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0.8
Probability
0.6
0.4
0.2
0
2
5
x 10
1
Time
0
0.55
0.5
0.65
0.6
Transmissibility
:;<2G=7M5 s v >=5k/061P3657M<251B/5O02FAS02S=G=HDA(8M:I061rTY:DCo897M:I?=G=89:;021Q890Ad1A2C9-4zS=890d8M:;/EH;:Iz:I8
Percentage Parasites
30
25
20
15
10
5
0
0
0.2
0.4
0.6
0.8
1
Virulence
0.8
1
0.6
0.4
0.2
0
Transmissibility
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14
12
10
8
6
4
2
0
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Percentage Parasites
60
50
40
30
20
10
0
0.2
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1
Virulence
1
0.6
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8y:;CEB7[CZ8y1=5!/5CMCMAd7M-e8M0QTY5B1=5@89>=5 D Y87RI 7:<MHA7cY8Q Q JVB 7 R[PGI 9 ? F#027E89>=53(AdH;G=5CX0dF <27M0 W 89>nV897[Ad1BCol
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C95/L8M:I061p89>B7955EA2CMC9G=zSY89:;021BC W :IH;H ?V5EzpA2TY5 8907M5T=GB/5 89>=5XCo: 5E0dF`89>B5ESBA27MA2z58957 C9SBA2/5 89>=5XTY5Ad89>
7[A(895 :DCC90C9zpAdH;HnA2C.8M0?5k:;<21=06795!TR5~=/5SY8O:;1Q89>=5kAd?VCo51B/5y0dFSBA27MA6Co:I895!C4:;1YF#5/895TQ>=0PCZ8[CAd7M5E1B0d8
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0dF HI0 W 57E3P:;7MG=HI51B/5QJ 8M>=5/[>VAd1B/50dF.TY-4:I1=<eF#7906z89>=5TY:DCo5!A2C95:DC H;5CMCMN 0d8M>=57y/A6Co5!CEC9GB/[>A6COF A6CZ8M57
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] J J N N ! v >=:DC:DCq02?434:I06GBCoH;-RA795!A2C9021BA2?=H;5EAdSBS=790{~Y:;zpA(89:;021R89089>B5yC9SBAd89:DAdH`z0YTY5H 8 Co>B02G=HDTe?5k1=02895Te89>VA(8
89>B5 8M>=50679589:D/AdHV795!CoGBH8 :DC./021P8[Ad:;1=5T?58 W 551p89>=5 A21BT 2 /021P8M02G=7[C&Ad1VTp?V5!/06z5!C A?V58o8957
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b
1
1
0.8
0.8
0.6
0.6
Ratio
Ratio
a
0.4
0.2
0.4
0.2
0
1
0
1
0.8
0.8
0.6
0.4
0.2
0
Virulence
0.02
0
0.04
0.06
0.08
0.6
0.1
0.4
0.2
0
Virulence
Growth Rate
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.02
0.06
0.08
Growth Rate
0.5
0.4
0
0
0
0.04
d
1
Virulence
Virulence
c
0.02
0.04
0.06
Growth Rate
0
0
0.08
0.02
0.04
0.06
Growth Rate
0.08
;: <2G=7M5 s v >=5@/02zSV589:I89:;021K?58 W 551,890r>=06Co8XC9S5/:;5C`Co>B0 W :I1=<R89>=5897[A2TY5@0 w?V5l
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v >=5!Co57M5C9G=HI8MC W 57M5q02?Y8[Ad:;1=5T021pA {d {d <27M:;T` W >=57M5 u 6 :8M57[A(89:;021VCCo>=06G=HDT@?5 C9G /:;51P8
8907M5z0(325XAd14-RC9SV5!/:;5C.89>VA(8OTY5/A{-YC.Ad8qA@F A6CZ8M57q7MAd895E89>BA21 , Ad1BT 62 :8M57[A(89:;021VC.C9>=06G=H;T
7M5z0(325A214-QC9SV5!/:;5C 89>BAd8yT=5/A{-YCOF A2Co8957 89>BA21 , M9 -258X/02lm5~Y:;Co8951B/5:DCXCo551K0(3657 ARF Ad:;79H;W :DTY5SBA27MA2z58957O7[Ad1=<65 8MGBTY-4:I1B<e5UPGBAd89:;021 =F#027 89>=5C9A2z5897[Ad1VCoz:DC9C9:I?B:IH;:8Z-QAd1BTK<27M0 W 89>7MAd895

?=GY8 A>B:I<6>=57q34:I7MG=H;51B/5RJ N >=0PCZ8EuTY5!/A{-YC F#7906z 5!U6GB:IH;:I?B79:;G=zjH;: 652s
J oN J N J NJ J oN N
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v >=5795F#027M5:I1K89>=5@>=02z02<651=502GBC /A2C95 W 5C9>=02G=HDT,061=HI-C955@/0dl 5~Y:DCZ8M51B/5A(F 8957:I8957MAd89:;021BC@J W :I98 >
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B 6 n F#027 ] 26
W >=:D/[>:;CAz027M5y795!CZ8M79:D/L8M:I365X7[Ad1B<25E89>BA21QC9551Q:;1QB<2GB795 ZqA21BFT 9
F061=5_/A2C95a:DCQ5~=Adz:;1=5T:I1 z027M5_TY58MA2:IHpJ :;<2G=7M5 u2NLX:I8Q:DCQ1=0d8M:;/5T89>BAd8AdF 8957Q8M7MA21BCo:;51P8
?5>BA{34:;02G=7E89>=5pTY5!/A{-,02F >=06Co8uR:DCyCoH;0 W A21BTKH;:I1B5Ad7 v >=5H;:I1B5Ad7yTY5!/A{-KAd1VTK:;1B/7M5A6Co5!T,89:;z5CX0dF
/02lm5~Y:;Co8951B/5?V5!/06z5pGB1BTY57[Co8MAd1VT=Ad?=H;5 W >=51w:I8:;Cy7M5A2HI:DCo5!T,8M>BA(8k8M>=5R/06zS58M:8M:I061w?V58 W 551a8 W 0
>=0PCZ8C9SV5!/:;5Ck/A21a061=H;-,04//G=7A2HI061=<Ae1BA2797M0 W ?02G=1BTBAd7M- v >=5HI:;1=5A27TY5/A{-KCoGB<2<25!CZ8[CE89>BAd8X8M>=5
3206HIGBz502FV8M>=:DC&?BA21BT0dF`/06zS58M:8M:I061p795zpAd:;1BCF Ad:;79H;-@/021BCo8MA21P889>=7M02G=<6>=02G=889>=5 C9:;zG=HDA(8M:I061BC v >=5
zpAd:;1RUPG=5Co89:;02189>VA(8 A279:DCo5!C1=0 W :;C W >4-:;C89>=5O>B02z02<651=502GBCz0YTY5HAy795!A2C9021BA2?=H;5.A2S=S=7M0{~4:;zpA(8M:I061
F#027>B06Co8O?V5>BA{34:I06G=7Ad1BTr1=0d8F#067SBAd7[A2C9:I895C A6/[>rSVAd7[A2C9:8M5X>VA2CA32579-rCo>=067o8OH;:IF#5y89:;z5rJ#GBC9GBAdH;H;-e021B5k:8M57[A(89:;021NqAd1BTC905~YSV579:;51B/5CHDAd7M<25
CM/AdH;5 BGB/89GBAd89:;021BC{AdHDCo0:I8MC 8M7MA21BC9z:DCMCo:;021X:;C 32579- T=5S51BTY51P8 021y:I8MCnH;0Y/A2Hd51434:I7M021Bz51P88M>=:DCn:DC C9551
:;1r8M>=5kCZ8M0Y/[>BA2Co89:D/ 1VA(89GB795y0dF 89>=5k7M5C9G=HI8MC 89>=5!Co5yF#5!A(89GB795!C.A2795X1=028qz0YTY5HIH;5TQ?4-p89>B5X>B02z02<651=502GBC
5UPGBAd89:;021BC v >=5>=06Co8MC&>=0 W 53257HI:;325qF#067F Ad7&H;021B<257&A21BTpC90X5~YSV579:;51B/5OAk1=5:I<6>P?02GB79>=040YT W >=:;/[>:DC
A 895zS027[AdHYA{3257MA2<256>=51B/5 89>B5.7M5C9G=HI8MC02?=8MAd:;1=5!TAd7M5&F A27Coz040d8M>=57 f >=51/02zS589:I89:;021@?V58 W 551
>=0PCZ8[C0Y//GB7O:8E:;COA2H W A{-YCq:I1,A1BA2797M0 W ?VAd1BT`BGBCoGVAdH;HI-r021=H;-rAF#5 W /5H;H;C A2/7M06CMC=Ad1BTC90p?V0289>,>=0PCZ8
8Z-4SV5!C5~YS57M:I51B/5kAC9:Iz:;H;A27S=7906SV067o8M:I061e02FSBA27MA6Co:I895!CqTY5C9S=:I895y89>B5kH;A279<65XCM/A2HI5kC9SBA(8M:;A2HnCZ8M79GB/89G=7M5
$
", / ,
14,.-0/ ,
FnAd1=0289>=57&HDA{-25702Fn/02zS=H;5~Y:I8Z-@:DC&A6T=TY5T8M0k89>B5 Co-YCo895zQ6?4-@>BA{34:;1=<y8 W 0kSBAd7[A2C9:8M5CA21BT8 W 0k>B06Co8MC
89>B51eAk7M:D/[>=57 C958 0dF`?V5>BA{34:I06G=7[C:;C&F#02G=1VT 0dl 5~Y:;Co8951B/502F`?V0289>R>=0PCZ8.CoS5/:I5!C Ad1VT?V0289>RSBA27MA6Co:I895
C9SV5!/:;5CE:;CX1=0 W S06CMCo:;?=H;5rJ A6C SB795!TY:;/895!TK?4-z5!Ad1,V5HDT,5UPGBAd89:;021BC[N W >=:D/[>K<6:I365CE79:DCo5890RF A27Ez06795
/06zS=HI5~RTY-41BAdz:D/C 5B1B5 8M0?5O89>B5ES=7M02S027989:;02102F /5H;HDC 0Y//G=S=:;5Tp?4-pAk>=5!AdHI89>4->=06Co8 [YAd1BT
H;58 ?5k8M>=5S=7M02S027989:;02102F /5H;HDC W :I89>Ap>B06Co8 q:I1YF#5!/L8M5T W :I89>KSVAd7[A2C9:8M5 . v >=5S=7M02S027989:;0210dF
>=0PCZ8[Cq:I1YF#5!/L8M5T W :I89>?V0289>SBA27MA6Co:I895!C :DCq32579-eHI0 W Ad1BTe89>=5795F#027M5k/Ad1Q?5k1=5<6HI5!/L895!T
?
CA215~=AdzS=H;5y0dF 8M>=5k?5>BA{34:;02G=7OC9551n489>=5yF#06HIH;0 W ;: 1=<pCo-YCo895zCo>BA2HIH`?5k5~=A2z:;1=5!T`
] J = 6 M= 6 N
] J M N
]
]

Proportions of Host 1, Host 2 and Parasite
0.6
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
Time
:;<2G=7M5u=s F 8957k:;1=:I89:DAdH8M7MA21BCo:;51V/5>VA2CyTY:;5Ta02GY8! A(8kAd7M02G=1VT d :8M57[A(8M:I061BC8M>=57M5
:DCAC9HI0 W H;:I1B5Ad7TY5/A{-w0dFO>=0PCZ8Ru W >=:;H;5e8M>=5eSBA27MA6Co:I895rSV06S=G=HDA(8M:I061795zpAd:;1BCF A2:I7MH;/021BCo8MA2168 ] ]= 6 ] ] B ]|B 2 ]=
a
b
0.65
0.025
0.6
0.02
Parasite1 and Parasite2
Host1 and Host2
0.55
0.5
0.45
0.4
0.35
0.3
0.015
0.01
0.005
0.25
0.2
0
1000
2000
3000
4000
0
0
5000
Time
1000
2000
3000
Time
4000
:;<2G=7M5nBs X7MA2S=>C9>=0 W :I1B<e/[>BAd0289:D/0PC9/:IH;H;Ad89:;021BCOF#027E89>=5@8 W 0R>=06Co8kCoS5/:I5!C@J A6NOA21BT
8M>=5y8 W 0SBA27MA6Co:I895!CqCoS5/:I5!CJ#?VN

5000
= = ]
] == u == u
= = v >=5795!CoGBH8[C W 57M506?Y8MA2:I1B5TK061A 2 2 H;Ad8o8M:;/52 W >=:D/[> J GBCo:;1=<r89>=5z58M>=0YT:;1 >BA2SY8957 o N
:DC&8M>=5X/02>=57951B/5EHI51=<d8M>eCM/AdH;5 v >=5yCoSVA(89:DAdH C9-4Co895z :;C./[>BAd0289:D/J#89>=:DC W A6C /[>=5!/ 25T?4-p/A2H;/G=HDA(89:;1=<
89>B5 -PAdS=GB1=0(3R5~YS021=51P8[N W :I89>KHDAd7M<25kA2zSBHI:I89GBT=5@/-Y/HI5!C0dF&Ad7M02GB1BT n :I8957MAd89:;021BC W >=5795!A2Cq8M>=5
z5A21rV5HDTe5!U6GVA(89:;021VCs
] 3 " t 3 3 t 3 t " 3 " t " " [ J J N (N
J N J N J J N N J J !N N ] 3 " t 3 3 t 3 t " 3 " t " " [ J J N N
J N J N J J N N J J N ] (J9J J J N N (N
] (J9J J J N N (N
] J9J J J N N (N
] 2J9J J J N !N (N
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D
D
Parasites
0.06
0.04
0.02
0
10
8
500
400
6
300
4
Range
200
2
100
Time
0
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d
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b
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0.8
Waveform
Waveform
a
1
0.6
0.6
0.4
0.4
0.2
0.2
0
-30
-25
-20
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Distance
0
5
10
0
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15
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1
0.8
0.8
0.6
0.4
0.2
0.2
-40
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Distance
5
10
0.6
0.4
0
-50
0
d
1
Waveform
Waveform
c
-10
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Distance
0
0
-100
10
-80
-60
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Distance
-20
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0.7
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Proportion
1
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
100
200
300
400
0
0
500
50
100
Time
Time

150
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140
Infectious
120
100
Pathogens
Proportion
a
1
Exposed
Recovered
80
60
40
20
0
0

5
10
15
20
25
30
35
40
45
Time
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436587 log Parasitoids
0
-50
-100
-150
-200
-30
-20
-10
0
log Hosts
10
20
30
+ GF H>ÌCD? l q V8?{CXEW8FG[ J _ ; F aOa X}ADF J =8_ JEU Y J ADV J _%A-XE=8[9
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:
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2
0
-2
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2
0
2
0
-2
-2
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-4
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-6
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Average Time to Extinction
10
4
10
3
10
2
10
1
10
+ FOHKÌC/?
0
10
20
30
40
50
Number of Cells
60
70
80
90
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1
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Percentage Abundance
10
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10
-2
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Lengths
-3
Times
10
-4
10
-5
c
10
0
10
20
30
40
Size of Lamina
50
60
70
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0.2
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0
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0.035
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Summed Errors D
0.04
Column 2 of US’
15
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=- 32 0
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I
!
a
b
3
3
10
Number of Exposed and Infectious
Number of Exposed and Infectious
10
2
10
1
10
0
10
2
10
1
10
0
0
5
10
Time (years)
15
10
20
0
5
10
Time (years)
c
15
20
d
0.14
3
0.12
0.1
Fourier Value
Number of Exposed and Infectious
10
2
10
0.08
0.06
1
10
0.04
0.02
0
10
0
5
4=E.,05+
10
Time (years)
15
0
1
20
2
2.5
3
3.5
Time (years)
4
& 69 - + ..3 +I0?69L4=.*P+ AT-!4=6./,H+6+24. F6)4.4#3-!4=69.R6 9+ 14
+=467. .VE909 :-!0569.,E- 05++2@9+90GA@*A+2$90!+ AB=+/0&=@+IN#4#+I.7- +05+) 4- <4=E -&@A"+=+0 A69.#-&7 AT-U03 -5+ +K- ++. <0!+ ! 5A 6*6LA 4
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J09
1.5
I 4.5
5
a
0
Proportion Susceptible, Infectious and Resistant
Proportion Susceptible, Infectious and Resistant
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
10
0
5
10
Time (years)
15
20
0
c
0
5
10
Time (years)
15
20
15
20
d
0
10
Proportion Susceptible, Infectious and Resistant
10
Proportion Susceptible, Infectious and Resistant
b
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
10
0
5
4=E.,05+
10
Time (years)
15
20
0
5
10
Time (years)
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5
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0
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3
10
00
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4#&A /69-!4BAX.<E94 /0 J05+.*P++ /. @ -
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- + A 96 #4 -
.<+0
//. C6 9+
0 = +":@*:-!+ ,+5AK054 + +I0!+".,.*P690!-<.<.3-5+=@ $ 90!E7+`.=. +0?6 +E70!+I+
69 P0!+I+ 6 ;- +I0!;+ /05+ 940 #4 :+ AK690505+#3-!4=69./$/./ :4=.,E)+C,0!6 4+=4-54+2 6 +N7+08*.,+2-!6
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D4
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10
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9
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+*H69.,+I.#-?A/. +94,056 4/-!+ @
= . + !5 ( = . +I. - + ;@)4,. .,63NJ+*H 69.,+I.#-Y4# A9B AB.
/ -!+ - +N:9
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0.25
Lyaponov Exponent
0.2
0.15
0.1
0.05
0
0
4=E.,05+
50
100
150
200
Time (days)
250
300
350
6 4.<ER- +A 9.,E9+4=. 8@)9.,.,63N +*F67.,+.#- 63N9+0 - +SAK6).,0&!+
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.,+ 4
94,056 4/-!+=@
32 $/./:6?- + !@:-!+
4
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Proportion of Hosts
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.,+ 6 -&94.,+2?.!4.<E 4#% E96*6*?94,056 4 /-!4=69.-!6 - + AK+ +
.4#/0
9.,-!6 /-!69.SN:9
.,+ .-&- +4
5AK05+//.<A@ -54+
;.,+I+4+ 94.,4=.,E
32
20
03 +I0!+/05+- 0!+I+4=.7AAB.<0&7AK4=+24.#-!056**.<A+ @Z- + <. AT-&.3 -5467. 1+- 6* + 0&:-14
2- 3RAK67.:-&/.#-"5A
90 4
.:+2O-!6Q05+,05+!+.#- 3-50!4O6A69050!+
/-!4=69. 4BA 4+
LN:/05@ 4- - +
.,+I4E 6<. 0 6*6* -@H+ 4#N:/05@*4.,E3-50!4 AK6).4
+J4=.<A690&F670&/-!+ 4.#-562- + .<AK-<./-!4=69. 6*+
. -Y- + + #-50&? AK6(4= +4-@7/ .> - +, 4 6AB. -@`4=.U6 -&/ 4=.,4=.,E%+7 - +9 0& 1+K-!+I0&8A/.<.,6/- + 0 .:-!4 +
@ - + " +H 4.FAK05+! +&4=.7 AAB<. 0&7 AK@ E)/ 4=.,+2 +8! +IA69. +I0!05690D4
C A.: +2 @ - +% ,. .<AK-!4=69. *4- 4#
): +2 69. - + +2/ . +#&<3. ( -!4=69. / ./ 6*+ .,6/- -& 9+?4=.#-!6;9 AA6,. .#- - + .,4-!+J.3 -&,. 05+?69 - +
.,+I4E 6<. 0 6*6* ' ; - +?-56/-& .,+4=E 6,. 0 6*6*1 4
69; -@H + #- +.A69.! 4# +054.<E`- + + N*4=6,. 0 69 AK+I.#-!05+JA+
H 6; -@F + Z- +>N1+<. +G6; - +>05+ / 4=.,4=.,EC<,. 050!6),. . 4=.,E AK+
# 6).4
+ 16* 4 + *,. +G-!6
N*4=.,E +I+.O0!+ 63N7+P 056 4#, 7 AT-!670?4#& 67.4 @4(F 670:-/ .#- +I.VAK67.: 4# +I0!4=.,E - +C7 AT-!4=69.
69] - +>.,+I4E 6),. 0 6#6* 69. 4-&! ++ 9 . 4;< 0!63N7+ 1+.#- 6).4
+ @ 05+4#9 A4.<E" ! 4=.+=.3 -5467.
@ ! ! -
2 ! + .Y
4=.7 AAB<. 0&7 AK@S4
*,. + -56 - + C.4 -!4=.,614#9> ,. .<AK-!4=69.
- + +4=E -!4=.,E E74N7+.R-56 - +
, 056 4
4-@ 4#: -!054 ,. -!4=69.> - 7+` .,6 7 AA6,. .#-U69 - + = 6*A9D A69050!+
/ -!4=69. +K- ++I. : 4-!+2 670U+ !
(4= + *,. +-56\- + , @#./5 14=A1
5
4-4
1
4(F 6=<! 4 +R4=. - +QA+
+.4#/ 0 . -!63 -!67. -56 . N7+A+
.,+I4E 6<. 0 6*6*\
AK6;< 0!4#: +2A: 6) + @O69 / 0! 4-!+ 6 +N9+I0>- + ".4 -!4=.,614#9 ,. .<AT-5467. 9 AK-!4=.,EQ69.
/ .*@`N7+IAT-5690 4- %H 6)! 4-54N7+9 0&): 4-!+CAK6;H 69.<+.#- 4+
* E94=N9+ ? .,67. ! +056 +4=E -54.,E?-!6J- + .4
= @/ 04!
! 4-!4#: +21
.,+4=E 6,. 0 6*6* +! +&+050!670& 4+
+%/ -D- +4=0 16): - + +-!+054=6. +. - +GA69050!+
/ -!4=69.
/ 05+7 -50!67.,E9+2 - / ./ - + ".4 -!4=.,6 4#9 4
: -!054 . -!4=69.S4# 4
, +( 4 + +. 4
4 9
# - 6),. E 9
- +! + / 05+ A6 169. 4.\A9 0!4BA/ -<,. 05+ 16* +
+I. 9
- +14=.7-5+09 AK-!4=69. / 05+169.O- + 16= -7= 6*A9 69
5 A9= + ;4=.\0!+29 670<#O
+45 (4= +G - +AK690505+#3 -!4=69./%9 0!+(.<.9
= @? / 0 + 9+0 / ./V
- + 4=.7-5+09 AK-!4=69.
-& 9+4
7 AK+63N9+0C
9 0!E7+.,+I4E 6<. 0 6*6*4 D: 6V- + , 056 = + ; 6).4#Z
.,6/- ++I.<AK6),. .#-!+05+Z
-!6Q- +
&5 1+7 +E70!+I+
:
,
' ' ! $ % & ! &(' * ' , - 6.,E - + 1+K- 6 6 6 -/4=.,4.<EC`N:9
.,+G69 ?.!+ 4=. - +$,05+N*4=6.L- 6C:+ AT-!4=69./LE94=N9+, /4=0<=@
9AIBA .,0 3-5+10!+25.4-& ;- +@O0!+@O69. - +.:+ 6 - +AK++
.4#/0".-!6 3-!67. N*4=.,ES-!6*.,6 - + 9.
+I0 B+ P670!+G- %+ 90& 1+K-!+I0&CA9. +7+K-5+0 4=.,+2 4#C.,6/- /0!-!4BAB.
90<=@+!40 +J+N7+.- 6.<E - +
9&0 +-!+I&0 26 - /4=.,+ AK)6 .4
+ .!+ -!6 =6#6 3-16/- +0105+#3-!+2 ,056 + ( /-14# 0!+ =.,405+
4# 1+- *6 96 . 4=.,
E 4BA #4 14=. + H+/. +.#- 96 G- +QA6 (4.- 3-!4=69/. 9
@Z+ H+./:4=N9+AK++
.4#/0
9.,-!6 /-!69. <- 3-J
4 8 1+K- *6 4BA A 9.VA =A .4#3-!+ AP0!6 4- 4.V- + .<AT&- .3-5467.Q+ =./-!4=69. E V+4 8- + 1*6 + 6*+ #4 9#:6 C- 7+ :N 9.7- /E7+&- 3-)4-&A /.A6 H+ 4- -!+ ;H690 9
=@ 1N 90!@*4=.,
.,69-Y-5+. -!76 8- 3-5
4 :-!4BA +=@ - 3-5467. /05@ 3-!-!0 9AT-5690 .- ) +4- +I0Y=6 !A+4 #3-!670!@>670 A /6/-54=A @*. 4BA &
!
9 9BAB.
/-!4=.,E
P056 - + .<AK-<./-!4=69. 1+- 6*6 + + 94<0!6=9A +2
s t
4=.- +&51+ (9.,.,+I0 )4.?!+IAK-!4=69. 310 10 0- 4
P6).,././- +.
s s s t
- 4
4
.!+1-56 16* 4+P@ J4N7+. N9+GA+
F.<+4=E 6).,0 6*6 6 -@=H+
s t
$ - - +.R- +C< 0!6 4
= 4-@ - / -?- +U- 0!+I+2AK++#$)10694=.,4=.,E67.,+U69
- +`.,+4=E 6,. 0 / 05+J6] -@H + $ - ! ++, 4
9 E905 L 4#
+ + - - 2 2 +I0!+ -O4#".,67044#:4=.,E?9 0& 1+K-!+I05F. A - / - " +<3. U 63N7+0 / ./ -&9+"+ F6=<!4 +C-3-!+2&6Y - +2A+.#-!05+ A+
/ ./S
- +269.,+U.<+4=E 6),. 0&4=.#-!67 AA6,. .#-<- .,P 690& 4=.,E
/.?)10694=.,4=.,E<N9+ÀK++ .<+4=E 6),. 0 6*6
69] -@F + " &:
:
a
b
0.025
1
0.9
0.8
Correlation and coupling
Proportion of Parasites
0.02
0.015
0.01
0.005
0.7
γ
2
0.6
0.5
0.4
0.3
0.2
µ
0.1
0
0.3
0.4
0.5
0.6
0.7
Transmissibility
0.8
0.9
0
1
0.3
0.4
0.5
0.6
0.7
Transmissibility
0.8
0.9
4=E.,05+ J09 ;/./ 8/05+P67080&9.,E9+ 69L-50&9. 14#<!4 4+=4-!4=+$P690J- +(!4 (4=+
6=- 90&):4-!+ 6*+ J09 ; 6 &- +(,056H690!-!4=69.R6 90&):4-!+2$,05+,4=AK-!+ @
- + .<AT&- .3-5467. 1+K- 6* J0& 6 L- +JN:9
.,+G6 /./1- +>AK670!05+<H69.4=.,E
O- /$- /05+À 9BBA .
/-!+ P056 4- 4=.S- + 16*+
"
056 - +$,056 4
4-!4=+LA9BAB.4#3-5+;P670L- +?N:/0546).N:9
.,+2D6 - +?A69050!+
/-!4=69. 90!+ P6).,.
/a
1
4=.+=./-!4=69.S 4E).,05+ ( 6 - +`N:9
.,+)6 /. /.- +UAK690505+<F67.4=.,EC<0!6)F670:-5467.
69 /0!4-5+ +<,4-!+G- +8=6 +I0 AK670!05+#3-5467.-<0!+2:+I.7-C4. - 4# :@*:-!+ A6(90!+21-!6 - +8,05+N*46).
- 6 ;- +05+ -54+
L + *4#: -& =6 +I0G- 05+ 6#Q-50&9.4 4#&:4 4
4-@R- 3-2A6(/05+ +
-!6- +AK++
.4#/0
9,. -!6/ -!69. 16* + / ./ - +05+U4#$
!6 .44H+0JAK054-54=A9 -50&9.4 4#&:4 4
4-@ 4#?A0!4-!4BA -!0/.414
"!
! 4 4
= 4-@\4# 4E +0 - 9 .Z- +S69.<+ P 6<. .^4=.Z- +SA+
+.4#/0;9.-563-569. 4BA 4#;*.,+-56Q- + &51+
05+! 69./ ) E94=N9+I. / -C- +`+./
6>! +IAK-!4=69. 310 10 :
a
b
0
10
0.05
-2
Proportion
Proportion
10
-4
10
0
-6
-0.05
0
0
10
2
2
0
4
0
4
2
2
4
4
Parasites
Parasites
Hosts
Hosts
c
d
0.1
0.1
Proportion
Proportion
0.05
0
0
-0.1
-0.05
0
0
2
2
0
4
0
4
2
2
4
Parasites
4
Parasites
Hosts
Hosts
=4 E.,05+ J09 C 6 )- +2N:+.,+2J69D .,.<AK-!4=69. 6 -/4=.,+?P0!6 - +1AK+
.4
90
.-!6 (/-!67. J0& 9. 6 - +%,+N*4
/-!4=69.P056 - +JA+
+.4#/0 9.-56 3-569.
N:9
.,2+ )96 ]- + .,.FAT-!4=69.6 -&94.,+2 @- +>- 0!+I+7 63N9+ 1+K- 6
,
!4(4=+ @ -!6 6 :+I0!N7+J- +",4 +05+.<A+ +K- ++I.S- + .<AT-&.3-5467. 1+- 6*?/./ - +"7AT-<./9
AK++
.4#/079,. -!6/-!69.O4
G-56 =6*6?3->- +; .,.<AT-5467. 4#G4#J- +(,+.!4-@R6 +29A -@H+26 .,+I4E !
6.,0 6*6* 4=A 4=. - 4
:4(4=+$:@*:-!+ A/. +?E94=N9+I. @ - +?..3 +I0D69 6=- 9.(/0!4-!+L4.
- + <N9+&A+
J09 6 4=E,. 05+ 6 Y- + ,056H690!-!4=69./ 69F+29A .,+4=E 6.,0 6*6*U-@H+ * 5
P 690 2 4-24
U A +2/ 0`- 3 - + AK05+): +27J - +.3. +0269 /0!4-5+>4=.\- + .<+4=E 6).,0 6*6
4=.<AK05+): +2 4- - + 6= --= 4 9+= @U.,+I4E 6),. 0 6#6* -@F + +4=.,E9
6): -& - 6).4# +).,6/-54=A+2- 3
- + 05+,/05+C.<6`.,+4=E 6.,0 6*6*4Y- /- AK67.#-&/4=.(:6)+@ <N9+ 90&):4-!+2D/.;=+& - 9. 6F- +&.,+I4E 6 .,0 6*6*4>AK69.#-/4=.Q- 6670 1670!+;/0!4-!+ J09 9. 6 - +1N:/054
/-!4=69. P056 - +
-!05
4 /. 2+0505690 1+- 6*? : :) :P0!6 90& 1+K-!+I0!4#!4.,E$P0!6 - +&AK++
.4#/0-9.,-!6 /-!69.
: ? /=a D//. (P0!6 /0 1+K-5+054
!4=.,E 4- 4=.- + 16*+> : : :7a) 05+<F+ AT-54N7+=@
056 E90 9 ?4-G#4 $9/05+.#-)- 3-?- + .<AT&- .3-5467. +- 6*R63N7+0$4,056 4 3-5+C- + ..
+I0?69
.,+I4E 6 .<0 6**6 4U- /(- 90!
+ .4+6 )6 :&- ;/. ^- /;- 90!+SAK6 (4=+K-5+ =@\+ (-@ L- #4 4 C,05+<. =@
+IA .!+ 6 +I0!05690 &4=.#-!056 4.<AK2+ @- + ".4-!4=.,6 1#4 9 .,.FAT-!4=69. 4BA 4=E9.<6905+ %6A LAK670!05+ #3-5467. , Y- +:N 9
.,+ 6 4=.<A0!2+ !+ PE90 9 /. 8- + +IN*
4 /-!4=69. P056 - +)AK+ +
.4#/D0 9.-56 3-569.21N +.<+ E9056 9. 1=6 -.,6/- =@1- +058+ 90!$+ /0 2+ < .,+I4E 6 .<0 6**6 4D- /,- /05+GAK6 (,0!#4 !+ + A +./:4=N9+ @196 + 3-!-!054 .,&- +`-56 /.R4=.<AK05+ ):4=.,E 6 4=. 9.<AK+ @ 2+ /. <+ -@ FC+ :)6 +.-5467. =6 - #4 @ "
C.4-!4=.,6 1#4 9D#4 -50!4 .-5467.V96 D- + 1+ /. <+ :)6 +.-54676
. 9a / : 7a :) ,+
N*
4 /-!2+ $P056 - +1AK+ .4
970 9.-56 3-569. #4 :-!054 .,-!4=69. @ N#4=.,E 9%0 =6 +I0G:N +.,C+ P6970 <N9+ =6 .,+I4E 6 .<0 6**6 +E905+ /-!+0 1N +.<+ 7P67C0 9.^+ (-@ .,+4=E 6 .,0 6**6 - /;- 90!+6 !+05N9+ 4=6
. 9
- 0!+I+È90 9 90!7+ *.<+>-!61- "+ 4 A .4-@ 4=.S2+ !A 9,4=.,(E P056 - #4 ,.,. :&- =+ + F674.#- .RE9+I.,+0 9
- 4F2+ /0 G- /-`- + +I. :4-@ .<.<AT-5467. ,05+ 4BAT-5+ @Q- + .FAT<- ./3-!4=69. 1+K- *6 4`#4 >4.
6 +N7+0J4"
E96**6 /E905++ 1+.#- 4- - +`0!2+ 9 4/&- !
20
310 20
310 10 5
2
310 3
3
3
3!
!
2
%
2
!
6
3
*"]$?* D$ D$ 3 , D$? ( (& "
GW .,+J69]- + /4=. )N:/.#-&9E9+2L69 +I4.<E( =+?-56 <./- + /0 1+K-5+0 P056 4- 4=.S- + 16*+ 4#
- 3- !@ :-!+ 4BA +IN96)N7+D-56 4 14-DA@*A+)670LAK6 ;+ C3-!-!09AT-5690LA/. + 1670!++):4
=@ 16,+
+2
, Y- +E =6 7:N +.,2+ 96 ,- + ,056 H690!-!4=69. ]96 F- +:N /0546).>5H+IA4+2A /.,E7+L- +. 4-4#D9#:6GN9+I0!@7=4 7+=@
- 3-U- + <3-5
4 -5<0 .FAT<- .<0!2+ /. \A69050!+ /-!4=69. 79. +.<A+ - + AK)6 .44=4.,E 4
+ 9#:6N:/05@ +
- 6 =6 -J- 6 /0 !4-5C+ !*@ -5+ @P0!6 :+ AT-5467.
4=. 9 -!+I0 9
+ + 4 14=.,+ .:4=.,E - 4#
-!+ A .,4 =.,+
5
0 10
5
056 - +AK+
.4
90"9.,-!6 /-!69. .:4=.,ER- + 1+K- 6*Z69,!+IAK-!4=69. : G- +N:9
.,+ 69 +I0!+
+:-!4 3-5+ -!6RN:90!@ +- +I+. 9. :4. ?:-!0569.,E)@ +K-5+0 4=.,4#-54=A (9.,.,+I0,05+5AK054 + @
- ++/. :4-!4=+`6- + 6=-/./ //0!4-5+ + .<AK-<./-!4=69. 16*+ )8P690 ".4#3-5+ .!4.,E- +
1+ 9. F+ #\+ =.3-5467.>E94=N9+I.O4=. 9 -!+I0 4BA 0!+ ).<405+G- + AK6 ;.-&/-!4=69.\69 /.*@
16905+ - 3-5+ - 3- B+ P670!+ C_
4 ).<6 + #-5+. ,+-!6 ! ! ! ! ! ! !61- +":+- .,6 AK67.7- /4=. ` - 3-5+ A ( P670`- + :4 (4=+ 6):- 90&):4-!+ 16*+ ]- 4#U6 N#4=6./5=@VE70!+23-<=@
4=.<AK05+ ):2+ H- +L. . +I086 *A 9BBA .4#3-5467. H+0;4-!+I&0 /-!4=69. .,+-56)- 4 8E905+ 3-5+0 . . +I0;69A=A.4
/-!4=69.
<. ,4.,
E 4BA 4 Y- + 1)6 :-DAK6 ;.&- /-!4=69. +=@U+ *H+. !4=N9+6 H+0 3-5467. 69. C@ F+I50 P670 1+ +IN9+05@
4M- 4-!+I&0 /-!4=69.
- :4 ".4#3-5467. ;.4 43-54.,
E 3-1+ 9A -5+ +S0!2+ 5.4&- +05+AK6 (90!2+ \-!6 690!
//. - +I0!+ +05+Q.,6 !4=E9.,4 FA 9.#- 4H+I0!+I.<AK2+ 4=. + N*4=6 .,0 4=E .,05+ BA 4)E .,05+ : 9 9-5+0
2
2
&
310 10
0 10
0
0
4%
"!
4
3
'
&%
&
#$
3
0
8( 6 - + + N*4=6.,016&- + .<AT-&.3-5467. !@:-!+ )E909 /. ( 6 - + <0!6)F670:-5467.
69 )6 :-&9. 90&):4-!+2`0!+25H+IAK-!4=N9+@ + 05+<.4- P690 - + 6):-&C:+I+ /0 670!+ 0!+IE.4#/0>- /.
- +205+ <.4&- G4. 9 9-5+0 H- 4
J4#$<0!+25. @ +IA9.!+ 67.4@?=6A .FAT-<./3-!4=69./?6L- +(!4I+ 69
- + .,+4=E 6 .,0 6*6* /05+ 4.<A690&F670&/-!+2Q4=.O- 4# 6*+ +05+)G- + 7AT-<./9LAK++
.4#/079.,-!6 /-!69.
A 9.^+ *H+054+I.<AK+ .<AK<- ./-!4=69. ;3(
- Q0 /.,E7+96 $5A+2 +.. +I026,/0!4-!+ 4.^- +AK++
.4#/0
9.,-!6 /-!69
. :*@ :-!+ 6*2+ L.,69- 056 ) =6 ,05+ 4BAT-5+ @ - + .<AK-<./-!4=69. 16*+ 9. - 4
@
+ *.,+ -!6 - ;+ !@#.FA 0!67.,6 ./. 3&- .,0!+ 96 05+ <3-5
4 !@ :-!+
C
6 0!+ AT-!+4 P@S- #4 %,0!6 =+ )6 .4
05+ =.,4=0!+ - /- - +I0!+ 9#:6 8+E70!+I+ 6 HAK)6 .44=4.<E?-56G- +C-!+ ;H690 9 12+ /. 4=A ; )6 .4# 4=.#-!056 4.<AK+
+N7+. 16905+>A6 (4=4=A 3-5467. 3
3
a
b
-1
0.6
10
Proportion of Parasite1 and Parasite2
0.55
Proportion of Host1 and Host2
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
-2
10
-3
10
-4
10
-5
10
0.1
1000
1500
2000
Time
2500
3000
1000
1500
2000
Time
2500
3000
c
0.14
0.12
Parsite1 + Parasite2
0.1
0.08
0.06
0.04
0.02
0
0.3
0.4
0.5
0.6
0.7
Host1 + Host2
0.8
0.9
1
=4 E.,05+ J0& /./ ( 6 2- +S-54 1+ !+054+2"P670 - +S- 6 6=- 9.^- 6
90&):4-!+ 5H+IA4+2 05+<F+ AT-54N7+=@ J09 ,+ 169.:-!03-5+- + + 4#-5+.<A+V69"66 4 1+. !4=69. A /6/-54=A?690 4- +I.- +J-!6/-9 ,056H690!-!4=69.6 6):-&C4# 469-:-5+9E)94.:- +>-!69&- 9 ,0!6)F670:-5467. 6>90&):4-!+2
0
+ + N#4=6.<069 - +%:@*:-!+ 4#LA /69-!4BA #E909 - 6 - +&+ *4#:-!+.FAK+&6 7=6 41+I.:4=69./9
A 96/-!4BA2690 4- - 4#73 -!-!09AK-!690 9F+2/0&-56 +- =+;) : 4".4#3-5467.J69D63N7+0G-!+I.V- 6. &/./Q4-!+04!
3 -5467.86 .,69-%=+)S-!6/.*@ !4;+I0%=414->A@AB=+U690 3+?H694=.#- - 6.4# 4F+2/0?- / -G4-J4#&- +
#/05E9+ !A9=+ .FAT-<./3-!4=69./4.- + / *- /-,/05+G05+<F67.!4 =+%P690C- +, +N*4
/-!4=69.P0!6 - +>05+<.4-&
,05+ 4BAT-5+ @ - + +2/. ! <+
+=./-!4=69.
9,
0
. - 4#)A 9,-!+0C- 6 /4=. +- 6*4 N9+ ++I.6).-<=4=.,+ - + <0:-C4 (44BAK4-<=@ 16*+
- + </-!4#9
AK670!05+#3-5467. /./- +;:+ AK67.S- +2+K +IAK-&G69D6A9 :-!6A )-54=A4-@ C6/- - +!+P+23-&.,0!+2&+H+ AT-?- +
@*. 4BA 296 ?- +:@*:-!+ +24=.,EV4=. E7+.,+I0&L-56 =6 +I0"=+N7+# 6,/0!4-54
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