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Epidemic Overdispersion Strengthens the

Eﬀectiveness of Mobility Restrictions

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. Human mobility is the fuel of global pandemics. In this sim-

ulation study, we analyze how mobility restrictions mitigate epidemic

processes and how this mitigation is inﬂuenced by the epidemic’s degree

of dispersion.

We ﬁnd that (even imperfect) mobility restrictions are generally eﬃcient

in mitigating epidemic spreading. Notably, the eﬀectiveness strongly de-

pends on the dispersion of the oﬀspring distribution associated with the

epidemic. We also ﬁnd that mobility restrictions are useful even when

the pathogen is already prevalent in the whole population. However, also

a delayed implementation is more eﬃcient in the presence of overdisper-

sion. Conclusively, this means that strategies based on mobility restric-

tions, like green zones, are easier to implement when the transmission

dynamics admits overdispersion (e.g., in the case of COVID-19).

To study these relationships at an appropriate level of abstraction, we

propose a spatial branching process model combining the ﬂexibility

of stochastic branching processes with an agent-based approach allowing

a conceptualization of locality, saturation, and interaction structure.

Keywords: COVID-19 ·Epidemic Simulation ·SIR ·Dispersion ·Overdis-

persion ·Spatial Branching Process ·Mobility Restriction ·Green Zone

·Herd Immunity Threshold ·Final Epidemic Size ·B117 ·501YV2.

1 Introduction

In 2020, the COVID-19 pandemic emerged and countries all over the world dis-

cussed which non-pharmaceutical interventions (NPIs) to implement in order

to suppress or mitigate the spread of the novel SARS-CoV-2 pathogen [1]. Cur-

rently, in early 2021, a similar situation arises due to the uncertainty surrounding

the appearance of novel variants. Mobility restrictions are an important class of

interventions, ranging from closures of international borders to complete stay-

at-home orders. The hope is that these measures constrain local outbreaks and

that the virus reaches fewer susceptible host populations. Mobility restrictions

could also help in creating so-called green zones within an infected population

[5]. However, the eﬀectiveness of NPIs depends on the epidemic’s properties.

Overdispersion is a property of an epidemic’s propagation. Speciﬁcally, each

infected individual creates a certain number of secondary infections (aka oﬀ-

spring). This number depends on many factors and can be modeled using a

2 G. Großmann et al.

Step 0 Step 5 Step 10 Step 20 Step 50

Fig. 1: Spatial branching process: Example trajectory with 104agents (suscep-

tible: blue, infected: red, recovered: green) using a Gaussian spatial interaction

kernel with σ= 0.008, two oﬀspring-candidates (deterministic), a travel proba-

bility pt= 0.1 (until 200 agents are recovered, then pt= 0.0), and 30 initially

infected agents.

discrete probability distribution called oﬀspring distribution. The mean of the

oﬀspring distribution at time point tis the eﬀective reproduction number Rt.

Traditionally, this distribution is the same for all agents and is independent of

time t[4]. The oﬀspring distribution can be characterized in terms of its disper-

sion (a measure of variance w.r.t. the mean). In the presence of high dispersion,

many individuals produce zero or very few oﬀspring while few individuals (so-

called supers-spreaders) infect many others. Typically, the oﬀspring number is

generated as follows (i): sample an individual reproduction number Rifrom a

Gamma distribution with mean R0and shape parameter kand (ii): sample the

oﬀspring number of agent iusing a Poisson distribution with mean Ri. Disper-

sion is typically quantiﬁed using the shape (aka dispersion) parameter k(smaller

kimplies higher dispersion) [4]. Combining a Gamma with a Poisson distribution

results in a negative binomial distribution (NBD). For COVID-19 it has been

reported that 80% of new infections can be traced back to only 15% of infected

individuals relating to a karound 0.3 [2]. Of particular importance for this work

is the association of high dispersion with an increased die-out probability. Hence,

it has been speculated that SARS-CoV-2 has to be introduced multiple times

to a susceptible population in order to ignite an outbreak [3]. This property

suggests a higher eﬃciency of mobility restrictions.

2 Spatial Branching Process

The default model type to study dispersion is a stochastic branching process [4]

(BP). The state of a BP is a tree that grows over time. The children (oﬀspring)

of a node correspond to the individuals infected by that node. In each step,

the oﬀspring number is sampled for each leaf. The epidemic is over when all

leaves have zero oﬀspring. An advantage of the BP model is the small number of

parametric assumptions (all relevant aspects of the pathogen/environment are

modeled with the oﬀspring distribution). The BP model has two disadvantages

for our purposes. There is no inherent form of locality in the model and no

natural saturation (typically, an epidemic slows down when more agents become

recovered/immune).

We account for this by placing the population (set of agents) in Euclidean

space. The number of oﬀspring (oﬀspring candidates to be more precisely) is still

Mobility Restrictions and Overdispersion 3

Dispersion

Fig. 2: F.l.t.r.: Results for Exp. 1to Exp. 3.x-axis: Locality/mobility measure.

y-axis: Final epidemic size/herd immunity threshold. Color indicates dispersion

parameter k(smaller kimplies higher dispersion).

sampled from an oﬀspring distribution but the spatial relationship inﬂuences

which individuals are chosen. An infection attempt is rejected if the sampled

oﬀspring candidate is already immune or infected. To model spatial movements

more explicitly, we can randomly reposition agents. Note that the set of agents is

ﬁxed from the beginning (we refer to the repository or Fig. 1 for an visualization).

Model State. The global state is given by a set of agents A. Each ai∈Ais

annotated with a position xi∈[0,1]2and a local state si∈ {S, I , R}(susceptible,

infected, recovered). We initialize xiaccording to a 2d-density ν.

Model Dynamics. The global state changes randomly in discrete-time according

to a discrete univariate oﬀspring distribution α, a spatial interaction kernel β:

R≥0−→ R≥0, and a travel probability pt∈[0,1]. In each time step, for each

infected agent ai:

1. Reposition aiwith probability ptaccording to ν.

2. Generate oﬀspring candidates, denoted Oi⊂A, and infect all susceptible

agents in Oi.

3. Set si=R.

Regarding (2): The oﬀspring-count |Oi|is sampled from α. Given |Oi|, choose

each aj(i6=j) to be in Oiwith a probability proportional to β(|| xi−xj||).

3 Experiments

We use 104agents, R0= 2.0, a Gaussian spatial kernel, and a mixture of 16 2d-

Gaussians to generate the spatial positions (thereby, we mimic some population

structure and reduce the probability of inter-cluster transmissions). Results are

given in Fig 2. We compare a ﬁxed oﬀspring distribution where the (unsaturated)

oﬀspring count is always R0= 2 with an NBD with varying k. Note that k=

∞leads to a Poisson oﬀspring distribution. In the ﬁrst experiment, we vary

the variance, σ, of the spatial interaction kernel and set pt= 0. Thereby, we

directly measure the inﬂuence of locality (smaller σimplies higher locality). In

4 G. Großmann et al.

the second experiment, we ﬁx σ= 0.007 and vary pt. Note that σ= 0.007 is

such that the epidemic still dies out early with high probability for all oﬀspring

distributions with mean 2.0 (as long as no traveling is happening). This way, we

measure mobility explicitly. In the third experiment, we study delayed mobility

restrictions. We wait until 200 agents are infected (over varying pt), and then

set pt= 0.

The experiments consistently show that reducing mobility (in terms of σor

pt) has a stronger impact on the ﬁnal epidemic the more dispersion is present in

an epidemic’s transmission dynamics. This applies even if the mobility restriction

is imperfect or implemented with some delay. Julia code is available1.

4 Conclusion

The relationship between dispersion, branching processes, and locality is under-

explored in literature. We believe our framework provides the right level of ab-

straction to study this relationship and hope to spark interest in theoretical and

practical work in this matter. Calibration to real-world data and comparisons to

other model types is still needed to deepen the understanding of dispersion and

locality. Moreover, it would be worthwhile to investigate if one can ﬁnd opti-

mal borders or levels on which mobility restrictions constitute the best trade-oﬀ

between social costs and eﬀectiveness. Understanding how to implement hier-

archical mobility restrictions is also largely an open problem. Conclusively, this

work can be seen as evidence that methods based on mobility restrictions are

more eﬀective in the presence of overdispersion. It is to be expected that this

also holds for green zones as new clusters are unlikely to emerge from single

pathogen introductions to a green zone.

Acknowledgements This work was partially supported by the DFG project

MULTIMODE.

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most don’t spread the virus at all. Science 10 (2020)

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the eﬀect of individual variation on disease emergence. Nature 438(7066), 355–359

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1github.com/gerritgr/SpatialBranchingProcess