In the case of exponentially distributed infectious periods and recovery from infection offering long-lasting immunity, the standard SIR equations provide an exact description of the mean behaviour of this system. Nevertheless, the existence of waning immunity, a latent period between an individual becoming infected and being able to transmit infection, and nonexponentially distributed recovery periods are also important for epidemiological applications [10, 42, 115] . These can often be incorporated into analytical approaches through the addition of extra disease compartments, which necessitates extra algebraic and computational effort but typically does not require a fundamental conceptual reevaluation. Sometimes, significant additional complexity does not even modify quantitative epidemiological results-for example, regardless of the rate of waning immunity, length of latent period, or infectious period distribution, if the mean infectious period is T, then the basic reproductive ratio is