Estimation in emerging epidemics: biases and remedies

Abstract

When analysing new emerging infectious disease outbreaks, one typically has observational data over a limited period of time and several parameters to estimate, such as growth rate, the basic reproduction number R₀, the case fatality rate and distributions of serial intervals, generation times, latency and incubation times and times between onset of symptoms, notification, death and recovery/discharge. These parameters form the basis for predicting a future outbreak, planning preventive measures and monitoring the progress of the disease outbreak. We study inference problems during the emerging phase of an outbreak, and point out potential sources of bias, with emphasis on: contact tracing backwards in time, replacing generation times by serial intervals, multiple potential infectors and censoring effects amplified by exponential growth. These biases directly affect the estimation of, for example, the generation time distribution and the case fatality rate, but can then propagate to other estimates such as R₀ and growth rate. We propose methods to remove or at least reduce bias using statistical modelling. We illustrate the theory by numerical examples and simulations.

Keywords: basic reproduction number; emerging epidemic; estimation bias; statistics.

Figure 1.

Initial stages of 10 simulated epidemics with R₀ = 1.7 and generation time G being Gamma distributed with mean = 15 days and standard deviation = 8.7 days, resulting in r = 0.03873. Cumulative incidence of notified cases over time in log-scale. Black line represents expected slope (line with equation y = rx). It is seen that incidence grows exponentially (linear on log-scale) but that there is a random time-shift before the epidemic takes off in the different simulations. As explained in the electronic supplementary material, simulations were continued until 4500 cumulated cases (=3.65 in log-scale) and then for further six weeks or until week 36. (Online version in colour.)

Figure 2.

Relationship between generation time G and serial interval S of an infector and its infectee. The infector is infected at time t₀ and then infects the infectee at t₁. The red circles indicate end of latent period and start of infectious period, the black circles indicate onset of symptoms, and black boxes end of infectious period (either by death or recovery). In the figure, the infectious period starts slightly before onset of symptoms, but, in general, the relationship between these event times is disease-dependent. In the illustration, the serial interval is shorter than the generation time, S < G, but the opposite relation could equally well happen. (Online version in colour.)