Modelling Covid-19 epidemic in Mexico, Finland and Iceland
Rafael A. Barrio, Kimmo K. Kaski, Gudmundur G. Haraldsson, Thor Aspelund, Tzipe Govezensky
MModelling Covid-19 epidemic in Mexico, Finland and Iceland
Rafael A. Barrio , Kimmo K. Kaski , , Guđmundur G. Haraldsson , Thor Aspelund , , and Tzipe Govezensky Instituto de Física, Universidad Nacional Autónoma de México, CP 01000 CDMX, Mexico Department of Computer Science, Aalto University School of Science, FI-00076 AALTO, Finland The Alan Turing Institute, 96 Euston Rd, Kings Cross, London NW1 2DB, UK Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland Centre for Public Health Sciences, University of Iceland, Reykjavik, Iceland. The Icelandic Heart Association, Iceland. and Instituto de Investigaciones Biomédicas, Universidad Nacional Autónoma de México,Apartado Postal 70228, Ciudad Universitaria, CP 04510 CDMX, Mexico (Dated: July 22, 2020)Over the past two decades there has been a number of global outbreaks of viral diseases. This hasaccelerated the efforts to model and forecast the disease spreading, in order to find ways to confinethe spreading regionally and between regions. Towards this we have devised a model of geographicalspreading of viral infections due to human spatial mobility and adapted it to the latest Covid-19pandemic. In this the region to be modelled is overlaid with a two-dimensional grid weighted with thepopulation density defined cells, in each of which a compartmental SEIRS system of delay differenceequations simulate the local dynamics (microdynamics) of the disease. The infections between cellsare stochastic and allow for the geographical spreading of the virus over the two-dimensional space(macrodynamics). This approach allows to separate the parameters related to the biological aspectsof the disease from the ones that represent the spatial contagious behaviour through different kindsof mobility of people acting as virus carriers. These provide sufficient information to trace theevolution of the pandemic in different situations. In particular we have applied this approach tothree in many ways different countries, Mexico, Finland and Iceland and found that the model iscapable of reproducing and predicting the stochastic global path of the pandemic. This study shedslight on how the diverse cultural and socioeconomic aspects of a country influence the evolution ofthe epidemics and also the efficacy of social distancing and other confinement measures.
I. INTRODUCTION
In today’s globalised world people easily travel longdistances to faraway places, where they can get infected,carry and consequently spread lethal and easily mutatingviruses that could subsequently lead to wide-spread epi-demic or even pandemic of catastrophic proportions. Re-cent examples of such globally spreading viral diseases arethe seasonal influenza, SARS, MERS, AH1N1 swine fluand currently COVID-19 that is turning out to be ratherlethal showing growing death tolls in various countries.This in turn causes a lot of concern among general pub-lic and huge strain in national healthcare and hospitalcapacities including their Intensive Care Units (ICUs),but also among public authorities and governments as towhat measures, restrictions, recommendations to put inplace and when, to keep the society running in balancewith the rest of its socioeconomic functions.In order to make decisions of these large scale, andfar reaching issues for mitigating the effects of the dis-ease, one would need to take into account the pathologi-cal aspects of the virus, epidemiological factors includingmechanisms of virus spreading, human sociality includ-ing their social contacting, behavioural, mobility, trav-elling and communication patterns, demographic factorsincluding age, gender, and regional density of popula-tion as well as concentration in built environments, espe-cially cities and public transportation volumes, terrestri-ally and aerially.The commonly accepted measures to counter the ef- fect of these factors are recommendations for individualsin terms of enhanced hygiene and keeping 1 to 2 me-ter social or in fact physical distance, or the governmentor local authority enforced restrictions for staying andworking at home in self-quarantine, requirement to wearface mask in public places and transportation or evencurfews, allowing gatherings with only a small numberof people, lock-down of countries or districts by closingschools, universities and other public institutions as wellas restaurants and shops, other than food markets, lim-iting public local, terrestrial and aerial transportation aswell as non-essential entries to the countries or regions.In addition, various countries have adopted compre-hensive testing of even asymptomatic individuals to-gether with tracking them with smart phone app. Itis evident that countries adopt, follow, and ease theserestriction and confinement measures in a number of dif-ferent ways and with different timings, as many of themtend to be strongly linked to the culture, politics, econ-omy and overall well-being of the country. Furthermore,there are collective effects due to news and social mediadepending on how lethal the disease is perceived to be.Over the past two decades a lot of effort and advancehave been made to get understanding of the populationlevel complex dynamics of infectious disease spreading [1–3]. In this various epidemiological models - often com-partmental SIRS (susceptible - infected - recovered - sus-ceptible) type - have proved very versatile in implement-ing the above described epidemically relevant factors forqualitative and quantitative or predictive insight into the a r X i v : . [ phy s i c s . s o c - ph ] J u l cause of infection spreading [4–7]. Models including geo-graphical spread and population heterogeneity have alsobeen developed [8–16], and recently, many models includ-ing different aspects of pandemics have been developed,see for example [17–20].Therefore, it is interesting to study how these socialrestrictions work and how efficient they are in countrieswith different cultures and social habits. In this work wedecided to do a comparative study of three very differ-ent countries, Mexico, Finland and Iceland. They differnot only in population size and density, but also in caseof pandemic in their behavioural habits, restriction andconfinement policies, testing and tracking as well as howindividuals follow and respond to them and to authori-ties. For this purpose we use a model [12] devised to dealwith the viral swine flu pandemics that started in Mexicoin 2009. The model is different from former studies [8–10, 15], the aims of which were to mimic the developmentof a particular epidemic by evaluating parameters fromspecific data. This model enables us to study not onlythe actual observed data, but also to predict the temporalevolution of the disease under hypothetical and changingscenarios of social distancing and confinement measuresin a given geographical region. II. MODELA. SEIRS epidemiological model (Microdynamics)
The model used in this study is an extension of the onepresented earlier by Barrio et al. [12], and it considers thegeographical spread of the epidemics at two levels: thelocal dynamics and more global transmission of the dis-ease from one place to another. For this purpose, a two-dimensional grid of a country or region is constructed insuch a way that within each cell ( i, j ) the population den-sity ρ ( i, j ) is considered to be homogeneous. For the localdynamics a Susceptible - Exposed - Infected - Recovered(-Susceptible) or SEIR(S) model [21] is used for eachcell, consisting of four compartments: not infected sus-ceptible ( X ), exposed yet not infectious ( E ), infectious( Y ), and recovered temporary immune ( Z ), representingan instantaneous local average state of the population.In the model the periods of latency ( (cid:15) ), infectious-ness ( σ ) and immunity ( ω ) are assumed to be constantand dimensionless by expressing them in units of a timescale τ of one day. Here we assume that the popula-tion size does not change throughout the simulation, i.e. N = X + E + Y + Z = constant and for the mortality weassume an exponential functional form with a constantrate µ = 1 /L, where L is the life expectancy, and thebirth rate is µN with all the newborn considered suscep-tible. As people who have recovered do not necessarilybecome susceptible again, because of long lasting immu-nity or changes in social behaviour, the actual fraction ofpopulation that becomes susceptible ( S ) is included.Considering all these assumptions, we can now express the flow rates per day for all the variables by the followingmap equations on each cell α = ( i, j ) : X t +1 ( α ) = q (cid:2) X t ( α ) − G t ( α ) + Sq b G t − − b ( α ) (cid:3) + µN,E t +1 ( α ) = q [ E t ( α ) + G t ( α ) − q (cid:15) G t − − (cid:15) ( α )] ,Y t +1 ( α ) = q [ Y t ( α ) + q (cid:15) G t − − (cid:15) ( α ) − q a G t − − a ( α )] ,Z t +1 ( α ) = q (cid:2) Z t ( α ) + q a G t − − a ( α ) − q b G t − − b ( α ) (cid:3) . (1)where q = (1 − µ ) , a = ( (cid:15) + σ ) , and b = ( (cid:15) + σ + ω ) .Unlike in case of other SEIRS models, in our modelthe probability of becoming infectious is not based onthe mass action principle, but instead we use a bio-logically more sensible incidence function: G t ( i, j ) = ρ ( i, j ) X t ( i, j )[1 − exp − βY t ( i,j ) ] [21], where β is a dimen-sionless constant representing the transmission parame-ter.The model exhibits a rich variety of behaviours, asexpected: regular oscillations of different period, dampedoscillations, quasi-periodicity, chaos and stable regimes.Therefore, the dynamics of other infectious diseases canbe modelled with the same system, since all parametersso far, except the population density, characterise thedisease, such as its virulence and lethality. In this studyWe shall give specific values to these parameters whenapplied to the specific country or region of COVID-19,as discussed below. B. Geographical disease spreading(Macrodynamics)
In each cell the relative population is represented bya population density matrix ρ ( i, j ) . The size of each cellis determined by the actual conditions of the country orregion to be simulated. The model in each cell representsan instantaneous local average state there, and the dis-ease can be transmitted from one cell to another becausepeople move distances larger than the cell size. There-fore, the size of each cell should be congruent with theaverage distance that a person travels every day, to workor shopping, etc.Here we consider three mobility mechanisms:1. Cell to cell transmission is often modelled as a dif-fusion process, but this implies that the susceptiblereceives a certain amount of infection, and the in-fectious one becomes healthier by the same amount.Other SEIR models consider anomalous diffusionwith fractional derivatives, which seems to be moreappropriate [17]. An alternative approach is to con-sider a parameter of average terrestrial mobility orvelocity, v t , which we assume to be stochastic innature. Therefore, we can model the probability ofspreading the disease from one cell to neighbour-ing cells by using a Metropolis Monte-Carlo algo-rithm. First, one locates the potential spreadercells, this means having Y t ( i, j ) ≥ η , where η isrelated to the infectiousness of the disease. Then,one chooses a random number from a uniform dis-tribution ( p ∈ [0 , ), and if p is smaller than v t ,a neighbour cell (denoted by α ) becomes infected,that is X ( α ) = 1 − η and Y ( α ) = η . It is clear thatwe know very little about the real meaning of v t ,since the reasons for people to making a trip arequite varied. In this sense, v t is related to the so-cial and cultural habits of the individuals, and it isrelated to the average number of personal contactsper day.2. Particular attention has been drawn on the in-fluence of air travelling upon the spread of dis-eases [20, 22–27]. A similar stochastic process isused to model transmission from a cell to othermore distant cells, connected by airlines. In thiscase, the probability of spreading the infection isproportional to the number of passengers per daytravelling from one airport to another airport. Thisshould be simulated by locating the airports in thegrid and defining an air mobility parameter v a . Theprobability of infection is represented by a weightedadjacency matrix ( A ) of the airline network, whoseelements represent the average number of passen-gers between linked airports. Then, we run onceagain the Metropolis Monte-Carlo algorithm by us-ing v a to decide infectiousness between the elementsof A (cells including the airports).3. Since people seemingly travel randomly betweendistant places, noise has to be considered. This willcause the spread of the epidemics to unexpectedplaces. In order to simulate this we introduce aMonte-Carlo procedure, similar to the geographi-cal spread of the illness. Therefore, we considercells ( i, j ) in which there are not many susceptible( X t ( i, j ) < η ) at random, and compare the value ofa random number p with a quantity of the form e − /kT , since these random displacements can beconsidered analogous to the “kinetic energy” kT or“temperature” T of the system. If the Monte Carlocondition is fulfilled, then one starts the disease inthat cell. III. APPLICATION TO THE COVID-19EPIDEMICS IN MEXICO, FINLAND ANDICELAND
As in this model the biological parameters are clearlyseparated from those related to social, cultural, and eco-nomic phenomena, one could adjust the biological param-eters according to present knowledge. For the COVID-19the latency, (cid:15) , is thought to be from 2 to 14 [19, 28] days,though interestingly, (cid:15) = 1 improved the adjustment ofthe data; the infectiousness σ = 14 is set to the standard âĂIJquarantineâĂİ time used in many countries; the im-munity ω is yet unknown, but for SARS-CoV antibodiesand memory T-cell response were detected 1 year or evenlonger after the infection [29, 30]. To be conservative,here we use ω = 140 . Currently there are no data abouttransmission parameter β and disease infectiousness pa-rameters η , so we estimated them to be β = 0 . and η = 0 . , by adjusting Mexican data from March 3 to April13. As we assume that these parameters to be epidemio-logically relevant for COVID-19, we use them unchangedfor the three countries studied here. However, the mobil-ity parameters were adjusted for each country taking intoaccount different strategies and measures used in them tomitigate the effect of the pandemic. A. The case of Mexico
In order to illustrate the results obtained from themodel, we use the geographical demographic data ofMexico from INEGI [31] and from the John Hopkins Uni-versity (J.H.) map [32] and apply it to the outbreak ofCOVID-19 in 2020. The frequency of air travel and thenumber of passengers per day in Mexico were obtainedfrom Dirección General de Aeronáutica Civil of the Sec-retaría de Comunicaciones y Transportes [33].In Mexico the first imported case occurred in Mex-ico City (central Mexico) on the 28th of February, andshortly after more imported cases were reported, onein Mexico City, other in Cancun (Yucatan peninsula) ,other in Baja California and another in Sinaloa (northernMexico). Mexican government together with Ministry ofHealth (SecretarÃŋa de Salud) implemented some mea-sures such as tracking COVID-19 positive people andtheir contacts. On March 14th, the Ministry of Pub-lic Education (SecretarÃŋa de EducaciÃşn PÞblica) ex-tended vacations in primary and high schools until April20th in the whole country; some days latter this periodwas extended until April 30th. On March 18th univer-sity students were sent home and social distancing wasinitiated. On March 23th, at national level, Universitieswere closed as well as public places such as concerts, the-atres, cinemas, churches, museums, gyms, zoos, and bars;home confinement was recommended. Airports were notclosed, but they diminished their operations to only asmall percentage of flights and passenger volumes. OnMarch 26th non-essential activities of the governmentwere suspended, and on March 30th non-essential eco-nomic activities at national level were suspended untilApril 30th.On April 13th we made a calculation adjusting themobility parameters to the data available from the JohnHopkins page [32]. The values which best fitted the datawere: v t = 0 . , v a = 0 . , and kT = 0 . . These valuesrepresent an overall mobility reduction of approximately35 % of the normal values. There is also the possibilitythat the survival parameter S changes in time, if thereare tests to detect potentially infectious people, or if thereare vaccines or effective medication. So far this has nothappened, so we used S = 0 . , throughout the calcula-tions.On May 25th the Mexican authorities announced aplan to lift the restrictions gradually, and asynchronously,considering the situation of the pandemic in each state.Consequently, the three mobility parameters and the fre-quency of flights would change with time slowly, fromMay to August assuming that the mobility will be re-stored to 50% of the value before the outbreak. FIG. 1. Geographical numerical calculations of the modelwith parameters appropriate for Covid-19. The predictionin this plot was made on April 13th, with the adjustmentsmade from data up to that date. The black line assumesthat mobility restrictions do not change during the year, whilethe red line was obtained by feeding the mobility strategyannounced by the government to lift restrictions gradually.
In Fig.1 we show examples of numerical calcula-tions performed with these two hypothetical conditions,namely without lifting the restrictions, and following theplan to restore normality. Observe that the effect of lift-ing the restrictions is very small, but noticeable: the peakis attained around the 9th of July, in both cases, buttaller when lifting, and the epidemic dies off faster thanin the case of doing nothing. However, the number ofinfected people would remain practically the same.In Fig. 2 we show an average of 50 realisations of thetime history of the pandemic and it is seen to modelvery closely the actual data. This means that althoughcostly economically, the social distancing measures areextremely effective in mitigating the effects of epidemics.It is remarkable that the prediction made in April stillholds tightly within the confidence interval more thantwo months later. A maximum of the number of cases ispredicted to occur between the 9th and the 14th of July.
B. The case of Finland
On January 29th Finland confirmed its first COVID-19case, a 32-year-old Chinese woman from Wuhan havingsought medical attention in Ivalo and tested positive forSARS-CoV-2. She was quarantined at Lapland Central
FIG. 2. Time history of the number of new cases per day.Continuous blue line: daily confirmed cases from February28th to April 13th used to adjust the model parameters. Con-tinuous red line: numerical prediction assuming no furtherchanges in societal conditions. Broken black lines : 99% con-fidence interval. Bars: actual daily data up to July 8th. Greenline is the 10-day average of the actual data.
Hospital in Rovaniemi. The woman recovered and wasdischarged on February 5th after testing negative on twoconsecutive days. Three weeks later, on February 26th,Finland’s health officials confirmed the second case, aFinnish woman, who made a trip to Milan and returnedon February 22th, tested positive at the Helsinki Uni-versity Central Hospital. On February 28th, a Finnishwoman who had also travelled to Northern Italy, testedpositive by the Helsinki and Uusimaa Hospital Districtand was advised to remain in home isolation [34].These two latter cases mark the beginning of COVID-19 epidemic in Finland, and the number of new cases wasthen followed closely by the National Institute of Healthand Well being (THL) [35], advising the Finnish Govern-ment the course of pandemic and the disease confinementmeasures [36]. As a result on March 12th the Governmentdecides on recommendations to curb the spread of coronavirus by public events and workplaces closures. This wassoon followed on March 16th by Government decision onlock down measures, to shut down all schools and mostgovernment-run public facilities (theatres, libraries, mu-seums etc.). In addition, at most 10 people were allowedto participate in a public meeting, outsiders were forbid-den from entering healthcare facilities and hospitals, andtravel cross the internal Schengen and EU borders werelimited to essential goods transportation, people havingto because of work, and citizens returning to Finland.These measures were scheduled to be in place until April13th, but were later extended to May 13th at which pointthe pre- and primary schools were opened for the last twoweeks of the spring semester, while other measures werekept in effect.On March 21st the first death, an elderly individualwho lived in the Helsinki and Uusimaa hospital district,was reported. On March 27th, the Parliament votedunanimously to temporarily close the borders of the Uusi-maa region (area surrounding the capital Helsinki), whichhad and still has the most confirmed cases. On April 15thtravel restrictions between Uusimaa region and the restof the country were lifted. In addition, from April 4th toMay 31st the Government decided to close all restaurantsand hotels. On June 1st onward the gradually opening ofcountry started by allowing the maximum number of peo-ple to be increased to 50, opening restaurants and publicplaces such as museums as well as allowing sports events,all of these "with special arrangements" to maintain suf-ficient physical or social distance. Then according to gov-ernment decision from June 15th on, travellers enteringfrom the Baltic countries and the other Nordic countriesexcept Sweden will no longer have to stay quarantined for14 days, but other international travel restrictions werekept. From July 1st onward Government decided thatthe outdoor events with more than 500 people will beallowed with the arrangements to keep sufficient physicaldistance. Overall it is worth mentioning that people haveso far followed all the confinement measures, restrictionand recommendations by the Government very closely,which is seen as a large reduction of reported daily infec-tion cases to less than 10.Due to the above mentioned travel restrictions airtravel has reduced to 2 - 3 % from mid-March to Mayand to 6 - 7 % in June in comparison to the same monthslast year (2019). The air travel volumes will very grad-ually start increasing from the beginning of July on, asthe EU internal border restrictions are gradually startedto be lifted. Due to strong recommendation by the Gov-ernment to avoid travelling within Finland during mid-March to June period even the passenger volumes by rail-ways have been running low, but expected to start pick-ing up from mid-June on as people prefer having theirvacations in Finland. We estimate the travel volumesonly to about 20% the value before the epidemic.Taking these measures into account we introducedthem in the model to modify the aerial, terrestrial andcasual mobilities along the time line of their introduction.For the model there are four major events: (i) the 12-14March closing down (with v t , reduced to 50 %, kT = 0 . ),(ii) the March 27th isolation of the Uusimaa region (caus-ing drastic reduction of v t to 25% and kT = 0 . ), (iii) theApril 15th lifting travel restrictions of the Uusimaa re-gion (with v t and kT restored to the March values), and(iv) the air travel reduction from March to June, (with v a changing from 2% in March - May to 7% in June).In Fig. 3 we show one example of the intricate cal-culation that resulted from the above described socialdistancing measures. Observe that the averaged valuesof daily cases follows remarkably well the averaged actualdata.We have also performed a set of forty realisations andaveraged them to calculate the of 99% confidence intervalof the model predictions. The results are shown in Fig. 4.Observe that the model works quite well over the wholetime interval, but little less so in between April 15th andMay 5th, when the Uusimaa area opened. This is possi- FIG. 3. Time history of the number of new cases per dayin Finland. Continuous blue line: Actual data averaged over7 days. Continuous red line: numerical prediction includingchanges in social distancing. Bars: actual data taken from [32]FIG. 4. Average over 20 realisations of the model calcula-tions (red line) showing the of 99% confidence interval (blackbroken kines) and the averaged data in green. bly due to people’s heightened willingness after openingto go visit their relatives and their summer cottages. Aswe have no data of the number of people that travelledby road during that lapse, we have not been able to takeit into account in the model. Just as a matter of curiositywe made a calculation to predict the effect of train travelpassenger volumes starting from June 15th to increasegradually up to 20% and air travel passenger volumesgradually up to 7% of the values before the pandemic.The results are depicted in Fig. 5 and show only a slightincrease in the number of daily cases to about a dozenand persisting at that level throughout summer.
FIG. 5. As Fig. 4 including in blue the prediction if trains andairport start running on June 15th (trains 20% and airportto 7%)
C. The case of Iceland
On January 29th, Chief Epidemiologist ThorolfurGuđnason advised against unnecessary travel to China,and recommended that people travelling from China un-dertake 14 days quarantine upon returning to Iceland.On the 31st of January, a meeting in the National Secu-rity Council was scheduled with the Minister of Healthand Chief Epidemiologist. The Department of Civil Pro-tection and Emergency Management (DCPEM) evokedthe National Crisis Co-ordination Center. By the 3rdof February, Iceland had defined high-risk areas, includ-ing Northern Italy and Tirol, earlier than other govern-ments, taking stricter measures with a 14-day quarantinerequirement for all residents returning from those areas.February 27th saw the first daily press conference, at-tended by IcelandâĂŹs Chief Epidemiologist, Directorof Health and Chief Superintendent. The first case ofCOVID-19 was confirmed in Iceland on February 28th.This was a person arriving from Northern Italy. DCPEMdeclared the alert phase. By March 6th over 30 importedcases had been confirmed and the first two transmissionswithin Iceland, traced to infected individuals who hadrecently traveled to Northern Italy. The alert level wasraised to the emergency phase.On March 13th, screening for the virus that causesCOVID-19 started among the general public. deCODEgenetics graciously offered to test everyone in the countrywho wanted to be tested. At the same time, a ban ongatherings of more than 100 people was announced andthen implemented on the 16th. High schools and Univer-sities were closed, and operations of kindergartens andprimary schools were limited.By March 31st, Iceland had limited gatherings to 20people or fewer, closed sports clubs, hair salons, bars, andsimilar establishments, implemented fines for breachingthe rules of quarantine, and became a party to an inter-national contract which enabled the Icelandic authorities to join European Union members in the procurement ofvarious healthcare equipment. April 2nd saw the tracingapp Rakning C-19 become available in the App Store andGoogle Play to track the virus.As a result of the quick action taken by the govern-ment, results from initial screenings indicated a low rateof infections among the general public. By April 21st, itwas announced that the ban on gatherings and school ac-tivities would be relaxed, effective on May 4th 2020. Thelimit on the number of people who may gather increasedfrom 20 to 50; pre-schools and primary schools reopened;athletic and youth activities became unrestricted again.On May 25th, gyms and swimming pools reopened andoperated with limitations, such as a maximum number ofguests. Gatherings of up to 200 people were allowed. Allrestaurants and bars re-opened with a curfew of 11 PM[37].Taking these measures into account in the model wechanged only the mobility parameters on the dates men-tioned. We considered domestic flights operating withvery few and diminishing number of passengers from thebeginning of the outbreak, while the international air-port was getting many imported cases until March 16th,when the aerial and terrestrial mobility were assumedto be reduced by 70%, reflecting the full obedience ofthe population to the confinement measures taken. Be-tween April 2nd and April 9th a 90% mobility reductionwas achieved, but on May 5th the mobility increased to60%. The random mobility was kept to the original valueof kT = 0 . , because of the uneven population densityin the island, as 62% of the population is concentratedaround the Reykjavik area, so it makes little difference if kT is reduced.In Fig.6 we show one realisation of the model and com-pare it with the actual data, both averaged over 7 days.We could say that in this case, the disease had been con-trolled in the early days of May. FIG. 6. (a) Time history of the number of new cases per dayin Iceland. Continuous blue line: Actual data averaged over7 days. Continuous red line: numerical prediction includingthe changes in social distancing mentioned in the text. Bars:actual data taken from [32]
The borders opened on June 15th. Passengers arrivingin the country were given the option of taking a COVID-19 test or undergoing quarantine for 14 days. Childrenborn in 2005 and later were exempt. Testing was offeredat KeflavÃŋk Airport and at other international portsof entry. Passengers were also required to answer a pre-arrival questionnaire, abide by sickness rules and wereencouraged to download the app, Rakning C-19. At thesame time, further relaxation of the ban on assembly tookeffect. The number of gatherings increased from 200 to500 and restrictions on the number of swimming poolsand fitness centers were reduced.Considering these relaxations of the social distancingmeasures, we performed a calculation on June 10th to trypredicting the effects that would result from this action.In Fig.7 we show the results from the model averagingover 20 realisations. We assume that one should expectaround 1.5 cases per 1000 passenger coming from abroad.
FIG. 7. (a) Average over 20 realisations of the time historyof the number of daily cases in Iceland. Continuous greenline: Actual data averaged over 7 days. Continuous red line:numerical prediction including the opening of the airport inKeflavik on June 15th. Broken lines: 99% confidence interval.
The results are encouraging, since the model is predict-ing a tail of 2 to 3 new daily cases on average. Observethat the confidence intervals are larger than in the casesof Finland and Mexico. This is expected, since the pop-ulation, and consequently, the number of cases is muchsmaller. Despite the small number of cases predictedwhen opening the airport, people in Iceland received awarning, they were too relaxed in early June, particu-larly young people thinking that the pandemic was overin June, but after some new cases appeared after June15th, they became more careful, scared and worried andstarted behaving accordingly.
IV. DISCUSSION
We have presented a stochastic model of geographi-cal spreading of infectious disease that differs in manyrespects from other compartmental SEIRS type models, commonly used to describe epidemic spreading. In par-ticular, our model separates the disease-defining epidemi-ological parameters from the ones that have to do withgeography, and people’s social habits. It also includes so-cietal traffic and travelling infrastructure and passengerflow networks based on roads, trains and airline routes.Furthermore, it allows one to define mobility quantitiesthat could vary in time, to make predictions of futurescenarios of the epidemic.There is a recent review [38] that discusses variousmodelling approaches to study particularly the COVID-19 pandemic, and the general comment is that for modelsto be more predictive, one needs much more and precisedata. The fact that with the model presented here onecan make quite accurate predictions is due to its intrinsicrandom nature, which allows to approach real situationswith very little information. This model has performedwell also with other viral diseases, as shown earlier withthe swine influenza [12]. With the present study we havedemonstrated that this model can be used for very differ-ent scales of population and their densities as well as forcountries with different social structures and situations.By studying the three countries presented here we areable to learn several important lessons. First of all thatit could be unwise to lift the country or regional levelconfinement measures, restrictions and recommendationscompletely when the pandemic is still active in othercountries or regions, as it was demonstrated in the caseof Iceland and Finland. It shows that going back to anormality as free as last year will not be possible for along time. Of course we hope that there is an effectivevaccine or treatment available soon. It should be notedthat in the case of Mexico the model prediction of the be-haviour of the pandemic was made early in the so calledexponential phase, i.e. April 13th when the amount ofavailable information was small, yet it has followed theactual number of cases very closely so far.The second issue is that of the so called second wave,to be expected around the end of this year or beginning ofnext year. So far we have made the calculations assum-ing that the countries remain very much isolated, withvery limited travel between them. In addition we haveused this model to foresee what could happen if the threecountries were made totally open by the end of this year.The results are shown in Fig.8.We observe that the rebound is greatest in Finland,small but noticeable in Iceland and extremely small inMexico. This is understandable on the grounds that inFinland a large portion of the population has not be-come exposed during the first wave due to obedientlyfollowed confinement measures, in Iceland this is not thecase, due to obediently followed confinement measuresand perhaps the most vivid population-wide testing andtracking policy and in Mexico the pandemics following anormal course infecting practically everybody and reach-ing even the most remote areas of the country during thefirst wave.Perhaps one of the most important points of this com-
FIG. 8. (a) Calculations comparing the number of daily casesfor the three countries in the hypothetical scenario that fromNovember 6th onward the mobility is reestablished as it wasin December 2019. Notice that the data for Mexico have beendivided by 10. (b) Same comparison for the total number ofinfections in logarithmic scale. parative study is that the timely social distancing andconfinement measures, as well as public restrictions andrecommendations are seemingly efficient ways to controlthe disease spreading, especially when people follow themtightly, as it is the case in Finland and Iceland both be-ing sparsely populated and socially relatively homoge-neous countries. In addition, in case of Iceland from thevery beginning of the epidemic the population-wide test-ing and tracking served as an enhanced control and life-saver. In case of Mexico imposing confinement measuresis more complicated, because of much larger populationand very large densely populated cities with huge socioe-conomic differences. Still the authorities have managedto slow down the speed of disease spreading. The price topay in all these three countries is that the pandemic willbe active for quite a while, but on the other hand ourunderstanding of confinement and prevention measureshas been tested and will allow to face the possible secondwave in a better way.
ACKNOWLEDGMENTS
RAB acknowledges support from The National Au-tonomous University of Mexico (UNAM) and AlianzaUCMX of the University of California (UC), through theproject included in the Special Call for Binational Col-laborative Projects addressing COVID-19. RAB was fi-nancially supported by Conacyt through project 283279.KK acknowledges support for Visiting Fellowship at TheAlan Turing Institute, UK, and the European Communi-tyâĂŹs H2020 Research Infrastructures "SoBigData++:Social Mining and Big Data EcosystemâĂİ project. [1] R. M. Anderson and R. M. May.
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