A Comparison of Methods for Calculating the Basic Reproductive Number for Periodic Epidemic Systems

When using mathematics to study epidemics, oftentimes the goal is to determine when an infection can invade and persist within a population. The most common way to do so uses threshold quantities called reproductive numbers. An infection's basic reproductive number (BRN), typically denoted [Formula: see text], measures the infection's initial ability to reproduce in a naive population and is tied mathematically to the stability of the disease-free equilibrium. Next-generation methods have long been used to derive [Formula: see text] for autonomous continuous-time systems; however, many diseases exhibit seasonal behavior. Incorporating seasonality into models may improve accuracy in important ways, but the resulting non-autonomous systems are much more difficult to analyze. In the literature, two principal methods have been used to derive BRNs for periodic epidemic models. One, based on time-averages, is simple to apply but does not always describe the correct threshold behavior. The other, based on linear operator theory, is more general but also more complicated, and no detailed explanations of the necessary computations have yet been laid out. This paper reconciles the two methods by laying out an explicit procedure for the second and then identifying conditions (and some important classes of models) under which the two methods agree. This allows the use of the simpler method, which yields interpretable closed-form expressions, when appropriate, and illustrates in detail the simplest possible case where they disagree. Results show that seasonality alone cannot affect disease persistence, but must act in conjunction with non-hierarchical heterogeneity in the infected population, in order to do so.