Estimating fine age structure and time trends in human contact patterns from coarse contact data the Bayesian rate consistency model

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Estimating fine age structure and time trends in human

contact patterns from coarse contact data: the Bayesian rate

consistency model

Shozen Dan1†Y*, Yu Chen1†Y, Yining Chen1, Melodie Monod1, Veronika K Jaeger2,

Samir Bhatt3,4, Andr´e Karch2, Oliver Ratmann1Y*on behalf of the Machine Learning

& Global Health network

1Department of Mathematics, Imperial College London, England, United Kingdom

2Institute of Epidemiology and Social Medicine, University of M¨unster, Germany

3School of Public Health, Imperial College London, England, United Kingdom

4Department of Public Health, University of Copenhagen, Denmark

†Joint first authors.

YThese authors contributed equally to this work.

¤Current Address: Imperial College London, Exhibition Road, London SW7 2AZ,

United Kingdom

* Corresponding author: Shozen Dan, shozen.dan21@imperial.ac.uk, Oliver Ratmann,

oliver.ratmann@imperial.ac.uk

Abstract

Since the emergence of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2),

many contact surveys have been conducted to measure the fundamental changes in

human interactions that occurred in the face of the pandemic and non-pharmaceutical

interventions. These surveys were typically conducted longitudinally, using protocols

that have important differences from those used in the pre-pandemic era. Here, we

present a model-based statistical approach that can reconstruct contact patterns at

1-year resolution even when the age of the contacts is reported coarsely by 5 or 10-year

age bands. This innovation is rooted in population-level consistency constraints in how

contacts between groups must add up, which prompts us to call the approach presented

here the Bayesian rate consistency model. The model also incorporates computationally

efficient Hilbert Space Gaussian process priors to infer the dynamics in age- and

gender-structured social contacts, and is designed to adjust for reporting fatigue

emerging in longitudinal surveys.

On simulations, we show that social contact patterns by gender and 1-year age

interval can indeed be reconstructed with adequate accuracy from coarsely reported

data and within a fully Bayesian framework to quantify uncertainty. We then

investigate the patterns and dynamics of social contact data collected in Germany from

April to June 2020 across five longitudinal survey waves. We estimate the fine age

structure in social contacts during the early stages of the pandemic and demonstrate

that social contact intensities rebounded in a structured, non-homogeneous manner. We

also show that by July 2020, social contact intensities remained well below

pre-pandemic values despite a considerable easing of non-pharmaceutical interventions.

The Bayesian rate consistency model provides a modern, non-parametric,

computationally tractable approach for estimating the fine structure and longitudinal

trends in social contacts, and is readily applicable to contemporary survey data as long

as the exact age of survey participants is reported.

October 21, 2022 1/40

arXiv:2210.11358v1 [stat.AP] 20 Oct 2022

Author summary

The transmission of respiratory infectious diseases occurs during close social contacts.

Hence, characterising social contact patterns within a population, encoded in contact

matrices, leads to a better understanding of disease spread. Contact matrices also

parameterise mathematical models, which played a pivotal role in informing health

policy during the coronavirus disease 2019 (COVID-19) pandemic. Unlike pre-pandemic

surveys, which recorded contacts’ age in one-year age intervals, COVID-era studies

recorded contacts’ age in discrete large age categories to facilitate reporting. Some

studies allowed participants to report an estimate for the total number of contacts for

which they could not remember age and gender information. Many studies were

partially longitudinal, which introduced the issue of reporting fatigue. Thus, directly

applying existing statistical methods for estimating social contact matrices may result

in a loss of age detail and confounded estimates. To this end, we develop a longitudinal

model-based approach which estimates fine-age contact patterns from coarse-age data

that also adjusts for the confounding effects of aggregate contact reporting and

reporting fatigue in a unified manner. We apply our approach to the COVIMOD study

to provide a detailed picture of how social contact dynamics evolved during the first

wave of COVID-19 in Germany.

Introduction

The transmission of human respiratory diseases such as influenza, tuberculosis, and

COVID-19 is directly driven by the rate of close social contact between individuals.

Social contact studies such as the pivotal POLYMOD study [1] have been widely

acknowledged as an effective method of obtaining social contact estimates to assess

infection risk and to parameterise mathematical infectious disease models [2–4].

Consequently, they provide critical epidemiological insights which inform the

implementation and evaluation of non-pharmaceutical interventions [5] as well as public

health policies such as vaccination schedules [6].

Since the outbreak of COVID-19, numerous contact studies have been conducted in

Europe and around the world, providing indispensable information on the evolving

patterns of human mixing behaviour during the pandemic [7, 8]. In Germany, the

COVIMOD study collected social contact data for close to two years, and initial

analyses [9] focused on data from April to June 2020 during the first partial lockdown in

Germany to quantify the scale of social contact reductions relative to pre-pandemic

contact patterns. High-resolution estimates of age- and gender-specific human contact

patterns and their trends over time are substantially more difficult to obtain, due to

limitations in available inference methods [2, 10, 11].

First, COVIMOD and other COVID-era social contact studies record the age of

contacts by large age categories of 5 to 10 years, reflecting that often it is challenging

for study participants to know the exact age of their contacts. In contrast, the

pre-pandemic POLYMOD surveys collected data on the exact age of contacts and

subsequently developed methods including thin-plate regressions [2] and Gaussian

Markov Random Field model approaches [10] relied on such data to estimate

high-resolution contact patterns. The bootstrap approach implemented in the

socialmixr library [11] provides a convenient and speedy way to estimate contact

matrices. But, it only provides contact estimates in discrete large age categories, which

is unsatisfactory as it may mask subtle but important age effects [12].

Second, most COVID-era studies adopted retrospective web-based survey protocols

and conducted longitudinal repeat surveys [7, 13

–

15]. Importantly, the survey waves are

typically inter-dependent because a varying number of participants were surveyed in

October 21, 2022 2/40

multiple waves, and additional participants were recruited to replenish the cohort size.

While this approach provides valuable longitudinal data, it also introduces the issue of

reporting fatigue, where participants tend to report fewer contacts in subsequent

participation due to becoming tired of filling out the survey. It follows that directly

applying existing methods, which do not incorporate adjustments to counter the

confounding reporting fatigue effects, is bound to lead to incorrect estimates.

Additionally, participants in the COVID-era surveys sometimes found it difficult to

recall specific age and gender information for all of their contacts. Instead, they were

allowed to report an estimate for the total number of contacts on that occasion [9],

which again may result in under-ascertainment of contact intensities if these data are

not accounted for in inference approaches of contact patterns.

In this work, we present a temporal Bayesian model to infer age- and gender-specific

contact patterns and trends at high 1-year resolution from longitudinal survey data.

The primary innovation of the model is the ability to infer contact patterns by 1-year

age bands even when the age contacts are reported in broad age categories. We call this

model-based approach the Bayesian rate consistency model for reasons that will be clear

soon. In addition, we use recently developed Hilbert Space Gaussian Process

approximations [16] to gain substantial advances in computational efficiency, which in

turn enable us to make full Bayesian inferences over time and uncover the dynamics in

social contact structure. We demonstrate that it is crucial to model contact patterns

over time to account for reporting fatigue effects in inter-dependent longitudinal survey

waves. The primary purpose behind developing the Bayesian rate consistency model is

its application to contemporary COVID19-era survey data, which we present for data

spanning the first five survey waves of the COVIMOD study in Germany. We present

high-resolution estimates of age- and gender-specific social contacts for each survey

wave and describe their time evolution. We also place the inferred contact dynamics

into a pre-pandemic context and quantify the differences in contact intensity change by

the age of contacts.

Methods

The COVIMOD study

The COVIMOD study was launched in April 2020 and continued until December 2021,

constituting 33 survey waves. Participants were recruited through email invitations to

existing panel members of the online market research platform IPSOS i-say [17]. To

ensure the sample’s broad representativeness of the German population, quota sampling

was conducted based on age, gender, and region. Participants were invited to

participate in multiple waves to track changes in social behaviour and attitudes toward

COVID-19. When the participant size did not meet the sampling quota due to study

withdrawals, new participants were recruited into the study. This approach enabled

COVIMOD study to obtain longitudinal samples, but it also introduced the issue of

response fatigue, where the number of detailed contacts reported decreased compared to

previous participation, irrespective of the survey wave. To procure information on

children, a subgroup of adult participants living with children under the age of 18 were

selected to be proxies. This procedure meant that middle-aged adults were

under-sampled as they completed the survey on behalf of their children.

The COVIMOD questionnaire was based on the CoMix study and includes questions

on demographics, the presence of a household member belonging to a high-risk group,

attitudes towards COVID-19 as well as related government measures, and current

preventative behaviors [9,18]. Participants were also asked to provide information about

their social contacts between 5 a.m. the preceding day to 5 a.m. the day of answering

October 21, 2022 3/40

the survey. Following the pre-pandemic POLYMOD study, a contact is defined as either

a skin-to-skin contact such as a kiss or a handshake (physical contact) or an exchange of

words in the presence of another person (non-physical contacts) [1]. Participants were

asked to report the age group, gender, relation, the contact setting (e.g. home, school,

workplace, place of entertainment, etc.), and whether the contact was a household

member. For survey waves 1 and 2, participants were asked to provide each contact’s

information separately. However, some participants reported contacts to groups of

individuals (e.g., customers, clients) for which a specific number of contacts was

assumed (Additional file 2 of [9]). From wave 3 onwards, participants were given the

possibility to record group contacts in addition to the recording of individual contacts.

Additionally, some participants could not recall or preferred not to answer the age or

gender information of some individual contacts. We treat these three types of entries

with missing age or gender equally and refer to them as missing &aggregate contact

reports. A copy of the COVIMOD questionnaire may be found in Additional file 1 of [9].

COVIMOD was approved by the ethics committee of the Medical Board

Westfalen-Lippe and the University of M¨unster, reference number 2020-473-f-s.

This current work concerns the first five survey waves of the COVIMOD study. In

Fig 1A and B, we show the sampling periods with the number of daily COVID-19 cases,

cumulative COVID-19-related deaths, and the OxCGRT Stringency Index [19]. The

following COVID-19 policy timeline is obtained from the ACAPS COVID-19

Government Measures dataset [20]. The first COVIMOD survey was administered from

April 30

to May 6

in year 2020, towards the end of the first partial lockdown and the

first wave of cases. Before the beginning of the first survey (April 20th), small stores,

auto dealers, and bookstores were allowed to reopen under strict hygiene regulations.

During the final few days of the survey period (May 4th to 6th), phase-out measures

were announced by the government, including the step-wise uptake of schools, the

reopening of hairdressers under strict hygiene regulations, lifting of the ban on public

gatherings of 30 people indoors and 50 outdoors, resumption of religious services, and

reopening of public services such as museums, botanical gardens, zoos, and playgrounds.

The second wave of the COVIMOD survey was administered from May 14th to May

. During this period, additional phase-out measures were announced, including the

resumption of all cross-country transport and the reopening of hotels and restaurants.

International travel to neighbouring countries was also slightly relaxed during this

period. The third, fourth, and fifth waves of COVIMOD surveys were taken from May

28th to July 4th, June 11th-22nd, and June 26th to July 1st, respectively. There was no

notable introduction or reduction of social contact restriction measures during this time,

but international travel restrictions were relaxed primarily for Schengen and EU

countries. COVID-19 cases and deaths remained stable during this period (Fig 1).

After excluding participants who prefer not to provide age or gender information

and 25 participants above the age of 84, there were 1549, 1345, 1076, 1881, and 1603

participants for waves 1 to 5. We observed 3244, 4852, 6344, 13471, and 8353 total

contacts for each wave. In Fig 1C, we show the proportion of participants who

consented to the survey multiple times. Most participants in waves 2 and 3 had

participated in wave 1, with only 6.8% and 16% of participants being new to the survey.

The proportion dropped sharply in wave 4, where only 35.1% of initial participants

remained. Hence the majority (57.7%) of wave 4 participants were first-time

participants. On the contrary, no new participants were enrolled for wave 5, and

individuals who participated for the second, third, and fifth time took up approximately

45%, 10.7%, 9.6%, and 34.6% of the sample.

October 21, 2022 4/40

Fig 1. Timeline and participants of the longitudinal COVIMOD study A. Daily COVID-19 case

counts in Germany (red bars), the OxCGRT Stringency Index (blue line), and COVIMOD survey administration

periods (grey ribbons). B. Cumulative COVID-19-related deaths in Germany (red line), the OxCGRT Stringency

Index (blue line), and COVIMOD survey administration periods (grey ribbons). C. Sample sizes and the

proportion of people repeatedly sampled in the COVIMOD survey for which zero repeats indicate first-time

participants.

Data processing

Following ethical guidelines, the participant age information for children is reported in

discrete categories, i.e., 0−4, 5−9, 10−14, 15−18. To obtain fine-age information for

participants under 18, we imputed their age by drawing from a discrete uniform

distribution with bounds set as the minimum and maximum age of the participant’s age

category. We excluded 20 participants (0.3% of the total) without age or gender

information as we could not estimate the information accurately. For total contact

counts Ygh

ab between all participants aged a∈ {0,1,...,84}of gender g∈ {M, F }and

contacted individuals aged

b∈ {

,...,

}

of gender

h∈ {M, F }

, we filled in missing

entries with zeroes if a participant of age aand gender gis present. If not, we treated

the entries as missing. We truncated group contacts at 60 (90th percentile of group

contacts) to remove the effects of extreme outliers. We show the distribution of

observed contacts in S1 Fig.

October 21, 2022 5/40

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Estimatingfineagestructureandtimetrendsinhumancontactpatternsfromcoarsecontactdata:theBayesianrateconsistencymodelShozenDan1yY*,YuChen1yY,YiningChen1,MelodieMonod1,VeronikaKJaeger2,SamirBhatt3,4,AndreKarch2,OliverRatmann1Y*onbehalfoftheMachineLearning&GlobalHealthnetwork1DepartmentofMathematics,Imp...

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Estimating fine age structure and time trends in human contact patterns from coarse contact data the Bayesian rate consistency model

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