SUPPLEMENTARY INFORMATION FOR IMPERFECT VACCINES AND THE EVOLUTION OF PATHOGEN VIRULENCE
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1 SUPPLEMETARY FORMATO FOR MPERFECT ACCES AD THE EOLUTO OF PATHOGE RULECE
2 PARASTE RULECE EOLUTO A HETEROGEEOUS HOST POPULATO. Deivion o evoluionil sble viulene Te nmis o esien n mun psie (e seisk ees o e mun sin) ompeing in eeogeneous os populion wi wo pes o oss (e pime ees o esisn oss) is given b (see min e n Fig. o eils): / / wi. Te ieion o evoluion n ulimel e evoluionil sble (ES) viulene n be eive om e nlsis o e e o e mun. Wen e mun is e we n negle is ee on e esien nmis n ous on is own nmis wi in mi om is given b : / / A wi A wee n. Tee e sevel ws o in e ES psie viulene om is moel. One is o mimise e ominn eigenvlue o A wi is e sn mesue o iness o e mun (e e o invsion wen e esien populion s ee n equilibium). is known in populions emogpi equilibium oe esuls n lso be obine b mimising e bsi epouive io o e mun seg given b equion 7 in e e.
3 3. Repouive vlues n e seng o seleion in ieen oss Tese mimisions mus be ie ou numeill. As bo ek on e numeil lulions n s mens o gining insig ino ei esuls ue ppo is useul. Mimising e ominn eigenvlue o A is equivlen o mimising e ollowing union: w[ α α]= v A u wee v n u e e le n ig eigenveos o A (wi is e sme s A bu wi e esien's pmees): T v n u. Tese wo eigenveos e espeivel e iniviul epouive vlues n e equenies o e wo lsses o esien psie ineions (i.e. e ineions o susepible o esisn oss). Te pou o ese wo quniies gives e lss epouive vlues 3 o e wo pes o psies. Tis nlsis o epouive vlues is piull insigul sine i povies e ppopie iness weigs ssoie wi e seleive oes ing in e ieen pes o oss. Fo insne e non-monooni ee o e ei o e ni-gow e vine on e ES viulene (Fig. ) n be epline wi e elp o lss epouive vlues o psies ineing ieen pes o oss. oe e epouive vlue o psies in vine oss is popoionl o. Wen is low iniviul epouive vlues e ve simil n e ieion o evoluion e sle o e wole os populion (wi bo susepible n vine oss) is minl govene b e elive equenies o e wo lsses o psie ineions. n is siuion sine n inese in vine ei seles o ige viulene in vine oss (equions n 5) is iels n inese in e ES viulene. Howeve wen is ig e epouive vlue o psies ineing vine oss n be ve low (wen 0 ). Tis mens o ve eiien vines psies ineing vine oss o no onibue o e uue o e psie populion. Even oug seleion o inese viulene is ve song in ose oss e
4 seleive pessues ing in susepible oss ive e evoluion o e psie. Tis eplins e op in ES viulene s.
5 MALARA MODEL Hee we esibe () e epiemiologil moel o mli une vinion llowing o nul immuni n veo nsmission () e evoluion nmis o viulene mun n (3) e eivion o e evoluionil sble viulene.. Epiemiolog We ssume ee e ee pes o oss - nïve null immune n vine enoe b subsips n espeivel - wi e inee b single (esien) sin o psie. Hoss slowl quie nul immuni oug epee ineions n o no lose is immuni one quie. Tis nul immuni ompises ll ou pes n is impee i.e. less n 00% eeive. inion wi single vine pe is ssume o immeiel one e sme level o immuni s nul immuni bu onl o e pe o immuni simule b e vine. We ue ssume e nmis o e ineion poesses o e veo ppen on elivel s ime sle so e veo nmis e pue in e equilibium ion o inee mosquioes 5. Finll we ssume onsn os populion size ieling e ollowing nmil equions wi vibles n pmees esibe below: oe s in equion 6 in e min e ee e no supeineion ems beuse e nel ou... ibles : popoions o uninee oss : popoions o inee oss : psie viulene wee ) )( ( n ) )( ( 5
6 6 : psie nsmission pobbiliies (ineivi om inee umns o uninee mosquioes) wee ] (see below) [ n ) ) ( 3 ( 3 : oes o ineion wee bmz ( ) n ( ) ẑ : equilibium popoion o ineious mosquioes given b z e : gow/immigion e o e os populion given b.. Pmees (ssume vlues in peneses bse on enemi mli in ig nsmission e 6-0 wi es given on n nnul bsis) : esisne o null immune oss (0.8) 3 3 : vine ei o ieen pes o vines (0.8). We noe is is subsnill ige n mos uen nie vines. : eove es wee 7. : vinion ovege (0 o ) : ion o e eoveing iniviuls beome null immune (0.03) : nul os e e (0.0) : biing e on umns b single mosquio (0) b : ineivi o inee mosquioes (0.) m : numbe o emle mosquioes pe umn os (5) : moli e o e mosquio (inee o no) (50) : len peio in e mosquio (0.0)
7 : susepibili o supeineion elive o n uninee os (0.). Compeiive suppession n ou in mli n supeineion evens ve been iel obseve in e iel 3.. nvsion nmis o mun Te invsion nmis o viulen mun n be esibe b e ollowing ssem o ieenil equions: wi bm z bm z As we ssume e nmis o e ineion poesses o e veo ppen on elivel s ime sle e equilibium ions o inee mosquioes wi e esien sin o wi e mun sin e: e z e z 7
8 Te bove ssem o equions n be use o ollow e invsion o e psie populion b viulene mun e e s o vinion mpign (Fig. 5). 3. Deivion o evoluionil sble viulene As in e simple moel pesene elie e long-em evoluion ouome o e psie populion (e evoluionil sble viulene) n be eive om e nlsis o e invsion o e mun wen e esien seg seles n epiemiologil equilibium. Te nmis o e mun sin ppeing in populion o e esien psie n be pu in mi om: A wi A n wee e inies e equilibium ensiies o esien oss. e bm is e oe o ineion o e mun on nïve oss (wi e bm ) n n e e oes o ineion o e mun on immune n vine oss. oe we n negle e ems in in e enomino o ẑ beuse e mun is ssume o be e. Anlsis poees s wi e genel moel ieling e ollowing epession o e iness o e viulene mun: 8
9 9 R 0. We pesen numeil emple (Fig. 3) wi onl e-o beween viulene n nsmission given b: e Howeve ing e-o beween viulene n eove oes no quliivel e ou onlusions. Te oie o e piul spe o e e-o we use ws bse on e ssumpion obseve levels o mli moli in non-immune iniviul 5 in enemi es 5 e e psie s ES viulene ( ). Tee e oweve mn ieen e-o unions wi m iel elisi vlues o e ES viulene. Ou moel iels e ollowing vlues o oe elevn vibles beoe vinion: n e ion o inee oss ie ue o mli s.% n 0.0% in nïve n null immune oss espeivel.. Gnon S. vn Blen M. & Jnsen. A. A. Te evoluion o psie viulene n os esisne. Am.. (Submie).. Cswell H. Mi populion moels: onsuion nlsis n inepeion (Sinue Suneln 00). 3. Tlo P. D. Allele-equen nge in lss-suue populion. Am (990).. Aneson R. M. & M R. M. neious Diseses o Humns (Oo Univesi Pess Oo 99). 5. M R. M. & Aneson R. M. Populion biolog o ineious iseses: P. ue (979).
10 0 6. Diez K. Molineu L. & Toms A. in Te Gki Poje. Rese on e Epiemiolog n Conol o Mli in e Sun Svnn o Wes Ai (es Molineu L. & Gmii G.) 6-89 (Wol Hel Ognizion Genev 980). 7. Aon J. L. & M R. M. in Te Populion Dnmis o neious Diseses: Teo n Appliions (Cpmn n Hll Lonon 98). 8. eelmn J. noulion e n eove es in e mli moel o Diez Molineu n Toms. M. Biosi (98). 9. Suine C. J. Hllon M. E. & Spielmn A. Moeling mli vines : new uses o ol ies. M. Biosi (989). 0. H S.. Roges D. J. Toome J. F. & Snow R. W. Annul Plsmoium lipum enomologil inoulion es (ER) oss Ai: lieue suve inene ess n eview. Tns. R. So. Top. Me. Hg (000).. Re A. F. & Tlo L.H. Te eolog o geneill ivese ineions Siene (00).. Meeeu-Puijlon O. Revisiing os/psie ineions: moleul nlsis o psies ollee uing longiuinl n oss-seionl suves in umns Psie mmunol (996). 3. Smi T. Felge. Tnne M. & Bek H.-P. Te epiemiolog o muliple Plsmoium lipum ineions -. Pemuniion in Plsmoium lipum ineion: insigs om e epiemiolog o muliple ineions. Tns. R. So. Top. Me. Hg. 93 S59-S6 (999).. Snow R. W. Cig M. Deimnn U. & Ms K. Esiming moli mobii n isbili ue o mli mong Ai s non-pegnn populion. Bull. WHO (999). 5. Alles H. K. Menis K.. & Ce R. Mli moli es in Sou Asi n in Ai: impliions o mli onol. Psiol. To (998).
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