Intrinsic Randomness in Epidemic Modelling Beyond Statistical Uncertainty Matthew J. Penn1. Daniel J. Laydon2 Joseph Penn1 Charles Whittaker2

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Intrinsic Randomness in Epidemic Modelling Beyond

Statistical Uncertainty

Matthew J. Penn1.*, Daniel J. Laydon2, Joseph Penn1, Charles Whittaker2,

Christian Morgenstern2, Oliver Ratmann2, Swapnil Mishra3, Mikko S. Pakkanen4,2,

Christl A. Donnelly1,2, and Samir Bhatt2,5.*

1University of Oxford, Oxford, UK

2Imperial College London, London, UK

3National University of Singapore, Singapore

4University of Waterloo, Ontario, Canada

5University of Copenhagen, Copenhagen, Denmark

∗corresponding authors: matthew.penn@st-annes.ox.ac.uk, s.bhatt@imperial.ac.uk

Abstract

Uncertainty can be classiﬁed as either aleatoric (intrinsic randomness) or epistemic (imper-

fect 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 explic-

itly 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

arXiv:2210.14221v2 [q-bio.PE] 8 Jun 2023

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.

Introduction

Infectious diseases remain a major cause of human mortality. Understanding their dynamics is

essential for forecasting cases, hospitalisations, and deaths, and to estimate the impact of interven-

tions. The sequence of infection events deﬁnes a particular epidemic trajectory – the outbreak –

from which we infer aggregate, population-level quantities. The mathematical link between individ-

ual events and aggregate population behaviour is key to inference and forecasting. The two most

common analytical frameworks for modelling aggregate data are susceptible-infected-recovered

(SIR) models [27] or renewal equation models [22, 40]. Under certain speciﬁc assumptions, these

frameworks are deterministic and equivalent to each other [11]. Several general stochastic analytical

frameworks exist [2, 40], and to ensure analytical tractability make strong simplifying assumptions

(e.g. Markov or Gaussian) regarding the probabilities of individual events that lead to emergent

aggregate behaviour.

We can classify uncertainty as either aleatoric (due to randomness) or epistemic (imprecise knowl-

edge of parameters) [29]. The study of uncertainty in infectious disease modelling has a rich history

in a range of disciplines, with many diﬀerent facets [9, 38, 44]. These frameworks commonly pro-

pose two general mechanisms to drive the infectious process. The ﬁrst is the infectiousness, which is

a probability distribution for how likely an infected individual is to infect someone else. The second

is the infectious period, i.e. how long a person remains infectious. The infectious period can also

be used to represent isolation, where a person might still be infectious but no longer infects others

and therefore is considered to have shortened their infectious period. Consider ﬁtting a renewal

equation to observed incidence data [40], where infectiousness is known but the rate of infection

events ρ(·) must be ﬁtted. The secondary infections produced by an infected individual will occur

randomly over their infectious period g, depending on their infectiousness ν. The population mean

rate of infection events is given by ρ(t), and we assume that this mean does not diﬀer between

individuals (although each individual has a diﬀerent random draw of their number of secondary

infections). In Bayesian settings, inference yields multiple posterior estimates for ρ, and therefore

multiple incidence values. This is epistemic uncertainty: any given value of ρcorresponds to a

single realisation of incidence. However, each posterior estimate of ρis in fact only the mean of

an underlying oﬀspring distribution (i.e. the distribution of the number of secondary infections an

infected person produces). If an epidemic governed by identical parameters were to happen again,

but with diﬀerent random draws of infection events, each realisation would be diﬀerent, thus giving

aleatoric uncertainty.

When performing inference, infectious disease models tend to consider epistemic uncertainty only

due to the diﬃculties in performing inference with aleatoric uncertainty (e.g. individual-based

models) or analytical tractability. There are many exceptions such as the susceptible-infected-

recovered model, which has stochastic variants that are capable of determining aleatoric uncertainty

[2] and have been used in extensive applications (e.g. [42]). However, we will show that this model

can underestimate uncertainty under certain conditions. An empirical alternative is to characterise

aleatoric uncertainty by the ﬁnal epidemic size from multiple historical outbreaks [12, 49] but

these are confounded by temporal, cultural, epidemiological, and biological context, and therefore

parameters vary between each outbreak. Here, following previous approaches [2], we analyse

aleatoric uncertainty by studying an epidemiologically-motivated stochastic process, serving as a

proxy for repeated realisations of an epidemic. Within our framework, we ﬁnd that using epistemic

uncertainty alone is a vast underestimate, and accounting for aleatoric uncertainty shows potential

risk to be much higher. We demonstrate our method using two historical examples: ﬁrstly the

2003 severe acute respiratory syndrome (SARS) outbreak in Hong Kong, and secondly the early

2020 UK COVID-19 epidemic.

Results

An analytical framework for aleatoric uncertainty

A time-varying general branching processes proceeds as follows: ﬁrst, an individual is infected,

and their infectious period is distributed with probability density function g(with corresponding

cumulative distribution function G). Second, while infectious, individuals randomly infect others

(via a counting process with independent increments), driven by their infectiousness νand a rate

of infection events ρ. That is, an individual infected at time l, will, at some later time while

still infectious t, generate secondary infections at a rate ρ(t)ν(t−l). ρ(t) is a population-level

parameter closely related to the time-varying reproduction number R(t) (see Methods and [40]

for further details), while ν(t−l) captures the individual’s current infectiousness (note that t−l

is the time since infection). We allow multiple infection events to occur simultaneously, and

assume individuals behave independently once infected, thus allowing mathematical tractability

[24]. Brieﬂy, we model an individual’s secondary infections using a stochastic counting process,

which gives rise to secondary infections (i.e. oﬀspring) that are either Poisson or Negative Binomial

distributed in their number, and Poisson distributed in their timing (see Supplementary Notes 3.3

and 3.4). We study the aggregate of these events (prevalence or incidence) through closed-form

probability generating functions and probability mass functions. Our approach models epidemic

evolution through intuitive individual-level characteristics while retaining analytical tractability.

Importantly, the mean of our process follows a renewal equation [1, 40, 41]. Our formulation

uniﬁes mechanistic and individual-based modelling within a single analytical framework based

on branching processes. Figure 1 shows a schematic of this process. Formal derivation is in

Supplementary Note 3.

Infection duration Infectiousness Rate of transmission

Time of infection (l) Time of infection (l) Calendar time (t)

g(⋅ )

ν(⋅)

ρ(⋅)

l+K1

l+K2

l+K3

Time

Density

Rate

End

Start

Figure 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 νand ρrespectively. 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 ∼ρ(t)ν(t−l)

lρ(u)ν(u−l)du . Each new infection then has its own infectious period and secondary

infections (thinner coloured lines).

Randomness occurs at individual level, and there is a distribution of possible realisations of the epi-

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IntrinsicRandomnessinEpidemicModellingBeyondStatisticalUncertaintyMatthewJ.Penn1.*,DanielJ.Laydon2,JosephPenn1,CharlesWhittaker2,ChristianMorgenstern2,OliverRatmann2,SwapnilMishra3,MikkoS.Pakkanen4,2,ChristlA.Donnelly1,2,andSamirBhatt2,5.*1UniversityofOxford,Oxford,UK2ImperialCollegeLondon,London,UK...

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Intrinsic Randomness in Epidemic Modelling Beyond Statistical Uncertainty Matthew J. Penn1. Daniel J. Laydon2 Joseph Penn1 Charles Whittaker2

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