Predictability in a highly stochastic system: final size of measles epidemics in small populations

Abstract

A standard assumption in the modelling of epidemic dynamics is that the population of interest is well mixed, and that no clusters of metapopulations exist. The well-known and oft-used SIR model, arguably the most important compartmental model in theoretical epidemiology, assumes that the disease being modelled is strongly immunizing, directly transmitted and has a well-defined period of infection, in addition to these population mixing assumptions. Childhood infections, such as measles, are prime examples of diseases that fit the SIR-like mechanism. These infections have been well studied for many systems with large, well-mixed populations with endemic infection. Here, we consider a setting where populations are small and isolated. The dynamics of infection are driven by stochastic extinction–recolonization events, producing large, sudden and short-lived epidemics before rapidly dying out from a lack of susceptible hosts. Using a TSIR model, we fit prevaccination measles incidence and demographic data in Bornholm, the Faroe Islands and four districts of Iceland, between 1901 and 1965. The datasets for each of these countries suffer from different levels of data heterogeneity and sparsity. We explore the potential for prediction of this model: given historical incidence data and up-to-date demographic information, and knowing that a new epidemic has just begun, can we predict how large it will be? We show that, despite a lack of significant seasonality in the incidence of measles cases, and potentially severe heterogeneity at the population level, we are able to estimate the size of upcoming epidemics, conditioned on the first time step, to within reasonable confidence. Our results have potential implications for possible control measures for the early stages of new epidemics in small populations.

Estimating enhanced prevaccination measles transmission hotspots in the context of cross-scale dynamics

“…At the most predictable scales, the Susceptible-Infected-Recovered (SIR) family of models has been used to analyze measles dynamics in city and country metapopulations (1)(2)(3)(4), as well as in allowing for heterogeneities in space (5,6), age (7,8), and population sizes (9,10). As a badge of simplicity, SIR models successfully assume well-mixed mass action in city populations (i.e., that the number of individuals who will become infected is proportional to the total number currently infected as well as those currently susceptible).…”

Climate drives spatial variation in Zika epidemics in Latin America

“…Here, the mixing parameter α indicates homogeneous frequency-dependent contact rate between cases in each province. We fixed the mixing parameter due to the tradeoff between capturing heterogeneity in transmission or mixing, a method commonly used in tSIR studies focused on transmission dynamics [10,53–56]. Given the similar transmission dynamics of chikungunya and Zika, we set the mixing parameter equal to 0.74, the value for chikungunya calculated in the Perkins et al study [2,51].…”

The practice of prediction: What can ecologists learn from applied, ecology-related fields?

“…The R 2 ranged from 0.98 -0.92 for large cities, and was still reasonably high (0.74) for small communities. Extensions of measles modeling to small communities that have highly stochastic dynamics still achieved R 2 of 0.86 to 0.55, with 5 out of 6 communities scoring higher than 0.73 (Caudron et al, 2015). A recent review gives a comprehensive account of the predictability of influenza outbreaks, comparing time series modeling, individual-based, compartmental and metapopulation models (Nsoesie et al, 2014).…”