Document [original]

RESEARCH ARTICLE

Predicting the effects of COVID-19 related

interventions in urban settings by combining

activity-based modelling, agent-based

simulation, and mobile phone data

Sebastian A. Mu¨llerID

, Michael Balmer

, William CharltonID

, Ricardo EwertID

Andreas NeumannID

, Christian Rakow

, Tilmann SchlentherID

, Kai NagelID

1Transport Systems Planning and Transport Telematics, TU Berlin, Berlin, Germany, 2Senozon AG, Zu¨rich,

Switzerland, 3Senozon GmbH, Berlin, Germany

*kai.nagel@tu-berlin.de

Abstract

Epidemiological simulations as a method are used to better understand and predict the

spreading of infectious diseases, for example of COVID-19. This paper presents an

approach that combines a well-established approach from transportation modelling that

uses person-centric data-driven human mobility modelling with a mechanistic infection

model and a person-centric disease progression model. The model includes the conse-

quences of different room sizes, air exchange rates, disease import, changed activity partici-

pation rates over time (coming from mobility data), masks, indoors vs. outdoors leisure

activities, and of contact tracing. It is validated against the infection dynamics in Berlin (Ger-

many). The model can be used to understand the contributions of different activity types to

the infection dynamics over time. It predicts the effects of contact reductions, school clo-

sures/vacations, masks, or the effect of moving leisure activities from outdoors to indoors in

fall, and is thus able to quantitatively predict the consequences of interventions. It is shown

that these effects are best given as additive changes of the reproduction number R. The

model also explains why contact reductions have decreasing marginal returns, i.e. the first

50% of contact reductions have considerably more effect than the second 50%. Our work

shows that is is possible to build detailed epidemiological simulations from microscopic

mobility models relatively quickly. They can be used to investigate mechanical aspects of

the dynamics, such as the transmission from political decisions via human behavior to infec-

tions, consequences of different lockdown measures, or consequences of wearing masks in

certain situations. The results can be used to inform political decisions.

Introduction

When COVID-19 took hold in Germany in February 2020, there was an urgent need for a dif-

ferentiated modelling capability to predict the consequences of interventions. We used our

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OPEN ACCESS

Citation: Mu¨ller SA, Balmer M, Charlton W, Ewert

R, Neumann A, Rakow C, et al. (2021) Predicting

the effects of COVID-19 related interventions in

urban settings by combining activity-based

modelling, agent-based simulation, and mobile

phone data. PLoS ONE 16(10): e0259037. https://

doi.org/10.1371/journal.pone.0259037

Editor: Itzhak Benenson, Tel Aviv University,

ISRAEL

Received: April 7, 2021

Accepted: October 12, 2021

Published: October 28, 2021

Peer Review History: PLOS recognizes the

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https://doi.org/10.1371/journal.pone.0259037

access article distributed under the terms of the

Creative Commons Attribution License, which

permits unrestricted use, distribution, and

reproduction in any medium, provided the original

author and source are credited.

Data Availability Statement: For computer code

see https://github.com/matsim-org/matsim-episim.

Simulations were computed with version

experience with person-centric modelling of traffic [1] to build a first prototype within two

weeks [2]. An advantage of using this starting point is that the whereabouts of all simulated

persons, including their overlapping time spent at facilities or in (public transport) vehicles,

are already given by the model, which is derived in part from mobile phone data. Since the

input data contains age as an attribute of each synthetic person, it was straightforward to

include agent-dependent disease progression into the model from the start. A short descrip-

tion of the different model variants over time is provided in S2 Text.

The model is regularly used to advise the German federal government (e.g. [3,4]). The

main contribution of those reports was and is to provide differentiated predictions of the influ-

ence of various interventions, such as reductions of activity participation, masks, or vaccina-

tions. For the present paper, we show the contributions of different activity types to the

infection dynamics as predicted by the model. We show how most activity types generate over

time fairly constant contributions to the reproduction number R, independent from the actual

level of R. In consequence it is structurally more stable to report reductions of Rcaused by

interventions as an additive term, rather than a term that is relative to the overall level of Ras

is usually done (e.g. [5]). The model also explains why there are decreasing marginal returns to

stay-at-home interventions [6]. Finally, the model makes a prediction concerning the magni-

tude of the difference between summer and winter, caused by moving activities indoors during

winter.

Related work

Compartmental models

The general dynamics of virus spreading are captured by compartmental models, most

famously the so-called SIR model, with S=susceptible,I=infected/infectious, and R=recovered

[7]. Every time a susceptible and an infectious person meet, there is a probability that the sus-

ceptible person becomes infected. Some time after the infection, the person typically recovers.

Variants include, e.g., an exposed (but not yet infectious) compartment between Sand I.

Instead of running these models with compartments, one can run them on a graph [8,9].

Persons are represented as vertices, connections between persons are denoted as edges. The

random interactions that are implied by the compartmental models are then replaced by inter-

actions with graph neighbors.

In reality, these interactions change from day to day; in particular, possible superspreading

events like weddings or other large gatherings cannot be encoded in a static graph. For this,

temporal networks have been investigated ([9], section VIII).

An advantage of compartmental models is that their runtime is independent from system

size; in that way, it is easily possible to run a model for a country or a continent. A disadvan-

tage is that one needs a separate compartment for each attribute combination (e.g.

age ×activity pattern ×disease state), and that mechanical aspects such as the reduction of

virus intake by masks, are difficult to include into the model. A special case is [10]: It treats

each census block as a subpopulation, computes how virus travels from one census block to

another via points-of-interest with visitors from both census blocks, and also has internal virus

dynamics in each census block. The differences to our work are discussed in more detail in S5

Text.

Person-centric epidemiological modelling

An alternative to compartmental models is to use synthetic persons as the starting point for

modelling, and to “consider nodes as entities where multiple individuals or particles can be

located and eventually wander by moving along the links connecting the nodes” [9]. Examples

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d16656f076640124de0361fc327d3803a80aa466

of the code, started with command java -jar

matsim-episim-1.0-SNAPSHOT.jar runParallel

–setup org.matsim.run.batch.BerlinSensitivityRuns

–params org.matsim.run.batch.

BerlinSensitivityRuns$Params. The input data

(including the synthetic mobility traces) are made

public here: https://doi.org/10.14279/depositonce-

11495. The output data used for the figures can be

retrieved at: http://dx.doi.org/10.14279/

depositonce-12113.

Funding: The work on the paper was funded by the

Ministry of research and education (BMBF)

Germany (01KX2022A) and TU Berlin. The BMBF

Grant also funded the data provided by the

commercial company senozon. MB and AN,

employed by senozon, worked together with the

rest of the team to iterate between input data and

simulations until the input data contained all

information needed to run the simulation. senozon

provided support in the form of salaries for MB and

AN, but did not have any additional role in the study

design, data collection and analysis, decision to

publish, or preparation of the manuscript. The

specific roles of these authors are articulated in the

‘author contributions’ section.

Competing interests: MB and KN own shares of

the commercial company senozon. MB and AN are

employees of senozon. None of this alters our

adherence to PLOS ONE policies on sharing data

and materials.

of such models can be found since approximately 2004 [11–14]. A model of this type by Impe-

rial College [15] had a large impact on policy in the UK. Other recent developments are [16,

17] on the global scale, or [18,19] on the urban scale. These models typically follow individual

synthetic persons. However, most of them, with the possible exception of the Virginia Biotech-

nology Institute model [12,20], have explicit person movements only for commute patterns;

all other infections are assumed to be in a local environment.

Aleta et al. [21] construct an agent-based model, similar to ours. Their data derives from

persons specifically recruited to collect their long-term trajectories. They have long trajecto-

ries, with high spatial precision, but for only 2% of the population. This is still an impressive

sample; however, with our work we aim for models where we have as many synthetic persons

in the model as there are persons in reality.

A special case is by Kucharski et al. [22], who use a pre-existing dataset with recorded social

contacts for 40 162 participants. This is close to our approach in that the persons who encoun-

ter each other for how long and in which context are microscopically specified. Differences

include that it is not a model for the full population of a region, and the study does not trace

behavioral changes throughout the pandemic.

Daily activity trajectories

Using daily activity chains as the basis for transport modelling is an established approach in

the transport modelling community. An activity chain is a sequence of activities of a person,

where activities have types such as home, work, shop, etc., starting and ending times, and loca-

tions. There are several ways to generate such activity chains, for example by using activity-

based demand generation models (e.g. [23,24]), by taking them from travel diaries (e.g. [25,

26]), by using mobile phone data (e.g. [27]), or by data fusion from open access data sources

(e.g. [28]).

In the present situation, we needed a technology that was readily available, allowed uniform

rollout at least in Germany, and that would allow updates along with changes in mobility

behavior during the unfolding of the COVID epidemics. For that reason, we used an estab-

lished process that generates activity chains mostly from mobile phone data [27]. The process

is described in more detail in S1 Text. The outcome of the process are activity chains, encoded

as events (cf. Fig 1), for as many synthetic persons as Germany has inhabitants. Since the activ-

ity chains stem from transport modelling, they also contain knowledge about trips between

activities, importantly trips by public transport, and in consequence also contain, for each syn-

thetic person, events when they enter or leave certain public transit vehicles.

Person-centric epidemiological models derived from transport simulations

From the section on person-centric epidemiological modelling above, one takes away that hav-

ing person trajectories, and in particular where persons meet, would be useful for an epidemio-

logical simulation. In consequence, the synthetic person trajectories from transport modelling

explained above seem like a good starting point, since they are already available. Smieszek

et al. [29,30] and Hackl and Dubernet [31] construct epidemiological models on top of such

pre-existing synthetic person trajectories; these are the main starting point for us. Najmi et al.

[32] start from a person-centric transportation planning model for Sydney, and add a disease

transmission model that computes possible infections based on co-locations during the simu-

lated day. The approach is similar to ours, but does not use mobile phone data to track the

actual mobility behavior. They also do not use an infection model that depends on the spatial

situation of the activity type.

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The model described in the present paper has always been open source, and earlier version

have been described in preprints [2,33,34]. This has been picked up by by Manout and Ciari

[35] for Montreal, and by Bossert et al. [36] for South Africa. It was also used as the “micro”

part by [37].

From reductions of mobility behavior to reductions of infections

There are many data sets that track or analyze mobility changes “during Corona” [38–43].

This is, however, not our primary focus; rather, we are interested in how the infection dynam-

ics can be better understood and possibly predicted with the help of mobility and other data. A

possible approach to achieve this is data mining [44,45]. We are, however, interested in mod-

els with more detail.

Jia et al. [46] and Xiong et al. [47] look at how long distance travel influences the disease

import; they find that a high inflow from areas with high incidences is positively correlated

Fig 1. Events for travel. TOP: By individual vehicle. BOTTOM: By public transport. Source: [1].

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with high infection numbers. They do not, however, look at disease spread within the urban

fabric, driven by daily movement patterns.

Fairly close to our work are Chang et al. [10], already mentioned earlier. They first con-

struct, based on mobile phone data, a mobility network between census block groups and

points of interest based on mobile phone data, and then use that model to investigate reopen-

ing strategies. They have a very detailed resolution of the facilities (they differentiate, e.g.,

between full-service restaurants, limited-service restaurants, and cafes/snack bars), but on the

other hand they do not simulate individual synthetic persons. Similarities and differences are

discussed in S5 Text.

Model details

Important sub-models of agent-based epidemics models are: contact model, infection model,

and disease progression model. These are described in more detail in the following sections.

Mobility model and resulting contact model

As stated, we take the synthetic persons and their movements from transport modelling, cf.

Fig 1. For the present study, the data is generated by a synthetic method developed by Senozon,

see S1 Text. We have used and are using the same data for other projects [48–51]. From these

activity chains, we extract how much time people spend with other people at activities or in

(public transport) vehicles. That is, infection opportunities are directly taken from the input

data. Details, for example of how multiple days or weekends are modelled, are provided in Sec

1 of S1 Appendix.

Infection model

Once two persons are identified to have contact, and one of them is contagious and the other

is susceptible, there is a probability of an infection. For this, we use the mechanical model by

Smieszek [45]: infected persons generate a “viral load” that they exhale, cough or sneeze into

the environment, and people close by are exposed. Overall, the probability for person nto

become infected by this process in a time step tis described as

pðinfectjcontactÞn;t¼1exp YX

shm;t�cinm;t�inn;t�tnm;t

! ð1Þ

where mgoes over all other persons with which the person has contact at time t,sh is the shed-

ding rate (*microbial load), ci the contact intensity, in the intake (reduced, e.g., by a mask), τ

the duration of interaction between the two individuals, and Θa calibration parameter. The

model of Smieszek has the advantage that it was specifically developed with our transport sim-

ulation in mind, but there are many models of the same type (e.g. [52,53]).

For small values of the exponent and just one contagious person in the room, one can

approximate Eq (1) as

pðinfectjcontactÞ � Y�sh �ci �in �t:ð2Þ

We do not use this approximation in our computer implementation, but it helps understand-

ing the following arguments. Fig 2 gives some intuition about when that approximation holds;

evidently, the effect of Eq (1) is to saturate when the infection probability becomes large.

All parameters can be given in arbitrary units as long as those units are always the same

since the units are absorbed by Θ. If one wanted to use physical units, then one could

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decompose sh ¼~

sh �out with ~

sh the, say, number of shedded virus particles per time. out (for

a mask on the shedding side), ci, and in would be correction factors, i.e. 1 for a standard situa-

tion. τwould denote the time duration, so that the result would be the number Vof virus parti-

cles that were inhaled during that time duration. Θ, or more precisely 1 −exp(−ΘV), would

translate that number of virus particles into an infection probability. If that translation was

known, one could attempt to calibrate the model from first principles. Practically, we use Θas

our main calibration parameter.

Contact intensities. For SARS-CoV-2, it is plausible to assume that a large share of the

virus material is shed as aerosol [54]. In consequence, the first relevant term to compute the

viral concentration in the air is the shedding rate, sh.

For such aerosols, it is plausible to assume that they mix quickly into the room, leading to

the same uniform concentration everywhere [55]. Evidently, that concentration is inversely

proportional to room size: if the room is twice as large, the resulting concentration is half as

large.

Next, air exchange plays a role [55]. One could, for example, assume that the windows are

opened once per hour, and all of the air is replaced with outside air. This would correspond to

an air exchange rate of 1/h. If one assumes a constant rate of virus emission, there would be a

linear increase of concentration up to the opening of the window, after which the virus con-

centration in the air would quickly go towards zero. The average virus concentration over this

process would be half as much as the maximum concentration just before window opening. In

consequence, the resulting average concentration is inversely proportional to the air exchange

rate: If the air is exchanged twice as often, the resulting average virus concentration is half as

large. This also holds for continuous air exchange, e.g. by mechanical means.

All of the above together replaces Eq 2 by

pðinfectjcontactÞ � Y�sh �in

rs �ae �t;ð3Þ

where rs is the size of the room, and ae is the air exchange rate. That is, it sets the contact inten-

sity ci from Eq (1) to

ci ¼1

rs �ae :ð4Þ

Fig 2. How duration (in hours) translates into the infection probability for two different contact intensities. The

linear approximation of Eq (2) is given for either curve by a dashed line.

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Again, the physical units are absorbed into Θ; note, however, that the air exchange rate ae is

defined as the frequency of exchanging the air of the full room, and not of, say, cubic meters.

Aspects such as loudness of speech or if persons perform a physical activity are not taken

into account in the present model although they are known to play important roles [56,57],

and they could, to some extent, be attached to the activity types. It is planned to include them

in a future version of the model.

Estimation of room sizes. As stated above, our data resolves down to the level of “facili-

ties”. These correspond roughly to buildings. In consequence, such a facility can be anything

from a single family home to a large office building or a sports arena. Since these facilities are

too large compared to typical rooms, we divide facilities into N

spaces

rooms. N

spaces

is set to 20;

the argument for this number is given in Sec 1 of S1 Appendix.

Since our simulation tracks when persons are at facilities, we can, for each facility, obtain

the maximum number of persons at that facility, NpersonsAtFacility

max , over the day. In addition, one

can obtain typical floor space per person, fs, from regulatory norms and other sources (see

Table 1). This leads to

facilityFloorSpace ¼NpersonsAtFacility

max �fs :ð5Þ

Divided by N

spacesPerFacility

, this leads for the room size to

rs ¼facilityFloorSpace

NspacesPerFacility ¼NpersonsAtFacility

max

NspacesPerFacility �fs ≕roomCapacity �fs ð6Þ

where roomCapacity is the maximum number of persons that are in the room during the day

(thus its “capacity”); note that N

spacesPerFacility

= 1 for home activities (cf. Sec 1 of S1 Appendix).

Air exchange rate and normalized contact intensities. Inserting Eq (6) into (4) results in

ci ¼1

roomCapacity �fs �ae ≕1

roomCapacity �ci0ð7Þ

Table 1. Normalized contact intensities ci0, relative to the contact intensity at home, ci0

home.

activity type area per person fs

]

air exchange rate old bldg ae

old

[1/

air exchange rate new bldg ae

new

[1/

share old

buildings

resulting ci0=ci0

home

home [58,59] 22 0.5 0.5 1

schools and day care

[60]

2 0.5 0.5 100% 11

universities 4 0.5 0.5 100% 5.5

public transport [61,

62]

0.33 2.0 10.0 50% 10

leisure [63] 1.25 0.5 10.0 50% 9.24

shop 10 0.5 1.5 10% 0.88

work [64–66] 10 0.5 1.5 50% 1.47

errands 10 0.5 1.5 50% 1.47

business 10 0.5 1.5 50% 1.47

Both the floor area per person and the air exchange rate come from building manuals or similar standards; note the given references in the table. The share of old

buildings/vehicles is an estimate. Universities are assumed to have twice as much space per student as schools. Shop, errands, and business are assumed to follow the

same characteristics as work. The contact intensities are computed separately for old and new buildings, and then averaged according to the assumed share of old

buildings.

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with the “normalized” contact intensity

ci0¼1

fs �ae :ð8Þ

See Table 1 for values of ci0.

ci0parameterizes the “closeness” of the interaction. This is, cf. Eq (7), divided by roomCapa-

city, which denotes the number of persons that fit into the room given typical usage. If we

share a room with one infectious other person, then our probability to become infected is, all

other things being equal, half as large if the room is twice as large. However, if the room is

twice as large, then there will presumably also be twice as many persons in it, doubling our

own risk, and thus in the average cancelling out the effect of the larger room size. This second

effect is computed directly by our contact model (Sec. From reductions of mobility behavior to

reductions of infections above), and thus does not have to be included into the conditional

infection probability. This has the additional advantage that if a person is in large container

outside its peak usage, the model will calculate a much reduced infection probability. Examples

for this are public transport vehicles, premises for large events, or restaurants.

A side effect of this model is that the above division by N

spacesPerFacility

has no effect in first

order. If npersons at the facility are all in one room, and one contagious person is added,

the expected number of newly infected persons is n�p, where pis the individual probability

to become infected according to Eq (1) with Eq (7). If the npersons are divided between

spacesPerFacility

, and one contagious person is added into one room, the expected number of

newly infected persons is (n/N

spacesPerFacility

)�p0, where p0in first order is p�N

spacesPerFacility

because roomCapacity in Eq (7) is divided by N

spacesPerFacility

. In consequence, in first order the

expected number of newly infected persons in the divided facility is the same, np, as in the

undivided facility.—Second-order corrections come from the fact that Eq (1) eventually satu-

rates when infection probabilities become large—then the smaller room sizes reduce the num-

ber of infections.

Masks. The effectiveness of different mask types is taken from from [67], i.e. cloth masks

reduce shedding and intake to 0.6 and 0.5 of their original values, surgical masks to 0.3 and

0.3, and N95 (FFP2) masks to 0.15 and 0.025. For some discussion of these values see Masks.

Children. Current research implies that the susceptibility and infectivity are reduced for

children compared to adults. We model this by including the susceptibility and infectivity into

Eq (1). For adults both parameters are set to one. For people below the age of twenty the infec-

tivity is reduced to 0.85 and the susceptibility to 0.45 [68,69]. Note that this does not mean

that the infection probability for children is necessarily lower than for adults, because children

are more likely to perform activities with a high contact intensity, as shown in Table 1.

Disease progression model

The disease progression model is taken from the literature [70–75] (also see [76]). The model

has states exposed, infectious, showing symptoms, seriously sick (= should be in hospital), critical

(= needs intensive care), and recovered. The durations from one state to the next follow log-

normal distributions; see Fig 3 (LEFT) for details. We use similar age-dependent transition

probabilities as [15], shown in Fig 3 (RIGHT).

Infecting another person is possible during infectious, and while showing symptoms, but no

longer than 4 days after becoming infectious. This models that persons are mostly infectious

relatively early through the disease [71], while in later stages the infection may move to the

lung [72], which makes it worse for the infected person, but seems to make it less infectious to

other persons.

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Time-dependent inputs and calibration

Simulation runs

Although the approach was designed with uniform rollout throughout Germany in mind, the

project, for reasons described in Sec 4 of S1 Appendix, mostly performed simulations for the

metropolitan area of Berlin in Germany, with approx. 5 million people. A typical simulation

run looks as follows:

1. One or more exposed (i.e. recently infected) persons are introduced into the population.

2. At some point, exposed persons become infectious. From then on, every time they spend

time together with some other person in a vehicle or at some activity, Eq (1) is used to cal-

culate the probability that the other person, if susceptible, can become infected (= exposed).

If infection happens, the newly infected person will follow the same progression.

3. Infectious persons eventually move on to other disease states, as described in Fig 3.

The model runs for many days, until no more infections occur and all persons have finished

their paths through the disease progression.

Calibration

The calibration procedure undertaken for the present paper is described in the following sec-

tions. Calibration is performed by visual comparison, with first priority against the time series

Fig 3. Disease progression model. LEFT: State transitions [70–75]. RIGHT: Age-dependent transition probabilities from

infectious to symptomatic, from symptomatic to seriously sick (= requiring hospitalisation), and from seriously sick to critical

(= requiring breathing support or intensive care). Source: [15], except that the numbers in the second column are divided by 2

(discussed in Under-reporting, and its variation over time).

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of the number of hospital patients in Berlin, and with second priority against the COVID case

numbers in Berlin. The calibration procedure, as described in the following, is as much about

which elements to include at all as about finding the right parameters. “Second priority” here

means that if calibration against hospital numbers is undecided between two alternatives, then

the case numbers are used in addition. The case numbers are only used with second priority

since the screening procedure has been changed multiple times, which means that the resulting

time series is not homogeneous and thus not useful for model calibration. In particular, under-

reporting in the initial phases was much larger than later. More information about the COVID

case numbers in Berlin can be found in Sec 2 of S1 Appendix. A formal calibration of Θcan be

found in Sec. Out of sample prediction. The calibration includes the following elements:

1. Calibration of the basic doubling time without reduction of activity participation

2. Integration of spring disease import

3. Calibration of the consequences of reduced activity participation

4. Calibration of an indoors/outdoors effect for leisure activities depending on the

temperature

5. Integration of contact tracing, masks, and summer disease import

All calibrations concern Θ(cf. Eq 1); item 4also involves defining threshold temperatures

at which activities are moved outdoors at the end of the winter, and indoors at the end of the

summer. All other aspects are data driven.

Unrestricted model

Most parameters of the model are taken from the literature, as explained earlier, in particular

Fig 3. The remaining free parameters are, from Eq (1),Θ,sh, and in. We have set the base val-

ues of sh =in = 1. As mentioned before, we use these parameters to model the wearing of

masks, meaning that they are reduced when masks are worn.

Fig 4 shows the unrestricted base case with four different values of Θ. One finds that the

aggregated behavior at this level corresponds to that of typical S(E)IR models, i.e. exponential

growth, followed by a maximum, followed by exponential decrease. Based on these plots, theta-

Factor values of 1.0 or 1.2 seem plausible to be consistent with the initial growth. A thetaFactor

of 1.0 corresponds to Θ= 0.000561.

Spring disease import

We take the disease import from abroad from data published by RKI ([77], always on Tues-

days). Currently, for Germany this data is only available on a nationwide aggregated level. For

Fig 4. Unrestricted base case. LEFT: Case numbers. The green and red dots denote case numbers as reported by

Robert Koch Institute [77]; the blue dots denote positive test fractions [78] multiplied by 200. RIGHT: Hospital

numbers. Each simulation curve is averaged over 10 independent Monte Carlo runs with different random seeds; the

shaded areas denote 5% and 95% percentiles of those 10 runs.

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this reason we scale it down to our Berlin model by using the population size. The data is

dated on the reporting date and not on the actual date of becoming sick. Since the infection

seeds are initiated into our model with the status exposed (cf. Fig 3) and it can be assumed that

the reporting date is significantly after the exposure date, we date the data from RKI back by

one week. The data provided by RKI is available as weekly values so we assign these values to

the respective Monday and then interpolate between them. Since we assume underreporting

in the RKI numbers, we multiply them by 4; this is discussed in Sec. Under-reporting, and its

variation over time. The initially infected persons are drawn randomly from the population.

The resulting disease import is shown in Fig 5. The description so far only concerns the spring

disease import; for summer disease import see Sec 3 of S1 Appendix.

An advantage about adding disease import is that the date of the first infection is no longer

a free parameter: As shown in Fig 6, the disease import is sufficient to drive the first wave. The

disease import data seems to lack some early cases, thus causing an initially nearly vertical

increase in the simulation. The dynamics then settles onto the exponential increase shown in

the previous section.

In terms of calibration, the initial growth is, within limits, insensitive against changes of Θ,

since it is dominated by the disease import. This can be explained by the fact that the exponen-

tial growth was running ahead in other areas, and in consequence the share of infected persons

from those areas also grew exponentially. Only after travel was stopped did disease import also

stop, and the dynamics in Berlin was dominated by internal processes.

Fig 5. Disease import over time. Based on data taken from [77] (always on Tuesdays), but multiplied by 4 in spring,

and divided by 2 in summer (see text for Discussion).

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Fig 6. Unrestricted base case, but with initial disease import from data. LEFT: new cases; RIGHT: hospital occupancies. One finds that the initial slope

dynamics is rather independent from the thetaFactor.

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Reductions of activity participation

During the unfolding of the epidemics, people decided or were ordered to no longer partici-

pate in certain activities. We model this by removing an activity from a person’s schedule, plus

the travel to and from the activity. For example, if a person in their original plan goes from

home to activity A and then back home, then the activity plus both trips are removed from the

schedule. If a person in their original plan goes from home via activity A to activity B and then

back home, and activity A is deleted, then the following elements are removed: (a) the trip

from home to activity A; (b) activity A; (c) the trip from activity A to activity B. In the current

model, the schedule is not repaired: neither is the home activity nor are other activities pro-

longed, and also the trip chain is not mended. See S6 Text for possible improvements here.

The consequence of those activity and trip removals is that the person no longer interacts with

people at that activity location, and in consequence neither can infect other persons nor can

become infected during that activity, or while in public transport vehicles to and from that

activity. Overall, this reduces contact options, and thus reduces epidemic spread.

A very important consequence of our modelling approach is that we can take that reduction

in activity participation from data. That data comes from the same source as our original activ-

ity patterns. However, the activity type detection algorithm is not very good for these unusual

activity patterns during the pandemics, as one can see in S1 Fig when knowing that all educa-

tional institutions were closed in Berlin after Mar/15. What is reliable, though, is the differenti-

ation between at-home and out-of-home time, as displayed in Fig 7. One clearly notices that

out-of-home activities are somewhat reduced after Mar/8, and dramatically reduced soon

Fig 7. Change in activity participation compared to the baseline for normal workdays. All out-of-home activities are combined into one number. (�) denotes

the first day of closures of schools, clubs, and bars; and (#) the first day of the so-called contact ban which came together with closures of all restaurants and non-

essential stores.

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after. After some experimentation, it was decided to take weekly averages of the activity non-

participation, and use that uniformly across all activity types in our model, except for educa-

tional activities, which were taken as ordered by the government.

To remove an activity with a certain probability, a random draw is made every time a syn-

thetic person has that activity type in its plan. This means that the model assumes that, say for

a 50% work reduction, there will be a different 50% subset of persons at work every day. This

intervention, in consequence, does not sever infection networks, but just slows down the

dynamics.

One takes from Fig 8 that the mobility reductions, as given by the mobility data, is by itself

not sufficient to explain the decreasing case numbers during spring. Evidently, one could now

reduce Θ, and this is what we have done in our early simulations. This, however, artificially

reduces the infection dynamics, and means that the simulation will miss the second wave in

fall.

Outdoors vs. indoors season

The probability of getting infected during an encounter depends on whether the encounter

takes place indoors or outdoors. Outdoors, the probability of infection is significantly reduced

compared to indoors. This is due to the fact that outdoors the air is constantly in motion and

therefore aerosols cannot accumulate. We assume that an encounter outdoors decreases the

infection probability by one magnitude [54,79]. In countries like Germany, seasonality has a

great influence on how much time people spend outdoors. In summer, people spend more

time outdoors, while in winter they tend to spend more time indoors.

We include into our model that up to 100% of leisure activities are undertaken outdoors

during summer, while that share reduces to 0% during winter. When an activity occurs out-

doors, the otherwise identical computation of the infection probability is divided by 10. The

model takes the actual temperatures as input; if the daily maximum temperature is larger than

T�+ 5C, then all leisure activities that can happen outdoors are outdoors; if the daily maximum

temperature is smaller than T�−5C, then all leisure activities happen indoors; in between,

probabilities are linearly interpolated. We use T�= 17.5Cin spring, linearly increasing to T�=

25Cin fall; using a lower T�in spring is behaviorally plausible in Germany, and yields a far

more plausible infection dynamics than keeping them the same.

The justification for this is as follows. A survey on physical activities [80] shows that, in

summer, people in Germany perform about 80% of their physical activities outdoors, while

this proportion shrinks to 10% in winter. We have assumed that other leisure activities (e.g.

restaurants, visit friends) behave similarly. We also extend our range to 0 and 100% since the

fluctuations of the temperature already lead to average values that are more than 0 and less

than 100% (cf. Fig 9).

Fig 8. Simulations with reductions of activity participation as obtained from mobility data. LEFT: new cases;

RIGHT: hospital occupancies.

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Fig 10 shows an example of the infection dynamics where both T�in spring and T�in fall

are 17.5C; as one can see, either the decrease of the first wave is not strong enough, or the sec-

ond wave comes too late; note in particular the hospital numbers, which for all values of theta-

Factor do not have enough slope in the second wave. The results with other T�, as long as they

are the same in spring and fall, are the same. Fig 11 shows instead using 17.5C for spring and

25C for fall; the second wave now is triggered earlier, and it is steeper. Fig 9 shows the outdoors

fractions for this model.

There were some restrictions concerning leisure activities in place in fall. They mostly con-

cerned large events. We know from our mobility data that all activities were at their normal

level in September 2020; in consequence, if anything, they were divided into smaller groups.

Masks, contact tracing, and summer disease import

From Fig 11 one takes away that a good calibration with the elements described so far would

be possible, with a thetaFactor between 0.6 and 0.8. Nevertheless, we add masks (in public

transport and shopping), contact tracing, and summer disease import, since they are plausible

elements of the dynamics. In particular, they result in the prediction of reduced infection num-

bers for public transport and shopping, which both is plausible. This is described in more

detail in Sec 3 of S1 Appendix.

Fig 9. Outdoors fraction for activities of type leisure, depending on the temperature of each day.

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Fig 10. Simulations that now also include a symmetric indoors/outdoors model, with a threshold temperature of 17.5C both in spring and in fall. LEFT: new

cases; RIGHT: hospital occupancies. A thetaFactor between 0.6 and 0.8 is most plausible, but the second wave would come too late (starting after September) and

would not be steep enough (compare slope of red dots in right plot after September) (cf. in particular the hospital numbers).

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Final model

The final model is shown in Fig 12, where the blue line traces the number of new cases with

state showingSymptoms from our simulation. Fig 12 (right) shows the cases in need of hospital

care and those in need of ICU care from our simulation compared to real data. As stated, we

find fitting to the hospital numbers more important; fully fitting to the case numbers is not

possible with just one Θthat is constant across the whole simulation. Note that this implies, as

stated, a strong deviation of the model curve (in blue) from the reported numbers (in red and

green) during the first months. Also see Under-reporting, and its variation over time in the

Discussion.

Methods and results

Infections per activity type

Evidently, in our microscopic models we can track how many infections happen at which

activity type. Fig 13 shows, on top, the absolute numbers of infections per activity type for the

simulation, and below the share of infections per activity type over time. To obtain these num-

bers, we evaluate what activity the infected person is performing at the time of infection and

date that to the date of infection.

Fig 11. Simulations that now also include an asymmetric indoors/outdoors model, with a threshold temperature of 17.5C in spring, and 25C in fall. LEFT: new

cases; RIGHT: hospital occupancies. A thetaFactor between 0.6 and 0.8 is most plausible, which would well reproduce the second wave (cf. in particular the hospital

numbers).

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Fig 12. Final model. LEFT: new cases; RIGHT: hospital occupancies. All simulation results are averaged over 10 runs with different Monte Carlo seeds; the shaded

areas denote 5% and 95% percentiles of those 10 runs. Evidently, the relative errors become larger with smaller case numbers. The simulation model can only be

fitted against the hospital numbers (right) when significant under-reporting is assumed in the early phase (left).

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Initially, all activity types play a role. After the closure of the universities, schools, and day

care in March, both their absolute numbers and their shares go to zero. At the same time, the

infections share of work (gray) in April and May reflects that persons were drifting back to

normal activity patterns (cf. Fig 7). Leisure (green) would have shown the same trend, but that

was counter-acted by the increasing shift of activities to outdoors. In the bottom plot, the pur-

ple line shows how the share of infections in public transit decreases significantly near the end

of April because of increased wearing of masks. (Recall that we use observed mask compli-

ance.) In July we see how day care (blue) picks up, because it was re-opened. Schools re-open

in the second week of August, and pick up accordingly (brown). Also, two weeks of school

vacation in October are clearly reflected in the brown curve. From September on we then see a

strong increase of the infections share of leisure activities—corresponding to moving leisure

activities from outdoors to indoors as explained earlier.

Fig 13. Infections per activity type. TOP: absolute numbers. Note logarithmic scale. BOTTOM: Share of infections

per activity type. The values are averaged over the same 10 runs as for the other figures, and in addition aggregated into

weekly bins. One can see, for example, the return to school near the beginning of August, and the fall vacations in

October.

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Fig 14. Reproduction number R(t) for the duration of the simulation. As explained in the text, we explicitly count

the reproduction number per agent, and then average them over all agents that turned contagious on a given day.

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Reproduction number

Since our method is person-centric, we can, for each infected person n, count the number of

persons that that person infects, i.e. its reproduction number, R

. When averaging over multi-

ple persons, one needs to make a decision to which date R

is assigned. We use the date when

nturned contagious, and in consequence

RðtÞ ¼ X

n2CðtÞ

Rn;

where C(t) refers to all persons who turned contagious on day t. An issue with this approach is

that the consequences of interventions become visible in R(t) before the interventions actually

start—since the infections that are suppressed happen later than t. This is also the reason why

we use the date when turning contagious and not the date when they got infected, since that

would increase that temporal gap even more. Fig 14 shows the resulting values, with R(t) much

larger than one in the initial phase, then lower than one until the end of summer, and then

increasing to above one in fall. We do not offer a comparison with the official Rvalues since

they have the same issues as the official case numbers.

Reproduction number per activity type

More insightful than the number or share of infections, as presented in Sec. Infections per

activity type above, is the average reproduction number in each activity type. The method

counts for each infected person the number of persons they infect at each activity context. As

in Sec. Reproduction number above, the numbers are dated back to the date when the person

became contagious, and then averaged over all those persons.

For example, an activity-specific R-value for school of R

school

(t) = 0.1 means that each per-

son that becomes contagious on day t, in the average, infects 0.1 other persons at school. Evi-

dently, if only 10% of persons turning contagious on day thave school anywhere in their

activity pattern, then each such person would have to infect one other person in the school

context in order to reach the population-average value of 0.1.

Adding up these activity-specific reproduction numbers leads to the overall reproduction

number. This explains why, in first order, the overall reproduction number can be additively

decomposed into the contributions of the different activity types.

One sees, in Fig 15, that the reproduction number at home remains roughly constant—a

person who gets infected in any way reinfects on average about 0.35 persons at home. Work is

related to the mobility data—if less time is spent out-of-home, then in the model less time is

Fig 15. Reproduction number per activity type.

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spent at work, leading to fewer infections. Schools were closed in the middle of March, and

not reopened until the second week of August. Also, there is a school vacation during the sec-

ond and third week of October. Day care according to the model has little effect. Day care was

already re-opened partially in June, and fully in July. The reproduction number at leisure is

strongly driven by the weather: If it is warm, the model assumes that most of the leisure activi-

ties take place outdoors, where they contribute little to the infection dynamics. In conse-

quence, this effect plays an important role in spring, where the warmer temperature played as

much a role as the reduction of the out-of-home activites. One also clearly sees the strong

growth of the leisure reproduction number in fall, which according to these simulations is

driving the second wave in Berlin. Public transport is strongly visible in March, until the obli-

gation to wear masks was introduced. All other infection contexts, e.g. errands or business

activities, are combined in the category “other”.

Table 2. Contributions to Rby activity type and intervention according to our model.

contribution to R

home 0.44

. . . with cloth / N95 masks 0.20 / 0.02

work 0.17

. . . @ 75% / 50% 0.10 / 0.04

. . . with cloth / N95 masks during work 0.06 / 0.01

. . . @ 50% with N95 masks during work <0.01

schools 0.15

. . . @ 75% / 50% 0.07 / 0.02

. . . with cloth / N95 masks during classes 0.05 / <0.01

. . . with N95 masks during classes and 50% attendance <0.01

day care 0.02

. . . @ 75% / 50% 0.01 / <0.01

. . . with cloth / N95 masks 0.01 / <0.01

universities 0.23

. . . @ 75% / 50% 0.11 / 0.03

. . . with cloth / N95 masks 0.06 / <0.01

retail and errands 0.09

. . . @ 75% / 50% 0.06 / 0.03

. . . with cloth / N95 masks 0.03 / 0.01

leisure (winter) 1.04

. . . @ 75% / 50% 0.52 / 0.21

. . . with cloth / N95 masks 0.38 / 0.03

leisure (summer) 0.2

public transport 0.12

. . . @ 75% / 50% 0.06 / 0.03

. . . with cloth / N95 masks 0.04 / <0.01

For these calculations we run the unrestricted model without any interventions and then introduced the

interventions described in the left column on 2020–04-01. The reductions to the Rvalues were calculated one week

after that, comparing the respective weekly averages. For the mask interventions, the compliance rate is 90%.

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Reductions of Rper intervention

Other papers, e.g. [5], report, for various interventions, corresponding percent reductions of

R. Our model clarifies that it is structurally more robust to report the additive reduction of the

reproduction number by a certain intervention. For example, according to our model closing

schools removes the school reproduction number from the dynamics, and in consequence

reduces Rby about 0.15. If Ris 1 when the intervention is introduced, this amounts to 15%; if

Ris 2, then this amounts to 7.5%.

Table 2 shows, based on simulations as explained in the previous section, the contributions

to Rof the different activity types. Adding up the boldface numbers leads to R= 2.26, i.e. a

strongly super-critical situation. In contrast, the 2020 Germany summer regime corresponds

to closed universities, schools and day care, and wearing masks in retail. Together with the lei-

sure summer number this leads to R= 0.88, i.e. makes the situation sub-critical.

It has been pointed out by other studies that the reproduction numbers at home play an

important role and reduce the remaining “space” one has available for infections outside home

[81]. The reproduction number at home can be reduced by moving persons showing symp-

toms, and more radically persons identified as contacts by contact tracing, into separate facili-

ties, sometimes called quarantine hotels.

One also notices that all infection contexts can be strongly reduced by wearing masks—this

(evidently) even holds for leisure. Clearly, they would need to be worn during the activities,

and not just during access and egress. Wearing masks during class at school has hesitantly

been adopted in Berlin during November; wearing masks during work, in particular in office

buildings, has never been pursued seriously in Germany and is still not obligatory if occupants

have at least 10 m

available per person—which is the value with which our simulations run

and which generate the numbers of Table 2.

Evidently, a tricky context is leisure. According to our simulations, leisure alone, in con-

junction with home, would be sufficient to keep Rabove one during winter, and thus needs to

be suppressed accordingly. Keeping other activity contexts open without masks implies that

leisure needs to be suppressed even further if R<1 is to be achieved.

Conversely, during summer achieving an R<1 is relatively easy. This explains why there

were few problems during summer in Germany (and most other European countries). Evi-

dently, all of this refers to the original variant of SARS-CoV-2, which was less transmissible

than later variants.

Decreasing marginal effect of interventions

In Table 2, for all activity types, a reduction of the participation by 50% reduces the contribu-

tion to R by far more than 50%: at work from 0.17 to 0.04, at school from 0.07 to 0.02, etc. In

consequence, the next 50% reduction of participation, i.e. closing the activity type completely,

will yield a much smaller reduction of infections. This is consistent with the empirical observa-

tion that the marginal effect of stay-at-home interventions decreases [6].

From our model, this can be explained as follows (see Fig 16): Assume, for example, that

each morning each school child throws a coin and goes to school only when it shows heads;

this means that school participation is reduced to 50%. In consequence, if there is an infectious

person at school, only half as many other persons have a chance to get infected. (This assumes

that they use the same classrooms as before, at half the density.) However, the probability that

an undetected infectious person goes to school is also reduced to 50%. Multiplying these two

probabilities means that only 50% �50% = 25% of the infections happen in this case. That is,

the first 50% of the reduction has already 75% of the possible effect.

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Fig 16. Effect of dividing a group of 10 persons into two groups of 5 persons each. In the original situation, each of the 9 susceptible persons (white and cyan)

has a probability to get infected of p

, resulting in a expected number of infected persons of 9p

. In the divided situation on the left, the expected number of

infections is 4p

. On the right, it is 0. Overall, this results in an expectation value of 1

2ð4p0þ0Þ ¼ 2p0. In consequence, when dividing classes and alternating their

attendance, the number of infections is reduced from 9p

to 2p

. For large group sizes, the reduction converges to 1/4. The same holds when each individual

attendance is decided randomly with probability 1/2 at the beginning of each day.

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More generally, consider an activity in an enclosed space, with Nparticipants, kof them

contagious, k�N, and conditions such that the linear approximation of Eq (2) for the infec-

tion probability holds. In this situation, the expected number of infections is proportional to

kN. Now assume that the participation probability at that activity, for each individual person,

is reduced from 1 to α<1. There are two consequences:

1. The expected number of contagious persons reduces to αk.

2. The expected number of participants reduces to αN.

The expected number of infections in consequence reduces to αkαN, i.e. α

as many as

before.

Evidently, this means that 1 −0.75

�44% of the effect is obtained with the first 25% of the

intervention, another 1 −0.5

−44% = 31% of the effect are obtained with the next 25% of the

intervention, and the remaining 25% of the effect need the remaining 50% of the stay-at-home

intervention for this particular activity.

In terms of the management of COVID-19, this implies that it is far better to include each

activity type/sector of the economy to some extent, rather than shutting down some sectors

completely while leaving some other sectors completely open.

Out of sample prediction

We show the predictive performance of our model by calibrating the simulation on a fixed

training set and comparing simulation results into the future against unused data. In this cali-

bration Θis calibrated such that the Root Mean Squared Logarithmic Error (RMSLE) between

hospital cases in the simulation compared to historic data is minimized. For this, the simula-

tion is run with eight different Monte Carlo seeds and then the results are averaged. Because

one simulation run is quite computationally expensive, a Tree-structured Parzen Estimator

[82], implemented by the Optuna package [83] in Python, is used to sample the parameter

space more efficiently.

RMSLE has the advantage that it is less sensitive to the scale of the data than RMSE. That is,

relative errors in valleys have as much weight as relative errors on the ridges. This also corre-

sponds to the visual impression of the logarithmic plots often used in epidemics and used

throughout the paper. Results using RMSE instead of RMSLE and some more discussion can

be found in S3 Text.

We run this calibration up to various dates. From there on, we perform two types of predic-

tion: (a) Fig 17 left: With activity participation levels frozen at the level of the last calibration

day (see the second column of Table 3; only during school vacations are school and work activ-

ities reduced for prediction dates); (b) Fig 17 right: With activity levels as given from the data

also for prediction dates. For both cases, the import is frozen at 4 imported cases per day,

while the daily maximum temperature is forecast based on the average over the last 10 years.

One finds that the correct activity level (Fig 17 right) is crucial especially for longer-term

predictions: Even with calibration only to the end of April, the model predicts the autumn

wave very well, while when the activity numbers are frozen (Fig 17 left), the forecasts have a

drift depending on whether the activity level is too low or too high on the day when it is frozen.

Particularly striking is the blue curve (“2020–05-01”): on the left, the activity participation level

is frozen at 71% (cf. Table 3), while on the right it varies mostly between 80% and 100% as

given by the data plotted in Fig 7. This is consistent with the theoretical argument (Sec.

Decreasing marginal effect of interventions) that an activity participation of 71% reduces

infections to 71% �71% �50% while an activity participation of 90% reduces infections only

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to 91% �90% �80%. That difference is sufficient to generate the difference between the two

curves.

Discussion

Intuition for these results

In an older version of the model [33], we had all contact intensities set to one. The contribu-

tions of each activity type to the infection dynamics then in first order corresponded to the

average weekly time consumption in the respective activity. For example, averaged over the

week including the weekend, school consumes about 5 hours per day for persons going to

school. However, since in Berlin only about 10% of the population are school children, https://

www.statistik-berlin-brandenburg.de/BasisZeitreiheGrafik/Bas-Schulen.asp?Ptyp=

300&Sageb=21001&creg=BBB&anzwer=5 the average time consumption for the school activ-

ity is only 0.43 hours per day when taken across the whole population (cf. Table 4). In contrast,

there are more persons going to work than to school, thus increasing the weight of work in the

infection dynamics (1.83 hours per day). A similar weight comes from the leisure activities,

which are not necessarily more hours per week for each individual person, but where all per-

sons contribute to this type of time consumption, resulting in an average of 1.67 hours per day.

In consequence, restricting leisure activities had a large effect in that model.

Fig 17. Hospitalized persons for different calibration runs compared to real data. Θis calibrated such that hospital numbers in the simulation match the real data

(red dots) until different points in time as indicated by the legend. After this date, an out of sample prediction is carried out. Until the calibration date real weather and

disease import data is used. After the calibration date average weather data from the past ten years is used and the disease import is set to 4 imported cases per day (1

agent per day) LEFT: Activity levels are frozen at the level of the last day of the period used for calibration. RIGHT: Real activity levels are used.—Results are averaged

over 30 Monte Carlo seeds.

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Table 3. Calibration parameter Θand activity participation for the different out of sample predictions shown in Fig 17.

run Θactivity participation (if activity level

frozen)

training error prediction error (frozen activity

levels)

prediction error (real activity

levels)

2020–05–01 1.20e-05 71% 0.372 5.500 0.34

2020–06–01 1.27e-05 88% 0.252 0.187 0.74

2020–07–01 1.29e-05 90% 0.232 0.047 0.11

2020–08–01 1.30e-05 90% 0.200 0.033 0.37

2020–09–01 1.32e-05 96% 0.181 0.442 0.52

RMSLE (= Root Mean Square Logarithmic Error) for the calibration interval (training error) as well as for prediction period between 09–01 and 10–31 (prediction

error). The Optuna package does not return confidence intervals for estimated parameters.

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In the present model, the time consumptions are now multiplied by the normalized contact

intensities in those activity types, cf. Table 4. In consequence, leisure, which already had a

large share before, is now multiplied with a large contact intensity, and in consequence now

gets even more weight. Work, despite occupying similar amounts of time, is weighted down

because of the multiplication by a much smaller contact intensity. On the other end of the

scale, public transport has, at full occupancy, a large contact intensity, but the times spent in

public transport are considerably smaller than, say, at work. Also, persons in public transport

are required to wear masks, while at work they are not.

A complicated case are schools and day care: They occupy large amounts of time, and have

a large contact intensity, both somewhat similar to leisure. In consequence, the re-opening of

day care in July and of the schools in August should have had strong consequences in the

infection numbers but did not. We took that observation as confirmation that their larger-

than-average contact intensity is compensated for by a smaller-than-average infectivity and

susceptibility (cf. Sec. Children).

For other diseases, for example influenza, all of the above may need to be adapted. For

example, children may have a larger infectivity/susceptibility than adults, which then multi-

plied with their large contact intensity would lead to a large contribution to the infection

dynamics. In consequence, these sub-models need to be understood and re-calibrated for each

individual communicable disease.

Robustness

The simulation uses one uniform Θ(cf. Eq 1) that remains the same over the whole simulation

period. In consequence, the dynamics is driven by other inputs. These are, after the initial dis-

ease import (Fig 5), primarily the activity participation (Fig 7) and the temperature-dependent

outdoors fraction (Fig 9).

The importance of the activity participation can be taken from Fig 17, where the blue curve

(“2020–05-01”) uses exactly the same setup on the left and on the right except for the activity

participation level, which on the left is frozen at 71%, and on the right varies mostly between

80% and 100% as given by the data plotted in Fig 7. That was already discussed at the end of

Sec. Out of sample prediction.

The importance of the temperature effect can best be taken from the calibration sequence:

Fig 6 showing that a Θsmaller than 0.6 is not plausible; Fig 8 showing that reduced activity

participation alone is not able to bring infections and resulting hospital levels down to the

observed level during summer; Fig 10 showing that a symmetric indoors/outdoors model is

Table 4. Average time consumption of out-of-home activities.

Activity average time consumption [hr] normalized contact intensity ci0time �ci0

day care 0.22 11 2.42

schools 0.43 11 4.73

university 0.13 5.5 0.75

work (incl. business) 1.83 1.47 2.69

shop 0.38 0.88 0.33

errands 0.77 1.47 1.14

leisure 1.67 9.24 15.43

home 16.45 1 16.45

Averaged over a full week including Saturday and Sunday. The remaining time is spent travelling between activities. Contact intensities are taken from Table 1.

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able to achieve that but misses the fall wave; and finally Fig 11 showing that the asymmetric

indoors/outdoors model, with a significantly higher threshold temperature at the end of the

summer, is able to also generate the fall wave. We have checked for other mechanisms driving

the fall wave but obvious possibilities, such as the return-to-school or the summer disease

import, both make the hospital numbers in fall start going up in the middle of August rather

than at the beginning of September. S4 Text also shows that a less asymmetric indoors/out-

doors model performs worse. Evidently, it is possible that in reality the virus seasonality is also

caused by other aspects [84], and our model absorbs them into the indoors/outdoors model.

Still, if one accepts the aerosol infection as major pathway, then the assumption that indoors

vs. outdoors activities play a strong role is plausible.

The effect of the contact intensities was already discussed in Sec. Intuition for these results.

From that discussion, it follows that the number of infections in an activity context depends,

in first order, linearly on the contact intensity of that activity type. In consequence, if a contact

intensity in Table 1 or Table 4 is, say, a factor of 2 too large, then the resulting infections (Fig

13 top), share of infections (Fig 13 bottom), and R-value per activity type (Fig 15) in first order

get divided by two. In second order, the calibration parameter Θwould have to get increased

to bring infections back to the previous level.

S4 Text also shows that the model fit degrades considerably when the mask model is

removed.

Comparison to other models

A comparison to compartmental models, in particular to the model of Chang et al. [10] which

is at the border between compartmental and agent-based, can be found in S5 Text. The same

text contains also a comparison with the model by Aleta et al. [21].

Comparison to other “reductions of R” studies

Table 5 extracts “additional reductions to R” from other studies and compares them to our

results. One immediately finds two issues: (A) The categories are not well aligned. For exam-

ple, “small gathering cancellation” refers to gatherings with 50 persons or less, while other

studies cancel gatherings larger than a certain number. Again other studies just consider a

“gathering ban”, but at the same time have “event ban” and “venue closure” as separate items.

(B) Even where the categories are well aligned, the resulting numbers vary significantly: for

example, “closing schools and universities” goes from 16% to 38%.

In part, this is a consequence of the fact that the interventions are not standardized: For

example, the number of exemptions in what is called a lockdown varied quite a lot between

countries.

Additionally, the transmission mechanisms from policy decision to execution vary, so even

if the concept may be the same, the effect may be quite different between countries. For exam-

ple, our reductions to Rcaused by school closures come out at the lower end of the range, and

lower than those of Brauner et al. [5]. We attribute this to the following two elements: First,

the model by Brauner et al. has no initial disease import which is then brought to a halt. In

consequence, their approach has to assign all changes in the infection dynamics to the school

closures. The school closures in Berlin, with Mar/12 (fri) or Mar/15 (mon) as the last day of

school, too late to explain that the infection numbers stopped in the middle of March. Also,

Dehning et al. [87] have an additional change point on Mar/7, corroborating that something

has changed before the school closures. Second, other than both Brauner et al. and Dehning

et al., we have the mobility data of Fig 7 at our disposal. It is clear that there was considerably

more societal adaptation around the weekend of Mar/13–14 than just keeping children at

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home. Brauner et al. themselves write that “the closure of schools . . . may have caused . . .

behaviour changes. We do not distinguish this indirect signalling effect from the direct effect”.

Additionally, in Germany, children staying at home will force their parents to stay at home,

thus forcing them into home office. In consequence, some of this may not be signalling, but

causal secondary effects. In consequence, our model is more differentiated: What Brauner

et al. attribute to the school closures alone is in our model attributed to a combination of

school closures, behavioral changes, and the reduction of various other out-of-home activities.

Thus, all of the values may be correct: The pure effect of school closures in western countries

(with relatively few young people) may not be larger than 7%, but the measurable consequence

for Rwhen governments closed schools as their first intervention presumably indeed was

much larger.

Clearly, data-driven mechanical models such as ours help clarifying the categories since we

can exactly specify what we mean by closing some activity type or wearing a mask at certain

activity types. Also, we can differentiate between the transmission from political decision to

behavioral execution vs. the consequences of the behavioral execution to the infection dynam-

ics. Finally, we can mechanically include organizational approaches such as contact tracing.

Table 5. Percent reduction of Rin other studies.

Measure Brauner et al. (1st

wave) [5]

Sharma et al. (2nd

wave) [85]

Haug et al.

“CC” [86]

Haug et al.

“Lasso” [86]

Our model (abs.)

(Table 2)

Our model

R = 2.24

Closing schools and universities 38% 16% 21% 0.38 21%

Closing educational institutions (after

implementing protective measures)

Closing schools 0.15 7%

Closures of businesses 35%

Closure of work sector 0.17 8%

Closing some high-risk face-to-face businesses 18%

Closing most nonessential face-to-face

businesses

27%

Closing retail and close contact services 12%

Closure of retail and errands sectors 12% 0.09 4%

Gatherings limited to �1000 23%

Mass gathering cancellation 33% 0%

Gatherings limited to �100 34%

Gatherings limited to �10 42%

Small gathering cancellation 35% 22%

Closures of gastronomy 12%

Closures of night clubs 12%

Leisure and entertainment venues 3%

Banning all leisure activities (including

gastronomy and private visits)

1.04 92%

Night time curfew 13%

Stricter mask policy (mandatory in most or all

shared/public spaces)

12%

Percentages are rounded to integers. To the right are our own results, first in absolute reductions of R, then in percent reductions of Rapplied to an Rof 2.24 (the overall

Rin the model where these values were taken). Evidently, for a smaller R, our percentage values would be higher.

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Masks

We have checked our relatively large reductions of Rfor masks in Table 2 multiple times. They

are a consequence of the assumption that N95 masks reduce intake to 2.5%, taken from [67].

The review article [88] comes up with about 5%, a factor of two larger, but still displaying a

very large reduction. The same paper [88] also shows that “masks” without a specification of

the type has much less of an effect. Finally, there may be the issue that lay people may not be

able to use N95 masks at full efficiency. In consequence, our results have to be interpreted

once more “mechanically”: They are plausible under the assumption that the fraction of people

specified in the model is indeed able to use N95 masks effectively.

Under-reporting, and its variation over time

A known issue with epidemiological data and thus the simulations that build on it is the issue

of under-reporting, i.e. that there are more infections in reality than are in the data. Looking at

Fig 12, it is clear that our current model assumes only little under-reporting during August to

October. This originally led to hospital numbers that were too large; since we cannot reduce

the number of infections below the case numbers, this justifies why we reduce simulated hospi-

tal numbers by a factor of 2 compared to [15] (cf. Fig 3). This, in turn, implies that, if we want

to get the spring hospital numbers right, our simulated infection numbers in spring need to be

about a factor of 8 larger than the reported case numbers.

Also note that our simulation includes non-symptomatic cases, which come on top of the

symptomatic cases that we show in our figures such as Fig 12; that is, the actual under-report-

ing is even larger. Still, it is entirely possible that Germany’s testing strategy is missing even

more cases, in which case the simulation would need to aim for even larger numbers of

infected persons. As long as the number of seropositive persons in Germany remains in the

single-digit percentage ranges [89], the predictions made by the simulation are not strongly

affected by this issue. Once the infections start to saturate, i.e. approach herd immunity, this

will become important. Hopefully, by then systematic antibody screenings will be available,

and we will be able to calibrate the model against the case numbers that must have been

infected in the past. Given that we have the hospital numbers for control, we expect this to be

straightforward.

Making the model more realistic

Evidently, the model can be made (even) more realistic. Important aspects are the adaptation

of the daily schedules to to restrictions, the dependence on income, and more realistic contact

structures. All three aspects are discussed in S6 Text.

Policy advice

The model was and is used for policy advice. Our regular reports to the government all have a

DOI, for example [3] or [4]. Again, see https://depositonce.tu-berlin.de/simple-search?query=

modus-covid.

Conclusions

We combine a person-centric human mobility model with a mechanical model of infection

and a person-centric disease progression model into an epidemiological simulation model.

Different from other models, we take the movements of the persons, including the intervening

activities where they can interact with other people, directly from data, which has already been

available for transport planning before the pandemics. For privacy reasons, we rely on a

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process that takes the original mobile phone data, extracts statistical properties, and then syn-

thesizes movement trajectories from the statistical properties; one could use the original

mobile phone trajectories directly if they were available. The model is used to replay the epi-

demics in Berlin. It is shown that the second wave in Berlin can be modelled well with an

explicit temperature dependency of the outdoors fraction for leisure activities. The model is

then used to evaluate different intervention strategies, such as closing educational facilities,

reducing other out-of-home activities, wearing masks, or contact tracing, and to determine

differentiated changes of the reproduction number Rper intervention.

Supporting information

S1 Appendix. Appendix.

(PDF)

S1 Fig. Reduced activity participation by activity type.

(PDF)

S1 Text. Senozon method.

(PDF)

S2 Text. Model history.

(PDF)

S3 Text. Error metric for calibration.

(PDF)

S4 Text. Robustness runs.

(PDF)

S5 Text. Comparison to other models.

(PDF)

S6 Text. Making the model more realistic.

(PDF)

S1 File.

(PDF)

Acknowledgments

We thank Kai Martins-Turner, Dominik Ziemke, Tim Conrad and Natasa Conrad for fre-

quent inputs and discussion. We are grateful to BVG (Berlin public transit operator) for pro-

viding the mask compliance rates which they surveyed on a daily basis. The work on the paper

was funded by the Ministry of research and education (BMBF) Germany (01KX2022A) and

TU Berlin; regular reports can be found through this search: https://depositonce.tu-berlin.de/

simple-search?query=modus-covid. Zuse Institute Berlin (ZIB) provided CPU time.

Author Contributions

Conceptualization: Sebastian A. Mu¨ller, Kai Nagel.

Data curation: Sebastian A. Mu¨ller, Michael Balmer, William Charlton, Ricardo Ewert,

Andreas Neumann, Christian Rakow, Tilmann Schlenther.

Formal analysis: Sebastian A. Mu¨ller.

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Funding acquisition: Kai Nagel.

Investigation: Sebastian A. Mu¨ller, Kai Nagel.

Methodology: Sebastian A. Mu¨ller, Kai Nagel.

Project administration: Ricardo Ewert.

Software: Sebastian A. Mu¨ller, Christian Rakow, Kai Nagel.

Supervision: Kai Nagel.

Validation: Sebastian A. Mu¨ller, Kai Nagel.

Visualization: Sebastian A. Mu¨ller, William Charlton.

Writing – original draft: Sebastian A. Mu¨ller, Kai Nagel.

Writing – review & editing: Sebastian A. Mu¨ller, Michael Balmer, William Charlton, Ricardo

Ewert, Andreas Neumann, Christian Rakow, Tilmann Schlenther, Kai Nagel.

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