Frequentist estimation of an epidemic's spreading potential when observations are scarce

We consider the problem of inferring the potential of an epidemic for escalating into a pandemic on the basis of limited observations in its initial stages. Classical results of Becker & Hasofer (J. R. Statist. Soc. B, 59, 415-29) illustrate that frequentist estimation of the complete set of parameters of an epidemic modelled as a birth and death process remains feasible even when one is able to observe only the deaths and the total number of births. These assumptions on the observation mechanism, however, are too strong to be met in practice. We consider a more realistic scenario where only temporally aggregated random proportions of the deaths are observed over time. We demonstrate that the frequentist estimation of the Malthusian parameter governing the growth of the epidemic is still feasible in this context. We construct explicit straightforwardly calculable estimators motivated heuristically by the martingale dynamics of the process, and show that they admit a rigorous quasilikelihood interpretation. We establish the consistency and asymptotic normality of these estimators, allowing for the construction of approximate confidence intervals that can be used to infer the spreading potential of the epidemic. A simulation study and an application to the initial outbreak data of the 2009 H1N1 influenza pandemic illustrate that the method can be expected to give reasonable results in practice.

Statistical Inference for Partially Observed Stochastic Epidemics

Andrea Kraus

This work is concerned with the estimation of the spreading potential of the disease in the initial stages of an epidemic. A speedy and accurate estimation is important for determining whether or not interventions are necessary to prevent a major outbreak. At the same time, the information available in the early stages is scarce and data collection imperfect. We consider an epidemic in a large susceptible population, and address the estimation based on temporally aggregated counts of new cases that are subject to unknown random under-reporting. We allow for an influence of the detection process on the evolution of the epidemic. While the proportion of infectious individuals in the population is small, the role of chance in the spread of the disease may be substantial. Therefore, stochastic epidemic models are applied. As these are difficult to analyse, the time evolution of the number of infectious individuals is approximated by branching processes. We study the estimation in a partially observed Galton–Watson process and in a partially observed linear birth and death process; and in each case focus on the parameter characterising the growth of the process. We aim at estimators that perform well in the asymptotic sense where a single trajectory is observed over a long period of time, and study the asymptotics conditionally on the eventual explosion of the process. The partially observed Galton–Watson process has been recently proposed in the literature as a model for the initial stages of an epidemic. Its probabilistic structure has been explored and estimation has been partially addressed, in that consistent estimators have been constructed. However, the estimation-related uncertainty has not been evaluated. We address this issue here by constructing estimators that are motivated from the asymptotic dependence structure of the process. We show that they are consistent and asymptotically normal, consistently estimate their asymptotic variances, and construct asymptotic confidence intervals. In addition, we evaluate their finite-sample performance in a simulation study and their practical performance on real data. The observation mechanism in the partially observed Galton–Watson process is inherently discrete. To allow for continuous-time dynamics, we incorporate partial observation in the linear birth and death process. In particular, we propose a model where the birth process is completely unobserved, while a random proportion of the death process is observed at discrete time points. We study the estimation in this model. Motivated by its counting process structure, we arrive at consistent and asymptotically normal estimators, consistently estimate their asymptotic variances, and construct asymptotic confidence intervals. We also evaluate the finite-sample and practical performance of the estimators in a simulation study and on real data.

EPFL2013

Statistical Inference for Partially Observed Stochastic Epidemics

Andrea Kraus

EPFL2013