Reproductive Numbers for Periodic Epidemic Systems

Abstract

When using mathematics to study epidemics, often times the goal is to deter- mine when an infection can invade and persist within a population. This can be done in a variety of ways but the most common is to use threshold quantities called reproductive numbers. For models with only one infection, the basic reproductive number (BRN) is used to determine the stability of the disease-free equilibrium. For many years this was done solely for autonomous systems; however, many diseases exhibit seasonal behavior. If this seasonality is incorporated into models, it gives nonautonomous systems, which while more accurate in their description, are much more difficult to analyze. The first chapter lays out methods to find the basic reproductive number for seasonal epidemic models. In the literature, two principal methods have been pro- posed to derive BRNs for periodic models. The first, using time-averages, does not always result in the correct threshold behavior. The more general one is also more complicated, and no detailed explanations of the necessary computations have yet been laid out. This chapter lays out such an explicit procedure and then identifies conditions (and some important classes of models) under which the two methods v agree. This allows the use of the more limited method, which is much simpler, when appropriate, and illustrates in detail the simplest possible case where they disagree. There are many cases within epidemiology where infections will compete to persist within a population. In studying these types of models, one of the goals is to determine when certain infections can invade a population and persist when other infections are already resident within the population. To study this, invasion repro- ductive numbers (IRN) are used, which can help determine the stability of certain endemic equilibria. Methods for both autonomous and nonautonomous systems are given for finding the IRNs, as well as examples which illustrate the often complex computations required. These methods are used for a single-host model of Chagas disease to determine if seasonality can explain why competitive exclusion does not seem to hold in certain sylvatic cycles of the disease. In this model there are two strains of the parasite, and studies show cross-immunity between strains. The single-host autonomous model predicts competitive exclusion, but there has been observed co-persistence in some host populations, in particular woodrats. To account for this, seasonality is added to the original model in the transmission parameters. For a set of biologically re- alistic parameters, seasonality even in just a single parameter is sufficient to make co-persistence possible.