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Exact simulation of hybrid stochastic and deterministic models for biochemical systems

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(1)Exact simulation of hybrid stochastic and deterministic models for biochemical systems Aurélien Alfonsi, Eric Cancès, Gabriel Turinici, Barbara Di Ventura, Wilhelm Huisinga. To cite this version: Aurélien Alfonsi, Eric Cancès, Gabriel Turinici, Barbara Di Ventura, Wilhelm Huisinga. Exact simulation of hybrid stochastic and deterministic models for biochemical systems. [Research Report] RR-5435, INRIA. 2004, pp.20. �inria-00070572�. HAL Id: inria-00070572 https://hal.inria.fr/inria-00070572 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(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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(30) „Šl~r‡’Œ‘k €lvk•&’Œll~‰S€iƒ–‘k0‰²¾%”igik‰²rF{=”„Š‘0°²’²ƒ‡„Š’Œˆi„³‹Š„³r¼{ Æ0g„Š‘–g&„†l5¾Ð’I°Œ‰Sƒ‡’Œˆi‹³k0„³&k°Œ‰Œ‹Š€}rv„Š‰ŒY’²ƒ‡{Ãrvkƒvj&/l )ô1ž 0 žSÏ + ¹,t¼ r‡gik+”Y’Œl~r,¾®kÆ7{ŒkI’²ƒ–l r‡gikƒ‡k¾®‰Sƒvk ’oŒƒ‡k’2r,|€ijˆQkƒ5‰²¾QlFr‡‰}‘–g’Œl~rv„†‘j‰}•}k‹Šl,g’Z°Œk-’²”i”Qk’ŒƒvkI•Ãrv‰‘‰Sƒvƒ‡k‘rv‹Š{ •}kI’²‹WÆ0„»r‡gºkÀ|rvƒ‡kjk‹³{ ‹³‰2Æ =€ijˆQkƒd‰Œ¾j‰Œ‹Šk‘€i‹Škl’Œ•‹†’²ƒ‡Œk , €‘rv€’²rv„Š‰Œl„Šƒ‡k’S‘¶r‡„³‰SÇ|„³ikrv„†‘l )20 0I™ .ž 3 › 0+ ¹4f-gikd•}{|’Œj„Š‘lɌ¾0’´l~rv‰i‘‡g’SlFr‡„Š‘&lv{}lFr‡kj'„†l•}kl‡‘ƒ‡„³ˆQkI•Åˆ={Or‡gikd‘–gikj „†‘’Œ‹Lj&’SlFr‡kƒ kI S€Y’2rv„Š‰Œ Æ0gi„†‘‡g‰Si‹³{¡lvk‹†•}‰Œj ”Q‰Sl‡l~kIl"’Œ’²‹Š{=rv„†‘’²‹il~‰S‹³€irv„Š‰Œl¹,Ôi‰Sƒ~r‡€i’2r‡k‹Š{ 54 „³‹Š‹³kIl~”„³k ˆ’Œ‘–Ç„Š6 03|Ï 7 •}k°|„Šlvk•¿rFƉ6kÀi’Œ‘¶r’Œ‹³S‰Œƒ‡„»r‡gij&lGrv‰W=€ijkƒ‡„†‘’²‹Š‹Š{¿l~„Šj€i‹†’2r‡k5r‡gikÂlFr‡‰}‘–g’Œl~rv„†‘,r‡„³jkk°Œ‰Œ‹Š€}rv„Š‰Œɲ¾‘‰Œ€”i‹³kI• ‘–gikj„†‘’²‹}ƒ‡k’S‘¶rv„Š‰ŒYl Æ0gi„†‘‡g’Œƒvk+k =€i„Š°2’Œ‹³kSr"rv‰Ãl~‰S‹³°|„ŠiWr‡gik0‘–gikj„†‘’Œ‹|j&’SlFr‡kƒ"kI S€Y’2rv„Š‰Œ )20Ÿ 00 ›Œ› + ¹ Ò ‹»r‡gi‰Œ€iSg4’¡j‰Œƒ‡k¿ k ‘„Šk=r0kÀ}’S‘¶r0j krvgi‰i•*g’Œl-ˆQkk”ƒv‰S”Q‰Slvk•ˆ={ „Šˆl~‰S’Œ•*s+ƒ‡€‘–Ç ˆ’Œlvk•d‰S ƒ‡k€lvk+‰Œ¾Qƒ‡’Œ•}‰Sj |€ijˆQkƒ–l5’Œ•¡„ŠSrvk‹³‹Š„ŠŒk=r•i’2r–’lFr‡ƒv€‘rv€iƒ‡k8l ) 3+ ”i€iƒ‡4 k‹Š{l~r‡‰}‘‡g’SlFr‡„†‘+l~„Šj€i‹†’2r‡„³‰S’²ƒ‡k °Œkƒv{‘‰Sj ”€}r‡’²rv„Š‰Œ’Œ‹³‹Š{¡kÀ|”Qklv„³°Œk Æ0gik&r‡gikƒ‡ko’²ƒ‡k6j&’²={ƒvkI’Œ‘¶r‡„Š‰Œl lv‰ŒjkWj ‰S‹³kI‘€i‹†’²ƒÂlv”Qk‘„³kIl’Œƒvk ”iƒ‡klvk=r„³ƒ‡k‹†’2rv„Š°ŒkL‹†’²ƒ‡Œk’Œj ‰S€iSr–l’²• 9Z‰Œƒ

(31) ƒ‡k’Œ‘rv„Š‰Œ֒2r‡kl’²ƒ‡kLgi„ŠŒg/¹5uoi‹Š{¿ƒ‡k‘kSr‡‹³{ j‰}•}„³ÍY‘’²rv„Š‰Œl r‡‰Wrvgik+‰Œƒ‡„³S„³Y’²‹S‘–gikj„†‘’Œ‹=j&’Œl~rvkƒ,k =€’²rv„Š‰Œg’Z°ŒkˆQkk”iƒ‡‰Œ”Q‰=l~kI•¿r‡‰6¾®€ƒ~r‡gikƒ"lv”QkkI•€i”¡lv„³j€‹Š’²rv„Š‰Œl¹ f-gikIl~kń³|°Œ‰Œ‹Š°ŒkOk„»r‡gikƒ´’Z°Œkƒ–’²Œ„Ši\‰2°Œkƒd¾Ð’SlFr4ƒ‡k’Œ‘rv„Š‰Œ:l )20<;+ ’Œ”i”i‹Š„Š‘’2r‡„³‰SC‰Œ¾¡ =€’Œlv„ ¼l~rvk’S•}{  lFr–’2rvk r‡gik‰Sƒv*{ )ô.ž 7+ ‰ŒƒŒƒ‡‰Œ€i”„³idr‡‰ŒŒkrvgikƒ¿ƒ‡k’S‘¶rv„Š‰ŒYl6r‡g’2rÉ}‘‘€iƒ„³±¾Ð’Œl~rÃlv€‘‘kl‡l~„Š‰Œ> =®rvgik&lv‰ ¼‘’Œ‹³‹Šk•´r‡’Œ€ ‹Šk’Œ”i„³´jkr‡gi‰}•il ?) › š 0š .ž ;+ ¹ Ò i‰²r‡gikƒ lFr‡ƒ‡’²rvkS{O„†lÃrv‰Oj ‰i•}k‹rvgi‰=l~k”iƒ‡‰}‘kIlvlvklrvg’²rk„»r‡gikƒ „Š=°Œ‰S‹³°Œk‹†’²ƒ‡Œk=€jˆQkƒ"‰²¾Y”’²ƒvrv„†‘‹Škl‰Œƒ,g’Z°Œk¾Ð’Œl~r"ƒ‡’²rvkl „Š ’o•}kr‡kƒ‡j„³i„†l~rv„†‘Æ-’I{ njkk”„³i¿l~r‡‰}‘‡g’SlFr‡„†‘ r‡gikƒ‡kj&’²„Ši„Ši&‰ŒikIlA¹ @5kƒ‡{*ƒvkI‘k=rv‹Š{ rFƉd’²‹ŠŒ‰Sƒv„³rvgjl-rv‰dlv„³j€i‹†’2r‡kˆi„³‰i‘‡gikj„Š‘’²‹

(32) lv{}lFr‡kj&l0„Šlv€‘–g g|{=ˆƒv„†•´¾®ƒ‡’ŒjkƉSƒvÇg’Z°Œk¡ˆQkkO”iƒ‡‰Œ”Q‰=l~kIB• )ôž²Ÿ ›S<ž + ¹f-gik{’²ƒ‡kˆ’Sl~k•‰S±’d”iƒ‡k•}„†‘¶r‡„³‰SO‘‰Œƒ‡ƒvkI‘¶r‡„³‰S rF{|”Qkgik€iƒv„†l~rv„†‘l0¾®‰Œƒ6rvgkƒ‡k’Œ‹³2„ C’²rv„Š‰Œ‰Œ¾,r‡gik¡lFr‡‰}‘–g’Œl~rv„†‘”’²ƒvrI¹+t¼´ˆQ‰Œrvg‘’Œlvkl r‡gikj&’Œ„³´„†•}k’&„†l0rv‰ ́ƒ–l~ro”iƒ‡k•}„†‘¶rorvgikr‡„³jk¡„ŠOÆ0gi„Š‘–g±’dlFr‡‰}‘–g’Œl~rv„†‘Ãk°Œk=r¿lvgi‰S€i‹Š•´‰i‘‘€iƒ¿’Œ•4rvgkOk°Œ‰S‹³°Œkrvgikl~{}l~rvkj ‰Œ¾,‰Œƒ–•}„³Y’²ƒ‡{&•i„»µGkƒ‡k=rv„†’²‹/k =€’²rv„Š‰Œl¹ Ò rWlv”Qk‘„³ÍY‘„³l~r‡’Œ‘kIl-„³r‡„³jk rvgikl~{}l~rvkj3„†l-€i”G•’2rvkI• ’Œ• „³r-„Šl-‘‡gikI‘‡ÇŒkI•Æ0gikr‡gikƒ+rvgikl~rv‰}‘–g’Œl~rv„†‘6k°ŒkSr-g’SlLr‡‰¡ˆQko”Qkƒ~¾®‰SƒvjkI•&‰Œƒ+i‰²rI¹5f-gik¿’Œ‹³S‰Œƒ‡„»r‡gij r‡gik ”iƒ‡‰}‘kk•il¡ƒ‡k‘€iƒ‡lv„Š°Œk‹Š{Œ¹ Ò ‹³rvgi‰S€iŒg7rvgkl~k´’²‹ŠŒ‰Œƒ‡„³rvgij&l¡lvkkj rv‰ÅS„³°Œk*S‰|‰}•½ƒ‡klv€i‹³r‡l ”iƒ‡‰Z°|„†•}k•½r‡gik r‡„³j k ¼l~rvk”4‰²¾rvgku¿Eœ Dºl~‰S‹³°Œkƒ-„Šl0njk”}rWl~j&’Œ‹³‹ rvgk{d‹†’Œ‘–Ç’ j&’²rvgikj’²rv„†‘’Œ‹|¼€YlFr‡„»ÍY‘’2r‡„³‰S ‰ŒÆ0gi„†‘‡g ˆ’Sl~„†lÂrvgk„Šƒ6’Œ‘‘€iƒ‡’S‘{&‘‰Œ€i‹†•dˆQk+F€•iŒk•/¹5t¼dr‡gik¿¾®‰S‹³‹Š‰ZÆ0„Ši ÆÂk”iƒ‡‰Œ”Q‰Slvk¿’¡ƒ‡„ŠŒ‰Œƒ‡‰Œ€lÂj’²rvgikj’²rv„†‘’Œ‹ Sƒv‰S€i• ¾®‰SƒÂg={|ˆiƒ‡„†•l~rv‰}‘–g’Œl~rv„†‘W’²••ikrvkƒvj„Ši„†lFr‡„Š‘Wj‰}•}k‹³‹Š„Ši„Šd’°Œkƒ‡{ k‹Šk=’²Sr+’Œ•&Y’2rv€ƒ‡’Œ‹YÆ+’I{S¹ Ì k4’Œ‹Šlv‰±”iƒ‡klvkSr r‡giƒ‡kk4’²‹ŠŒ‰Sƒv„³rvgij&l¾®‰Œƒ„³r‡l„³j”i‹ŠkjkSr‡’²rv„Š‰Œ’²•\́’Œ‹³‹Š{\lvgi‰2Æ1́ƒ‡l~r„ŠSr‡ƒv„ŠŒ€i„Ši |€ijkƒv„†‘’Œ‹%ƒ‡kl~€‹»r–l+¾®‰Œƒ-r‡gikȁ’Œ‘rvkƒ‡„Š‰Œ”ig’ŒŒk¿fWϝj‰}•}k‹/lv{}lFr‡kj4¹ F G d!")0¶ %&" % qÃ! #)+)-!5 .   %  .#)0 „Š=°Œ‰S‹³°ŒkI•7„Š M ƒ‡k’S‘¶r‡„³‰Sl R , . . . , R ¹xÕÂgikj„†‘’²‹ Õ‰Sl~„†•}kƒ ‘–gikj„†‘’²‹Wl~”QkI‘„Škl lv”Qk‘„Škl-’²ƒ‡kWNj‰}•}k‹³‹Šk•d„³dr‡kƒ‡jl+‰²S¾

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(34) j‰Œ‹Šk‘€i‹³kIl X(t) ¹5f-gik¿ƒ‡k’Œ ‘  r‡„³‰S ƒ–’2r‡k+¾®‰ŒƒLk’S‘‡gƒvkI’Œ‘rv„Š‰Œ R „Šll~”Qk‘„»ÍYk• ˆ={ ’lv‰ ¼‘’Œ‹³‹Šk•”ƒv‰S”Q=k(Xlv„»rF{(t), ¾®€i.‘¶.r‡.„³‰S, X a (t)) Æ0gi„†‘–g½„ŠlkI S€Y’²‹Lrv‰´rvgikd”iƒ‡‰}•}€‘r‰Œ¾0’´ƒ‡’²rvk‘‰Œl~r‡’ŒSr c ’Œ•Årvgikd=€ijˆQkƒ‰²¾0”Q‰Sl‡l~=„Šˆi‹Šak‘(X(t), ‰Sjˆi„Št)’ '. (. 1. N. 1. 1. j. j. j. ßß ì.Hÿ–þ ¶ÿ. M. N. j.

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(36). . . . 1# ! 5. . . r‡„³‰Slo‰²¾ƒvkI’Œ‘¶r–’²=rWj‰S‹³kI‘€i‹Šklo„³|°Œ‰Œ‹Š°Œk•´„ŠOƒ‡k’Œ‘rv„Š‰Œ ¹ÃÔi‰SƒWr‡gik¡j‰Sl~ro¾®ƒvkI S€kSr‡‹³{´€lvk•ƒvkI’Œ‘rv„Š‰Œ rF{|”Qkl S → ∗ S + S → ∗ ’Œ• S + S → ∗ RÆÂkSkr a = c X (t) a = c X (t)X (t) ’Œ• ƒvkIl~”QkI‘¶r‡„³°Œk‹³{:)20IŸ 00+ ¹0uoY‘k’ƒ‡k’S‘¶rv„Š‰Œ „†l6”Qkƒv¾®‰Œƒ‡j kI• r‡gik |€ijˆQakƒ=‰²¾GcjX‰Œ‹Šk(t)(X ‘€i‹Škl"¾®(t)‰ŒƒLkI−’Œ‘–1)/2 glv”Qk‘„³kIl5„ŠlL€i”G•i’2r‡k•&’Œ‘‘‰Œƒ–•}„Šior‡‰rvgk6l~r‡’²rvk6‘–Rg’ŒiŒk0°Œk‘¶r‡‰Œƒ ν „?¹ kŒ¹ ¹ X(t) ← X(t) + ν *É ¥ É ¦

(37) \¬ ¯ ¬Ð¤¥I¬Ð¨

(38) « Ü É Ë s+’Sl~kI•d‰Œdr‡gikˊ’ZÆ·‰²¾,j&’Œl‡l-’Œ‘rv„Š‰Œ ’l~{}l~rvkj؉Œ¾,‘‰Œ€i”i‹Šk•*‰Œƒ–•}„Š’²ƒ‡{ •}„³µGkƒ‡kSr‡„Š’Œ‹SkI =€’2r‡„³‰Sl = u¿œ DÂl ?

(39) „Šlkl~r‡’Œˆi‹³„†lvgik•¾®‰Œƒ

(40) r‡gikrv„ŠjkÂk°Œ‰S‹³€}r‡„³‰SɌ¾}rvgiEk  1-‰²¾ij‰Œ‹Šk‘€i‹³kIl =?l~kk kŒ¹ ¹ )ô™ 0<3+!? X(t) ∈ R j. a. j. a. j. a. b. a. a. j. j. a. a. j. j. a. b. j. j. j. . . . N +. d X(t) dt. M X. =. = 0?. νj aj (X(t), t). j=1. Æ0„³rvgÁl~‰Sj k „³i„³rv„†’Œ‹"°2’Œ‹³€ik X(t ) ∈ R ¹ Ì g„³‹Šk¡r‡gik&l~{}l~rvkj lvgi‰S€i‹Š•OˆQk•}kIlv‘ƒv„ŠˆQk•O’Sl¿’*°Œk‘rv‰Œƒ¿‰Œ¾ „ŠSr‡kŒkƒ‡l rvgi„†lj‰}•}k‹ikk•lƒvk’Œ‹L°2’²‹Š€ikIl¾®‰Œƒ X(t) ¹4f-gi„†l„Šlgi‰2ÆÂk°Œkƒ’S‘‘k”}r‡’Œˆi‹Šk&€i•}kƒr‡gikd’Sl  lv€ij”}rv„Š‰Œɲ¾|‹†’²ƒ‡Œk"=€jˆQkƒ

(41) ‰²¾|j‰Œ‹Šk‘€i‹Škl-= X (t) À 1?%lv‰-rvg’²r%r‡gikLƒ‡k‹†’2rv„Š°Œk"kƒvƒ‡‰Œƒ/‘’²ˆQkLikS‹³kI‘¶r‡k•%¹ %¥ « ¨ §i¤¥¬?¨

(42) « Ü É Ë s+’Sl~kI•Å‰Œ½”ig={}lv„Š‘’²‹‹Š’ZÆ6l’Œ•Årvgikd„Š•ik’4rvg’²r¡‘–gikj„†‘’Œ‹ƒvkI’Œ‘¶r‡„³‰Sl’²ƒ‡k kIlvlvkSr‡„†’²‹Š‹³{Áƒ–’²•i‰Œj ”iƒ‡‰}‘kIlvlvkl rvgk4lFr‡‰}‘‡gY’Œl~rv„†‘&¾®‰Œƒ‡j€i‹†’2r‡„³‰S\‰Œ¾o‘‡gkj„Š‘’²‹-ƒvkI’Œ‘¶r‡„³‰Sl„†l¡S„³°Sk7„³ r‡kƒ‡jl‰²¾-’p’Œƒvnj‰2° F€ij”Á”iƒ‡‰}‘kIlvl X(t) ∈ N )20› ›Œ› + ¹ t r–lчgY’²ƒ–’Œ‘¶r‡kƒ‡„ CI’2rv„Š‰Œ„†lˆ’Sl~k•±‰ŒOr‡gik ”iƒ‡‰Œˆ’Œˆi„Š‹³„³rF{ a (X(t), t)dt ‰²¾ƒ‡k’S‘¶rv„Š‰Œ R ‰}‘‘€iƒvƒ‡„Ši½„Šºrvgik±ikÀ=r„³íi„³rvklv„Šj’Œ‹Wr‡„³jk±„ŠSrvkƒv°²’²‹ ¹5œok‰²rv„Ši¡ˆ={ T (t) rvgikorv„Šj k’2r+Æ0gi„Š‘–gdƒvkI’Œ‘rv„Š‰Œ R ÍYƒ‡l~r‰}‘‘€iƒ–l’²¾ rvkƒ t rvgi„†l+’²j‰S€iSr–l [t, t + dt] r‡‰Æ0ƒv„³rvkrvg’²r P[T (t) ∈ [t, t + dt] |X(t)] = a (X(t), t) dt. t rL„Šl5€l~€’Œ‹Š‹³{’Œl‡lv€ijk•r‡g’2rLƒvkI’Œ‘¶r‡„Š‰Œl"’²ƒ‡k+‹³‰i‘’²‹Š‹Š{ijY•}k”QkY•}k=rL„Šj ”i‹Š{|„Ši¿rvg’²r P[{T (t), T (t)} ∈ ¹ [t, t + dt] |X(t)] = a (X(t), t)a (X(t), t) (dt) t¼¡‰Sƒ‡•ikƒr‡‰¿kIlFr–’²ˆi‹Š„Šlvg ’² k°Œ‰Œ‹Š€}r‡„³‰SkI =€’2r‡„³‰S¾®‰Sƒ X(t) ’Sl,ÆÂk‹Š‹i’ŒlQF€lFr‡„»¾®{|„Šig|{|ˆiƒv„†•¡l~rv‰i‘‡g’SlFr‡„Š‘ ’Œ••}kr‡kƒ‡j„³i„†l~rv„†‘¿j‰}•}k‹Šl r‡gik¾®‰Œ‹Š‹³‰2Æ0„Ši¡r‡„³jkr‡ƒ‡’ŒlF¾®‰Sƒvj&’²rv„Š‰Œ N +. 0. i. . . . N. j. j. j. j. j. j. j. j. 2. k. gj (s|t). Z. =. k. s. aj (X(τ ), τ ) dτ. „†l*‰²¾njk{„Šj”Q‰Œƒvr‡’²Y‘kŒ¹ f-gikO¾®€i‘rv„Š‰Œ „†l*i‰Œ ¼•}kI‘ƒvkI’Œlv„³Á¾®‰Œƒ l~„Š‘kOrvgik ”iƒ‡‰Œ”Qkl~„³rv„Škl a ’Œƒvk i‰Œ  ik=’2r‡„³°Œk¡ˆ={O•}ksÍYi7→„»r‡„Š‰Œg/¹(s|t) œWki‰²r‡k ˆ|{ Exp(1) rvgk kÀ}”Ys‰Si>k=trv„†’²‹"ƒ–’²•}‰Sj °2’Œƒv„†’Œˆi‹³k‰²¾”Y’²ƒ–’²jkr‡kƒ 1 „A¹ kŒ¹ ξ ∼ Exp(1) „³¾ P[ξ ∈ [x, x + dx]] = e dx ¾®‰Œƒ’Œ‹³‹ x ≥ 0 =Ðgikƒ‡k ∼ •}k‰²rvkIlWkI =€’²‹Š„»rF{4„³O‹†’IÆ ˆQkrFÆÂkkrFƉdƒ–’²•i‰Œj °2’Œƒv„†’²ˆi‹Škl? ξ g’Œl¿l~€iƒ‡°|„³°²’²‹

(43) ”ƒv‰Sˆ’²ˆi„Š‹Š„»rF{ ¹m6‰2Æ ‹Škr¡€l¡‘‰Slv„Š•}kƒ’lvk =€ik‘k (ξ ) ‰Œ¾0„³Y•}k”QkY•}k=r¡ƒ–’²•i‰Œj'°2’²ƒ‡„†’²ˆi‹Škl P[ξ ≥ x] = e 0 Æ » „ ‡ r g ’²Y• ¹5œoḱkWrvgikoƒ–’²Y•}‰Œj °2’Œƒv„†’²ˆ‹³kIl S (n) = P ξ ξ ∼ Exp(1) ¾®‰Sƒ j = 1, . . . , M ’²j• = 1, . . . , M k ∈ N X =Až ? 1 . N (t) = t. j. j. −x. . −x. jk. jk. j. n k=1 jk. ∞. j. {Sj (n)≤gj (t|t0 )}. n=1. á ä,ß/á é.

(44) ™. 

(45)         !"#   $

(46)  %&. f-gik „³r6‘’²ˆQkÃk’Œlv„Š‹³{dlvgi‰ZÆ·rvg’²r0¾®‰Œƒ6’Œ‹³‹ j ∈ {1, .., M } P[Nj (t + dt) − Nj (t) = 1|X(t)] = aj (X(t), t)dt. vl ‰r‡g’2r+r‡gik¿ikÀ|r,F€ij” Arv„Šj k‰Œ¾/rvgik¿”iƒ‡‰}‘kIlvlvkl ÆÂk͌kr-rvgikĊj”Q‰Œƒvr‡’²=r0ƒvk‹Š’²rv„Š‰Œ ∼. Tj (t). Nj (t). g’I°ŒkWr‡giklv’Œj ko‹†’ZÆr‡g’². gj−1 (Exp(1)|t),. Tj (t). ’Œ•drvgikˆ’ZÆ·‰Œ¾rvgikÃk°Œ‰Œ‹Š€}rv„Š‰Œk =€’²rv„Š‰Œ*‰Œ¾rvgikÃ|€ijˆQkƒ6‰Œ¾,j ‰S‹³kI‘€i‹Škl-„†l0Œ„Š°Œkˆ={ dX(t). M X. =. ¹Lf-gikƒ‡k¾®‰Œƒ‡k =Л5? =К5?. νj dNj (t).. mW‰²rvkorvgY’2r N (t) ‘‰Œ€Sr‡lÂrvgik=€ijˆQkƒ-‰Œ¾/rv„ŠjklÂr‡g’2r-ƒ‡k’S‘¶rv„Š‰Œ R ‰i‘‘€ƒvƒ‡k•¾®ƒv‰Sj r‡gik¿„Ši„³rv„†’²‹Qrv„Šjk €i”rv‰rv„Šj k t ¹ D°Œ‰Œ‹Š€}rv„Š‰ŒkI =€’2r‡„³‰S =К5?-‘‰Œƒ‡ƒ‡klv”Q‰Œ•ilr‡‰ r‡gikij}ÍYi„»r‡klv„³j&’Œ‹/Œkkƒ–’2rv‰Sƒ 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*•}€’Œ‹Â”Q‰Œ„ŠSr¡‰²¾6°=„ŠkÆ A= ÕÂg’Œ”ij’Œ W‰Œ‹Šj‰ŒŒ‰Œƒv‰Z°?o‹Šk’S•ilÃr‡‰´r‡gikdÆÂk‹³‹ AÇ|i‰2Æ0\‘‡gikj„Š‘’²‹j&’SlFr‡kƒ kI S€Y’2rv„Š‰Œ ) 0Zž ›Œ›+ 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] .. gi‰S‹Š•i„³i ¾®‰Sƒ6’²‹Š‹ x ∈ N ¹   !, # %," / '*#(#)+)-!5 . *  %  .#)0 ԁ€i‹³‹Š{dlFr‡‰}‘–g’Œl~rv„†‘ol~„Šj€i‹†’2r‡„³‰Sl+ˆQk‘‰Sjk¿ =€i„³rvklv‹Š‰ZÆCÆ0gik*j&’²|{&j‰Œ‹Šk‘€‹³kIl+’Œ•&¾Ð’SlFr0ƒvkI’Œ‘¶r‡„³‰Sl+’²ƒ‡k „Š=°Œ‰S‹³°ŒkI• •}€ikÂr‡‰Wr‡gik¾ВS‘¶r,r‡g’2r,r‡gik+kµQ‰Sƒ~r"„Šl,”ƒv‰S”Y‰Sƒ~r‡„Š‰Œ’²‹Œrv‰or‡gik+=€jˆQkƒ"‰Œ¾ƒ‡k’Œ‘rv„Š‰Œl”Qkƒv¾®‰Œƒ‡j kI•%¹ ÄW‰ZÆÂk°Œkƒ ¾®‰Sƒ0k’Œ‘–g*rv„Šj k ¼‘‡g’ŒiŒkI•dŽ"‰Œ„†lvlv‰Œ”iƒv‰i‘kl‡l N (t) Æ+kg’Z°Œk N. j. E[Nj (t)]. E[(Nj (t) − gj (t|t0 ))2 ] Z t = E[aj (X(s), s)]ds. =. 0. ßß ì.Hÿ–þ ¶ÿ.

(47) 7. ¾®‰Sƒ. E% "  . t > t0. . ¹y|„Š‘krvgkÃvk‹Š’²rv„Š°Œk , € ‘rv€’²rv„Š‰Œ4ˆYkr¼Æ+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-Œ„Š°Œk*ˆ|{. 1 , E[Nj (t)]1/2. =. ³„ r„†l ƒ‡k’Œlv‰ŒY’²ˆi‹³kr‡‰ÅikS‹³kI‘¶r„»r’Œ•7’Œ”i”iƒ‡‰IÀ}„Šj&’2rvkdrvgik´lFr‡‰}‘–g’Œl~rv„†‘*•}{|’²j„†‘l ˆ={\„»r–l‘‰ŒSr‡„³|€i‰Œ€Yl ‘‰Œ€iSr‡kƒ‡”’²ƒvr =?’Œl„³ r‡gik´•}krvkƒvj„Ši„Šl~rv„†‘j‰}•}k‹#?¡Æ0gik7r‡gik4”ƒv‰S”Yklv„»rF{ „Šl‹†’²ƒ‡Œk’Œ• r‡gik=€ijˆQkƒ–l0g‰Œ¾"(t|tj‰Œ)‹Šk‘€i‹Škl0„Š=°Œ‰S‹³°ŒkI•*„Šr‡gikƒvkI’Œ‘rv„Š‰Œ4’²ƒ‡ki‰Œr0rv‰|‰lvj&’²‹Š‹?¹Âf-gi„†l0ja‰Œrv„Š°2’2r‡kl0j„»À}k• l~rv‰}‘–g’SlFr‡„Š‘’²••}krvkƒvj„Ši„Šl~rv„†‘j ‰i•}k‹†l¹ Õ‰Œlv„Š•ikƒÂ’”Y’²ƒvrv„³rv„Š‰Œ‰²¾%r‡gikoƒvkI’Œ‘rv„Š‰Œl „ŠSrv‰r‡gi‰SlvkWj‰}•}k‹Š‹Šk•l~rv‰}‘–g’Œl~rv„†‘’Œ‹Š‹³{ =ÐÆ0„»r‡g „Š•}kÀ j ∈ S ?6’Œ•r‡gi‰Slvkj ‰i•}k‹Š‹³kI••}krvkƒvjR„Ši„†,lF.r‡.„Š‘. ’²R‹Š‹³B{ =ÐÆ0„»r‡g„Š•}kÀ j ∈ D ?¶¹oԁ‰Œƒo’Œ„Š°Œk´j ‰i•}k‹ r‡gikƒ‡kd’²ƒ‡k&’²r‹³kI’Œl~rÃr‡giƒvkk‰Œ”}r‡„³‰Sl¾®‰SƒŒkrvrv„ŠiOl~€‘–g\’4”Y’²ƒvrv„³rv„Š‰Œ =®!„ ?Ãv€i\’4¾®€i‹³‹Š{Ál~rv‰}‘–g’SlFr‡„Š‘ƒ‡ k  ’Œ‹³2„ C’²rv„Š‰Œ’²•’²Y’²‹Š5{ Ckrvgik´¾®ƒvkI =€ik‘„³kIl 92”iƒv‰S”Qklv„»r‡„Škl ‰²¾k’S‘‡gƒvkI’Œ‘rv„Š‰Œ =®i‰Œrvk4r‡g’2rrvgikj&’2F‰Œƒ ‘‰Œj”i€}r‡’²rv„Š‰ŒY’²‹‘‰Sl~rW„†l6„ŠO‘‰Œj”i€}r‡„³j&’²|{*ƒ‡k’Œ‹³2„ C’²rv„Š‰Œl ? =Є³!„ ?0€Yl~kˆi„³‰S‹³‰SŒ„†‘’Œ‹%„Šlv„³SgSrI¹0t r¿l~kkj&l ƒ‡k’Sl~‰S’²ˆi‹Šk ¾®‰ŒƒL„ŠlFr–’²‘k rv‰Ãj‰}•}k‹}Skik0ƒvkŒ€i‹†’2r‡‰Œƒv{ԁ’Œƒ~r–l5lFr‡‰}‘–g’Œl~rv„†‘’Œ‹³‹Š{ Æ0gi„Š‹³k0jkr–’²ˆQ‰S‹³„†‘+ƒ‡k’Œ ‘  r‡„³‰Sl•}kr‡kƒ‡j„³i„†l~rv„†‘’²‹Š‹Š{ =Є³„Š!„ ?5¾®‰ŒƒÂk’S‘‡gƒvkI’Œ‘¶r‡„³‰S&‘–gi‰|‰=l~kW’Œ•’²”}r‡„³°Œk‹³{ ˆQkrFÆÂkk&r‡gik6rFƉ¡’Œ”i”iƒ‡‰S’Œ‘–gikl €lv„Ši±’‘ƒv„³rvkƒv„Š‰Œ½ˆ’Sl~kI•Á‰S½rvgik*=€ijˆQkƒ¡‰Œ¾6j‰Œ‹Šk‘€i‹Škl¡’Œ•½„³r‡l¡”iƒv‰S”Qklv„»rF{±¾®€i‘rv„Š‰Œ =Ðlvkk4y|k‘  r‡„³‰S6 7 ?¶¹Lf-gkk°Œ‰S‹³€irv„Š‰Œk =€’2r‡„Š‰Œd¾®‰Sƒ X(t) ∈ R „†l-i‰2ÆxŒ„Š°Œkˆ={r‡gikÃg={|ˆiƒ‡„Š•lv{}lFr‡kj 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. Æ0„³rvgd„³i„³rv„†’Œ‹Y°2’Œ‹³€k ) ∈ R ¹5œW€ikWr‡‰rvgiko”’²ƒvrv„³rv„Š‰Œ‰²¾%rvgikoƒ‡k’Œ‘rv„Š‰Œl ’lv”Qk‘„Škl‘’Œ&ˆQk‹³‰SiÃrv‰ r‡gik¡l~rv‰i‘‡g’SlFr‡„Š‘’ŒlWX(t ÆÂk‹Š‹,’Sl0r‡gik •}kr‡kƒ‡j „Ši„†lFr‡„†‘”’²ƒvro‰²¾5rvgkg|{|ˆiƒv„†•l~{}l~rvkj¹ Ò lo’d‘‰Sl~kI =€ik‘k „³r j&’Z{Wrvgik‰Œƒ‡krv„†‘’Œ‹³‹Š{Wg’Œ”i”Qkr‡g’2r,lv‰Œjk X (t) ˆQk‘‰Œjkl/ik=’2r‡„³°ŒkS¹ Ì kLi‰Œrvk5ÍYƒ‡l~r/rvg’²r/r‡gi„Šllv„»r‡€’2r‡„³‰S ik°Œkƒ"‰}‘‘€iƒ‡ƒvkI•Ã„Š ‰Œ€iƒ5lv„³j€‹Š’²rv„Š‰Œl ’Œ•„³r5„Šl"ƒ–’2rvgkƒ"•i€ik+rv‰’Œ¡€ilv€i„³r‡’²ˆ‹³k-j‰}•}k‹Š‹Š„³i‘–gi‰S„Š‘k = rvgk •}krvkƒ‡j„Ši„Šl~rv„†‘¿kI S€Y’2rv„Š‰Sl-lvgi‰Œ€i‹†•*i‰²r6’S‘¶r0‰Slvj&’²‹Š‹/ S€Y’²Sr‡„»r‡„³kIl ?¹y|k‘‰Œ•}‹Š{ rvgi„†l6l~„³rv€Y’2rv„Š‰Œ4‘’Œ*ˆQk lv‰Œ‹Š°Œk•dˆ={*’²4’Œ•’²”}r‡„³°ŒkÃlFr–’2rvk  •ik”Qk•ikSr6‘–gi‰S„Š‘k¿‰Œ¾

(49) r‡gikƒ‡k’S‘¶r‡„³‰Slr‡‰ˆQkj‰}•}k‹Š‹Šk•lFr‡‰}‘–g’Œl~rv„†‘’Œ‹³‹Š{ ‰Sƒ0•}krvkƒ‡j„³„Šl~rv„†‘’Œ‹³‹Š{S¹Ly|€‘–g4’Œ4’²”i”ƒv‰=’Œ‘–gd„†l0‘€ƒvƒ‡k=rv‹Š{€i•}kƒ6l~rv€•i{Œ¹ f-gikg|{|ˆiƒv„†•lv{|l~rvkj =A™ ?-‘‰Sƒvƒ‡klv”Q‰Œ•ilÂrv‰ r‡gikijíi„³rvklv„Šj’Œ‹/Œkikƒ–’2r‡‰Œƒ 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-gikÕ}€’²‹%”Q‰Œ„ŠSr0‰²¾°|„³kÆ = ÕÂg’²”ij&’Œ  o‰Œ‹Šj S‰ Œ‰Œƒv‰Z°1?L‹³kI’Œ•ilÂrv‰ rvgikіgikj„Š‘’²‹Gj&’SlFr‡kƒ-k =€’2r‡„³‰S*‘‰S€  ”i‹Šk•*rv‰’  „Š‰Œ€i°|„Š‹³‹Šk¿r¼{|”QkÃk =€’²rv„Š‰ŒB=Б¶¾F¹ ) ; ÂÕ g’Œ”/¹L›i¹ š.+ ?¶¹ j∈ S. á ä,ß/á é.

(50) Ï. 

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(53) ‰„³‹Š‹³€YlFr‡ƒ‡’²rvkor‡gikˆ’Sl~„†‘o„†•}k’ ‘‰Œlv„†•}kƒ0’ lv„³Œ‹Šk‘‡gkj„Š‘’²‹/lv”YkI‘„Škl ˆQk„Ši„³|°Œ‰Œ‹Š°Œk•dˆQ‰²r‡g„³ ’ƒ‡k’Œ‘rv„Š‰Œ R j ‰i•}k‹Š‹³kI•dlFr‡‰}‘–g’Œl~rv„†‘’Œ‹³‹Š{’²•d’ƒ‡k’S‘¶rv„Š‰Œ R j‰}•}k‹Š‹Šk•dS•}kr‡kƒ‡j„³i„†l~rv„†‘’Œ‹³‹Š{Œ¹Ly|kr+r‡gik „Ši„³rv„†’²‹"rv„Šjk τ = t ’Œ•rvgik|€ijˆQkƒÃ‰Œ¾Âj‰Œ‹Šk‘€i‹Škl X(τ ) = X (t ) ¹ œoƒ‡’ZÆΒ*ƒ‡’Œ•}‰Œj =€jˆQkƒ ’Œ•O„³i„³rv„†’²‹Š2„ Ck ¹ Ò l¿‹Š‰Œi4’Œl ‰l~rv‰i‘‡g’SlFr‡„Š‘k°Œk=r‰}‘‘€iƒ–l ξ ∼ Exp(1) ’Œ•drvgik•i{=Y’²j„Š‘l-„Šl6lv„³j”i‹Š{dgŒ(τ„Š°Œk|t)ˆ=={0r‡gik•}kr‡kƒ‡j „Ši„†lFr‡g„Š‘(τ”|t)’Œƒ~r < ξ . 1. 1. 2. 0. 1. 1. 1. 1. dX (τ ) dτ. =. 0. 1. = 75?. ν2 a2 (X(τ ), τ ). =?‘¶¾F¹,k QA¹ =?™ ? ?¶¹Ot¼Ár‡gik*‘‰Œ€iƒ–l~k‰²¾-r‡gik =?•}kr‡kƒ‡j „Ši„†lFr‡„†‘?k°Œ‰Œ‹Š€}r‡„³‰S gi‰2ÆÂk°Skƒ r‡gikd°2’Œ‹³€k‰²¾ „Š‘ƒ‡k’Sl~kIl¾®‰S‹³‹Š‰2Æ0„³¡rvgk•}„»µGkƒ‡k=rv„†’²‹/k =€’²rv„Š‰Œ d g1 (τ |t) dτ. =. g1 (τ |t). =AÏ?. a1 (X(τ ), τ ).. ÄWk‘k ’²‹ŠŒ‰Œƒ‡„³rvgij„†‘’Œ‹³‹Š{ Æ+k4lv„³j”i‹Š{7l~„Šj€i‹³r‡’Œik‰S€l~‹Š{\lv‰Œ‹Š°Œkdr‡gik´l~{}l~rvkj‰Œ¾Ãu¿œ Dl = 7 ? ’²Y• =AÏ.?¹ f-gikW=€ijkƒ‡„†‘’²‹Ylv‰Œ‹Š€}r‡„³‰S‰²¾/lv€‘–g&‰Œƒ–•}„Š’²ƒ‡{¡•}„»µGkƒ‡k=rv„†’²‹Yk =€’2r‡„³‰SlL„Šl°Œkƒ‡{ÆÂk‹³‹Q•}‰}‘€ijkSr‡k• =®kS¹ Y¹ )ô™ 0<7 0ZÏ žŒ™+!?¹ Ì kWr‡gikƒ‡k¾®‰Œƒ‡kWlv€i”i”Q‰=l~k6rvg’²rÂÆ+ko’²ƒ‡kW’²ˆ‹³k6rv‰ ‘‰Œj”i€}r‡k6r‡gik¿l~‰S‹³€irv„Š‰Œ&‰Œ¾%rvgikÃu¿œEDl €i” rv‰Á’Œ={\•ikl~„Šƒ‡k• ’Œ‘‘€iƒ–’Œ‘{½’²•7kŒ‹Šk‘r rvgik´•}„†l‡‘ƒ‡krv2„ C’²rv„Š‰Œ7kƒ‡ƒ‡‰Œƒ¡„Š Æ0g’²r ¾®‰Œ‹Š‹Š‰ZÆ6l¹7œo€ikrv‰ kI Y¹ =Л ? ÆÂkÇ|i‰2Æxr‡g’2rWrvgiḱƒ–lFrol~rv‰}‘–g’SlFr‡„Š‘ƒ‡k’S‘¶rv„Š‰S‰}‘‘€iƒ–l6’Œ‘‘‰Œƒ–•}„Ši rv‰rvgikƒ‡’Œ•}‰Sj3°2’Œƒv„†’²ˆ‹³k ¹Õ‰Œlvk =€ikSrv‹Š{ Æ+k „³=rvkSƒ‡’²rvkr‡giklv{}lFr‡kj ‰²¾0u¿Eœ Dl = 7 ?¿’²>• =A.Ï ?o€iSr‡„³‹"r‡„³jk lv€‘‡g T (t ) r‡g’2r g (s|t) = ξ ¹Of-g=€l T (t ) = s ’²•Árvgik*l~rv‰i‘‡g’SlFr‡„Š‘&ƒ‡k’S‘¶rv„Š‰Œ½„Šl”Qkƒv¾®‰Œƒ‡jk•%¹τf-=gikdsk=rv„Šƒvk ”iƒ‡‰}‘kI•}€iƒ‡k¿„Šl-rvgik4ƒvk”Qk’2r‡k•*€iSr‡„³‹’lv”Qk‘„³Ík•*́’²‹%rv„ŠjkŒ¹ f-gik+’ŒˆQ‰Z°Œk+’²‹ŠŒ‰Œƒ‡„³rvgij„†‘lv‘–gikjk+ƒvkI S€„³ƒ‡kl/r‡gik+€lvk‰²¾Y=€ijkƒ‡„†‘’²‹=„ŠSr‡kSƒ‡’²rv‰Œƒ–l%r‡g’2r5’Œ‹³‹Š‰2ÆÁrv‰¿l~rv‰S” „ŠSr‡kŒƒ–’2r‡„³‰SÆ0gik±lv‰Œjk l~‰  ‘’²‹Š‹³kI•4k°ŒkSr¿¾®€i‘¶r‡„³‰S =®„ŠO‰S€iƒ¿‘’Œlvk τ 7→ g (τ |t) − ξ ?W°2’²i„†lvgikl¹ Ò ¾®kƺ|€ijkƒ‡„Š‘’²‹Gu¿Eœ D\„ŠSr‡kŒƒ–’2r‡‰Œƒ–l5’Œ‹³ƒ‡k’S•}{„Š‘‹Š€•}kok°Œk=rÂg’²•i‹³„Ši¹5Ä6‰2ÆÂk°Œkƒ rvgik¿l~”QkI‘„†’²‹’²rv€iƒ‡k ‰Œ¾W‰S€iƒk°ŒkSr¾ €‘¶r‡„³‰Sj’ŒÇŒkl „»r&kI’Œlv{År‡‰Å„Šj”i‹³kj kSr&rvgik4k°ŒkSr•}krvk‘rv„Š‰Œ lv„³‘k g (τ |t) − ξ „Š‘ƒ‡k’Sl~kIld„³ =®„³r4„†lik=’2r‡„³°Œk±ˆQk¾®‰Œƒ‡kOrvgkÅk°Œk=rrv„Šj k½’²•C”Q‰=l~„³rv„Š°ŒkOrvgikƒvkI’2¾ r‡kƒ ?¶¹Øf-g=€l‰Sik |€ijkƒv„†‘’Œ‹Š‹³{*„ŠτSr‡kSƒ‡’²rvkl0rvgk¡l~{}l~rvkj ‰Œ¾Âu¿Eœ DlW€i=rv„Š‹rvgikÍYƒ‡l~r6r‡„³jk τ Æ0gk´rvgik¡k°Œk=rW¾®€i‘¶r‡„³‰S ˆQk‘‰Œjklo”Q‰Slv„»r‡„³°Sk Æ0gi„³‹Šk¡„³r„ŠllFr‡„³‹Š‹,kS’²rv„Š°Œk¡’2ror‡gik ”iƒvk°|„³‰S€l6rv„Šjk g (τ |t) − ξ = δt ˆQk„Šir‡gikÃr‡„Šj k¡lFr‡k” ?¹0f-gik ’&”Q‰Œ‹Š{|i‰Œj„†’²‹/„ŠSr‡kƒ‡”Y‰S‹†’2rv„Š‰Œ4ˆQkrFÆÂkk (t , g (τ τ|t) =− ξτ ) −’Œδt• ‘’²\ˆQk*€Yl~kI•½„Š7‰Œƒ–•}kƒr‡‰Ok°2’Œ‹³€’²rvkrvgk*k°ŒkSr¡r‡„³jk s ∈ [τ , τ ] ’2r Æ0gi„†‘‡g (t , g (τ |t) − ξ ) L ¹ f i g Š „ l-”ƒv‰}‘k•}€iƒ‡ko„†l+rvgiklv’Œj k€l~kI•r‡‰ Skikƒ‡’²rvk¿’•}klvk‰Œ€}r‡”i€}r )20<7 0Z<Ï + ’²•*„Šl g(s|t) − ξ = 0 ’ÆÂk‹Š‹kIlFr–’²ˆi‹Š„ŠlvgikI•4jkrvg‰}•%¹ Ò ‹³rvkƒv’²rv„Š°Œk‹Š{ Æ0„»r‡gOlv‰Œjk’Œ••}„»r‡„Š‰Œ’²‹,‘‰Sj”i€}r‡’²rv„Š‰Œ’Œ‹‘‰=lFr ’&́ikƒ r‡„³jkl~rvk”´‘‰S€i‹Š•*ˆQkÀlvk•dr‡‰&lv‰Œ‹Š°Œkor‡gik¡u¿œ D„Š*r‡gikĊSrvkƒv°²’²‹ [τ , τ ] ¹ 1. 0. 1. 1. 1. 0. 1. 1. 1. 1. +. 1. +. 1. −. −. +. 1. +. 1. −. −. 1. −. ßß ì.Hÿ–þ ¶ÿ. 1. +. +. 1. +.

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(57)  . -f gik•}„Šƒ‡k‘¶rWjkr‡gi‰}•4kÀ}”i‹Š„†‘„³rv‹Š{‘’Œ‹Š‘€i‹Š’²rvkIl-Æ0g„Š‘–g´ƒvkI’Œ‘¶r‡„Š‰Œ‰}‘‘€iƒ–l-kÀ|ro’²Y•Æ0gik4„³rW‰i‘‘€iƒ–l ) 0IŸ P 0 0+A¹Of-gikdƒvkI’Œ‘¶r‡„Š‰ŒÅrv„Šjkd„†lS„³°Œk½ˆ={±rvgkF€ij”\‰Œ¾-rvgik*Ž,‰Œ„†l‡l~‰SÁ”iƒ‡‰}‘kIlvl Æ0„³rvgńŠSr‡klv„»rF{OŒ„Š°Œk±ˆ={´r‡gik&‘€ij€i‹†’2rv„Š°Œk rv„Šjk rvƒ–’²l~¾®‰Œƒ‡j&’2rv„Š‰S g (τ |t) =NP(t) =g (τ |t) ¹ Nf-g|(t)€l r‡gik’²‹ŠŒ‰Sƒv„³rvgij„†‘oƒvkI’²‹Š„ CI’2rv„Š‰Sd‰²¾r‡gik•}„ŠƒvkI‘¶r0g|{=ˆƒv„†•*jkr‡gi‰}•*„†l6’Œl+¾®‰Œ‹Š‹³‰2Æ6l 0Œ¹6y|kr6„³„»r‡„Š’Œ‹Gr‡„Šj k t = t ’²Y•*„³„»r‡„Š’Œ‹%|€ijˆQkƒ–l-‰Œ¾,j‰Œ‹Šk‘€‹³kIl X(t ) ž}¹ 4 kikƒ–’2r‡k’ ƒ–’²•i‰ŒjØ°2’²ƒ‡„†’²ˆi‹Šk ξ ∼ Exp(1) ›i¹6y|kr g (t|t) = 0 ’Œ•lv‰Œ‹Š°Œkor‡gikl~{}l~rvkj؉²¾u¿œ Dl6l~r‡’Œƒ~r‡„³’2r0rv„Šj 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„Š‹%r‡„³jk τ = s l~€‘–g*r‡g’2r g (s|t) = ξ š¹ 4 kikƒ–’2r‡k’6•}„†lv‘ƒvkrvkLƒ–’²Y•}‰Œjx°²’²ƒ‡„Š’Œˆi‹³k5Æ0„³r‡g°2’²‹Š€ikIl/„³ S ’²•Ã”iƒv‰Sˆ’²ˆ„³‹Š„»r‡„Škl (a (X(s), s)) „Š‰Sƒ‡•ikƒ+rv‰•}kr‡kƒ‡j„³ikrvgikÇk’S‘¶r‡„³‰S R rv‰&ˆQk”Qkƒ~¾®‰Sƒvjk• ™}¹0nW”G•i’2r‡k X(s) ’S‘‘‰Sƒ‡•}„Širv‰dƒvkI’Œ‘¶r‡„³‰S R gik‘k¡l~kr X(s) ← X(s) + ν Glvkr t ← s ’²Y• S‰¡rv‰dy=rvk”ž}¹ j∈S. . σ. j. . j∈S. m. . m. "!. m. $# %&('*) +)

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(62) °²’²ƒ‡„Š’ŒSr5‰²¾Qrvgk-́ƒ–l~r ƒ‡k’S‘¶rv„Š‰Œ jkrvgi‰}•%¹"t r„†l5ˆ’Sl~kI• ‰Œƒ‡k€lvk0‰²¾Gƒ–’²•}‰Sj °2’²ƒ‡„†’²ˆi‹Škl5’²Y• ‰Œ”}r‡„³j2„ CkI••i’²r‡’Ãl~rvƒ‡€‘¶r‡€iƒ‡kl ) 3+ ¹ f-gi„†l¿ƒ‡k€lvk ’²‹Š‹³‰2Æ6lo€lorv‰4l‡’²j”i‹Šk¡‰Œ‹³{´‰Œik ƒ–’²•}‰SjÓ°2’²ƒ‡„†’²ˆi‹Šk¡’2rk’S‘‡gO„³r‡kƒ–’2rv„Š‰Œ> =Є³l~rvkI’Œ•O‰²¾r¼Æ+‰ ‹Š„³ÇŒk„Š±r‡gik&•}„Šƒ‡k‘¶rÃjkrvgi‰}• ?¶¹&f-gik&’Œ‹ŠŒ‰Œƒ‡„³rvgij„†‘ƒ‡k’²‹Š2„ C’²rv„Š‰Œlo‰Œ¾r‡gik ÍYƒ‡l~r’²•Orvgik&ikÀ=rƒvkI’Œ‘rv„Š‰Œ g|{=ˆƒv„†•*jkrvg‰}•il6’²ƒ‡k¿°Œkƒ‡{dl~„Šj„³‹†’²ƒ rvg=€Yl0ÆÂkÃl~r‡’²rvk¿r‡gikj „Š4’‘‰Œj”’Œ‘r-¾®‰Œƒ‡j 0Œ¹6y|kr6„³„»r‡„Š’Œ‹Gr‡„Šj k t = t ’²Y•*„³„»r‡„Š’Œ‹%|€ijˆQkƒ–l-‰Œ¾,j‰Œ‹Šk‘€‹³kIl X(t ) ž}¹ 4 kikƒ–’2r‡k„³Y•}k”Qk•}k=rƒ–’²•}‰Sj °2’Œƒv„†’²ˆ‹³kIl (ξ ) ‰Œik¾®‰Œƒ¿k’S‘‡gOƒvkI’Œ‘rv„Š‰Œ R „Š S Æ0„³rvg . . 0. ξj ∼ Exp(1). ›i¹6y|kr. . gj (t|t) = 0. 0. j j∈S. ¾®‰Œƒ. j∈S. j. . á ä,ß/á é.

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(64)         !"#   $

(65)  %&. 3. m6‰Y¹ vƒ kI’Œ‘¶r‡„³‰S ”iƒ‡‰Œ”Qklv„³r¼{ ƒ–’2r‡k=5•i’Z{ ? Œk −−→ r‡kj Œk R a =c · c = 0.025 v r  k j v r  k j R a =c · c = 0.25 rvkj −−−−→→ ∅rvkjdÖ Œk rvkj R a =c · c = 1.0  °|„Šƒv€l  Œ   k G  ¿ Ö F l ‡ r v ƒ  € ‘ Œ   k  ~ l v r ‡ ƒ  € ‘ R c = 7.5 · 10 rvkj −−→ rvk−jd−→Ö¿lFr‡ƒv€‘ aa == cc ·· rvkj · R c = 1000 lFr‡ƒv€‘ −−→ ∅ lFr‡ƒv€‘ R a =c · c = 1.99  ð !Zïô ë   ýû IëFï=ë

(66) í2ð‡ö?ò ¶ìIó š¹6y|‰S‹³°Œk¿r‡gikl~{}l~rvkj؉Œ¾Lu¿œ Dl6l~r‡’Œƒ~r‡„³’2r0rv„Šjk τ = t. l~r‡’²rvkіg’ŒiŒk = 0  0 5Ÿ ? ν =  0 Ÿ 5Ÿ ? ν =П 0 5Ÿ ? ν =?Ÿ  0 0<? ν =?Ÿ Ÿ 0<? ν =?Ÿ Ÿ  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. = 0Ÿ5?. νj aj (X(τ ), τ ). j∈D. aj (X(τ ), τ );. = 0 0<?. j∈S. i€ =rv„Š‹%r‡„³jk τ = s l~€‘–g*r‡g’2r0¾®‰Œƒ0rvgiḱƒ–lFr0rv„Šjk g (s|t) = ξ ¾®‰Œƒ6lv‰Œjk m ∈ S ™}¹0nW”G•i’2r‡k X(s) ’Œ‘‘‰Œƒ–•}„Šir‡‰ƒ‡k’Œ‘rv„Š‰Œ R gik‘kÃlvkr X(s) ← X(s) + ν 7i¹6y|kr t ← s ’Œ• =Ð!„ ?*Ô,„Šƒ‡l~r0ƒ‡k’S‘¶rv„Š‰Œ*g|{|ˆiƒv„†•*jkr‡gi‰}• 5S‰¡r‡‰y=rvk”ž|¹ =®„Š#„ ?mWkÀ|rWƒ‡k’S‘¶r‡„³‰Sg|{=ˆƒv„†•4j krvgi‰i• +ԁ‰ŒƒWƒvk’S‘¶r‡„³‰S R Skikƒ‡’²rvk’kÆ׃‡’Œ•}‰Sj3°2’Œƒv„†’²ˆ‹³k ’²Y•dlvkr Æ0gi„Š‹³knjkk”„³i’²‹Š‹G‰²r‡gikƒ-°2’²‹Š€ikIl ¾®‰Sƒ ξ ∼ Exp(1) =0 ’Sl-„³i„³r‡„Š’Œ‹%°²’²‹Š€ikl+¾®‰Œƒ-r‡gigklv(t|t) {|l~rvkjØ ‰Œ¾Lu¿œ DÂl = 0Ÿ ?-’²• = 00<? iS‰¡r‡‰y=r‡kg”4(t|t) š¹ j 6= m Ôi‰Œƒ’²O’Œ‹»r‡kƒ‡’2r‡„³°Œk¡’²‹ŠŒ‰Sƒv„³rvgij„†‘ÃvkI’²‹Š2„ C’2r‡„³‰S4‰Œ¾5rvgk¡g={|ˆiƒ‡„Š•ikÀ|r¿ƒ‡k’Œ‘rv„Š‰Œ´jkrvgi‰}•ˆ’Sl~kI•4‰S ƒ‡kl‡‘’²‹Š„Ši¡‰Œ¾rvgik֒²Y•}‰ŒjØ°2’Œƒv„†’²ˆi‹Škl ξ Æ+kÃlv„³j”i‹Š{dƒvk”i‹Š’S‘ky=rvk6” 7 ˆ|{ 7  =Є³!„ ?+Ôi‰Sƒ+ƒ‡k’Œ‘rv„Š‰Œ R Skikƒ‡’²rvk¿’ikÆ·ƒ‡’Œ•}‰Œj °²’²ƒ‡„Š’Œˆi‹Šk ξ ∼ Exp(1) |¾®‰SƒÂr‡gik¿ƒ‡kj&’Œ„³i„Ši ƒ‡k’S‘¶rv„Š‰ŒYl R Æ0„³rvg j ∈ S ’²• j 6= m ƒ‡kl‡‘’²‹Šk ξ ’S‘‘‰Œƒ–•}„³dr‡‰ ξ ← ξ − g (s|t) ¹*y|kr ’²•Œ‰¡r‡‰y=r‡k”´›i¹ t←s )  .#)0   4 .)-!" ½ /*# )0 32Ã!Ø Ì k*i‰2Æ1Æ0„†lvg\rv‰Å•}kj‰SlFr‡ƒ‡’²rvkr‡gik*”Q‰2ÆÂkƒ¡‰²¾W‰Œ€ƒ¡g={|ˆiƒ‡„†•½jkr‡gi‰}•\€Yl~„ŠiÅ’j‰}•}k‹-•}kƒ‡„Š°Œk•½ˆ|{ y|ƒ‡„Š°2’Œl~r‡’Z°2’4kr ’Œ‹?¹ )ô.ž 3+0•}kl‡‘ƒ‡„Šˆi„³´rvgik*„³=rvƒ–’Œ‘k‹³‹Š€i‹†’²ƒŒƒ‡‰ZÆ-r‡gÁ‰Œ¾0ˆY’Œ‘¶r‡kƒ‡„³‰S”ig’²SkfWÏ|¹ Ì k*‘‡g‰|‰Slvk r‡gi„Šlj ‰i•}k‹5lv„³‘k¡„³rÑ‹³kI’²ƒ‡‹³{´lvgi‰ZÆ6lWr‡gik•}„³µGkƒ‡k‘k¡ˆQkr¼ÆÂkkŕ}krvkƒ‡j„³„Šl~rv„†‘ ’²•±lFr‡‰}‘–g’Œl~rv„†‘j ‰}•  k‹³‹Š„³B =ÐÔ,„ŠŒ€iƒ‡k 0?¿’Œ•rvg|€lҌ‹³‹Š‰ZÆ6lorv‰4‘–gikI‘‡ÇÆ0gikr‡gikƒrvgikg|{|ˆiƒv„†•Ojkr‡gi‰}•O„†lÒ²ˆi‹Šk¡rv‰´l~”QkkI•O€i” lv„³j€‹Š’²rv„Š‰ŒÆ0„³rvgi‰S€}rW‘‰Œj”iƒ‡‰Œj„†l~„Ši¡rvgikÇklv€i‹³r‡l¹ . m. m. . m. m. m. m. m. j. . j. . m. j. ßß ì.Hÿ–þ ¶ÿ. m. j. j. j. j.

(67) 0IŸ. E% "  . . . . 1!   $ " 1 .   . 

(68). . . . 1# ! 5. . . f-gik*ˆ’Œ‘rvkƒ‡„Š‰Œ”ig’ŒŒkdfWÏrvkIlFr j‰}•ik‹-‘‰Sj ”ƒv„†l~kIlÃr‡giƒvkk‘‡gikj „†‘’Œ‹-‘‰Œj”Q‰ŒkSr–l &°|„Šƒ‡’Œ‹Â=€‘‹³k„Š‘ S’ ‘„†•il+‘‹†’Œl‡lv„»ÍkI•&„ŠSr‡‰ Œk‰Œj„Š‘ =ЍŒk ?’²•rvkj ”i‹†’²rvk= r‡kj ?+’²Y•°|„³ƒ–’²‹Gl~rvƒ‡€‘¶r‡€iƒ–’²‹Q”iƒ‡‰²rvk„³Yl =Ðl~rvƒ‡€‘?¹ f-gik„Š}¾®k‘rv„Š‰Œ½”iƒ‡‰}‘kIlvl„†lj‰}•}k‹Š‹Šk•Åˆ|{Ål~„³ÀŃvkI’Œ‘¶r‡„³‰Sl

(69) =?f’Œˆi‹³k 0<?¶¹t¼Árvgikdlvk =€ik‹ ÆÂk&Æ0„Š‹Š‹L¾®‰}‘€l ‰Sdr‡gik‹Š‰2Æ·„³i¾®k‘¶r‡„³‰S‹Šk°Œk‹/‘‰Sƒvƒ‡klv”Q‰Œ•}„Širv‰ rvgk„Ši„³rv„†’²‹/=€ijˆQkƒ–l-‰Œ¾j‰S‹³kI‘€i‹Škl+rvkj = 1 Œk = l~rvƒ‡€‘ = 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!Zð–Ý%óA ¶ë

(70) û0 ü2 ¶ð–ì ùÐòôó ¶ì ùÐ ë~=ð–öAôï òúZvë ð‡ö?IòëF ¶ö?ìZëFóù? û0 òôìIòôó?ö?òôõ,û +ZëFï

(71) ò öAú-öAúZë,û0ë~ð–ìZó =öAúZë,³íZïôï +ó?ö õ ú2ð–ó?ö?òôõ5ð–ì Zò ŒëFù?ëFìö

(72) ú  !Zù?ò Ò lÆ+’Sllvgi‰2Æ0Á„Š ) ž3+ rvgkd•}krvkƒ‡j„³„Šl~rv„†‘&j‰}•}k‹‰²¾-rvgik*’²ˆQ‰2°Œk&lv{}lFr‡kj'”Q‰=lvlvkl‡l~kIl¿rFƉOlFr–’ r‡„³‰S’²ƒ‡{O”Q‰Œ„ŠSr–l = 0<?rvgikd”Q‰Œ„ŠSrrvkj = Œk = lFr‡ƒ‡€‘ = 0 Æ0gi„†‘–g½„†l€il~r‡’²ˆ‹³:k =®rvgik*lv{}lFr‡kj'Æ0„Š‹Š‹ j‰2°Œk ’ZÆ+’Z{¾®ƒ‡‰Œj „»r’²¾ rvkƒl~j&’²‹Š‹"”Qkƒvr‡€iƒvˆ’²rv„Š‰Œl ?W’Œ>• =?ž ?6r‡gik”Q‰Œ„ŠSrrvkj = 20 Œk = 200 ’²• l~rvƒ‡€‘ = 10000 Æ0gi„†‘–gO„†l¿lFr–’²ˆi‹Šk = r‡gikl~{}l~rvkjÓÆ0„Š‹³‹"ƒvkrv€iƒ‡´r‡‰*„»rÒ²¾ rvkƒlvj&’²‹Š‹,”Qkƒvrv€iƒ‡ˆ’2r‡„³‰Sl ?6’²• ’²r~rvƒ–’Œ‘rv„Š°Œk =®rvgiklv{}lFr‡kjØrvk•il-rv‰&ƒvkI’Œ‘–gdrvgi„†l6l~r‡’2r‡k „»¾"”Q‰Sl‡lv„³ˆi‹Š<k ?¹ Ò l’²&’²’Œ‹³{}lv„Šl"‰²¾Qr‡gik6”iƒv‰S”Qklv„³rv„Škl =ÐkŒ¹ ¹ ˆ|{ƒ‡€iii„Ši’Ãlv„³iS‹³k0ƒvkI’²‹Š2„ C’2r‡„³‰S ?ƒvk°Œk’²‹†l ƒ‡k’S‘¶r‡„³‰Sl ’Œ• g’²”i”Qkj€‘‡gOj‰Œƒ‡kþ®ƒvkI =€ik=rv‹Š{*rvgY’²´r‡gik‰Œrvgikƒ‡l l~€ŒŒkIlFr‡„³rvgY’2roÆ+kj ‰i•}k‹

(73) r‡gikj R •}krvkƒ‡j„ŠiR„Šl~rv„†‘’Œ‹³‹Š{ Æ0gi„³‹Šk¡rvƒ‡k’2r‡„³r‡gik ƒvkj’Œ„³„³i*‰ŒikIl¿lFr‡‰}‘‡gY’Œl~rv„†‘’²‹Š‹Š{Œ¹ԁ‰Œƒ¿‰Œ€iƒfWÏ&j‰}•}k‹ rvgi„†l ‘’²k°ŒkˆQkj&’Œ•ikj‰Œƒ‡k¿”iƒ‡k‘„Šlvk 5f-gikÔiƒ‡‰Œ”QkYl~„³rv„Škl0’2r0rvgikl~r‡’²ˆ‹³kÃl~rvkI’Œ•}{*lFr–’2rvk’²ƒ‡k = 0Iž ? a =5 a =5 a = 20 a = 15 a = 20000 a = 19900 Æ0gi„†‘–g*‘‹³kI’²ƒ‡‹³{&l~g‰ZÆ·’¡l~k”’²ƒ–’2r‡„³‰S&‰Œ¾/rv„Šjkl‡‘’²‹ŠkoˆYkr¼Æ+kkdr‡gikor¼Æ+‰ lvkr–l‰Œ¾

(74) ƒvkI’Œ‘¶r‡„Š‰Œl¹,f-gik¿ƒ‡klv€i‹»r–l ¾®‰Sƒ 10 ƒvkI’²‹Š2„ C’2r‡„³‰Sl¿‰²¾+rvgik‘‰Œƒ‡ƒvkIl~”Q‰ŒY•}„³ig={|ˆiƒ‡„†•±j‰}•}k‹’²ƒ‡klvgi‰2Æ0ń³½Ô,„ŠŒ€iƒ‡kž}¹ Ò lĊ )ô.ž 3+ ÆÂk&•}„Šlv”i‹†’I{ƒvkIl~€i‹³r‡l¿ƒ‡k‹†’2r‡k•´r‡‰*rvgk=€ijˆQkƒÃ‰²¾r‡kj j‰S‹³kI‘€i‹Škl¹t rÑ’²±ˆQklvkkOrvg’²r¿r‡gik&•}„†lFr‡ƒv„  ˆi€}r‡„Š‰ŒlÉSˆ}r‡’Œ„³ikI•±Æ0„³rvgÅrvgk&g={|ˆiƒ‡„Š•Åj‰}•}k‹’Œƒvk&’²‹Šj‰Sl~rij•i„Šl~rv„ŠiŒ€„Šlvg’²ˆ‹³k¾®ƒv‰Sj r‡gi‰Slvk‰Sˆ}r‡’Œ„³k• Æ0„³rvgrvgikW¾®€i‹Š‹³{lFr‡‰}‘‡gY’Œl~rv„†‘0j ‰i•}k‹A¹"f-gik¿’Œ•}°²’²Sr–’²Sk0‰²¾%rvgkWg|{=ˆƒv„†•&j ‰i•}k‹Y„†l’‘‰Sl~„†•}kƒ‡’Œˆi‹³kWl‡’I°|„Ši ‰Œ¾ÂÕŽn rv„Šj k¡„Š|€ijkƒ‡„Š‘’²‹,l~„Šj€i‹†’2r‡„³‰Sl¹oÔi‰Sƒ6r‡gik¡l‡’²ÇŒk‰²¾‘‰Œj”’Œƒv„†lv‰Œ ÆÂk¡g’I°Œk¡lv„³j€i‹†’2r‡k•4r‡gik ¾®€i‹Š‹Š{½l~rv‰i‘‡g’SlFr‡„Š‘d’Œ•Ár‡gik*g|{|ˆiƒv„†•½j‰}•ik‹†lˆ’Sl~kI•½‰Œ7k„³rvgikƒ¡rvgik4•}„³ƒ‡k‘r¡jkrvgi‰}• )20Ÿ 00+-‰Œƒr‡gik ikÀ|r ƒvkI’Œ‘¶r‡„Š‰Œ½jkr‡gi‰}• ) 3+A¹½f-gik*‰ŒˆYl~kƒ‡°Œk• ÕŽn3rv„Šjkl ’Œƒvkdƒ‡k”Q‰Sƒ~r‡k•Á„Š f,’²ˆi‹Škž}¹Át r’²””Qk’²ƒ–l r‡g’2rrvgikg={|ˆiƒ‡„†•Ålv„Šj€i‹Š’²rv„Š‰Œl’Œƒvk&’ŒˆY‰S€}r 100 rv„Šjkl’Œl¾Ð’Œl~r’Sl¿r‡gik¾®€‹³‹Š{Ål~rv‰}‘–g’Œl~rv„†‘ ‰Sikl¹df-gik g|{=ˆƒv„†•4l~„Šj€i‹†’2rv„Š‰Œ4„†l0ˆ’Sl~kI•*‰S4’&“0€iŒk €}rvr‡’„ŠSr‡kŒƒ–’2r‡‰Œƒ0‰²¾"‰Œƒ–•}kƒ 4 Æ0„³rvg‘‰ŒYlFr–’²Sr0r‡„³jkl~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¶ð–ë~íZõ¼ïôï ù?ö?ë$òã ¶ìùÐë~ð–Zõ¼òôóÐöAöAò ¶!ZùÐòìZùÐ!Zòó íIö?òò ¶ôìZìã ë¼ ö?òôiõFö?óû ëFû. ¶Zù%ûëFöAïôúIïô ¶ë

(78) ë ïô ëF!2õFIíZð–ëFïôõ¼ö?ëFöAëFó ëFù?ù?û0 ò!2 ¶òôð–üZìIó?ú2òôë

(79) ó?ð ö?¶òôõvë ¶ð–ì ïôï  ýû õ

(80) ¶ +ùÐû0ëvZð–ëFü2ïôï òð–~Šù?ð‡ùÐë

(81) ö?ë~0ò𖠶õ¼ö ìZöA Ló ò ¶ö? ìZúZó ë,ùÐë

(82) ³ëFã ù?ëFìZõFã ë³íZïôð–ï +ì  óÐö õ ú2ð–û óÐöA +òôZõ,ëFû ïôïôë WIë~ó?ïö| õAü ú2 ¶ð–óÐó?ö?çç R1 R2 R3. R4. R5 R6. 104. •i’Z{ )ô™ 0ZÏ 07+ ¹½t¼7‰Sƒ‡•}kƒrv‰±°Œkƒv„³¾®{År‡gik4’Œ‘‘€iƒ‡’S‘{ʼn²¾6r‡gikk°Œk=r•ikrvkI‘¶r‡„³‰S ÆÂkkÀ  ”i‹Š‰Œ„³r6rvgikkÀ}„Šl~rvk‘k‰²¾5’Œ´’²Y’²‹Š{Sr‡„Š‘’²‹

(83) l~‰S‹³€irv„Š‰Œ¾®‰ŒƒWg={|ˆiƒ‡„Š•4k =€’2r‡„³‰Sl-¾®‰Œƒ S = {1, 2, 3, 4} ¹-ÄWk‘k lv‰Œ‹Š°|„³i¡rvgikk°ŒkSr0kI S€Y’2rv„Š‰ŒYl = 3 ?‰Œƒ = 00?-‘’²*ˆQkҌ‘‘‰Sj ”‹³„†l~gk•ˆ|{d’¾®kƸm6kÆ-rv‰S*„³rvkƒ‡’²rv„Š‰Œl ) 7+ ¹ f-gikƒ‡klv€i‹»r–l+‰Sˆ}r‡’Œ„³ikI•*’²ƒ‡ko°Skƒv{dl~„Šj„³‹†’²ƒ0lv€iŒSkl~rv„ŠirvgY’2r-rvgkk°ŒkSr6•}kr‡k‘rv„Š‰Œ„Šl-”Qkƒ~¾®‰Sƒvjk•dÆ0„³rvg lv€ ‘„ŠkSrW’S‘‘€iƒ‡’S‘{S¹. δt = 0.01. ßß ì.Hÿ–þ ¶ÿ.

(84) 0Zž. E% "  . . . . 1!   $ " 1 .   . 

(85). p´‰}•}k‹ œW„Šƒ‡k‘¶r6jkr‡gi‰}• m6kÀ=r6ƒ‡k’S‘¶rv„Š‰Sdjkrvgi‰}• Ôi€‹³‹Š{*l~rv‰}‘–g’Œl~rv„†‘ 0I™7ŒŸŒŸ¡l ž0Iž²ŸSŸl Ä6{|ˆiƒ‡„†•*j ‰i•}k‹ 0;S›¡l žŒŸ0l S = {1, 2, 3, 4}  õ

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