Impact of the infectious period on epidemics

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 (R_{0}), 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 R_{0} 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.

Epidemic Models of Contact Tracing: Systematic Review of Transmission Studies of Severe Acute Respiratory Syndrome and Middle East Respiratory Syndrome

“…The nature of memory-less free characteristics of gamma distribution ensured the biology in the transmission process to be more realistic. Previous work [ [35] , [36] , [37] ] suggested the importance of realistic distribution of these quantities facilitated more understanding in the epidemic size, the disease progression and how their distributions be adjusted in response to different intervention strategies [ 38 ]. These choices are especially relevant to models of contact tracing, as was evident in the early 2000s when two apparently similar models [ 39 , 40 ] came to very different conclusions about likely efficacy of contact tracing because of their different implicit assumptions about the variance of the latent period distribution.…”

Evolutionary bet-hedging in structured populations

“…This information is captured in the higher order moments of the distributions, such as the skew and kurtosis. A more comprehensive representation of variation is given by the convex order (Shaked and Shanthikumar 2007 ; Wilkinson and Sharkey 2018 ), such that if one distribution is greater than another in convex order then it is more variable. Convex order describes variability by ordering the expected values of convex functions, which are sensitive to the variation.…”

A monotonic relationship between the variability of the infectious period and final size in pairwise epidemic modelling

“…Remark 3 An example when [27] can not be applied but our methodology works. Let I 1 ∼ Exp(1) (exponential distribution with parameter 1).…”

“…In light of Theorem 3.A.1b in [23], in this case the random variables I 1 , I 2 are not convex ordered, thus [27] does not apply. For sufficiently large τ, we have L [ f I 1 ](τ) > L [ f I 2 ](τ), hence R p 0,I 1 < R p 0,I 2 , and the discrete random variable, which has the smaller variance, generates a larger epidemic outbreak.…”