Survival properties and spread rates in non-autonomous spread models.

As time progresses, the transmission pattern of a disease may change. To more precisely determine the spread behaviors of the disease, we develop non-autonomous topological and random spread models. In this article, we validate the survival characteristics of these spread models and elucidate their connection with mixing properties using the associated ξ-matrices or spread mean matrices. We also introduce the concept of spread rates for both periodic topological and random spread models and provide rigorous formulas for calculating these rates. Additionally, numerical examples and simulation results are provided as supporting evidence for the theory in both topological and random models.