Effect of Crop Growth and Canopy Filtration on the Dynamics of Plant Disease Epidemics Spread by Aerially Dispersed Spores

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1 Analyial and Theoeial lan ahology Effe of Cop Gowh and Canopy Filaion on he Dynamis of lan Disease Epidemis Spead by Aeially Dispesed Spoes F. J. Feandino Depamen of lan ahology and Eology, The Conneiu Agiulual Expeimen Saion,.O. Box 6, ew Haven 654. Aeped fo publiaion 5 Januay 8. ABSTRACT Feandino, F. J. 8. Effe of op gowh and anopy filaion on he dynamis of plan disease epidemis spead by aeially dispesed spoes. hyopahology 98: Mos mahemaial models of plan disease epidemis ignoe he gowh and phenology of he hos op. Unfounaely, epos of disease developmen ae ofen no aompanied by a simulaneous and ommensuae evaluaion of op developmen. Howeve, he ime sale fo ineases in he leaf aea of field ops is ompaable o he ime sale of epidemis. This simulaneous developmen of hos and pahogen has many amifiaions on he esuling plan disease epidemi. Fis, hee is a simple diluion effe esuling fom he inoduion of new healhy leaf aea wih ime. Ofen, measuemens of disease levels ae made po aa (pe uni of hos leaf aea o oal oo lengh o mass. Thus, hos gowh will edue he appaen infeion ae. A seond, elaed effe, has o do wih he so-alled oeion fao, whih aouns fo inoulum falling on aleady infeed issue. This fao aouns fo muliple infeion and is given by he faion of he hos issue ha is susepible o disease. As an epidemi develops, less and less issue is open o infeion and he iniial exponenial gowh slows. Cop gowh delays he impa of his limiing effe and, heefoe, ends o inease he ae of disease pogess. A hid and ofen negleed effe aises when an inease in he densiy of susepible hos issue esuls in a oesponding inease in he basi epoduion aio, R, defined as he aio of he oal numbe of daughe lesions podued o he numbe of oiginal mohe lesions. This ous when he anspo effiieny of inoulum fom infeed o susepible hos is songly dependen on he spaial densiy of plan issue. Thus, op gowh may have a majo impa on he developmen of plan disease epidemis ouing duing he vegeaive phase of op gowh. The effes ha hese op gowh-elaed faos have on plan disease epidemis spead by aibone spoes ae evaluaed using mahemaial models and hei impoane is disussed. In paiula, plan disease epidemis iniiaed by he inoduion of inoulum duing his sage of developmen ae shown o be elaively insensiive o he ime a whih inoulum is inodued. Addiional keywods: mass-aion, spoe deposiion. Duing he vegeaive gowh phase of an annual field op, gound ove o ligh ineepion ineases logisially wih ime, and he leaf aea, afe an iniial exponenial gowh peiod, ineases appoximaely linealy wih ime (,8. The imum gowh ae is suh ha a squae mee of leaf aea pe uni of gound aea is podued evey 5 o 7 days (9. This vegeaive phase, depending on he op, lass fom beween 3 and 5 days, duing whih ime leaf aea densiy ofen ineases by a fao of 4 o 6. Towad he end of he gowing season, leaves gow moe slowly as abohydae is alloaed o he soage and epoduive pas of he plan and senesen leaves ae shed. One way o desibe he oveall behavio of a gowing op is o use he logisi equaion asympoially appoahing a imum leaf aea (36. Waggone (38 epoed logisi op gowh ae paamees, (d, vaying fom.6 o.4 d. Fo ompaison, logisi fis o disease pogess uves yield a ange fo Vandeplank s (d vaying fom.5 o.5 d (37. Thus, he elaive ae of gowh and iming of disease developmen ompaed wih op gowh will pobably play an impoan ole in he empoal developmen of plan diseases. Almos all mahemaial models desibing he ime ouse of an epidemi ae based on some fom of he following podu law: Coesponding auho: F. J. Feandino addess: fanis.feandino@po.sae..us doi:.94/ HYTO The Ameian hyopahologial Soiey α S H S ( d whee Y is some measue of he umulaive oal infeed issue and α is a ae onsan. The undelying assumpion is ha he numbe of new infeions pe uni of ime is diely popoional o some measue of he amoun of infeive (spoulaing issue, S, muliplied by some measue of he amoun of healhy susepible issue, H S (,,37. The analogy has been dawn beween equaion and he piniple of mass-aion used o desibe he dynamis of hemial eaions (,3,39. The ae onsan, α, in equaion epesens a numbe of diffeen poesses: popagule poduion, dispesal, deposiion o new hos sies, suvival ove he ouse of his poess, and he evenual poduion of new infeions. Hos gowh may affe any o all of hese poesses. Fuhemoe, i is well known ha he abiliy of a pahogen o infe a hos an be a song funion of he phenologial age of he aaked issue (,4,8,4 6,5. Due o his onogeni esisane, a poion of he sanding healhy plan aea may no be susepible o disease. Theefoe, he numbe of susepible and sill healhy sanding sies, H S, will expliily depend on he age disibuion of he hos sies (, whih is deemined by he ime ouse of op gowh. If S and H S in equaion ae expessed in ems of numbe of infeion sies, hen he podu S H S epesens he oal numbe of spoulaing sie-healhy sie pais in he populaion, whih epesen evey possible pahway by whih a popagule of disease an esul in a new infeion. Fo a field of fixed aea, suh a fomulaion is onsisen wih obsevaions showing ha appaen infeion aes ae dependen on iniial plan densiy (5,7. The above assumpion is equivalen o wha Andeson e al. ( and de Jong e al. ( all pseudo mass-aion. Thee ae many easons why his simple assumpion may lead us asay. Fo plan diseases disseminaed by aibone spoes, he deposiion of po- 49 HYTOATHOLOGY

2 pagules o plan foliage is definiely no a simple linea funion of susepible sies as implied above (3,. On he ohe hand, if H S in equaion is expessed as he faion of he oal numbe of infeion sies ha ae healhy and susepible, hen he esuling epidemi model is based on wha Andeson e al. ( and de Jong e al. ( all ue mass-aion (,. In his ase, any effe of plan gowh on disease pogess is implii in he fao α. Thee has been some disussion in he lieaue (7,8 as o appopiae use of he above nomenlaue. In he eologial lieaue (8, he pseudo mass-aion ase is haaeized by densiydependen ansmission, wheeas he ue mass-aion ase is dependen on fequeny-dependen ansmission. Iespeive of he names given o hese models, hey ae based on, pehaps, unaepably simple mahemaial absaions fom ealiy. MCallum e al. (8 sugges ha a moe ealisi epidemi model should efle he physial mehanisms of he ansmission of disease. The fae of aibone spoes is he esul of a hee-way ompeiion beween deposiion o he gound, deposiion o plans, and esape fom he field. The elaive aes of hese ompeing poesses ae govened by leaf aea densiy, he gaviaional seling speed of he spoes, as well as wind speed and ubulene wihin and above he plan anopy. In fa, as I will show, his ompeiive sysem follows Mihaelis-Menen kineis (9 wih espe o leaf aea densiy, so ha he pobabiliy of infeion pe hos sie deeases as he numbe of hos sies ineases. This is due o he fa ha leaves, in a dense anopy, shield one anohe fom he flux of aibone spoes. Fuhemoe, if spoe suvival is dependen on anopy densiy, he spoe suvival faion is also expliily dependen on he numbe of hos sies. The ne esul is ha he paamee α in equaion has a song funional dependene on op gowh and is definiely no onsan in ime. A low folia densiies, he amoun of spoes deposied ineases wih ineasing leaf aea and pseudo mass-aion is he appopiae model. Lae in he season, fo dense plan anopies, spoe ah on leaves appoahes a imum value and ue mass-aion desibes he siuaion. The pupose of his pape is o demonsae ha pseudo massaion and ue mass-aion epidemi models ae eally exeme ases of a ange of moe ealisi epidemi models based on op densiy-dependen inoulum ansmission. To ahieve his goal, I inopoae a op gowh model (,8 and an esimae fo anopy filaion effiieny ino he deivaion of an epidemi model. Leaf aea as a funion of ime is assumed o follow a logisi law. The elaionship of leaf aea o anopy filaion effiieny is esimaed using dimensional analysis and a gadien diffusion appoximaion (K-heoy fo ubulen anspo (3. The esuls of model alulaions ae hen used o evaluae he amifiaions of op gowh on he basi epoduion aio, R (defined as he aio of he oal numbe of daughe lesions podued o he numbe of oiginal mohe lesions, of aeially dispesed pahogens and on he empoal developmen of he ensuing epidemi. The vaiables and paamees used ae lised in Table. THEORY AD AROACHES Epidemi models: pevious wok and pesen oulook. Mass-aion models. The podu law (equaion has hee populaion vaiables: Y, S, and H S, as well as a ae paamee α, whih is usually assumed o be a onsan. Thus, a losed fom soluion o equaion will depend on ou abiliy o expess wo of hese vaiables in ems of he emaining one. The speifiaion of S involves some assumpion abou he phenology of infeed issue and will, in geneal, depend on he ime ouse of infeion. By analogy, he speifiaion of H S involves some assumpion abou he phenology and susepibiliy of plan issue o disease and will depend on he ime ouse of op gowh. Vandeplank (37 solved his poblem by ignoing op gowh and assuming ha a newly infeed hos sie is laen fo a disee ime peiod p and hen emains infeious fo a ime i. Sine he oal numbe of hos sies,, was assumed o be onsan, equaion ould be divided hough by his value and expessed in ems of faional values. Alenaively, ompamen models haaeize he oal numbe of hos sies,, as eihe being healhy, H, laenly infeed, L, spoulaing, S, o emoved, R, suh ha H L S R and Y L S R (9,,6,3. The developmen of he epidemi is hen expessed as a seies of oupled linea diffeenial equaions (35. Reenly, i has been shown ha boh he Vandeplank disee ime model and he ompamenal models fi ino a lage heoeial famewok of he lassi Kemak-MKendik epidemi model (3,7,35. Following Diekman e al. ( and Heesebeek (, we now define he basi epoduion aio, R, as he expeed numbe of seonday ases podued, in a ompleely susepible populaion, by a ypial infeed individual duing is enie peiod of infeiousness. In addiion, we assume ha he ae of spoe poduion pe lesion an be expessed as a funion τ (spoes d lesion of lesion age, τ (d. Wih he above assumpions and definiions, he Kemak-MKendik epidemi model esuls in a disease pogess uve ha is oally deemined by R, he shape of he spoulaion uve (τ, and he iniial ondiions (9,35. Howeve, wha happens o he above analysis when he op gows and he ae paamee α is a funion of ime? Explii models. We an, of ouse, examine he poblem numeially and inlude he ime dependene of hos issue susepibiliy o disease as well as op gowh and vaiable α. Thee ae many numeial plan disease models in he lieaue based on eihe he ue mass-aion o he pseudo mass-aion assumpion (6,,4,5,34,4. These models aoun fo he effe of op gowh as an inease in he amoun of susepible healhy issue, bu do no inlude he ineased effiieny by whih aibone spoes ae apued by plan issue as folia densiy ineases. Feandino ( pesened a ombined analyial/numeial model fo he spead of lae bligh on poao and is effes on yield, whih inluded he udimens of anopy filaion. In wha follows, I seek analyial soluions fo disease developmen, whih inlude he effes of op densiy on disease ansmission o illuminae boh he physial and biologial naue of he infeion poess. To his end, following Vandeplank (37, I will make he following simplifying assumpions. S, he numbe of spoulaing sies, is a onsan faion, f R, of Y, he oal numbe of infeed sies. All healhy sies, H, ae susepible o disease. Ignoing emovals, he oal numbe of sies,, is equal o he sum of H and Y. In his desipion, sies ould be infeion ous, leafles, leaves, sems, o whole plans. The above assumpions lead o hee elaions among Y, S, H, H S, and : S fr Y H Y H H S whih when ombined wih equaion yield ( αf R Y( Y (3 d The geneal soluion o equaions of he above fom (Appendix equaion A an be obained by quadaue o die inegaion if he quaniies αf R and ae given as funions of ime (Appendix equaions A3 o A5. Fo he speial ase whee is a onsan, equaion 3 an be divided by o yield dy y( y (4 d whee he diseased faion, y, is defined by y Y/ and Vandeplank s ae onsan, (d, is se equal o he podu, Vol. 98, o. 5, 8 493

3 αf R. If is assumed onsan, hen he soluion o equaion 4 esuls in a logisi epidemi (Appendix equaion A7. In wha follows, I will assume ha he oal numbe of hos sies,, is similaly given by a logisi equaion whih asympoially appoahes a imum value of wih he logisi ae of op gowh denoed by (d, suh ha d (5 d Tue mass-aion. In ode o aoun fo logisi op gowh, by simulaneously solving equaions 3 and 5, Waggone (38 assumed ha he quaniy W (d αf R emains onsan fo a TABLE. Symbols Symbol Unis a Definiion a m The aea of a sie A D Leaf aea index (leaf aea pe uni gound aea A H D Healhy hos leaf aea pe uni gound aea A D Maximum leaf aea pe uni gound aea in logisi op gowh model A Y D Infeed hos leaf aea pe uni gound aea b [X s Rae paamee in equaion A b [X Limi paamee in equaion A, oesponding o he imum value of he abiay funion X C sp m 3 Aeial spoe onenaion Aonym fo anopy filaion COFR D The oeion fao (H/, lieaue iaion 4 C, C, C 3 D aamees defined o simplify he equaions A6 o A6 f h D Hoizonally pojeed faion of leaf aea f p D Canopy filaion effiieny, defined as he pobabiliy ha an aibone spoe is deposied on foliage f R D Faion of infeed sies whih ae epoduive F A m The aea of he field unde sudy h m Cop heigh H s Toal numbe of healhy sies H s Iniial numbe of healhy sies H S s Toal numbe of susepible healhy sies τ sp s d The ime ae of spoe poduion pe lesion as a funion of lesion age IF( D Inegaing fao whih endes equaion A exa (Appendix J E sp m s Flux of spoes esaping fom he plan anopy J G sp m s Flux of spoes deposied o he gound J sp m s Flux of spoes deposied o he plan anopy k D von Kaman s onsan (3, k.4 K M m s Eddy diffusiviy fo momenum (3 L le umbe of laenly infeed sies s and le Toal umulaive numbe of sanding sies, inluding healhy and infeed sies s and le Assumed onsan numbe of sanding sies, inluding healhy and infeed sies s and le Maximum numbe of sanding sies, inluding healhy and infeed sies n D omalized leaf aea pe uni gound aea (n A/A MA Aonym fo pseudo mass-aion d Vandeplank s ae onsan d Logisi ae paamee fo op gowh model d Rae onsan fo anopy filaion model d Rae onsan fo pseudo mass-aion model W d Rae onsan fo Waggone s ue mass-aion model R s Toal umulaive numbe of emoved sies R D The basi epoduion aio, defined as he aio of he oal numbe of daughe lesions podued o he numbe of oiginal mohe lesions S le umbe of infeive aively spoulaing sies sa d Time a whih a disease lesion is iniiaed T E s Chaaeisi ime fo esape fom he plan anopy T G s Chaaeisi ime fo deposiion o he gound TMA Aonym fo ue mass-aion T s Chaaeisi ime fo deposiion o plan maeial u(z m s Hoizonal wind speed u * m s The fiion veloiy (3 v g m s Gaviaional seling speed of aibone spoes X [X Abiay funion of ime in equaion A Y D The diseased faion (y Y/ Y le Toal umulaive numbe of infeed sies, inluding laen infeions, infeious sies, and emovals: Y L S R z m Heigh z m Roughness sale (33 α s s Rae onsan fo he law of mass aion (equaion β D The assumed onsan value fo he anopy filaion effiieny, f, in he ue mass-aion model γ D The assumed onsan of popoionaliy in he equaion: f γ/ in he pseudo mass-aion model δ D The aio of he numbe of spoes los o boh he gound and he ai above he anopy o he numbe of spoes deposied on plan issue ξ s obabiliy pe healhy hos sie ha a spoe is libeaed, beomes aibone, and is physially anspoed and deposied o suh a sie τ d Age of lesion τ sa d Time a whih inoulum is inodued ino a field ψ le sp Faion of spoes ha having landed on susepible hos ause a new lesion a D no dimension, kg mass, s numbe of sies, le numbe of lesions, sp numbe of spoes, m disane, MJ enegy (Mega Joules and s and d ime. 494 HYTOATHOLOGY

4 logisially gowing op. The above assumpion is equivalen o wha Andeson e al. ( and de Jong e al. ( all ue massaion. Heeafe, his model is denoed by TMA. In essene, he paamee α in equaions and 3 is assumed o be invesely popoional o he size of he hos populaion,. Wih hese assumpions, equaion 3 beomes ( Y αfry( Y W Y (6 d The developmen of disease in suh a siuaion is modeled using wo oupled nonlinea diffeenial equaions (equaions 5 and 6 whih an be solved analyially (Appendix equaions A8 and A9; Feandino s Appendix in lieaue iaion 38. seudo mass-aion. Alenaively, one an assume ha he α in equaions and 3 is a onsan. The above assumpion is equivalen o wha Andeson e al. ( and de Jong e al. ( all pseudo mass-aion. Heeafe, his model is denoed by MA. Leing (d αf R, equaion 3 beomes ( Y αfry( Y Y (7 d The developmen of disease in suh a siuaion is one again modeled using wo oupled nonlinea diffeenial equaions (equaions 5 and 7. The ime inegals in he quadaue soluion o equaion 5 oupled o equaion 7 an be wien in losed fom when is an inege muliple of (Appendix equaions A o A. oe ha if he oal numbe of sies,, is onsan hen boh equaion 6 and equaion 7 eve o equaion 3. As will be shown below fo aeially dispesed pahogens, Waggone s ue massaion appoah (equaion 6 is appopiae fo dense anopies o low wind speeds when spoe deposiion is anspo-limied (3. In his ase, mos of he aibone spoes ae deposied o foliage, independen of leaf aea. The pseudo mass-aion ompaible appoah (equaion 7 is appopiae fo spase anopies o high wind speeds when spoe deposiion is sink-limied (3. In his ase, mos of he spoes ae eihe deposied o he gound o esape fom he plan anopy, and spoe deposiion o foliage ineases linealy wih leaf aea. In ealiy, spoe deposiion in a gowing op will lie somewhee beween he above wo exemes depending on wind, wihin anopy ubulene, and hanging folia densiy. Repeussions of he finie naue of spoe poduion. Conside he pseudo mass-aion fom of equaion wih all populaion vaiables (Y, S, H H S expessed as sies o sies pe uni of gound aea. As long as vegeaion is spase we would expe he pe diem ae of new infeions o inease wih he podu of he numbe of spoulaing sies, S, and he numbe of uninfeed susepible sies, H. Howeve, as folia densiy ineases a diffiuly wih his simple podu law aises. To illusae he poblem, assume ha only one sie in an enie field of plans is infeed and spoulaing. This spoulaing sie (S sie libeaes a finie numbe of spoes pe day, I (spoe d (S sie. Thus, even unde he mos opimal ondiions, suh ha evey spoe podued esuls in a new infeed sie, a imum of I new infeed sies an be podued pe diem. Howeve, aoding o equaion, if we oninue o inease he size of he field, he numbe of age sies, H (T sie, will inease and he numbe of lesions podued pe day, αh (emembe S, will oninue o inease and evenually beome lage han I. Obviously, he plans anno ah moe spoes han ae eleased, so ha he law of pseudo mass-aion beaks down a high plan densiies, and he paamee α anno emain onsan independen of he numbe of healhy sies, H. Models based on he ue mass-aion assumpion avoid his diffiuly hough he assumpion ha he paamee α is invesely popoional o, he oal numbe of sies. Unde his unealisi assumpion, disease pogess is no expliily dependen on op gowh and he plan anopy ineeps a onsan faion of he aibone inoulum independen of leaf aea densiy. As saed above, he ae onsan, α, in equaion epesens a numbe of diffeen poesses. Fo aeially dispesed spoes, α an be expessed as he podu of hee quaniies: he ime ae of aeial popagule poduion pe infeious sie (I [spoe (S sie d, he pobabiliy of an aibone spoe being deposied on a healhy sie pe healhy sie (ξ [(T sie, and he faion of he spoes deposied on healhy sies ha go on o podue a new infeed sie (ψ [lesion spoe. To analyze he impliaions of nononsan α, we will fis eexamine he dimensionaliy of equaion and he ole of he paamee α: α S H d lesion lesion day day S sie T sie α I [ S sie [ T sie lesion spoe lesion (8 day S sie T sie day S sie T sie spoe The nomenlaue of equaion 8 is onsisen wih ha of Segaa e al. (35. The dimensionaliy of he paamee ξ is poblemaial. Fo an epidemi in a gowing op he numbe of healhy sies is simulaneously deeased by new infeions and ineased by op gowh. Sine ξ is defined pe healhy sie, his paamee is impliily dependen on boh he sage of he epidemi and he sage of op gowh. We need o make hese dependenies explii. If spoes ae deposied equally o evey sie iespeive of whehe he sie is infeed o no, hen he pobabiliy of an aibone spoe being deposied on any sie, f (no dimensioned D, divided by he oal numbe of sies,, mus equal ξ (i.e., ξ f /. The podu ξh in equaion, he faion of spoes whih ae deposied on healhy sies, an hen be expessed as he podu of f and he faion of all sies ha ae healhy, H/. The anopy filaion effiieny, f, is dependen on he physial popeies of he anopy and he mehanial popeies of wind and ubulene and independen of epidemi developmen. The aio, H/, he faion healhy sies, is wha Zadoks (4 alled he oeion fao (COFR, whih aouns fo he effe of disease pogess on he faional spoe deposiion o healhy sies. A ompaison of he above models (equaions 6 and 7 wih equaion 8 leads us o he following onlusions. Tue mass-aion. Equaion 6 is based on he assumpion ha f p is a onsan, so ha α Iψ/ and W Iψf R. Fo lae efeene, his onsan value fo he anopy filaion effiieny will be denoed by β. seudo mass-aion. Equaion 7 is based on he assumpion ha he quaniy, f p, is popoional o he oal numbe of sies,. Leing f p gives α Iψ and Iψf R. Fo lae efeene, he popoionaliy onsan fo he anopy filaion effiieny will be denoed by γ suh ha f p γ/. Thus, he majo diffeene beween he above wo models (equaions 6 and 7 involves he assumed elaionship beween effiieny wih whih foliage apues aibone spoes and he numbe of hos sies. Canopy filaion model. The infomaion povided by (, he numbe of hos sies as a funion of ime, is inadequae o povide a mehanisi view of op gowh and spoe disseminaion. Ohe infomaion abou he hos sies, he op, and he field mus be supplied, e.g., Wha is he sie densiy pe uni volume? How lage ae hese hos sies? How ae hey physially oiened? Wha is hei spaial disibuion? How lage is he field? How all is he op? Assuming ha he aea of a sie is given by a (m and ha hee ae a oal of ( sies a ime,, wihin a field of aea F A ξ ψ ( Vol. 98, o. 5, 8 495

5 (m, hen he oal leaf aea pe uni of gound aea, A (he leaf aea index, he infeed leaf aea pe uni of gound aea, A Y, and he healhy leaf aea pe uni of gound aea, A H, ae given by ( a Y( a H( a A( ; AY ( ; AH ( (9 FA FA FA The piniple of mass-aion, upon whih equaions and ae based, is no appopiae fo aeially dispesed pahogens wihin a apidly gowing field op (3,,. Mehanially, he faion of aibone spoes deposied on leaves is dependen on he leaf aea index, whih is a funion of ime. Fo ha eason, in wha follows all vaiables enumeaing he numbe of hos sies (, Y, H, e. ae muliplied by he aio a/f A o expess hem as leaf aea indies (equaion 9. The fae of aibone spoes eleased fom infeive issue wihin a op depends on he elaive aes of esape fom he field, deposiion o he gound, and deposiion o plan maeial. Following Aylo (3, I daw he analogy o an elei iui, whee J, J G, and J E ae spoe fluxes assumed diely popoional o he invese of he ime sales T, T G, and T E (esisanes and oesponding o deposiion on he plan anopy, deposiion on he gound, and esape fom he plan anopy, espeively. These hee fluxes ae in paallel, so ha he faion of spoes landing on plan maeial, f (he anopy filaion effiieny, is equal o he aio of he spoe flux deposied on he plans, J, divided by he oal spoe flux, J J G J E, suh ha J ( J JG J E f T T T The evaluaion of he anopy filaion effiieny, f, above depends on he deposiion of aibone spoes o folia elemens in he ubulen ai wihin he plan anopy. If we assume ha he aibone spoes ae disibued evenly houghou he plan anopy wih aeial onenaion, C (spoes m 3, and ha deposiion ono foliage is dominaed by gaviaional seling, hen he spoe flux ono plan sufaes and he gound ae given by J vg fh A C ( JG vg C whee v g (m s is he gaviaional seling speed of spoes and he podu f h A is he hoizonally pojeed leaf aea pe uni of gound aea. The ime sale fo esape fom he plan anopy will depend on he wind and ai ubulene wihin and above he op. Above he anopy wind speed and ubulene ae govened by Monin- Obukhov similaiy heoy (3. Above he plan anopy unde neually sable ondiions, wind speed, u (m s, vaies logaihmially and eddy diffusiviy fo momenum, K M (m s, vaies linealy wih aeodynami heigh, (z d (m: u * z d u ( z ln k z and ( K M ( z ku* T ( z d whee k (D is von Kaman s onsan (k.4, u * (m s is he fiion veloiy, d (m is he zeo plane displaemen heigh, and z (m is he oughness sale (33. In wha follows, I assume ha he eddy diffusiviy fo momenum, K M, also applies o aibone spoes (33. Thus, he ime sale fo esape fom he plan anopy an be appoximaed as h TE K M z h h ku G h E *( h d. u * (3 whee h is he heigh of he plan anopy and he zeo-plane displaemen heigh, d, is assumed equal o.3 h (33. Simple dimensional analysis yields J E C h. u* C, (4 T E and he above appoximaions (equaions and 3 ombined wih equaion yield whee f f fha. u* A v h δ g vg. u v f A g h n δ n δ n * (5 and n A/A /. Equaion 5 has he same fom as he Mihaelis-Menen equaion (9, wheein, he ae of eaion fo a subsae of onenaion A is popoional o A divided by he sum of a onsan and A. The paamee δ is he aio of he numbe of spoes los o boh he gound and he ai above he anopy o he numbe of spoes deposied on plan issue. The value of δ is of he ode of uniy, whih an be seen by inseing easonable values ino he above definiion (fo v g. m s, u *. m s, f h.5, A 6, and δ.3. When he leaf aea is vey small, pseudo mass-aion applies, i.e., f is appoximaely popoional o leaf aea. Howeve, as he op gows, leaf aea ineases, and f appoahes he imum value of ( δ. So ha, fo dense anopies, f is a onsan and ue mass-aion applies. Heeafe, his anopy filaion model is denoed by. Combining equaions 3, 8, 9, and 5 yields day IψfRn ( A AY n( δ ( A AY AY AY (6 d ( n δ A ( n δ A whee (d Iψf R /( δ. Alenaively, using equaion 9, he above equaion (equaion 6 an be expessed in ems of sies as IψfRn ( Y ( δ n ( Y Y Y (7 d ( n δ ( n δ The ime inegals in he quadaue soluion o equaion 5 oupled o equaion 7 an be wien in losed fom when is an inegal muliple of (Appendix equaions A7 and A8. RESULTS Model alulaions. A ompaison of models. The above hee models (TMA [equaion 6, MA [equaion 7, and [equaion 7 diffe only in he assumed expession fo f (Table. The wo abiay paamees, β and γ, fo he MA and TMA models, espeively, an be hosen suh ha all hee models give he same esuls fo lae season epidemis (Table : β γ ( δ. The ole of he paamee δ (equaion 5 in deemining he anopy filaion effiieny fo he model is illusaed in Figue. oe f fo he model is bounded by he assumpions of he ohe wo models (MA and TMA. The soluions of he hee models (TMA [equaion 6, MA [equaion 7, and [equaion 7 wih W, Y /.67, n., δ ae ploed in Figue. Due o he low spoe ah when leaf aea is small fo boh he MA and models, plan disease epidemis ae delayed unil he anopy sas o gow. These mass-aion-based epidemis ae haaeized by aeleaing logisi gowh aes and ae negaively skewed in ime, as illusaed in Figue. oe ha all hee models degeneae o he same simple logisi equaion fo epidemis saing a a ime, τ sa /, afe he vegeaive peiod fo whih n is vey lose o uniy. 496 HYTOATHOLOGY

6 The diseased faion y. Expeimenally, one does no diely measue he numbe of infeed sies, Y. Usually, a sample (a olleion of leaves, sems, o plans is assayed fo disease o obain he diseased faion, y Y/. Diseased faions fo he epidemis shown in Figue ae ploed in Figue 3. The iniial deease in he diseased faion fo he MA and models is due o he diluion effe of he gowing op. This effe is also wha auses he ime lag in hese epidemis. Basially, he op mus gow o a eain size befoe enough spoes ae apued so ha disease ineases a leas as fas as he op gows. To pu he above saemen ino moe fomal ems we mus eexamine he model equaions. Ou nex ask is o desibe he ime ouse of disease developmen in ems of y when he op is gowing. Combining equaions 5 and 6 wih he definiion of he deivaive of he quoien, Y/, yields ( dy d Y Y d y( y y( n W (8 d d d d whee n / and he negaive em on he igh side of equaion 8 aouns fo diluion due o op gowh. Seing he deivaive in equaion 8 equal o zeo, we define he iial disease seveiy, y i, as a funion of ime: ( n ( exp( yi ( (9 W W exp( whee he logisi soluion fo op gowh (Appendix equaion A7 has been subsiued fo n(. Analogous analyses fo he pseudo mass-aion ase (MA: equaions 5 and 7 yield and ( dy d Y Y d ny( y y( n p ( d d d d y i ( n ( exp n p p ( ( and fo he anopy filaion ase (: equaions 5 and 7, he esuls ae dy ( δ n y( y y( n ( d ( n δ whee y Y/ A Y /A, n / A/A, and y i is given by y i ( n( n δ ( δ n [ exp δ[ n( ( exp( ( δ n( [ exp( (3 The soluion o equaion is illusaed in Figue 4A fo he model whee y is ploed vesus nondimensional ime. The sho line segmens in Figue 4A epesen he deivaive in equaion and he hahed line is a plo of y i (equaion 3. Fo disease seveiies less han he iial value (equaion 3, he diseased faion ineases wih ime. Fo disease seveiies geae han he iial value (equaion 3, he op gows fase han he epidemi and he faional disease seveiy deeases wih ime. A he iial value, boh disease levels and op densiy gow a he same ae, so ha he diseased faion emains onsan. The elaionships among he values of y i fo all of he models (TMA, equaion 9; MA, equaion, and, equaion 3 ae illusaed in Figue 4B. oe ha he anopy filaion model smoohly ansfoms fom he MA ase (δ o he MA ase (δ as he paamee δ ineases. The ime lag. The iniial behavio of he hee models is examined in he Appendix. The TMA model oesponds o an inease of Y ove ime (equaion A. Howeve, he behavio of he ohe wo models is moe omplex. Due o he low effiieny of folia spoe apue in he ealy sages of op gowh, disease pogess is sifled fo he MA and models when ompaed wih he TMA model. The ne esul (Appendix equaions A9 o A6 is ha he iniial epidemi uves fo he MA and models appoah exponenial uves afe peiods of ime, τ and τ, espeively, given by ln[ n ( τ (4 and τ ( δ ln δ ln ( C (5 whee C is defined in equaion A6. This behavio is independen of he iniial level of disease [Y( o y(. Fo ompaison, onside he ime i akes fo a logisially gowing op o aain half is imum leaf aea, τ ½, given by τ ln (6 If he iniial value of n is small (n <., he wo ime sales τ (equaion 4 and τ ½ (equaion 6 ae almos idenial, wih τ ~ τ ½ /. The ime lags fo models (equaion 5 ae smalle by a fao depending on he value of he paamee δ. Fo example, if. and δ, hen τ ½ 4.595, τ 4.65, and τ.3, 3.53, 3.9, and 4.43 fo δ.,.5,., and 5., espeively (Fig. 5A. One again, as δ ineases, he model appoahes he behavio of he MA model. The effe of ime of iniial infeion. Fo mos of he above epidemis, we have assumed ha hee was some iniial level of diseased issue a ime zeo. Conside he siuaion whee he op is allowed o gow uninfeed fo a eain ime peiod, τ sa, a whih ime a eain amoun of disease, Y(τ sa, is inodued. This siuaion is illusaed in Figue 5B fo he paamees of he TABLE. A ompaison of he hee models pesened in he ex a Model f f ; n α Tue mass-aion (equaion 6 β β seudo mass-aion (equaion 7 γn γ Canopy filaion (equaion 7 n n δ βiψ γiψ Iψn ( δ ( n δ ( Y ( Y W βiψf R γiψf R IψfR δ a All hee models ae desibed by he same diffeenial equaion: αfry IψfR fy wih suiable values fo he anopy filaion d effiieny, f. In ode fo he hee models o give he same esuls as n, β γ ( δ. oe n A/A /. Vol. 98, o. 5, 8 497

7 epidemi illusaed in Figue 4A (τ sa,., Y(τ sa /.667, /, and δ. Fo values of τ sa < τ, he esulan epidemis ae almos idenial. To illusae his fa, noe ha he dimensionless ime a whih he diseased faion y is.5 ae 7., 7.3, 7., 7.4, 7.83, 8.48 fo values of τ sa,,, 3, 4, 5, 6, espeively. In his ase, τ 3.9 (equaion 5 and he dimensionless ime lag beween τ sa and τ sa 3 is only.3. To pu his infomaion in pespeive, a ypial value fo is. d, so ha a 3 day delay in he inoduion of inoulum esuls in a mee 3. day lag in he ime fo he epidemi o eah he 5% poin (y.5. Basially, disease developmen is delayed unil he flush of op gowh. Fig.. The effe of op gowh on anopy filaion. The faion of aibone spoes ha land on plans, f, divided by is imum value ( δ, is ploed vesus nomalized leaf aea index (n A/A fo diffeen values of he paamee δ (equaion 5. The paamee δ is defined as he aio of he numbe of spoes los o boh he gound and he ai above he anopy o he numbe of spoes deposied on plan issue. Fo ompaison, he assumed anopy filaion fo he ue mass-aion model (TMA, dashed line, equaion 6 and he pseudo mass-aion model (MA, heavy solid line, equaion 7 ae also shown. Smalle values of δ oespond o dense anopies, fase spoe seling veloiies, and lowe wind speeds. opulaion effes and lesion phenology. The above models of disease developmen ae based on ahe simplisi assumpions onening he populaion of lesions and he populaion of hos sies. Fis spoulaing o infeive sies, S, ae assumed o be a onsan faion, f R, of infeed sies. Due o he laeny peiod and he dependene of spoulaion on he age of a lesion, he age disibuion of he populaion of infeed sies mus be aken ino aoun in ode o oely elae infeive sies, S, o infeed sies, Y. Canopy filaion and he basi epoduion aio, R. Segaa e al. (35: equaion 4 defined he basi epoduion aio, R, as ψ τ R ξh dτ (7 whee H is he oal numbe of healhy sies (assumed susepible a ime, ξ and ψ ae defined as in equaion 8, and τ is he ime ae of aeial spoe poduion pe lesion, as befoe. Howeve, equaion 7 is no longe valid fo a gowing op whee he oal numbe of hos sies, (, is an explii funion of ime. Cop gowh (equaion 5 and he anopy filaion effiieny, f (equaion 5, mus be inopoaed ino he definiion of R. oe he faion of deposied spoes geneaing lesions, ψ (lesion spoe, was also assumed onsan in equaion 7 (35: equaion 4. If ψ is a funion of ime, i mus be bough inside he ime inegal. The iniial podu, ξh, on he igh side of equaion 7, by definiion, is he faion of he oal numbe of spoes whih end up on he imal numbe of susepible hos sies, H, he iniial value. The analog of ξh, fo a gowing op, is ξ( f [n(, δ, whih is an explii funion of ime. Theefoe, in his ase, he imal numbe of available susepible hos sies is no iniially pesen. In ode o aoun fo he ineasing availabiliy of hos sies, he podu, ξ(, mus be also be bough inside he ime inegal in equaion 7. Realling ha ξ f /( and assuming ha all sies ae healhy and susepible, he oal numbe of daughe lesions pe mohe lesion infeed a ime, sa, is given by sa f[ n( sa τ, δ ψ( sa τ τ τ (8 R( d whee f and δ ae defined in equaion 5. In wiing equaion 8, I have, inenionally, lef ou he zeo subsip on R( sa in Fig.. lo of he epidemis esuling fom he hee models disussed in he ex (ue mass-aion [TMA, equaion 6; pseudo mass-aion [MA, equaion 7; and anopy filaion [, equaion 7 wih W, Y /.67, n., δ. Y is he umulaive numbe of infeed sies and is he imum numbe of oal sies. All hee models give he same esul when he epidemi is saed as he op appoahes imum gound ove ( ; lag TMA, lag MA, lag. Fo ompaison, nomalized op gowh is also ploed (n / ; heavy line. Fig. 3. Diseased faion, y, vesus dimensionless ime,, fo he epidemis ploed in Figues and 3 ( W, Y /.67, n., δ. oe ha y deeases fo he pseudo mass-aion (MA and anopy filaion ( models duing he ealy sages of he epidemi as he op gows fase han disease ineases. All hee models give he same esul when he epidemi is saed afe he op has gown ( τ sa, ylag dashed line. Fo ompaison, nomalized op gowh is also ploed (n / ; heavy line. 498 HYTOATHOLOGY

8 equaion 8 o emind he eade ha his quaniy is a funion of he ime of lesion iniiaion, sa. Fo mos pahosysems, due o he favoable ondiions fo infeion assoiaed wih ineasing leaf aea, he fis wo ems wihin he inegand of equaion 8 will boh, on aveage, be monoonially ineasing funions of ime houghou he vegeaive phase of op gowh. Thee is an addiional sohasi vaiaion in ψ, he spoe suvival-infeion pobabiliy, due o episodi meeoologial evens suh as ainfall. Sine a suessful invasion of he pahogen equies ha R > (, hee may be a heshold value fo sa befoe whih an epidemi is no likely o ou. In ohe wods, lesions iniiaed befoe his iial ime may no ause a self-susainable epidemi (3. An example: spoulaion given by Vandeplank s sep funion. Assume ha ψ is onsan and is given by he podu of a onsan ae of spoe elease, q (spoes day, and a uni sep funion in ime, whih has he value of fo p > > p i and is zeo a all ohe imes (37. Thus, afe a laen peiod of duaion, p (d, a new lesion podues spoes a a onsan ae q fo a Fig. 4. A, Sho line segmens epesening dy/d alulaed fo / and δ using equaion in ex, ogehe wih equaion A7 in he Appendix, ae ploed vesus dimensionless ime,. The U shaped line oesponds o he / soluion ploed in Figues o 5. (d is he imum logisi ae of disease inease and (d is he logisi ae of op gowh. The hahed line is he iial diseased faion, y i, given by equaion 3 in he ex. oe slopes ae negaive above and o he lef of his line and posiive below and o he igh. The logisi uve desibing he faional leaf aea, n(, is also shown fo ompaison (bold sigmoidal line. Figue was alulaed assuming., y( B, The effe of op gowh on he iial diseased faion, y i, alulaed fo he anopy filaion model (, equaion 7, whih is ploed vesus nomalized op size (n / A/A fo diffeen values of he paamee δ (equaion 3. The paamee δ is defined as he aio of numbe of spoes los o boh he gound and he ai above he anopy o he numbe of spoes deposied on plan issue. Fo ompaison, he assumed filaion fo he ue mass-aion model (TMA, dashed line, equaion 9 and he pseudo mass-aion model (MA, heavy solid line, equaion ae also shown. Smalle values of δ oespond o dense anopies, fase falling spoes, and lowe wind speeds. These alulaions wee aied ou assuming W. Fig. 5. A, Log plos of appoximae soluions fo Y<< (Appendix equaions A9 o A6. A saigh line indiaes an exponenial inease of disease. Fo hese alulaions i was assumed ha., W, and he asympoi slopes of all lines is. The pseudo mass-aion (MA soluion (heavy line has he imum dimensionless delay ime, τ, equal o he naual logaihm of o, appoximaely, 4.6 (noe τ ln(., equaion A5. The asympoi line fo he MA soluion is also shown (do-dash line. B, lo of diseased faion, y, vesus dimensionless ime,, illusaing he effe of he ime of iniial infeion (τ sa on he anopy filaion ( epidemi illusaed in Figue 6 (τ sa,., Y(τ sa /.667, /, and δ. The dimensionless delay ime, τ, fo his epidemi is 3.9 (equaion 5. As he iniial inoulum is inodued lae and lae, vey lile happens unil τ sa > τ. Vol. 98, o. 5, 8 499

9 peiod of ime equal o he infeious peiod, i (d. The oal amoun of spoes podued ove his ime peiod is equal o he podu iq. Then, using equaion 5 fo f, equaion 8 an be inegaed o yield R( sa p i dτ ψq p δ n( sa τ i R e C 3 exp ln i C3 exp [ ( [ ( sa p sa p (9 whee he imum value fo f is ( δ, whih when inseed ino equaion 7, yields R iψq/( δ, and δ( ( δ C 3 (3 whih was defined in equaion A6. The aio R( sa /R is ploed vesus sa in Figue 6A and B fo δ. In Figue 6A, he value of he aio of he infeious peiod o he laen peiod, i/p, is se equal o and p is vaied fom.5 o. In Figue 6B, he value of he aio p is se equal o and i/p is vaied fom. o 4. In geneal, an inease in eihe p o i edues he impa of he lesion sa ime on he effeive epoduive aio (R( sa. The Kemak-MKendik epidemi model. The above speifiaion of he elaions beween lesion age, spoe poduion, anspo, and deposiion used in he deivaion of R (equaion 7 suggess a esaemen of he mass-aion equaion (equaion. In paiula, he podu αs an be expessed as ( τ αs ξψ τ dτ d (3 whee τ is he age of a lesion, τ is he ime ae of spoe poduion pe lesion as a funion of lesion age, ξ is he pobabiliy pe susepible uninfeed hos sie ha a spoe is physially anspoed o suh a sie, and ψ is he faion of spoes ha having landed on susepible hos ause a new lesion. The deivaive in equaion 3 epesens he ae a whih lesions wee being iniiaed a ime τ. Wih all he above assumpions, equaions and 3 an hen be ombined o yield ( ( τ ξh( ψ τ dτ g( d d (3 whee g( is he spoe poduion of lesions iniiaed befoe and all healhy issue is assumed susepible. Equaion 3, as wien, sill inludes wo populaion vaiables, Y and H. Thus, an explii soluion of equaion 3 involves some assumpion abou he elaion beween hese wo vaiables. Following Segaa e al. (35, we assume ha he oal numbe of hos sies emains onsan and is given by H Y, suh ha dh/d /d, whih, when ombined wih equaion 3, gives he lassi fom of he Kemak-MKendik epidemi model (KM model: 3: dh( dh( τ ξh( ψ τ dτ g( d d (33 The soluion of equaion 33 esuls in a disease pogess uve ha is oally deemined by he value of R (equaion 7, he spoulaion uve (τ, and he ime ae of spoe poduion fom he iniial infeion (g(. Ou ask a hand is o modify he above analysis o inlude a gowing op fo whih he oal numbe of sanding sies, (, is an explii funion of ime. Fo equaion 3, his poess is simple. Expessing he paamee ξ in equaion 3 as an explii funion of ime, ξ f /(, and ignoing emovals suh ha H Y, yields ( τ Y( ( f ψ τ dτ g( d ( d (34 The deivaion of equaion 33 involved an assumpion onening he ime deivaives of H and Y whih is no longe valid fo a gowing op. Insead, ignoing emovals, we have d( dh( ( ( H( Y( (35 d d d so ha, equaion 33 an be expessed as [ ( H( f ψh( d[ ( τ H( τ d τ dτ g( d ( d o (36 dh( f ψ τ ψ τ τ τ H( dh( d( f H( d( d g( τ dτ d ( d d ( d Die ompaison of equaion 36 wih equaion 33 eveals wo addiion ems involving he ime deivaive of he oal numbe of sies, d/d. Of ouse, if emains onsan, hese ems ae zeo and equaion 36 eves o equaion 33. The seond posiive em on he igh side of equaion 36 epesens he effe of op gowh on he numbe of healhy sies, H, and he negaive las em on he igh side of equaion 36 is a oeion o he spoe Fig. 6. The nomalized effeive epoduion aio, R( sa /R (equaion 9, is ploed vesus dimensionless lesion sa ime, sa (δ,.. A, The value of he aio of he infeious peiod o he laen peiod, i/p, is se equal o and p is vaied fom.5 o. An inease in he laen peiod, p, edues he impa of he lesion sa ime on he effeive epoduive aio (R( sa. B, The value of he aio p is se equal o and i/p is vaied fom. o 4. An inease in he infeious peiod, i, edues he impa of he lesion sa ime on he effeive epoduive aio (R( sa. 5 HYTOATHOLOGY

10 poduion em (bakeed onvoluion inegal in he fis em on he igh side of equaion 36 aouning fo ineases in H due o op gowh, whih is independen of Y and has nohing o do wih spoe poduion. As he above analysis indiaes, he expession of epidemi developmen in ems of healhy sies is ompliaed by op gowh. In his siuaion, equaion 34 is moe useful han he modified KM model above (equaion 36, sine i is expessed in ems of infeed sies, whih ae no diely affeed by he inoduion of a new hos. Fo equaion 34, op gowh enes expliily in he funional dependene of f on anopy densiy (equaion 5 and he diluion effe ouing in he oeion fao ( Y/. Thee is no explii dependene on d/d. Lesion poduion days. The fom of equaion 8 fo R and he nondimensional pemuliplie, f ψ, in equaions 34 and 36 sugges he inoduion of a ansfomed ime vaiable, (d, defined as I ( f ψd (37 whih I shall all lesion poduion days. If spoes ae podued a a onsan ae of I pe day, and all hos issue is assumed susepible so ha hee ae no muliple infeions, hen he oal numbe of lesions podued in ime,, is I. Dividing boh sides of equaion 34 by he podu, f (ψ(, and muliplying and dividing he inegand in equaion 34 by he podu, f ( sa ψ( sa, yields ( I Y( I ( I di ( I di sa I I sa di g( I (38 whee τ sa, and all ime vaiables have been ansfomed using equaion 37. Thus, a simple ime ansfomaion (equaion 37 an aoun fo he ime dependene in spoe deposiion and spoe suvival. Thee sill emains he effe of diluion due o op gowh in he oeion fao of equaion 38 (fis em on he igh side, whih aouns fo he inoduion of new healhy issue. Howeve, ealy in an epidemi, he em Y/ is small and an be ignoed. If we assume ha spoe infeion pobabiliy, ψ, is onsan in ime, hen he ime dependene of I is idenial o he ime dependene of ln(if fo he model (equaions A5 and A4. This funion, illusaed in Figue 5A, is haaeized by he ime lag, τ, given by equaion 5. This ime lag is independen of he spoe poduion ae and is oally given by op-based paamees. DISCUSSIO The epidemiologial piniple of pseudo mass-aion (equaion 7, wheein he numbe of new lesions pe uni of ime is popoional o he numbe of spoulaing sies muliplied by he numbe of susepible hos sies, has been eexamined. Fo spoes spead by wind, his simple linea hypohesis beaks down a high plan anopy densiies. On he ohe hand, he simple modifiaion of his ule, whih leads o ue mass-aion models (equaion 6, does no popely aoun fo he vaiable effiieny wih whih a gowing plan anopy ineeps aibone spoes. A simple physial model of spoe anspo and deposiion povided a easonable and moe ealisi appoximaion fo he hanging anopy filaion effiieny. Soluions of he esuling diffeenial equaion (equaions 6 and 7 fo disease seveiy as a funion of ime in a gowing op povide some ineesing insighs. Disease pogess is deemined by wo ime sales, one desibing op gowh and he ohe desibing spoe poduion ae. The basi epoduion aio, R, defined as he numbe of daughe lesions pe mohe lesion when all hos issue is susepible, ineases damaially duing he vegeaive sage of op gowh. This inease is due o ineased anopy spoe filaion. As he op gows, fewe and fewe spoes ae deposied o he gound and/o esape fom he field. The inease in he value of R, desibed above, an be aouned fo by a ansfomaion in he ime vaiable. The developmen of plan disease epidemis spead by aibone spoes duing he vegeaive peiod is songly dependen on anopy gowh. The oveall effe of he ineasing anopy filaion effiieny is o delay he nomal iniial exponenial gowh of he epidemi. The magniude of his ime delay is expliily dependen on he op gowh ae and he wind dependen anopy filaion pobabiliy. One of he amifiaions of his ime delay is he synhonizaion of ealy epidemis wih he flush of op gowh, whih is elaively independen of he ime of he iniial inoduion of inoulum. AEDIX The quadaue soluion o he law of mass-aion fo a logisially gowing op. We seek he soluion o he following se of oupled nonlinea equaions: αf R Y( Y (3 d d (5 d The geneal poedue is o fis solve equaion 5 fo ( as an explii funion of ime, whih is hen inseed ino equaion 3. Boh of he above diffeenial equaions have he same geneal fom, ha is dx b ( X( b( X (A d whee b ( and b ( ae abiay funions of ime. We sa by ansfoming equaion A by seing Z( X( suh ha o dx dz b( b( (A d Z d Z Z dz b ( b( Z b( (A d Muliplying boh sides of equaion A by he inegaing fao, IF(, defined by ( b ( b ( d ( exp (A3 IF endes equaion A exa, suh ha d d [ IF( Z( b( IF( (A4 yielding he following quadaue soluion fo equaion A: IF( X( (A5 b ( IF d X ( ( A die ompaison of equaions 5, 6, 7, and wih equaion A yields b W b ( b ( δ f b ( ; b ( ; b ( ; b fo Eq. 5 fo Eq.6 fo Eq.7 ; b ( fo Eq. 7 (A6 Vol. 98, o. 5, 8 5

11 Sine equaions 6, 7, and 7 expliily depend on (, equaion 5 mus be solved fis. Combining equaions A3, A5, and A6 yields he following logisi equaion soluion fo equaion 5: ( ( ( ( o n( exp( (A7 ( exp ( whee n /. Fo his ase, he inegaing fao (equaion A3 is faily simple (IF( exp(. The soluion fo ue mass-aion (α. Fo equaion 6, we noe he podu b.b is a onsan and IF( exp( W (equaion A6. So ha, he inegal in he denominao of equaion A5 an be diely evaluaed: W ( b ( IF( d exp( W exp( [ exp( W W ( { exp[ ( W } ( W [ exp( W W ( d fo W W fo (A8 Inseing equaion A ino equaion A5, we obain he soluion o equaion 6 (equaion A9 in 38: exp( W [ exp( W W ( ( ( ( ( Y e e Y W n - exp( W [ exp( W W exp( W ( Y( Y( ( W - fo W W fo (A9 The soluion o he law of pseudo mass-aion wih onsan α. In ode o wie he soluion o equaion 7 oupled wih equaion 5, we need o evaluae IF( (equaion A3 subje o he value of he podu b. b (equaion A6: ln [ IF( b ( b ( exp( [ IF( exp( n IF( [ exp( ln d αf R ( d ln ( exp ( [ exp( d (A whee (d αf R. Thus, he inegal in he denominao of equaion A5 an be wien as b ( IF d [ n ( n d ( ( exp ( whih an be inegaed, in losed fom, if he aio / is an inege, i: j i i i i! ( ( [ ( [ ( i j b IF d n j n ( ( ( exp ( j i j! j! j (A The soluions o equaion 7 ae obained by inseing equaions A8 and A9 ino equaion A5. Fo he speial ases MA / i and, he soluions ae shown below and he soluions fo MA / i,, 3, and 4 ae ploed in Figue : exp( Y( ; i ( [ exp( Y( [ exp( Y( ; i ( 4 ( [ exp( [ exp( Y( (A In he geneal ase when he aio MA / is no an inege, he soluion an be obained by numeial inegaion. The soluion wih vaiable anopy filaion. Fo equaion 7, he fom of he inegaing fao is a bi moe omplex. The paamee b is given by ( ( e ( ( δ δ e Iψf Rn IψfR n b (A3 ( n δ ( and podu b. b (equaion A6 is given by b b Iψf Iψf R R ( δ ( R [ ( ( ( δ δ e (A4 n δ exp Iψf In ode o wie he soluion o equaion 7 oupled wih equaion 5, we need o evaluae IF( (equaion A3 subje o he value of he podu b.b (equaion A4. The esul is [ e [ IF( Iψf ( δ δ( ln R Iψf R ln ( δ hus : IF( ( δ e δ( ( δ e δ( δ δ d (A5 whee (d Iψf R /( δ. Thus, he inegal in he denominao of equaion A5 an be wien as ( ψ ( δ δ( I fr e e b ( IF d d ( ( δ δ( e δ IψfR [ e ( [ ( δ e δ( d ( δ o b C ( IF( d [ e C [ e C whee ( δ ( δ( e e δ ( δ 3 d ( δ ( δ( ; C ; C3 δ ( δ C (A6 whih an be inegaed in losed fom when he aio / is an inege i: i i ic i ( i! i j C3 C j ( i b 3 3 [ ( IF( d C C C e e (A7 j ( i j! j! i j j The soluions o equaion 7 ae obained by inseing equaions A5 and A7 ino equaion A5. Fo he speial ases / i and, he soluions ae shown below and he soluions fo / i,, 3, and 4 ae ploed in Figue 3: i ; Y ( i ; Y( { } [ Y( C( e C3 ( C e ( e C3 ( ( ( ( ( e Y C e C3 CC 3 C C3 e ( e C3 d (A8 In he geneal ase when he aio / is no an inege, he soluion an be obained by numeial inegaion. Esimaing he ime lag. Le us eexamine equaion A: dividing by he podu b b, we obain b dz b ( b( Z b( (A d dz Z (A9 ( b ( d b ( 5 HYTOATHOLOGY

12 Fo equaions 6, 7, and 7, b (, he oal numbe of hos sies. If we assume a vey low inidene of disease a he sa of he epidemi, hen Y( << ( and hus Z( >> /b (. egleing he seond em on he igh side of equaion A9, and seing Z /Y, we obain d [ ln( Z d whih implies [ ln( Y d[ ln( IF d b ( b( (A d d Y ln Y ( ( ln [ IF( (A sine IF( (equaion A3. Equaion A is valid as long as disease inidene is low (Y <<. Fo he above hee models, equaion A beomes ( ( YW ln W (A YW Y ln Y MA MA ( ( ln [ exp( (A3 Y( ( δexp( δ( ln ln[ C exp( C (A4 Y ( δ ln whee C was defined in equaions A6. Equaion A is linea in ime and equaions A3 and A4 appoah empoal lineaiy asympoially (Fig. 5A afe ime lags, τ (d and τ (d, espeively, given by ln[ n ( τ (A5 and τ ( δ ln δ ln ( C LITERATURE CITED (A6. Andeson, R. M., and May, R. M The invasion, pesisene, and spead of infeious diseases wihin animal and plan ommuniies. hil. Tans. R. So. Lond. B 34: Aylo, D. E The aeobiology of apple sab. lan Dis. 8: Aylo, D. 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