doi 10.1098/rspb.2000.1180

Amodellingapproachtovaccinationa nd contraceptionpr ogrammesfor rabiescontrol infox ChristelleSuppo 1,Jean-MarcNaulin 2*,MichelLanglais 2 and MarcArtois 3 1IRBI-UMR CNRS6 035,Universite ¨ deTours,37200Tours,France ([email protected] ) 2UMR CNRS5 466,` Mathe ¨ matiques Applique ¨ es deBordeaux’,BP26, Universite ¨ Victor Segalen Bordeaux 2, 33076 Bordeaux Cedex, France ([email protected],[email protected] ) 3AFSSAN ancy,Laboratoire d’Etudes sur la Rage et la Pathologie,des Animaux Sauvages,B.P.9,54220Malzeville, France ([email protected] ) Ina previousstudy ,three ofthe authorsdesigned a one-dimensionalmodel to simulate the propagation ofrabies within a growingfox ; the in£uence of various parameters onthe epidemic model wasstudied, includingoral-vaccination programmes. In this work,a two-dimensionalmodel of a fox populationhaving either anexponential or a logisticgrowth pattern wasconsidered. Using numerical simulations,the e¤ciencies oftwo prophylactic methods (foxcontraception and vaccination against rabies) wereassessed, used either separatelyor jointly.Itwasconcluded that farlower rates ofadministra- tionare necessary toeradicate rabies, and that the undesirableside-e¡ ects ofeach programme disappear, whenboth are used together. Keywords: discrete modelling;rabies;foxes; oral vaccination ;contraception

highfox- areas ( Artois et al. 1997). This 1.INTRODUCTION modelemphasized that avaccinationrate lowerthan 70% Foxrabies is amajorveterinary public-health problem in willallow the epidemic topersist, a¢gurealready severalcountries ofthe world(Blancou et al. 1991). Oral described bySmith (1995).Inthis study,the focuswas on vaccinationof foxes carried out by distribution of vaccine fertility controlthrough the use ofbaits¢ lledwith a contra- baitshas had a clearimpact onthe prevalenceof the ceptivevaccine in conjunction with a rabiesvaccine as a virusin W estern Europe(StÎ hr &Meslin 1997;Pastoret possiblemethod of controlling rabies when vaccination &Brochier1 999). aloneis notsu¤ cient fordisease eradication(Smith 1995). Datafrom fox-hunting records indicatethat the Europeanfox population tends toincrease (Artois 1997). 2.DESCRIPTION OFTHE MODEL Thishas been observed in both rabies-free (Great Britain, Tapper1 992;J .-A.Reynolds, personal communication) Thepresent modelwas based on the one-dimensional andrabies-infected areas(Belgium, De Combrugghe discrete deterministic modelof Artois et al. (1997). The 1994;Germany ,MÏller 1995).Thisdoes not mean that foxpopulation has been structured inspace (a two- foxpopulations are not regulated over the longterm, but dimensionalmodel in this paperwith N homeranges) ,in simplythat overthe short term the populationincome ^ age(young and adult, i.e. dispersing foxes or residents outcomeratio is notbalanced for unknown reasons oneyear old and more) ,insex (female andmale) andin (increase ofresources and/ordecrease inmortality) . disease state (healthy,exposedand vaccinated) .Thisgave Whateverthe causeof fox-population increase, it could 12classes offoxes per cell throughwhich rabies propa- eventuallyimpede the success offurther oral-vaccination gated(¢ gure 1 ).Thedensity of healthy young females in campaignswhen the number ofnon-immunized foxes cell n at time t hasbeen denoted by HYF( n,t),withanalo- becomes highenough to carry on the infection gousnotations for the 11otherfox classes. (Breitenmoser et al. 1995;Vuillaume et al. 1997).Asu¤- Thetime-step, ¢t 10days, chosen in the simulations ˆ cient levelof culling to achieve a sustainablecontrol of is longerthan the lifeexpectancy of clinically ill indivi- the populationis di¤cult toobtain if the rabiesthreshold duals( 1^4days) (Blancou et al. 1991).Thusno speci¢ c densityis much lowerthan that ofthe population class ofinfectious individuals has been considered. carryingcapacity (Anderson et al. 1981).Thecombination Instead,the number ofinfectious individuals in the time ofculling and vaccination is still amatter ofdebate intervalfrom t to t +¢t is proportionalto the number of (Smith 1995;Barlow 1996) .Apromisingsolution could be exposedindividuals; the proportionalitycoe¤ cient ¼(t) is the limitationof ofhealthy foxes through the inverse ofthe latencyperiod. fertility control.Increasing e¡ orts havebeen focused on this techniquefor red foxpredation control in Australia (a) Demography (Bradley1 994). Asa further contrast withArtois et al. (1997),this Ina previousmodel, vaccination by the oralroute papersimulates the demographyof the foxpopulation as alonewas examined as a wayof controlling rabies in either exponentiallyincreasing or density dependent. Ina density-dependentfox population the natural *Author forcorrespondence. mortalityis di¡erent foryoung foxes and adult ones,

Proc. R.Soc.Lond. B (2000) 267, 1575^1582 1575 © 2000The RoyalSociety Received 6 March 2000 Accepted 12April 2000 1576C. Suppoand others Rabies vaccination and sterilization

mn mn mn mn x a m

vaccinated v healthy d healthy v vaccinated n adult adult young young males males males males d c c mn mn infected d infected adult young males males mr mr reproduction

mn healthy healthy mn d m max v adult young v females females Figure2. Structure of thetwo-dimensiona ldomain.The c d c shadedarea represents the set of cellsthat a youngfox living in cell n canreach through dispersal. vaccinated infected d infected vaccinated adult adult young young n (i71) n + j, (3) females females males females ˆ £ max where nmax is the maximalnumber ofcells ona line mn mn mr mn mr mn (¢gure 2). Conversely,the location( i, j)onthe gridof a hexagon Figure1. Interaction between the 12 classesof foxes.mn, havingnumber n couldquickly be foundfrom naturalmortality; mr, mortality induced by rabies;v, vacci- n 1 nation;d, dispersal;c, contamination. i I ¡ 1, I r ˆ nmax ‡ ‰ Š is theinteger partof the realnumber r, (4) accordingto season. Survival is therefore alsodensity dependent:for adult female foxes it hasbeen determined j n (i 1) n . (5) as ˆ ¡ ¡ £ max 1 Duringthe dispersal process, youngfoxes leave their saf(t) saf^ (t) ; (1) ˆ £ 1 ¯(n,t)P(n,t) parentalhome range to become territorial. Inour model, ‡ weassumed that ayoungfox disperses oneway along a where saª f(t)is the naturalsurvival rate ofadult female straightpath and crosses atmost L homeranges before foxes, P(n,t)isthe totaldensity of foxes living at time t in settling down(Lloyd 1 980;Macdonald & Bacon1 982; home range n and ¯(n,t)is anon-negativeparameter . Trewhella et al. 1988).Thusit canreach 1 + 3L(L + 1) When ¯ 0there is nodensity dependence, while a posi- ˆ di¡erent cells (¢gure 2) .Inthis model D(n,L) was de¢ned tive ¯ willyield a logistice¡ ect. Similarformulae have asthe set ofcells that ayoungfox living in cell n can beenused forother age and sex classes. Insimulations, ¯ reachthrough dispersal andas the set ofcells fromwhich is aconstant,numerically evaluated to supply an average ayoungfox arriving in cell n started from.Finally ,the foxdensity of 0.01 ha 71 (Artois 1989).Onlyhealthy and radialdistance between two cells wasdetermined through vaccinatedfemales wereable to reproduce as incubation asimple algorithm. periodwas shorter thangestation and weaning duration; Theprobability of a youngfox settling ina givenhome henceinfected cubs hadno chance of survival. The rangewas assumed todepend only on the number ofcells densityof healthy young females incell n is it crossed, i.e.the radialdistance between the end-points ofits path.As a modelwe took a linearlydecreasing HYF(n,t +¢t) b(t) saf(t) HAF(n,t), (2) functionof the distancetravelled: the probabilityof ˆ £ £ reachinga cell locatedat a radialdistance d is wherethe birth function, b(t), satis¢es b(t) b on 1 April ˆ 0 » (L 1) d 2 and b(t) 0otherwise(Artois 1989),with2 b being the C(d) £ ‰ ‡ ¡ Š , with » , ˆ 0 ˆ 6 d ˆ (L 1)(L 2) (6) averagenumber ofcubs per litter per femaleand b £ ‡ ‡ 0 for d 1, . . . . referred toas the halfbirth rate. ˆ Asanexample, in a rabies-free situation,the densities of (b) Two-dimensional spatial structure healthyyoung and adult females aregiven by the anddispersion followingequations: Arectangulardomain is subdividedinto cells havinga hexagonalshape, each cell correspondingto the size ofan HYF(n,t +¢t) (17F(t)) syf(t) HYF(n,t), (7) averagefox’ shomerange. Cells have been numbered ˆ £ £ from 1 to N;the hexagonlying at the intercept ofline i and column j is numbered HAF(n,t + ¢t) F(t) hydf (n,t) + saf (t) HAF(n,t), (8) ˆ £ £

Proc. R.Soc.Lond. B (2000) Rabies vaccination and sterilization C.Suppoand others 1577 where F(t)is the proportionof young foxes that disperse, Table 1. Data used insimulations syf(t)is the survivalrate ofyoung females, saf( t) is the survivalrate ofadult females andhydf( n,t)isthe number studiedarea n 61; N n m 3721 max ˆ ˆ max £ max ˆ n dispersaldistance l 5,or 91reachablehome ranges ofhealthyyoung females that arrivein cell . ˆ At time t survivalrate (Artois et al.1997)adult young hydf(n,t) ’(k,n) syf (t) HYF(k,t), (9) summer 0.99 0.97 ˆ £ £ k D(n,1) winter 0.98 0.98 2X latencyperiod where ’(k,n)is the probabilityof a foxlocated in cell k (Blancou et al.1991)21 daysor ¼ 0:48 ˆ cominginto cell n ata radialdistance d from cell k: birth rate b(t) b on 1 April; b(t) 0 otherwise ˆ 0 ˆ transmissionrate ­ (t) ­ a(t) ­ ’(k,n) Á(d). (10) ˆ ˆ ˆ Asin the one-dimensionalmodel, there wasa problem foryoung foxes leaving a homerange close to the saf(t) IAF(n,t) + ­ (t)+ (syf(t) HAF(t)) I(n,t) £ j £ £ boundaryof the domain.Here weconsidered that young + F(t) iydf(n,t), (14) foxesthat wouldhave left the domainthrough dispersal £ remainedin their parentalterritory ,therefore nofox where iydf(n,t)is the number ofinfectious young crossed the boundary.Underthis assumption,numerical females that couldarrive in cell n at time t: simulationsshow that inarabies-free situation,the global iydf(n,t) ¼(t) ’(k,n) syf(t) IYF(k,t). (15) ˆ £ £ dynamicsis that ofalargeisolated population. N umerical k D(n,l) simulationsin a disease-free environmentshow no 2X densityincrease atthe edges.Since the aimof this paper wasto compare the e¤ciency of di¡ erent control (d) Vaccination andsterilization programmes strategies withinthe centre ofthe domain,the edgee¡ ect Twovaccination campaigns per yearwere simulated in atthe boundarycould be neglected. ourmodel ( Aubert1 995):one in spring to target adult animals,and one in autumn to target allage classes. The (c) Transmission number ofhealthy and vaccinated young females ina cell Transmission ofrabiesoccurs throughbites andlicking, n at time t +¢t hasbeen determined as asthe rabiesvirus is transmitted viathe saliva(Blancou et al. 1991).Itis thoughtthat rabiescan be propagatedby HYF(n,t +¢t) (17vay(t)) syf(n,t) HYF(n,t), (16) ˆ two modes: £ £ VYF(n,t +¢t) vay(t) syf(n,t) HYF(n,t), (17) ˆ (i) Outside the dispersal ofjuveniles, between foxes £ £ where vay(t)is the vaccinationrate attime t. livingin the same oradjacent home ranges ( Artois Infertility-control campaigns,each autumn only 1989).Fora givencell, n,these homeranges have females areconcerned and contraception is onlye¡ ective numbers duringone breeding season. T otakecontraception into v (n) n7 n , n7 n + 1, n71, n, n + 1, accountthe equationsof the previousmodel were modi- j ˆ max max ¢ed:the number ofhealthyyoung females, ormales, ina n + nmax, n+ nmax + 1, (11) cell n at time t +¢t wasdetermined as the densityof infected youngfemales beingdeter- HYF(n,t +¢t) HYM(n,t +¢t) ˆ mined as b(t) (17st(t)) HAF(n,t) saf(n,t), (18) ˆ £ £ £ IYF(n,t +¢t) syf(t) IYF(n,t) + ­ (t) where st(t)isthe sterilizationrate attime t. ˆ £ j (syf(t) HYF(n,t)) I(n,t); (12) £ £ £ 3.SIMULATION RESULTS where ­ j(t)is the transmission rate froman infectious foxto a healthyyoung fox at time t, and I(n,t) is the Numericalsimulations were performed ona worksta- number ofinfectious foxes living in cell n and in the tionusing a Fortran77 code on the datashown in table 1 . sixsurrounding cells determined as Variousnumerical values of the birth andtransmission rates havebeen used. More precisely valuesfor ­ and b0 I(n,t) ¼(t) IYF(v ,t) IAF(v ,t) IYM(v ,t) weredetermined that wereconsistent witheither an ˆ £ ‰ j ‡ j ‡ j Xvj(n) endemic state oradisease-free state. Thee¤ ciency of fox contraception,dependent on or independent of vaccina- IAM(vj,t) . (13) ‡ Š tionagainst rabies was then easier toanalyse. Asindicatedearlier ¼(t)isthe inverse ofthe latency In Artois et al. (1997)birth rate, b0,was2.5. Here b0 period. variedfrom 1 .02(see ½3(a))to a maximumof 3.5, which (ii) Duringdispersal (October andN ovember)( Artois correspondedto seven cubs per litter per female,with a 1989),infected youngindividuals carry the infection balancedsex ratio (Artois 1989). further thanone home range (see ¢gure2) .The In Garnerin et al. (1986)and Artois et al. (1997)trans- densityIAF( n,t +¢t)ofinfected adultfemales incell mission rate, ­ ,was0. 18.Here ­ variedfrom 0.04 to a n at time t+¢t is given by maximumof 0.20(see ½3(b)).

Proc. R.Soc.Lond. B (2000) 1578C. Suppoand others Rabies vaccination and sterilization

Table 2. Malthusian parameters for di¡erent birth rates 0.2 birth rate Malthusianparameter 0.15 e t

1.02 0.0001 a r

1.5 0.005 n o i

2.0 0.0095 s s 0.1 i

2.5 0.0133 m s

3.0 0.0166 n a r 3.5 0.0182 t 0.05

50 0 1.5 2 2.5 3 3.5 e g

n birth rate a r

40 e

m Figure4. Maximum transmission rate ­ max(b0,¯) for ¯ 0.003

o ˆ h

(opendiamonds), 0.002 (dashed line), 0.001 (asterisks) and 0 r 30 e (opensquares). p

s e x o

f 20

f 35 000 o

r e b 10 30 000 m u n

s 25 000 e 0 x o f 0.001 0.003 0.005 0.007 0.009

f 20 000 o

d r e

b 15 000

Figure3. Number of youngand adult foxes per cell as a m u functionof thedensity-depende ntparameter ¯ for b 3.5 n 10 000 0 ˆ (topcurve), 3, 2.5and 2 (bottomcurve). 5000

Actually, ­ isunknown,but b0 variedwithin a narrow 0 range(V oigt& Macdonald1 984). 6 12 18 24 30 36 time (years) (a) Populationequilibrium Figure5. Dynamics of rabiesfor b 2.5, ­ 0.08 and Simulationswere carried out using one healthy pair of 0 ˆ ˆ ¯ 0.003:healthy individuals (top curve) and infected foxesper cell andno youngas initialdistribution levels. ˆ First, fora constantsurvival rate ( ¯ 0),there wasa individuals(bottom curve). ˆ thresholdvalue bmin(0)close to 1.02;if b0 5 bmin(0)the popu- lationgoes extinct andif b0 4 bmin(0)the populationfollows offoxes on 1 April.Assuming this maximumto be 13and aMalthusiangrowth pattern. Fordi¡erent birth rates, the b 2.5,we ¢ rst put apairof healthy adults and no 0 ˆ correspondingMalthusian parameters havebeen deter- youngin each cell andran the programuntil a yearly mined(table 2) .Asin the one-dimensionalmodel (Suppo periodicdistribution of individuals was achieved. Using a 1996),fora birth rate b 2.9this parameter is0.016. dichotomymethod we estimated ¯ 0.0025,the average 0 ˆ ˆ Second,with a density-dependentsurvival rate ( ¯ 4 0), transient time being1 2years.W enextmodi¢ ed the initial after atransient perioda maximumyearly periodic distributionof healthyfoxes and re-ran the programwith distributionof individuals will be observed for ¯ 0.0025until a yearlyperiodic distribution of indivi- ˆ b0 4 bmin(¯),whilethe populationwill become extinct for dualswas achieved ;weagain found 1 3tobe the b0 5 bmin(¯).Eventually,the maximumnumber offoxes maximumnumber, with variable transient times. willbe achievedon 1Apriland can be determined witha suitablydesigned ¯.Typically,afoxgroup on 1Aprilwill (b) Rabies-endemic equilibrium becomposed of one male, two fertile females andtheir Inthis section, weintroduced a pairof exposed adult litters, i.e.an averageof 1 3individualsfor b 2.5 (Artois foxesin a singlecell locatedat the centre ofthe domainand 0 ˆ 1989).Inthe modelthis willoccur when ¯ 0.0025 and assumed that eachcell containedone pair of healthyfoxes. ˆ b 2.5. For ¯ 0.003the minimum thresholdbirth rate, Assuminga Malthusiangrowth trend ( ¯ 0), we ¢rst 0 ˆ ˆ ˆ bmin(0.003),neededto prevent the populationfrom going determined the set ofpairs ( ­ ,b0)neededto yield an extinct is closeto 1 .3.F orbirth rates varyingup to endemic state. Results showthat foreach ¢ xedbirth rate, b 3.5,the number offoxes per homerange was deter- b ,rabieswill not remain if the transmission rate islarger 0 ˆ 0 minedfor di¡ erent valuesof ¯ (¢gure 3) .Atotalof 1 3 thana maximumthreshold ­ max(b0,0)(¢ gure 4) . foxesper homerange can be obtained with di¡ erent Furthermore, the transmission rate must alsobe larger combinationsof ¯ and b0. thana more orless constantvalue to further anendemic From anumericalpoint of view ,weproceeded along state, ­ (b ,0) 0.04. For b 2.5 and ­ 0.08numerical min 0 ˆ 0 ˆ ˆ twolines inorder to get a prescribed maximumnumber simulationsshow similar results tothose obtainedin the

Proc. R.Soc.Lond. B (2000) Rabies vaccination and sterilization C.Suppo and others 1579

60

50 e t 40 a r

80 ) n o % i (

s

s 30 e i t 60 a m r

s n n o a i r

20 t t 40 a n i c c

10 a 20 v

0 0 2 2.2 2.4 2.6 2.8 3 3.2 3.4 0.04 birth rate 0.06 2.9 0.08 Figure6. Sterilization rate required to eradicaterabies for 2.5 0.10 birth rate ¯ 0.003,transmission rates ­ 0.06(top curve), 0.08, 0.10, ˆ ˆ transmission rate 2 0.12and 0.14 (bottom curve); variable birth rates. 0.11 one-dimensionalmodel (Suppo 1 996):between two waves, Figure7. Vaccination rate required to eradicaterabies for di¡erent pairs of ­ and b for ¯ 0. the populationresumes aMalthusiangrowth trend asina 0 ˆ disease-free situation;during the ¢rst ten yearsthe occur- rence offour successive waveswith high prevalence of and ­ max(b0,¯),forrabies to disappear an initialbirth rate infectionwas observed ;duringthe next30 years,the inthe range2^3.5 must bedecreased toa minimum value number ofsuccessive wavesof rabiesincreased, the growth bopt(¯,­ ) 4 bmin(¯).Consequently,ane¤ cient sterilization ofthe healthypopulation was regulated by rabies and a rate couldbe determined fordi¡ erent birth rates anda periodicendemic state emerged (¢gure 5) . ¢xedtransmission rate (¢gure 6): for ¯ 0.0025, b 2.5 ˆ 0 ˆ Fora logisticsituation, the same pairs( ­ ,b ) were and ­ 0.08a sterilization rate closeto 35% is required. 0 ˆ determined fordi¡ erent valuesof ¯ (see ¢gure4) .For b0 5 1.7rabiescould be sustained withhigher transmis- (d) E¤ciency ofvaccination against rabies sionrates when ¯ 4 0 than when ¯ 0;this threshold Inour computations, vaccination programmes would ˆ ­ max(b0,¯)increasedwith ¯ and b0.Again,the transmis- beginafter three years,corresponding to the controlof an sionrate hada minimum threshold ­ min(b0,¯) to further unexpectedoutbreak of rabies spreading quickly across anendemic state, butthe latter wasstrictly largerthan the spatialdomain. The vaccination rate wasconsidered 0.04,increased with ¯ anddecreased with b0. Finally, ase¤ cient whenrabies was totally eradicated ;numeri- there existed anoptimum transmission rate ­ opt(b0,¯), callythis means the totalnumber infected inthe whole ­ min(¯)5 ­ opt(b0,¯)5 ­ max(¯ ),atwhich the prevalence domainequal to zero for at least 20years. wasmaximal (see ½3(d)).Theseresults showthat fora First, fora Malthusiangrowth trend, aminimum giventransmission rate, ifthe birth rate decreased below e¤cient vaccinationrate wasdetermined forvarious pairs 1.7,anendemic state couldnot be obtained and rabies (­ ,b0)inorder to eradicate rabies (¢ gure 7) .This disappeared.Thus, depending on the size ofthe birth minimum rate waslarger in high-density populations and rate, asterilization methodcould decrease this rate and decreased when ­ increased.F or b 2.5 and ­ 0.06, 0 ˆ ˆ leadto eradication of rabies. In addition, for a birth rate simulationsgave the minimum e¤cient rate ofvaccination b 2.5and a transmission rate ­ 0.07rabies waves will necessary toeradicate rabies as 70%,which is closeto the 0 ˆ ˆ occurevery six or seven years. upperlimit achievedin the ¢eldduring actual vaccination If ¯ is closeto zero, results wouldbe similar tothe campaigns(Aubert 1995).Inother words, both numerical Malthusiangrowth model. simulationand ¢ eld¢ ndingsshow that there isadensityat whichvaccination fails to eradicate rabies. (c) E¤ciency offertility control Second,for a logisticgrowth trend, the same computa- Thee¤ ciency of sterilization programmescould be tionswere carried out to emphasize the di¡erence for deducedfrom the previouscomputations. W ecandraw lower-densitypopulations; see ¢gure8 for ¯ 0.003. ˆ di¡erent conclusionsfrom Malthusian and logistic growth Accordingto the results, inorder for a vaccinationto be trends. e¤cient the rate must behigher for high birth rates First, wewill consider a Malthusiangrowth trend (see (correspondingto a largerpopulation) ,withmaximum ¢gure4) .Fora¢xedtransmission rate lyingbetween 0.04 valuesat the optimum transmission rate ­ opt(b0,¯). and0. 12,in order to eradicate rabies an initial birth rate Itis worthnoting that inboth cases (logisticand inthe range2^3.5 must bedecreased toa valueclose to Malthusian)the dynamicsof rabies was modi¢ ed by low bmin(0).Thus,it wouldbe di¤ cult toemploy a steriliza- vaccinationrates. Fora vaccinationrate between20 and tionmethod alone because a healthypopulation would go 30%(¢ gure 9) ,the ¢rst waveof rabies was delayed, but extinct beforerabies was eradicated. afterwardthe number ofsuccessive wavesincreased and a Let us alsoconsider a logisticgrowth with ¯ 4 0, (see newone could occur every year with the same preva- ¢gure4) .Foratransmission rate lyingbetween ­ min(b0,¯) lence.F ora rate lowerthan, and close to, the e¤cient

Proc. R.Soc.Lond. B (2000) 1580C. Suppoand others Rabies vaccination andsterilization

60 (a) 50 ) % ( 40 70 e t a ) r

60 % n (

o 30 i e

50 t t a a r z

i

40 l n i 20 r o i e t 30 t a s n i

c 20 10 c a

v 10 0 0 0.08 3.5 60 (b) 0.10 3 50

0.12 2.5 birth rate ) %

transmission rate ( 40 0.14 e 2 t a r

n

Figure8. Vaccination rate required to eradicaterabies for o 30 i t di¡erent pairs of ­ and b for ¯ 0.003. a 0 z ˆ i l

i 20 r e t 3000 s 10 2500 0 0 10 20 30 40 50 60 70 80 s

e 2000 vaccination rate (%) x o f

f

o Figure10. The shadedarea contains combinations of 1500 r

e vaccinationand sterilization rates required, that could be b

m achievedin the¢ eld,to eradicaterabies for ( a) ¯ 0 and u 1000 ˆ n ­ (3,0) 0.04(dashed line) 4­ 4 0.11 ­ (3,0)(solid min ˆ ˆ max line) and (b) ¯ 0.003 and ­ (3.5,0.003) 0.08(dashed ˆ min ˆ 500 line)4 ­ 40.14 ­ (3.5,0.003)(solid line). ˆ max 0 Fora logisticgrowth trend, wesaw that asuccessful 0 2 4 6 8 10 sterilization rate couldbe determined but wouldbe time (years) di¤cult toobtain. A combinationof bothcontrol methods Figure9. Dynamics of rabiesfor a vaccinationrate close to is alsobene¢ cial (see ¢gure8) .For ¯ 0.003,¢ gure1 0 ˆ 25%:healthy individuals (dotted line) and infected indivi- showsthe successful combinationsof sterilization and duals(solid line). vaccinationfor ­ (3.5,0.003) 0.08 and ­ (3.5,0.003) min ˆ max 0.14.W eobserveda linearrelationship between the ˆ rate, the ¢rst waveappeared later butthe prevalenceof vaccinationand sterilization rates; linearregression followingwaves increased continuously . yieldsa slopeequal to 7 0.76 for ­ min(b0,0.003). Nowwe come to the keypoint of ouranalysis. F orsome pairs of b and ­ ,the vaccinationrate neededto eradicate 0 4.DISCUSSION rabieshad to be larger than 70%, which is di¤cult to achievein the ¢eld(Breitenmoser et al. 1995;V uillaume Over time, more than1 5models havebeen devoted to et al. 1997).Inthese cases contraceptionis required to foxrabies (reviews in Barlow ( 1995)and Pech & Hone improvethe e¤ciency of anti-rabiesvaccination. (1992)).Themain value of the oneherein presented lies in the use ofrecent andactual data from fox baiting in (e) Vaccination andfertility control combined France(Aubert 1995).Additionally,the use ofcontra- Fora Malthusiangrowth trend, wesaw that contra- ceptionto manage rabies in fox populations is considered ceptionalone could lead to extinction of foxes (see ½3(c)). (Artois &Bradley1995) .Aswith many models ofthe Acombinationof both methods couldbe e¤ cient ifbirth same type,ours is oversimpli¢ed in several regards and rates weredecreased toa valuerequiring a lowervaccina- some ofthe results obtainedcould be consequences of tionrate. Itis straightforwardto observe from ¢ gure7 these oversimpli¢ed choices. that couplingvaccination and sterilization couldbe Wehaveomitted di¡erences betweenthe dispersal successful. Thus,we could ¢ xseveralvaccination rates modes ofmale and female animals. Also we do not take less thanor equal to that neededto be successful when intoaccount the e¡ect ofculling,because its e¤ciency has vaccinationalone was used. Wecouldthen deducethe notbeen fully demonstrated onthe Europeancontinent minimum sterilization rate required toeradicate rabies. (Aubert 1994).Therefore,it wasconsidered that, to a Figure1 0showsthe required combinationsfor largeextent, foxcontrol by various methods constituted a ­ (3,0) 0.04 and ­ (3,0) 0.11. partof the density-dependentmortality .Additionally, min ˆ max ˆ

Proc. R.Soc.Lond. B (2000) Rabies vaccination and sterilization C.Suppoand others 1581 densitydependence in this studyacted only on survival Inthe conditionsdescribed byour model of anisolated andnot on reproduction. This is closeto what has been host population,one observes that astableendemic equi- observedin nature: a lackof variability in the fertility libriumemerges withrabies regulating the foxpopulation rate withina widerange of natural conditions suggests inboth demographic settings, i.e.logistic and Malthusian. that fecundityis astable demographicparameter in Thisoccurs whendemographic and epidemiological Europe.Finally ,the dispersal modeused inthis model parameters liewithin a reasonablyrealistic range.U nder enabledus toestimate the populationsize after yearling ourassumptions this stableequilibrium between the virus dispersal,but did not simulate apreferred settlement of andthe host requires afast turn-overof the healthyfoxes. youngin less denselyoccupied areas. Knowledge about Sincesurvival of the populationis assured bydispersal dispersal patterns that includethis behaviouris currently (October) andreproduction (April) ,it is understandable solimited (see Lloyd1 980;Harris 1981;Macdonald& that fora small transmission rate the virusdoes not Bacon1 982;T rewhella et al.1988;Allen & Sargeant1 993) propagateat a su¤cient speed tosurvive, while for a that this simpli¢cation is worthkeeping. largetransmission rate the mortalitydue to rabies cannot Obvioustrends infox demographic indices suggesta becompensated for in time. Stillunder our assumptions, steadypopulation increase overthe longterm (MÏller it followsthat the propagationof the epidemic disease is 1995;Artois et al. 1997).Ecologicalreasons for this popula- notvery sensitive todispersal, whilebeing more sensitive tionincrease remainunclear ,butlinks with a decrease in tobirth rate. Thisshould moderate biological considera- humancontrol seem the most likelyexplanation (Aubert tionsthat couldbe drawn from our model. N evertheless, 1994;Szemethy &Heltai 1997).Thesetrends weresimu- providedthat these simpli¢cations can be accepted, the latedin this modelthrough a Malthusiangrowth process modelsuggests that sterilization turns outto be a strong obtainedby a constantthat ensured reproductiongreater complement forcontrolling fox rabies. Additionally ,our thanmortality .Trends inthe modelare similar butnot modelsuggests that densitydependence smoothes out precisely adjustedto those observedunder ¢ eldconditions. £uctuationsat equilibrium between the host andthe Socio-spatialadjustment withincreasing density was not virus.In contrast withintuitive predictions, the successful consideredin this model.This could have an in£ uence in a rate ofrabies eradication is higherwhen the host popula- spatialmodel of rabiesdi¡ usion: in brief, foxes are regarded tionis notregulated, i.e. Malthusian growth. as`contractors’(Kruuk & Macdonald1 984)maintaining Nevertheless, fora foxpopulation experiencing a the smallest economicallydefensible home range. Addi- Malthusiangrowth curve, vaccination alone would be tionalresidents wouldbe tolerated as long as su¤ cient ine¤cient toeradicate rabies, as expected, so sterilization resources areavailable (compatible with the resources turns outto be speci¢cally helpful here. dispersionhypothesis, see Carr& Macdonald( 1986)). Finally,asan alternative approach we compared our Accordingto ¢ eldobservations, the number ofadult indivi- results toa deterministic andtime-continuous model, dualswithin a socialgroup is nevertheless limited tofour or givenin an electronic appendix(h ttp://durandal.mass.u- ¢ve(constant territory size hypothesis(CTH) versus terri- bordeaux2.fr/~naulin/appendix/appendix.html),basedon toryinheritance hypothesis (TIH) ,see LindstrÎm et al. asystem ofordinary di¡ erential equations such asthat 1982;V onSchantz 1 984;LindstrÎ m 1986).Fewbehavioural used inAnderson et al.(1981)andBarlow ( 1996).Asa studies havebeen recently devotedto this aspect ofthe ¢rst di¡erence this continuousmodel does not predict spatialbehaviour of foxes. Therefore, the response toa self-eradicationof rabies when the transmission rate is decrease inmortality within a situationof stable accessi- large.Also, for a Malthusianpopulation growth trend, bilityto resources is unknown.The model herein presented the vaccinatione¤ ciency cannot be predicted bythe accepts the unveri¢ed hypothesis that underthese condi- modeland depends on the parameters de¢ning the initial tionsthe number ofindividualswithin a socialgroup could state. Nevertheless, similar conclusionsconcerning reacha limit transgressing the CTH^TIHhypotheses. sterilization canbe drawn from both continuous and Further ¢eldresearch is neededto clarify this aspect. discrete models.Discrete-time modellingappears, then, Concerningthe propagationof the virus,a uniformtrans- tobe more appropriatefor our purpose. Predictions mission rate wasused; there isthen novariation due to sex obtainedfrom both models areencouraging in consid- orage (dispersers couldbe less exposedthan resident eringimmunocontraception as a possiblemethod of adults,see Artois& Aubert( 1985)),andno variation in controllinga re-emerging outbreakof rabies in highly contactrate betweenfoxes living within the same orinadja- dense foxpopulations. N evertheless, additionalbiological cent territories (see Artois& Aubert1985). hypothesesthat needto be taken into account in further Our modelis consideredin a constantenvironment, studies includefox culling considered as a non-density- unlikethat ofPech et al.(1997),whohave studied the e¡ect dependentmortality factor ,changesin spacing strategies ofenvironmentalvariability on the use offertility control whendensity increases, andthe in£uence of dispersal in offoxes in arid Australia. N ocompensatoryphenomena the recoveryof healthy populations. (Hone1 994)to contraception, such asan increase inthe birth rate ofnon-sterilized females (Newsome1 995),an Supportedby the CNRS under the grant `Mode ¨lisationde la increase inthe survivalrate offoxes, or immigration circulationde parasites dans des populations structure ¨es’. (Seagle& Close1 996),wereintroduced in our model. Therefore,no side e¡ects orretarded e¡ects couldbe REFERENCES expectedin our short-term analysis;with the purposeof Allen, S. H.&Sargeant,A. B. 1993Dispersal patterns of this projectbeing the fast controlof an outbreak of rabies, redfoxes relative to population density . J.Wildl. Mgmt 57, long-term e¡ects didnot need to be considered. 526^533.

Proc. R.Soc.Lond. B (2000) 1582C. Suppoand others Rabies vaccination and sterilization

Anderson,R. M., Jackson,H. C., May,R.M. &Smith, LindstrÎm, E., Poulsen, O. & VonSchantz, T .1982Spacing of A.D.M. 1981Population dynamics of fox rabies in Europe. the red fox Vulpesvulpes L.in relationto food supply .In Nature 289, 765^770. Populationecology of the red fox in relation to food supply (ed. E. Artois,M. 1989 Lerenardroux . Encyclope¨ diedes de France . LindstrÎm) ,pp.82^107 .PhDthesis, University of Stockholm, 3.Puceul,F rance:Socie ¨ te¨ Franc°aisepour l’Etude et la Sweden. Protectiondes Mammif e©res. Lloyd,H. G.1980 The red fox.London,UK: B. T.BatsfordLtd. Artois,M. 1997Managing problem wildlife in the`Old W orld’:a Macdonald,D. W .&Bacon,P .J.1982 F oxsociety ,contactrate veterinaryperspective. Reprod.F ertil.Dev . 9, 17^25. andrabies epizootiology . Comp.Immunol. Microbiol. Infect. Dis. Artois,M. &Aubert,M. 1985Behaviour of rabid foxes. Rev. 5, 247^256. Ecol. (Terre etVie) 5, 171^176. MÏller, W .W.1995Oral vaccination and high density fox popu- Artois,M. &Bradley,M. 1995U nvaccincontre les renards. lations. RabiesBull. Eur . 19, 14^15. Pourenrayer la prolife ¨rationdes animaux inde ¨sirables,un Newsome,A. E.1995Socio -ecologicalmodels for the red fox appaª tcontraceptif. LaRecherche 281, 40^41. populationssubject to fertility control in Australia. Ann. Zool. Artois,M., Langlais,M. &Suppo,C. 1997Simulation of rabies Fenn. 32, 99^110. controlwithin anincreasing fox population. Ecol.Model. 97, Pastoret,P .P.&Brochier,B. 1999Epidemiology and control of 23^34. rabiesin Europe. Vaccine 17,1750^1754. Aubert,M. 1994Control of rabies in foxes:whatare the Pech,R. P.&Hone,J. 1992 Models of wildlife rabies . In appropriatemeasures ? Vet. Rec. 134, 55^59. Wildliferabies contingency planning in Australia (ed. P. H. Aubert,M. 1995Epide ¨miologieet lutte contre la rage en F rance O’Brien& G.Berry) ,pp.147^157.NationalWildife Rabies eten Europe. Bull.Acad. N at.Me ¨ d. 179,1033^1054. Workshop,1 2^16March 1990. Canberra: Australian Barlow,N.D. 1 995Critical evaluation of wildlife diseasemodels. GovernmentPublishing Service. In Ecologyof infectious diseases in natural populations (ed. B. T. Pech,R., Hood, G. M., McIlroy,J.&Saunders,G. 1 997Can Grenfell& A.P.Dobson),pp.230^259. Cambridge University foxesbe controlled by reducing their fertility ? Reprod.F ertil. Press. Dev. 9, 41^50. Barlow,N.D .1996The ecologyof wildlife diseasecontrol: Seagle,S. W.&Close,J. D .1996Modeling white-tailed deer simplemodels revisited. J.Appl.Ecol. 33, 303^314. (Odocoileusvirginianus )populationcontrol by contraception. Blancou,J., Aubert, M. F.A.&Artois,M. 1991F oxrabies. In Biol.Conserv. 76, 87^91. Thenatural history of rabies, 2ndedn (ed. G. M. Baer), Smith,G. C. 1995Modelling rabies control in theUK: the pp.25 7^290.Boca Raton, FL: CR CPress. inclusionof vaccination. Mammalia 59, 629^637. Bradley,M. P.1994Experimental strategies for the development StÎhr, K. &Meslin,F .X.1997Oral vaccination of wildlife in ofan immunocontraceptive vaccine for the European fox Europe. In Rabiescontrol in Asia (ed.B. Dodet& F.X.Meslin), Vulpesvulpes . Reprod.F ertil.Dev. 6, 307^317. pp.27^34. Amsterdam, The Netherlands:Elsevier. Breitenmoser,U .,Kaphegyi,T .,Kappeler,A. &Zanoni,R. Suppo,C. 1996Mode ¨lisationet analyse mathe ¨matiquede la 1995Signi¢ cance of young foxes for the persistence of rabies propagationdes viroses dans les populations de carnivores. in northwesternSwitzerland. In Proceedingsof the Third Congress Thesisin Mathematics,U niversite¨BordeauxI, F rance. ofthe European Society of V eterinaryVirology ,pp.391^396. F rance: Szemethy,L.&Heltai,M. 1997E¡ ectsof per-oral vaccination FondationMe ¨ rieux. againstrabies on red fox in Hungary.In Carr,G. M. &Macdonald,D. W .1986The socialityof solitary 23rdCongress of the International Union of Game Biologists , Lyon, foragers:a modelbased on dispersion. Anim.Behav . France,1 September1 997.Re¨sume¨sdesposters. 34,1540^1549. Tapper,S. 1992 Gameheritage . Anecological review from shooting and DeCombrugghe,S. A.1994Statut des mammif e ©ressauvages en gamekeepingrecords .Fordingbridge,UK: Game Conservancy Ltd. Wallonie. Annls Me¨ d.Ve¨ t. 138, 229^235. Trewhella,W .J.,Harris, S. &Macallister,F.E.1988Dispersal Garnerin,P .,Hazout,S. &Valleron,A. J.1986Estimation of distance,home-range size and population density in thered twoepidemiological parameters of fox rabies: the length of fox (Vulpesvulpes ):a quantitativeanalysis. J.Appl.Ecol. 25, incubationperiod and the dispersal distance of cubs. Ecol. 423^434. Model. 33, 123^135. Voigt,D. R. &Macdonald,D. W .1984V ariationin thespatial Harris,S. 1981An estimationof the number of foxes ( Vulpes andsocial behaviour of the red fox, Vulpesvulpes . Acta Zool. vulpes)in thecity of Bristol,and some possible factors a¡ ecting Fenn. 171, 261^265. theirdistribution. J.Appl.Ecol. 18, 455^465. VonSchantz, T .1984Carnivore social behaviour ödoes it need Hone,J .1994 Analysisof vertebrate pest control . Cambridge patches? Nature 307, 388^390. UniversityPress. Vuillaume,P .,Aubert,M., Demerson,J .M., Cliquet,F .,Barrat, Kruuk,H. &Macdonald,D. W .1984Group territories of carni- J.&Breitenmoser,U .1997V accinationdes renards contre la vores:empires and enclaves. In Behaviouralecology: ecological ragepar de ¨poªtd’appaªtsvaccinauxa ©l’entre¨edesterriers. Ann. consequencesof adaptative behaviour (ed.R. M. Sibly& R.H. Med.Vet. 141, 55^62. Smith),pp.52 1^536.Oxford, UK: Blackwell Scienti¢ c Publications. LindstrÎm, E. 1986T erritoryinheritance and the evolution of Asthispaper exceeds the maximum length normally permitted, group-livingin carnivores. Anim.Behav. 34,1825^1835. theauthors have agreed to contribute to production costs.

Proc. R.Soc.Lond. B (2000)