Abstract

In this talk I will review deterministic epidemic models that do not have an explicit spatial structure. The most ubiquitous of these is the extit{SIR} model, which is a special case of the extit{Kermack-McKendrick} model. Many properties of these models can be deduced from the well-known basic reproduction number, $mathcal{R}_0$. Following the introduction of a typical primary infectious case in an otherwise susceptible population, $mathcal{R}_0$ measures the expected change in prevalence from one infection generation to the next. There is a one-to-one correspondence between $mathcal{R}_0$ and $r$, the Malthusian parameter or initial rate of increase in infection incidence, directly linking generation time and chronological time. The value of $mathcal{R}_0$ determines the final size of the epidemic, which is independent of temporal dynamics. It also provides a measure of the control effort required to prevent an epidemic, or to eliminate an existing infection from a population. Where the modeled populations are structured, for example by sex, species, or groups at high risk of infection, $mathcal{R}_0$ can be determined from the extit{Next Generation Matrix}. However, it is not always sensible to average over different host types or states at infection, so an alternative threshold quantity the extit{Type Reproduction Number} $mathcal{T}$ has been defined. The value of $mathcal{T}$ provides a measure of the effort required when control is targeted. For macroparasite life cycles there is only one state at infection, as pathogen development proceeds through prescribed stages. Here, $mathcal{R}_0$ measures the change in parasite population density from one infection generation to the next. Finally, in periodic environments the number of secondary cases depends on the timing of the primary case. Careful averaging is then necessary, and the value of $mathcal{R}_0$ can be determined as the spectral radius of the extit{Next Generation Operator}.