Real time bayesian estimation of the epidemic potential of emerging infectious diseases

Abstract

Background: Fast changes in human demographics worldwide, coupled with increased mobility, and modified land uses make the threat of emerging infectious diseases increasingly important. Currently there is worldwide alert for H5N1 avian influenza becoming as transmissible in humans as seasonal influenza, and potentially causing a pandemic of unprecedented proportions. Here we show how epidemiological surveillance data for emerging infectious diseases can be interpreted in real time to assess changes in transmissibility with quantified uncertainty, and to perform running time predictions of new cases and guide logistics allocations.

Methodology/principal findings: We develop an extension of standard epidemiological models, appropriate for emerging infectious diseases, that describes the probabilistic progression of case numbers due to the concurrent effects of (incipient) human transmission and multiple introductions from a reservoir. The model is cast in terms of surveillance observables and immediately suggests a simple graphical estimation procedure for the effective reproductive number R (mean number of cases generated by an infectious individual) of standard epidemics. For emerging infectious diseases, which typically show large relative case number fluctuations over time, we develop a bayesian scheme for real time estimation of the probability distribution of the effective reproduction number and show how to use such inferences to formulate significance tests on future epidemiological observations.

Conclusions/significance: Violations of these significance tests define statistical anomalies that may signal changes in the epidemiology of emerging diseases and should trigger further field investigation. We apply the methodology to case data from World Health Organization reports to place bounds on the current transmissibility of H5N1 influenza in humans and establish a statistical basis for monitoring its evolution in real time.

Figure 1. Time series of new cases for an emerging infectious disease vs. a standard epidemic.

(a) Laboratory confirmed new human H5N1 avian influenza cases, from WHO reports in Vietnam (from January 2004 to June 2006); (b) Number of isolates for seasonal H3N2 influenza in the USA, over the 2004–2005 season. Note the 100-fold difference in case numbers (y-axis) between panel (a) and (b). For an emerging infectious disease such as H5N1 influenza in humans, case numbers are small, very stochastic, and alternate short outbreaks with long quiet periods.

Figure 2. Epidemic time delay diagrams for different R ₀.

(a) Relation between new cases at consecutive time periods (weeks) for H3N2 isolates in the US 2004–05 season, and for simulated data with (b) R ₀ = 1.7, (c) R ₀ = 1.0 and (d) R ₀ = 0.8. For these simulations, the introduction of new cases from the reservoir follows the Vietnam case history, Figure 1a. New cases are then generated using expression [11], according to a Poisson distribution. The trajectories connecting new cases at consecutive times (red arrows) eventually return to the origin because depletion of susceptibles reduces the effective reproduction number (i.e. the actual number of secondary cases produced by an infectious individual). Dashed lines in (a) and (b) are the tangents at the origin to the case number trajectories (red arrows), with slope b(R).

Figure 3. Evolution of R estimates over time (weeks) for single realization simulated data with R ₀ = 0.8, 1.0, 1.4 and 1.7 (left to right, top to bottom).

Dashed lines indicate the value of R ₀ in the simulation. The decay of R estimates over time in standard epidemics is due to the depletion of susceptibles. For R ₀ = 1.0, 1.4 and 1.7 the mean is indistinguishable from the estimate of R with maximum probability and is not shown.

Figure 4. Sequential Bayesian estimation of the posterior mean R (red dots) and 95% credible intervals (solid lines) for the time series of H5N1 avian influenza in (a) Vietnam and (b) Indonesia, under the pessimistic assumption that 29% of reported cases are due to human-to-human transmission (see Table 1); and (c) for seasonal H3N2 human influenza isolates in the USA during the 2004–2005 season.

(Note that isolates represent only a small fraction of total cases, and may contain reporting biases.) The estimate of the effective reproduction number for an epidemic outbreak asymptotes to unity at late times because initial growth and long-term decay in new case numbers (due to depletion of susceptibles) average out over the history of the outbreak.

Figure 5. Prediction for new cases of avian influenza (simulated data R ₀ = 0.8, with infections from reservoir taken from Vietnam time series, Fig. 1a) vs. realized new cases (blue dots).

Between weeks 74 and 75, the reproduction number is shifted R ₀ = 0.8→1.3 to create an epidemic. Although we continued to iterate the R distributions via the Bayesian procedure described in the text, note that the shift in R upwards leads to many statistical anomalies (indicated by black arrows). The anomaly is detected immediately, on weeks 75 and 76. Anomalies here are defined as observed numbers of new cases that fall outside the expected 95% credible interval. These anomalies indicate a violation of the hypothesis that R is unchanged, and could be used to trigger alerts in surveillance.