in Hong Kong Modelling the skip-and-resurgence of Japanese encephalitis epidemics Journal of Theoretical Biology

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Journal of Theoretical Biology

journalhomepage:www.elsevier.com/locate/jtb

Modelling the skip-and-resurgence of Japanese encephalitis epidemics in Hong Kong

Shi Zhao, Yijun Lou, Alice P.Y. Chiu, Daihai He

^∗

Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong (SAR) China

a rt i c l e i n f o

Article history:

Received 12 October 2017 Revised 14 May 2018 Accepted 16 May 2018 Available online 21 May 2018 Keywords:

Japanese encephalitis virus Mathematical modelling Skip-and-resurgence Vector-free transmission

a b s t r a c t

Japaneseencephalitisvirus(JEV)isazoonoticmosquito-bornevirus,persistinginpigs,Ardeidbirdsand Culexmosquitoes.ItisendemictoChinaandSoutheasternAsia.Thecase-fatalityratio(CFR)ortherate ofpermanentpsychiatricsequelaeis30%amongsymptomaticpatients.TherewerenoreportedlocalJEV humancasesbetween2006and2010inHongKong,butitwasfollowedbyaresurgenceofcasesfrom 2011to2017.Themechanismbehindthis“skip-and-resurgence” patterns is unclear.

Thisworkaimstorevealthemechanismbehindthe“skip-and-resurgence” patternsusing mathemati- calmodellingandlikelihood-basedinferencetechniques.Wefoundthatpig-to-pigtransmissionincreases thesize ofJEVepidemicsbutisunlikelyto maintainthe samelevel oftransmissionamongpigs.The disappearanceofJEVhumancasesin2006–2010couldbeexplainedbyasuddenreductionofthepopu- lationoffarmpigsasaresultoftheimplementationofthevoluntary“pig-rearinglicencesurrendering”

policy.Theresurgencecouldbeexplainedbyofanewstrainin2011,whichincreasedthetransmissibility ofthevirusorthespill-overratiofromreservoirtohostorboth.

1. Introduction

Japanese encephalitis virus (JEV) is a zoonotic andmosquito- borne virus that is the major causeof viral encephalitis inAsia.

The annual total conﬁrmed human cases hasdecreased substan- tially from 12,594 cases to 3429 cases between 2006 and 2012, and then resurged to 5399 in 2016 (Fig. 1). The case-fatality ratio andthe rateof permanent neurologic orpsychiatric sequelae ofpatientswithencephalitiscanbe30%(Araietal.,2008;Centers forDiseaseControl,2017; CentersforDiseaseControlandPreven- tion, 2017; Libraty etal., 2002; Mackenzieet al., 2004; Solomon and Winter, 2004; World Health Organization, 2017; Lam et al., 2005)andover35%inchildren(Kumaretal.,1990).JEVpersistsin atransmissioncycleofpigs,Ardeidbirdsandmosquitoes.Itcould infect humans through mosquito bites by Culex tritaeniorhynchus species(CentersforDiseaseControlandPrevention, 2017; Center forHealthProtection,2017;WorldHealthOrganization,2017).Hu- mans are dead-end hosts where they cannot develop viremia to infect mosquitoes. Population sizes of farm pigs and the size of riceland,whichfavorstheCulexmosquitoes’growth,arethetwo keyfactorsaffectinglocaltransmission(CentersforDiseaseControl andPrevention,2017;Impoinviletal.,2018;WorldHealthOrgani- zation,2017).

∗ Corresponding author.

E-mail address: [email protected] (D. He).

VerticaltransmissionofJEVinmosquitoesandpig-to-pigtrans- missionarethe majordeterminantsofthe followingyear’strans- mission. Vertical transmission exists between mosquitoes and their e.g. Rosenet al.(1989, 1978);Takashima andRosen(1989). Mosquitopopulationincreasesduringspring,peaksinsummerand decreasesduringfallannually(Riley etal., 2007,2012). According toa recentstudyby Ricklinetal., pig-to-pig transmissionis also present(Ricklinetal.,2016a).

JEVtransmissionviabloodtransfusionhasrecentlybeenfound in Hong Kong, which was probably the ﬁrst case worldwide.

The transmission was reported to come from an asymptomatic viremic donortotwo hospitalized patients(SouthChina Morning Post,2017).

Sero-prevalence of JEV antibodies varied by season among swinesandamonghumanpopulation groupsinHong Kong.Dur- ingrainyseasonbetweenMayandJuly,thesero-prevalenceamong swines reached 91% compared with 34% that is reported in dry season (Scientiﬁc Committee on Vector-borne Diseases, 2017). It was approximately 80%–90% among swines in July and August from2000 to 2004 (Riley et al., 2007). Another local serological surveyfoundthat23.5%ofpigfarmersand5.9%ofabattoirwork- ersare seropositive to JEV antibodies, incontrast to 0% reported among30blooddonors(ScientiﬁcCommitteeonVector-borneDis- eases,2017).

JEVvaccineprotectionrateshasbeeninvestigated.Thevaccine wasreportedtohave ahigheffectiveprotection rateof93.3%by https://doi.org/10.1016/j.jtbi.2018.05.017

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2006 2008 2010 2012 2014 2016 Bangladesh Cambodia Mainland China India Nepal Vietnam others

annual JEV confirmations

0 2000 4000 6000 8000 10000 12000

Fig. 1. Annual JEV conﬁrmations from 2006 to 2016 among the top six JEV endemic countries and the total of other countries. Data are obtained from World Health Organization (2018) .

ﬁve years and a predicted protection rate of 85.5% by 10 years (Desai etal., 2012). Arecent studyby Cao etal.investigated the current JEV vaccine derived from G3 JEV genotype against the emergingG5genotypeinmice,andfoundthatthelethalchallenge protectionratewas50%. The samestudyalso reportedthat neu- tralizingantibodiesagainstG5 JEVwere detectedin35%ofvacci- natedhealthychildren(Caoetal.,2016).

Riley etal. examined the skip-and-resurgence patterns of JEV from1969 to 2004 inHong Kong (Riley etal., 2007). They sug- gestedthat theskip from 1990 to 2002,except for one case re- portedin 1996,was likelydue to the lack ofrice production,as Culexspeciesbreedprincipallyinriceﬁelds(Tianetal.,2015).They proposedthattheresurgencefrom2003to2004waslikelydueto theheighteningofinfectiousdisease notiﬁcationsystemafterthe SevereAcuteRespiratorySyndrome(i.e.,SARS)outbreakin2003in HongKong(Rileyetal.,2007).

In Hong Kong, no locally-acquired JEV casewas reported be- tween2006 and2010, but17 caseswere reportedbetween 2011 andJune,2017.Consideringthedeclininglocallivepigpopulation from350,000to60,000between2004and2017(HongKong,2017;

LegislativeCouncil ofHong Kong,2017a;2017b; The Government ofHongKong,Agriculture,FisheriesandConservationDepartment, 2017a;2017c),thedisappearanceoflocalJEVcasesbetween2006 and2010areexpected,buttheresurgenceoflocalcasesisintrigu- ing.

Our work aims to identifythe mechanism underlyingthe JEV skip-and-resurgence patterns between December 2003 and May 2017inHongKong.Wehypothesizedsuch behaviorcould bedue to the surrendering of pig licenses during a pig rearing policy change in 2006 and/or a new JEV strain invasion around 2011.

These hypotheses are tested using mathematical modelling and likelihood-basedinferencetechniques.

2. Dataandmethods 2.1.Data

ThemonthlyJEVcasesbetweenDecember2003andMay2017 were retrieved online from the Centre for Health Protection in

Hong Kong(Scientiﬁc Committee onVector-borne Diseases,2017;

TheGovernmentofHongKong,CenterforHealthProtection,2017).

TheregionalmonthlymosquitoovitrapindexfromDecember2003 toDecember2016 wereretrievedonlinefromtheFoodandEnvi- ronmentalHygieneDepartmentinHongKong(TheGovernmentof Hong Kong, FoodandEnvironmental Hygiene Department, 2017).

The annual pattern of reported JEV cases and regional mosquito ovitrap indexare shownin Fig. 2.We presentthe time seriesof JEVcasesandregionalmosquitoovitrapindexinFig.3.

The population sizes of live pigsin Hong Kong (Fig. 4) were obtained fromlocal government reports, localnews articles, and reports from Department of Agriculture in the United States (HongKong,2017;LegislativeCouncilofHongKong,2017a;2017b;

Ta Kung Pao, 2017; The Government of Hong Kong, Agricul- ture,FisheriesandConservationDepartment,2017a;2017c;FEHD, 2017a; 2017b; United States Department of Agriculture, 2017).

Sincethepigrearinglicensesurrenderingpolicywasimplemented in May2006, the numberof locallive pigshas rapidlydeclined.

243 out of 265pig farms ownershad surrendered their licenses (LegislativeCouncilofHongKong,2017b).

2.2. JEVcompartmentalmodel

Ricklinet al.recently reportedthat pig-to-pig transmission of JEV can also occur without the mosquito vectors (Ricklin et al., 2016a). After the infectious period in pigs where JEV in swine serumareinfectioustomosquitoes,thereisalsoaconvalescentpe- riodinwhichpigshedsJEVvirusintheiroronasalsecretions.Thus, thepigpopulationcouldbeclassiﬁedintoﬁvecompartments:susceptible,exposed,infectious,convalescentandrecoveredwhichare denoted asS_p, E_p,I_p,C_p andR_p respectively. We considered pig- to-pig transmission and vector-borne transmission. Fig. 5 shows themodeldiagramconsideringpigs, mosquitoesandhumans.JEV transmission can be described by the following system of equa- tions(Eq.(1)).

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩

S_p=

(

¹−

η )

·Bp

(

^t

)

·Np−

ν

^p^S^p−

λ

v^p+

β

^p·^C_N^p_p

Sp, E_p=

λ

vp+

β

p·^C_N^p_p

Sp−

( σ

p+

ν

p

)

^Ep,

I_p=

η

^·^B^p

(

^t

)

·Np+

σ

^p^E^p⁻

( γ

^p⁺

ν

^p

)

^Ip, C_p=

γ

^p^I^p⁻

( δ

^p⁺

ν

^p

)

^Cp,

R_p=

δ

^p^C^p⁻

ν

^p^R^p^.

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Table1summarizesthemodelparametersinEq.(1).Theeffectsof B_p(t)and

ν

parepresentedinthedynamicsoflocallivepigpopula- tionbecauseN_p=[Bp(^t)−

ν

^p^]^N^p^.^In^this^model,^the^total^pig^pop-

ulationis:

Np=Sp+Ep+Ip+Cp+Rp,

where N_p is the observed live pig populations in Hong Kong and is a time-dependent parameter (see purple dashed line in Fig. 4). Bp(t) is the time-dependent birth rate of local live pigs.

Humansare dead-end hostsandcannot further transmitthe disease(CentersforDiseaseControlandPrevention,2017; Impoinvil etal., 2018;WorldHealthOrganization,2017),thuswemodelhu- mancases using a variable spill-over ratio(

ρ

⁾ ⁱⁿ^week ⁱ

ρ

i (see Eq.(2)):

Zi=

weeki

ρ

ⁱ

γ

^p^I^p^dt. ⁽²⁾

PleaseseeS.2.2formorereasoningonmodelstructure.

2.3. Modelframework

JEV cases were modelled asa Partially ObservedMarkov Pro- cess(POMP),alsoknownasHidden Markovmodel,using

R

pack-

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0 1 2 3 4 5 6

no. of JEV cases

annual local cases annual imported casese

1980 1985 1990 1995 2000 2005 2010 2015

(c)

SARS

resurge skip resurge skip resurge

0 100 200 300 400

no. offarm pigs (x 1000)

(b)

stable no. of pig. of pigffarms reduced no. of pig. of pigffarms

0 0.1 0.2 0.3

self−supportingratio ofvegetables 0

15 30

size of localrice fields (a)

self−supporting ratio of vegetables size of local rice fields

rice production ceasing of rice production

Fig. 2. Skip-and-resurgence of JEV epidemic from 1980 to 2017 in Hong Kong. Panel (a) shows the area of local rice production and self-supporting vegetable ratio. Panel (b) the orange line shows the number of local live pigs. Panel (c) shows the reported annual (i.e. both local and imported) JEV cases in Hong Kong. The arrow indicates the timing of SARS outbreak. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

Table 1

Model input parameters. We denote JEV transmitted from vectors to pigs as “v → p”, and from pigs to humans as “p → h” respectively. Please see S.2.1 for more information of the initial proportions.

Parameters Notations Values Ranges Remarks/Units Sources

Force of infection λvp - time-dependent v → p, per year Eq. (3)

Latent period in pigs σp⁻¹ 1.5 1–2 days Kodama et al. (1968) ;

Ricklin et al. (2016a)

Infection period in pigs γp⁻¹ 3 2–4 days Centers for Disease Control (2017) ;

Kodama et al. (1968) ; Ricklin et al.

(2016a,b) ; Williams et al. (2001)

Convalescent period in pigs δ⁻¹p 2.5 1–4 days Ricklin et al. (2016a)

Proportion of infected among imported pigs η ^1.0% 0.43%–1.45% pigs, Nil Eq. (8)

Effective contact rate βp estimate 0.0–0.4 pigs, per days Eq. (10)

Lifespan of pigs ν⁻¹p 234.0 234.0 days Eq. (7)

Population size of pigs N p - time-dependent see Fig. 4 Hong Kong (2017) ; Legislative Council of Hong Kong (2017a,b) ; Scientiﬁc Committee on Vector-borne Diseases (2017) ; The Government of Hong Kong, Agriculture, Fisheries and Conservation Department (2017a,c)

Spill-over ratio ρ - time-dependent p → h, Nil Eq. (5)

Initial proportion of susceptible S p0 estimate 45–75% Nil Center for Health Protection (2017) ; Konno (1969) ; Riley et al. (2007)

Initial proportion of exposed E p0 0.1% – Nil assumed

Initial proportion of infectious I p0 0.1% – Nil assumed

Initial proportion of convalescent C p0 0.1% – Nil assumed

Initial proportion of recovered R p0 estimate 25–55% Nil Center for Health Protection (2017) ; Konno (1969) ; Riley et al. (2007)

age “

POMP

” (King, 2017). The iteratedﬁltering andplug-and-play likelihood-based inference frameworks were employed to ﬁt the time series (Gao et al., 2016; He et al., 2010; King et al., 2016).

Furthermore,theMaximumLikelihoodEstimate(MLE)wasusedto estimate the modelparameters.To quantifythetradeoff between thegoodness-of-ﬁtofamodelanditscomplexity(Schwarz,1978),

Bayesian Information Criterion (BIC) was used for model com- parison.Simulations were performedby implementing the Euler- multinomialintegrationmethodwithaﬁxedtime-stepofoneday (Allenetal.,2008;Heetal.,2010).

The modelwas ﬁrstvalidated withthe observed JEV cases in Hong Kong, based on information about the size of pig popula-

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Fig. 3. Panel (a) shows the monthly reported local JEV cases in Hong Kong, and panel (b) shows the average monthly ovitrap index of regions (i.e., including Yuen Kong, Yuen Long and Tin Shui Wai) around Yuen Long district in Hong Kong.

For both panels, darker lines represent annual averages, lighter lines represent smoothed data. Dots represent the annual reported data from 2004 to 2016.

Fig. 4. Local live pig populations and their daily consumptions from January 2004 to May 2017 in Hong Kong. Purple line represents the annual live pig populations, connected by their reported and estimated numbers, which are depicted in ﬁlled and hollow circles respectively ( N p). Violet red line and dots represent the daily local live pig consumptions ( νpN p). Vertical grey dashed line denotes the time when the pig rearing license surrendering policy was implemented in Hong Kong. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

tion.Mosquito abundance is a time-dependentparameter, which was smoothed over time based on the ovitrap index (

ω

^). ^The

forceofinfection(

λ

^vp⁾^from^vectors^to^reservoirs ^is^another^time-

dependentparameter.Thespill-overratio(

ρ

⁾^is^estimated^through

ω

^.

Themonthlyobservedcases,C_i,wereassumedtofollowaPois- son distribution (Poi), with a mean Z_i, the underlying monthly casesmodelledbyEq.(2).Hence,wehave:

Ci∼Poi

( λ

=Zi

)

^with ^mean^:

μ

i=Zi.

Thus,theoveralllog-likelihoodfunction,l,wasgivenby:

l

( |

^C¹^,^.^.^.^,^Cⁿ

)

=

n

i=1

lnf

(

^Ci

|

^C1:(i−1),

)

where ^denotes ^the ^parameter ^vector ^being ^estimated, f(^Ci

|

^C1:(i−1),) ^was ^the ^posterior probability measurement function forC_i givenC_1:₍_i₋₁₎,which were then numerically computed by Sequential Monte Carlo (SMC, also known as particle ﬁltering) (He et al., 2010), and n denotes the total number of monthsduringthestudyperiod.

Fig. 5. JEV model diagram. Infectious classes are denoted in red, and JEV human cases are in grey (i.e., Z h, or Z iin Eq. (2) ). The transition paths are represented in black arrows. Red dashed arrows represent paths of transmission. Births and deaths (including slaughtering) of pigs are represented in light blue arrows. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

Using the proﬁle likelihood method, the conﬁdence intervals (C.I.)ofthemodeloutputparameterswereestimatedbasedonthe modelinputparameterrangesasdescribedinTable1.Parameters estimationandstatisticalanalyses areconductedusing

R

(version 3.4.1)(R,2018).

2.4. Parameterestimation

2.4.1. Forceofinfectionfromvectorstoreservoirs(

λ

^vp⁾

Wecanexpress

λ

^vp^as

λ

vp=a

ϑ

vp·_N^I^v_p,whereaisthemosquito bitingrate;ϑvpisthetransmissionprobabilityofJEVpermosquito bite;andIvisthenumberofinfectedmosquitoes.However,inthis study, we simplify

λ

vp as a function of ovitrap indexover time, since we are employing a vector-free modelling framework. This isjustiﬁablebyassumingthat aandϑvpareconstant,while _N^I^v

p is roughlyproportionaltotheovitrapindex.Brieﬂy,weassume:

λ

v^p=k·

ω (

^t

)

+b (3)

where

ω

îs^the^time^seriesôfôvitrapîndexⁱⁿ^Hong^Kong,^kând^b

aremodelparametersunderestimation.Constantbrepresentsthe contributionoftheverticaltransmissionfromadultmosquitoesto theireggs.TheverticaltransmissionratioofCulextritaeniorhynchus isvaried from12%to nearly 100%(Rosenetal., 1989; Takashima andRosen, 1989). By usingEq. (3),we also incorporate the case thatdespitetheovitrapindexisclosetozeroduringadryseason, thetransmissionratecanstillbepositiveduetoverticaltransmis- sionofvectors.

ToinvestigatethemechanismoftheobservedresurgenceofJEV after2011inHongKong,wepartitionedtheforceofinfectioninto timesegmentsbasedonthehypothesisthatthere-emergencewas due to the invasion ofa new JEV strain. This hypothesis is then validatedusingstatisticalapproaches.We assumethe forceofinfection(

λ

^vp⁾^takes^the^following^form:

λ

vp=

k1·

ω (

^t

)

+b, t<T0

k2·

ω (

^t

)

+b, tT0

(4)

whereT₀isthetimewhenthenewJEVstraininvaded.

Biologically,theforceofinfection(

λ

vp)underthenewstrainin- vasion scenarioishigherthan theno-invasionscenario,since the pigpopulationisimmunologicallynaiveintheﬁrstfewyearsafter invasion.Thus,underthenew-straininvasionhypothesis,wehave k₂>k₁>0.Theaveragespill-overratio

λ

^vp

^after^invasion^should

bemuchhigherthanthatbeforeinvasion.Withoutnewstrainin- vasion,wehavethespecialcasewherek₁=k₂inEq.(4).

2.4.2. Spill-overratiofromreservoirstohumans(

ρ

⁾

InEq.(2),weassumethathumansaredead-endhosts(Centers for Disease Control and Prevention, 2017; Impoinvil et al., 2018;

World HealthOrganization, 2017). Thus, the reported JEV human

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cases are proportional to pig infections according to the time- dependent spill-over ratio(

ρ

^). ^Since ^the ^number ^of ^human ^cases

arerelatedtothetotalnumberofvectors, wecanfurtherassume thespill-overratioasafunctionofovitrapindex:

ρ

⁼

ξ

^·

ω (

^t⁻

τ )

⁽⁵⁾

where

ξ

^is^the^strength^ofinfectivityparameterunderestimation and

τ

îs ^the ^sum ôf încubation ^period ôf ^mosquitoes ^(i.e. ^6–12

days) (Konno, 1969; Takashima and Rosen, 1989; Van den Hurk etal.,2009), latentperiodinhumans (i.e.5–13days) (Centersfor DiseaseControl,2017;CentersforDiseaseControlandPrevention, 2017;CenterforHealthProtection,2017)andcase-reportingdelay.

Forsimplicity,weﬁx

τ

^to^be¹⁵^days^for^this^study.

ToinvestigatethemechanismoftheobservedresurgenceofJEV after 2011 inHong Kong,we partitioned the spill-over ratiointo timesegmentsbasedonthehypothesisthatthere-emergencewas duetotheinvasionofanewJEVstrain.Thishypothesisisthenval- idatedusingstatisticalapproaches.Weassume thespill-overratio (

ρ

⁾^takes^the^following^form^from^Tien^et^al.⁽²⁰¹⁰⁾^:

ρ

⁼

ξ

1·

ω (

^t−

τ )

, t<T0

ξ

²^·

ω (

^t−

τ )

, tT0

(6)

whereT₀isthetimewhenthenewJEVstraininvaded.

Biologically,thespill-overratio(

ρ

⁾ ^under^the^new^strain^inva-

sionscenario ismuchhigherthan theno-invasionscenario,since thepig populationisimmunologicallynaive intheﬁrstfewyears afterinvasion.Thus,underthenew-straininvasionhypothesis,we have

ξ

2>

ξ

1>0. The average spill-over ratio

ρ

^after ^invasion

should be much higher than that before invasion. Without new straininvasion,wehavethespecialcasewhere

ξ

1=

ξ

2inEq.(6).

2.4.3. Lifespanofpigs(

ν

⁻p¹)

Hong Kongpeople consumedapproximately265livedomestic pigs per day during2016–17 (FEHD, 2017a;2017b), and roughly 275livedomesticpigsperdayinaround2012(TaKungPao,2017), whereastheconsumptionwasaround1450livedomesticpigsback in 2004 (Legislative Council of Hong Kong, 2017b). The total pig populationhasfallenfrom350,000in2004–05(LegislativeCouncil ofHongKong,2017a;2017b;TheGovernmentofHongKong,Agri- culture,FisheriesandConservationDepartment, 2017c), to65,000 in2012(TheGovernmentofHongKong,Agriculture,Fisheriesand ConservationDepartment,2017a)andfurtherdroppedto60,000in 2016–17(HongKong,2017).Thustheaveragelifespanofpigs

ν

⁻p¹

canbeestimatedas:

During2004-05:

ν

⁻p¹

= Np

dailyconsumptions

= 350000

1450 ≈241.38 days, In2012:

ν

⁻¹p

⁼ _dailyconsumptions^N^p

= 65000

275 ≈236.36 days, During2016-17:

ν

⁻p¹

⁼ _dailyconsumptions^N^p

= 60000

265 ≈226.42 days. (7) Byaveragingtheabove,wecomputedtheaveragelifespanofpigs

ν

⁻p¹tobeapproximately234days.

2.4.4. Pigs’population(Np)

Giventhattheaveragelocallivingpigconsumptionis650live pigs per day in 2007 (Legislative Council of Hong Kong, 2017b), we approximatethe totalnumberof pigs(Np) to be234×650= 152,100. The sudden drop in the number of live pigs between

2006 and 2007 was due to the surrendering of pig rearing licenses in early 2006, which result in 243 out of 265 pig farm owners turning over their licenses (Legislative Council of Hong Kong,2017b). Although the daily livepig consumption is not in- cludedasamodellingparameter,giventhepig’saverage lifespan, we could inferthe totalnumberof livepigsfromthe amountof dailyconsumption.

2.4.5. Infectionratioamongimportedpigs(

η

⁾

η

^can^be^computed^as:

η

=

^IARp

·

γ

p⁻¹+

δ

⁻¹p

ν

⁻p¹

^, ⁽⁸⁾

where

^IAR^p

îs^the âverage âttack ^rateôver^the âverage^lifespan

ofpigs

ν

⁻p¹

^. Âccording ^to ^the ^value ôf ^parameter ⁱⁿ ^Table ¹^,îf

γ

p⁻¹is1.5day,

δ

⁻¹p is2.5days,

ν

p⁻¹

^is²³⁴^days^and

^IAR^p

∈[25%, 85%](CenterforHealthProtection,2017;Khanetal.,2014),wees- timatedthat

η

∈[0.43%,1.45%].

3. Results

3.1. Modelﬁttingresults

Inadditiontothesimulatedmedian, wealsopresentthesim- ulatedannualmeans ofthemodel predictionusingthe approach describedinCamachoetal.(2014),sincesimulatedmeansdemon- strated ﬁttingresults more consistently when thedata are being restrictedasintegersandaresubjecttostochasticnoise.

ThemodelﬁttingresultsunderthenewJEVstraininvasionsce- nario are shown in Fig. 6. The estimated model parameters are summarizedinTable2.Althoughthelong-termﬁttingsuffersfrom severestochasticnoise(i.e.zero,oneortwocasespermonth),the 95% simulatedquantile intervalcovers all observeddata, andthe simulatedaverageannualpatternis consistentwiththe observed pattern.

BICreducesmorethan28unitswhenwewentfromthebase- line(i.e.noinvasion,seeS.1.1)scenariotothenewstraininvasion scenario(see Fig. 6and explanationI3 in Table3). We modelled anothernew strain invasionscenario whereonly theforce ofinfection

λ

^vp ^ispartitioned(seeexplanationI2inTable3andS.1.3).

TheBICincreasesapproximately1.0unitwhichimpliesanalmost equivalentﬁttingperformance to themainresults(see Fig.6and explanationI3inTable 3).Thepartitioned forceofinfectionwith nopartition onspill-over ratioispresented inS.1.3(see explana- tionI2inTable3).Anotherinvasionscenariowithtime-dependent

λ

^vp^and

ρ

^areinvestigatedinS.1.2(seeexplanationI1inTable3), andBICincreases4.55unit. Thedetailedmodelperformance and explanationofareinTable3andSupplementaryInformation.

3.2.Basicreproductionnumberofpig-to-pigtransmission

Usingthe next generationmatrixmethod (Brauer etal., 2016;

Van denDriesscheand Watmough,2002), the basicreproduction number,Rpp,ofpig-to-pigtransmissioniscomputedas:

Rpp=

β

^p

γ

^p^·

( ην

^p⁺

σ

^p

)

[

(

¹−

η )( γ

^p⁺

δ

^p⁺

ν

^p

) ν

^p⁺

γ

^p

δ

^p^]

( ν

^p⁺

σ

^p

)

^. ⁽⁹⁾

We consider that imported infections are rare (

η

≈0⁺), and bothincubationperiodandinfectionperiodofJEVinpigsareneg- ligible(i.e.,

σ

p⁻¹≈0⁺ and

γ

p⁻¹≈0⁺). Then, itcould be seenthat Rpp≈_δ_p^β₊^p_ν_p, whichis consistent withthe standardSIR compart- mentalmodel(Allenetal., 2008).WeestimatedRpp tobe0.0013 (95%C.I.:[0.00,0.31])undertheinvasionscenario(seeFig.7).Fur- thermore,therangeofeffectivecontactrate,

β

^p^∈^[0^.⁰^,⁰^.^4]^, ⁽¹⁰⁾

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Fig. 6. Model fitting results of JEV local cases in Hong Kong from 2004 to 2016 under new JEV strain invasion scenario with variable ρ. Panel (a) and (b) are the scaled force of infection (from vectors to pigs, scaled by the population size of pigs) and simulation results from 2004 to 2016 respectively. Panel (c) and (d) are the one-year- average scaled force of infection and simulation results from 2004 to 2016 respectively. In panel (a) and (b), black dashed lines are the scaled force of infection. In panel (b) and (d), blue lines are the simulation results, shaded regions are 95% quantile interval from simulation, pink dots are the reported (i.e., observed) JEV local cases and red lines are the smoothed (by loess function) reported JEV cases. The vertical grey dashed line marks the time point when Hong Kong government triggered the pig rearing licences surrender policy. The vertical dark green dashed line marks the time point when the new JEV strain introduced to the pigs’ population. The inset panel shows the maximum log-likelihood (MLL) values of different k 1s and k 2s , the red dot with the highest MLL are selected for fitting in main panels. The model scenario is associated with explanation I3 in Table 3 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 2

Summary table of model parameters’ estimates under new JEV strain invasion scenario with both variable λvp

and ρ. X p0denotes the initial proportion of class X p.

Parameter Notation Value Type Initial status Unit/Remarks

Average force of infection 2004–10: λvp 0.0043 estimated time-dependent before invasion Average force of infection 2011–16: λvp ^0.0071 ^estimated time-dependent after invasion

Pig latent period σp⁻¹ 1.5 ﬁxed 1–2 days

Pig infection period γp⁻¹ 3 ﬁxed 2–4 days

Pig convalescent period δ⁻¹p 2.5 ﬁxed 1–4 days

Imported infection ratio η 1.0% ﬁxed 0.43%-1.45% Nil

Effective contact rate βp 0.0058 estimated 0.0-0.4 per days

Pig living period ν⁻¹p 234 ﬁxed 234 days

Pig population N p – – time-dependent pigs

Average spill over ratio 2004–10: ρ 0.0 0 02 estimated time-dependent before invasion Average spill over ratio 2011-16: ρ 0.0013 estimated time-dependent after invasion Average ovitrap index ω ^0.0564 ^given time-dependent Nil

Initial susceptible S p0 0.6470 estimated 45–75% Nil

Initial exposed E p0 0.001 assumed 0.0-0.25% Nil

Initial infectious I p0 0.001 assumed 0.0-0.25% Nil

Initial convalescent C p0 0.001 assumed 0.0-0.25% Nil

Initial recovered R p0 0.3400 estimated 25–55% Nil

BIC BIC 144.8174 estimated – Nil

(seeTables1and2)issetcorrespondingtoRpp∈[0.0,1.0]. TherangeofRppcouldbeinferredasfollows:JEVsero-positive ratesamongpigs quicklydecreases inwinter (Riley etal., 2007).

Mosquitoabundanceisalmost zeroduringthesametime period.

Thus,thisindicatesthevector-freetransmissionofJEVamongpigs cannotpersist.Therefore,Rpp isless1.0 andpositive(i.e.,greater than0).

3.3.Criticalcommunitysize

The critical communitysize (CCS)is deﬁnedas“the minimum sizeofaclosedpopulationwithinwhichahost-to-host,non-zoonotic pathogencanpersistindeﬁnitely” (Bartlett,1960).Nåsell(2005)pre- senteda methodofapproximatingtheCCSfromsimplecompart-

mentalmodelsofdirectly-transmitteddiseases.Thisisrelevantto our studyconcerning pig-to-pig transmission.CCScan be formu- lated(obtainedfromEq.(12.1)inNåsell,2005)as:

CCS≈ 2

π

ln2· R0·

α

¹p^.⁵

(

R0−1

)

¹^.⁵^, ⁽¹¹⁾

where

α

^p=^γ^p_ν⁺_p^ν^p denotes the ratio of average lifespan of pigs (

ν

⁻p¹) to the average duration of infection. The basic reproduction number of pig-to-pig transmission, Rpp in Eq. (9) is rela- tively smallcompared to that of vector-borne transmission, Rv^p. Amodellingstudyby Khan etal.estimatedRvp to be1.2among pigs(Khan et al., 2014). Also, Rvp isbelieved to be greater than 1.0, as JEV doesspread during every rainy season. Applying the parameters under the new strain invasion scenario in Table 2

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Table 3

Summary of model ﬁtting results and associated trait(s) under different scenarios.

Label Scenario and its trait(s) BIC BIC ^a Remarks B baseline (no invasion) 168.7009 28.4376 see S.1.1 I1 invasion ( ρchanged since 2011) 140.2633 0.0 see S.1.2 I2 invasion ( λvpchanged since 2011) 141.2743 1.0110 see S.1.3

I3 invasion ( ρand λvpchanged since 2011) 144.8174 4.5541 see Fig. 6 and Table 2

a BIC = BIC −BIC min, where BIC minis the minimum of BICs of all scenarios. Here, BIC min= 140 . 2633

(see S.1.2).

Fig. 7. The results of estimation of the basic reproduction number of pig-to-pig transmission ( R pp) under new JEV strain invasion scenario with variable ρ. The horizontal blue dashed line is the 95% conﬁdence threshold. The model scenario is associated with explanation I3 in Table 3 . (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

(explanationI3inTable3)toEq.(9),Rpp ismerely0.0013,which ismuchsmallerthanRvp.Byusingthenextgenerationmatrixap- proach(Braueretal.,2016;Gaoetal.,2016),thebasicreproduction numberis:

R0= Rpp+

R²pp+4R²_vp

2 .

Ifwe ignoretheeffects ofRpp (i.e.,Rpp≈0⁺asinGao etal., 2016),we haveR0≈Rv^p.Ifweﬁx

ν

⁻¹p =234.0(see Eq.(7)),and set R0∈[1.10,1.40] and

γ

p⁻¹∈[2.0,4.0] (thus,

α

^p∈[59.5, 118.0]), therelationshipamong

α

^p^,R0andCCS(inEq.(11))areillustrated inFig.8.

The number of local live pigs was reduced from 80,000 to 60,000since 2008afterthe pig licensesurrendering policy came intoeffect(Fig.4) (LegislativeCouncilofHongKong,2017b).With relativelylowR0,CCSisgreaterthan150,000(seetheareabelow theblue lineinFig.8) overabroadrangeof

α

^p^’s.^This^could ^ex-

plainthelocalJEVcasedisappearancefrom2006to2010inHong Kong.

Moreover, for the invasion scenario, we found that the force ofinfection,

λ

^vp^,^could încrease âfter^theintroductionofnewJEV strainwhileassumingafixedspill-overratio,

ρ

^.^The^ﬁtting^results

of the partitioned force of infection, which did not partition on the spill-overratio,are presented intheS.1.3 (see explanationI2 inTable 3).The goodness-of-ﬁtisalmost equivalentto theprevi- ous scenario.We modelledthe partitioned

λ

^vp ^and

ρ

^scenarioⁱⁿ

S.1.2 (see explanation I1 in Table 3). Although the ﬁttingresults are not asgood asin the invasion scenario,(i.e.the main results inFig.6andexplanationI3inTable3),itisstilla signiﬁcantim- provementfromthebaseline(noinvasion)scenario.Theestimated

force ofinfection,

λ

vp, alsoincreasesafter introducinga newJEV strain(seeS.1.2andexplanationI1inTable3).Withanincreased R0duetoanincreased

λ

^vp^,^the^CCS^level^could^become^lower^than

thelocallivepig’spopulationlevel(seeFig.8),whichexplainsthe resurgenceofJEVcases.FurtherdiscussiononR0 andCCScanbe foundinS.2.3

4. Discussion

In thisstudy, we argue that the resurgence of JEV after 2011 waslikelyduetonewstraininvasionthathasahighertransmissi- bility.someindirectoverseasevidencedoesexist:(i)JEVgenotype 1(G1)strainsince2000:InSoutheast Asia,studies reportedthat Genotype 3 waspredominant during the late 20th century, and then genotype1 started to replacegenotype 3 around 2000 and become dominant thereafter (Gao et al., 2015; Mackenzie et al., 2004;Panetal.,2011).Onegeneticstudyfoundthatthegenotype 1strain was not observeduntil 2008 inthe regions ofMainland China surrounding Hong Kong (Gao etal., 2013). Thus, it is very likely that there is a newly introduced JEV strain frompigs im- portedfromtheseMainland Chinese regions(The Governmentof Hong Kong, Agriculture, Fisheries and Conservation Department, 2017b); (ii) JEV genotype 5 (G5) strain: Similar JEV resurgences were observed in South Korea in 1998 and 2010 (Sunwoo etal., 2016). The resurgencein 1998waslikely duetothe introduction ofG1strain inthemid-1990’s(Gao etal., 2013;Mackenzieetal., 2004). The ﬁrst isolated local G5 strain was reported in2010 in SouthKorea, whichcoincidedwiththe resurgenceofJEV in2010 (Takhampunya etal., 2011),where theaverage numberofannual JEVcasesincreasedapproximatelysix-toeight-fold(Sunwooetal., 2016).ThisisalsoconsistentwithourresultsofexplanationI1and I3inTable3.

We achievedalmost equivalentgoodness-of-ﬁt underthe two invasionscenarios. There are three potential explanations for the JEV resurgence since 2011: (I1) The newly introduced JEV strain hasslightlyincreasedthetransmissibilityfromvectorstopigs,and considerably increased the spill-over ratio from pigs to humans;

(I2) The newly introduced JEV strain has considerably increased the transmissibility,butthe spill-over ratio is held constant; (I3) The main results: the newly introduced JEV strain has increased both the transmissibility andthe spill-over ratio.Please also see thesummaryofI1-I3inTable3.

The symptomatic ratio ofJEV could be further used to reﬁne the mathematical model. As the majority of JEV infections are asymptomatic, and the mortality ratio for clinical JEV cases are approximately 30% (Table 4). We assume the symptomatic ratio ofJEV amongpigs are thesame as that ofhumans, and asymp- tomaticpigshavenegligibleJEVtransmissibilitytovectorsdueto lowwithin-hostviralloads.

Ourmain results are derived fromI3 (Fig. 6). In Table 2, we estimatedtheannualforceofinfection

λ

^vp

^to^be^0.0042^and^the

proportionof susceptible pigsSp0 to be 68%at the beginning of eachyear (Center forHealthProtection, 2017;Konno, 1969;Riley et al., 2007), the annual average JEV infection attack rate (IARp)

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50000

75000

150000

350000

60 70 80 90 100 110 120

1.10 1.15 1.20 1.25 1.30 1.35

350000 3 3

Basic Reproduction Number

αp=

(

^γ^p^{+ ν}^p

)

^ν^p

Fig. 8. Contour plot of the relationships between critical community size (CCS), αpand the basic reproduction number R 0. Numbers indicating the different CCS levels.

Table 4

Symptomatic JEV infection ratio and case-fatality ratio (CFR) of JEV with clinical illness from various sources. The numbers in brackets are the geometric average of the upper and lower ranges.

Symptomatics

TotalInfections (Symptomatic%) CFR = _Clinical^Mortality_Illness Source(s)

(0.48%) 0.4%–0.5% 30% Centers for Disease Control (2017) ; World Health Organization (2017)

< 1% 20%–30% Arai et al. (2008) ; Centers for Disease Control and Prevention (2017)

(0.81%) 0.33%–2% 25% Libraty et al. (2002)

(0.63%) 0.1%–4% 25%–30% Mackenzie et al. (2004) ; Solomon and Winter (2004) 4% (occasionally) – Van den Hurk et al. (2009)

(0.63%) 0.1%-4% – Chakraborty et al. (1980) ; Grossman et al. (1973, 1973) ; Halstead and Grosz (1962) ; Konishi and Suzuki (2002)

– 35% Monath et al. (20 0 0)

– 36.4% (children) Kumar et al. (1990)

amongpigscouldbecomputedas IARp=

λ

v^p

·S_p0

Symptomatic%

whereSymptomatic%is theJEV symptomaticrate, bywhich IAR_p isestimatedfrom 35.26%to 59.50%(with Symptomatic%∈[0.48%, 0.81%]), which is consistent with (Center for Health Protection, 2017;Khanetal.,2014;Rileyetal.,2007).Theresultscorrespond- ingtoI2arepresentedinS.1.3,wheretheannualtransmissionrate

λ

^vp

îs^set^to^be^0.0044ând^0.1763^beforeândâfterînvasion,^re-

spectively.Thelarger annualforceofinfection afterinvasionpro- ducesunreasonableIARp. Withparameter valuesinS.1.3,

λ

v^p

= 0.1763 and S_p0=57.67% lead to IARp∈[1255%, 2118%], which is largerthan 100%.ThemechanismI3 impliesincreasein both

λ

^vp

and

ρ

^after^invasion^(see ^S.1.2).^With^Sp0=64%,IAR_p isincreased from[33.98%,57.33%]to[56.10%,94.67%]afterinvasion,within ac- ceptableranges.Therefore,accordingto therangeofIARp,I1and I3are likelyto be the possibleexplanations ofthe resurgence of JEVepidemicsin2011,whileI2isunlikelysinceitpredictsunrea- sonablevaluesofIARp.

Furthermore,I3could bemorebiologically reasonablethanI1. InI1,weassumedthe forceofinfection(i.e.,

λ

^vp⁾^unchanged^but

onlychange the value ofspill-over ratebefore/after the newJEV invasion.However, withthe force of infectionin I1, theJEV epi- demicisunlike tomaintainaccordingtotheestimationresultsof CCS(i.e.,inthiscase,thepigpopulationislowerthantheCCS).On

the contrary,withincreased force ofinfectionin I3,the JEV epi- demicisverylikelytocomeback (i.e.,resurgent)since2011.Fur- ther workis neededin orderto identifythe biological evidences andmechanismsofI3.

Inthiswork,weonlyinvestigatedthescenariosofnewJEVin- vasion, more possible causes (or scenarios) due to various other factorscanbefurtherexplored.

5. Conclusions

Wedevelopeda simplemathematicalmodeltoinvestigatethe mechanisms behind the skip-and-resurgence patterns of JEV human cases in Hong Kong. The critical community size (CCS)estimated through the modelindicates that the pig rearing license surrenderpolicyonMay2006couldberesponsibleforthedisap- pearanceofJEVhuman during2006–10,assuming that otherfac- torsremainunchanged.Comparedwiththeresultsofbaseline(no invasion)scenario(seeS.1.1),ourﬁttingresultsinthehypothetical scenarioimplythattheresurgenceofJEVin2011waslikelydueto theintroductionofnewstrainswhichhasahighertransmissibility and/oraspill-overratio.

The basic reproduction number (Rpp) of pig-to-pig transmis- sionisestimatedtobe0.0013(95%C.I.:[0.00,0.30]).Althoughthe vector-freeJEVtransmissionrouteexists(Ricklinetal.,2016a)and itcouldincreasetheepidemicsizeandprolongstheoutbreak,JEV isunabletospreadamongpigswithoutvectors.