Time‐dependent sensitivity of R0 and other threshold parameters in seasonal ecological and epidemiological dynamics

Periodically driven systems are ubiquitous in nature, and their stability against perturbations such as the introduction of a novel pathogens or invasive species is often a major concern. To better understand and control these systems, it is common to assess the sensitivity to changes in the model parameters of stability threshold quantities such as the basic reproduction number , or the Floquet multiplier , which determines stability against invasion of the disease/invader‐free system. However, traditional methods often assume that these parameter changes are permanent and constant in time, so they may over‐ or underpredict the cost‐effectiveness of management activities that can only be maintained for a limited time during each period, such as culling when most individuals are at an especially vulnerable life‐stage. Moreover, they cannot highlight important time junctures in the model dynamics. In this article, we address this problem by providing efficient algorithms to compute the sensitivities of objectives like and to periodic, time‐localised perturbations in the model parameters. These sensitivities provide information about when during each period interventions should have the largest effect on the objective, and can also be used to calculate the sensitivities to more general periodic perturbations. Hence, this article extends previous work on time‐dependent sensitivity analysis (TDSA) to a new class of problems relevant to stability analysis. Using empirically parameterised models for Lyme disease and hantavirus, we demonstrate that TDSA can provide useful and sometimes surprising insights that nonetheless have intuitive biological interpretations. Specifically, we find that culling at the optimal times of susceptible and infected non‐human hosts has additive effects in the hantavirus model, but opposite and nearly cancelling effects in the Lyme disease model, so that indiscriminate culling is predicted to be effective for hantavirus but ineffective for Lyme disease.