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A Modelling Approach to Vaccination and Contraception Programmes For doi 10.1098/rspb.2000.1180 Amodellingapproachtovaccinationa nd contraceptionpr ogrammesfor rabiescontrol infox populations 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 population; 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-population density 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 Whatever the 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 11other foxclasses. (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 recruitment 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 straight pathand 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 ageand sex classes. Insimulations, ¯ reachthrough dispersal andas the set ofcells from which 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 tothe saf(t) IAF(n,t) + ­ (t)+ (syf(t) HAF(t)) I(n,t) £ j £ £ boundaryof the
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