Ecological Archives E A1. Meghan A. Duffy, Spencer R. Hall, Carla E. Cáceres, and Anthony R. Ives.
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1 Ecological Archives E9-95-A1 Meghan A. Duffy, pencer R. Hall, Carla E. Cáceres, and Anhony R. ves. 29. Rapid evoluion, seasonaliy, and he erminaion of parasie epidemics. Ecology 9: Appendix A. upplemenary mehods and resuls. Derivaion of evoluionary effecs on ransmission (Eq. 1c) The equaions we use o model epidemics, repeaed from he main ex, are (A.1a) o e T T (A.1) 1 v, (A.1c) where, and are he densiies of suscepiles and infeceds in he populaion a ime, is he ransmission rae, is he nominal ransmission rae ha is independen of emperaure T, and v is a measure of he clonal geneic variaion in resisance o infecion. Exponens and allow infecion raes o scale nonlinearly wih densiies of suscepile and infeced hoss. Transmission rae can depend on emperaure T, as governed y he exponenial funcion in Eq. A.1; larger values of T resul in larger declines in ransmission wih decreasing emperaure, as compared o is nominal, emperaure independen value,. Here, we derive Eq. A.1c.
2 Alhough no used for fiing daa, we need o specify a general equaion for he dynamics of suscepile Daphnia o derive Eq. A.1c. e assume he dynamics of he suscepile populaion are governed y he equaion f( ), (A.2) where f() is some funcion giving he dynamics of an uninfeced Daphnia populaion, and is he oal populaion densiy ( + ). The per capia finess, individual eween imes and as a funcion of ransmission rae is, for a given f ( ). (A.3) Here, we have used o denoe he ransmission rae of a paricular individual. Assume ha he ransmission rae among individuals is disriued wih mean and addiive geneic variance ( ). e allow he variance o depend on he mean, ecause he ransmission rae canno e negaive; herefore, as he ransmission rae decreases, he variance mus as well. n paricular, we assume ha ( ) is proporional o he mean, ( ) v. Assuming ha he disriuion of ransmission raes is symmerical and he addiive geneic variance is no oo large (wasa 1991, Arams e al. 1993, Arams 21), we can use a common quaniaive geneic ased recursion equaion o represen change in mean phenoype of
3 a rai (here, ransmission rae, ) ha is under selecion. Therefore, change in mean ransmission rae equals he addiive geneic variance of ha rai, ( ), imes he slope of an individual s finess wih respec o is own ransmission rae when evaluaed a he value of he populaion mean, all divided y mean finess ( ), or. Δ (A.4) f we comine Eq. A.2 and Eq. A.3, we find ha mean finess ( ) equals he densiy of suscepile hoss a ime divided y oal hos densiy a he previous lagged ime, or. Using his simplificaion, Eq. A.1, and he assumpion ha geneic variance in ransmission rae ( ) is proporional o he mean, ( ) v,, we can derive a dynamic equaion for ransmission rae. Below, we show he seps of he derivaion for ineresed readers: Δ + (A.5a) (A.5) (A.5c) 1 (A.5d) v (A.5e)
4 e sar wih a very general form of he equaion (Eq. A.5a). Then, we susiue in our derived, quaniaive-geneics ased expression for Δ (from Eq. A.4) o produce Eq. (A.5), which is furher simplified wih he derived expression for mean finess, (yielding Eq. A.5c). Afer a lile i of algera (moving o Eq. A.5d) and susiuion of our assumpion aou geneic variance (yielding Eq. A.5e), he derivaion is complee. oe ha he final equaion provided y his process (Eq. A.5e) differs from Eq. A.1c y he inclusion of he lag in on he righhand side of he equaion, raher han 1. For he saisical analyses, we lagged he changes in ransmission rae y only a single sample raher han 3 samples in order o assure smooh changes in ransmission raes. This will no change he general resuls. Esimaion of wo ypes of R 2 using he Kalman Filer The Kalman filer uses a wo-sep procedure for esimaing he log densiy of infeced individuals. Firs, i akes he esimaed value of, denoed ˆ, along wih he associaed esimae of he ransmission rae, o projec he dynamics 3 samples forward using Eqs. 1, herey giving ˆ p. hen updaes his prediced value y comparing i o he oserved value. This produces he updaed esimae ˆ ha moves closer o o he exen allowed y he measuremen error η. Thus, here are wo esimaes of : ˆ p and ˆ. Because ˆ p predics using only informaion from generaion, he errors ε ˆ p include process errors ε i, and measuremen error η(). Because ˆ predics afer facoring ou measuremen error, he errors ˆ ε ˆ ˆ p include only process error. The wo R 2 values are: (i) predicion R 2 for 1 var ε /var[ l ] (Harvey 1989), and (ii) process error predicion R 2 for ˆ 1 var ˆ ε /var[ ˆ ˆ ].
5 The former is he equivalen of he R 2 for -sep-ahead predicions, and he laer is equivalen o he -sep-ahead predicions afer facoring ou unavoidale measuremen error. Transmission of Meschnikowia Our analysis indicaes ha ransmission is nonlinear in oh and. n Eq. 1a, he exponens and allow for ransmission ha is no direcly proporional o or, respecively. Our esimae for is.7 indicaing ha he densiy of infeced individuals is slighly negaively relaed o he densiy of suscepile hoss when infecion would have occurred, ha is, a ime where is he ime lag required for he incuaion of infecions (9-12 d). Our esimae of.91 indicaes ha he densiy of infeced individuals increases almos linearly wih he densiy of infeced individuals a a lag of. nfeced individuals a ime are unlikely o live o ime, so his suggess individuals who are infeced a ime infec oher (suscepile) hoss a ha ime, and hose hoss hen appear as infeced a ime. Effec of emperaure on ransmission rae n an earlier laoraory sudy, we showed ha emperaure influences ransmission rae (Hall e al. 26). Tha sudy included wo experimens wih four emperaure reamens, and we fi he infecion daa produced from each o an Arrhenius funcion. Because hose Arrheniusfuncion-ased resuls evenually poined o a role for emperaure in disease dynamics, our presen model includes an effec of emperaure on ransmission rae -- u using an exponenial raher han an Arrhenius funcion ( T ; Eq. 1). Our curren esimae of T for he model ha includes emperaure u no evoluion is.17 (Tale A1). ince his exponenial funcion differed from he one used in he original paper (Arrhenius), we re-esimaed T from hose la
6 experimens. More specifically, we fi an exponenial emperaure-ased differenial equaion model using a inomial likelihood funcion (see Hall e al. 26 Appendix B for more mehods). The esimaes for he experimens from Hall e al. (26) are.2 (for all emperaure reamens) and.23 (excluding he 1 C reamen) from experimen 1, and.18 (all emperaure reamens) and.75 (excluding he 1 C reamen) from experimen 2. Thus, here is a remarkale concordance eween our la and field esimaes of his exponen relaing ransmission rae wih emperaure. TERATURE CTED Hall,. R., A. J. Tessier, M. A. Duffy, M. Huener, and C. E. Cáceres. 26. armer does no have o mean sicker: Temperaure and predaors can joinly drive iming of epidemics. Ecology 87: Harvey, A. C The economic analysis of ime series. econd ediion. MT Press, Camridge, Massachuses, UA.
7 TABE A1: Parameer esimaes for he hree models of epidemic dynamics. TR ransmission rae. Full Excluding Excluding Parameer ymol model evoluion (v ) emperaure ( T ) TR exponen, infeced class TR exponen, suscepile class TR exponen, emp. funcion T Geneic (clonal) variance in TR v Baseline TR, Baker 3 1, Baseline TR, Basse 3 2, Baseline TR, Basse 4 3, Baseline TR, Brisol 4 4, Baseline TR, arner 3 5, Process errors, infeced class 2 σ
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