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Deterministic approximation of stochastic metapopulation models

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(144)   "! $# % &!. 6K13;M13@k6H=G;>8,@”° ™ ;>/5Qm1HA3;>/G6K4N1H47LU/AH;M13@C6“Z>4N[U@C/¬^f‚ ‰KG yHG *  O ‰$G ( ,

(145) yLG *  ;M1“A3;M1H@ G:

(146) (M  ‰KG yHG *  O ‰$G yHG * ( ,

(147)  ;M1“A3;M1H@ G:*

(148)  (NG7

(149) ‰ (M 0 ‰KG yHG *  O ‰$G D,

(150) yHG * ( ,

(151)  ;M1“A3;M1H@ G:

(152) OM (PFG:

(153) 0‰Kw!"5 ‰KG yHG *  O ‰$G PG7

(154) ,

(155) yHG * (QG:

(156) ,

(157) ·;M1“A3;M1H@ & ‰KG yHG *  O ‰$G D,

(158) ( , $ yLG *  ;M1“A3;M1H@ "R ™ 0 G7

(159) duLUAF;>V7VŸw > tH'y R > o~}"c2G@—=5AHLa8,@,6K6SG 476F„q;>AH ML[?47;>/J2G;4N/GZƒ1H2G@_6H;>R@“ZU@C/G@,AH;M13LUAF;>6= ³a4Nd G * ‰ ¢ žv ¢ 1H2G@k=GAHLa8,@C6H6PG * 476)=bL>6H4N1H4N[>@ˆ‰132G;M1)476TG:*

(160) ¡ ¢ duLUActžvxw>y{z{z{z3o EduLUA“;>V7VTš L 3 ¢ y=<>7X} . Ud G ‰ ¢ “2G;>6—132G@96H;>R-@\QG476p13AK47^GO51H47LU/q;> 6 ‰ ¢ :Br1H2G@,/'1325@9=GAHLa8,@C6H6 : ™ v ¨

(161) ™ ‰KG:

(162) (VG:*

(163) ‹“4N6 ;m=5OGAH@^G47Ap132Ž=GAHLa8,@C6H6kD)4N132 *-,/. 02/*35492¸A3;M1H@ †W³ ² * ™ > : ™ B;>/GX Q ² * ™ ‰ ¢ _© 2G0 ;>6”1H2G@-63;>R-@ QG476p13AK47^GO51H47LU/LMLd : ™ ‰ ¢ :} B`@,8,;>OG6K@ƒL>C d YjAHLU=TLU6K4N134NLU/¸A‰ .a2} .?}swM3B5D`@”2<;{[U@ e'ª bd6HOGc=bfe ² * ™ ‰4grA7 >  3ª'bd6KOGc_=bfe : ™ ‰4gŸ/7 2 5h 4 ,%5TiK7f5h,/.j#&7 .Jk0 *]# 4 ,(!3,!144 , ,%2)*3M.302/.2 ,265%²€™ ‰ ¢ A 2&!LlH5V354 , ), m *-,B0 42R4 3n#&5ko 6 w ,qp e `$‰ : ™ ‰ ¢ Hž`v `‰‹² ™ ‰ ¢ H ,qp e  36HO5= ² ™ n‰ grA 7 > o ª'bdcbfe. Z

(164) 8

(165) 8[[\] _^`a.  3. Z

(166) 8

(167) 8[[\] _^`artsuwv. 3,! 21*]# !354 3x , )0 #&5y! 492654z 44 ,%5 w F‰‹² ™ ‰ ¢ H , p,| {'3 ² ™ ‰š3 ( v 7 > v.

(168) w.  

(169)   $ . `@C8{;>OG6K@ƒLMd .?}F.?}swUB]{'3 ² ™ ‰uš3 ( v@7 > { 3 : ™ ‰š3 ( vƒo 7 } <LUAƒ;¬’5–?@,Q‘š|BžV7@C1€OG69;>=5=GVN‚'1H2G@„q;>AK >L[47/G@C±aOG;>V74s10‚Ž1HL¬1H2G@ =TLU6H4s134N[>@AH;>/GQGL>R [M;>AK4];>^GVN@ : ™ ‰uš3 {'3 :ƒ™‰uš3 ( v€J o 7 > vƒw o `$‰ :”™q‰uš3H+v vƒw o `$‰ :”™q‰ ¢ K , p,| v v w F‰X²\™q‰ ¢ H , p,| > v w `‰X² ™ ‰ ¢ K , p duLUA);MV7Vbš LD3 ¢ =y <>7X} ?A@FB B DE. B. . . c2G@kR;>47/=GO5AH=TLU6H@”L>de132G4N6c8:2<;>=51H@,A“476g1325@”=5AHLaL>deL>d

(170) . a suˆ˜ ™ ª ¹ ª 35 h 3,! 45.3n9%7-453n#&5yz < 3544Ž6HO5= ™ ¨œ)b 0   ´ ˜ œ™ ‰ ¢ 4& * +›zF,Uz z 4 ,%5q˜ ™ ¹ 35‰ 9 3 ¢ =y <>7 § 35 h 3,! 4 .3n9%7 4 3n#&5yz <  ,/.,g¹ 3,! 44 ,-7f5V3 7L, *]# !35453x ,-!#6.17 453n#&5 # u 4 , 7267L0%k01*.'#k9. ,%2 ¹

(171) ‰š3+v  3‰B

(172) (WO

(173) (PŸIt (W&#(P"R ¨ §   ¹ œ ‰š3 q¹L

(174) ‰š3 t ¡ w œH©Gª + ( B

(175) %N 0 ‰t SwM (W"R ¨ §   ¹ œ ‰uš3 L¹

(176) %N 0 ‰uš3 œK©Gª ( 3 O

(177) 0 (P 7U‰‹. t ( wM†-¹

(178) 0 ‰uš3 ¹ ª ‰š3+v &r‰p!w ¦ q¹ ª ‰š3K ( ‰  0 (P¦ Ÿ7¹ 0 ‰š3 "R ¨ §   ¹ œ ‰š3 q¹ ª ‰š3 œH©Gª ¹ ‰ ¢ žv ¹ ª. .

(179) . . . . . .   . . .        . .  . . . .   . . . . . . . . . . . . . . .

(180) 

(181)       !"#$  ¢   \!+"*,   +##? µ OGA`;M47R47/ˆ1325476`6K@,8P1347L>/J476j13Lƒ6H2GLD£132G;M1`6KLUR-@Š=GAKL?8C@,6K6H@,6CB5duOG/G8P134NLU/G6`LMdŸLUOGAE=GAHLa8,@C6H6g˜ ™ B ;>AH@YR;>Ap1347/5Zf;>V7@C6`D)4N1H2%AH@,6K=b@C8C1“13L91325@”/G;M13OGAH;>Vi’<Vs13AH;M134NLU/ | ™ v ­†˜ ™ ‰ % :y ¢ > % > š:®I} ¤W@\AK@,8,;>V7V6HL>R@ƒAH@,6KOGVN1H6—;>^TLUO51—„q;>AK >L=5AHLa8,@,6K6H@C6k;>/GQqR;>AK1H47/GZU;>V7@C6,} ž47AK6K1“D`@€2<;{[U@ 132G@YduLUVNV7LD)47/5Z\@P–?=GAK@,6H6K47LU/mduLUAg1325@”Z>@,/G@CA3;M1HLUA SL>dž;ˆ„q;>AH ML[€‡KOGR=%=GAKL?8C@,6KA6 G | h ž‰ a+v a‰ a 3 ‰ <  ž‰ IA 7 j‰ 5y  <:z /‰ .?2} Ef D)2G@,AK@ q^T@,VNLU/GZU6g1HLˆ1H2G@kQGLUR;>47/LMd1H2G@kZU@,/G@CA3;M1HLUA,} .0/132G@_duL>V7V7LD)4N/GZƒ13;>^GV7@MB?DF@YR@C/f1347L>/ 132G@—=TLU6K4N134s[U@_[M;>VNOG@,6`L>d132G@Y=GO5AH@,Vs‚ ;Œ13LUR-478ŠR@,;>6HOGAK@ WAK@,V7;M13@CQ%1HLˆLUOGAc=GAKL?8C@,6H6“˜ ™ a‰ a "‰ 5y  < .. . . . . .

(182) . . .  .   .  .  .  . . . . .  .  . !. ™ 0 /‰ ,

(183) ¦ 0 D,

(184) . o ˜L

(185) 5B

(186) 7t t ¡ w. ™ 0 ‰/,

(187) N 0 D,

(188) . o ˜L

(189) 0‰O

(190) ]t(ž‰pw!"50tp t ¡. ™ 0 /‰ ,

(191) ¦ 0  ,

(192) ( ™ 0 ‰/, U/N 0  , U . oQ"RTS?˜VU˜L

(193) tHyBS ¡ w. ™ 0 /‰ , ª D,

(194) . o ˜L

(195) M& t ¡. ™ ‰, ª  , 0. o ˜ 0 ‰&#(P 0 ( ‰Kw!"5 Ÿ. 0. .. .. . 1“476c I/GLD)/¬1H2<;M1{B54Nd ¸4N6);9^bLUO5/GQG@,Q%duOG/G8P134NLU/rBG1325@”=5AHLa8,@,6K6 v| ž$‰ G |   žK‰ G ª   | žK‰ G e   % ‰/.?}FI ª 476”;¬R;>Ap1347/5Zf;>V7@M}.0/ _;>R {;†l YV7@C^<;>/G@CCA 3  EG7F;>/G#Q 6ŸOG8:256H47/5ZU@,A3 6ŸOG8/87XB1325@duLUVNV7LD)47/GZ 6HO -8C47@,/f1c8CLU/GQG4s13LU/56g;>AK@—ZU4N[>@,/JduL>AF1325@_47/f1H@,ZUAH;>^G47VN4N10‚9LMdŸ132G@Š=GAHLa8,@C6H6 ž$‰ G | ";>/GQJ1H2G@—R;>Apl 134N/GZf;>VN@—=GAHL>=b@CAK10‚L>d | BGduLUAcO5/a^TLUOG/5QG@,QmduOG/G8C1H47LU/56 } ". #%$ &. ('. ). +*. . . . ,'-*. .. /. "#. $. /.

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