PreprintPDF Available
Preprints and early-stage research may not have been peer reviewed yet.

Abstract

Human mobility is the fuel of global pandemics. In this simulation 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 depends 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 overdispersion. Conclusively, this means that implementing green zones is easier for epidemics with overdispersed transmission dynamics (e.g., 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.
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 aiAis
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:
R0R0, 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 OiA, 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(|| xixj||).
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.
References
1. et al., B.: Inferring the effectiveness of government interventions against covid-19.
Science 371(6531) (2021)
2. et al., S.: Transmission heterogeneities, kinetics, and controllability of sars-cov-2.
Science 371(6526) (2021)
3. Kupferschmidt, K.: Why do some covid-19 patients infect many others, whereas
most don’t spread the virus at all. Science 10 (2020)
4. Lloyd-Smith, J.O., Schreiber, S.J., Kopp, P.E., Getz, W.M.: Superspreading and
the effect of individual variation on disease emergence. Nature 438(7066), 355–359
(2005)
5. Siegenfeld, A.F., Bar-Yam, Y.: The impact of travel and timing in eliminating covid-
19. Communications Physics 3(1), 1–8 (2020)
1github.com/gerritgr/SpatialBranchingProcess
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
Governments are attempting to control the COVID-19 pandemic with nonpharmaceutical interventions (NPIs). However, the effectiveness of different NPIs at reducing transmission is poorly understood. We gathered chronological data on the implementation of NPIs for several European, and other, countries between January and the end of May 2020. We estimate the effectiveness of NPIs, ranging from limiting gathering sizes, business closures, and closure of educational institutions to stay-at-home orders. To do so, we used a Bayesian hierarchical model that links NPI implementation dates to national case and death counts and supported the results with extensive empirical validation. Closing all educational institutions, limiting gatherings to 10 people or less, and closing face-to-face businesses each reduced transmission considerably. The additional effect of stay-at-home orders was comparatively small.
Article
Full-text available
A long-standing question in infectious disease dynamics concerns the role of transmission heterogeneities, driven by demography, behavior and interventions. Based on detailed patient and contact tracing data in Hunan, China we find 80% of secondary infections traced back to 15% of SARS-CoV-2 primary infections, indicating substantial transmission heterogeneities. Transmission risk scales positively with the duration of exposure and the closeness of social interactions and is modulated by demographic and clinical factors. The lockdown period increases transmission risk in the family and households, while isolation and quarantine reduce risks across all types of contacts. The reconstructed infectiousness profile of a typical SARS-CoV-2 patient peaks just before symptom presentation. Modeling indicates SARS-CoV-2 control requires the synergistic efforts of case isolation, contact quarantine, and population-level interventions, owing to the specific transmission kinetics of this virus.
Article
Full-text available
While the spread of communicable diseases such as coronavirus disease 2019 (COVID-19) is often analyzed assuming a well-mixed population, more realistic models distinguish between transmission within and between geographic regions. A disease can be eliminated if the region-to-region reproductive number—i.e., the average number of other regions to which a single infected region will transmit the disease—is reduced to less than one. Here we show that this region-to-region reproductive number is proportional to the travel rate between regions and exponential in the length of the time-delay before region-level control measures are imposed. If, on average, infected regions (including those that become re-infected in the future) impose social distancing measures shortly after experiencing community transmission, the number of infected regions, and thus the number of regions in which such measures are required, will exponentially decrease over time. Elimination will in this case be a stable fixed point even after the social distancing measures have been lifted from most of the regions.
Article
Full-text available
Background: A novel coronavirus disease (COVID-19) outbreak has now spread to a number of countries worldwide. While sustained transmission chains of human-to-human transmission suggest high basic reproduction number R 0 , variation in the number of secondary transmissions (often characterised by so-called superspreading events) may be large as some countries have observed fewer local transmissions than others. Methods: We quantified individual-level variation in COVID-19 transmission by applying a mathematical model to observed outbreak sizes in affected countries. We extracted the number of imported and local cases in the affected countries from the World Health Organization situation report and applied a branching process model where the number of secondary transmissions was assumed to follow a negative-binomial distribution. Results: Our model suggested a high degree of individual-level variation in the transmission of COVID-19. Within the current consensus range of R 0 (2-3), the overdispersion parameter k of a negative-binomial distribution was estimated to be around 0.1 (median estimate 0.1; 95% CrI: 0.05-0.2 for R0 = 2.5), suggesting that 80% of secondary transmissions may have been caused by a small fraction of infectious individuals (~10%). A joint estimation yielded likely ranges for R 0 and k (95% CrIs: R 0 1.4-12; k 0.04-0.2); however, the upper bound of R 0 was not well informed by the model and data, which did not notably differ from that of the prior distribution. Conclusions: Our finding of a highly-overdispersed offspring distribution highlights a potential benefit to focusing intervention efforts on superspreading. As most infected individuals do not contribute to the expansion of an epidemic, the effective reproduction number could be drastically reduced by preventing relatively rare superspreading events.
Article
Full-text available
Models can help us determine how to stop the spread of COVID-19, but it is important to distinguish between that which models can and cannot predict. All models’ assumptions fail to describe the details of most real-world systems, but these systems may possess large-scale behaviors that do not depend on all these details. A simple model that correctly captures these large-scale behaviors is useful; a complicated model that gets some details correct but mischaracterizes the large-scale behaviors is misleading at best. Large-scale behaviors of the COVID-19 pandemic include the rate of exponential growth/decay in the number of active infections in each region, as well as the transmission rates between regions. The values of these parameters, both of which can be controlled with interventions, determine whether the large-scale behavior of COVID-19 is that of exponential spread until saturation or exponential decay until elimination. We may not be able to precisely predict the trajectory of the epidemic under any given set of interventions, but we know that a strong enough set of interventions can ensure we are in the latter regime.
Article
Full-text available
COVID-19 caused rapid mass infection worldwide. Understanding its transmission characteristics, including heterogeneity and the emergence of super spreading events (SSEs) where certain individuals infect large numbers of secondary cases, is of vital importance for prediction and intervention of future epidemics. Here, we collected information of all infected cases (135 cases) between 21 January and 26 February 2020 from official public sources in Tianjin, a metropolis of China, and grouped them into 43 transmission chains with the largest chain of 45 cases and the longest chain of four generations. Utilizing a heterogeneous transmission model based on branching process along with a negative binomial offspring distribution, we estimated the reproductive number R and the dispersion parameter k (lower value indicating higher heterogeneity) to be 0.67 (95% CI: 0.54–0.84) and 0.25 (95% CI: 0.13–0.88), respectively. A super-spreader causing six infections was identified in Tianjin. In addition, our simulation allowing for heterogeneity showed that the outbreak in Tianjin would have caused 165 infections and sustained for 7.56 generations on average if no control measures had been taken by local government since 28 January. Our results highlighted more efforts are needed to verify the transmission heterogeneity of COVID-19 in other populations and its contributing factors.
Article
Full-text available
Population-level analyses often use average quantities to describe heterogeneous systems, particularly when variation does not arise from identifiable groups. A prominent example, central to our current understanding of epidemic spread, is the basic reproductive number, R(0), which is defined as the mean number of infections caused by an infected individual in a susceptible population. Population estimates of R(0) can obscure considerable individual variation in infectiousness, as highlighted during the global emergence of severe acute respiratory syndrome (SARS) by numerous 'superspreading events' in which certain individuals infected unusually large numbers of secondary cases. For diseases transmitted by non-sexual direct contacts, such as SARS or smallpox, individual variation is difficult to measure empirically, and thus its importance for outbreak dynamics has been unclear. Here we present an integrated theoretical and statistical analysis of the influence of individual variation in infectiousness on disease emergence. Using contact tracing data from eight directly transmitted diseases, we show that the distribution of individual infectiousness around R(0) is often highly skewed. Model predictions accounting for this variation differ sharply from average-based approaches, with disease extinction more likely and outbreaks rarer but more explosive. Using these models, we explore implications for outbreak control, showing that individual-specific control measures outperform population-wide measures. Moreover, the dramatic improvements achieved through targeted control policies emphasize the need to identify predictive correlates of higher infectiousness. Our findings indicate that superspreading is a normal feature of disease spread, and to frame ongoing discussion we propose a rigorous definition for superspreading events and a method to predict their frequency.
Emergence and rapid spread of a new severe acute respiratory syndrome-related coronavirus 2 (sars-cov-2) lineage with multiple spike mutations in south africa
  • H Tegally
  • E Wilkinson
  • M Giovanetti
  • A Iranzadeh
  • V Fonseca
  • J Giandhari
  • D Doolabh
  • S Pillay
  • E J San
  • N Msomi
Tegally, H., Wilkinson, E., Giovanetti, M., Iranzadeh, A., Fonseca, V., Giandhari, J., Doolabh, D., Pillay, S., San, E.J., Msomi, N., et al.: Emergence and rapid spread of a new severe acute respiratory syndrome-related coronavirus 2 (sars-cov-2) lineage with multiple spike mutations in south africa. medRxiv (2020)