Chronic Wasting Disease in Deer and Elk: a Critique of Current Models and Their Application

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1 Souhern Illinois Universiy Carbondale OpenSIUC ublicaions Deparmen of Zoology 2003 Chronic Wasing Disease in Deer and Elk: a Criique of Curren Models and Their Applicaion Eric M. Schauber Souhern Illinois Universiy Carbondale, schauber@siu.edu Alan Woolf Follow his and addiional works a: hp://opensiuc.lib.siu.edu/zool_pubs Recommended Ciaion Schauber, Eric M. and Woolf, Alan. "Chronic Wasing Disease in Deer and Elk: a Criique of Curren Models and Their Applicaion." Wildlife Sociey Bullein 31, No. 3 ( Jan 2003): doi:hp:// This Aricle is brough o you for free and open access by he Deparmen of Zoology a OpenSIUC. I has been acceped for inclusion in ublicaions by an auhorized adminisraor of OpenSIUC. For more informaion, please conac opensiuc@lib.siu.edu.

2 13 May 2003 Eric M. Schauber Cooperaive Wildlife Research Laboraory Souhern Illinois Universiy Carbondale, IL ; Fax: RH: Criique of CWD models Schauber and Woolf Chronic wasing disease in deer and elk: a criique of curren models and heir applicaion Eric M. Schauber, Cooperaive Wildlife Research Laboraory and Deparmen of Zoology, Souhern Illinois Universiy, Carbondale, IL 62901, USA, schauber@siu.edu Alan Woolf, Cooperaive Wildlife Research Laboraory and Deparmen of Zoology, Souhern Illinois Universiy, Carbondale, IL 62901, USA Absrac: Chronic wasing disease (CWD), a faal ransmissible spongiform encephalopahy of deer (Odocoileus spp.) and elk (Cervus elaphus), presens a challenge o wildlife managers because lile is known abou is ransmission, ye i could severely hreaen wildlife populaions if acion is no aken rapidly. ublished mahemaical models predic ha CWD could devasae populaions of free-living deer and elk, promping wildlife managers o aemp large-scale eradicaion of deer in hopes of conaining CWD oubreaks. Our objecive is o criically examine he heoreical and empirical suppor for curren models of CWD epizooiology, in ligh of herd healh managemen acions. We idenify a criical, unesed premise (i.e., sricly frequency-dependen ransmission) which underlies he dire model predicions. We reevaluae published comparisons of model oupu wih field daa and find lile suppor for published model srucures. Given he uncerainy surrounding he fuure effecs of chronic wasing disease

3 Schauber and Woolf 2 on deer and elk populaions, and he poenial coss of unnecessarily culling large numbers of charismaic and valuable animals, we propose ha consideraion of alernaive models and managemen acions in a decision-heoreic framework is necessary for wildlife managemen acions o reain heir scienific basis. Key words: assumpions, Cervus elaphus, chronic wasing disease, deer, densiy-dependence, elk, epizooiology, eradicaion, frequency-dependence, models, ransmission Wildlife Sociey Bullein 00(0): 000B000 Chronic wasing disease (CWD) has recenly emerged as a major concern of wildlife managers, biologiss, and sakeholders hroughou Norh America (Enserink 2001, Williams e al. 2002). Chronic wasing disease is a faal ransmissible spongiform encephalopahy (TSE; Williams and Young 1980) ha has been observed in free-living and capive deer (Odocoileus spp.) and elk (Cervus elaphus), and is he only TSE known o persis in free-living wildlife populaions (Spraker e al. 1997, Miller e al. 2000). Alhough CWD appears o have persised for decades a relaively low prevalence in an enzooic region in pars of Colorado and Wyoming, recen daa sugges ha is prevalence in free-living mule deer (O. hemionus) and elk may be increasing (Miller e al. 2000, Gross and Miller 2001). Ouside his enzooic area, CWD has been deeced in free-living mule deer, whieail deer (O. virginianus), or elk in Illinois, Nebraska, New Mexico, Saskachewan, Souh Dakoa, and Wisconsin, and appears o be spreading (Williams e al. 2002). No link beween CWD and disease in humans or noncervid livesock has been found, bu hese risks canno be dismissed wih absolue cerainy (Barz e al. 1998, Raymond e al. 2000, Hamir e al. 2001). Compounding he poenial impac on wildlife and human healh, CWD hreaens o erode favorable public percepion of wildlife resources and he fundamenal imporance of spor huning as boh a ool for managemen of free-ranging deer

4 Schauber and Woolf 3 and elk populaions and a major moneary source for wildlife managemen agencies and local economies. While much is known abou he empirical epizooiology of CWD (Williams e al. 2002), criical parameers and processes relaed o modes and paerns of ransmission are unknown. Also, CWD epizooics in wild herds have no been observed long enough o know wha heir ulimae populaion-level effecs will be. Therefore, mahemaical models are criical ools for assessing he poenial impac of CWD on deer and elk populaions and weighing his risk agains he coss of alernaive managemen acions. Mahemaical models have been developed o synhesize exising knowledge of CWD epizooiology in wild mule deer; hese models uniformly predic ha CWD can cause exincion of hos populaions (Miller e al. 2000, Gross and Miller 2001). In he face of dire model predicions, scarce daa, and uncerainy, expers have recommended srong and rapid seps o conain and eradicae CWD oubreaks (Gross and Miller 2001, Williams e al. 2002). The Wisconsin Deparmen of Naural Resources has begun an aemped eradicaion of all whie-ailed deer wihin a >900-km 2 area where CWD has been deeced (Nolen 2002), represening a prominen managemen philosophy and a sraegy likely o be considered by many agencies responsible for managing populaions a risk for CWD. We believe i useful o criically examine he premises and empirical suppor of published CWD models, because models are currenly he bes available ools for assessing he poenial impac of CWD on hos populaions and weighing ha risk agains he poenial coss of managemen alernaives. While we acknowledge ha rapid managemen acion o conrol CWD may be warraned and ha wildlife managers invariably mus ac wihou he luxury of complee knowledge, we propose ha science-based wildlife managemen will advance if compeing models and managemen alernaives are carefully explored in a decision-heoreic framework.

5 Schauber and Woolf 4 All scienific knowledge is enaive and provisional, and science advances by repeaedly confroning hypoheses and models wih logic and daa. I is in his spiri ha we offer he following criique of he premises and suppor of curren CWD models. Theoreical foundaion A model is a formal consruc ha illuminaes he logical consequences of he assumpions upon which i is based, and he validiy of he model as a represenaion of realiy depends on how closely is assumpions reflec he characerisics of he real sysem. The published models of CWD epizooiology in wild mule deer (Miller e al. 2000, Gross and Miller 2001) share a common assumpion: he number of effecive conacs beween an infecious individual and oher individuals per uni ime () is consan and independen of populaion size or densiy. This premise resuls in frequency-dependen ransmission, where he force of ransmission is a funcion of he frequency (i.e., proporion) of infecious individuals wihin he populaion (Appendix A). The idea of frequency-dependen ransmission is based on he premise ha opporuniies for conac beween an infecious individual and suscepible individuals are unaffeced by populaion size (de Jong e al. 1995). An imporan disincion exiss beween populaion size and densiy, depending on he scale over which ransmission occurs. For example, if ransmission occurs exclusively wihin social groups and he number of individuals per group is consan, changes in he number of groups inhabiing an area will change he populaion densiy on a large scale bu may no affec he local densiy or conac rae experienced by an individual wihin a group. The assumpion of frequency-dependen ransmission for CWD has been jusified on he basis of he aggregaive, migraory, and habia-selecion behaviors of wild deer (Gross and Miller 2001), bu is imporance o model oupu has no been discussed.

6 Schauber and Woolf 5 Epidemiological heory indicaes ha pure frequency-dependen ransmission srongly promoes unsable hos-pahogen dynamics (Gez and ickering 1983), such ha he disease eiher dies away ( <, where is a hreshold value) or i drives he hos and iself o exincion ( > ). However, if is no consan bu decreases as he hos densiy decreases, known as densiy-dependen ransmission, he disease and is hos can reach a sable equilibrium or exhibi regular cycles (Anderson and May 1978, May and Anderson 1978). The presumpion of frequency-dependen versus densiy-dependen ransmission is criical o he prediced oucome of an epizooic: hos-pahogen exincion versus hos-pahogen coexisence. Gross and Miller (2001:213) repor A disurbing resul of his modeling exercise was our inabiliy o idenify a se of realisic parameers ha permis susained coexisence of CWD in a wild deer populaion. We emphasize ha his dire oucome of CWD models is enirely a predicable consequence of he frequency-dependen assumpion and does no sem from any paricular known characerisics of CWD. Wheher ransmission is frequency-dependen or densiy-dependen is deermined by he primary mechanism of ransmission and he spaial srucure of hos populaions. Frequency-dependen ransmission is paricularly likely in cases of venereal or vecor-borne ransmission (May and Anderson 1978, Gez and ickering 1983) because he number of maes per individual hos or hos-bies per vecor may be essenially independen of hos densiy in many species. Frequency-dependen ransmission also is promoed when a hos populaion is subdivided ino groups of nearly consan size, so ha changes in overall populaion size or densiy do no affec he local densiy wihin groups (de Jong e al. 1995). Due o heir marilineal social srucure (Hawkins and Klimsra 1970, Geis 1981, 1982, Nelson and Mech 1999), deer and elk appear o be candidaes for frequency-dependen ransmission. However, he

7 Schauber and Woolf 6 CWD agen is mos likely ransmied via bodily fluids boh hrough direc conac and indirecly because he agen appears o persis in he environmen (Miller e al. 1998). We argue ha his combinaion of direc and indirec ransmission is unlikely o be sricly frequency-dependen. Also, mule deer and elk (paricularly females) congregae on winer range (Geis 1982, 1998), where he exudaes of an infeced animal poenially can conac a larger number of animals if more animals congregae in or migrae along he same area, suggesing ha some form of densiy-dependen ransmission is feasible. Group size and social srucure of deer and elk also respond o changes in populaion densiy (Kie and Bowyer 1999, Hebblewhie and lecher 2002). Finally, if CWD ransmission is sricly frequency-dependen, oher diseases of deer or elk ransmied by he oral-fecal roue should exhibi similar dynamics and should cause hos exincion. This has no been observed. Classical epidemiological models have been based on he premise ha is direcly proporional o he populaion densiy of hoss (Kermack and McKendrick 1927, Anderson and May 1978), and such sricly linear densiy dependence is clearly unrealisic for wild deer and elk. However, i also is unrealisic o presume ha is compleely independen of hos densiy. For example, CWD has become much more prevalen in capive cervid herds mainained a high densiies han in free-living herds (Williams and Young 1980, Williams e al. 2002), suggesing ha populaion densiy has a posiive influence on he probabiliy of ransmission. For group-living species like cervids, may be approximaely consan over a range of moderae populaion densiies bu is likely o change when populaion densiy is very low or high. Thus, migh rise o an asympoe as populaion densiy increases (Diez 1982, Heeserbeek and Mez 1993, Anonovics e al. 1995, Heeserbeek and Robers 1995, Ramsey e al. 2002) or could vary as a power funcion or some oher nonlinear funcion of densiy (Figure 1). Incorporaing hese

8 Schauber and Woolf 7 nonlinear forms of densiy-dependen ransmission ino he model of Miller e al. (2000) would resul in persisen hos-pahogen coexisence if drops below * as hos densiy drops. Oupu of CWD models is very sensiive o changes in he value of (Miller e al. 2000, Gross and Miller 2001), so even weakly densiy-dependen ransmission may enable hos-pahogen coexisence. The many unknown aspecs of CWD ransmission prohibi robus predicion of he populaion impac. Empirical suppor Miller e al. (2000) esed he validiy of heir frequency-dependen model by comparing is predicions wih empirical daa relaing o changes in CWD prevalence over ime and paerns of CWD prevalence across and age classes in mule deer in Colorado. However, he apparen concordance repored by Miller e al. (2000) beween observed and prediced CWD prevalence across age- classes is in error. As he model is described by Miller e al. (2000), i is incapable of producing a paern in which infecion prevalence differs across age classes >4 years old or beween es (Appendix B). This conrass wih heir field daa and heir purpored model oupu (Miller e al. 2000: figure 4). The empirical age-prevalence relaionship of CWD in free-living mule deer, paricularly is rariy in older male deer, canno be explained by he model and indicaes ha some imporan biological processes are missing from he model. The concordance wih empirical daa is also quesionable for he model of Gross and Miller (2001), who used he same age-prevalence daa for mule deer as Miller e al. (2000) bu lumped he daa beween es. The auhors claimed ha he nearly fla age-prevalence relaionship ha emerged from he model closely mached independen field observaions, (Gross and Miller 2001:210) despie he prominen differences in observed CWD prevalence among age groups. revalence of CWD, averaged across age classes, was similar in model

9 Schauber and Woolf 8 oupu and field daa, bu wheher ha similariy ruly represens concordance wih independen daa is unclear. In he model of Gross and Miller (2001:210), CWD prevalence generally increased over ime during simulaion runs, so he model oupu would closely mach observed CWD prevalence only a cerain ime seps. The auhors neglec o indicae from wha ime sep in he simulaion he model oupu came, and wheher ha ime sep was chosen based on crieria oher han similariy o observed daa. Implicaions for managemen Culling has been used ofen in aemps o conain or eliminae wildlife diseases by driving hos populaions below a hreshold densiy (Barlow 1996, Wobeser 2002). However, unlike densiy-dependen ransmission, sric frequency-dependen ransmission does no permi he exisence of a hreshold hos densiy below which he pahogen canno persis (Gez and ickering 1983). Therefore, incomplee hos eradicaion (i.e., parial culling) can deerminisically cause eliminaion of a disease wih densiy-dependen bu no a frequency-dependen ransmission, unless infeced individuals can be idenified and culled selecively (Gross and Miller 2001). If ransmission is ruly frequency-dependen, incomplee eradicaion migh only hasen he ulimae exincion of ha hos populaion wihou prevening disease spread o oher populaions. In oher words, even if he frequency-dependen assumpion upon which eradicaion programs are based is rue, i implies ha eradicaion is unlikely o successfully conrol he spread of he disease unless nearly 100% of hoss are eliminaed. Given ha he assumpion of sric frequency-dependen ransmission is boh criically imporan and unesed, i seems pruden o consider wha managemen opions migh be appropriae if his assumpion is unrue. If ransmission is no frequency-dependen, hen a hreshold hos densiy may exis. If so, ha hreshold hos densiy may be high or low, relaive

10 Schauber and Woolf 9 o curren densiies of deer and elk. If he hreshold densiy is high, he disease will no subsanially reduce wild populaions, and CWD does no endanger deer herds. If i is low, he hos populaion mus be reduced o even lower densiies o locally eliminae he disease. I remains an open quesion wheher such exreme culling programs will be logisically or poliically feasible, paricularly if CWD inroducion is no a one-ime occurrence or he CWD agen persiss in he environmen. Complee eliminaion of CWD from all Norh American deer and elk herds is unlikely, despie he bes effors of humans, suggesing ha i could be reinroduced relaively frequenly ino disease-free populaions (because boh CWD epidemiology and proposed managemen acions occur on he scale of decades, relaively frequenly migh mean once per 20 years). If CWD inroducion in a region is no a 1-ime occurrence, hen CWD esablishmen could only be prevened in he long erm by suppressing hos densiies for an indefinie period below he densiies ha would resul if he disease ook is course. Even if necessary and successful, defeaing CWD via hos eradicaion would come a a cos, no only economic bu also in erms of public percepion of wildlife resources, accepance of managemen paradigms, and inerrupion of he huning radiion. These coss of success and he uncerainy surrounding he necessiy of hos eradicaion should be accouned for when weighing alernaive managemen acions. We do no inend o imply ha any aemp o manage wildlife diseases by reducing hos densiy is wrong or undesirable. However, he fac ha all wildlife diseases are no inensively managed implies ha managers implicily weigh he coss of various acions agains he risks of inacion. For CWD models are he bes available ools for esimaing he risks of inacion and, herefore, he appropriae magniude of response. The exisence of a model of CWD epizooiology ha predics cerain exincion of he hos could be inerpreed as jusificaion for

11 Schauber and Woolf 10 whaever managemen acion is deemed mos likely o preven his dire oucome. However, any modeler wih undersanding of fundamenal ecological heory and no consideraion for validiy of assumpions could produce models predicing cerain exincion of one or all species for any hosbpahogen (Gez and ickering 1983), hosbparasie (Hassell and May 1973), predaorbprey (Murdoch and Oaen 1975), or oher ecological ineracion. Therefore, predicions of alernaive models need o be considered and judged on he validiy of heir assumpions and concordance wih daa in order o evaluae he appropriae magniude of response. Our objecive is o call aenion o a criical unesed premise (i.e., frequency-dependen ransmission) of curren CWD models and o emper accepance of model predicions wih he uncerainy surrounding he validiy of ha premise and he weakness of empirical suppor. Gross and Miller (2001:213) claimed ha To he exen ha modeled mechanisms of CWD ransmission appear o offer a leas a reasonable approximaion of disease processes occurring in naure, i follows ha his model provides plausible forecass of fuure epidemic rends. We agree wih he logic of he firs par of his saemen bu quesion of wheher heory or daa permi he accepance of hese models as a reasonable approximaion of CWD ransmission dynamics. The predicions of frequency-dependen models of CWD epizooiology (Miller e al. 2000, Gross and Miller 2001) represen a small se of possible oucomes of CWD epizooics in wild populaions. Oher oucomes are also plausible, and heir acualiy depends on he rue (bu unknown) relaionships beween ransmission and populaion densiy, and age srucure, and spaial srucure. Curren frequency-dependen models are consisen wih he observed long-erm persisence of CWD a low prevalence in free-living deer and elk, bu his observaion also is consisen wih oher hypoheses (e.g., CWD will remain a relaively low prevalence indefiniely). The range of possible realiy saes and he poenial benefis and coss of

12 Schauber and Woolf 11 alernaive managemen acions may be bes analyzed in a decision-heoreic framework where poenial coss of inacion and alernaive acions are explicily weighed and poin o he urgen need for research ino he ransmission dynamics of CWD o firmly base managemen decisions on he bes possible science. Acknowledgmens. reparaion of his manuscrip was suppored by he Illinois Deparmen of Naural Resources. We hank. Shelon, J. Roseberry, and wo anonymous reviewers for commens ha subsanially improved earlier drafs of his manuscrip. Lieraure cied Anderson, R. M., and R. M. May Regulaion and sabiliy of hos-parasie populaion ineracions: I. Regulaory processes. Journal of Animal Ecology 47: Anonovics, J., Y. Iwasa, and M.. Hassell A generalized model of parasioid, venereal, and vecor-based ransmission processes. American Nauralis 145: Barlow, N. D The ecology of wildlife disease conrol: Simple models revisied. Journal of Applied Ecology 33: Barz, J. C., R. F. Marsh, D. I. McKenzie, and J. M. Aiken The hos range of chronic wasing disease is alered on passage in ferres. Virology 251: de Jong, M. C. M., O. Diekmann, and H. Heeserbeek How does ransmission of infecion depend on populaion size? ages in D. Mollison, edior. Epidemic models: heir srucure and relaion o daa. Cambridge Universiy ress, Cambridge, Unied Kingdom. Diez, K Overall populaion paerns in he ransmission cycle of infecious disease agens. ages in R. M. Anderson and R. M. May, ediors. opulaion biology of infecious diseases. Springer-Verlag, Berlin, Germany.

13 Schauber and Woolf 12 Enserink, M rion diseases US ges ough agains chronic wasing disease. Science 294: Geis, V Behavior: adapive sraegies in mule deer. ages in O. C. Wallmo, edior. Mule and black-ailed deer of Norh America. Universiy of Nebraska ress, Lincoln, Nebraska, USA. Geis, V Adapive behavioral sraegies. ages in J. W. Thomas and D. E. Toweill, ediors. Elk of Norh America: ecology and managemen. Wildlife Managemen Insiue, Harrisburg, ennsylvania, USA. Geis, V Deer of he world: heir evoluion, behavior, and ecology. Sackpole Books, Machanicsburg, ennsylvania, USA. Gez, W. M., and J. ickering Epidemic models: hresholds and populaion regulaion. American Nauralis 121: Gross, J. E., and M. W. Miller Chronic wasing disease in mule deer: disease dynamics and conrol. Journal of Wildlife Managemen 65: Hamir, A. N., R. C. Culip, J. M. Miller, E. S. Williams, M. J. Sack, M. W. Miller, K. I. O'Rourke, and M. J. Chaplin reliminary findings on he experimenal ransmission of chronic wasing disease agen of mule deer o cale. Journal of Veerinary Diagnosic Invesigaion 13: Hassell, M.., and R. M. May Sabiliy in insec hos-parasie models. Journal of Animal Ecology 42: Hawkins, R. E., and W. D. Klimsra A preliminary sudy of he social organizaion of he whie ailed deer. Journal of Wildlife Managemen 34:

14 Schauber and Woolf 13 Hebblewhie, M., and D. lescher Effecs of elk group size on predaion by wolves. Canadian Journal of Zoology 80: Heeserbeek, J. A.., and J. A. J. Mez The sauraing conac rae in marriage- and epidemic models. Journal of Mahemaical Biology 31: Heeserbeek, J. A.., and M. G. Robers Mahemaical models for microparasies of wildlife. ages in B. T. Grenfell and A.. Dobson, ediors. Ecology of infecious diseases in naural populaions. Cambridge Universiy ress, Cambridge, Unied Kingdom. Kermack, W. O., and A. G. McKendrick A conribuion o he mahemaical heory of epidemics. roceedings of he Royal Sociey of London, Series A 115: Kie, J. G., and R. T. Bowyer Sexual segregaion in whie-ailed deer: densiy dependen changes in use of space, habia selecion, and dieary niche. Journal of Mammalogy 80: May, R. M., and R. M. Anderson Regulaion and sabiliy of hos-parasie populaion ineracions: II. Desabilizing processes. Journal of Animal Ecology 47: McCary, C. W., and M. W. Miller A versaile model of disease ransmission applied o forecasing bovine uberculosis dynamics in whie-ailed deer populaions. Journal of Wildlife Diseases 34: Miller, M. W., M. A. Wild, and E. S. Williams Epidemiology of chronic wasing disease in capive Rocky Mounain elk. Journal of Wildlife Diseases 34: Miller, M. W., E. S. Williams, C. W. McCary, T. R. Spraker, T. J. Kreeger, C. T. Larsen, and E. T. Thorne Epizooiology of chronic wasing disease in free-ranging cervids in Colorado and Wyoming. Journal of Wildlife Diseases 36:

15 Schauber and Woolf 14 Murdoch, W. W., and A. Oaen redaion and populaion sabiliy. Advances in Ecological Research 9: Nelson, M. E., and L. D. Mech Tweny-year home-range dynamics of a whie-ailed deer mariline. Canadian Journal of Zoology 77: Nolen, R. S Wisconsin mobilizes o bale chronic wasing disease. Journal of he American Veerinary Medical Associaion 220: Ramsey, D., N. Spencer,. Caley, M. Efford, K. Hansen, M. Lam, and D. Cooper The effecs of reducing populaion densiy on conac raes beween brushail possums: implicaions for ransmission of bovine uberculosis. Journal of Applied Ecology 39: Raymond, G. J., A. Bossers, L. D. Raymond, K. I. O'Rourke, L. E. McHolland,. K. Bryan III, M. W. Miller, E. S. Williams, M. Smis, and B. Caughey Evidence of a molecular barrier limiing suscepibiliy of humans, cale and sheep o chronic wasing disease. EMBO Journal 19: Spraker, T. R., M. W. Miller, E. S. Williams, D. M. Gezy, W. J. Adrian, G. G. Schoonveld, R. A. Spowar, K. I. O'Rourke, J. M. Miller, and. A. Merz Spongiform encephalopahy in free-ranging mule deer (Odocoileus hemionus), whie-ailed deer (Odocoileus virginianus) and Rocky Mounain elk (Cervus elaphus nelsoni) in norhcenral Colorado. Journal of Wildlife Diseases 33: 1 6. Williams, E. S., M. W. Miller, T. J. Kreeger, R. H. Kahn, and E. T. Thorne Chronic wasing disease of deer and elk: a review wih recommendaions for managemen. Journal of Wildlife Managemen 66:

16 Schauber and Woolf 15 Williams, E. S., and S. Young Chronic wasing disease of capive mule deer: a spongiform encephalopahy. Journal of Wildlife Diseases 16: Wobeser, G Disease managemen sraegies for wildlife. Revue Scienific e Technique de L'Office Inernaional des Epizooics 21: Associae edior: Krausman

17 Schauber and Woolf 16 Appendix A: If he number of effecive conacs per individual per year () is consan and populaion size (N) is a finie ineger, hen he probabiliy of an uninfeced hos becoming infeced during a ime sep (i.e., force of infecion) is given by 1 1 1, where I is he N number of infecious hoss (McCary and Miller 1998, Miller e al. 2000; noe ypographical error in Gross and Miller 2001:208). As N approaches infiniy (e.g., if a large geographic area is I examined) bu infecion prevalence ( N I ) remains consan, his probabiliy converges o he zero I erm of a oisson disribuion subraced from uniy: 1Bexp( N ). Numerical analyses indicae ha he finie-n and oisson ransmission probabiliies differ by a facor of ~0.05 or less for N > 10. Thus, he force of ransmission is primarily a funcion of infecion prevalence, and no he absolue of number of infeceds. For ime seps <1 year, as seps (and hence ) become small, I he oisson probabiliy converges o. N

18 Schauber and Woolf 17 Appendix B: Model oupu repored in figure 4CBD in Miller e al. (2000) canno be produced by he model hey describe; see figure 4 in Gross and Miller (2001) for an age-prevalence relaionship consisen wih model srucure. Below, we describe he model of Miller e al. (2000), which represens he spread of CWD in a free-living populaion of mule deer, and prove ha heir model necessarily produces - and age-independen prevalence for above age class 4. Gross and Miller (2001) use a sligh modificaion of he same model srucure. Yearly survival of uninfeced yearling and adul deer is assumed o differ beween es bu no age classes (excep ha no deer survives pas age class 15), and he probabiliy of a suscepible deer becoming infeced in a given year is equal for all and age classes excep fawns (which may receive verical ransmission from heir mohers). Afer a deer is infeced, i spends 1 year in he laen sage before becoming infecious. Afer an animal becomes infecious, survival is successively halved in each of 3 subsequen years irrespecive of and age. No infecious deer are allowed o live >3 years in his model. For each he number of infeced individuals in age class j > 4 in year (I,j) is a funcion of he number of uninfeced individuals in age class j-4 in year -4 (S-4,j-4), he -specific yearly survival rae (ψ), and he yearly probabiliies of becoming infeced for he previous 4 years (-4, -3, -2, -1): I, j S 4, j S 4, j4 ( 1 4 ) S S 4, j4 ( 1 4 ) (1 3) 2 2 4, j4 ( 1 4 ) (1 3 ) (1 2 ) 1

19 Schauber and Woolf , j S The oal number of individuals of ha in age class j > 4 in year (N,j) is given by he sum of I,j and hose individuals ha were uninfeced in year -4, avoided infecion enirely, and survived for 4 years: ) )(1 )(1 )(1 1 ( ,,, j j j S I N The produc 4 4, 4 j S can be facored ou of boh I,j and N,j, so infecion prevalence (I,j/N,j) is funcion only of -4, -3, -2, and -1, which are consan across and age classes. We conclude ha, given he assumpions upon which his model is based, infecion prevalence canno differ beween es or among age classes above age class 4.

20 Schauber and Woolf 19 Figure 1. lausible relaionships beween hos populaion densiy and he number of effecive conacs per uni ime beween each hos individual and ohers. Curren published models of chronic wasing disease assume no relaionship (consan), whereas Anderson-May ype models assume a linear relaionship. However, here exiss a range of inermediae and more complex relaionships ha are biologically feasible.

21 Schauber and Woolf 20 ersonal Daa: Eric M. Schauber is a wildlife ecologis wih he Cooperaive Wildlife Research Laboraory (CWRL) and an assisan professor in he Deparmen of Zoology a Souhern Illinois Universiy Carbondale (SIUC). He obained a B.S. in wildlife from Universiy of Massachuses Amhers, an M.S. in wildlife from Oregon Sae Universiy, and a h.d. in ecology from Universiy of Connecicu. Eric s research has focused on epizooiology, ecooxicology, populaion dynamics, and predaorbprey ineracions of small mammals, wih an emphasis on quaniaive mehods and modeling. Alan Woolf is he direcor of he CWRL and a professor in he Deparmen of Zoology a SIUC. He obained a B.S. Cornell Universiy, an M.S. from Colorado Sae Universiy, and reurned o Cornell for his h.d. His research ineress are varied, alhough wildlife disease has been a major componen. He has served he CWRL for he pas 23 years and has been direcor since 1987.