Shape theorems and nesting of limiting infected sets

Establish shape theorems for the discrete-time infection models under consideration in order to determine whether their limiting deterministic shapes are nested under the comparison inequalities.

Background

The discrete-time local-to-global comparison theorem orders probabilities of prescribed infection-path events and, in particular, can compare survival probabilities and first-hitting-time events. The authors note that such first-hitting comparisons would imply nesting of the limiting deterministic shapes if shape theorems were available for the infection models being compared.

The unresolved task is therefore to establish shape theorems for the relevant non-standard oriented-percolation or contact-process-type models and use them to obtain a nesting relation for their asymptotic infected regions. The paper does not prove these shape results, although it points to existing shape theorems for some non-standard oriented-percolation models.

References

This in turn would guarantee a nesting of the limiting deterministic shapes, once shape theorems for the models under consideration are established. This last task depends on the models at hand and we leave it to future work, not without mentioning that such shape theorems have been established for non-standard oriented percolation models for example in.

— Comparison inequalities for discrete- and continuous-time infection processes  (2609.09892 - Jahnel et al., 9 Sep 2026) in Section 2.1, immediately after Theorem 2.1 (Local-to-global in discrete time)