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Abstract

In the recent COVID-19 pandemic, mathematical modeling constitutes an important tool to evaluate the prospective effectiveness of non-pharmaceutical interventions (NPIs) and to guide policy-making. Most research is, however, centered around characterizing the epidemic based on point estimates like the average infectiousness or the average number of contacts. In this work, we use stochastic simulations to investigate the consequences of a population's heterogeneity regarding connectivity and individual viral load levels. Therefore, we translate a COVID-19 ODE model to a stochastic multi-agent system. We use contact networks to model complex interaction structures and a probabilistic infection rate to model individual viral load variation. We observe a large dependency of the dispersion and dynamical evolution on the population's heterogeneity that is not adequately captured by point estimates, for instance, used in ODE models. In particular, models that assume the same clinical and transmission parameters may lead to different conclusions, depending on different types of heterogeneity in the population. For instance, the existence of hubs in the contact network leads to an initial increase of dispersion and the effective reproduction number, but to a lower herd immunity threshold (HIT) compared to homogeneous populations or a population where the heterogeneity stems solely from individual infectivity variations.
Why ODE models for COVID-19 fail: Heterogeneity shapes
epidemic dynamics
Gerrit Großmann1*, Michael Backenk¨ohler1, Verena Wolf1
1Saarland Informatics Campus, Saarland University, Saarbr¨ucken, Germany
* gerrit.grossmann@uni-saarland.de
Abstract
In the recent COVID-19 pandemic, mathematical modeling constitutes an important
tool to evaluate the prospective effectiveness of non-pharmaceutical interventions (NPIs)
and to guide policy-making. Most research is, however, centered around characterizing
the epidemic based on point estimates like the average infectiousness or the average
number of contacts.
In this work, we use stochastic simulations to investigate the consequences of a
population’s heterogeneity regarding connectivity and individual viral load levels.
Therefore, we translate a COVID-19 ODE model to a stochastic multi-agent system.
We use contact networks to model complex interaction structures and a probabilistic
infection rate to model individual viral load variation.
We observe a large dependency of the dispersion and dynamical evolution on the
population’s heterogeneity that is not adequately captured by point estimates, for
instance, used in ODE models. In particular, models that assume the same clinical and
transmission parameters may lead to different conclusions, depending on different types
of heterogeneity in the population. For instance, the existence of hubs in the contact
network leads to an initial increase of dispersion and the effective reproduction number,
but to a lower herd immunity threshold (HIT) compared to homogeneous populations or
a population where the heterogeneity stems solely from individual infectivity variations.
Author summary
Computational modeling can support decision-making in the face of pandemics like
COVID-19. Models help to understand transmission data and predict important
epidemiological properties (e.g., When will herd immunity be reached?). They can also
examine the effectiveness of certain measures, and—to a limited extent—extrapolate the
dynamics under specific assumptions. In all these cases, the heterogeneity of the
population plays an important role. For instance, it is known that connectivity
differences in (and among) age groups influence the dynamics of epidemic propagation.
Here we focus on two types of differences among individuals: their social interactions
and on how infectious they are. We show that only considering population averages
(e.g., What is the average number of contacts of an individual?) may lead to misleading
conclusions, because the individual differences (such as those related to the epidemic
(over-)dispersion) play an important role in shaping the epidemic dynamics. Many
commonly used model classes, such as SEIR-type ODE compartmental models, ignore
differences within a population to a large extent. This omission bears the potential of
misleading conclusions.
March 25, 2021 1/21
Introduction 1
At the beginning of 2020, the world was struck by the coronavirus (SARS-CoV-2) 2
pandemic. Faced with the approaching overload of healthcare systems, the international
3
community turned to non-pharmaceutical interventions (NPIs) in an attempt to contain
4
the spread of the pathogen [1]. Computational epidemiological modeling became an 5
important asset to predict the propagation and to evaluate the prospective effectiveness
6
of various measures such as school closings and travel restrictions [2,3]. For an overview
7
of COVID-19 models and their successes and failures, we refer the reader to [4,5]. 8
In this work, we highlight the importance of population heterogeneity for 9
computational epidemiology and explain why many models used in the current 10
COVID-19 pandemic do not adequately capture it. In particular, we analyze how 11
individual variations in contact numbers and infectiousness influence the evolution of a
12
pandemic. We use the term infectiousness as a property of the host to denote the 13
probability of passing a pathogen, given an established contact to a susceptible 14
individual. We qualitatively study the dynamical evolution based on different properties
15
like the height of the infection peak and the fluctuations of the effective reproduction 16
number Rt(average number of secondary infections at time point t). Furthermore, we 17
study how this heterogeneity influences the dispersion during an epidemic’s evolution. 18
COVID-19 is associated with an exceptional high dispersion and understanding how it 19
emerges is a crucial asset in controlling the pandemic [6]. 20
A noticeable example of heterogeneity in a population’s interaction structure are 21
individuals with extraordinary many contacts, so-called hubs. Similarly, super-spreader
22
events refer to temporary gatherings where one infected individual (potentially) infects
23
many others. Early on in the pandemic, it was pointed out that hubs are not accurately
24
captured by many models [7]. The evidence for the importance of hubs and 25
super-spreader events became increasingly conclusive over time [8–11]. 26
The concept of (over-)dispersion is closely related [12–14] and is consistently 27
reported for the COVID-19 pandemic [15–19]. In short, this concept reflects that a 28
small number of infected individuals infect many others while most infected individuals
29
infect no one or only very few. Overdispersion can be caused by hubs (many contacts, 30
average transmission probability) but also by individuals with high infectiousness 31
(average contact number, high transmission probability). Different individual levels of 32
infectiousness are a source of further heterogeneity in the population. 33
Viral load levels (and other properties that determine a host’s infectiousness) differ 34
between individuals and within individuals over time [20–23]. While many models 35
include the temporal aspect, the effects of individual variations are not well explored. 36
Moreover, some virus variants are associated with higher infectiousness [24]. 37
In order to study the effects of heterogeneity, we translate an ODE model for the 38
spread of COVID-19 to a stochastic network-based model. More specifically, this work 39
uses contact networks (i.e., graphs) to model individual variations in the number of 40
social contacts. To model the individual viral load variations, we use a randomly drawn
41
infectiousness parameter for each individual. Epidemic spreading on networks is well 42
understood and contact networks are a universal and well-established paradigm for the
43
analysis of complex interaction structures. 44
This paradigm allows us to study population heterogeneity while keeping population
45
averages fixed. In particular, we only compare networks with the same connectivity (i.e.,
46
number of edges). We also fix the mean infectiousness (i.e., we only modulate how much
47
an individual may deviate). On this premise, we investigate the effects of population 48
heterogeneity on the emergence of dispersion and the dynamical evolution of the 49
epidemic. We find, for instance, that power-law networks admit a natural early growth
50
of Rtand a very high dispersion. Individual differences in infectiousness increase the 51
dispersion even more while they generally weaken the epidemic. 52
March 25, 2021 2/21
SIR ODE NetworkBranching Process
d
dt s(t)=s(t)i(t)
d
dt i(t)=s(t)i(t)i(t)
d
dt r(t)=i(t)
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S
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R
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dtr(t)=i(t)
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Offspring Distribution
# Secondary Cases
Sample
Sample Next State
Fig 1. Schematic overview of model types.
Our contributions are as follows: 53
1. We give an overview of popular COVID-19 models used in the literature and 54
discuss their (implicit) assumptions about a population’s heterogeneity. 55
2. We show that imposing common interaction structures drastically changes an 56
epidemic’s evolution. 57
3. We analyze the additional effects of individual viral load variations. 58
4. We propose a novel method to quantify time-dependent dispersion based on an 59
empirical analysis of simulation runs. 60
Contribution (2) is based on preliminary results that were published during the first
61
COVID-19 wave in the spring of 2020 [25]. 62
Organization. Our manuscript is structured as follows: We present relevant 63
literature in Section Related work. In Section Method, we show how to translate ODE 64
models to network-based models and discuss their relationship. Section Experiments 65
provides numeral experiments based on synthetically generated random contact 66
networks. Conclusions completes the manuscript. 67
Related work 68
Mathematical modeling of epidemics is a wide research area to control, predict, and 69
understand epidemics or similar types of propagation processes (like opinions, or 70
computer viruses). Here, we mostly focus on the network-based spreading paradigm [26]
71
and its relation to other model types. In particular, we study which types of population
72
heterogeneity can be expressed and how models are used in the current COVID-19 73
pandemic. Note that currently all models that study COVID-19 quantitatively suffer 74
from the poor quality of data and uncertainty about parameters [27, 28]. We focus on 75
the three model types, we consider most relevant for epidemiological modeling in 76
general. For a comparative analysis of models specific to COVID-19, we refer the reader
77
to [4, 29]. 78
ODE models 79
The most wide-spread epidemiological model type is based on a system of ordinary 80
differential equations (ODEs) in which coupled fractions of individuals in disease 81
compartments change deterministically and continuously over time [30]. Compartments
82
refer to different disease stages (e.g., susceptible (
S
), infected (
I
), recovered (
R
), exposed
83
(E), dead (D)). Most commonly used is the three-compartment SIR-model (cf. Fig. 1). 84
Note that ODE models use a single parameter (
λ
) to model the chance to meet someone
85
(interaction structure) and the probability to transmit the infection. 86
March 25, 2021 3/21
Population heterogeneity.
Expressing population heterogeneity is only possible to
87
a very limited degree. The typical way is to introduce additional compartments that 88
encode a membership to a certain group (e.g., susceptible and “younger than 20”). 89
These extended models are often referred to as meta-population [31] models. Apart from
90
that, a homogeneous interaction structure is assumed. Effects like super-spreaders (or 91
overdispersion) are practically not expressible. The same holds for local die-outs. 92
Moreover, the deterministic nature makes it difficult to conceptualize risk and 93
uncertainty. ODE models arise as the mean-field limit of a well-mixed Markov 94
population model (corresponding to a complete graph in the network-based 95
paradigm) [32]. 96
COVID-19. Literature abounds with ODE-based COVID-19 models, some methods 97
and applications a summarized in [33]. For instance, Dehning et al. [34] use a model, 98
where the infection rate may change over time to predict a suitable time point to loosen
99
NPIs in Germany. Jos´e Louren¸co et al. infer epidemiological parameters [35]. Khailaie et
100
al. analyze how changes in the reproduction number affect the epidemic dynamics [36]. 101
Stutt et al. evaluate the effectiveness of NPIs [37]. The spread of COVID-19 was studied
102
for many more countries and settings [38–42]. Other studies use a meta-population 103
approach and group individuals based age [43–46] or region [47–49]. Moreover, [50] 104
and [51] modify ODE models to account for individual variation in susceptibility. 105
Roda et al. use an ODE model to illustrate the general difficulty of predicting the 106
spread of COVID-19 data [52]. Limitations in the applicability of ODE models 107
regarding data from Italy is reported in [53]. Similar results are found by Castro et 108
al. using COVID-19 data from Spain [54]. General concerns are articulated in [55]. 109
Branching processes 110
Stochastic branching processes operate in discrete or continuous-time and are useful 111
when studying the underlying stochastic nature of an epidemic. They are based on a 112
tree that grows over time and represents the infected individuals. The children 113
(offspring) of each node represent an individual’s secondary infections and the number 114
of children is drawn from an offspring distribution with mean
R0
that is provided by the
115
modeler [14, 56–58]. 116
Population heterogeneity. The offspring distribution makes it straight-forward to 117
encode individual variations in infectiousness or connectivity. The paradigm allows to 118
study random extinction probabilities of the epidemic and the effects of 119
super-spreaders/overdispersion [12]. However, branching processes do not admit a 120
(model intrinsic) saturation due to growing immunity in the population. Moreover, the
121
high level of abstraction makes it difficult to study the effects of NPIs and the 122
characteristics of the spatial diffusion of the pathogen. 123
COVID-19. Zhang et al. use a branching process to measure the dispersion of 124
COVID-19 inside China [59] and Endo et al. estimate the dispersion based on local 125
clusters outside China [16]. Moreover, [60] use a branching process to infer 126
epidemiological values and [22] study the influence of temporal viral load variation. 127
Alternative branching process models to study dynamical properties specific to 128
COVID-19 were proposed in [61–63]. 129
Network-based models 130
Network-based epidemic models use graphs to express the interactions (edges) among 131
individuals (nodes). They are stochastic in nature and can be formulated in discrete or
132
March 25, 2021 4/21
continuous time. Each node occupies a compartment (node state) at each point in time
133
and infected nodes can (randomly) infect their susceptible neighbors [26, 64, 65]. 134
Generalizations to multi-layer and weighted networks have been suggested [66]. 135
Population heterogeneity.
The network-based paradigm decouples the connectivity
136
of the population from the infectiousness of the virus. Moreover, each individual is 137
represented by an autonomous agent which adds flexibility and makes it straightforward
138
to include individual variations of the population. The key advantage of networks is 139
that they represent a universal way of encoding different types of complex interaction 140
structures like hubs, communities, households, small-worldness, different mixing within
141
in population-groups, etc. The contact network can also represent spatial or 142
geographical constraints. Network-based models relate to ODE models in the sense that
143
the ODE model represents the mean propagation of an epidemic on an infinite complete
144
graph (all nodes are directly connected), assuming that all nodes are attributed with 145
exponentially distributed jump times. Conceptually, the completeness “removes” the 146
heterogeneity from the interaction structure and the infinite size prevents artifacts from
147
the stochasticity. 148
COVID-19.
Effects of different contact networks were studied in [25,67
69]. Contact
149
networks are being used to build realistic simulations of a society, for instance by 150
creating household-structures with various types of inter-household connections [70
73].
151
The flexibility to control networks modeling NPIs easy [71, 72, 74, 75]. Moreover, [76
78]
152
use a network-based approach for spatial properties (e.g., flow between geographical 153
regions). Although the importance of hubs was recognized very early [79], the concrete
154
relation to overdispersion as it is studied in branching processes remains under-explored.
155
Networks, where the contact structure changes over time, are particularly well-suited to
156
study quarantine measure and social distancing [80–82]. 157
Method 158
In this section, we show how to translate a ODE model to a network-based model in 159
order to impose variation in connectivity and infectiousness while keeping the 160
population averages of clinical and transmission parameters fixed. We use a COVID-19
161
ODE model that is heavily inspired by the SIR-extension of [71]. A summary of the 162
model is depicted in Fig. 2 and Table 1. We do note upfront that we are only interested
163
in qualitative results and do not rely on exact parameter values. 164
ODE model 165
Our model contains six disease stages or compartments (cf. Fig. 2): susceptible (S), 166
exposed (
E
) (infected but not yet infectious), removed (
R
) (immune or dead), as well as
167
mild,severe, and critical infection stages (
I1
,
I2
,
I3
). In contrast to [71], we merge dead
168
and recovered stages to a single removed stage, as both do not influence the infection 169
dynamics further (we assume immunity after recovery). The fraction of individuals in 170
each compartment evolves according to a system of ordinary differential equations 171
(ODEs) given in Fig. 2b. Unlike network-based models, ODE models admit a semantics
172
that is invariant to the population size. Thus, we assume that the population is 173
normalized. A further difference to [71] is that we only have a single infection parameter
174
γ. All other parameters have a meaningful clinical interpretation and can be specified 175
accordingly (cf. Table 1). The set of transition parameters γ,µj,βjgives raise to a 176
specific R0. Thus, we can fix R0and thereby control γ[71]. We use R0= 2.5 which 177
leads to γ0.394. 178
March 25, 2021 5/21
S
E
I1
Susceptible
(healthy)
Exposed
(but not infectious)
Mild
Infection
I2
Severe
Infection
I3
Critical
Infection
R
Removed
(recovered + immune
or dead)
S
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3
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µ1
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µ2
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µ3
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ifor all neighbours iin I1+i/z for all neighbors iin I2,I3
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<latexit sha1_base64="tNP4pMPhf06tzck08vSd3wRE3H0=">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</latexit>
ifor all neighbours iin I1+i/z for all neighbors iin I2,I3
<latexit sha1_base64="tNP4pMPhf06tzck08vSd3wRE3H0=">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</latexit>
<latexit sha1_base64="tNP4pMPhf06tzck08vSd3wRE3H0=">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</latexit>
<latexit sha1_base64="tNP4pMPhf06tzck08vSd3wRE3H0=">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</latexit>
<latexit sha1_base64="tNP4pMPhf06tzck08vSd3wRE3H0=">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</latexit>
(a)
(b)
d
dts(t)=s(t)i1(t)+
zi2(t)+
zi3(t)
d
dte(t)=s(t)i1(t)+
zi2(t)+
zi3(t)µ1e(t)
d
dti1(t)=µ1e(t)1i1(t)
d
dti2(t)=µ2i1(t)µ3i2(t)2i2(t)
d
dti3(t)=µ3i2(t)3i3(t)
d
dtr(t)=1i1(t)+2i2(t)+3i3(t),
(1)
<latexit sha1_base64="88I/Sz4Mk+854Z5w3G/GT+3t7kI=">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</latexit>
<latexit sha1_base64="88I/Sz4Mk+854Z5w3G/GT+3t7kI=">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</latexit>
<latexit sha1_base64="88I/Sz4Mk+854Z5w3G/GT+3t7kI=">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</latexit>
<latexit sha1_base64="88I/Sz4Mk+854Z5w3G/GT+3t7kI=">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</latexit>
d
dts(t)=s(t)i1(t)+
zi2(t)+
zi3(t)
d
dte(t)=s(t)i1(t)+
zi2(t)+
zi3(t)µ1e(t)
d
dti1(t)=µ1e(t)1i1(t)
d
dti2(t)=µ2i1(t)µ3i2(t)2i2(t)
d
dti3(t)=µ3i2(t)3i3(t)
d
dtr(t)=1i1(t)+2i2(t)+3i3(t),
(1)
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I2
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I3
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Fig 2. COVID-19 model. (a) Transitioning system of the network model with
subject-level infectiousness (
λi
for subject
i
). The transition rates refer to exponentially
distributed residence times. The expected residence time in each disease sage is the
inverse of the sum of the outgoing transitions (e.g. for I1it is 1/(β1+µ2) = 6 (days)).
Likewise, the probability to go to I2is 0.2.). (b) Corresponding ODE model with
infection rate γ, where γencodes connectivity and infectiousness.
Table 1. Model Parameters
Parameter Value Meaning
λ0.0706 Infection rate when fixed (for R0= 2.5 and kmean = 8).
λiInfection rate (when variable) of subject i,λiν
νDensity of λiwith E[λi] = λ. E.g., ν= Exp(λ).
z5.0 Reduction in infectivity in disease stages I2,I3
R02.5 Basic reproduction number (for fixed λ).
kmean 8 Mean number of neighbors (by construction).
µ11/5 Disease progression rate in E
µ20.2/6 Disease progression rate in I1
µ30.25/6 Disease progression rate in I2
β10.8/6 Recovery rate in I1
β20.75/6 Recovery rate in I2
β31/8 Recovery/death rate in I3
γ0.394 Infection/connectivity rate for ODE model (R0= 2.5).
We refer to [71] for clinical justification of µj,βj.
March 25, 2021 6/21
R
R
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I3
I3
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E
E
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µ2
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µ3
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Start Goal
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Start Goal
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Fig 3. Computation of R0for fixed λ.Right: The probability that an I1Sedge
transmits the infection,
pI
, is the sum of
pI1
: probably that the infection is transmitted
while the infected node is in I1;pI2: probability that I1transitions to I2(before
transmitting the infection) and transmits while in I2; and pI3: probability that the
infection happens in
I3
(and not earlier). For individually varying
λi
,
R0kmeanE
[
pI
]
is based on an integral over ν.Left: Representation of pIas a reachability probability
(from Start to Goal) in a CTMC.
Network-based dynamics 179
We consider a static, undirected, unweighted, strongly connected graph. At each point 180
in time, each node (individual) is annotated with a compartment. The dynamics is 181
specified as a continuous-time Markov chain (CTMC) [26] where the labeling changes 182
randomly over time. We use the compartments described in Fig. 2. Nodes change their
183
compartment following exponentially distributed residence times corresponding to 184
specific instantaneous rates. For the transition from susceptible top exposed, the rate 185
depends on the neighborhood of the node (cf. Fig. 2a). We consider two cases: (i) all 186
nodes have the same infection rate λand (ii): each node ihas an individual infection 187
rate λi, sampled from a probability distribution with density ν. We start with the 188
former case. 189
Case (i): Homogeneous infectiousness Each SI1can transmit an infection 190
with rate
λ
. If the infected node is in compartment
I2
or
I3
the infectiousness decreases
191
(e.g., because people stay home) and is given by λ/z. Note that we use exponentially 192
distributed residence times which are potentially less realistic than, for instance, 193
beta-distributions [71], but these relate directly to ODE models. Hence, observed 194
differences in the dynamics can be attributed to the connectivity/stochasticity not the 195
shape of the residence time. 196
R0is defined as the expected number of neighbors that a randomly chosen patient 197
zero infects in a susceptible population, thus
R0
cannot be larger than the mean degree
198
(number of neighbors). In the case of a fixed infection rate λ, fixing kmean also 199
determines
R0
. We can approximate
R0
as shown in Fig. 3. We use that each infection
200
happens independently and approximate
R0pIkmean
where
pI
denotes the probability
201
that patient zero infects a random neighbor (while potentially transitioning to I2,I3). 202
The approximation comes from the fact that an already infected neighbor can infect 203
another neighbor of patient zero violating the independence assumption, rendering this
204
an over-approximation. Note that
pI
is the conceptually similar to the secondary attack
205
rate in a completely susceptible population. We construct the networks such that 206
kmean = 8 (except for the complete graph where kmean is the number of nodes minus 207
one). Like in ODE-approach, we fix R0= 2.5 and determine λ= 0.0706 (cf. Fig. 3). 208
March 25, 2021 7/21
Power-law Household Watts–Strogatz
Fig 4.
Schematic visualizations of random graph models with 80 nodes and
kmean
= 8.
Case (ii): Individual differences in infectiousness. In the case of individually 209
varying infectiousness, we associate each node iwith infection rate λithat is drawn 210
from a distribution with density ν. Again, our goal is to introduce variation while 211
keeping the population mean. Hence, we construct
ν
such that
λ
=
E
[
λi
] = 0
.
0706. We
212
define
Ri
0
as the basic reproduction number when the infection starts in node
i
and find
213
that R0=E[Ri
0]. Interestingly, different ν(with the same mean) can lead to different 214
R0. Theoretically, this follows from the computation of pIwhich is now based on an 215
integral over ν. In the next section, we set νto be an exponential distribution and 216
study the resulting changes in the dynamics. A key takeaway of our study is that 217
increasing the variance in the degree distribution does not change R0, increasing the 218
variance in the individual infectiousness does so (in fact, it decreases R0). For the 219
evaluation, we use an exponentially distributed λiwith mean 0.0706. 220
Human-to-human contact networks 221
We test different types of contact networks that highlight different characteristics of 222
real-world human-to-human connectivity. To this end, we describe the contact networks
223
using random graph models. In each simulation, we create a specific realization (variate)
224
of such a random graph model. A schematic visualization of example networks is 225
provided in Fig. 4. We use a complete graph (each possible pair of nodes is 226
connected) as a baseline to study the evolution of the epidemic when no contact 227
structure is present. Thereby, we can mimic the effects of stochasticity and variation in
228
infectiousness while keeping the simulation as close as possible to the assumptions 229
underlying the ODE. We also compute results for Watts–Strogatz (WS) random 230
networks. They are based on a ring topology with random re-wiring. Each node has 231
exactly kmean neighbors. We use a small re-wiring probability of 5% to highlight the 232
locality of real-world epidemics. We use
Power-law Configuration Model
networks
233
to study the effects of hubs (potential super-spreaders). These networks are—except 234
from being constrained on having power-law degree distribution—completely random. 235
The power-law degree distribution is omnipresent in real-world networks and entails a 236
small number of nodes with a very high degree. We fix the minimal degree to be two 237
and choose the power-law parameter numerically such that the network admits the 238
desired mean degree. We also test a synthetically generated Household network that 239
was loosely inspired by [83]. Each household is a clique, the edges between households 240
represent connections stemming from work, education, shopping, leisure, etc. We use a
241
geometric network to generate the global inter-household structure. The household size
242
is 4. In the case of kmean = 8, each node has 3 edges within its household and (on 243
average) 5 outgoing edges. 244
March 25, 2021 8/21
Deterministic
Homogeneous
Interaction Structure
Homogeneous
Infectiousness
Stochastic
Homogeneous
Interaction Structure
Homogeneous
Infectiousness
Stochastic
Heterogeneous
Interaction Structure
Homogeneous
Infectiousness
Stochastic
Heterogeneous
Interaction Structure
Heterogeneous
Infectiousness
S
S
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R
R
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I3
I3
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Fig 5. Overview of how population heterogeneity shapes an epidemic. f.l.t.r.: ODE
model, a complete graph, a power-law network, and a power-law network with
exponentially distributed infectiousness. The mean fraction of the population in each
compartment at each point in time is shown. Shaded areas indicate standard deviations,
not confidence intervals.
Dispersion in networks 245
Given a set of independent simulation runs, we measure dispersion by analyzing the 246
empirical offspring distribution at day t(averaged over all nodes). Specifically, we 247
consider the offspring distributions of the nodes that were exposed within day t(the 248
actual secondary infections may happen later). We also perform a discretization of time
249
over intervals of one day. We quantify dispersion in three ways: 250
CoV: Together with the mean of the offspring distribution Rt, we report the 251
coefficient of variation (CoV), that is the ratio of the standard deviation to mean.
252
The CoV is a widely-used measure of dispersion of a probability distribution. 253
Top-k
: We explicitly report how many new infections within day
t
are caused by
254
which fraction of infected nodes (e.g., 80% of the new infections are caused by 255
20% of the nodes). We report which fraction of new infections can be traced back
256
to (the most harmful) 10%, 20%, and 30% of infected nodes. 257
Offspring
: We report the fraction of nodes (that were infected within day
t
) that
258
lead to 0,1,2, . . . children. 259
Experiments 260
We compare the evolution and dispersion of the four network models. We have two 261
main experiments. In the first experiment (overview in Fig. 7), we study a fixed 262
infection rate (mimicking the case that there is only variation in the connectivity). In 263
the second experiment (Fig. 8), we additionally impose individual variation in the 264
infectiousness λi. Recall that the networks (aside from the complete graph) have 265
approximately the same density (number of edges) and that nodes approximately admit
266
the same mean infectiousness. Thereby, ensure that the resulting differences are solely a
267
consequence of the corresponding variation. 268
Fig. 5 and Fig. 6 summarize some of our findings. Fig. 5 visualizes the effects of 269
adding stochasticity and individual variation to a population. Fig. 6 highlights the 270
different dynamics emerging on different contact networks. 271
Python code is available at www.github.com/gerritgr/Covid19Dispersion.272
March 25, 2021 9/21
Homogeneous Infectiousness Heterogeneous Infectiousness
Fig 6. Fraction of nodes in I1(y-axis) over time Left: Fixed infection rate λ.Right:
Node-based infection rate λidrawn from an exponential distribution. Note the large
difference between the two evolutions on the Watts–Strogatz (WS) networks.
Setup. For each network, we perform 1000 simulation runs on a network with 1000 273
nodes. Networks are generated such that the mean degree is approximately eight. For 274
network models where we cannot directly control kmean, we start by generating sparse 275
networks and increase the density until
kmean
has the desired value. In each simulation
276
run, we start with three randomly chosen infected nodes in
I1
(to reduce the likelihood
277
of initial instantaneous die-outs). The ODE (cf. Fig. 6) starts with a value of 3
/
1000 in
278
I1. The number of simulation runs is enough to estimate the mean fractions (and the 279
standard deviations) corresponding to each compartment with high accuracy. 280
Quantities of interest. We characterize epidemics in terms of the evolution of 281
population fractions, that is, mean fraction of nodes in compartment S,I1,Rfor each 282
time point t(the remaining compartments evolve approximately proportional to I1,283
thus, we leave them out for clarity). This evolution informs about the time point and 284
the height of the infection peak and the final epidemic size (or HIT) that is equivalent 285
to the (mean) fraction of recovered nodes when the epidemic is over (which is mostly 286
the case at 200 time units). Note that the final death toll is proportional to the final 287
epidemic size. 288
Moreover, we study the effective reproduction number Rt(2nd row in Fig. 7 and 289
Fig. 8). We define
Rt
to be the average number of secondary infections for a node that
290
got exposed at day tN0(we discretize time for this purpose). Thereby, we also 291
report an empirical
R0
based on the three initially infected nodes that diverges slightly
292
from the theoretical R0in Table 1. Dispersion is quantified using the three techniques 293
explained in Section (2nd to 4th row in Fig. 7 and Fig. 8). 294
Experiment 1: Network-based connectivity heterogeneity 295
Results for a fixed
λ
on different network types are shown in Fig. 7. In most simulation
296
runs, the power-law dynamics admits a very early peak and the epidemic dies out early
297
with a comparably small final epidemic size. This effect can directly be attributed to 298
the hubs that get infected very early (because of their high centrality) and jump-start 299
the epidemic. In contrast, in household networks—and even more so in WS 300
networks—the infection peak is lower and happens at a later time point. This is no 301
surprise as the connectivity in both networks imposes a level of locality that slows down
302
the propagation. For better visibility, the differences in the infection curve (based on
I1
)
303
are summarized in Fig. 6. We also see that the complete graph behaves very similarly to
304
the ODE model. 305
The effective reproduction Rtstarts from around 2.5 (the theoretical 306
March 25, 2021 10/21
Complete Household Watts–StrogatzPower-law
S
S
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Det ohne geo
Fig 7. Experiment 1: Epidemic dynamics of different network types. Row 1:
Evolution in terms of mean fractions (and standard deviation) in each compartment
over time. Row 2: Effective reproduction number Rt(the empirical R0is shown as a
black triangle) and coefficient of variation of the offspring distribution (with 95% CI,
note a significant amount of noise in the power-law case). Row 3: Top-kplots: The
fraction of new infections that can be attributed to a particular fraction of infected
nodes.
Row 4
: Characterization of the offspring distribution in terms of the fraction of
nodes that cause a specific number of secondary infections.
March 25, 2021 11/21
overapproximation) and decreases in most cases monotonically. The exception is again 307
the power-law network where hubs cause a huge jump of
Rt
in the first day from 2
.
5 to
308
around 12. This jump is also reflected in the dispersion measure, most noticeable in the
309
offspring plot (4th row). The power-law topology generally admits a higher dispersion 310
than the other networks. For instance, the fraction of nodes with zero offspring is much
311
higher. Moreover, on most days, the top 20% of the infected nodes account for more 312
than 80% of the new infections which would roughly fit the estimations for COVID-19 313
in a typical population. In the other network types, there is less temporal variation of 314
the dispersion. The dispersion is the lowest in the WS networks which is unsurprising as
315
all nodes have degree 8 which provides an upper bound to the offspring number. 316
Generally speaking, we see that dispersion can be measured robustly using the empirical
317
offspring distribution. 318
Experiment 2: Individual differences in infectiousness 319
In this experiment, we draw in each simulation run for each node
i
a random
λi
that is
320
distributed according to
ν
. Here we use an exponential distribution. The effects on the
321
evolution and dispersion are reported in Fig. 8. In all networks, the epidemic becomes 322
“weaker” in terms of final epidemic size and infection peak height (see also Fig. 6). This
323
effect is strongest in the WS network (where the epidemic dies out almost immediately)
324
and weakest in the complete graph. This is also mirrored in the difference of R0325
compared to Experiment 1 (as explained in Fig.-3, the relationship between
λ
and
R0
is
326
now non-linear). The complete graph leaves R0almost unchanged (i.e., around 2.5) 327
while it goes down to around 2.0 in the WS network. Effects on the household and 328
power-law network are less drastic but still evident. 329
Regarding the dispersion, the differences to Experiment 1 are expectable. The 330
variance in the empirical offspring distribution generally increases. Interestingly, this 331
happens in all networks by roughly the same amount regardless of whether the 332
dispersion was high or low in the first case. We can also consistently see the change in 333
dispersion in all three dispersion measures, but it is especially evident in the top-
k
plots
334
(3rd row). It is also interesting to see that all networks admit a characteristic signature
335
in the histogram of the offspring distribution (4th row). The infection rate variation 336
shifts these plots (in particular, because the number of nodes without offspring 337
increases) but they still entail a clear distinction between networks. 338
We also tested uniform and gamma distributions for ν(results not shown), and 339
found that the epidemic generally becomes weaker with higher variance. We expect that
340
this is due to an increased likelihood of local and global die-outs. 341
March 25, 2021 12/21
Complete Household Watts–StrogatzPower-law
S
S
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I1
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Expo
ohne geo
Fig 8. Experiment 2: Epidemic dynamics with individual infectiousness λiRow 1:
Evolution in terms of mean fractions in each compartment over time
Row 2
: Effective
reproduction number Rtand coefficient of variance of the offspring distribution. Row
3: Top-kplots: what fraction of new infection can be attributed to which fraction of
infected nodes.
Row 4
: Characterization of the offspring distribution in terms of what
fraction of nodes have how many offspring (infect how many neighbors).
March 25, 2021 13/21
Discussion 342
The two experiments show that heterogeneous interaction structures and variations in 343
infectiousness strongly influence key quantities of an epidemic. However, there are 344
important differences between the sources of variations: 345
The existence of hubs in the network can cause Rtto increase, variations in λ346
generally do not cause this behavior. 347
Different networks with the same
kmean
and a fixed
λ
will (approximately) admit
348
the same R0. However, choosing densities νof different form (while keeping the 349
mean of the distribution fixed) changes R0.350
Varying infectiousness generally weakens the epidemic’s impact. Varying the 351
connectivity can make some aspects of the epidemic worse (e.g., height of the 352
infection peak in power-law networks). 353
λhas the weakest influence when the interaction structure is homogeneous (i.e., 354
on a complete graph) and the strongest when the interaction structure is based on
355
locality (the average distances in the graph increase) as in the Watts–Strogatz 356
network. 357
Varying λincreases the stochasticity of the epidemic. 358
The interaction structure has a large influence on the dispersion. Individual 359
infectiousness variations induces a smaller but consistent increase of dispersion. 360
Hubs influence how the dispersion changes over time. Infectiousness variations 361
increase the dispersion consistently for all time points. 362
Our results underline that networks are a feasible tool to encode a wide variety of 363
different features of a population’s interaction structure. Generally speaking, it is not 364
surprising that some networks support the formation of epidemics better than others. 365
To some extent, this has been studied in terms of the epidemic threshold of graphs [84].
366
However, the variety of the influence of the networks and the interplay between 367
heterogeneity in the degree of infectiousness and dispersion is remarkable and, to our 368
knowledge, underexplored in literature. 369
There are even further possibilities to adjust population heterogeneity, e.g., by 370
adding non-Markovian residence times in the compartments, by varying the remaining 371
transition rates, or by imposing more temporal variability in infectiousness. Our results
372
already show that models, based on point estimators of population averages (i.e., most
373
mean-field ODE models), are not adequate for analyzing or predicting the dynamics of
374
an epidemic. 375
Regarding the dispersion, we see that no network structure by itself leads to a 376
dispersion where 80% of the infections are caused by only 15% of the infected nodes [19]
377
(at least initially). However, the differences between networks are remarkable. From 378
branching process theory, it is known that a higher dispersion increases the die-out 379
probability [12]. Generally, this effect also holds for networks. For a fixed network, 380
increasing dispersion by using a probabilistic infection rate does, in fact, increase the 381
die-out probability. However, the network topology strongly modulates the strength of 382
this effect. 383
Conclusively, we find that in most cases population diversity makes an epidemic less
384
harmful but increases the dispersion and the variability of the evolution. Hubs in the 385
contact networks are an exception to this rule. These are drivers of the epidemic as they
386
become infected very early and infect many others. This distinguishes them from very 387
March 25, 2021 14/21
infectious people (due to a high viral load) with an average number of contacts who also
388
potentially infect many others. However, a high infectiousness alone does not make 389
them more likely to be infected early enough (i.e., on average earlier than other nodes)
390
to cause an early explosive surge of the epidemic. Hubs also highlight that the effective
391
reproduction number can change significantly while the environmental conditions 392
remain the same simply because the prevalence shifts towards highly connected 393
individuals in the beginning. 394
Considering that an exponentially distributed λican be considered a fairly strong 395
assumption about individual differences, our work can—with necessary caution—be 396
seen as further evidence that the network structure plays a more important role for the
397
dispersion than individual viral load variability. 398
Transferring these characteristics to NPIs, our work indicates that reducing 399
long-range connections (e.g., by corresponding mobility restrictions) and keeping 400
degree-variability small (to avoid hubs) are particularly effective control strategies. 401
Reducing mobility seems to be especially effective for overdispersed epidemics. Note 402
that the differences between the WS networks (that admit a high level of locality) and 403
the other networks become even more evident when we vary infectiousness. This can be
404
explained by the observation that in overdispersed epidemics the virus has to be 405
introduced to a susceptible population multiple times before an outbreak becomes likely.
406
Conclusions 407
We tested the influence of heterogeneity in the population’s interaction structure and 408
degree of individual infectiousness on the dynamical evolution of an epidemic. We find
409
that the dynamics depends strongly on these properties and that there is an intriguing
410
interplay between these two sources of variation. Our work also highlights the role of 411
small-worldness, local die-outs, and super-spreaders for an epidemic. Discussing 412
epidemics in terms of population averages does not adequately reflect the complexity of
413
the emerging dynamics. Specifically, the widely-used class of ODE models cannot 414
accurately capture population heterogeneity. Naturally, mathematical modeling is based
415
on assumptions and abstractions. However, heterogeneity seems to be particularly vital
416
and excluding it should only be done with great caution. 417
On a high-level, this work highlights limitations of certain model classes and shows 418
that subtle differences in assumptions can make important differences. For future work,
419
it would be interesting to study the effects of heterogeneity and their implications for 420
NPIs empirically. Moreover, we plan to test implications of further sources of 421
heterogeneity, for instance regarding compliance with NPIs, susceptibility to infections,
422
or whether people tend to meet indoors or outdoors. 423
In a broader spectrum, population heterogeneity is only one aspect that may cause 424
models to perform much worse in the real-world than one might expect. This simulation
425
study is a reminder that models are prone to hidden assumptions, and that we should 426
be cautious with their interpretation. 427
Acknowledgments 428
This work was partially supported by the DFG project MULTIMODE. 429
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... Next, it might be interesting to connect information spreading in active matter with corresponding results in other research fields, such as network theory or epidemiology. Especially in the latter, the effects of network heterogeneity on the spreading of diseases is a widely studied aspect [6,[61][62][63]. Finally, it would seem interesting to pursue the question how the interaction rules in the VM can be optimized to facilitate information transfer. ...
Chapter
This concluding chapter presents a summary of the research findings in the previous chapters, along with some reflections for each of the five themes of the book and a discussion of necessary future responses (post-pandemic or in the event of a new pandemic) and topics that require further exploration. The pandemic brought into sharp relief pre-existing social disparities and affected vulnerable populations the most. The economic impacts of the pandemic were diverse and varied by geography, but again certain geographies and economic sectors were more buffered from negative outcomes than others. A lesson and a challenge for policymakers is to find ways to understand and reduce these disparities, instead of pushing them under the rug. The impacts on mobility and travel were dramatic as total trips decreased, transit usage fell dramatically, and telecommuting and active modes of transportation increased. Some positive impacts included an improved air quality, a reduced number of traffic crashes, and a proliferation of walking and biking in some neighbourhoods. As cities are slowly recovering from the pandemic, the challenge is to keep the positive impacts but also find ways to help the transit industry rebound from its plunge. Long-term impacts of the pandemic in terms of changing patterns of work and work arrangements, shopping, recreation, and other human activities that will affect travel need additional time and more research to discern.
ResearchGate has not been able to resolve any references for this publication.