Mathematical analysis of delayed spread models: Spread rate and spread speed

This paper extends the spread-model framework developed in our previous work by incorporating delay mechanisms representing waiting or activation periods before spreading. The contribution is not to introduce delay as a new modeling concept, but to formulate it within the symbolic-dynamical and projected-spread structures of spread models and to study the resulting spectral and asymptotic properties. In the topological setting, delay states are encoded through an expanded type space, yielding an induced 1-spread model that admits a natural projected-spread representation. Using the associated rome matrix, we derive an explicit characteristic polynomial for the delayed ξ-matrix under suitable structural assumptions and show that its Perron eigenvalue decreases as the delay increases. In the random setting, an analogous age-expanded construction transforms the delayed model into a multitype Galton–Watson process whose projection recovers the original delayed process, allowing its asymptotic spread rates to be characterized. Together, these results provide a unified extension of the spread-model framework to deterministic and stochastic systems with delays.