpha,d)^- \leq \lambda \leq \lambda_w(\alpha,d)^+\) and \(\lambda_w(\alpha,d)^+ \leq \lambda \leq \lambda(\alpha,d)^+\), including whether the process becomes extinct, survives weakly, or survives strongly.

Background

The contact process with avoidance lacks the attractiveness and parameter monotonicity available for the classical contact process. Consequently, the paper defines separate lower, weak-survival, and strong-survival critical values rather than a single critical threshold. The main theorems establish extinction below the lower critical value and strong survival above the upper critical value, but do not determine the process’s behavior in the two intermediate parameter ranges.

References

However, we are unable to say precisely what happens for $\lambda(\alpha,d)- \leq \lambda \leq \lambda_w(\alpha,d)+$ and $\lambda_w(\alpha,d)+ \leq \lambda \leq \lambda(\alpha,d)+$.

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