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Importance of Interaction Structure and

Stochasticity for Epidemic Spreading:

A COVID-19 Case Study

Gerrit Großmann ,1 , Michael Backenk¨ohler1, and Verena Wolf1

Saarland Informatics Campus, Saarland University, 66123 Saarbr¨ucken, Germany

{gerrit.grossmann,michael.backenkoehler,verena.wolf}@uni-saarland.de

Abstract. In the recent COVID-19 pandemic, computer simulations are

used to predict the evolution of the virus propagation and to evaluate the

prospective eﬀectiveness of non-pharmaceutical interventions. As such,

the corresponding mathematical models and their simulations are central

tools to guide political decision-making. Typically, ODE-based models

are considered, in which fractions of infected and healthy individuals

change deterministically and continuously over time.

In this work, we translate an ODE-based COVID-19 spreading model

from literature to a stochastic multi-agent system and use a contact

network to mimic complex interaction structures. We observe a large

dependency of the epidemic’s dynamics on the structure of the underly-

ing contact graph, which is not adequately captured by existing ODE-

models. For instance, existence of super-spreaders leads to a higher in-

fection peak but a lower death toll compared to interaction structures

without super-spreaders. Overall, we observe that the interaction struc-

ture has a crucial impact on the spreading dynamics, which exceeds the

eﬀects of other parameters such as the basic reproduction number R0.

We conclude that deterministic models ﬁtted to COVID-19 outbreak

data have limited predictive power or may even lead to wrong conclusions

while stochastic models taking interaction structure into account oﬀer

diﬀerent and probably more realistic epidemiological insights.

Keywords: COVID-19 ·Epidemic Stochastic Simulation ·SEIR Model

·SARS-CoV-2 ·2019–2020 coronavirus pandemic.

1 Introduction

On March 11th, 2020, the World Health Organization (WHO) oﬃcially declared

the outbreak of the coronavirus disease 2019 (COVID-19) to be a pandemic. By

this date at the latest, curbing the spread of the virus became a major worldwide

concern. Given the lack of a vaccine, the international community relied on

non-pharmaceutical interventions (NPIs) such as social distancing, mandatory

quarantines, or border closures. Such intervention strategies, however, inﬂict high

costs on society. Hence, for political decision-making it is crucial to forecast the

spreading dynamics and to estimate the eﬀectiveness of diﬀerent interventions.

2 G. Großmann et al.

Mathematical and computational modeling of epidemics is a long-established

research ﬁeld with the goal of predicting and controlling epidemics. It has devel-

oped epidemic spreading models of many diﬀerent types: data-driven and mech-

anistic as well as deterministic and stochastic approaches, ranging over many

diﬀerent temporal and spatial scales (see [50,15] for an overview).

Computational models have been calibrated to predict the spreading dynam-

ics of the COVID-19 pandemic and inﬂuenced public discourse. Most models and

in particular those with high impact are based on ordinary diﬀerential equations

(ODEs). In these equations, the fractions of individuals in certain compartments

(e.g., infected and healthy) change continuously and deterministically over time,

and interventions can be modeled by adjusting parameters.

In this paper, we compare the results of COVID-19 spreading models that are

based on ODEs to results obtained from a diﬀerent class of models: stochastic

spreading processes on contact networks. We argue that virus spreading models

taking into account the interaction structure of individuals and reﬂecting the

stochasticity of the spreading process yield a more realistic view on the epi-

demic’s dynamics.

If an underlying interaction structure is considered, not all individuals of

a population meet equally likely as assumed for ODE-based models. A well-

established way to model such structures is to simulate the spreading on a net-

work structure that represents the individuals of a population and their social

contacts. Eﬀects of the network structure are largely related to the epidemic

threshold which describes the minimal infection rate needed for a pathogen to

be able to spread over a network [38]. In the network-free paradigm the basic

reproduction number (R0), which describes the (mean) number of susceptible

individuals infected by patient zero, determines the evolution of the spreading

process. The value R0depends on both, the connectivity of the society and the

infectiousness of the pathogen. In contrast, in the network-based paradigm the

interaction structure (given by the network) and the infectiousness (given by the

infection rate) are decoupled.

Here, we focus on contact networks as they provide a universal way of en-

coding real-world interaction characteristics like super-spreaders, grouping of

diﬀerent parts of the population (e.g. senior citizens or children with diﬀerent

contact patterns), as well as restrictions due to spatial conditions and mobility,

and household structures. Moreover, models based on contact networks can be

used to predict the eﬃciency of interventions [39,35,5].

Here, we analyze in detail a network-based stochastic model for the spreading

of COVID-19 with respect to its diﬀerences from existing ODE-based models and

the sensitivity of the spreading dynamics on particular network features. We

calibrate both, ODE-models and stochastic models with interaction structure

to the same basic reproduction number R0or to the same infection peak and

compare the corresponding results. In particular, we analyze the changes in the

eﬀective reproduction number over time. For instance, early exposure of super-

spreaders leads to a sharp increase of the reproduction number, which results

in a strong increase of infected individuals. We compare the times at which the

Importance of Interaction Structure and Stochasticity for COVID-19 3

number of infected individuals is maximal for diﬀerent network structures as well

as the death toll. Our results show that the interaction structure has a major

impact on the spreading dynamics and, in particular, important characteristic

values deviate strongly from those of the ODE model.

2 Related Work

In the last decade, research focused largely on epidemic spreading, where in-

teractions were constrained by contact networks, i.e., a graph representing the

individuals (as nodes) and their connectivity (as edges). Many generalizations,

e.g. to weighted, adaptive, temporal, and multi-layer networks exist [32,45]. Here,

we focus on simple contact networks without such extensions.

Spreading characteristics on diﬀerent contact networks based on the Susceptible-

Infected-Susceptible (SIS) or Susceptible-Infected-Recovered (SIR) compartment

model have been investigated intensively. In such models, each individual (node)

successively passes through the individual stages (compartments). For an overview,

we refer the reader to [36]. Qualitative and quantitative diﬀerences between net-

work structures and network-free models have been investigated in [23,2]. In

contrast, this work considers a speciﬁc COVID-19 spreading model and focuses

on those characteristics that are most relevant for COVID-19 and which have,

to the best of our knowledge, not been analyzed in previous work.

SIS-type models require knowledge of the spreading parameters (infection

strength, recovery rate, etc.) and the contact network, which can partially be

inferred from real-world observations. Currently, inferred data for COVID-19

seems to be of very poor quality [25]. However, while the spreading parame-

ters are subject to a broad scientiﬁc discussion. Publicly available data, which

could be used for inferring a realistic contact network, practically does not exist.

Therefore real-world data on contact networks is rare [31,46,24,33,44] and not

available for large-scale populations. A reasonable approach is to generate the

data synthetically, for instance by using mobility and population data based on

geographical diﬀusion [47,18,37,3]. For instance, this has been applied to the

inﬂuenza virus [34]. Due to the major challenge of inferring a realistic contact

network, most of these works, however, focus on how speciﬁc network features

shape the spreading dynamics.

2.1 COVID-19 Spreading Models

Literature abounds with proposed models of the COVID-19 spreading dynamics.

Very inﬂuential is the work of Neil Ferguson and his research group that regularly

publishes reports on the outbreak (e.g. [11]). They study the eﬀects of diﬀerent

interventions on the outbreak dynamics. The computational modeling is based

on a model of inﬂuenza outbreaks [20,12]. They present a very high-resolution

spatial analysis based on movement-data, air-traﬃc networks etc. and perform

sensitivity analysis on the spreading parameters, but to the best of our knowledge

not on the interaction data. Interaction data were also inferred locally at the

4 G. Großmann et al.

beginning of the outbreak in Wuhan [4] or in Singapore [41] and Chicago [13].

Models based on community structures, however, consider isolated (parts of)

cities and are of limited signiﬁcance for large-scale model-based analysis of the

outbreak dynamic.

Another work focusing on interaction structure is the modeling of outbreak

dynamics in Germany and Poland done by Bock et al. [6]. The interaction struc-

ture within households is modeled based on census data. Inter-household in-

teractions are expressed as a single variable and are inferred from data. They

then generated “representative households” by re-sampling but remain vague on

many details of the method.

A more rigorous model of stochastic propagation of the virus is proposed

by Arenas et al. [1]. They take the interaction structure and heterogeneity of

the population into account by using demographic and mobility data. They

analyze the model by deriving a mean-ﬁeld equation. Mean-ﬁeld equations are

more suitable to express the mean of a stochastic process than other ODE-based

methods but tend to be inaccurate for complex interaction structures. Moreover,

the relationship between networked-constrained interactions and mobility data

remains unclear to us.

Other notable approaches use SIR-type methods, but cluster individuals into

age-groups [40,29], which increases the model’s accuracy. Rader et al. [42] com-

bined spatial-, urbanization-, and census-data and observed that the crowding

structure of densely populated cities strongly shaped the epidemics intensity and

duration. In a similar way, a meta-population model for a more realistic inter-

action structure has been developed [8] without considering an explicit network

structure.

The majority of research, however, is based on deterministic, network-free

SIR-based ODE-models. For instance, the work of Jos´e Louren¸co et al. [30] infers

epidemiological parameters based on a standard SIR model. Similarly, Dehning

et al. [9] use an SIR-based ODE-model, where the infection rate may change

over time. They use their model to predict a suitable time point to loosen NPIs

in Germany. Khailaie et al. analyze how changes in the reproduction number

(“mimicking NPIs”) aﬀect the epidemic dynamics [26], where a variant of the

deterministic, network-free SIR-model is used and modiﬁed to include states

(compartments) for hospitalized, deceased, and asymptomatic patients. Other-

wise, the method is conceptually very similar to [30,9] and the authors argue

against a relaxation of NPIs in Germany. Another popular work is the online

simulator covidsim1. The underlying method is also based on a network-free SIR-

approach [51,52]. However, the role of an interaction structure is not discussed

and the authors explicitly state that they believe that the stochastic eﬀects

are only relevant in the early stages of the outbreak. A very similar method

has been developed at the German Robert-Koch-Institut (RKI) [7]. Jianxi Luo

et al. proposed an ODE-based SIR-model to predict the end of the COVID-19

pandemic2, which is regressed with daily updated data. ODE-models have also

1available at covidsim.eu

2available at ddi.sutd.edu.sg

Importance of Interaction Structure and Stochasticity for COVID-19 5

SIR

λ·#Neigh(I)β

a) Networked SIR model

I

I

I

II

I I

II

I

2λλββ

b) CTMC semantics

Fig. 1: Networked SIR model. (a) Compartments with instantaneous transition

rates. Each node successively passes through the three compartments/states:

susceptible (S), infected (I), and recovered/removed (R). (b) Four possible tran-

sitions on a 4-node contact network based on the CTMC semantics.

been used to project the epidemic dynamics into the “postpandemic” future by

Kissler et al. [28]. Some groups also resort to to branching processes, which are

inherently stochastic but not based on a complex interaction structure [22,43].

3 Translating SIR-type Models for Epidemic Spreading

A very popular class of epidemic models is based on the assumption that dur-

ing an epidemic individuals are either susceptible (S), infected (I), or recov-

ered/removed (R). The mean number of individuals in each compartment evolves

according to the following system of ordinary diﬀerential equations

d

dts(t) = −λODE

Ns(t)i(t)

d

dti(t) = λODE

Ns(t)i(t)−βi(t)

d

dtr(t) = βi(t),

(1)

where Ndenotes the total population size, λODE and βare the infection and

recovery rates. Typically, one assumes that N= 1 in which case the equation

refers to fractions of the population, leading to the invariance s(t)+i(t)+r(t)=1

for all t. It is trivial to extend the compartments and transitions.

3.1 Network-Based Spreading Model

A stochastic network-based spreading model is a continuous-time stochastic pro-

cess on a discrete state space. The underlying structure is given by a graph,

where each node represents one individual (or any other entity of interest). At

each point in time, each node occupies a compartment, for instance: S,I, or R.

Moreover, nodes can only receive or transmit infections from neighboring nodes

6 G. Großmann et al.

(according to the edges of the graph). For the general case with mpossible com-

partments, this yields a state space of size mn, where nis the number of nodes.

The jump times until events happen are typically assumed to follow an exponen-

tial distribution. Note that in the ODE model, residual residence times in the

compartments are not tracked, which naturally corresponds to the exponential

distribution in the network model. Hence, the underlying stochastic process is a

continuous-time Markov Chain (CTMC) [27]. The extension to non-Markovian

semantics is trivial. We illustrate the three-compartment case in Fig. 1. The tran-

sition rates of the CTMC are such that an infected node transmits infections at

rate λ. Hence, the rate at which a susceptible node is infected is λ·#Neigh(I),

where #Neigh(I) is the number of its infected direct neighbors. Spontaneous

recovery of a node occurs at rate β. The size of the state space renders a full

solution of the model infeasible and approximations of the mean-ﬁeld [14] or

Monte-Carlo simulations are common ways to analyze the process.

General Diﬀerences to the ODE model. The aforementioned formalism

yields some fundamental diﬀerences from network-free ODE-based approaches.

The most distinct diﬀerence is the decoupling of infectiousness and interaction

structure. The infectiousness λ(i.e., the infection rate) is assumed to be a param-

eter expressing how contagious a pathogen inherently is. It encodes the prob-

ability of a virus transmission if two people meet. That is, it is independent

from the social interactions of individuals (it might however depend on hygiene,

masks, etc.). The inﬂuence of social contacts is expressed in the (potentially

time-varying) connectivity of the graph. Loosely speaking, it encodes the possi-

bility that two individuals meet. In the ODE-approach both are combined in the

basic reproduction number. Note that, throughout this manuscript, we use λto

denote the infectiousness of COVID-19 (as an instantaneous transmission rate).

Another important diﬀerence is that ODE-models consider fractions of in-

dividuals in each compartment. In the network-based paradigm, we model ab-

solute numbers of entities in each compartment and extinction of the epidemic

may happen with positive probability. While ODE-models are agnostic to the

actual population size, in network-based models, increasing the population by

adding more nodes inevitably changes the dynamics.

A key link between the two paradigms is that if the network topology is a

complete graph (resp. clique) then the ODE-model gives an accurate approxima-

tion of the expected fractions of the network-based model. In systems biology

this assumption is often referred to as well-stirredness. In the limit of an inﬁnite

graph size, the approximation approaches the true mean.

3.2 From ODE-Models to Networks

To transform an ODE-model to a network-based model, one can simply keep

rates relating to spontaneous transitions between compartments as these tran-

sitions do not depend on interactions (e.g., recovery at rate β). Translating the

infection rate is more complicated. In ODE-models, one typically has given an

Importance of Interaction Structure and Stochasticity for COVID-19 7

infection rate and assumes that each infected individual can infect all suscepti-

ble ones. To make the model invariant to the actual number of individuals, one

typically divides the rate by the population size (or assumes the population size

is one and the ODEs express fractions). Naturally, in a contact network, we do

not work with fractions but each node relates to one entity.

Here, we propose to choose an infection rate such that the network-based

model yields the same basic reproduction number R0as the ODE-model. The

basic reproduction number describes the (expected) number of individuals that

an infected person infects in a completely susceptible population. We calibrate

our model to this starting point of the spreading process, where there is a single

infected node (patient zero). We assume that R0is either explicitly given or can

implicitly be derived from an ODE-based model speciﬁcation. Hence, when we

pick a random node as patient zero, we want it to infect on average R0susceptible

neighbors (all neighbors are susceptible at that point in time) before it recovers

or dies.

Let us assume that, like in the aforementioned SIR-model, infectious node

infect their susceptible neighbors with rate λand that an infectious node loses its

infectiousness (by dying, recovering, or quarantining) with rate β. According to

the underlying CTMC semantics of the network model, each susceptible neighbor

gets infected with probability λ

β+λ[27]. Note that we only take direct infections

from patient zero into account and, for simplicity, assume all neighbors are only

infected by patient zero. Hence, when patient zero has kneighbors, the expected

number of neighbors it infects is kλ

β+λ. Since the mean degree of the network is

kmean, the expected number of nodes infected by patient zero is

R0=kmean

λ

β+λ.(2)

Now we can calibrate λto relate to any desired R0. That is

λ=βR0

kmean −R0

.(3)

Note that R0will always be smaller than kmean which follows from Eq. (2),

considering that kmean ≥1 (by construction of the network) and β > 0. In

contrast, in the deterministic paradigm this relationship is given by the equation

(cf. [30,9]):

λODE =R0β . (4)

Note that the recovery rate βis identical in the ODE- and network-model. We

can translate the infection rate of an ODE-model to a corresponding network-

based stochastic model with the equation

λ=λODE

kmean −R0

,(5)

while keeping R0ﬁxed. In the limit of an inﬁnite complete network, this yields

limn→∞ λ=λODE

n, which is equivalent to the eﬀective infection rate in the ODE-

model λODE

Nfor population size N(cf. Eq. (1)).

8 G. Großmann et al.

Example. Consider a network where each node has exactly 5 neighbors (a 5-

regular graph) and let R0= 2. We also assume that the recovery rate is β= 1,

which then yields λODE = 2. The probability that a random neighbor of patient

zero becomes infected is 2

5=λ

(β+λ), which gives λ=2

3.

Extensions of SIR.It is trivial to extent the compartments and transitions,

for instance by including an exposed compartment for the time-period where an

individual is infected but not yet infectious. The derivation of R0remains the

same. The only requirement is the existence of a distinct infection and recovery

rate, respectively. In the next section, we discuss a more complex case.

4 A Network-based COVID-19 Spreading Model

We consider a network-based model that is strongly inspired by the ODE-model

used in [26] and document it in Fig. 2. We use the same compartments and

transition-types but simplify the notation compared to [26] to make the intuitive

meaning of the variables clearer3.

We denote the compartments by C={S,E,C,I,H,U,R,D}, where each node

can be susceptible (S), exposed (E), a carrier (C), infected (I), hospitalized (H), in

the intensive care unit (U), dead (D), or recovered (R). Exposed agents are already

infected but symptom-free and not infectious. Carriers are also symptom-free but

already infectious. Infected nodes show symptoms and are infectious. Therefore,

we assume that their infectiousness is reduced by a factor of γ(γ≤1, sick

people will reduce their social activity). Individuals that are hospitalized (or in

the ICU) are assumed to be properly quarantined and cannot infect others.

Accurate spreading parameters are very diﬃcult to infer in general and the

high number of undetected cases complicates the problem further in the current

pandemic. Here, we choose values that are within the ranges listed in [26], where

the ranges are rigorously discussed and justiﬁed. We document them in Table 1.

We remark that there is a high amount of uncertainty in the spreading param-

eters. However, our goal is not a rigorous ﬁt to data but rather a comparison

of network-free ODE-models to stochastic models with an underlying network

structure.

Note that the mean number of days in a compartment is the inverse of the

cumulative instantaneous rate to leave that compartment. For instance, the mean

residence time in compartment His 1

(1−rh)µh+rhµh=1

µh. As a consequence of the

race condition of the exponential distribution [48], rhmodulates the probability

of entering the successor compartment. That is, with probability rh, the successor

compartment will be Rand not U.

Inferring the infection rate λfor a ﬁxed R0is somewhat more complex than

in the previous section because this model admits two compartments for infec-

tious agents. We ﬁrst consider the expected number of nodes that a randomly

3At the time of ﬁnalizing this manuscript, the model of Khailaie et al. seems

to be updated in a similar way. However, it also became more complex (see

gitlab.com/simm/covid19/secir/-/wikis/Report).

Importance of Interaction Structure and Stochasticity for COVID-19 9

chosen patient zero infects, while being in state C. We denote the corresponding

basic reproduction number by c

R0. We calibrate the only unknown parameter

λaccordingly (the relationships from the previous section remain valid). We

explain the relation to R0when taking Cand Iinto account in the Appendix

(available at [16]). Substituting βby µcgives

λ=λODE

kmean −c

R0

=λODE

kmean −λODE

µc

.(6)

4.1 Human-to-Human Contact Networks

Naturally, it is extremely challenging to reconstruct large-scale contact-networks

based on data. Here, we test diﬀerent types of contact networks with diﬀerent

features, which are likely to resemble important real-world characteristics. The

contact networks are speciﬁc realizations (i.e., variates) of random graph models.

Diﬀerent graph models highlight diﬀerent (potential) features of the real-world

interaction structure. The number of nodes ranges from 100 to 105. We only

use strongly connected networks (where each node is reachable from all other

nodes). We refer to [10] or the NetworkX [19] documentation for further informa-

tion about the network models discussed in the sequel. We provide a schematic

visualization in Fig. 3.

We consider Erd˝os–R´enyi (ER) random graphs as a baseline, where each

pair of nodes is connected with a certain (ﬁxed) probability. We also compute

results for Watts–Strogatz (WS) random networks. They are based on a ring

topology with random re-wiring. The re-wiring yields to a small-world property

of the network. Colloquially, this means that one can reach each node from each

other node with a small number of steps (even when the number of nodes in-

creases). We further consider Geometric Random Networks (GN), where

nodes are randomly sampled in an Euclidean space and randomly connected

such that nodes closer to each other have a higher connection probability. We

also consider Barab´asi–Albert (BA) random graphs, that are generated using

a preferential attachment mechanism among nodes, as well as graphs gener-

ated using the Conﬁguration Model (CM-PL) which are—except from being

constrained on having power-law degree distribution—completely random. The

latter two models models contain a very small number of nodes with very high

degree, which act as super-spreaders. We also test a synthetically generated

Household (HH) network that was loosely inspired by [2]. Each household is a

clique, the edges between households represent connections stemming from work,

education, shopping, leisure, etc. We use a conﬁguration model to generate the

global inter-household structure that follows a power-law distribution. We also

use a complete graph (CG) as a sanity check. It allows the extinction of the

epidemic, but otherwise similar results to those of the ODE are expected.

10 G. Großmann et al.

λ·#Neigh(C) + λγ ·#Neigh(I)

SECIHUD

R

µe(1 −rc)µc

rcµc

(1 −ri)µi

riµi

(1 −rh)µh

rhµh

(1 −ru)µu

ruµu

Fig. 2: Multi-agent compartment model for COVID-19 with instantaneous tran-

sition rates. The infection rate is λ. Exposed (E) nodes are newly infected. Car-

riers (C) are already infectious but still symptomless, infected nodes (I) develop

symptoms and reduce their social activity (modulated by γ). Nodes that are

hospitalized (H) and in the ICU (U) are properly quarantined. Recovered (R)

nodes remain recovered.

Table 1: Model Parameters

Parameter Value Meaning

λ−Infection rate, set w.r.t. Eq. (6) (using

c

R0)

γ0.2 How much social contact is maintained when sick

#Neigh(x)∈Z≥0Current number of neighbors in compartment x

λODE 0.29 Infection rate in ODE-model (denoted by R1in [26])

c

R01.8R0assuming γ= 0

R0≈2.05 R0, when assuming γ= 0.2 (cf. Appendix [16])

x(t)−Number of (expected) nodes in x∈ C at time t

Itotal(t)−e(t) + c(t) + i(t) + h(t) + u(t)

rx∈[0,1] Recovery probability when node is in compartment x

µx>0 Instantaneous rate of leaving x

µe

1

5.2Rate of transitioning from Eto C

rc0.08 Recovery probability when node is a carrier

µc

1

5.2Rate of leaving C

ri0.8 Recovery probability when node is infected

µi

1

5Rate of leaving I

rh0.74 Recovery probability when node is hospitalized

µh

1

10 Rate of leaving H

ru0.46 Recovery probability when node is in the ICU

µu

1

8Rate of leaving U

Importance of Interaction Structure and Stochasticity for COVID-19 11

a) ER b) BA c) CM-PL

d) HH e) GN f) WS

Fig. 3: Schematic visualizations of the random graph models with 80 nodes.

4.2 Parameter Calibration

We are interested in the relationship between the contact network structure, R0,

the height and time point of the infection-peak, and the number of individuals

ultimately aﬀected by the epidemic. Therefore, we run diﬀerent network models

with diﬀerent c

R0. For one series of experiments, we ﬁx c

R0= 1.8 and derive the

corresponding infection rate λand the value for λODE in the ODE model. In the

second experiments, calibrate λand λODE such that all infection peaks lie on

the same level.

4.3 Interventions

In the sequel, we do not explicitly model NPIs. However, we note that the

network-based paradigm makes it intuitive to distinguish between NPIs related

to the probability that people meet (by changing the contact network) and NPIs

related to the probability of a transmission happening when two people meet (by

changing the infection rate λ). Political decision-making is faced with the chal-

lenge of transforming a network structure which inherently supports COVID-19

spreading to one which tends to suppress it. Here, we investigate how changes

in λaﬀect the dynamics of the epidemic in Section 5 (Experiment 3).

5 Numerical Results

We compare the solution of the ODE model (using numerical integration) with

the solution of the corresponding stochastic network-based model (using Monte-

Carlo simulations). Code will be made available4. We investigate the evolution

4github.com/gerritgr/StochasticNetworkedCovid19

12 G. Großmann et al.

of mean fractions in each compartment over time, the evolution of the so-called

eﬀective reproduction number, and the inﬂuence of the infectiousness λ.

Setup. We used contact networks with n= 1000 nodes (except for the complete

graph where we used 100 nodes). To generate samples of the stochastic spreading

process, we used event-driven simulation (similar to the rejection-free version

in [17]). Speciﬁcally, we utilized a simulation scheme, where all future events

(i.e., transitions of nodes) were sorted in a priority queue (according to their

application time). In each simulation step, the next event is drawn from the

queue, the event is applied to the network, and the time is updated accordingly.

Depending on the type of transition (a) new event(s) is (are) generated and

pushed to the queue. Some events already in the queue might become irrelevant

and are removed. The event queue is initialized by generating one event for each

node.

The simulation started with three random seed nodes in compartment C(and

with an initial fraction of 3/1000 for the ODE model). One thousand simulation

runs were performed on a ﬁxed variate of a random graph. We remark that

results for other variates were very similar. Hence, for better comparability, we

refrained from taking an average over the random graphs. The parameters to

generate a graph are: ER: kmean = 6, WS: k= 4 (numbers of neighbors), p= 0.2

(re-wire probability), GN: r= 0.1 (radius), BA: m= 2 (number of nodes for

attachment), CM-PL: γ= 2.0 (power-law parameter), kmin = 2, HH: household

size is 4, global network is CM-PL with γ= 2.0, kmin = 3. The CPU time for

a single simulation on a standard desktop computer was in the range of a few

hours.

Experiment 1: Results with Homogeneous c

R0.In our ﬁrst experiment,

we compare the epidemic’s evolution (cf. Fig. 4) while λis calibrated such that

all networks admit an c

R0of 1.8. and λis set (w.r.t. the mean degree) according

to Eq. (6). Thereby, we analyze how well certain network structures generally

support the spread of COVID-19. The evolution of the mean fraction of nodes

in each compartment is illustrated in Fig. 4 and Fig. 5.

Based on the Monte-Carlo simulations, we analyzed how Rt, the eﬀective

reproduction number, changes over time. The number Rtdenotes the average

number of neighbors that an infectious node (who got exposed at day t) infects

over time (cf. Fig. 6). For t= 0, the estimated eﬀective reproduction number

always starts around the same value and matched the theoretical prediction.

Independent of the network, c

R0= 1.8 yields R0≈2.05 (cf. Appendix [16]).

In Fig. 6 we see that the evolution of Rtdiﬀers tremendously for diﬀerent

contact networks. Unsurprisingly, Rtdecreases on the complete graph (CG), as

nodes, that become infectious later, will not infect more of their neighbors. This

also happens for GN- and WS-networks, but they cause a much slower decline

of Rtwhich is around 1 in most parts (the sharp decrease in the end stems from

the end of the simulation being reached). This indicates that the epidemic slowly

“burns” through the network.

Importance of Interaction Structure and Stochasticity for COVID-19 13

a) ODE b) CG c) ER d) BA

e) CM-PL f) HH g) GN h) WS

Fig. 4: Exp. 1: Evolution of the mean fractions in each compartment over time

with 95% conﬁdence intervals (barely visible).

In contrast, in networks that admit super-spreaders (CM-PL, HH, and also

BA), it is principally possible for Rtto increase. For the CM-PL network, we

have a very early and intense peak of the infection while the number of individ-

uals ultimately aﬀected by the virus (and consequently the death toll5) remains

comparably small (when we remove the super-spreaders from the network while

keeping the same R0, the death toll and the time point of the peak increase, plot

not shown). Note that the high value of Rtin Fig. 6c in the ﬁrst days results

from the fact that super-spreaders become exposed, which later infect a large

number of individuals. As there are very few super-spreaders, they are unlikely

to be part of the seeds. However, due to their high centrality, they are likely to

be one of the ﬁrst exposed nodes, leading to an “explosion” of the epidemic. In

HH-networks this eﬀect is way more subtle but follows the same principle.

Experiment 2: Calibrating c

R0to a Fixed Peak. Next, we calibrate λ

such that each network admits an infection peak (regarding Itotal) of the same

height (0.2). Results are shown in Fig. 7. They emphasize that there is no direct

relationship between the number of individuals aﬀected by the epidemic and

the height of the infection peak, which is particularly relevant in the light of

limited ICU capacities. It also shows that vastly diﬀerent infection rates and

basic reproduction numbers are acceptable when aiming at keeping the peak

below a certain threshold.

5The number of fatalities in the ﬁgures is diﬃcult to see, but it is (in the time limit) proportional

to the number of recovered nodes.

14 G. Großmann et al.

a) Height of the Infection Curve b) Fraction of Susceptible Individuals

Fig. 5: Exp. 1: Same data as in Fig. 4 but only the evolution Itotal and Sare

shown to highlight diﬀerences between networks.

a) CG b) HH c) CM-PL d) GN e) WS

Fig. 6: Exp. 1, Eﬀective Reproduction Number: Evolution of the (mean)

eﬀective reproduction number, Rt, over time, empirically evaluated. x-axis: Day

at which a node becomes exposed, y-axis: (mean) number of neighbors this node

infects while being a carrier or infected. Note that at later time points results

are more noisy as the number of samples decreases. The ﬁrst data-point is the

simulation-based estimation of R0and is shown as a blue square.

Experiment 3: Sensitivity Regarding λ.Assume we have an estimate of the

infectiousness, λ, of COVID-19. How do changes of λ(e.g., by better hygiene)

inﬂuence epidemic’s properties and what is the impact of uncertainty about the

value? Here, we investigate how the height of the infection-peak and c

R0scale

with λfor diﬀerent topologies. Our results are illustrated in Fig. 8.

Noticeably, the relationship is concave for most network models but almost

linear for the ODE model. This indicates that the networks models are more

sensitive to small changes of λ(and R0). This suggests that the use of ODE

models might lead to a misleading sense of conﬁdence because, roughly speaking,

it will tend to yield similar results when adding some noise to λ. That makes them

seemingly robust to uncertainty in the parameters, while in reality the process is

much less robust. Assuming that BA-networks resemble some important features

of real social networks, the non-linear relationship between infection peak and

infectiousness indicates that small changes of λ(which could be achieved through

proper hand-washing, wearing masks, keeping distance, etc.) can signiﬁcantly

“ﬂatten the curve”.

Importance of Interaction Structure and Stochasticity for COVID-19 15

a) ODE b) ER c) HH d) CM-PL e) WS

Fig. 7: Exp. 2: Evolution of mean-fractions in each compartment over time with

infectiousness calibrated such that the peak has the same height.

a) ODE b) BA c) WS d) GN

Fig. 8: Exp. 3:c

R0and maximal expected height of Itotal (expected refers to

samples, maximal refers to time) w.r.t. λ. For the network-models, R0(given by

Eq. (7)) is drawn as a scatter plot. Note the diﬀerent scales on x- and y-axis

5.1 Discussion

In the series of experiments, we tested how various network types inﬂuence an

epidemic’s dynamics. The network types highlight diﬀerent potential features

of real-world social networks. Most results do not contradict with real-world

observations. For instance, we found that better hygiene and the truncation of

super-spreaders will likely reduce the peak of an epidemic by a large amount. We

also observed that, even when R0is ﬁxed, the evolution of Rtlargely depends

on the network structure. For certain networks, in particular those admitting

super-spreaders, it can even increase. An increasing reproduction number can

be seen in many countries, for instance in Germany [21]. How much of this

can be attributed to super-spreaders is still being researched. Note that super-

spreaders do not necessarily have to correspond to certain individuals. It can

also, on a more abstract level, refer to a type of events. We also observed that

CM-PL networks have a very early and very intense infection peak. However, the

number of people ultimately aﬀected (and therefore also the death toll) remain

comparably small. This is somewhat surprising and requires further research.

We speculate that the fragmentation in the network makes it diﬃcult for the

virus to “reach every corner” of the graph while it “burns out” relatively quickly

in region of the more central high-degree nodes.

16 G. Großmann et al.

6 Conclusions and Future Work

We presented results for a COVID-19 case study that is based on the translation

of an ODE model to a stochastic network-based setting. We compared several

interaction structures using contact graphs where one was (a ﬁnite version of)

the implicit underlying structure of the ODE model, the complete graph. We

found that inhomogeneity in the interaction structure signiﬁcantly shapes the

epidemic’s dynamic. This indicates that ﬁtting deterministic ODE models to

real-world data might lead to qualitatively and quantitatively wrong results. The

interaction structure should be included into computational models and should

undergo the same rigorous scientiﬁc discussion as other model parameters.

Contact graphs have the advantage of encoding various types of interaction

structures (spatial, social, etc.) and they decouple the infectiousness from the

connectivity. We found that the choice of the network structure has a signiﬁcant

and counterintuitive impact and it is very likely that this is also the case for

the inhomogeneous interaction structure among humans. Speciﬁcally, networks

containing super-spreaders consistently lead to the emergence of an earlier and

higher peak of the infection. Moreover, the almost linear relationship between

R0,λODE, and the peak intensity in ODE-models might also lead to misplaced

conﬁdence in the results. Regarding the network structure in general, we ﬁnd

that super-spreaders can lead to a very early “explosion” of the epidemic. Small-

worldness, by itself, does not admit this property. Generally, it seems that—

unsurprisingly—a geometric network is best at containing a pandemic. This is

an indication for the eﬀectiveness of corresponding mobility restrictions. Sur-

prisingly, we found a trade-oﬀ between the height of the infection peak and the

fraction of individuals aﬀected by the epidemic in total.

For future work, it would be interesting to investigate the inﬂuence of non-

Markovian dynamics. ODE-models naturally correspond to an exponentially dis-

tributed residence times in each compartment [49,17]. Moreover, it would be

interesting to reconstruct more realistic contact networks. They would allow

to investigate the eﬀect of NPIs in the network-based paradigm and to have a

well-founded scientiﬁc discussion about their eﬃciency. From a risk-assessment

perspective, it would also be interesting to focus more explicitly on worst-case

trajectories (taking the model’s inherent stochasticity into account). This is es-

pecially relevant because the costs to society do not scale linearly with the char-

acteristic values of an epidemic. For instance, when ICU capacities are reached,

a small additional number of severe cases might lead to dramatic consequences.

Acknowledgements. We thank Luca Bortolussi and Thilo Kr¨uger for helpful

comments regarding the manuscript. This work was partially funded by the DFG

project MULTIMODE.

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A Inferring R0for Networks

S−CS−I

E−C

S−R

S−H

λ

rcµc

(1 −rc)µc

riµi

γλ

(1 −ri)µi

Fig. 9: The reachability probability from S−Cto E−Cis related to R0. It

expresses the probability that an infected node (patient zero, in C) infects a

speciﬁc random (susceptible) neighbor. The infection can happen via two paths.

Furthermore, we assume that this happens for all edges/neighbors of patient

zero independently.

Assume a randomly chosen patient zero that is in compartment C. We are

interested in R0in the model given in Fig. 2 assuming γ > 0. Again, we con-

sider each neighbor independently and multiply with kmean. Moreover, we have

to consider the likelihood that patient zero infects a neighbor while being in

compartment Cand the possibility of transitioning to Iand then transmitting

the virus. This can be expressed as a reachability probability (cf. Fig. 9) and

gives raise to the equation:

R0=kmean ·λ

µc+λ+(1 −rc)µc

µc+λ·γλ

µi+γλ .(7)

In the brackets, the ﬁrst part of the sum expresses the probability that patient

zero infects a random neighbor as long as it is in C. In the second part of the

sum, the ﬁrst factor expresses the probability that patient zero transitions to I

before infecting a random neighbor. The second factor is then the probability of

infecting a random neighbor as long as being in I. Note that, as we consider a

ﬁxed random neighbor, we need to condition the second part of the sum on the

fact that the neighbor was not already infected in the ﬁrst step.