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Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2013 Article ID 576381 9 pageshttpdxdoiorg1011552013576381

Research ArticleRisk of Infectious Disease Outbreaks by Imported Cases withApplication to the European Football Championship 2012

Attila Deacutenes1 Peacuteter Kevei2 Hiroshi Nishiura34 and Gergely Roumlst1

1 Bolyai Institute University of Szeged Aradi vertanuk tere 1 Szeged 6720 Hungary2MTA-SZTE Analysis and Stochastics Research Group Bolyai Institute Aradi vertanuk tere 1 Szeged 6720 Hungary3 School of Public Health The University of Hong Kong 100 Cyberport Road Hong Kong4 PRESTO Japan Science and Technology Agency Saitama Japan

Correspondence should be addressed to Attila Denes denesamathu-szegedhu

Received 28 September 2012 Accepted 27 November 2012

Academic Editor Charles J Mode

Copyright copy 2013 Attila Denes et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The European Centre for Disease Prevention and Control called the attention in March 2012 to the risk of measles in Ukraineamong visitors to the 2012 UEFA European Football Championship Large populations of supporters travelled to various locationsin Poland and Ukraine depending on the schedule of Euro 2012 and the outcome of the games possibly carrying the disease fromone location to another In the present study we propose a novel two-phase multitype branching process model with immigrationto describe the risk of a major epidemic in connection with large-scale sports-related mass gathering events By analytic means wecalculate the expected number and the variance of imported cases and the probability of a major epidemic caused by the importedcases in their home country Applying ourmodel to the case study of Euro 2012 we demonstrate that the results of the football gamescan be highly influential to the risk of measles outbreaks in the home countries of supporters To prevent imported epidemics itshould be emphasized that vaccinating travellers would most efficiently reduce the risk of epidemic while requiring the minimumdoses of vaccines as compared to other vaccination strategies Our theoretical framework can be applied to other future sporttournaments too

1 Introduction

The European Centre for Disease Prevention and Controlreported ameasles outbreak inUkrainewithmore than 11000cases from the beginning of 2012 until the end of June 2012[1 2] The 2012 UEFA European Championship (Euro 2012)took place in Ukraine and Poland between 8 June and 1 July2012 attracting several hundreds of thousands of football fansto these countries [3] Susceptible visitors not only had a highrisk of being infected but also geographically propagating theepidemic to other countries

We introduce a discrete timeMarkov chain model whichis an adaptation of a multitype Galton-Watson process withimmigration to give a mathematical model for the evolutionof the epidemic Thus we calculate the risk of epidemicsconnected to sports-related mass gathering events Ourmodel consists of two parts the first one describing the spreadof the disease during the championship in the host country

while the second partmodels the spread of the disease by fansreturning to their home countries

We apply our model to the special case of measlesepidemics in Ukraine during the Euro 2012 Four of theeight host cities of this championship are in Ukraine (KievKharkiv Lviv and Donetsk) one of these Lviv is situatedin the western region where the prevalence is the highestand vaccination coverage remained the lowest in the countryGames of the group phase took place in the four Ukrainiancities for groups B and D including Denmark GermanyNetherlands Portugal andUkraine England France Sweden[4] Two of the quarterfinals one of the semifinals as well asthe final took place in Ukraine so Spain and Italy also playedsome games in Ukraine The suboptimal measles vaccinationcoverage in many European countries poses a risk of measlesepidemics caused by fans returning from Euro 2012 Herewe study the impact of different outcomes of Euro 2012on the probability of post-tournament measles epidemics in

2 International Journal of Stochastic Analysis

the participating countries and compare the effectiveness ofdifferent vaccination strategies by target host in reducing therisk of imported epidemics in other countries after Euro 2012We discuss the applicability of our approach to other futureevents as well

The rest of the paper is organized as follows In Section 2we describe the general mathematical model In Section 3we compute the probability of major epidemic in Franceafter Euro 2012 while in Section 4 we compare the resultswith Euro 2008 Finally we close with a discussion onthe applicability of our model for other sports-related massgathering events In the appendix we calculate explicitlythe expected number and the variance of infectious casesimported to the home country by supporters

2 Methods

Since the supporter group spends a relatively short time inthe infected area it is possible that nobody gets infectedin which case there is no increased chance for epidemicin the home country It is also clear that the risk of ahuge epidemic is larger when five infected individuals arrivehome (maybe to different parts of the country) than inthe case when only one infectious supporter arrives Thefact that the number of infected supporters is zero one orfive is just a matter of chance thus a deterministic modeldoes not serve for our purposes in this case It is wellknown (see [5]) that early stages of an epidemic in a largepopulation can be approximated by branching processeswhere having a descendant means infecting somebody Thisexactly fits to our model because in the host country thesupporters spend short time (up to a month say) and afterreturning to the home country we are only interested in theprobability of a major epidemic that is in the early stageof a possible epidemic To determine the final number ofinfectious individuals andor the duration of the epidemica mixture of a stochastic and a deterministic model is moreappropriate For general use of stochastic epidemicmodelswerefer to a recent survey by Britton [5]

To describe the importation dynamics in the simplestmanner as a mathematical model we propose a branchingprocess with immigration For simplicity consider a singlesupporter population 119878 from a country 119865 which follows thematches of the team during the tournament and ignore theinteraction with other supporter groups They can contractthe disease from the local population or from each otherWe define a discrete time Markov chain model which isan adaptation of a multitype Galton-Watson process withimmigration We say that an individual is of type-119895 if heshecontracted the infection exactly 119895 days ago The model isdivided into two phases the first phase takes 119879 days andcorresponds to the time spent in the host country while thesecond phase describes the process upon returning to thehome country Let 119898 be the mean latent period and 119896 themean infectious period of the disease (in days) that is anewly infected individual becomes infectious only after 119898

days and remains infectious for additional 119896daysWe assumethese as constants Denote byX

119905the integer vector of infected

individuals in population 119878 on day 119905 le 119879 where 119883119905(119895)

the number of type-119895 individuals is the number of infectedindividuals in population 119878 who got infected 119895 days ago 119895 =

1 2 119898+119896minus1 The evolution is the following On day 119905+1the newly infected individuals that is type-1 individuals canoriginate from the local population (immigrants) or from an119878-individual who is infectious on day 119905 + 1 (which means thatheshe got infected at least 119898 + 1 and at most 119898 + 119896 daysago) and thus is of type ge119898 on day 119905 (offsprings) We assumethat the force of infection from the local population to 119878 isconstant during the first phase and that the daily incidenceproduced by an infectious member of 119878 is also constantFinally for 119895 gt 1 type-119895 individuals arise only by getting oneday older After Phase 1 the infected vector X

119879returns to the

home country and each infected individual independentlystarts a simple single type Galton-Watson process

In the following we describe the exact mathematicalmodel

21 Phase 1 Let X119905

= (119883119905(1) 119883

119905(119889)) be a multitype

Galton-Watson process with immigration defined by

X119905=

119883119905minus1(1)

sum

119896=1

1205851199051198961

+ sdot sdot sdot +

119883119905minus1(119889)

sum

119896=1

120585119905119896119889

+ 120576119905 119905 isin 1 2 119879

X0= 0

(1)

where 120585119905119896119894

120576119905 119905 isin 1 2 119879 119896 isin N 119894 isin 1 2 119889

are independent random vectors with nonnegative integercoordinates such that 120585

119905119896119894 119905 isin 1 2 119879 119896 isin N are

identically distributed and 120576 120576119905 119905 isin 1 2 119879 are also

identically distributed Here the offsprings 120585rsquos correspond tothe new infections originated from an 119878-individual while theimmigrants 120576rsquos correspond to new infections originated fromthe local population Introduce the generating functions

119865119905(z) = EzX119905 119866

119894(z) = Ez12058511119894

G (z) = (1198661(z) 119866

119889(z)) 119867 (z) = Ez120576

(2)

where xk = 1199091198961

1sdot sdot sdot 119909119896119889

119889 In the following boldface symbols

x y z stand for 119889-dimensional vectorsIt is easy to show that the recursion 119865

119905(z) =

119865119905minus1

(G(z))119867(z) holds Let G119896 denote the 119896-fold iteration ofG that is G0(z) = z and G119905+1 = G119905 ∘ G Then an inductionargument shows (see Quine [6]) that the generating functionof the 119905th generation is

119865119905(z) =

119905minus1

prod

119896=0

119867(G119896 (z)) (3)

Up to now we did not use any particular property ofthe branching structure However note that in our case wehave the followingThe immigrants are always of type-1 thusthe generating function is in fact a one-variable functionthat is 119867(z) = 119867(119911

1 119911

119889) = 119867(119911

1) We also have 119889 =

119898 + 119896 minus 1 For 119895 = 1 2 119898 minus 1 every type-119895 particle has

International Journal of Stochastic Analysis 3

exactly one descendant of type-(119895+1) (the individual alreadyinfected is still not infectious only getting one day older)thus119866

119895(z) = 119911

119895+1 while for 119895 = 119898119898+1 119898+119896minus1 the type-

119895 individuals are already infecting and also getting one dayolder so 119866

119895(z) = 119911

119895+1119866119878(1199111) with 119866

119878(119911) being the generating

function of the infected individuals on one day by a singleinfectious individual in 119878 Without the vector notation wehave

119883119905(1) =

119898+119896minus1

sum

119894=119898

119883119905minus1(119894)

sum

119895=1

120585119905119894119895

+ 120576119905

119883119905(119895) = 119883

119905minus1(119895 minus 1) 119895 = 2 119898 + 119896 minus 1

(4)

where 120585119894119895119905

119894 = 119898 119898 + 119896 minus 1 119895 isin N 119905 = 1 2 119879 areiid random variables with generating function 119866

119878

22 Phase 2 Phase 2 starts with the infected vector X119879

arriving home In this stage there is no immigration andsince the infected individuals stay home there is no point onregistering the different types hence instead of counting thedays we count the generation119884

0is the number of individuals

who are infected by X119879 1198841is the number of individuals who

are infected by1198840 and so forthThat is the process now canbe

described by a single type Galton-Watson process Howeverthe first step is different because the different types havedifferent meanings Individuals of type-119895 119895 le 119898 spend alltheir infectious days in the home country while individualsof type-(119898+ 119895) 119895 = 1 2 119896 minus 1 spend only 119896minus 119895 infectiousdays in the home country Let 119866

119865(119911) denote the generating

function of the infected individuals on one day by a singleinfectious individual in the home country Let 119884

0be the

number of individuals whowere infected byX119879 thenwe have

1198840=

119898

sum

119894=1

119883119879(119894)

sum

119895=1

120585119894119895+

119898+119896minus1

sum

119894=119898+1

119883119879(119894)

sum

119895=1

120585119894119895 (5)

where 120585119894119895

119894 = 1 2 119898 + 119896 minus 1 119895 isin N are independentrandom variables and 120585

119894119895 119894 = 1 2 119898 119895 isin N are iid

with generating function119866119896119865 and for 119894 isin 119898+1 119898+119896minus1

120585119894119895

119895 isin N are iid with generating function 119866119898+119896minus119894

119865 Using

the representation above for the generating function of 1198840we

obtain

ℎ (119911) = E1199111198840

= 119865119879(119866119896

119865(119911) 119866

119896

119865(119911) 119866

119896minus1

119865(119911) 119866

119865(119911))

(6)

Now all who were infected after this step spend theirinfectious days in the home country so the process now isa simple single type Galton-Watson process with offspringgenerating function 119892(119911) = 119866

119896

119865(119911) starting from random

initial state 1198840

If this simple Galton-Watson process is critical or subcrit-ical that is 1198921015840(1) le 1 then the process dies out almost surelyregardless of the distribution of 119884

0 that is there is no major

epidemic in this case In the supercritical case when 1198921015840(1) gt

1 the probability that starting from a single individual the

process dies out is the unique root in (0 1) of the equation119892(119909) = 119909 Let 119902 denote this extinction probabilityThe processstarting from 119884

0dies out if all the 119884

0branches die out which

has probability 1199021198840 That is the probability of extinction of thewhole process is

P extinction = E1199021198840 = ℎ (119902) (7)

with ℎ as in (6)

3 Computations for the European FootballChampionship 2012

31 Risk of Measles Outbreak Depends on the Results ofthe Football Games In this section we apply the results tothe measles epidemic in Ukraine during the 2012 UEFAEuropean Football Championship For illustratory purposeswe have chosen France as a prototype for describing theresults In fact as being amongst the four favourites for theEuropean championship title [7] France was likely to beamongst the teams with the most supporters while havinglow vaccination coverage against measles posing an elevatedrisk of imported epidemic caused by supporter cases afterEuro 2012 We compare the following three scenarios one ofwhich is the real situation in Euro 2012 while the two othersare hypothetical cases representing the extremes for Franceby means of total time spent in Ukraine (see also Figure 1)

(a) France is eliminated in the group stage thus playingonly three games inUkraine between June 11 and June19 (hypothetical case)

(b) France finishes second in the group and is eliminatedin the quarterfinals playing four games in Ukrainebetween June 11 and June 23 (this is what actuallyhappened)

(c) France finishes second in the group and gets into thefinal thus playing six games between June 11 and July1 all in Ukraine (hypothetical case)

We assume that the supporter population is staying inUkraine as long as the team continues to play gamesThe totallength of stay would be the length of games plus one extra daydue to international travel and thus in the three cases we have119879 = 10 (a) 119879 = 14 (b) and 119879 = 21 (c)

For our computations we set 119898 = 9 119896 = 9 [8] Sincemeasles is generally rare in Europe the effective reproductionnumber in France 119877

119865= 1198921015840(1) is determined by the basic

reproduction number 1198770of measles and the effective vacci-

nation coverage 119907119865in France (eg the fraction of population

that is immunized and protected) due to 119877119865= 1198770(1 minus 119907

119865)

The basic reproduction number 1198770is estimated between 12

and 18 For computations we use 1198770

= 15 Thus 119877119865is

realistically assumed to be in the range 1ndash3 [9ndash12]The contactpattern within the supporter group might be different fromthe general population but still the effective reproductionnumber in Ukraine 119877

119878= 119896120573 = 119896119866

1015840

119878(1) is expected to be of

the samemagnitude as119877119865providing a reasonable range for 120573

Parameter 120582 = 1198671015840(1) which represents the expected number

of daily infected individuals infected by members of the local


Kiev

19060107

Kharkiv

11061506

Donetsk23062706

2406

Warsaw

Wroclaw

2806

1006

14061806

Lviv

Gdansk

Poznan

Figure 1Themovement of France during Euro 2012 and the dates of gamesThe solid arrow corresponds to the group stage the dashed arrowcorresponds to additional games in scenarios (b) and (c) and the dot-dashed arrow corresponds to the hypothetical case of getting into thefinal (scenario (c)) The dotted arrows represent the movement of Italy (chosen randomly for illustratory purposes) during the tournament

population is the most difficult to estimate as this is given bya combination of several factors the morbidity of measles inUkraine during the tournament the contact patterns betweenand within local and supporter populations the size of thesupporter group and the level of susceptibility in this groupWe scanned a large domain [0 0125] for 120582 A person havingmeasles changes his mixing and contact patterns due tothe infection but it should be noted that generally such achange in social behaviour is accounted for the estimate of1198770 Here we assume that individuals in the host country

home country and visitor populations modify their socialbehaviour similarly after contracting the disease thus ourthree key parameters 120582 120573 and 119877

119865are all proportional to 119877

0

By the nature of the immigration and the offspringdistributions it is natural to assume that these are Poisson orcompound Poisson distributed We calculate the extinctionprobabilities in two cases when the offspring and immi-gration distributions are Poisson distributions and whenthey are negative binomial distributions In the appendix weexplicitly calculate some relevant quantities We assume thatthe expectations of the total number of daily new infectionsfrom the local population (120582) the expectation of daily newinfections by one infectious individual from the supporterpopulation (120573) and the expectation of daily new infectionsby one infectious individual in the home country (120574) areknown and choose the parameters of the generating functionsaccordingly Note that 119877

119865= 9120574

Assuming that both the immigration and the offspringdistributions are Poisson we have

119867(119911) = 119890120582(119911minus1)

119866119878(119911) = 119890

120573(119911minus1)

119866119865(119911) = 119890

120574(119911minus1) 119892 (119911) = 119890

9120574(119911minus1)

(8)

A random variable 119883 has negative binomial distributionwith parameters 119903 gt 0 and 119901 isin (0 1) if P119883 = 119896 =

( 119896+119903minus1119903minus1

) (1 minus 119901)119903119901119896 119896 = 0 1 2 where the binomial

coefficient is defined by ( 119896+119903minus1119903minus1

) = (119896+119903minus1)(119896+119903minus2) sdot sdot sdot 119903119896The generating function is

E119911119883 = (1 minus 119901

1 minus 119901119911)

119903

(9)

so the expectation is E119883 = 119903119901(1 minus 119901) In the special casewhen 119903 = 1 we obtain the geometric distribution Assuminggeometric distribution for the immigration and the one-dayinfections

119867(119911) = [1 + 120582 minus 120582119911]minus1 119866

119878(119911) = [1 + 120573 minus 120573119911]

minus1

119866119865(119911) = [1 + 120574 minus 120574119911]

minus1

(10)

(the parameters are chosen to make the correspondingexpectations to be 120582 120573 and 120574 resp) and using that 119866119896

119865= 119892

we necessarily have

119892 (119911) = [1 + 120574 minus 120574119911]minus9

(11)

Figure 2 shows that the risk in scenario 119879 = 21 can betwice as large as in scenario 119879 = 10

Comparing Figures 2(a) and 2(b) we see that thereis no much difference in the behaviour of the extinctionprobabilities In the Poissonian case the extinction proba-bility is slightly larger than in the negative binomial casecorresponding to the sameparameter valuesTherefore in thefollowing we assume the Poissonian setup

The extinction probabilities cannot be computed explic-itly This is because 119902 the probability of extinction starting


from one individual cannot be calculated explicitly neitherin the Poissonian nor in the negative binomial case Wenumerically solve the equation for different values of 120574

running from 0111 up to 0334 that is 119877119865varies in the range

1ndash3 (recall that the expected value of offsprings in Phase 2 is9120574)Then we substitute these values into the explicitly knowngenerating function ℎ given in (6)

32 Vaccination and the Risk of Epidemics after Euro 2012We compare the effectiveness of three potential vaccinationstrategies in reducing the risk of imported major epidemic

(i) vaccination of the general population in France

(ii) vaccination of the general population in Ukraine

(iii) vaccination of football-associated travellers betweenFrance and Euro 2012 venues

To consider (i) note that increasing the vaccination rate119907119865decreases each of the parameters in our model That is

9120574 = 119877119865= 1198770(1 minus 119907

119865) 120582 = 120582

1(1 minus 119907

119865) and 120573 = 120573

1(1 minus 119907

119865)

We plotted the risk of major epidemic 119901 as a function of 119907119865

in Figure 3 and the expected number of imported cases inFigure 4

Figure 3(a) shows amildermeasles epidemics in Ukraineand Figure 3(b) corresponds to a more severe situation Wemay notice that it is particularly worthwhile to increase theimmunization rate in France if the epidemic is severe inUkraine because in this case we could observe a steep declinein the risk as 119907

119865increased beyond 084 (which is roughly

speaking consistentwith the reported present coverage in thecountry) Increasing 119907

119865has the benefit of decreasing the risk

of outbreaks by imported cases unrelated to Euro 2012On the other hand elevating the vaccination level 119907

119880of

the local Ukrainian population decreases 120582 = 1205822(1 minus 119907

119880)

The current value of 119907119880is reported to be about 05 [13]

Given the difference between the total populations ofUkraineand France increasing 119907

119865by one unit requires the same

amount of vaccines as increasing 119907119880by 14 units However the

computations show that 119901 is much less sensitive to 119907119880than to

119907119865(Figure 5) because small reduction in the risk of infection

during travel may only slightly reduce the imported casesTargeted vaccination of football visitors reduces both 120582 =

1205821(1 minus 119907

119879) and 120573 = 120573

1(1 minus 119907

119879) where 119907

119879is the level of

immunization in 119878 (ie in the absence of targeted vaccinationof travellers it is assumed that 119907

119879= 119907119865) Figure 6 shows the

efficiency of this strategy in the case of a milder and a moresevere Ukrainianmeasles epidemic If the vaccination historywas perfectly known elevating from 084 to 094 would beachieved by vaccinating 10 of the travellers from France(targeting the unvaccinated ones) Such an interventioncan halve the risk with relatively small efforts It shouldbe noted that elevating the coverage 119907

119879would require the

smallest number of doses (as compared to conducting massvaccinations in other scenarios) as vaccinating the supportersrequires only a couple of thousands of doses

4 Comparison with Euro 2008

In contrast to Euro 2012 here we descriptively review themeasles outbreaks which are likely associated with Euro 2008and other mass gathering events The 2008 UEFA EuropeanFootball Championship (Euro 2008) took place in Austriaand Switzerland from 7 to 29 June 2008 Significant measlesoutbreaks were reported in both of the host countries beforethe championship [14] However that situation was differentfrom this yearrsquos in several aspects First of all the vaccinationcoverage is much higher in Switzerland and Austria thanin Ukraine and consequently as the morbidity data showthe measles outbreak in 2012 in Ukraine is of significantlylarger scale than the one in the two host countries fouryears ago [15] It is also likely that Euro 2008 did not elevatethe relative number of travellers as much as Euro 2012 inUkraine as the two host countries of Euro 2008 are close toseveral of the participating countries and most host citiesare popular tourist destinations hosting a large number ofvisitors even without the football championship As it hasbeen pointed out in [16] a large scalemass gathering can evendiscourage regular tourists to visit the given cities to avoidthe crowdedness as happened in 2008 during the OlympicGames in BeijingThemedia reported a similar phenomenonin London during the 2012 Olympic Games In other cases(eg Sydney 2000) there was a surge of travellers and wecan assume the same for Ukraine as well

For Euro 2008we chose Germany as theGerman nationalteam reached the final of the championship whichmeans thattheir supporters spent 21 days in Austria and Switzerlandand WHO reports a suboptimal coverage of 83ndash89 forthe second dose of measles-containing vaccine in Germany[17] Taking into account the number of measles cases inAustriaSwitzerland in 2008 and in Ukraine in 2012 and thepopulation of these countries we can expect the parameter120582 to be approximately ten times smaller for Euro 2008 thanfor Euro 2012 Assuming Poisson distribution calculatingwith 120582 = 0004 and 120573 = 027 formula (A4) says thatthe probability of no imported infection is 092 that is theprobability of major epidemic is less than 008 which issignificantly smaller than the probabilities for Euro 2012

Data from 2008 show that in several participating coun-tries (eg France Germany Spain and Switzerland) therewere increases in the number of measles cases after Euro2008 compared to the same period of the year in 2007 [18ndash21] However based on available data a direct link cannot beestablished between Euro 2008 and these outbreaks

5 Other Sports-Related Mass Gatherings

As pointed out in [22] the last twoEuropean football champi-onships are not unique in the sense that curiously the footballchampionships seem to coincide with measles outbreaksApart from the two cases mentioned above during the FIFAWorld Cup 2006 a large measles outbreak was ongoing inGermany (host country) while there was an outbreak inSouth Africa during the FIFAWorld Cup 2010 Furthermorethe Winter Olympic Games in 2010 held in Vancouver werefollowed by a measles outbreak in British Columbia of about


0

01

02

03

04

05

1 15 2 25 3Effective reproduction number

in France RF

Prob

abili

ty o

f maj

orep

idem

icp

(a)

0

01

02

03

04

05

1 15 2 25 3

Prob

abili

ty o

f maj

orep

idem

icp

Effective reproduction number

in France RF

(b)

Figure 2 The probability of a major epidemic as the function of the effective reproduction number in France in the Poissonian case (a) andin the negative binomial case (b) The parameters are 120582 = 004 120573 = 027 The solid curve is for 119879 = 21 the dashed is for 119879 = 14 and thedot-dashed is for 119879 = 10

08 082 084 086 088 09 092 0940

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp

Effective vaccination coveragein France vF

(a)

0

02

04

06

08

1

08 082 084 086 088 09 092 094

Prob

abili

ty o

f maj

orep

idem

icp


(b)

Figure 3 The probability of a major epidemic as the function of the immunization rate in France The parameters are 1205731= 18 and 120582

1= 01

in (a) and 1205821= 05 in (b) The solid curve is for 119879 = 21 the dashed is for 119879 = 14 and the dot-dashed is for 119879 = 10

0

2

4

6

8

Num

ber

of im

port

ed c

ases

08 082 084 086 088 09 092 094

Effective vaccination coverage in France vF

Figure 4 The solid curve is the expectation of the total number ofimported cases in scenario (b) in the function of the immunizationrate in France At least with probability 075 the number of importedcases is smaller than the dashed curve and with probability 09 issmaller than the dot-dashed curve (calculated from Chebyshevrsquosinequality) The parameter values are the same as in Figure 3(b)

80 cases following three separate importations two of whichwere linked to the Olympic Games [23]

04 045 05 055 06 065 070

01

02

03

04

05

06

07

Effective vaccination coverage in Ukraine vu

Prob

abili

ty o

f maj

or e

pid

emic

p

Figure 5The probability of a major epidemic as the function of theimmunization rate in UkraineThe parameters are 119877

119865= 2 120573

1= 18

and 1205821= 01 The solid curve is for 119879 = 21 the dashed is for 119879 = 14

and the dot-dashed is for 119879 = 10

After Euro 2012 another sports related mass gatheringevent followed the Summer Olympic Games in LondonThere were several alerts about measles in connection withthe Olympic Games [24 25] However there are severaldifferences between football championships and the Olympic


08 085 09 095 10

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp

Level of immunization among

football-related travellers vT

(a)

0

02

04

06

08

1

08 085 09 095 1

Prob

abili

ty o

f maj

orep

idem

icp



(b)

Figure 6 The probability of a major epidemic as the function of the immunization rate in the supporter group The parameters are 119877119865= 2

1205731= 18 and 120582

1= 01 in (a) and 120582

1= 05 in (b) The solid curve is for 119879 = 21 the dashed is for 119879 = 14 and the dot-dashed is for 119879 = 10

Games Football championships have a special tournamentstructure and huge groups of fans moving together followingtheir national teams which is not typical for the OlympicGames Football championships are hosted by several citieswhile the Olympic Games are held (apart from some minorevents) in one city This means that our model fits ratherfor sport events which have the tournament structure likefootball World Cup and European Championship With anefficient monitoring after Euro 2012 it may be possibleto refine our parameters and prepare more realistic riskassessments using our approach for the forthcoming majorchampionships such as FIFA World Cup 2014 in Brazil andEuro 2016 in France The low vaccination rate and the recentand ongoingmeasles epidemics in France [26 27] suggest thatthere will be a risk of measles during Euro 2016 as well

6 Discussion

We constructed and applied a stochastic model to investigatethe risk of imported epidemics caused by visitors returningfrom a sports related mass gathering event to their homecountries after the tournament For the sake of simplicitywe considered a single supporter population while a realisticsituation of course involves many additional complicatingfactors including movements within the host country andinteractions between supporters and local population Weintroduced a discrete time Markov chain model with twophases which is an adaptation of a multitype Galton-Watsonprocess with immigration as a mathematical model andderived several analytical relations for the expectations vari-ances and probabilities regarding key aspects of the process

We applied our theoretical model to the measles epi-demics in Ukraine during the 2012 UEFA European FootballChampionship selecting the national team of France forillustratory purposes Due to the uncertainties in socialparameters we considered a wide interval for the transmis-sion rate between local and visitor populations Our approachclearly demonstrated that the travel patterns depend on theschedule and the results of the football games showing thatthe probability of a major measles epidemic in France couldbe greatly elevated by the successful outcomes of French

games Namely the more successful the national team is in afootball tournament the higher the risk of a post-tournamentimported measles epidemic would be in the home countryMore importantly we have compared different vaccinationstrategies and our study theoretically demonstrated that therisk of an imported measles epidemic by the visitors to Euro2012 and other mass gatherings would be most efficientlyreduced by vaccinating the visitors (travellers) Of coursevaccinating the entire French population would also beeffective (which actually prevents the country from not onlythe risk from Euro 2012 but also any other epidemics to beimported) but in theory this option requires us to securemillions of dosesThe optimal control by effectively targetingtravellers is novel both in practical and theoretical sensebecause the condensed interventions among travellers havebeen shown not to be very effective in preventing an epidemic(eg pandemic influenza) as long as there are arbitrarily largenumber of travellers We have shown that it is worth focusingon travellers when the number is finite and in themanageableorder Unvaccinated travellers would likely be covered withina few thousand doses and thus any country to respond to theassociated risk is suggested to consider this option

Appendix

We compute explicitly the expectation and variance of theoverall number of infectious individuals arriving home afterday 119879 that is 119885 = sum

17

119894=1119883119879(119894) and also we compute the

probability that there is no imported infection that is P119885 =

0 To do this we compute the generating function119865119879(z) given

in (3) and then we use that

P 119885 = 0 = 119865119879(0) E119885 =

17

sum

119894=1

120597119865119879

120597119911119894

(1)

Var119885 = sum

119894119895

(1205972119865119879

120597119911119894120597119911119895

(1) minus E119883119879(119894)E119883

119879(119895) + 120575

119894119895E119883119879(119894))

(A1)

with 120575119894119895

= 1 for 119894 = 119895 and 0 otherwise Note that oneminus the probability of no imported infection is a trivial


upper bound for the probability of a major epidemic and it isindependent of 120574

Assuming Poissonian offspring and immigration distri-bution we have the following

(i) 119879 = 10 in this case everything is relatively easy tocompute For example there is no imported infectionif and only if each day the number of immigrants is 0which has probability 119890minus10120582 We have

P 119885 = 0 = 119890minus10120582

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 + 3120573 + 1205732)

(A2)

(ii) 119879 = 14

P 119885 = 0 = 119890minus14120582

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 + 45120573 + 551205732)

(A3)

(iii) 119879 = 21

P 119885 = 0 = 119890minus17120582minus4(1minus119890

minus9120573)120582 E119885 = 120582 (17 + 72120573 + 10120573

2)

Var119885 = 120582 (17 + 144120573 + 5581205732+ 200120573

3+ 46120573

4)

(A4)

In the negative binomial case for the different scenarioswe have

(i) 119879 = 10

P 119885 = 0 = (1 + 120582)minus10

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 (1 + 120582) + 1205732(2 + 120582) + 120573 (3 + 2120582))

(A5)

(ii) 119879 = 14

P 119885 = 0 = (1 + 120582)minus14

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 (1 + 120582) + 15120573 (3 + 2120582) + 1205732(70 + 55120582))

(A6)

(iii) 119879 = 21

P 119885 = 0 = (1 + 120582)minus17

(1 + 120582 minus120582

(1 + 120573)9)

minus4

E119885 = 120582 (17 + 72120573 + 101205732)

Var119885 = 120582 (17 (1 + 120582) + 72120573 (2 + 120582) + 61205732(105 + 88120582)

+101205733(23 + 18120582) + 120573

4(66 + 46120582))

(A7)

Also note that in both cases the variance is large comparedto the expectation implying that the probability of noimported cases is large

Acknowledgments

A Denes and G Rost were supported by the EuropeanResearch Council Starting Investigator Grant no 259559the Hungarian Scientific Research Fund OTKA K75517 andBolyai Scholarship of the Hungarian Academy of SciencesP Kevei was supported by the TAMOP-421B-091KONV-2010-0005 Project and the Hungarian Scientific ResearchFundOTKAPD106181 H Nishiura received funding supportfrom the JST PRESTO Program and The University of HongKong Seed Funding Program (Grant Code 10208192)

References

[1] ldquoOutbreak of measles in Ukraine and potential for spreadin the EU Rapid Risk Assessmentrdquo ECDC 13 March 2012httpecdceuropaeuenpublicationsPublications20120314RA Measles Ukrainepdf

[2] ldquoCommunicable disease threats reportrdquo ECDCWeek 26 24ndash30June 2012 httpecdceuropaeuenpublicationsPublicationsCDTR web 2012 6 28pdf

[3] B R Humphreys and S Prokopowicz ldquoAssessing the impactof sports mega-events in transition economies EURO 2012 inPoland and Ukrainerdquo International Journal of Sport Manage-ment and Marketing vol 2 no 5-6 pp 496ndash509 2007

[4] ldquoUEFA webpagerdquo httpwwwuefacomuefaeuroseason=2012tournament-calendarindexhtml

[5] T Britton ldquoStochastic epidemic models a surveyrdquo Mathemati-cal Biosciences vol 225 no 1 pp 24ndash35 2010

[6] M P Quine ldquoThe multi-type Galton-Watson process with im-migrationrdquo Journal of Applied Probability vol 7 no 2 pp 411ndash422 1970

[7] ldquoBwinrdquo httpwwwbwincomEuro2012[8] R D Feigin J Cherry G J Demmler-Harrison and S L

Kaplan Feigin and Cherryrsquos Textbook of Pediatric InfectiousDiseases Saunders 6th edition 2009

[9] LrsquoInstitut de Veille Sanitaire (InVS) httpwwwinvssantefrDossiers-thematiquesMaladies-infectieusesMaladies-a-pre-vention-vaccinaleRougeolePoints-d-actualites

[10] L Fonteneau J-M Urcun C Kerneur et al ldquoCouverturevaccinale des enfants gs de 11 ans scolariss en CM2 France2004-2005rdquo Bulletin Epidemiologique Hebdomadaire vol 51-52pp 493ndash497 2008 (French)

[11] I P du Chatelet D Antona F Freymuth et al ldquoSpot-light on measles 2010 update on the ongoing measles out-break in france 2008ndash2010rdquo Euro Surveillance vol 15 no36 pp 1ndash4 2010 httpwwweurosurveillanceorgViewArticleaspxArticleId=19656

[12] I Bonmarin and D Lvy-Bruhl ldquoMeasles in France the epi-demiological impact of suboptimal immunisation coveragerdquoEuro Surveillance vol 7 no 4 p 322 2002 httpwwweuro-surveillanceorgViewArticleaspxArticleId=322

[13] ldquoWHO epidemiological briefrdquo no 21 2012 httpwwweurowhoint dataassetspdf file0004159475WHO EPIBrief Feb 2012epdf

[14] R Strauss P Kreidl M Muscat et al ldquoThe measles situa-tion in Austria a rapid risk assessment by an ECDC teamand the outcome of an International Meeting in ViennaAustriardquo Euro Surveillance vol 13 no 17 2008 httpwwweurosurveillanceorgViewArticleaspxArticleId=18852


[15] P Kreidl P Buxbaum F Santos-OrsquoConnor et al ldquo2008 Euro-pean Football ChampionshipmdashECDC epidemic intelligencesupportrdquo Euro Surveillance vol 13 no 32 2008 httpwwweurosurveillanceorgViewArticleaspxArticleId=18946

[16] K Khan S J N McNabb Z A Memish et al ldquoInfectiousdisease surveillance and modelling across geographic frontiersand scientific specialtiesrdquoThe Lancet Infectious Diseases vol 12no 3 pp 222ndash230 2012

[17] WHO ldquoReported estimates of MCV coveragerdquo httpappswhointimmunization monitoringenglobalsummarytimeseriestscoveragemcvhtm

[18] I Parent du Chtelet D Floret D Antona and D Lvy-Bruhl ldquoMeasles resurgence in France in 2008 a preliminaryreportrdquo Euro Surveillance vol 14 no 6 2009 httpwwweuro-surveillanceorgViewArticleaspxArticleId=19118

[19] Robert Koch Institut SurvStatRKI Abfrage der Meldedatennach Infektionsschutzgesetz (IfSG) ber das Webhttpwww3rkideSurvStat

[20] ldquoResultados de la vigilancia epidemiolgica de las enfer-medades transmisiblesrdquo Informe Anual 2008 httpwwwisciiiesISCIIIescontenidosfd-servicios-cientifico-tecnicosfd-vigilancias-alertasfd-enfermedadesInformeanual2008pdf

[21] E Delaporte E Jeannot P Sudre C A W Lazarevic JL Richard and P Chastonay ldquoMeasles in Geneva between2003 and 2010 persistence of measles outbreaks despite highimmunisation coveragerdquo Euro Surveillance vol 16 no 39 2011httpwwweurosurveillanceorgViewArticleaspxArticleId=19980

[22] Editorial team ldquoSpotlight on measles 2010rdquo Euro Surveillvol 15 no 17 2010 httpwwweurosurveillanceorgView-ArticleaspxArticleId=19559

[23] ldquoBC Centre for Disease Controlrdquo httpwwwbccdccaresourcematerialsnewsandalertshealthalertsBefore+travell-ing+ensure+your+measles+vaccination+is+up+to+datehtm

[24] ldquoSurveillance reportrdquo European Monthly Measles Monitoring(EMMO) Issue 10 2012 httpecdceuropaeuenpublicationsPublications1205-SUR-Measles-monthly-monitoringpdf

[25] ldquoCenters for Disease Control and Preventionrdquo httpwwwnccdcgovtravelnoticesoutbreak-noticemeasleshtm

[26] C Huoi J S Casalegno T Bnet et al ldquoA report onthe large measles outbreak in Lyon France 2010 to 2011rdquoEuro Surveillance vol 17 no 36 2012 httpwwweuro-surveillanceorgViewArticleaspxArticleId=20264

[27] ldquoMeasles and rubella monitoringrdquo Surveillance reportECDC 2012 httpwwwecdceuropaeuenpublicationsPub-lications2012Sept SUR measles-rubella-monitoringpdf

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the participating countries and compare the effectiveness ofdifferent vaccination strategies by target host in reducing therisk of imported epidemics in other countries after Euro 2012We discuss the applicability of our approach to other futureevents as well

The rest of the paper is organized as follows In Section 2we describe the general mathematical model In Section 3we compute the probability of major epidemic in Franceafter Euro 2012 while in Section 4 we compare the resultswith Euro 2008 Finally we close with a discussion onthe applicability of our model for other sports-related massgathering events In the appendix we calculate explicitlythe expected number and the variance of infectious casesimported to the home country by supporters

2 Methods

Since the supporter group spends a relatively short time inthe infected area it is possible that nobody gets infectedin which case there is no increased chance for epidemicin the home country It is also clear that the risk of ahuge epidemic is larger when five infected individuals arrivehome (maybe to different parts of the country) than inthe case when only one infectious supporter arrives Thefact that the number of infected supporters is zero one orfive is just a matter of chance thus a deterministic modeldoes not serve for our purposes in this case It is wellknown (see [5]) that early stages of an epidemic in a largepopulation can be approximated by branching processeswhere having a descendant means infecting somebody Thisexactly fits to our model because in the host country thesupporters spend short time (up to a month say) and afterreturning to the home country we are only interested in theprobability of a major epidemic that is in the early stageof a possible epidemic To determine the final number ofinfectious individuals andor the duration of the epidemica mixture of a stochastic and a deterministic model is moreappropriate For general use of stochastic epidemicmodelswerefer to a recent survey by Britton [5]

To describe the importation dynamics in the simplestmanner as a mathematical model we propose a branchingprocess with immigration For simplicity consider a singlesupporter population 119878 from a country 119865 which follows thematches of the team during the tournament and ignore theinteraction with other supporter groups They can contractthe disease from the local population or from each otherWe define a discrete time Markov chain model which isan adaptation of a multitype Galton-Watson process withimmigration We say that an individual is of type-119895 if heshecontracted the infection exactly 119895 days ago The model isdivided into two phases the first phase takes 119879 days andcorresponds to the time spent in the host country while thesecond phase describes the process upon returning to thehome country Let 119898 be the mean latent period and 119896 themean infectious period of the disease (in days) that is anewly infected individual becomes infectious only after 119898

days and remains infectious for additional 119896daysWe assumethese as constants Denote byX

119905the integer vector of infected

individuals in population 119878 on day 119905 le 119879 where 119883119905(119895)

the number of type-119895 individuals is the number of infectedindividuals in population 119878 who got infected 119895 days ago 119895 =

1 2 119898+119896minus1 The evolution is the following On day 119905+1the newly infected individuals that is type-1 individuals canoriginate from the local population (immigrants) or from an119878-individual who is infectious on day 119905 + 1 (which means thatheshe got infected at least 119898 + 1 and at most 119898 + 119896 daysago) and thus is of type ge119898 on day 119905 (offsprings) We assumethat the force of infection from the local population to 119878 isconstant during the first phase and that the daily incidenceproduced by an infectious member of 119878 is also constantFinally for 119895 gt 1 type-119895 individuals arise only by getting oneday older After Phase 1 the infected vector X

119879returns to the

home country and each infected individual independentlystarts a simple single type Galton-Watson process

In the following we describe the exact mathematicalmodel

21 Phase 1 Let X119905

= (119883119905(1) 119883

119905(119889)) be a multitype

Galton-Watson process with immigration defined by

X119905=

119883119905minus1(1)

sum

119896=1

1205851199051198961

+ sdot sdot sdot +

119883119905minus1(119889)

sum

119896=1

120585119905119896119889

+ 120576119905 119905 isin 1 2 119879

X0= 0

(1)

where 120585119905119896119894

120576119905 119905 isin 1 2 119879 119896 isin N 119894 isin 1 2 119889

are independent random vectors with nonnegative integercoordinates such that 120585

119905119896119894 119905 isin 1 2 119879 119896 isin N are

identically distributed and 120576 120576119905 119905 isin 1 2 119879 are also

identically distributed Here the offsprings 120585rsquos correspond tothe new infections originated from an 119878-individual while theimmigrants 120576rsquos correspond to new infections originated fromthe local population Introduce the generating functions

119865119905(z) = EzX119905 119866

119894(z) = Ez12058511119894

G (z) = (1198661(z) 119866

119889(z)) 119867 (z) = Ez120576

(2)

where xk = 1199091198961

1sdot sdot sdot 119909119896119889

119889 In the following boldface symbols

x y z stand for 119889-dimensional vectorsIt is easy to show that the recursion 119865

119905(z) =

119865119905minus1

(G(z))119867(z) holds Let G119896 denote the 119896-fold iteration ofG that is G0(z) = z and G119905+1 = G119905 ∘ G Then an inductionargument shows (see Quine [6]) that the generating functionof the 119905th generation is

119865119905(z) =

119905minus1

prod

119896=0

119867(G119896 (z)) (3)

Up to now we did not use any particular property ofthe branching structure However note that in our case wehave the followingThe immigrants are always of type-1 thusthe generating function is in fact a one-variable functionthat is 119867(z) = 119867(119911

1 119911

119889) = 119867(119911

1) We also have 119889 =

119898 + 119896 minus 1 For 119895 = 1 2 119898 minus 1 every type-119895 particle has



119895(z) = 119911



119895(z) = 119911

119895+1119866119878(1199111) with 119866



119883119905(1) =

119898+119896minus1

sum

119894=119898

119883119905minus1(119894)

sum

119895=1

120585119905119894119895

+ 120576119905

119883119905(119895) = 119883


(4)

where 120585119894119895119905


119878









0be the


1198840=

119898

sum

119894=1

119883119879(119894)

sum

119895=1

120585119894119895+

119898+119896minus1

sum

119894=119898+1

119883119879(119894)

sum

119895=1

120585119894119895 (5)

where 120585119894119895


119894119895 119894 = 1 2 119898 119895 isin N are iid


120585119894119895


119865 Using


obtain

ℎ (119911) = E1199111198840

= 119865119879(119866119896

119865(119911) 119866

119896

119865(119911) 119866

119896minus1

119865(119911) 119866

119865(119911))

(6)


119896











P extinction = E1199021198840 = ℎ (119902) (7)

with ℎ as in (6)












119865)



= 15 Thus 119877119865is


119878= 119896120573 = 119896119866

1015840






Kiev

19060107

Kharkiv

11061506

Donetsk23062706

2406

Warsaw

Wroclaw

2806

1006

14061806

Lviv

Gdansk

Poznan





0


119865= 9120574


119867(119911) = 119890120582(119911minus1)

119866119878(119911) = 119890

120573(119911minus1)

119866119865(119911) = 119890

120574(119911minus1) 119892 (119911) = 119890

9120574(119911minus1)

(8)


( 119896+119903minus1119903minus1




E119911119883 = (1 minus 119901

1 minus 119901119911)

119903

(9)


119867(119911) = [1 + 120582 minus 120582119911]minus1 119866

119878(119911) = [1 + 120573 minus 120573119911]

minus1

119866119865(119911) = [1 + 120574 minus 120574119911]

minus1

(10)


119865= 119892

we necessarily have

119892 (119911) = [1 + 120574 minus 120574119911]minus9

(11)













9120574 = 119877119865= 1198770(1 minus 119907

119865) 120582 = 120582

1(1 minus 119907

119865) and 120573 = 120573

1(1 minus 119907

119865)








119880of


119880)








1205821(1 minus 119907

119879) and 120573 = 120573

1(1 minus 119907

119879) where 119907



119879= 119907119865) Figure 6 shows the











0

01

02

03

04

05


in France RF

Prob

abili

ty o

f maj

orep

idem

icp

(a)

0

01

02

03

04

05

1 15 2 25 3

Prob

abili

ty o

f maj

orep

idem

icp


in France RF

(b)


08 082 084 086 088 09 092 0940

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp


(a)

0

02

04

06

08

1

08 082 084 086 088 09 092 094

Prob

abili

ty o

f maj

orep

idem

icp


(b)


1= 01


0

2

4

6

8

Num

ber

of im

port

ed c

ases

08 082 084 086 088 09 092 094




04 045 05 055 06 065 070

01

02

03

04

05

06

07


Prob

abili

ty o

f maj

or e

pid

emic

p


119865= 2 120573

1= 18





08 085 09 095 10

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp



(a)

0

02

04

06

08

1

08 085 09 095 1

Prob

abili

ty o

f maj

orep

idem

icp



(b)


1205731= 18 and 120582

1= 01 in (a) and 120582



6 Discussion




Appendix


17





P 119885 = 0 = 119865119879(0) E119885 =

17

sum

119894=1

120597119865119879

120597119911119894

(1)

Var119885 = sum

119894119895

(1205972119865119879

120597119911119894120597119911119895

(1) minus E119883119879(119894)E119883

119879(119895) + 120575

119894119895E119883119879(119894))

(A1)

with 120575119894119895






P 119885 = 0 = 119890minus10120582

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 + 3120573 + 1205732)

(A2)

(ii) 119879 = 14

P 119885 = 0 = 119890minus14120582

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 + 45120573 + 551205732)

(A3)

(iii) 119879 = 21


minus9120573)120582 E119885 = 120582 (17 + 72120573 + 10120573

2)

Var119885 = 120582 (17 + 144120573 + 5581205732+ 200120573

3+ 46120573

4)

(A4)


(i) 119879 = 10

P 119885 = 0 = (1 + 120582)minus10

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 (1 + 120582) + 1205732(2 + 120582) + 120573 (3 + 2120582))

(A5)

(ii) 119879 = 14

P 119885 = 0 = (1 + 120582)minus14

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 (1 + 120582) + 15120573 (3 + 2120582) + 1205732(70 + 55120582))

(A6)

(iii) 119879 = 21

P 119885 = 0 = (1 + 120582)minus17

(1 + 120582 minus120582

(1 + 120573)9)

minus4

E119885 = 120582 (17 + 72120573 + 101205732)

Var119885 = 120582 (17 (1 + 120582) + 72120573 (2 + 120582) + 61205732(105 + 88120582)

+101205733(23 + 18120582) + 120573

4(66 + 46120582))

(A7)


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119895(z) = 119911



119895(z) = 119911

119895+1119866119878(1199111) with 119866



119883119905(1) =

119898+119896minus1

sum

119894=119898

119883119905minus1(119894)

sum

119895=1

120585119905119894119895

+ 120576119905

119883119905(119895) = 119883


(4)

where 120585119894119895119905


119878









0be the


1198840=

119898

sum

119894=1

119883119879(119894)

sum

119895=1

120585119894119895+

119898+119896minus1

sum

119894=119898+1

119883119879(119894)

sum

119895=1

120585119894119895 (5)

where 120585119894119895


119894119895 119894 = 1 2 119898 119895 isin N are iid


120585119894119895


119865 Using


obtain

ℎ (119911) = E1199111198840

= 119865119879(119866119896

119865(119911) 119866

119896

119865(119911) 119866

119896minus1

119865(119911) 119866

119865(119911))

(6)


119896











P extinction = E1199021198840 = ℎ (119902) (7)

with ℎ as in (6)












119865)



= 15 Thus 119877119865is


119878= 119896120573 = 119896119866

1015840






Kiev

19060107

Kharkiv

11061506

Donetsk23062706

2406

Warsaw

Wroclaw

2806

1006

14061806

Lviv

Gdansk

Poznan





0


119865= 9120574


119867(119911) = 119890120582(119911minus1)

119866119878(119911) = 119890

120573(119911minus1)

119866119865(119911) = 119890

120574(119911minus1) 119892 (119911) = 119890

9120574(119911minus1)

(8)


( 119896+119903minus1119903minus1




E119911119883 = (1 minus 119901

1 minus 119901119911)

119903

(9)


119867(119911) = [1 + 120582 minus 120582119911]minus1 119866

119878(119911) = [1 + 120573 minus 120573119911]

minus1

119866119865(119911) = [1 + 120574 minus 120574119911]

minus1

(10)


119865= 119892

we necessarily have

119892 (119911) = [1 + 120574 minus 120574119911]minus9

(11)













9120574 = 119877119865= 1198770(1 minus 119907

119865) 120582 = 120582

1(1 minus 119907

119865) and 120573 = 120573

1(1 minus 119907

119865)








119880of


119880)








1205821(1 minus 119907

119879) and 120573 = 120573

1(1 minus 119907

119879) where 119907



119879= 119907119865) Figure 6 shows the











0

01

02

03

04

05


in France RF

Prob

abili

ty o

f maj

orep

idem

icp

(a)

0

01

02

03

04

05

1 15 2 25 3

Prob

abili

ty o

f maj

orep

idem

icp


in France RF

(b)


08 082 084 086 088 09 092 0940

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp


(a)

0

02

04

06

08

1

08 082 084 086 088 09 092 094

Prob

abili

ty o

f maj

orep

idem

icp


(b)


1= 01


0

2

4

6

8

Num

ber

of im

port

ed c

ases

08 082 084 086 088 09 092 094




04 045 05 055 06 065 070

01

02

03

04

05

06

07


Prob

abili

ty o

f maj

or e

pid

emic

p


119865= 2 120573

1= 18





08 085 09 095 10

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp



(a)

0

02

04

06

08

1

08 085 09 095 1

Prob

abili

ty o

f maj

orep

idem

icp



(b)


1205731= 18 and 120582

1= 01 in (a) and 120582



6 Discussion




Appendix


17





P 119885 = 0 = 119865119879(0) E119885 =

17

sum

119894=1

120597119865119879

120597119911119894

(1)

Var119885 = sum

119894119895

(1205972119865119879

120597119911119894120597119911119895

(1) minus E119883119879(119894)E119883

119879(119895) + 120575

119894119895E119883119879(119894))

(A1)

with 120575119894119895






P 119885 = 0 = 119890minus10120582

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 + 3120573 + 1205732)

(A2)

(ii) 119879 = 14

P 119885 = 0 = 119890minus14120582

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 + 45120573 + 551205732)

(A3)

(iii) 119879 = 21


minus9120573)120582 E119885 = 120582 (17 + 72120573 + 10120573

2)

Var119885 = 120582 (17 + 144120573 + 5581205732+ 200120573

3+ 46120573

4)

(A4)


(i) 119879 = 10

P 119885 = 0 = (1 + 120582)minus10

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 (1 + 120582) + 1205732(2 + 120582) + 120573 (3 + 2120582))

(A5)

(ii) 119879 = 14

P 119885 = 0 = (1 + 120582)minus14

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 (1 + 120582) + 15120573 (3 + 2120582) + 1205732(70 + 55120582))

(A6)

(iii) 119879 = 21

P 119885 = 0 = (1 + 120582)minus17

(1 + 120582 minus120582

(1 + 120573)9)

minus4

E119885 = 120582 (17 + 72120573 + 101205732)

Var119885 = 120582 (17 (1 + 120582) + 72120573 (2 + 120582) + 61205732(105 + 88120582)

+101205733(23 + 18120582) + 120573

4(66 + 46120582))

(A7)


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Kiev

19060107

Kharkiv

11061506

Donetsk23062706

2406

Warsaw

Wroclaw

2806

1006

14061806

Lviv

Gdansk

Poznan





0


119865= 9120574


119867(119911) = 119890120582(119911minus1)

119866119878(119911) = 119890

120573(119911minus1)

119866119865(119911) = 119890

120574(119911minus1) 119892 (119911) = 119890

9120574(119911minus1)

(8)


( 119896+119903minus1119903minus1




E119911119883 = (1 minus 119901

1 minus 119901119911)

119903

(9)


119867(119911) = [1 + 120582 minus 120582119911]minus1 119866

119878(119911) = [1 + 120573 minus 120573119911]

minus1

119866119865(119911) = [1 + 120574 minus 120574119911]

minus1

(10)


119865= 119892

we necessarily have

119892 (119911) = [1 + 120574 minus 120574119911]minus9

(11)













9120574 = 119877119865= 1198770(1 minus 119907

119865) 120582 = 120582

1(1 minus 119907

119865) and 120573 = 120573

1(1 minus 119907

119865)








119880of


119880)








1205821(1 minus 119907

119879) and 120573 = 120573

1(1 minus 119907

119879) where 119907



119879= 119907119865) Figure 6 shows the











0

01

02

03

04

05


in France RF

Prob

abili

ty o

f maj

orep

idem

icp

(a)

0

01

02

03

04

05

1 15 2 25 3

Prob

abili

ty o

f maj

orep

idem

icp


in France RF

(b)


08 082 084 086 088 09 092 0940

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp


(a)

0

02

04

06

08

1

08 082 084 086 088 09 092 094

Prob

abili

ty o

f maj

orep

idem

icp


(b)


1= 01


0

2

4

6

8

Num

ber

of im

port

ed c

ases

08 082 084 086 088 09 092 094




04 045 05 055 06 065 070

01

02

03

04

05

06

07


Prob

abili

ty o

f maj

or e

pid

emic

p


119865= 2 120573

1= 18





08 085 09 095 10

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp



(a)

0

02

04

06

08

1

08 085 09 095 1

Prob

abili

ty o

f maj

orep

idem

icp



(b)


1205731= 18 and 120582

1= 01 in (a) and 120582



6 Discussion




Appendix


17





P 119885 = 0 = 119865119879(0) E119885 =

17

sum

119894=1

120597119865119879

120597119911119894

(1)

Var119885 = sum

119894119895

(1205972119865119879

120597119911119894120597119911119895

(1) minus E119883119879(119894)E119883

119879(119895) + 120575

119894119895E119883119879(119894))

(A1)

with 120575119894119895






P 119885 = 0 = 119890minus10120582

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 + 3120573 + 1205732)

(A2)

(ii) 119879 = 14

P 119885 = 0 = 119890minus14120582

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 + 45120573 + 551205732)

(A3)

(iii) 119879 = 21


minus9120573)120582 E119885 = 120582 (17 + 72120573 + 10120573

2)

Var119885 = 120582 (17 + 144120573 + 5581205732+ 200120573

3+ 46120573

4)

(A4)


(i) 119879 = 10

P 119885 = 0 = (1 + 120582)minus10

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 (1 + 120582) + 1205732(2 + 120582) + 120573 (3 + 2120582))

(A5)

(ii) 119879 = 14

P 119885 = 0 = (1 + 120582)minus14

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 (1 + 120582) + 15120573 (3 + 2120582) + 1205732(70 + 55120582))

(A6)

(iii) 119879 = 21

P 119885 = 0 = (1 + 120582)minus17

(1 + 120582 minus120582

(1 + 120573)9)

minus4

E119885 = 120582 (17 + 72120573 + 101205732)

Var119885 = 120582 (17 (1 + 120582) + 72120573 (2 + 120582) + 61205732(105 + 88120582)

+101205733(23 + 18120582) + 120573

4(66 + 46120582))

(A7)


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















9120574 = 119877119865= 1198770(1 minus 119907

119865) 120582 = 120582

1(1 minus 119907

119865) and 120573 = 120573

1(1 minus 119907

119865)








119880of


119880)








1205821(1 minus 119907

119879) and 120573 = 120573

1(1 minus 119907

119879) where 119907



119879= 119907119865) Figure 6 shows the











0

01

02

03

04

05


in France RF

Prob

abili

ty o

f maj

orep

idem

icp

(a)

0

01

02

03

04

05

1 15 2 25 3

Prob

abili

ty o

f maj

orep

idem

icp


in France RF

(b)


08 082 084 086 088 09 092 0940

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp


(a)

0

02

04

06

08

1

08 082 084 086 088 09 092 094

Prob

abili

ty o

f maj

orep

idem

icp


(b)


1= 01


0

2

4

6

8

Num

ber

of im

port

ed c

ases

08 082 084 086 088 09 092 094




04 045 05 055 06 065 070

01

02

03

04

05

06

07


Prob

abili

ty o

f maj

or e

pid

emic

p


119865= 2 120573

1= 18





08 085 09 095 10

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp



(a)

0

02

04

06

08

1

08 085 09 095 1

Prob

abili

ty o

f maj

orep

idem

icp



(b)


1205731= 18 and 120582

1= 01 in (a) and 120582



6 Discussion




Appendix


17





P 119885 = 0 = 119865119879(0) E119885 =

17

sum

119894=1

120597119865119879

120597119911119894

(1)

Var119885 = sum

119894119895

(1205972119865119879

120597119911119894120597119911119895

(1) minus E119883119879(119894)E119883

119879(119895) + 120575

119894119895E119883119879(119894))

(A1)

with 120575119894119895






P 119885 = 0 = 119890minus10120582

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 + 3120573 + 1205732)

(A2)

(ii) 119879 = 14

P 119885 = 0 = 119890minus14120582

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 + 45120573 + 551205732)

(A3)

(iii) 119879 = 21


minus9120573)120582 E119885 = 120582 (17 + 72120573 + 10120573

2)

Var119885 = 120582 (17 + 144120573 + 5581205732+ 200120573

3+ 46120573

4)

(A4)


(i) 119879 = 10

P 119885 = 0 = (1 + 120582)minus10

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 (1 + 120582) + 1205732(2 + 120582) + 120573 (3 + 2120582))

(A5)

(ii) 119879 = 14

P 119885 = 0 = (1 + 120582)minus14

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 (1 + 120582) + 15120573 (3 + 2120582) + 1205732(70 + 55120582))

(A6)

(iii) 119879 = 21

P 119885 = 0 = (1 + 120582)minus17

(1 + 120582 minus120582

(1 + 120573)9)

minus4

E119885 = 120582 (17 + 72120573 + 101205732)

Var119885 = 120582 (17 (1 + 120582) + 72120573 (2 + 120582) + 61205732(105 + 88120582)

+101205733(23 + 18120582) + 120573

4(66 + 46120582))

(A7)


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

01

02

03

04

05


in France RF

Prob

abili

ty o

f maj

orep

idem

icp

(a)

0

01

02

03

04

05

1 15 2 25 3

Prob

abili

ty o

f maj

orep

idem

icp


in France RF

(b)


08 082 084 086 088 09 092 0940

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp


(a)

0

02

04

06

08

1

08 082 084 086 088 09 092 094

Prob

abili

ty o

f maj

orep

idem

icp


(b)


1= 01


0

2

4

6

8

Num

ber

of im

port

ed c

ases

08 082 084 086 088 09 092 094




04 045 05 055 06 065 070

01

02

03

04

05

06

07


Prob

abili

ty o

f maj

or e

pid

emic

p


119865= 2 120573

1= 18





08 085 09 095 10

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp



(a)

0

02

04

06

08

1

08 085 09 095 1

Prob

abili

ty o

f maj

orep

idem

icp



(b)


1205731= 18 and 120582

1= 01 in (a) and 120582



6 Discussion




Appendix


17





P 119885 = 0 = 119865119879(0) E119885 =

17

sum

119894=1

120597119865119879

120597119911119894

(1)

Var119885 = sum

119894119895

(1205972119865119879

120597119911119894120597119911119895

(1) minus E119883119879(119894)E119883

119879(119895) + 120575

119894119895E119883119879(119894))

(A1)

with 120575119894119895






P 119885 = 0 = 119890minus10120582

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 + 3120573 + 1205732)

(A2)

(ii) 119879 = 14

P 119885 = 0 = 119890minus14120582

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 + 45120573 + 551205732)

(A3)

(iii) 119879 = 21


minus9120573)120582 E119885 = 120582 (17 + 72120573 + 10120573

2)

Var119885 = 120582 (17 + 144120573 + 5581205732+ 200120573

3+ 46120573

4)

(A4)


(i) 119879 = 10

P 119885 = 0 = (1 + 120582)minus10

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 (1 + 120582) + 1205732(2 + 120582) + 120573 (3 + 2120582))

(A5)

(ii) 119879 = 14

P 119885 = 0 = (1 + 120582)minus14

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 (1 + 120582) + 15120573 (3 + 2120582) + 1205732(70 + 55120582))

(A6)

(iii) 119879 = 21

P 119885 = 0 = (1 + 120582)minus17

(1 + 120582 minus120582

(1 + 120573)9)

minus4

E119885 = 120582 (17 + 72120573 + 101205732)

Var119885 = 120582 (17 (1 + 120582) + 72120573 (2 + 120582) + 61205732(105 + 88120582)

+101205733(23 + 18120582) + 120573

4(66 + 46120582))

(A7)


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










08 085 09 095 10

02

04

06

08

1

Prob

abili

ty o

f maj

orep

idem

icp



(a)

0

02

04

06

08

1

08 085 09 095 1

Prob

abili

ty o

f maj

orep

idem

icp



(b)


1205731= 18 and 120582

1= 01 in (a) and 120582



6 Discussion




Appendix


17





P 119885 = 0 = 119865119879(0) E119885 =

17

sum

119894=1

120597119865119879

120597119911119894

(1)

Var119885 = sum

119894119895

(1205972119865119879

120597119911119894120597119911119895

(1) minus E119883119879(119894)E119883

119879(119895) + 120575

119894119895E119883119879(119894))

(A1)

with 120575119894119895






P 119885 = 0 = 119890minus10120582

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 + 3120573 + 1205732)

(A2)

(ii) 119879 = 14

P 119885 = 0 = 119890minus14120582

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 + 45120573 + 551205732)

(A3)

(iii) 119879 = 21


minus9120573)120582 E119885 = 120582 (17 + 72120573 + 10120573

2)

Var119885 = 120582 (17 + 144120573 + 5581205732+ 200120573

3+ 46120573

4)

(A4)


(i) 119879 = 10

P 119885 = 0 = (1 + 120582)minus10

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 (1 + 120582) + 1205732(2 + 120582) + 120573 (3 + 2120582))

(A5)

(ii) 119879 = 14

P 119885 = 0 = (1 + 120582)minus14

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 (1 + 120582) + 15120573 (3 + 2120582) + 1205732(70 + 55120582))

(A6)

(iii) 119879 = 21

P 119885 = 0 = (1 + 120582)minus17

(1 + 120582 minus120582

(1 + 120573)9)

minus4

E119885 = 120582 (17 + 72120573 + 101205732)

Var119885 = 120582 (17 (1 + 120582) + 72120573 (2 + 120582) + 61205732(105 + 88120582)

+101205733(23 + 18120582) + 120573

4(66 + 46120582))

(A7)


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













P 119885 = 0 = 119890minus10120582

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 + 3120573 + 1205732)

(A2)

(ii) 119879 = 14

P 119885 = 0 = 119890minus14120582

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 + 45120573 + 551205732)

(A3)

(iii) 119879 = 21


minus9120573)120582 E119885 = 120582 (17 + 72120573 + 10120573

2)

Var119885 = 120582 (17 + 144120573 + 5581205732+ 200120573

3+ 46120573

4)

(A4)


(i) 119879 = 10

P 119885 = 0 = (1 + 120582)minus10

E119885 = 120582 (10 + 120573)

Var119885 = 120582 (10 (1 + 120582) + 1205732(2 + 120582) + 120573 (3 + 2120582))

(A5)

(ii) 119879 = 14

P 119885 = 0 = (1 + 120582)minus14

E119885 = 120582 (14 + 15120573)

Var119885 = 120582 (14 (1 + 120582) + 15120573 (3 + 2120582) + 1205732(70 + 55120582))

(A6)

(iii) 119879 = 21

P 119885 = 0 = (1 + 120582)minus17

(1 + 120582 minus120582

(1 + 120573)9)

minus4

E119885 = 120582 (17 + 72120573 + 101205732)

Var119885 = 120582 (17 (1 + 120582) + 72120573 (2 + 120582) + 61205732(105 + 88120582)

+101205733(23 + 18120582) + 120573

4(66 + 46120582))

(A7)


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article risk of infectious disease outbreaks by ...internationaljournalofstochasticanalysis...

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