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Urban mobility needs alternative sustainable travel modes to keep our pandemic cities in motion. Ride-pooling, where a single vehicle is shared by more than one traveller, is not only appealing for mobility platforms and their travellers, but also for promoting the sustainability of urban mobility systems. Yet, the potential of ride-pooling rides to serve as a safe and effective alternative given the personal and public health risks considerations associated with the COVID-19 pandemic is hitherto unknown. To answer this, we combine epidemiological and behavioural shareability models to examine spreading among ride-pooling travellers, with an application for Amsterdam. Findings are at first sight devastating, with only few initially infected travellers needed to spread the virus to hundreds of ride-pooling users. Without intervention, ride-pooling system may substantially contribute to virus spreading. Notwithstanding, we identify an effective control measure allowing to halt the spreading before the outbreaks (at 50 instead of 800 infections) without sacrificing the efficiency achieved by pooling. Fixed matches among co-travellers disconnect the otherwise dense contact network, encapsulating the virus in small communities and preventing the outbreaks.
Methodology at glance: We consider travel demand for ride-pooling trips (a), for which we compute a shareability network (b) with a given behavioural parameters β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}, system design λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} and alternatives’ attractiveness ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document}. We simulate the day-to-day evolution of spreading until the virus is halted. Each day we obtain the daily demand (c), consisting of those who want and can travel (decided to travel with probability p and are not quarantined). Daily trip demand is optimally assigned to shared rides, which forms the contact network (d) on which virus spreading is then modelled (e). Starting from initially infected travellers, each day we simulate epidemic transitions: susceptible travellers are infected by infected co-travellers who quarantine after 7 days and return immune to the system after 14 days.
… 
Number of infected travellers over the course of epidemic outbreaks, with various settings of initially infected (rows) and demand stability (columns), bold lines denote averages over all experiments (shown individually using thin lines). With an unstable demand (0.65), 20 initially infected always triggers transmission reaching at least 60 travellers (out of 2000) and lasting at least 60 days. Yet in most other configurations results are less stable and actual outbreaks strongly depends on the location of initially infected, revealing a strongly heterogeneous structure of the underlying contact network. In most cases we can observe a smooth, log-normal shape with a strong outbreak in the first phase, exponentially decaying in the latter phases. Mean temporal profiles of outbreaks are consistently following the trend of decreasing when the number of initial infections is lower and a demand pattern becomes more stable . For stable demand patterns, we can observe that the number of infected drops when initially infected quarantine, followed by a smooth transmission in the second phase when demand still fluctuates (p=0.85\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=0.85$$\end{document}) or halted immediately (p>0.95)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>0.95)$$\end{document}. Typically, stabilising the demand halts the epidemic faster. For p>0.85\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>0.85$$\end{document} epidemic is over in less than 50 days, while for p = 0.65 it can remain active after 100 days (regardless number of initially infected). Despite a clear and strong trend, some simulated outbreaks do not follow the same patterns. We can observe for example an exceptionally high number of infections for p = 0.8 starting from 2 infections when a highly connected hub got infected; quickly halted spreading from 10 infections at p = 0.65; or second wave at p = 0.75 and 10 initial infections.
… 
(a) Number of eventually infected travellers for varying demand stability p and initially infected travellers. Distributions based on 20 replications (mean within interquantile box and whiskers from min to max values). Initially infected 20 travellers may spread the virus to almost 40% of the population (800 out of 2000 travellers). Yet as long as demand becomes stabilised, outbreaks start being contained. Even a large number of initially infected does not reach more than 10% of the population if demand stability is set to 90% and is eventually contained below 60 travellers (3%) for fully stable demand. The variability of outbreaks also decreases as the demand stabilises: 10 initial infections may reach between 40 and 100 travellers if p=0.9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=0.9$$\end{document}, while the range expands from 50 to over 400 for p=0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=0.7$$\end{document}. The lower bound increases when the number of initially infected is high, making outbreaks more predictable, unlike the ones starting from a small number of infections, for which variability is greater. Importantly, stabilizing the demand does not reduce the efficiency, as we report in (c), where the mean occupancy (key efficiency indicator of ride-pooling) remains stable as demand stabilizes. Notably, the importance of demand stabilization increases with the demand level as we demonstrate on panel (b) which shows share of infected travellers changing with a demand for various p’s. Each dot is the average from 20 replications. For all the values of p share of infected individuals scales with the demand level as Aexp(αQ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\exp (\alpha Q)$$\end{document}, marked as trendlines on (b), with p-values and R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2$$\end{document} reported in the text. For demand levels below 1500, the virus rarely reaches more than 4% of the travellers. In contrast, when the demand level is 2500, the epidemic may reach up to 10% when p=0.8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=0.8$$\end{document} or be contained below 2% for a stable demand (p>0.9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>0.9$$\end{document}), which underlines the importance of control measures for higher demand levels.
… 
(a) Average node degree in the evolving contact networks. Regardless demand stability p, an average traveller is linked to 1.7 other travellers each day. Yet if the demand is unstable it evolves, after 10 days it reaches 1.9 if p=0.99\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=0.99$$\end{document}, 2.5 if p=0.95\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=0.95$$\end{document} and goes beyond 3 if p<0.8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p<0.8$$\end{document}. (b) Mean transmission rate r (number of new infections per infected) distributions. The long tails for low demand stability reveal the super-spreaders (transmitting to 5 and more travellers). For a stable demand initial infections does not manage to transmit a disease effectively, eventually reducing transmissivity below 1 when p>0.9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>0.9$$\end{document}. (c) Insights into the first phase of the epidemic outbreak in the case of 10 initial infections. When first infected travellers are diagnosed after 7 days, their accumulated contact network may vary from 18 to over 60 infected travellers. If contact tracing and mitigation strategies are put in place, already infected travellers may be identified and quarantined before the second outbreak after day 7.
… 
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
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Modelling virus spreading
in ride‑pooling networks
Rafał Kucharski1,2*, Oded Cats1 & Julian Sienkiewicz3
Urban mobility needs alternative sustainable travel modes to keep our pandemic cities in motion.
Ride‑pooling, where a single vehicle is shared by more than one traveller, is not only appealing for
mobility platforms and their travellers, but also for promoting the sustainability of urban mobility
systems. Yet, the potential of ride‑pooling rides to serve as a safe and eective alternative given the
personal and public health risks considerations associated with the COVID‑19 pandemic is hitherto
unknown. To answer this, we combine epidemiological and behavioural shareability models to
examine spreading among ride‑pooling travellers, with an application for Amsterdam. Findings
are at rst sight devastating, with only few initially infected travellers needed to spread the virus
to hundreds of ride‑pooling users. Without intervention, ride‑pooling system may substantially
contribute to virus spreading. Notwithstanding, we identify an eective control measure allowing
to halt the spreading before the outbreaks (at 50 instead of 800 infections) without sacricing the
eciency achieved by pooling. Fixed matches among co‑travellers disconnect the otherwise dense
contact network, encapsulating the virus in small communities and preventing the outbreaks.
In the era of widespread concerns about personal safety and exposure to virus transmission, urban mobility faces
an unprecedented challenge13. While mass transit, a crowded backbone of pre-pandemic megacities’ mobility
systems, is under societal pressure due to health concerns related to its potential role in virus spreading4,5, peo-
ple search for other travel alternatives that reduce one’s exposure. e natural reaction of risk-averse travellers
is to opt for individual transport modes, such as private cars6, which can be devastating for the sustainability
of pandemic urban mobility systems7. To counteract this, we explore whether shared mobility may oer an
attractive alternative by eciently serving travel demand using a shared eet while allowing users to avoid the
crowd. Ride-pooling8,9, available via two-sided mobility platforms (such as UberPool and Lyft), has recently
emerged as a travel alternative in cities worldwide and gained attention or researchers studying both specic
systems (like Singapore10 and New York City11) as well as uncovering universal scaling laws governing cities1214
which in turn allow for generalizing the results to any urban system. Travellers, requesting rides are oered
a pooled ride, where they share a single vehicle with co-travellers riding in a similar direction. Despite being
perceived by policymakers as a solution for improving mobility and sustainability by leveraging on the platform
economy revolution, the COVID pandemic led to safety concerns among sharing travellers (worried about their
health), policymakers (concerned about public health and epidemic outbreak) and operators (uncertain about
the future of their business).
While preliminary ndings on COVID-1915 suggest transmission taking place in proximity (e.g. among co-
travellers within the same vehicle), evidence on how and if the virus transmits beyond a single vehicle is lacking.
Indeed, the potential of shared rides to serve as an alternative, in-between the mass transit (where - perceived
or real - virus exposure may be high) and private cars (which generate negative externalities) remains largely
unknown. Will the random infected passenger spread the virus across a large number of travellers across the
network, or will it be encapsulated and thus conned to a distinct community? How many other travellers will
get infected and how will the epidemiological process evolve? Finally, can we mitigate it by eective control and
design measures and thus introduce it to policymakers as a safe alternative? Such questions are valid as COVID-
19 pandemic constantly challenge our current policies16,17 calling for a long-term preparedness18.
e propagation of dierent types of epidemics (biological and social) has long been a playground of network
science community (summed up in the seminal work of Pastor-Satorras etal.19 and reaching as far as to propose
the idea of “physics of vaccination20). Addressing also mobility network (e.g. public transport) structures21,22 with
their complex topology and temporal evolution studies23. Recent COVID-19 propagation studies either follow
a coarse-grained level of aggregated cases2426, adopt purely synthetic network structures27,28 or lack emerging
mobility modes (like ride-pooling) in the picture3,6. is study brings to the front the network evolution (crucial
OPEN
Department of Transport and Planning, Delft University of Technology, Delft, The Netherlands. Department of
Transport Systems, Cracow University of Technology, Cracow, Poland. Faculty of Physics, Warsaw University of
Technology, Warsaw, Poland. *email: r.m.kucharski@tudelft.nl
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in the context of epidemic spreading29,30) and couples it with empirical, behaviour driven contact network9,
specic to ride-pooling, yielding a simulation framework (see Methods for details) capable to provide rich and
realistic insights into possible epidemic outbreaks specic to ride-pooling networks.
We model the evolution of virus spreading in ride-pooling systems through an extensive set of experiments
with demand sampled from actual mobility patterns of aernoon commuters in Amsterdam. e underlying
shareability network (see Methods for details how such network is set up) is the outcome of travellers’ willing-
ness to share, which depends on whether they are suciently close to each other in terms of induced detour
(compatibility of origins and destinations) and delay (compatibility of departure times) compared to a private
ride-hailing. e resulting dynamic, time-dependent contact network31 is subject to day-to-day variations as well
as the results of the iterative SIQR epidemiological model32 (see Methods section for explanation). We exam-
ine the resulting spreading process, i.e. number of infections along with its temporal and spatial evolution. To
instantly show the methodology at glance, Fig.1 presents all steps of our framework and their inter-dependencies,
further detailed in Methods section.
Findings from our extensive simulation study are on rst sight devastating, with only few initially infected
travellers needed to spread the virus to hundreds of ride-pooling users. Even under conservative assumptions
where the driver is not a spreader and mobility pattern is restricted to only two trips per day, the virus makes
its way to infect the majority of the giant component. Introducing natural stochasticity and non-recurrence
inherent to travel demand triggers a virus transmission. Despite slow temporal evolution virus gradually makes
its way towards new communities, neglecting natural spatial barriers. ere seems to be no epidemic threshold
and even two initial infections may trigger an outbreak and reach high transmissivity. is is a very alarming
nding, suggesting that ride-pooling system without intervention may substantially contribute to virus spread-
ing. Nonetheless, we identied eective control measure allowing to halt the spreading before the outbreaks.
Namely, if we trade-o spontaneity of platform-based ride-pooling service and let the operator x matches with
co-travellers, radically dierent image appears. Such setting disconnects the otherwise dense contact network,
containing the virus in small communities and preventing the outbreaks. Notably, this trade-o is not at the cost
of system eectiveness, most importantly not at the cost of occupancy rate, which may remain at the original level.
We argue that under strict demand control measures, mobility platforms may provide an appealing alternative
service in-between public and private transport modes for pandemic reality. Universal properties of ride-pooling
networks12,14,33 allow us to generalise our Amsterdam ndings to a generic systems for which the critical mass
needed to induce sharing comes along with a highly connected shareability network, whereas xing the matches
disconnects it into isolated communities.
Application and Results
To understand how the virus spreads among travellers sharing rides, we conduct a series of experiments within
a realistic travel demand setting of Amsterdam. Aernoon commuters, sampled from the actual trip demand
dataset35, hail a ride from a mobility platform to reach their destination. ey may opt for a shared ride if
reduced trip fare will compensate for any detour and delay imposed by sharing. We consider a system where a
30% discount is oered for sharing and we specify the private ride-hailing ride as an alternative
ǫ
. We employ
behavioural parameters (value-of-time and willingness-to-share
β
s) in-line with recent ndings36,37 and apply
ExMAS algorithm9 to reproduce a behaviourally rich shareability network connecting 3200 travellers to 11,000
Figure1. Methodology at glance: We consider travel demand for ride-pooling trips (a), for which we compute
a shareability network (b) with a given behavioural parameters
β
, system design
and alternatives’ attractiveness
ǫ
. We simulate the day-to-day evolution of spreading until the virus is halted. Each day we obtain the daily
demand (c), consisting of those who want and can travel (decided to travel with probability p and are not
quarantined). Daily trip demand is optimally assigned to shared rides, which forms the contact network (d)
on which virus spreading is then modelled (e). Starting from initially infected travellers, each day we simulate
epidemic transitions: susceptible travellers are infected by infected co-travellers who quarantine aer 7 days and
return immune to the system aer 14 days.
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feasible shared rides (see Methods for algorithm description). e size of travel demand sample is such that,
on one hand, the critical mass needed to induce sharing can be attained and on the other hand, it represents a
relatively low demand levels reached by ride-pooling services so far38. Notably, we model demand for shared
rides which is non-stable and uctuating from one day to the other39, hence each aernoon is comprised of a
slightly or signicantly dierent pool of travellers, controlled through a demand stability parameter p (i.e., the
participation probability, see Methods). To allow for comparisons, while experimenting with p we keep the total
daily number of travellers in the system xed (to 2000), yet we adjust the pool of passenger from which we draw
them on any given day. In our series of experiments we explore demand stability varying from
p=0.65
(where
each day we draw from the pool of
3075 =2000/0.65
travellers) up to
p=1
(where the total demand is assumed
constant). We vary the number of initially infected travellers from 2 to 20. To assess the impact of demand level,
we conduct experiments where we gradually increase it up to 3200 travellers. In order to account for the impact
of their random location, we replicate each scenario 20 times.
We present the results through epidemic evolution plots (Fig.3) for various settings of demand stability and
number of initial infections, summarised with boxplots of the total number of infections in Fig.4a. On Fig.4c we
trace the eciency of ride-pooling across scenarios. In Fig.4b we explore spreading for increasing demand levels
and reveal exponential growth of the share of infected characterized by tting coecients scaling linearly with
participation probability p (Fig.4c). To understand the impact of demand stability on the course of outbreaks
we plot node degree evolution in Fig.5a and the distribution of transmissivity in Fig.5b, further visualised in
terms of its spatial distributions in Fig.6. To demonstrate the potential to control the outbreak, we display its
rst phase in Fig.5c.
Outbreaks. As long as the demand is unstable and varies considerably from one day to the other, the virus
may outbreak even when only a small number of passengers are initially infected. Outbreaks starting from two
infections are highly variable (Fig.3 lower le). For initial spreaders located centrally in a highly connected giant
component of the contact network (Fig.2) the virus outbreaks and eventually infects over 250 travellers during
the course of the spreading, whereas outbreaks from disconnected part of network can be naturally contained
and halted already aer 7 infections (Fig.4a). An outbreak starting from 20 initial spreaders is always devastat-
ing and reaches from 450 up to almost 800 travellers, it needs only few days to reach 100 cases (Fig.3 upper
Figure2. Shareability graph linking 3 200 travellers to 11,000 pooled rides feasible for them (a). Size of nodes
is proportional to degree (number of travellers for shared rides and number of feasible rides for travellers). e
graph structure includes a giant component and a high degree nodes, which may become a super-spreaders, as
well as isolated peripheral nodes, where travellers either cannot nd a feasible match or form a small, isolated
communities from which virus will not outbreak. e actual matching of travellers to shared rides on a single
day (b) has a substantially dierent structure. Here (b) nodes denote travellers, linked if they share a ride.
Single dots are unmatched travellers riding alone, while lines, triangles and squares denote pooled rides of
higher degree (2, 3 and 4, respectively). While the potential shareability (a) is densely connected, matching on a
single day (b) is disintegrated. Each pooled-ride forms an isolated community (i.e. co-travellers within a single
vehicle), with a clique of size bounded with vehicle capacity (four in our case). e virus will spread within
each clique but will not reach beyond it on a given day. However, infected traveller may be assigned to a new
ride on successive day, resulting with virus propagation beyond the single vehicle. Networks visualized with
newtulf34.
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le). Such fast and prevalent spreading can be attributed to a gradually evolving contact network, where each
additional day may bring opportunities to be pooled with a new set of travellers, extending the accumulated
contact network (Fig.5a). Consequently, despite having a low mean node degree (i.e. sharing with few travellers
at the time), some travellers become hubs, spreading the virus to over 10 travellers, resulting with a long tail of
the transmission distribution (Fig.5b). With low demand stability even two infections may spatially penetrate to
all parts of the Amsterdam area, whereas for stable demand the virus may be contained spatially and not spread
beyond its original community (Fig.6).
Scaling for the demand level. We experiment with changing the demand levels, gradually increasing it
from from 100 to 3000 travellers (Fig.4b). For low demand levels the virus cannot spread since the potential
shareability network remains disconnected, i.e. with no giant component (few matches between sparse travellers
are found and trips are rarely pooled). us, below the critical mass of ride pooling, stabilizing demand has a
limited impact. However, as soon as the shareability network includes a larger number of connections (thanks
to more compatible trip groups in the demand set) spreading is triggered and the importance of controlling
becomes evident.
e relation between the share of infected individuals
ni
and the demand level Q can be tted with an expo-
nential function
, allowing to make predictions about the number of people reached by the virus
for higher values of Q than those explored in our study. For all values of p shown in Fig.4b we get high statisti-
cal signicance of coecients A and
α
(in all cases p-value
<.001
), with the coecient of determination
R2
of
0.92,0.93,0.87,0.81,0.73 and 0.52 for demand stabilities p of 0.8,0.85,0.9,0.95,0.99 and 1 respectively. For less
stable demand spreading is ubiquitous and thus less variable, while for stable demand spreading can still remain
contained, leading to signicant variability in the results and lower goodness-of-t.
Ride‑pooling eciency. Ride-pooling needs a critical mass of demand to become ecient and sustain-
able. We report ride-pooling eciency by means of the average occupancy o, i.e. ratio of passenger-kilometer
hours to vehicle kilometer hours. In line with previous studies9, we nd that occupancy is a function of demand
Figure3. Number of infected travellers over the course of epidemic outbreaks, with various settings of initially
infected (rows) and demand stability (columns), bold lines denote averages over all experiments (shown
individually using thin lines). With an unstable demand (0.65), 20 initially infected always triggers transmission
reaching at least 60 travellers (out of 2000) and lasting at least 60 days. Yet in most other congurations results
are less stable and actual outbreaks strongly depends on the location of initially infected, revealing a strongly
heterogeneous structure of the underlying contact network. In most cases we can observe a smooth, log-normal
shape with a strong outbreak in the rst phase, exponentially decaying in the latter phases. Mean temporal
proles of outbreaks are consistently following the trend of decreasing when the number of initial infections
is lower and a demand pattern becomes more stable . For stable demand patterns, we can observe that the
number of infected drops when initially infected quarantine, followed by a smooth transmission in the second
phase when demand still uctuates (
p
=
0.85
) or halted immediately (
p>0.95)
. Typically, stabilising the
demand halts the epidemic faster. For
p>0.85
epidemic is over in less than 50 days, while for p = 0.65 it can
remain active aer 100 days (regardless number of initially infected). Despite a clear and strong trend, some
simulated outbreaks do not follow the same patterns. We can observe for example an exceptionally high number
of infections for p = 0.8 starting from 2 infections when a highly connected hub got infected; quickly halted
spreading from 10 infections at p = 0.65; or second wave at p = 0.75 and 10 initial infections.
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levels, yet, notably, stabilizing the demand with our control parameter p does not aect it. As long as the same
number of travellers participates in pooling everyday, the eciency remains more or less stable, as we report in
Fig.4c, where 2000 travellers participating daily in the system yield the same occupancy regardless of the replica-
tion (dots) and demand stability (x-axis).
Control and mitigation. Results show that the virus may easily spread through the ride-pooling networks
infecting the majority of the population with a low epidemic threshold (20 initial spreaders may infect up to 800
out of 2000 travellers). While the ride-pooling service provider cannot control for the initial share of infected
traveller, nor the incubation and recovery periods, the ride-pooling demand may be controlled to mitigate the
virus spreading. Specically, we show that imposing xed matches by means of a more stable demand level -
solely by controlling for p, without making amendments to the matching algorithm itself - can mitigate the
Figure4. (a) Number of eventually infected travellers for varying demand stability p and initially infected
travellers. Distributions based on 20 replications (mean within interquantile box and whiskers from min to max
values). Initially infected 20 travellers may spread the virus to almost 40% of the population (800 out of 2000
travellers). Yet as long as demand becomes stabilised, outbreaks start being contained. Even a large number
of initially infected does not reach more than 10% of the population if demand stability is set to 90% and is
eventually contained below 60 travellers (3%) for fully stable demand. e variability of outbreaks also decreases
as the demand stabilises: 10 initial infections may reach between 40 and 100 travellers if
p
=
0.9
, while the
range expands from 50 to over 400 for
p
=
0.7
. e lower bound increases when the number of initially infected
is high, making outbreaks more predictable, unlike the ones starting from a small number of infections, for
which variability is greater. Importantly, stabilizing the demand does not reduce the eciency, as we report in
(c), where the mean occupancy (key eciency indicator of ride-pooling) remains stable as demand stabilizes.
Notably, the importance of demand stabilization increases with the demand level as we demonstrate on panel
(b) which shows share of infected travellers changing with a demand for various ps. Each dot is the average from
20 replications. For all the values of p share of infected individuals scales with the demand level as
AexpQ)
,
marked as trendlines on (b), with p-values and
R2
reported in the text. For demand levels below 1500, the virus
rarely reaches more than 4% of the travellers. In contrast, when the demand level is 2500, the epidemic may
reach up to 10% when
p
=
0.8
or be contained below 2% for a stable demand (
p>0.9
), which underlines the
importance of control measures for higher demand levels.
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spreading and bring it to halt (Figs.3, 4, 5, 6). We demonstrate that matching and its stability is key to halt
epidemics in ride-pooling networks. It can be used proactively in the the design of the real-time matching algo-
rithm.
Moreover, if contact tracing apps are used, when a traveller is diagnosed not only s/he has to quarantine, but
also his/her traced contact network over the relevant period of time can be identied and eventually isolated.
As we show in Fig.5c, 10 initial infections will spread to a maximum of 60 travellers prior to diagnosis, which
seems to be feasible to trace back, isolate and halt spreading.
Study limitations and caveats. We aim at revealing the universal patterns characterising the spread-
ing of a virus in ride-pooling networks, yet our ndings shall be considered with caveats. Namely, we simulate
only a subset of daily mobility patterns (aernoon commute), from many sources of non-recurrence present in
travel patterns we picked-up one (participation probability), which we found sucient to reproduce its impact,
while role of others (varying and uctuating travel modes, destinations, departure times etc.) may be similar
or potentially even stronger. While the shareability network in the morning commute is likely to be similar
(inverse of aernoon), other, non-commuting trips, will likely yield a dierent shareability network, catalysing
the spreading to the new co-travellers. Nonetheless, our results are valid for systems with regular users with
symmetric demand patterns in the morning and aernoon. Moreover, drivers are assumed not to be spread-
ers (which seems plausible in the context of ad-hoc made shields isolating many of ride-sourcing drivers from
travellers). We applied a xed and deterministic epidemiological model in terms of the infection probability,
incubation period and quarantining, since reliable estimates of those parameters distribution are not reported
yet. Despite, we claim that the main message holds true for general urban networks: without intervention ride-
pooling signicantly contributes to virus spreading, while xing matches between co-travellers dramatically
reduces transmission.
Discussion and conclusions
Sharing a single vehicle with co-travellers during pandemics induces a risk to become exposed to viruses. Sadly,
the risk extends beyond the fellow travellers one shares the ride within a single vehicle, mainly due to the accu-
mulated contact graph resulting from day to day variations. Regardless of the number of initial infections, the
Figure5. (a) Average node degree in the evolving contact networks. Regardless demand stability p, an average
traveller is linked to 1.7 other travellers each day. Yet if the demand is unstable it evolves, aer 10 days it
reaches 1.9 if
p
=
0.99
, 2.5 if
p
=
0.95
and goes beyond 3 if
p<0.8
. (b) Mean transmission rate r (number of
new infections per infected) distributions. e long tails for low demand stability reveal the super-spreaders
(transmitting to 5 and more travellers). For a stable demand initial infections does not manage to transmit a
disease eectively, eventually reducing transmissivity below 1 when
p>0.9
. (c) Insights into the rst phase of
the epidemic outbreak in the case of 10 initial infections. When rst infected travellers are diagnosed aer 7
days, their accumulated contact network may vary from 18 to over 60 infected travellers. If contact tracing and
mitigation strategies are put in place, already infected travellers may be identied and quarantined before the
second outbreak aer day 7.
Figure6. An illustration of the spatial extent of epidemic outbreaks originating from two initial infections. A
major part of Amsterdam becomes infected for spontaneous demand (le), while it remains spatially contained
as the demand stabilises (right). For stable demand (
p>0.8
) the geographical boundaries are conned, while
otherwise, the virus crosses the river Ij and reaches also the north parts of Amsterdam.
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upper bound of outbreaks in spontaneous ride-pooling networks is high. Even two initial infections may lead
to hundreds of cases across the network. We did not observe spatial nor topological limits to the spreading and
disease starting from only two initial infections managed to reach most parts of the Amsterdam’s area.
In plausible demand and behavioural settings, if a generic ride-pooling system reaches a critical mass, the
travellers become densely connected through the shareability network so that the virus transmits easily through
the giant component without clearly visible epidemic thresholds. Only travellers belonging to isolated communi-
ties are le unaected. e pace at which it spreads, however, is low, requiring a long time until virus penetrates
across the network. Nonetheless, the daily contact network with its low node degree and hub-free, will evolve
due to spontaneity in the demand patterns. If each day, a slightly dierent pool of travellers decides to travel,
this will yield a new matching and resulting with a new shareability, contributing to the accumulated number
of contacts steadily growing over time. e slow pace evolution of contact networks becomes benecial when
tracking measures are applied and we can trace past co-travellers for each diagnosed spreader. Even in a highly
spontaneous ride-pooling networks, 10 initial infections manage to transmit to only 60 within the 7 day incuba-
tion period, which seems to be feasible to trace, isolate and halt the spreading, specically given the app-based
operations of the mobility platform, presumably storing travellers’ traces anyhow. Otherwise, if not halted early,
epidemics may evolve unhampered and randomly. Depending on the location of initial infections, the epidemic
may die-out as well as outbreak, making it potentially risky and uncertain.
Notably, we can substantially limit spreading by sacricing the spontaneity oered by the ride-pooling service.
If we enforce the same matching and x the pools of co-travellers, spreading is eciently mitigated and even 20
initial infections remain manageable to be contained. With one, clearly controllable parameter we can reduce the
outbreak of viruses. If translated to platform operations, this can become an ecient management and control
measure, adjustable along with other country-wide pandemic measures. is may contribute to the provision of
a safe shared-mobility alternative in the presence of public health fears and risks. Future research may modify
the matching algorithm itself so as to favor the matching of travellers that have already travelled together in
past rides. Such an approach is expected to allow for virus spreading reductions even when the demand pattern
is subject to large day-to-day variations. Finally, if tracing is combined with xed pooling, the system’s safety
may further improve, making ride-pooling a promising intermediate mobility solution for the pandemic world.
Although the presented methodology has been illustrated on the case of Amsterdam ride-pooling it is essen-
tial to emphasise its general applicability to examine, in a non-invasive way, the likely outcomes of dierent
underlying topologies on the way the virus spreads through the network. In this sense it opens space for discus-
sion of potential alterations of practical ride-pooling systems on one side and theoretical studies on the other
one. Although our study considers a limited number of rides, it is widely acknowledged that cities and their
properties are connected with scaling laws40 even if the form and the details of the methodology behind these
relations is questioned41,42. Typically such laws21 may associate a certain index x with city population size M by
an allometric scaling
xMα
. More importantly the existence of scaling laws has also been proven in the case
of ride-sharing networks with respect to shareability12, visitation frequency13 or lately ride-sharing eciency33.
In view of this we may assume that our results regarding the number of infected individuals as a function of the
demand level Q should hold for larger systems emphasising the necessity to stabilize the demand.
Methods
Travel demand data. We run a series of experiments on a travel dataset available for Amsterdam from a
nation-wide activity-based model35, with a single trip dened as a combination of its origin
oi
, destination
di
and
desired pick-up time
tp
i
:
e dataset contains over 240 thousand trips conducted within the boundaries of Amsterdam on a repre-
sentative working day, which we lter to aernoon (2PM–6PM) trips longer than one kilometer. We use 3200
passenger trip requests for the experiments, 2000 of which participates in the pooling on any given day. e pool
of travellers from which we sample the daily demand is controlled using p, based on which each day we draw
from the pool of 2000/p travellers.
Ride‑pooling algorithm. To identify attractive pooled rides we use the ExMAS9 algorithm (publicly avail-
able python library), which for a given network (osmnx graph), travel demand, behavioral parameters (like
willingness-to-share) and system parameters (pooling discount) identies all feasible shared rides and then con-
structs a shareability network (Fig.2a) to nally optimally match trips into shared rides (Fig.2b).
It generates the so-called shareability network, linking two kinds of nodes: travellers and rides. Traveller i is
linked to a feasible ride r if and only if s/he nds it attractive, which we express as the probability that ride utility
Ui,r
- reecting the extent to which delays and detours
δr,i
imposed by sharing are compensated by a discounted
ride fare
under traveller’s behavioural parameters
βi
(value of time and willingness-to-share) - is greater than
travellers’ attractiveness threshold
ǫi
. e theoretical number of shared-rides explodes combinatorically with
the number of travellers (e.g. 2000 travellers can be matched into
4.65 ×1020
theoretically feasible trips shared
by up to ve passengers). is can be made tractable by considering only attractive rides, which is governed on
one hand by travellers preferences
βi
(i.e. individual trade-os between longer ride and discounted price) and
on the other hand by service design
(controlled through the discount oered by the platform for sharing) and
ǫi
expressing the quality of alternatives for ride-pooling (private ride-hailing, or public transport and/or bike),
further detailed in9. Importantly, the shareability network is composed of feasible rides only, expressed with
Fr
,
being one if the ride is attractive for all travellers sharing it and zero otherwise. We formalize the shareability
network with a link formation
li,r
formula, combining ride feasibility and attractiveness as follows:
(1)
Qi=(oi,di,ti).
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Matching travellers to attractive shared rides. Each traveller may be linked to multiple rides and the
resulting shareability network is typically highly connected, characterized by the formation of communities and
hubs (Fig.2a). While the shareability network denotes the potential to share a ride, on any given day travellers
are matched to exactly one particular shared-ride (Fig.2b).
To address this, we formulate a binary traveller-ride assignment problem, where each traveller i is unilaterally
assigned to a ride r and the assignment yields the minimal costs. It is formulated as a problem of determining a
binary vector
xr
, an assignment variable equal to one if a ride is selected and zero otherwise (eq.3c). e objec-
tive of this deterministic assignment are ride costs
cr
, multiplied by the assignment variable
xr
, aggregated for
all rides (eq.3a).
Such an assignment satises the constraint of assigning each traveller to exactly one ride, obtained through
the row-wise sum for assignment variable
xr
and traveller-ride incidence matrix
Ii,r
. e latter is a binary matrix,
where each entry is one if ride r serves traveller i and zero otherwise (eq.3b). Eventually, the solution to the
problem (eq.3a) is the subset
R
of feasible rides
R
such that
xr=1
rR
. We express the shareability problem
as the following program:
Although matching problem (Eq.3a) can be read as the set cover problem43, which is known to be NP-Hard,
real-life ride-pooling situations usually yield congurations managed by standard solvers (like in8,9).
Contact network. On any given day, the contact graph is composed of connecting each ride to all travellers
that have shared (part of) it. Notably, the contact network evolves over time, primarily due to the dierent pool
of travellers being matched on any given day. Hence, this representation allows simulating an epidemic outbreak
by analyzing potential transmissions between travellers that have shared rides with other travellers over the
course of the analysis period. In our model the contact network changes from day to day due to one or more of
the following reasons: (i) infected travellers quarantine (which may catalyse spreading as quarantined travel-
lers are replaced by susceptible ones, who will get infected) (ii) recovered travellers return to the system (which
impedes spreading as recovered travellers restore to previous, optimal matches, already penetrated by the virus)
or (iii) daily variations in travel demand as travellers decide not to use ride-hailing on a given day (for example
because they opt for an alternative mode). We represent the daily participation, central endogenous variable of
the model, through the demand stability parameter p in our experiments. Each day we update the pool of travel-
lers (using the daily participation formula
Fd
i
=
Pr
(
p
)·(
1
Kd
i)
which combines the participation probability
p and quarantined travellers on day
Ki
d
). is, in turn, results with updating the pool of rides feasible on a given
day (composed only of travellers present in the daily pool). e contact network will then evolve as travellers are
matched to new rides when their co-travellers are quarantined or absent.
Epidemic model. We adopt a SIQR model to represent the four compartments characteristic of the COVID-
19 pandemic: Susceptible (S), Infected (I), Quarantined (Q) and Recovered (R), recently, directly applied to
tackle COVID-19 propagation in other studies (Italy44 and Japan45). Following the argumentation of Pedersen
and Meneghini44 we do not explicitly designate the E state, given the evidence suggesting that the COVID-19
virus can be propagated without rst exhibiting visual symptoms. e SIQR model was rst introduced by Feng
and ieme in 199532 and then examined in detail by Hethcote et al.46. Previous studies focused on mathematical
aspects of the model (e.g., oscillations32, stability analysis46 or the role of stochastic noise47). While the aggregate
epidemiological properties of the SIQR model are well studied, studies taking into account the underlying net-
work structure and its evolution are scarce.
e phenomenon central to this paper is driven by the structure and evolution of the contact network, rather
than by the parameters of the epidemic model. We, therefore, adopt a deterministic model where infected travel-
lers infect all of their co-travellers with a probability of 1. For the sake of clarity, unlike SEIR models, we assume
that all exposed inevitably become infected, all of which quarantine and recover aer certain incubation and
recovery periods (we use here the latest reliable ndings suggesting, respectively, 7 and 14 for COVID-1948).
is ubiquitous spread over the contact network may be seen as a pessimistic upper-bound of the spreading
process, yet in the view of recent pandemics15, sharing a vehicle with infected co-traveller is expected to yield a
high contagion risk. Furthermore, the focus of this study is on spreading across the network and over multiple
vehicles and rides rather than within vehicle transmission probabilities. Future medical estimates of the latter
can be embedded into the analysis performed in this study as soon as those are made available to rene our
model specications and thus obtain more precise estimates. Our ndings should therefore be considered an
upper-bound of the epidemiological consequences of virus spreading in ride-pooling systems.
(2)
li,r=Fr·Pr(Ui,r=U(i,βi,δr,i,)>ǫ
i)
(3a)
min
rR
crx
r
(3b)
subject to
iQ
Ii,rxr=1,
i
(3c)
xr∈{0, 1}.
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Modelling framework. e ExMAS ride-pooling algorithm is embedded within the day-to-day loop char-
acteristic to epidemiological model. e simulation initializes with a trip demand set composed of all the travel-
lers that may consider ride-pooling on any given day during the course of the simulated epidemic outbreak, to
further allow embedding the participation probability p. Before entering the main epidemic loop, we identify
all feasible pooled rides (Fig.2a)—to determine potential co-travellers that any given traveller may encounter
during the course of an epidemic outbreak. We create the complete shareability graph by applying equation (2)
with
ǫ
corresponding to a private, non-shared ride alternative in a deterministic model.
Following this initialisation phase, we then enter the main simulation loop. We start with assigning initial
infectors - drawn in random by sampling a pre-dened number of initially infected, which is treated as random
input and vary from one replication to another. Next, we enter the day-to-day simulation: every day we rst
determine the daily ride-pooling demand. We assume that only a subset of travellers actually participates in the
ride-pooling system on any given day, i.e. every day we sample a given number of travellers from the total latent
demand. We x the demand to 2000 everyday in our experiments to ease comparisons (except Fig.4b where we
experiment with various demand levels). ose travellers are then matched to identify the realization of shared-
rides on a given day. Everyday we apply the SIQR model with transitions taking place when:
(a) infected travellers infect their susceptible co-riders (
SI
),
(b) infected travellers are quarantined aer the incubation period (
IQ
),
(c) travellers recover aer the quarantine and acquire complete immunity to the virus (
QR
).
For any given day, the model outputs information about the number of travellers in each state (S-I-Q-
R) and newly infected travellers, based on which we can reproduce epidemic spreading proles. e loop
terminates when all the infected travellers are quarantined (there are no active infections).
Code and data availability
e code to generate the shareability network from a given demand pattern and then to reproduce the epidemic
simulations is available at the public GitHub repository (http:// www. github. com/ rafal kucha rskiPK/ ExMAS).
e experimental results data is available under https:// doi. org/ 10. 4121/ 14140 616. v1. e network data was
obtained from Open Street Map with osmnx, Amsterdam travel demand was derived from Albatross data set35.
Received: 30 October 2020; Accepted: 17 March 2021
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Acknowledgements
is research was supported by the CriticalMaaS project (Grant Number 804469), which is nanced by the
European Research Council and Amsterdam Institute for Advanced Metropolitan Solutions and the Transport
Institute of TU Del. is research was funded by National Science Centre in Poland program OPUS 19 (Grant
Number 2020/37/B/HS4/01847) and by IDUB against COVID-19 project granted by Warsaw University of
Technology under the program Excellence Initiative: Research University (IDUB).
Author contributions
R.K. and O.C. conceived the experiment(s), R.K. conducted the experiment(s), R.K O.C and J.S. analysed the
results. All authors reviewed the manuscript.
Competing interests
e authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to R.K.
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