Intrinsic randomness in epidemic modelling beyond statistical uncertainty

Abstract

Uncertainty can be classified as either aleatoric (intrinsic randomness) or epistemic (imperfect knowledge of parameters). The majority of frameworks assessing infectious disease risk consider only epistemic uncertainty. We only ever observe a single epidemic, and therefore cannot empirically determine aleatoric uncertainty. Here, we characterise both epistemic and aleatoric uncertainty using a time-varying general branching process. Our framework explicitly decomposes aleatoric variance into mechanistic components, quantifying the contribution to uncertainty produced by each factor in the epidemic process, and how these contributions vary over time. The aleatoric variance of an outbreak is itself a renewal equation where past variance affects future variance. We find that, superspreading is not necessary for substantial uncertainty, and profound variation in outbreak size can occur even without overdispersion in the offspring distribution (i.e. the distribution of the number of secondary infections an infected person produces). Aleatoric forecasting uncertainty grows dynamically and rapidly, and so forecasting using only epistemic uncertainty is a significant underestimate. Therefore, failure to account for aleatoric uncertainty will ensure that policymakers are misled about the substantially higher true extent of potential risk. We demonstrate our method, and the extent to which potential risk is underestimated, using two historical examples.

Keywords: Applied mathematics; SARS-CoV-2; Statistics.

Fig. 1. Schematic of a time-varying general branching process.

a Shows schematics for the infectious period, an individual’s time-varying infectiousness (both functions of time post infection t^*), and the population-level mean rate of infection events. The infectious period is given by probability density function g. For each individual their (time-varying) infectiousness and rate of infection events are given by ν and ρ respectively. In an example (b), an individual is infected at time l, and infects three people (random variables K, purple dashed lines) at times l + K₁, l + K₂ and l + K₃. The times of these infections are given by a random variable with probability density function $~\frac{\rho \left(t\right)\nu \left(t-l\right)}{{\int }_l^t\rho \left(u\right)\nu \left(u-l\right)du}$. Each new infection then has its own infectious period and secondary infections (thinner coloured lines).

Fig. 2. Aleatoric uncertainty without overdispersed offspring distribution.

Plots show simulated epidemic where ρ(t) = 1.4 + sin(0.15t), with a Poisson offspring distribution. We use infectiousness ν ~ Gamma(3, 1), and infectious period g ~ Gamma(5,1). a Overlap between g and the infectiousness ν, where g controls when the infection ends e.g. by isolation. b Predicted mean and 95% aleatoric uncertainty intervals for prevalence. Note there is no epistemic uncertainty as the parameters are known exactly c Phase plane plot showing the mean plotting against the variance. d Proportional contribution to the variance from the individual terms in Eq. (9). Compounding uncertainty from past events is the dominant contributor to overall uncertainty.

Fig. 3. The 2003 SARS epidemic in Hong Kong^,.

a ρ(t) with 95% epistemic uncertainty. b Fitted incidence mean, 95% epistemic uncertainty with observational noise from using Eq. (4). Data is daily incidence of symptom onset. c Aleatoric uncertainty from the start of the epidemic under an optimistic and pessimistic ρ(t). d Epistemic (blue) and epistemic and aleatoric uncertainty (red) while keeping ρ constant at the forecast data (dotted line). Forecasting is from day 60.

Fig. 4. Early 2020 COVID-19 pandemic in the UK.

a shows a simulated epidemic using parameters available on March 16th 2020 (Table 1), for a plausible range of ρ = R₀ between 2 and 4. Blue bars indicate actual COVID-19 deaths, which we assume no knowledge of. The purple line is March 17th 2020, we set transmission to zero i.e. ρ = 0, to simulate an intervention that stops transmission completely. The grey envelope is the epistemic uncertainty and the red envelope the aleatoric uncertainty. b is the same as the top plot, except time is extended past March 17th with transmission being zero. Note aleatoric uncertainty is presented but is very close to zero.