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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.
