Exact simulation of hybrid stochastic and deterministic models for biochemical systems
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Exact simulation of hybrid stochastic and deterministic models for biochemical systems Aurélien Alfonsi — Eric Cancès — Gabriel Turinici — Barbara Di Ventura — Wilhelm Huisinga. N° 5435 December 2004. N 0249-6399. ISRN INRIA/RR--5435--FR+ENG. Thèmes NUM et BIO. apport de recherche.
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(46) %&. f-gik ³r6²QkÃklv³{dlvgiZÆ·rvg²r0¾®6³ j ∈ {1, .., M } P[Nj (t + dt) − Nj (t) = 1|X(t)] = aj (X(t), t)dt. vl rg2r+rgik¿ikÀ|r,Fij Arvj k¾/rvgik¿i}kIlvlvkl ÆÂkÃkr-rvgikÃjQvr²=r0vk²rv ∼. Tj (t). Nj (t). gI°kWrgiklvj koZÆrg². gj−1 (Exp(1)|t),. Tj (t). drvgikÃZÆ·¾rvgikÃk°}rvk =²rv*¾rvgikÃ|ijQk6¾,j S³kIikl-l0°k={ dX(t). M X. =. ¹Lf-gikk¾®k =Ð5? =Ð5?. νj dNj (t).. mW²rvkorvgY2r N (t) SrlÂrvgik=ijQk-¾/rvjklÂrg2r-kS¶rv R ivk¾®vSj rgik¿i³rv²Qrvjk irvrvj k t ¹ D°}rvkI =2r³S =Ð5?-klvQilr rgikó}ÍYi»rklv³j&/kk2rvS t j=1. j. j. 0. Af (x). = =. lim. s→t+ M X. d E[f (X(s)) |X(t) = x] ds. f (x + νj )aj (x, t) − aj (x, t)f (x).. f-gik*}ÂQSr¡²¾6°=kÆ A= ÕÂgij WjvZ°?okSilÃr´rgikdÆÂk³ AÇ|i2Æ0\gikj²j&SlFrk kI SY2rv ) 0Z + j=1. d P[X(t) = x] dt. =. M ³ X. aj (x − νj , t)P[X(t) − νj = x] −. j=1. ´ aj (x, t)P[X(t) = x] .. giSi³i ¾®S6² x ∈ N ¹ !, # %," / '*#(#)+)-!5 . * % .#)0 Ôi³{dlFr}gl~rvol~ji2r³Sl+QkSjk¿ =i³rvklvZÆCÆ0gik*j&²|{&jk³kIl+&¾ÐSlFr0vkI¶r³Sl+²k =°S³°kI }ikÂrWrgik¾ÐS¶r,rg2r,rgik+kµQS~r"l,vSYS~r²rvorgik+=jQk"¾krvlQkv¾®j kI%¹ ÄWZÆÂk°k ¾®S0kg*rvj k ¼gikId"lvlvivikll N (t) Æ+kgZ°k N. j. E[Nj (t)]. E[(Nj (t) − gj (t|t0 ))2 ] Z t = E[aj (X(s), s)]ds. =. 0. ßß ì.Hÿþ ¶ÿ.
(47) 7. ¾®S. E% " . t > t0. . ¹y|krvgkÃvk²rv°k , rv²rv4Ykr¼Æ+kk E[(Nj (t) − gj (t|t0 ))2 ] E[Nj (t)]. . . 1! $ " 1 . . Nj (t). 1/2. ².
(48). gj (t|t0 ). . . . 1# ! 5. . . l-°k*|{. 1 , E[Nj (t)]1/2. =. ³ rl klvY²i³krÅikS³kI¶r»r7iiIÀ}j&2rvkdrvgik´lFr}gl~rv*}{|²jl ={\»rlSr³|iYl iSrk²vr =?l³ rgik´}krvkvjil~rvj}}k#?¡Æ0gik7rgik4vSYklv»rF{ l²k rgik=ijQkl0g¾"(t|tj)kikl0=°S³°kI*rgikvkIrv4²kir0rv|lvj&²?¹Âf-gil0jarv°22rkl0j»À}k l~rv}gSlFr²}krvkvjil~rvj i}kl¹ ÕÂlvikÂY²vrv³rv²¾%rgikovkIrvl SrvrgiSlvkWj}}kkl~rv}gl~rv³{ =ÐÆ0»rg }kÀ j ∈ S ?6rgiSlvkj i}k³kI}krvkvjRi,lF.r.. ²R³B{ =ÐÆ0»rg}kÀ j ∈ D ?¶¹oÔo°k´j i}k rgikkd²k&²r³kIl~rÃrgivkk}r³Sl¾®SkrvrviOl~g\4Y²vrv³rv =®! ?Ãvi\4¾®i³{Ál~rv}gSlFr k ³2 C²rv²²Y²5{ Ckrvgik´¾®vkI =ik³kIl 92ivSQklv»rkl ²¾kSgvkIrv =®irvk4rg2rrvgikj&2F ji}r²rvY²Sl~rWl6Oji}r³j&²|{*k³2 C²rvl ? =г! ?0Yl~ki³S³S%lv³SgSrI¹0t r¿l~kkj&l kSl~S²ik ¾®LlFr²k rvÃj}}k}Skik0vki2rv{Ã~rl5lFr}gl~rv³{ Æ0gi³k0jkr²QS³+k r³SlÂ}krkj³il~rv²{ =г! ?5¾®ÂkSgvkI¶r³S&gi|=l~kW²}r³°k³{ QkrFÆÂkk&rgik6rFÆ¡iiSgikl lvi±v³rvkv½Sl~kIÁS½rvgik*=ijQk¡¾6jkikl¡½³rl¡ivSQklv»rF{±¾®irv =Ðlvkk4y|k r³S6 7 ?¶¹Lf-gkk°S³irvk =2rd¾®S X(t) ∈ R l-i2Æx°k={rgikÃg={|ilv{}lFrkj X X =A ? ν dN (t) ν a (X(t), t) dt + dX(t) = j. 0. j. 1. M. . . N. j. j∈ D. j. j. j. j∈ S. Æ0³rvgd³i³rvY°2³k ) ∈ R ¹5WikWrrvgiko²vrv³rv²¾%rvgikokrvl lvQkklÂ&Qk³SiÃrv rgik¡l~rvigSlFrlWX(t ÆÂk,Sl0rgik }krkj ilFr²vro²¾5rvgkg|{|ivl~{}l~rvkj¹ Ò lodSl~kI =ikk ³r j&Z{Wrvgikkrv³{WgiQkrg2r,lvjk X (t) Qkjkl/ik=2r³°kS¹ Ì kLirvk5ÍYl~r/rvg²r/rgillv»r2r³S ik°k"}ivkIà i5lv³j²rvl ³r5l"2rvgk"iik+rv¡ilvi³r²³k-j}}k³igiSk = rvgk }krvkjil~rv¿kI SY2rvSl-lvgii*i²r6S¶r0Slvj&²/ SY²Sr»r³kIl ?¹y|k}{ rvgil6l~³rvY2rv4*Qk lv°kd={*²4²}r³°kÃlFr2rvk ikQkikSr6giSk¿¾
(49) rgikkS¶r³SlrQkj}}kklFr}gl~rv³{ S0}krvkj³l~rv³{S¹Ly|g44²iv=gdl0vk=rv{i}k6l~rvi{¹ f-gikg|{|ivlv{|l~rvkj =A ?-SvklvQilÂrv rgikóiÍi³rvklvj/kik2r 0. N +. j. Af (x). = =. d E[f (X(s))|X(t) = x] ds X d aj (x, t) f (x) + dx j∈ D X f (x + νj )aj (x, t) − aj (x, t)f (x). lim. s→t+. f-gikÃ}²%QSr0²¾°|³kÆ = ÕÂg²ij& oj S vZ°1?L³kIilÂrv rvgikÃgikj²Gj&SlFrk-k =2r³S*S ik*rv i°|³k¿r¼{|QkÃk =²rvB=ж¾F¹ ) ; ÂÕ g/¹Li¹ .+ ?¶¹ j∈ S. á ä,ß/á é.
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(52) %&. Ã!" 4 . '!")0 Â, $ *) *!" # .&#*)0 Ò r6³kIl~r+rvgvkki»µGkkSrW²Sv³rvgj ¿²iiSSgikIl¾®6l~ji2rvi rvgkÃl~rv}gl~rvo²vr0³*rgikg|{ i±j}}k =A ?²kZ°2³²³k =Ð# ?¿rvgikÍYl~rÃkS¶rvÅj krvgii> )20 00+ =®#?¿rvgk}³krÃjkrvgi} )20 00+,²B =г# ?+rgikikÀ|r6kS¶rvjkrvg} ) 3+ ¹t¼rvgikþ®³2Æ0i ÆÂkÃÆ0³}lYlvl6²Sv³rvgj k³2 C²rvl+²¾rvgkÃg={|i*j}}k%Sl~kI*kSg²¾rgiklvk¹ f
(53) ³³YlFr²rvkorgikSl~o}k lv}k0 lv³kgkj²/lvYkIkl Qki³|°°kdQ²rg³ krv R j i}k³kIdlFr}gl~rv³{²dkS¶rv R j}}kkdS}krkj³il~rv³{¹Ly|kr+rgik i³rv²"rvjk τ = t rvgik|ijQkþÂjkikl X(τ ) = X (t ) ¹ oZÆÎ*}j =jQk O³i³rv²2 Ck ¹ Ò l¿i4l l~rvigSlFrk°k=r}il ξ ∼ Exp(1) drvgiki{=Y²jl-l6lv³ji{dg(τ°k|t)=={0rgik}krkj ilFrg(τ|t)~r < ξ . 1. 1. 2. 0. 1. 1. 1. 1. dX (τ ) dτ. =. 0. 1. = 75?. ν2 a2 (X(τ ), τ ). =?¶¾F¹,k QA¹ =? ? ?¶¹Ot¼Árgik*il~k²¾-rgik =?}krkj ilFr?k°}r³S gi2ÆÂk°Sk rgikd°2³k²¾ kSl~kIl¾®S³2Æ0³¡rvgk}»µGkk=rv²/k =²rv d g1 (τ |t) dτ. =. g1 (τ |t). =AÏ?. a1 (X(τ ), τ ).. ÄWkk ²³rvgij³{ Æ+k4lv³ji{7l~ji³rikSl~{\lv°kdrgik´l~{}l~rvkj¾Ãu¿ Dl = 7 ? ²Y =AÏ.?¹ f-gikW=ijk²Ylv}r³S²¾/lvg&}²{¡}»µGkk=rv²Yk =2r³SlLl°k{ÆÂk³Q}}ijkSrk =®kS¹ Y¹ )ô 0<7 0ZÏ +!?¹ Ì kWrgikk¾®kWlviiQ=l~k6rvg²rÂÆ+ko²kW²³k6rv ji}rk6rgik¿l~S³irv&¾%rvgikÃu¿EDl i rvÁ={\ikl~k i{½²7kkr rvgik´}lkrv2 C²rv7k¡ Æ0g²r ¾®ZÆ6l¹7oikrv kI Y¹ =Ð ? ÆÂkÇ|i2Æxrg2rWrvgikÍlFrol~rv}gSlFrkS¶rvS}il6}i rvrvgik}Sj3°2v²³k ¹ÕÂlvk =ikSrv{ Æ+k ³=rvkS²rvkrgiklv{}lFrkj ²¾0u¿E Dl = 7 ?¿²> =A.Ï ?oiSr³"r³jk lvg T (t ) rg2r g (s|t) = ξ ¹Of-g=l T (t ) = s ²Árvgik*l~rvigSlFr&kS¶rv½lQkv¾®jk%¹τf-=gikdsk=rvvk i}kI}ik¿l-rvgik4vkQk2rk*iSr³lvQk³Ík*Ͳ%rvjk¹ f-gik+QZ°k+²³rvgijlvgikjk+vkI S³kl/rgik+lvk²¾Y=ijk²=SrkS²rvl%rg2r5³2ÆÁrv¿l~rvS Srk2r³SÆ0gik±lvjk l~ ²³kI4k°kSr¿¾®i¶r³S =®OSi¿lvk τ 7→ g (τ |t) − ξ ?W°2²ilvgikl¹ Ò ¾®kƺ|ijk²Gu¿E D\Srk2rl5³kS}{}kok°k=rÂg²i³i¹5Ä62ÆÂk°k rvgik¿l~QkI²²rvik ¾WSik°kSr¾ ¶r³SjÇkl »r&kIlv{ÅrÅji³kj kSr&rvgik4k°kSr}krvkrv lv³k g (τ |t) − ξ kSl~kIld³ =®³r4lik=2r³°k±Qk¾®kOrvgkÅk°k=rrvj k½²CQ=l~³rv°kOrvgikvkI2¾ rk ?¶¹Øf-g=lSik |ijkv³{*τSrkS²rvkl0rvgk¡l~{}l~rvkj ¾Âu¿E DlWi=rvrvgikÍYl~r6r³jk τ Æ0gk´rvgik¡k°k=rW¾®i¶r³S QkjkloQSlv»r³°Sk Æ0gi³k¡³rllFr³,kS²rv°k¡2rorgik ivk°|³Sl6rvjk g (τ |t) − ξ = δt QkirgikÃrj k¡lFrk ?¹0f-gik &Q{|ij²/SrkYS2rv4QkrFÆÂkk (t , g (τ τ|t) =− ξτ ) −δt ²\Qk*Yl~kI½7}krOk°2³²rvkrvgk*k°kSr¡r³jk s ∈ [τ , τ ] 2r Æ0gig (t , g (τ |t) − ξ ) L ¹ f i g l-v}k}ikol+rvgiklvj kl~kIr Skik²rvk¿}klvk}ri}r )20<7 0Z<Ï + ²*l g(s|t) − ξ = 0 ÆÂkkIlFr²ilvgikI4jkrvg}%¹ Ò ³rvkv²rv°k{ Æ0»rgOlvjk}»r²,Sji}r²rv=lFr &Íik r³jkl~rvk´Si*QkÃlvkdr&lv°korgik¡u¿ D*rgikÃSrvkv°²² [τ , τ ] ¹ 1. 0. 1. 1. 1. 0. 1. 1. 1. 1. +. 1. +. 1. −. −. +. 1. +. 1. −. −. 1. −. ßß ì.Hÿþ ¶ÿ. 1. +. +. 1. +.
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(57) . -f gik}k¶rWjkrgi}4kÀ}i³rv{i²rvkIl-Æ0gg´vkI¶r}il-kÀ|ro²YÆ0gik4³rWiil ) 0I P 0 0+A¹Of-gikdvkI¶rÅrvjkdlS³°k½={±rvgkFij\¾-rvgik*,ll~SÁi}kIlvl Æ0³rvgÅSrklv»rF{O°k±={´rgik&iji2rv°k rvjk rv²l~¾®j&2rvS g (τ |t) =NP(t) =g (τ |t) ¹ Nf-g|(t)l rgik²Sv³rvgijovkI² CI2rvSd²¾rgik}vkI¶r0g|{=v*jkrgi}*l6l+¾®³2Æ6l 0¹6y|kr6³»rGrj k t = t ²Y*³»r%|ijQkl-¾,jk³kIl X(t ) }¹ 4 kik2rk ²ijØ°2²²ik ξ ∼ Exp(1) i¹6y|kr g (t|t) = 0 lv°korgikl~{}l~rvkjز¾u¿ Dl6l~r~r³2r0rvj k τ = t σ. σ. j. j∈S. j. j∈S. . 0. 0. . σ. dX (τ ) dτ. =. dgσ (τ |t) dτ. =. X. νj aj (X(τ ), τ ). = ;5?. X. aj (X(τ ), τ ). = 35?. j∈D. i=rv%r³jk τ = s l~g*rg2r g (s|t) = ξ ¹ 4 kik2rk6}lvvkrvkL²Y}jx°²²i³k5Æ0³rg°2²ikIl/³ S ²ÃivS²³»rkl (a (X(s), s)) Sik+rv}krkj³ikrvgikÃkS¶r³S R rv&QkQk~¾®Svjk }¹0nWGi2rk X(s) SS}irvdvkI¶r³S R gikk¡l~kr X(s) ← X(s) + ν Glvkr t ← s ²Y S¡rvdy=rvk}¹ j∈S. . σ. j. . j∈S. m. . m. "!. m. $# %&('*) +)
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(61) %. t¼\rvgikdÍlFr¡kS¶r³SÁjkrgi} ´}r²rv°kdkS¶rvÁr³jk*lkik2rkž®¡kIg½kS¶rv "rgik kS¶rvvkIl~Qi³iÅrv½rvgikOlvj&²³kIlFr&r³jkldgSlvkrv\}i rgikOl~r2rk´°kI¶rvS X(t) l SS}iS³{ÁiGi2rk²\rgik4i}kIlvl¡kQk²rvkI )20 0 0+A¹f-gik4ikÀ|r&krv7jkrgi}7l k&³kSr² kIiSj =®¾®vSjrgik6YSSrL¾Q°|kƲ¾Glvk0²¾G}j °2v²ikl ?
(62) °²²Sr5²¾Qrvgk-Íl~r kS¶rv jkrvgi}%¹"t rl5Sl~kI klvk0²¾G²}Sj °2²²ikl5²Y }r³j2 CkIi²rÃl~rv¶rikl ) 3+ ¹ f-gil¿klvk ²³2Æ6lolorv4l²jik¡³{´ik ²}SjÓ°2²²ik¡2rkSgO³rk2rv> =гl~rvkIO²¾r¼Æ+ ³Çk±rgik&}k¶rÃjkrvgi} ?¶¹&f-gik&³rvgijk²2 C²rvlo¾rgik ÍYl~r²Orvgik&ikÀ=rvkIrv g|{=v*jkrvg}il6²k¿°k{dl~j³² rvg=Yl0ÆÂkÃl~r²rvk¿rgikj 4jr-¾®j 0¹6y|kr6³»rGrj k t = t ²Y*³»r%|ijQkl-¾,jk³kIl X(t ) }¹ 4 kik2rk³Y}kQk}k=r²}Sj °2v²³kIl (ξ ) ik¾®¿kSgOvkIrv R S Æ0³rvg . . 0. ξj ∼ Exp(1). i¹6y|kr. . gj (t|t) = 0. 0. j j∈S. ¾®. j∈S. j. . á ä,ß/á é.
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(65) %&. 3. m6Y¹ v kI¶r³S iQklv³r¼{ 2rk=5iZ{ ? k −−→ rkj k R a =c · c = 0.025 v r k j v r k j R a =c · c = 0.25 rvkj −−−−→→ ∅rvkjdÖ k rvkj R a =c · c = 1.0 °|vl k G ¿ Ö F l r v k ~ l v r R c = 7.5 · 10 rvkj −−→ rvk−jd−→Ö¿lFrv aa == cc ·· rvkj · R c = 1000 lFrv −−→ ∅ lFrv R a =c · c = 1.99 ð !Zïô ë ýû IëFï=ë
(66) í2ðö?ò ¶ìIó ¹6y|S³°k¿rgikl~{}l~rvkjؾLu¿ Dl6l~r~r³2r0rvjk τ = t. l~r²rvkÃgik = 0 0 5 ? ν = 0 5 ? ν =Ð 0 5 ? ν =? 0 0<? ν =? 0<? ν =? 0? ν. −1. c1. 1 2. c2 c3. 3 4. c4. c5. 5. c6. 6. 1. 1. 1. 2. 2. 2. 3. 3. 3. 4. 4. 4. 5. 5. 5. 6. 6. 6. dX (τ ) dτ. =. dgj (τ |t) dτ. =. X. 1. 2. 3. −6. 4. 5. 6. = 05?. νj aj (X(τ ), τ ). j∈D. aj (X(τ ), τ );. = 0 0<?. j∈S. i =rv%r³jk τ = s l~g*rg2r0¾®0rvgikÍlFr0rvjk g (s|t) = ξ ¾®6lvjk m ∈ S }¹0nWGi2rk X(s) }irkrv R gikkÃlvkr X(s) ← X(s) + ν 7i¹6y|kr t ← s =Ð! ?*Ô,l~r0kS¶rv*g|{|iv*jkrgi} 5S¡ry=rvk|¹ =®# ?mWkÀ|rWkS¶r³Sg|{=v4j krvgii +ÔWvkS¶r³S R Skik²rvkkÆ×}Sj3°2v²³k ²Ydlvkr Æ0gi³kÇkk³i²G²rgik-°2²ikIl ¾®S ξ ∼ Exp(1) =0 Sl-³i³r%°²²ikl+¾®-rgigklv(t|t) {|l~rvkjØ ¾Lu¿ DÂl = 0 ?-² = 00<? iS¡ry=rkg4(t|t) ¹ j 6= m Ôi²O»rk2r³°k¡²Sv³rvgijÃvkI²2 C2r³S4¾5rvgk¡g={|iikÀ|r¿krv´jkrvgi}Sl~kI4S kl²i¡¾rvgikòY}jØ°2v²ikl ξ Æ+kÃlv³ji{dvkiSky=rvk6 7 |{ 7 =г! ?+ÔiS+krv R Skik²rvk¿ikÆ·}j °²²ik ξ ∼ Exp(1) |¾®SÂrgik¿kj&³ii kS¶rvYl R Æ0³rvg j ∈ S ² j 6= m kl²k ξ S}³dr ξ ← ξ − g (s|t) ¹*y|kr ²¡ry=rk´i¹ t←s ) .#)0 4 .)-!" ½ /*# )0 32Ã!Ø Ì k*i2Æ1Æ0lvg\rvÅ}kjSlFr²rvkrgik*Q2ÆÂk¡²¾W¡g={|i½jkrgi}\Yl~iÅj}}k-}k°k½|{ y|°2l~rZ°24kr ?¹ )ô. 3+0}kli³´rvgik*³=rvk³i²ZÆ-rgÁ¾0Y¶rk³Sig²SkfWÏ|¹ Ì k*g|Slvk rgilj i}k5lv³k¡³rókI²³{´lvgiZÆ6lWrgik}³µGkkk¡Qkr¼ÆÂkkÅ}krvkj³l~rv ²±lFr}gl~rvj } k³³B =ÐÔ,ik 0?¿rvg|lóZÆ6lorv4gikIÇÆ0gikrgikrvgikg|{|ivOjkrgi}Olòik¡rv´l~QkkIOi lv³j²rvÆ0³rvgiS}rWjijl~i¡rvgikÃklvi³rl¹ . m. m. . m. m. m. m. m. j. . j. . m. j. ßß ì.Hÿþ ¶ÿ. m. j. j. j. j.
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(69) =?fi³k 0<?¶¹t¼Árvgikdlvk =ik ÆÂk&Æ0L¾®}l Sdrgik2Æ·³i¾®k¶r³Sk°k/SvklvQ}irv rvgki³rv²/=ijQkl-¾jS³kIikl+rvkj = 1 k = l~rv = 0 ¹ number of tem molecules. 20. fully stochastic hybrid S={1,2,3,4} hybrid S={1,2,3} hybrid S={1,2} hybrid S={1,3} deterministic. 15. 10. 5. 0 0. 50. 100. 150. 200. time (days). iû ò ¶IíZëFù?ïñë ó Y!ZðÝ%óA ¶ë
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(72) ú !Zù?ò Ò lÆ+Sllvgi2Æ0Á ) 3+ rvgkd}krvkj³l~rv&j}}k²¾-rvgik*²Q2°k&lv{}lFrkj'Q=lvlvkll~kIl¿rFÆÂOlFr r³S²{OQSrl = 0<?rvgikdQSrrvkj = k = lFr = 0 Æ0gig½lil~r²³:k =®rvgik*lv{}lFrkj'Æ0 j2°k ZÆ+Z{¾®j »r²¾ rvkl~j&²"Qkvriv²rvl ?W> =? ?6rgikQSrrvkj = 20 k = 200 ² l~rv = 10000 Æ0gigOl¿lFr²ik = rgikl~{}l~rvkjÓÆ0³"vkrvi´r*»rò¾ rvklvj&²,Qkvrvi2r³Sl ?6² ²r~rvrv°k =®rvgiklv{}lFrkjØrvkil-rv&vkIgdrvgil6l~r2rk »¾"QSllv³i<k ?¹ Ò l²&²³{}lvl"²¾Qrgik6ivSQklv³rvkl =Ðk¹ ¹ |{iiiiÃlv³iS³k0vkI²2 C2r³S ?vk°k²l kS¶r³Sl g²iQkjgOjkþ®vkI =ik=rv{*rvgY²´rgikrvgikl l~kIlFr³rvgY2roÆ+kj i}k
(73) rgikj R }krvkjiRl~rv³{ Æ0gi³k¡rvk2r³rgik vkj³³i*ikIl¿lFr}gYl~rv²{¹Ô¿ifWÏ&j}}k rvgil ²k°kQkj&ikjk¿iklvk 5f-gikÃiQkYl~³rvkl02r0rvgikl~r²³kÃl~rvkI}{*lFr2rvk²k = 0I ? a =5 a =5 a = 20 a = 15 a = 20000 a = 19900 Æ0gig*³kI²³{&l~gZÆ·¡l~k²2r³S&¾/rvjkl²koYkr¼Æ+kkdrgikor¼Æ+ lvkrl¾
(74) vkI¶rl¹,f-gik¿klvi»rl ¾®S 10 vkI²2 C2r³Sl¿²¾+rvgikvkIl~QY}³ig={|i±j}}k²klvgi2Æ0ų½Ô,ik}¹ Ò là )ô. 3+ ÆÂk&}lviI{vkIl~i³rl¿k2rk´r*rvgk=ijQkò¾rkj jS³kIikl¹t rò±QklvkkOrvg²r¿rgik&}lFrv i}rlÃS}r³ikI±Æ0³rvgÅrvgk&g={|iÅj}}kvk&²jSl~róil~rvilvg²³k¾®vSj rgiSlvkS}r³k Æ0³rvgrvgikW¾®i³{lFr}gYl~rv0j i}kA¹"f-gik¿}°²²Sr²Sk0²¾%rvgkWg|{=v&j i}kYlÂSl~}ki³kWlI°|i ¾ÂÕÂn rvj k¡|ijk²,l~ji2r³Sl¹oÔiS6rgik¡l²Çk²¾jvlv ÆÂk¡gI°k¡lv³ji2rk4rgik ¾®i{½l~rvigSlFrdÁrgik*g|{|iv½j}iklSl~kI½7k³rvgik¡rvgik4}³kr¡jkrvgi} )20 00+-rgik ikÀ|r vkI¶r½jkrgi} ) 3+A¹½f-gik*Yl~k°k ÕÂn3rvjkl vkdkQS~rkÁ f,²ik}¹Át r²Qk²l rg2rrvgikg={|iÅlvji²rvlvk&YS}r 100 rvjkll¾Ðl~rSl¿rgik¾®³{Ål~rv}gl~rv Sikl¹df-gik g|{=v4l~ji2rv4l0Sl~kI*S4&0ik }rvrSrk2r0²¾"}k 4 Æ0³rvgYlFr²Sr0r³jkl~rvk . . 104. 5. 6. 1. 2. 3. 4. 5. 6. 4. á ä,ß/á é.
(75)
(76) !"# $
(77) %&. 00. 25 days post-infection. 100 days post-infection hybrid S={1,2,3,4} fully stochastic. hybrid S={1,2,3,4} fully stochastic. 0.3. 0.5. relative frequencies. relative frequencies. 0.6. 0.4 0.3 0.2. 0.2. 0.1. 0.1 0. 0. 10. 20. 30. 0. 40. number of tem molecules 50 days post-infection. 20. 30. hybrid S={1,2,3,4} fully stochastic. 0.3 0.2 0.1. 0. 10. 20. 30. 0.2. 0.1. 0. 40. 0. number of tem molecules 75 days post-infection. 10. 20. 30. hybrid S={1,2,3,4} fully stochastic. relative frequencies. relative frequencies. hybrid S={1,2,3,4} fully stochastic. 0.2. 0.1. 0. 10. 20. 40. number of tem molecules 200 days post-infection. 0.3. 0. 40. number of tem molecules 150 days post-infection. 0.3. relative frequencies. relative frequencies. 10. hybrid S={1,2,3,4} fully stochastic. 0.4. 0. 0. 30. 40. number of tem molecules. 0.2. 0.1. 0. 0. 10. 20. 30. 40. number of tem molecules. Aöòôiìòôòõ~³¶ðë~íZõ¼ïôï ù?ö?ë$òã ¶ìùÐë~ðZõ¼òôóÐöAöAò ¶!ZùÐòìZùÐ!Zòó íIö?òò ¶ôìZìã ë¼ ö?òôiõFö?óû ëFû. ¶Zù%ûëFöAïôúIïô ¶ë
(78) ë ïô ëF!2õFIíZðëFïôõ¼ö?ëFöAëFó ëFù?ù?û0 ò!2 ¶òôðüZìIó?ú2òôë
(79) ó?ð ö?¶òôõvë ¶ðì ïôï ýÂû õ
(80) ¶ +ùÐû0ëvZðëFü2ïôï òð~ù?ðùÐë
(81) ö?ë~0òð ¶õ¼ö ìZöA Ló ò ¶ö? ìZúZó ë,ùÐë
(82) ³ëFã ù?ëFìZõFã ë³íZïôðï +ì óÐö õ ú2ðû óÐöA +òôZõ,ëFû ïôïôë WIë~ó?ïö| õAü ú2 ¶ðóÐó?ö?çç R1 R2 R3. R4. R5 R6. 104. iZ{ )ô 0ZÏ 07+ ¹½t¼7S}krv±°kv³¾®{Årgik4iS{Ų¾6rgikk°k=rikrvkI¶r³S ÆÂkkÀ i³r6rvgikkÀ}l~rvkk²¾5´²Y²{Sr²
(83) l~S³irv¾®Wg={|i4k =2r³Sl-¾® S = {1, 2, 3, 4} ¹-ÄWkk lv°|³i¡rvgikk°kSr0kI SY2rvYl = 3 ? = 00?-²*QkÃSj ³l~gk|{d¾®kƸm6kÆ-rvS*³rvk²rvl ) 7+ ¹ f-gikklvi»rl+S}r³ikI*²ko°Skv{dl~j³²0lviSkl~rvirvgY2r-rvgkk°kSr6}krkrvl-Qk~¾®SvjkdÆ0³rvg lv kSrWSiS{S¹. δt = 0.01. ßß ì.Hÿþ ¶ÿ.
(84) 0Z. E% " . . . . 1! $ " 1 . .
(85). p´}}k Wk¶r6jkrgi} m6kÀ=r6kS¶rvSdjkrvgi} Ôi³{*l~rv}gl~rv 0I7¡l 0I²Sl Ä6{|i*j i}k 0;S¡l 0l S = {1, 2, 3, 4} õ
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