Using asymptotics for efficient stability determination in epidemiological models

Local stability analysis is an important tool in the study of dynamical systems. When the goal is to determine the effect of parameter values on stability, it is necessary to perform the analysis without explicit parameter values. For systems with three components, the usual method of finding the characteristic polynomial as $ \det(J-\lambda I) $ and applying the Routh-Hurwitz conditions is reasonably efficient. For larger systems of four to six components, the method is impractical, as the calculations become too messy. In epidemiological models, there is often a very small parameter that appears as the ratio of a disease-based timescale to a demographic timescale; this allows efficient use of asymptotic approximation to simplify the calculations at little cost. Here, we describe the tools and a set of guidelines that are generally useful in applying the method, followed by two examples of efficient stability analysis.