The impact of the infectious period on epidemics

The duration of the infectious period is crucial in determining the ability of an infectious disease to spread. We consider an epidemic model that is network-based and non-Markovian, containing classic KermackMcKendrick, pairwise, message passing and spatial models as special cases. For this model, we prove a monotonic relationship between the variability of the infectious period (with fixed mean) and the probability that the infection will reach an arbitrary subset of the population by time t. The striking importance of this relationship, even under standard assumptions, is demonstrated. In a homogeneously mixing large population, under standard assumptions, the central epidemiological quantity R0 (this being the expected number of secondary cases per typical primary case in an otherwise susceptible population) only depends on the infectious period through its mean [1]. However, other important quantifiers such as the probability of a major outbreak and the initial growth rate can depend on the variability of the infectious period; with higher variability tending to decrease these quantities [1]. When incorporating a much greater degree of realism such that individuals can only make contacts to their neighbours in a contact network [2], R0 typically depends on the variability of the infectious period and, even when R0 is held fixed, the probability that any given individual will eventually get infected is still dependent on the variability of the infectious period [3]. In this letter, we extend these results to a much more general epidemic model and consider the effect of the infectious period distribution on the fundamental probability P (A, t) that the disease will spread to an arbitrary subset A of the population by an arbitrary time t. This probability is relevant to the likelihood of an epidemic, and the speed and extent of its propagation. It is commonplace to assume that the infectious period is exponentially distributed because this leads to greater mathematical tractability. In choosing the parameter for this distribution, the modeller may try to replicate the estimated average infectious period or the estimated value for R0. In any case, data shows that the exponential distribution is not very realistic for this variable. For example, it has been suggested that gamma, Weibull and degenerate Liverpool John Moores University (email: [email protected]) The University of Liverpool