Describing, modelling and forecasting the spatial and temporal spread of COVID-19 -- A short review
DDescribing, modelling and forecasting the spatialand temporal spread of COVID-19 – A shortreview
Julien Arino
Abstract
SARS-CoV-2 started propagating worldwide in January 2020 and has nowreached virtually all communities on the planet. This short review provides evidenceof this spread and documents modelling efforts undertaken to understand and forecastit, including a short section about the new variants that emerged in late 2020.
On 21 February 2003, a disease now known as the severe acute respiratory syn-drome (SARS) arrived in Hong Kong when a physician from Guandong Provincein Mainland China bearing the SARS coronavirus (SARS-CoV), checked in at theMetropole Hotel [54]. The primary case in Hong Kong was by no means the indexcase: the virus had been circulating in Guandong since at least November 2002 [52].However, that patient triggered a chain of infections that, together with earlier casesin China, led to 8,098 known cases and 774 deaths in 28 countries [42] and wasdeclared a pandemic by the World Health Organisation (WHO).Similarly, it is not certain at the time of writing that SARS-CoV-2 and its associ-ated disease COVID-19 had its index case in Wuhan, Hubei Provice, China. What iscertain, on the other hand, is that it is in Wuhan that COVID-19 underwent its firstnoticeable amplification phase, following which it spread rapidly across the world,to the point that there are now very few top level jurisdictions not having reportedCOVID-19 cases.SARS-CoV-2 is the third novel Coronavirus to emerge in the 21st century (afterSARS and the Middle East respiratory syndrome – MERS [28]) and the second togenerate a pandemic (a third pandemic was triggered by the H1N1 influenza outbreak
Julien ArinoDepartment of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada, e-mail:
[email protected] a r X i v : . [ q - b i o . P E ] F e b Julien Arino in 2009). COVID-19 is also the most devastating pandemic in over a century in termsof its death toll as well as its economic and societal impact.Here, I review some aspects of the spatio-temporal spread of COVID-19. Somecaveats are in order. Firstly, while spatial epidemiology is not the most popular topicamong modellers, it does remain a vast field where a myriad of approaches coexist;surveying the work done on the topic would require an entire monograph. While Ihave tried to be inclusive, it is certain that I have omitted some topics or techniques. Iam for instance making the choice to describe mostly mechanistic models of spread,be they deterministic or stochastic, mathematical or computational. Some very goodstatistical work has been published on the topic of COVID-19, but I focus here to alarge extent on models that can explain reality perhaps at the detriment of forecastingpower. Secondly, I am aware that many modellers who have worked or are workingwith public health authorities may not have had the time yet to publish their work.Except for my work, I report here only on papers already published or available onrecognised preprint servers. Thirdly, new variants of concern (VoC) were detectedwhile this paper was under review. A short section at the end of this document todescribe what little is known in terms of spatio-temporal spread of these variants.Finally, even though some work makes use of data at a very fine spatial resolution,in keeping with the philosophy of some prior work [13], I focus on models that canbe used with publicly available data.This review is organised as follows. First, in Section 2, I provide an overview ofthe mechanisms that lead to the spatial spread of infections and three of the majortypes of models that have been used to study it. In Section 3, I then describe thespread of COVID-19 from a chronological point of view. Finally, in Section 4, Idiscuss modelling work specific to COVID-19.
Before considering work specific to COVID-19, let me spend some time on thespatialisation of infectious diseases in general. Indeed, while COVID-19 presentsspecific challenges, it is by no means the first spatial epidemic that humanity isconfronted to; for instance, a simple description of the spatio-temporal trajectoryof the Plague of Athens can be found in the History of the Peloponnesian War[176], which was written almost 2,500 years ago; on a more local scale, spatialepidemiology can be traced to the cholera epidemic of London in 1854 [171]. Thereis therefore much understanding to be gained about the current crisis by consideringwhat was known prior to its start. patio-temporal spread of COVID-19 3
Different conceptual models explain the mechanisms that lead to the spatialisationof an infectious disease, leading to potentially different modelling paradigms.Working at the level of the individual, one can envision spatial spread as therepetition of inter-individual spread events. Individuals are mobile in space and itis their movement while bearing the infectious pathogen that leads to the diseasebecoming spatial, when they come into contact with susceptible individuals whoare also mobile. This description falls mostly into what have been called Markoviancontact processes [146]. When considered at the population level, this leads to modelsusing partial differential equations and is particularly appropriate for describing thespread of a disease where the hosts can move freely, such as epizooties. Such adescription can also lead to network or agent-based models.Work with my collaborators usually instead focuses on locations and adopts avision of spatial spread articulated in [17]: an infectious disease becomes spatialisedby the repetition of processes summarised as importation , amplification , exportation and transport . Importation itself is the event when an individual infected with thedisease reaches a new location. Importation is successful if the imported case leadsto at least one local transmission event. This way of thinking about spread is easy toreconcile with data, since locations are jurisdictions in the context of public health.It also matches the cones of resolution that some geographers use when they thinkabout the spatial spread of epidemics; see [11] and references therein. See also[110, 111], which consider the roles of the different levels of mobility on the spreadof SARS in and to and from Beijing. Whatever the way one conceptualises the spatialisation process, the main driver ofspatial spread is human mobility. Long range fast movements using air travel haveconsiderably changed the way diseases spread and while amplification in a locationremains driven by population effects, the initial spread is to a large extent driven byair travel. This was shown for SARS [42], the 2009 H1N1 influenza pandemic [119]and MERS [90], for instance. Long distance high speed train travel has also beenassociated to spread; see, e.g., [47]. It is interesting to note, though, that despite thehighly heterogeneous nature of spatio-temporal spread brought on by modern travelmodalities, continental-level effects can still be observed [93].
As mentioned in the Introduction, I focus here on mechanistic models. There aremany ways to model the spatio-temporal spread of infectious diseases. Let me present
Julien Arino the main contenders; see an interesting and more complete list in [161] or [147].I do not detail reaction-diffusion equations, because, to the best of my knowledge,they have seen very little use in modelling the spatio-temporal spread of COVID-19;readers are referred to [158], for instance, for more details on deterministic aspectsinvolving such systems. See [145] for a seminal review of the link between stochasticand deterministic spatial models, as well as interesting overviews in [61].
Agent-based models (ABM) consider populations of autonomous agents that interactfollowing some rules [76]. Agents have a set of characteristics that can be modifiedthrough their interactions with other agents. Although there are some attempts tomathematise some of the properties of such systems, they remain for the most partcomputational tools that need to be studied using a large number of simulations.Their strength lies in their realism: an agent can be given realistic behaviouralcharacteristics (schedule, place of residence or of work, etc.). ABM are also easier toimplement because they require very little mathematical background. Agent-basedmodels have proved most useful when considering the effect of individual behaviourson the spread of infection in smaller populations. For instance, they have been used tostudy individual protective behaviour [116], the effect of presenteism while infectedwith a disease [126], the risk in small isolated communities [49, 129], the effect ofsocial distancing [169] or the role of avoidance behaviour when vaccines have loweffectiveness [177]. Examples of spatialised problems (all about influenza) that werestudied using agent-based models include its spread in slums of Delhi [5], the use ofa hybrid approach involving networks to describe the social structure and ABM todescribe inter-individual spread in Forsyth Country, NC [98], the potential for socialstructure to generate inequalities in incidence in different areas of a county [127] orthe spread within an airport terminal [165]. See also [154, 155], which use a detailedlocation survey to conduct a simulation of the spread of influenza in Japan. ABMare also useful as a means to model evolution; see, for instance, [96], where an ABMis used to model evolution of virulence at the front line of a spreading epidemicThe area where ABM have proved most informative is when considering spreadof infections within areas where movement is constrained and generalised contactis impossible, such as buildings or cruise ships. For instance, when consideringnosocomial infections, it is possible to monitor health care personnel movement anduse this data to parametrise an ABM of spread within a hospital [109], or to formulatea model of spread between beds within an intensive care unit [109].However, ABM lose in value when populations become larger, except in rareinstances where unexpected emergent behaviour occurs. Where the law of largenumber applies, it is indeed less computationally onerous to use “classic” determin-istic or stochastic models. For instance, the model in [18] reproduces almost exactlythe behaviour of the agent-based model in [134], but furthermore gives access toexplicit expressions of the basic reproduction number R and the final size of theepidemic. See also the comparisons in [7, 66]. patio-temporal spread of COVID-19 5 Altogether, agent-based models are powerful tools of investigation at the hy-perlocal scale. Considering agents consisting of groups of individuals instead ofindividuals also allows to operate at a higher spatial scale, although models then losesome of the interesting properties they have at the finer scale.
Network models are very similar to agent-based models, of which they were, essen-tially, the inspiration. The two types of models are sometimes difficult to distinguish.In epidemiology, work on networks and ABM was popularised in particular by theNIGMS Models of Infectious Disease Agent Study, which led for instance to EpiSimS[56, 77]. In network models, nodes are typically simpler than agents in agent-basedmodels; the most straightforward example would be a network consisting of nodes(individuals) that can be in two states: susceptible to the disease or infected andinfectious with it. If there is an edge in the network between two nodes, this meansthe two nodes came into contact; in the case of a network used to model diseasespread, this indicates that a contact took place, which could lead to the transmissionof the disease. One promising direction of research that has been explored usingnetworks is that of the link between network structure and shape of the epidemiccurve; see, e.g., [48, 57]. This is particularly important during the early spread of adisease and has been considered in a variety of contexts using network models.Because they are simpler than ABM, network models are more amenable to anal-ysis; see, e.g., [34, 38]. Originally, tools used to study the dynamics of networkepidemic models originated in statistical mechanics [139, 140]. Because networksallow to incorporate a more realistic description of the contact process while main-taining some level of analytic tractability, comparing their dynamics with that ofclassical models is useful. In [8], this is done for instance for a two-strains influenzamodel with vaccination. [156] There has been a move lately towards characterisingthe dynamics of smaller networks using the properties of individual nodes ratherthan through distributions of these properties; see, e.g., [32].While examples of use of network models in mathematical epidemiology abound,their use in situations that are specifically spatial are not as common. Instancesinclude [79], who considered an SEIR model set in a lattice and simulated using aMonte Carlo process, to incorporate both stochasticity and space, [33], who considerthe spread of dengue in city blocks or [82], who consider the spread of equineinfluenza. The latter paper illustrates the strength of the method, in that they haveaccess to extensive data on horse movement between locations and are able to assessthe effect of the topology of the networks, both for long range movement and shortercontact patterns with locations, on the spread of the infection.Networks are also a natural candidate for considering the spread of infectionsusing the air transportation network as a conduit. This was done for instance withSARS [42].
Julien Arino
Also known as patch models, metapopulations couple together (typically similar)models, with each model encoding for the dynamics of the disease in a populationand coupling representing the movement of individuals between the populations[16], the average time spent in remote locations [37] or the interactions betweenpopulations. Metapopulation models had been used computationally since the early1970s, for instance to consider the spread of influenza within countries [83, 164].They have known a resurgence since the beginning of the 21st century, because, onthe one hand, computing resources made simulating them easier and, on the otherhand, papers such as [15, 19] showed that linear algebra techniques could be usedto render the study of such systems very similar to that of their constituting systems.See [31] for a list of problems the authors identify as interesting challenges in thefield.Metapopulations are now quite popular and have been used in a variety of settings.A lot of work concerns investigation of properties of spatial models. For instance,with coauthors, I investigated the effect of lowering travel rates between locations[20] and of interconnection between a large urban centre and smaller satellite cities[22]. Other interesting issues studied include the effect of vaccination targeted athigh risk areas [29], cooperation between governments on vaccination policy [123].Geographically targeted vaccination has also been considered at smaller spatialscales; see, e.g., [26, 29, 101, 118, 121, 135]. Other spatial control issues have beenconsidered in [41, 92, 95, 102, 122, 130, 136].Papers addressing issues that are present also with COVID-19 have consideredinfection during transport [25], in particular in relation to entry screening [133] aswell as exit and entry screening [132]. Exit and entry screening were also consideredin [178]. Some work has also considered the effect of media-induced social distancing[87, 173].Metapopulation models were used to consider specific diseases as well; the spreadof SARS [162], age-structured contact patterns during the 2009 H1N1 pandemic [12],chikungunya [50], dengue [141], cholera [73] or malaria [14, 88, 89]; see also [160].Standard metapopulation models are not well suited to consider the hyperlocalscale, because they assume homogeneity within the constituting units. In [12], aninteresting approach is used that allows more heterogeneous contacts within patches.Similarly, in [148], the authors consider behaviour at the hyperlocal scale but stillwithin a metapopulation model. In [7], the behaviour of an ABM is compared withthat of a stochastic metapopulation model.
To describe the spatio-temporal spread of COVID-19, I use the previously discussedframework of [17]. There is some discrepancy in reporting units, but to some extent,one can think along the lines of the ISO 3166 standard [114].
Global spread occurs patio-temporal spread of COVID-19 7 between ISO 3166-1 codes (countries, dependent territories and special areas ofgeographical interest).
Local spread within ISO 3166-1 codes occurs between ISO3166-2 codes (provinces in Australia and Canada, départements in France, statesin Brazil and the USA, etc.). Many countries also report at a finer geographicalscale, which I still call local : counties, regional health authorities or cities. Anythingbelow the city level is hyperlocal ; typically, this corresponds to commuting to work,school or shopping, but can even be mobility within a building. In keeping with myavowed preference for publicly available data, my description involves higher leveljurisdictions rather than spread at the hyperlocal scale, which is typically associatedto confidential data.Throughout the description that follows, one should bear in mind that data usedto describe the spread is very likely wrong in some instances, or rather, somejurisdictions might be reporting with a delay because of a lower capacity to detectcases. See for instance [100], in which two health security indices, the Global HealthSecurity Index and the Joint External Evaluation, are used to assess the likelihoodthat countries detected COVID-19 early. (The work also shows that countries withhigher values of these indices also saw reduced mortality from the disease to 1 July2020, although this is likely not true anymore.)
There is evidence that COVID-19 could have started its global spread in December2019, with reports of a case in France [68] as well as suspicious cases [39] and positivewastewater samples [128] in Italy. However, these retrospective analyses have yetto be confirmed, so at the time of writing, the first ten locations to have confirmedimportations are those listed in Table 1. The remainder of January saw cases beingconfirmed in several other countries. Of note is that China imposed a cordon sanitaire in Wuhan on 23 January 2020 and that the first successful importation (in theterminology of [17], i.e., a local transmission event) was reported by Vietnam on 24January 2020 [182].Starting in February 2020 and with more and more of the locations having reportedimportations earlier seeing local transmission chains, global spread accelerated.Figure 1 shows the number of ISO 3166-1-alpha3 codes (top level jurisdictions)reporting their first confirmed case as a function of the time since the first confirmedinternational exportation event.While I do not detail them in the modelling section because of my focus thereon models able to provide explanations of the phenomena, it is worth noting thatinteresting time series analyses were performed during the early stages of spread. Forinstance, [58] used ARIMA analysis of travel data together with disease propagationdata to forecast future destinations. The authors find that uncertainty as to the per-centage of asymptomatic cases makes previsions complicated; this conclusion is inline with personal work [23]. Likewise, [99, 105] considered spatial autoregressivemodels. Also, although not global, continental-level spread as documented for Africa
Julien ArinoDate Location Note Source13 Jan. Thailand Arrived 8 Jan. [181]16 Jan. Japan Arrived 6 Jan. [168, 179]20 Jan. Republic of Korea Airport detected on 19 Jan. [180]20 Jan. USA Arrived Jan. 15 [107]23 Jan. Nepal Arrived 13 Jan. [35]23 Jan. Singapore Arrived 20 Jan. [2]24 Jan. France Arrived 22 Jan. [1]24 Jan. Vietnam Arrived 13 Jan. [60, 152]25 Jan. Australia Arrived 19 Jan. [27]25 Jan. Malaysia Arrived 24 Jan. [151, 74]
Table 1
First ten international locations having reported imported COVID-19.
Date refers to thedate the case was reported. All dates are in 2020. All cases in this table were imported from China,except for Vietnam, which concerned both an imported case and a local contact.
Days post start of global spread C / T w i t h c on f i r m ed c a s e s SARSCOVID−19
Fig. 1
Number of countries and territories (ISO-3166-1-alpha3 codes) having reported confirmedcases of SARS-CoV (red) and SARS-CoV-2 (blue) as a function of the number of days sinceimportation in HKG (SARS-CoV) and importation in THL (SARS-CoV-2). in [91] is included here because the focus is not on the transition between the globallevel to the continental level but on spread within the continent. Finally, [138] useself-organising maps to look for similarities in epidemic curves to identify countriesseeing propagation of the same type.
Using the terminology of the conceptual model of spatialisation, when COVID-19started its international spread, there were very few jurisdictions that were exporters patio-temporal spread of COVID-19 9 of COVID-19 and an immense majority of potential importers. Public health au-thorities in those jurisdictions that did not have cases at that point therefore tookmeasures to try to stop or at least delay importations. To this end, they used threemain types of measures: restriction or suspension of travel, entry screening andpost-arrival self-isolation measures.
Mar Apr May Jun Jul Aug Sep OctMonth T r a v e ll e r s sc r eened Fig. 2
Daily number of passengers processed by the United States Transport Security Agency (TSA)in 2019 and 2020; data from . Starting early on and ongoing at the time of writing, various jurisdictions tookmeasures to curtail or even interrupt travel. Passengers themselves also abstainedfrom travelling. The result of this was a precipitous drop in travel volumes. Theintensity of this effect can be seen in Figure 2, which shows the daily numberof passengers processed by the United States Transport Security Agency, i.e., thenumber of individuals undertaking a trip originating in the USA, in 2019 (red)and 2020 (blue). The data shown for 2019 is for a year earlier, but shifted so itcorresponds to the same day in the week. At the lowest point, on Thursday 16April 2020, TSA screened 3.63% of the number of travellers they had screened onThursday 18 April 2019. The same trend can be observed for instance in tourism,with the United Nations World Tourism Organisation Tourism Dashboard ( ) reporting that thenumber of international tourists arrivals in April and May 2020 was 97% less thanthe same months in 2019.In [6], an analysis of the global air transportation network is undertaken usingthe network distance defined in [44], attempting to tease out the effect of travelinterruptions on the spread. While both of these studies provide very interestinginsights into the issue, a precise quantification of the effect of such fundamentalchanges to travel is hard.Entry screening is typically implemented at ports of entry (ports, airports, bordercrossings) and seeks to identify individuals who are bearing the disease of concern,in order to isolate them and thereby avoid potential transmission of the disease in the local population. There is some debate about the usefulness of entry screening,especially during the early stages of a global spread event. See [149] for an extensivereview. The sensitivity and specificity of the thermal detection equipment used isquestionable [40]. In the case of COVID-19, it has also been argued that this lowsensitivity would combine with the fact that fever detection would often fail becauseof the frequency of asymptomatic cases [45]. Entry screening at the beginning ofa health crisis also means looking for a needle in a haystack, since the volume ofincoming passengers from all locations vastly dominates the volume of passengerscoming from the location of interest [120]. Since prevalence is low at the beginningof the event, this further compounds the lack of efficacy and results in poor charac-teristics for the method [67]. Also, screening protocols themselves vary widely fromlocation to location [86], rendering a general evaluation of the value of a protocoldifficult. Despite these reservations, in the case of COVID-19, some of the evidenceof early international spread comes through entry screening, so there seems to havebeen some limited benefit to thermal imaging entry screening.After the initial few days during which testing was thermal imaging-based, screen-ing switched to using much more reliable PCR tests. This became possible becausesequencing of the virus genome was performed remarkably quickly. As a conse-quence, currently, there are four main attitudes towards screening: no screening atall; “soft” screening, i.e., verbal or written questionnaires; testing on entry; testingprior to entry. Some countries use a combination of approaches, for instance re-quiring testing only for individuals arriving from regions considered particularly atrisk.A jurisdiction still has one option to combat the risk that successful importationstake place: it can recommend or impose that individuals arriving from anotherjurisdiction spend some time in quarantine . Canada, for instance, has insisted on atwo-weeks quarantine period for all incoming travellers since the beginning of thecrisis, with exemptions.
The first step in switching from a purely global vision of spread to a local one isto consider when COVID-19 could arrive in “one’s backyard”. In the early stagesof the pandemic, before most top-level jurisdictions reported reported human-to-human transmission chains, it was of interest to those jurisdictions having no or fewlocal cases to understand the risks that their connectedness to other jurisdictionscarried. Because of evidence gathered during past pandemics and other notablepublic health events (see Section 2.2), this evaluation was mostly carried out byinvestigating a given jurisdiction’s connection to the rest of the world by means ofthe air transportation network. This was the method used for instance in Mexico[64], India [97] or Europe [157]. Much practical work on this aspect has come torely on global airline transportation data such as that provided by the InternationalAir Transport Association (IATA). However, it should be noted that this dataset patio-temporal spread of COVID-19 11 quickly became unreliable because of the dramatic fall in travel volumes discussedin Section 3.2.
Most of the very early work on local spread concerned China, since it was the firstcountry to experience this. The same type of method was used in [78] as is detailedat the start of this section, but with past data on mobility during the Spring Festivalof millions of migrant workers residing in Wuhan. The aim was to assess the riskto locations visited by the migrant workers. Since Chinese New Year was on 25January 2020, while the cordon sanitaire was imposed in Wuhan on 23 January, a lotof individuals did make the trip. This allowed the authors to venture which placeswere probably under-reporting cases. See also [184], which uses GIS techniques tostudy the spread within China and the factors contributing to this spread. As do theauthors of the previous paper, they find that connection to Wuhan, both in terms ofpopulation flow and economically, was the main driver of the initial spread.Tracing transmission chains originating from importations allowed to better un-derstand the consequences of importations. See, for instance, [36], which breaksdown such a transmission chain that started on 27 January 2020 in Bavaria (Ger-many). In [46], the early spread in Brazil is documented, from importation fromEurope (as evidenced by genome typing of the strains) to local spread within states,finally followed by exportation from urban centres. This is confirmed by [84], whoconsider spread among 604 cities in São Paulo State, Brazil. They show that in theheterogeneous setting they consider, there are two patterns of spread: one spatial,where the disease spreads to the nearest spatial component; the other hierarchical,where within one unit, spread starts with the top level urban centre then makes its wayto smaller cities. In [80], the authors use genomic and transportation data to considerthe spread within the USA and conclude that quite early on in the spread, impor-tations into uninfected locations in the country were much more likely to originateelsewhere in the country than abroad. Propagation within the USA was also studiedby [106], in which the occurrence of space-time clusters is studied. This interestinglyshows that as the epidemic took hold, there occurred more and more smaller clusters,confirming in some sense the similar observations in Brazil. Another investigationof continental spread in the USA is carried out in [144] using a multilayer percep-tron neural network. The authors use the Moran index computed on the incidencerates and a large number (57) of explanatory variables: socioeconomic, behavioural,environmental, topographic, demographic, age-adjusted mortality rates from severaldiseases, both infectious and chronic. They find that some of the most importantfactors predicting COVID-19 incidence rates are the age-adjusted mortality rates ofischemic heart disease, pancreatic cancer and leukemia, median household incomeand total precipitation. In [3], the time evolution in several countries and the timeevolution within France are investigated using time series methods incorporating spatial components. While mostly methodological, this provides interesting tools toconsider the spatio-temporal evolution of the disease across multiple jurisdictions.Other authors considered mechanisms for slowing down the spatial spread withina country. The authors of [142] advocate for a disconnection between locked-downurban centres and rural areas in India as a means to avoid complete country-widelockdown. This position is justified; indeed, authors in [63], for instance, foundstrong correlation between population density and the spread of SARS-CoV-2 inChina, so it could be that forbidding movement between locations at high risk (thecities) and those at lower risk is a valid approach. However, to the best of ourknowledge, no nation implemented such a system; indeed, during the initial wave,most countries implemented country-level lockdowns that also relied on severelylimiting or completely interrupting mobility within their territory. In [70], the authorsconsider the effect of containment measures on the spread of COVID-19 betweenprovinces in Italy. In [124], the authors used human mobility data in China to considerthe spread of COVID-19 within China, in particular in relation to the impact of controlmeasures.
Hyperlocal spread was documented early on during the course of the pandemicbecause of cases that happened onboard cruise ships that were under quarantine.These events, while unfortunate for those involved, have provided a wealth of data. Inparticular, they were extremely helpful in finding out key epidemiologic parameterssuch as reproduction number [185], prevalence of asymptomatic infections [75],incidence [153], transmissibility of the disease [143] or case fatality ratio [163].Because cruise ships have records of who was infected together with the room theywere in, it should become possible to build a good understanding of spatial aspects,although to the best of our knowledge, this data has not yet been released.Many countries faced and are facing outbreaks in long-term care facilities (LTC).There are a variety of reasons for this elevated risk; see, e.g., [172, 175]. This led totrememdous effort to control such outbreaks [175]. Movement within LTC can bedocumented (and modelled) accurately; see, e.g., [51, 150]. The health of residents isalso monitored (usually) well. As a consequence, nosocomial COVID-19 outbreaksalso provide valuable data at the hyperlocal level. See documented outbreaks in[113, 125, 137, 166, 167].Note an interesting “twist” on hyperlocal spread: in [108], the authors conducta wide-ranging analysis at the hyperlocal scale, in the sense that the consider themovements of individuals at the local scale but over the entire territory of the UnitedStates of America. This allows them to consider the effect of spatial heterogeneityof public health orders. patio-temporal spread of COVID-19 13
I replicate here the hierarchical spatial structure in Section 3 rather than the method-ological one in Section 2.3. Indeed, while most work detailed here falls within oneof the three classes of methods in Section 2.3, I also report on other methods thatgave interesting results.
To the best of my knowledge, most models for the global spread had in their objectivesto study how to slow down the global spread of the infection or considered spreadwithin specific countries or groups of countries; these are discussed in the relevantlater sections.In [21, 24], we set the 𝑆𝐿 𝐿 𝐼 𝐼 𝐴 𝐴 𝑅 model of [23] in a metapopulationcontext and focused on the risk of importation in different countries. The model wasrun daily to provide the Public Health Agency of Canada with an assessment of themost likely countries to import the disease in the coming days. The model includestravel at different levels, which, as pointed out in Section 3.3, was a documentedfeature of spread.In [170], an SIR-type metapopulation model in the GLEAM framework [30] isused that combines population densities, commute patterns and long-range travel.Used at the early stage of the spread, the authors find that it is likely that the value ofthe basic reproduction number R and the prevalence are badly estimated in somelocations, with estimates in the literature at the time driven by locations with a largepopulation. They conclude that the number of cases was probably underestimated. In [55], a metapopulation model for the global spread of COVID-19 is used toconsider in particular the role of international travel bans. The authors show thatwhile the cordon sanitaire in Wuhan did little to slow spread within China, its impactinternationally was more pronounced. The combined effect of travel restrictions andcommunity effort is also studied, with the interesting finding that travel restrictionsalone do not suffice to have an effect on propagation. In [4], a stochastic SEIRmetapopulation model is used, together with Official Airlines Guide (OAG) data, toconsider the role of travel restrictions taking place after 24 January 2020. The authorsfound good adequation with the number of imported cases in several countries asof the end of January. They focused in particular on Australia and establish that thetravel ban there might have delayed the onset of widespread propagation by fourweeks.
A stochastic simulation model is used in [69] to consider different scenariosregarding testing (rather than screening) of incoming individuals and the duration ofquarantine periods. Similarly, [59] use a stochastic model to quantify the effectivenessof screening and so-called sensitisation of travellers, i.e., the provision of healthinformation in an effort to trigger compliance with self-isolation recommendations.In [159], the example of air transportation in Brazil is considered using an SIR-type metapopulation. The speed of spread in relation to network measures such ascentrality was explored, with closeness centrality shown to be a good predictor ofthe vulnerability of a city.In [17], we considered the risk of disease importation in a location that is seeinglittle to no local transmission chains. As with most of our work on the subject, we useda modified version of the model in [23]. In this case, we used a stochastic version,which we subjected to stimulations to represent the inflow of infected individuals intoa location. The model also allowed us to quantify precisely the effect of quarantinein terms of its effect on the inflow rate.
In [94], the risk of importation of COVID-19 in African countries was consideredusing air travel data as well as data from the Monitoring Evaluation Framework(MEF) of the WHO International Health Regulations. The model is quite simple andcomprises no dynamic components, meaning that it provides a snapshot evaluationof the risk of importation. As it was formulated at the beginning of the spread event,when most of the exportation was assumed to come from China, it does nonethelessprovide meaningful results.In the already cited [17], we focused on the risk of importation of COVID-19 in locations that are seeing little to no local transmission, thereby consideringthe interface between the rest of the world and such locations. We showed thatthe probability of importation was most dependent on the rate at which cases areimported in the locations, but that the outcome of a successful importation wasthen determined to a large extent by the intensity of public health measures in thelocations.
The location for which data became readily available the soonest was China. SinceChina is also a very large country, some very interesting work was carried out in thecontext of spread within that country. The authors of [183] considered the spread ofCOVID-19 within China using an interesting idea: they estimated the size of the out-break in Wuhan from known international exportations, then used a metapopulationmodel with Wuhan as the source of infection to estimate spread within China. In patio-temporal spread of COVID-19 15 [131], the role of undocumented infections is investigated in relation with the spreadof COVID-19 between 372 Chinese cities using a metapopulation SEIR model incor-porating documented and undocumented infections. In [112], spread within HubeiProvince and in the rest of China is investigated using statistical tools (the Moranindex and a logistic model). Then an ODE SEIR model is used to compute R in thedifferent locations. In [174], an SEIR-type model with additional compartments fordiagnosed and confirmed, suspected and infected as well as suspected but uninfectedindividuals is set in a metapopulation framework with two patches: Hubei Provinceand the rest of China. The model is used to consider the effect of lifting lockdownmeasures.In [10], a simulation platform is used to consider the spread in France. The modeloperates at the level of subregions ( départements ) and supposes that individuals canbe susceptible, asymptomatic, symptomatic, recovered, hospitalised and diseased.An interesting feature of the paper is a comparison between the results of continuoustime deterministic and discrete stochastic methods, with the latter showing betteradequation with observed data.The authors of [46] considered spread of SARS-CoV-2 within Brazil. This colos-sal endeavour considers actual genotyping of the virus and prior to modelling workproper, details importations and the spatio-temporal spread of various genomesof the virus. Spatio-temporal modelling then uses a continuous phylogeographicmodel. The model is not predictive but sheds light on the spread process: theyfind that spread was mostly local, i.e., within state borders. Both within-state andbetween-state spread was also found to have decreased after the implementation ofNPI.In [9], the effect of heterogeneity of policies in the USA is investigated. A model isformulated that is a metapopulation in essence; based on data on people movementto places of gathering such as churches, the model allows the redistribution ofindividuals between locations following different types of policies. They observethat spatial heterogeneity in measures tends to increase the likelihood of subsequentinfection waves. Spatial heterogeneity is also investigated in [65], which uses ametapopulation model to probe the impact of disparity of healthcare capacity inOhio. In [53], the effect of changing travel rates within and between locations isinvestigated, with data for Taiwan.Finally, note that because COVID-19 is spreading globally and that nationallevel jurisdictions (and sometimes even lower level ones) implemented a variety ofresponses, it is useful to compare the situation in different jurisdictions. Even thoughthis is not spatial modelling stricto sensu , such works are worth mentioning here asthey provide the underpinning to spatial models. In [104], the authors use an SEIRmodel to compare transmission patterns in China, South Korea, Italy and Iran. In[103], an age-structured SEIR model is used to compare the dynamics of diseasespread in Hubei Province and six European regions. The focus is on the estimationof the case-fatality (CFR), symptomatic case-fatality (sCFR) and infection-fatality(IFR) ratios. The authors find that the latter two indicators are better suited todescribe the potential impact of the pandemic and note that they find geographicheterogeneity of the estimated values. This heterogeneity is not only between Hubei Province and the European locations under consideration, but also between theEuropean locations themselves. With collaborators, we used the model in [23] toprovide a daily forecast of spread in several Canadian provinces [21] and found thatestimates for some parameters were consistent across provinces while estimates forothers varied widely, in particular, the proportion of asymptomatic cases.
A lot of work during this pandemic has focused implicitly on the hyperlocal level,but recall that here the object is models in which I found an explicit reference tospatial aspects.In [115], a network model is used to model the spread of SARS-CoV-2 onboardthe Diamond Princess cruise ship, with nodes representing individual passengersand crew members. Age-structure was used as well. The model was calibrated toknown transmission data and the effect of control measures was then considered.See also the already cited [75, 143, 153, 185] for more modelling work related tospread aboard the Diamond Princess.The authors of [43] used an SLIAR agent-based model to consider the effect ofsocial distancing, viral shedding and what they call the social distance threshold.They find that the three lead to threshold behaviour (“phase transitions”) that havedifferent effects on the course of the epidemic.In [81], ABM are used to consider in particular the effect of testing policies.Agents are distributed on a map depending on the population density in the areasunder consideration. They are also assigned movement patterns that can cover thewhole map, a medium range or a small one. Some interesting observations are thatwhen tests have low reliability or that the ability to trace contact is low, a large fractionof the testing capacity remains unused despite an increasing incidence. They alsofind that mixed testing policies are useful to contain spread.
SARS-CoV-2 is an RNA virus and as such is subject to high mutation rates leadingpotentially to variants [72, 117]. Thus the emergence of new variants was expectedfrom the onset of the crisis. At this point, there are several major variants to the orig-inal variant that have been detected. This number can be expected to rise: detectionof most variants requires genome sequencing, which is performed at different ratesin different countries [85], meaning that capacity to detect variants varies greatlyglobally. Of particular interest at the time of writing is B.1.1.7, which was first de-tected in the United Kingdom in early December 2020 but is presumed to have beenspreading since as early as September 2020. This variant is particularly concerningas it appears to be more transmissible than the original variant. It seems that this patio-temporal spread of COVID-19 17 variant should not, however, be detrimental to the ongoing vaccination efforts [62].Many countries took preventive measures in order to delay the arrival of the variant,essentially forbidding all travel originating from the United Kingdom, but given thatcirculation probably started several months before these measures, their efficacy isdebatable. For instance, [71] estimates that, of 19 countries evaluated, 16 had at leasta 50% chance of having already imported the variant by 7 December 2020. Thenovel variants led some countries to consider exit control measures; some EuropeanUnion (EU) countries (Belgium and France, for instance) decided late January 2021or early February 2021 to forbid both entry from and exit to non-EU countries fornon-essential travel. In the context of pandemic H1N1 influenza, exit screening wasshown to have the potential to be more an efficacious control measure than entryscreening [120]. It is therefore interesting to see this type of control finally beingapplied, although the intent is not the one we were advocating in [120].Modelling the spatio-temporal spread of these novel variants can be conductedin very much the same manner as was done for the original variant. For instance,metapopulation models for multiple species such as those considered in [19, 20] canbe readily adapted to a multiple variant situation. However, it is important to bear inmind that because of the detection issues mentioned earlier, these models are hardto parametrise when considering the initial spread of the variants.
This is but a brief and very incomplete snapshot of the state of knowledge about thespread of COVID-19 at the time of writing in December 2020, with a few additionaldetails about the new variants added in January 2021. As indicated, it is likely that Iomitted a lot of publications on the subject, given the immense amount of literatureCOVID-19 has generated.From the perspective of the spatio-temporal spread of the disease, although thereis still much to learn, I think we also now have the luxury of hindsight : manygroups, mine included, have produced a variety of models in the first few months ofthe crisis, which can and should now be confronted to the reality of the outbreak.Because COVID-19 is so widespread, there is less urgency to consider its spatialspread in the perspective of emergency response and the focus could now evolve,at least in part, to the evaluation of the models we produced. The problem of re-importation of the disease in locations having managed to drive it away remains animportant one, so I am not advocating to stop all work regarding spatial spread; Iam only pointing out that understanding what worked and what did not during theinitial spread would actually help for these subsequent importation events.Going forward, though, I believe that there is still a lot to be done on one key aspectof spatio-temporal models: most of the work carried out by those of us working inthis area has come to rely on one particular dataset, the so-called IATA air transportdata. Figure 2 shows that in the particular instance of COVID-19, the quality ofthis data leaves a lot to be desired. When the data for 2020 becomes available in
Acknowledgements
I am supported in part by NSERC and by CIHR through the Canadian COVID-19 Mathematical Modelling Task Force. I acknowledge support both financial and logistical fromthe Public Health Agency of Canada.
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