Effects of distribution of infection rate on epidemic models

Abstract

A goal of many epidemic models is to compute the outcome of the epidemics from the observed infected early dynamics. However, often, the total number of infected individuals at the end of the epidemics is much lower than predicted from the early dynamics. This discrepancy is argued to result from human intervention or nonlinear dynamics not incorporated in standard models. We show that when variability in infection rates is included in standard susciptible-infected-susceptible (SIS) and susceptible-infected-recovered (SIR) models the total number of infected individuals in the late dynamics can be orders lower than predicted from the early dynamics. This discrepancy holds for SIS and SIR models, where the assumption that all individuals have the same sensitivity is eliminated. In contrast with network models, fixed partnerships are not assumed. We derive a moment closure scheme capturing the distribution of sensitivities. We find that the shape of the sensitivity distribution does not affect R_{0} or the number of infected individuals in the early phases of the epidemics. However, a wide distribution of sensitivities reduces the total number of removed individuals in the SIR model and the steady-state infected fraction in the SIS model. The difference between the early and late dynamics implies that in order to extrapolate the expected effect of the epidemics from the initial phase of the epidemics, the rate of change in the average infectivity should be computed. These results are supported by a comparison of the theoretical model to the Ebola epidemics and by numerical simulation.

Heterogeneous social interactions and the COVID-19 lockdown outcome in a multi-group SEIR model

“…Modeling heterogeneous mixing in infectious disease dynamics when the population is subdivided by characteristics other than those that are disease-related, such as risk status or age, is not new. This has been considered for instance from the dynamical point of view in [28] or in the case of sexually transmitted diseases and particularly in HIV/AIDS models, with groups that are not all defined by disease related properties. In this perspective, contact matrices have been considered, which involve a detailed analysis of the transmission mechanisms.…”

Initial growth rates of malware epidemics fail to predict their reach

“…These results may be understood from the perspective of a random mixing infection model with a distribution of susceptibility as in Fig. 1 A 17 , or of network infection models with a similar degree of distribution. For the random mixing model, we ran stochastic SIR simulations with each susceptible node having a constant yet different probability of being infected as well as a constant probability of infecting others (See Supp.…”

“…The lack of correlation between the epidemics’ early spread rates and their later dynamics was systematically confirmed by several out-of-sample studies and has been a source of concern for the relevant authorities. This variability in the epidemics’ spread dynamics may be due to the extreme sensitivity of the contagion to the volume of the susceptibility-distribution tail 17 – 19 .…”

“…Recent publications suggest that the failure of existing diffusion models to extrapolate the behavior of future epidemics from their early growth rates could be related to these models’ failure to take extreme population heterogeneity into account 17 – 19 . The general idea is that pathogens insufficiently contagious to survive and spread in the general population may still spread at a very high rate in a heterogeneous population, specifically among exceptionally susceptible targets.…”

Epidemic dynamics in inhomogeneous populations and the role of superspreaders

“…Several recent works (see, e.g., [25][26][27]) have addressed the issue of heterogeneity in the population, but they either concentrate on specific distributions or treat the variability in infectivity and susceptibility separately, without considering the effect of a possible correlation between the two.…”