Abstract
The duration of the infectious period is a crucial determinant of the ability of an infectious disease to spread. We consider an epidemic model that is network based and non-Markovian, containing classic Kermack-McKendrick, 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 any given subset of the population by any given time. For certain families of distributions, this result implies that epidemic severity is decreasing with respect to the variance of the infectious period. The striking importance of this relationship is demonstrated numerically. We then prove, with a fixed basic reproductive ratio (), a monotonic relationship between the variability of the posterior transmission probability (which is a function of the infectious period) and the probability that the infection will reach any given subset of the population by any given time. Thus again, even when is fixed, variability of the infectious period tends to dampen the epidemic. Numerical results illustrate this but indicate the relationship is weaker. We then show how our results apply to message passing, pairwise, and Kermack-McKendrick epidemic models, even when they are not exactly consistent with the stochastic dynamics. For Poissonian contact processes, and arbitrarily distributed infectious periods, we demonstrate how systems of delay differential equations and ordinary differential equations can provide upper and lower bounds, respectively, for the probability that any given individual has been infected by any given time.
Highlights
In a homogeneously mixing large population, under certain common assumptions, the epidemiological quantity R0 depends on the infectious period only through its mean [1]
We prove a monotonic relationship between the variability of the infectious period and the probability that the infection will reach any given subset of the population by any given time
We show how our results apply to message passing, pairwise, and Kermack-McKendrick epidemic models, even when they are not exactly consistent with the stochastic dynamics
Summary
In a homogeneously mixing large population, under certain common assumptions, the epidemiological quantity R0 (this being the expected number of secondary cases per typical primary case near the start of an epidemic) depends on the infectious period only through its mean [1]. When accounting for the more realistic scenario where individuals can only make direct contacts to their neighbor in a contact network [3], 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 [4] We extend these results to a much more general epidemic model and consider the effect of the infectious period distribution on the probability P (A,t) that the disease will spread to an arbitrary subset A of the population by an. We show how our results carry over to wellknown message passing, pairwise, and Kermack-McKendrick models
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