1
Supplementary Material for Hall, Altizer and Bartel:“Greater migratory propensity in
2
hosts lowers pathogen transmission and impacts”
3
4
Data S1: Examples of migratory animals where pathogen transmission occurs at one
5
stage of migratory cycle
6
7
Table 1: Examples of migratory animals and their parasites/pathogens for which
8
transmission occurs primarily at one migratory stage (breeding, migration or wintering).
Species
Infectious disease(s)
Migratory
stage(s) of
transmission
Monarch butterfly 1
(Danaus plexippus)
One-way
migration
distance
(km)
up to
2500
Neogregarine protozoan
(Ophryocystis elektroschirra)
Breeding
Black-legged kittiwake2
(Rissa tridactyla)
10003000
Lyme disease spirochaete
(Borrelia burgdorferi)
Breeding
Grey whale3
(Eschrichtius robustus)
9000
Whale lice
(Cyamus spp)
Calving
Reindeer 4
(Rangifer tarandus)
up to
2500
Warble fly (Hypoderma
Calving
Harbor Seal (Phoca
vitulina) 5
Up to 800
Phocine Distemper Virus
Calving
Bar-tailed godwit
(Limosa lapponica) 6
up to
11,000
Low pathogenic avian
influenza virus
Breeding/
Migration
Bottlenose dolphin 7
(Tursiops truncatus)
>1000
Morbillivirus
Migration
Chinook salmon 8
(Oncorhynchus
tshawytscha)
Great reed warbler9
(Acrocephalus
arundinaceus)
Purple finch10
(Carpodacus purpureus)
Little brown bat 11
(Myotis lucifigus)
Asian lady beetle 12
(Harmonia axyridis)
up to
1500
Sea lice
(Lepeophtheirus sp)
Migration/
wintering
6000
Haemosporidian
(Haemoproteus payevski)
Migration/
Wintering
up to
2200
200-800
Mycoplasmal conjunctivitis
(Mycoplasma gallisepticum)
White nose syndrome
(Geomyces destructans)
Parasitic fungus
(Hesperomyces virescens)
Wintering
up to 100
tarandi)
Wintering
Wintering
1
9
References
10
11
1. Altizer, S. M., K. S. Oberhauser, and L. P. Brower. 2000. Associations between
12
host migration and the prevalence of a protozoan parasite in natural populations of
13
monarch butterflies. Ecological Entomology 25: 125-139.
14
15
2. Chambert, T., Staszewski, V., Lobato, E., Choquet, R., Carrie, C., McCoy, K. D., ... & Boulinier,
16
T. 2012. Exposure of black‐legged kittiwakes to Lyme disease spirochetes: dynamics of the
17
immune status of adult hosts and effects on their survival. Journal of Animal Ecology, 81(5),
18
986-995.
19
20
3. Rice, D. W. 1998. Marine Mammals of the World: Systematics and Distribution. D.
21
Wartzok, Ed. Society for Marine Mammalogy, Special Publication Number 4, Lawrence,
22
Kansas.
23
24
4. Folstad, I., F. I. Nilssen, A. C. Halvorsen, and O. Andersen. 1991. Parasite avoidance:
25
the cause of post-calving migrations in Rangifer? Canadian Journal of Zoology 69: 2423-
26
2429.
27
28
5.Harkonen, T., R. Dietz, P. Reijnders, J. Teilmann, K. Harding, A. Hall, S. Brasseur, U.
29
Siebert, S. J. Goodman, P. D. Jepson, T. D. Rasmussen, and P. Thompson. 2006. The 1988 and
30
2002 Phocine distemper virus epidemics in European harbour seals. Diseases of Aquatic
31
Organisms 68: 115–130.
2
32
33
6. Hansbro, P. M., S. Warner, J. P. Tracey, K. E. Arzey, P. Selleck, K. O’Reilly, E. L. Beckett, C.
34
Bunn, P. D. Kirkland, D. Vijaykrishna, B. Olsen and A. C. Hurt. 2010. Surveillance and
35
analysis of avian influenza viruses, Australia. Emerging Infectious Diseases 16: 1896-1904.
36
37
7. Duignan, P. J., C. House, D. K. Odell, R. S. Wells, L. J. Hansen, M. T. Walsh, D. J. St.
38
Aubin, B. K. Rima and J. R. Geraci. 1996. Morbillivirus infection in bottlenose dolphins:
39
Evidence for recurrent epizootics in the western Atlantic and Gulf of Mexico. Marine
40
Mammal Science 12:499-515.
41
42
8. Boyce, N.P., Z. Kabata, and L. Margolis. 1985. Investigation of the distribution,
43
detection, and biology of Henneguya salminicola (Protozoa, Myxozoa), a parasite of the flesh
44
of Pacific Salmon. Canadian Technical Report of Fisheries and Aquatic Sciences 1450: 1-53.
45
46
9. Hasselquist, D., Östman, Ö., Waldenström, J., & Bensch, S. 2007. Temporal patterns of
47
occurrence and transmission of the blood parasite Haemoproteus payevskyi in the great
48
reed warbler Acrocephalus arundinaceus. Journal of Ornithology, 148(4), 401-409.
49
50
10. Hartup, B. K., Dhondt, A. A., Sydenstricker, K. V., Hochachka, W. M., & Kollias, G. V. 2001.
51
Host range and dynamics of mycoplasmal conjunctivitis among birds in North America.
52
Journal of Wildlife Diseases, 37(1), 72-81.
53
3
54
11. Blehert, D. S., A. C. Hicks, M. Behr, C. U. Meteyer, B. M. Berlowski-Zier, E. L. Buckles, J. T.
55
H. Coleman, S. R. Darling, A. Gargas, R. Niver, J. C. Okoniewski, R. J. Rudd, and W. B. Stone.
56
2009. Bat white-nose syndrome: an emerging fungal pathogen? Science 323: 227.
57
58
12. Riddick, E. W. and P. W. Schaefer. 2005. Occurrence, density, and distribution of
59
parasitic fungus Hesperomyces virescens (Laboulbeniales: Laboulbeniaceae) on
60
multicolored Asian lady beetle (Coleoptera: Coccinellidae). Annals of the Entomological
61
Society of America 98:615‒624.
62
63
Data S2: Winter mortality rate
64
65
How does a species choose where to winter? Since long-distance migrations are costly, the
66
risk of mortality during migration must be countered by the benefit of reaching winter
67
habitat where survival is relatively high; indeed, empirical evidence (Sillett and Holmes
68
2002) suggests that Neotropical migrant passerines choose wintering sites where
69
overwintering mortality is comparable to that at breeding sites during the breeding season.
70
Therefore we assume that a species experiences relatively high mortality (approaching that
71
of the mortality at the breeding site in the unfavourable season) if it chooses to winter close
72
to the breeding site, and that the mortality reduces with increasing distance migrated (d),
73
and that at some characteristic distance db, the species experiences the same mortality rate
74
as it does during the favourable season at the breeding site. The final ingredient is a tunable
75
shape parameter, n, which determines how rapidly winter mortality drops with increasing
76
distance from the breeding site. The functional form of this expression is given by equation
4
77
(6) in the manuscript, and is illustrated in supplementary Fig. 1 below as a function of
78
migration distance and the shape parameter (the shape of the mortality function chosen for
79
our parameterisation, with n=5, is depicted by the thick line).
80
winter mortality rate (mw)
mnb
Fig. 1
n=1
n=2
n=5
n=10
m
b
81
d
b
distance from breeding to wintering ground (d )
82
83
Data S3: Deriving default parameters for the migratory host species
84
85
One of the best-studied life histories of any migratory bird is that of the Black-throated
86
Blue Warbler (Setophaga caerulescens). All life history parameter values are derived from
87
the species account at Birds of North America Online (Holmes et al. 2005) while stage-
88
specific migratory survival parameters are taken directly from Sillett and Holmes (2002).
89
5
90
Stage-dependent survival
91
92
For each migratory stage i (i=b, w, m), the instantaneous mortality rate miis assumed to be
93
related to the monthly survival probability as follows
s i,month = exp(-mi /12) .
94
(S3.1)
95
Rearranging this expression, we can estimate mi from estimated monthly survival
96
probabilities, yielding
mi = -12 ln (s i,month ).
97
(S3.2)
98
According to Sillett and Holmes (2002), the monthly survival probabilities for adult females
99
is 0.99 on both the breeding and wintering site; using expression (A3.2) yields estimates of
100
mb= mw= 0.12. The monthly survival probability on migration ranges from 0.77 to 0.81, so
101
taking the midpoint of 0.79 yields mm= 2.8. There is no data on survival probability of
102
individuals remaining at the breeding site during the winter, presumably reflecting the
103
extremely low chances of survival in the absence of invertebrate prey. We conservatively
104
assume a monthly survival probability of 50%, yielding mnb=8.3.
105
106
Fecundity
107
108
For convenience, we assume a continuous function for the per capita birth rate during the
109
breeding season; since there is intraspecific variation in the timing of nesting due to
110
variable success in establishing territories and mating success, the ability of the birds to re-
111
nest following nestfailure, and multiple brooding, this approximation may be more
6
112
appropriate than including synchronized birth pulses in our deterministic model.
113
Individuals typically have 2 broods per season (range 1-3), and lay 4 eggs per clutch.
114
Therefore we assume that the maximum (i.e. density-independent) per capita fecundity is 8
115
juveniles reared over the 4 month (= 1/3 of a year) breeding season. In the absence of
116
density dependence, the expected number of young per individual over the breeding
117
season can be approximated by
Y = exp(b0 / 3) .
118
(S3.3)
119
Setting Y= 8 in (A3.3) and rearranging, we obtain b0= 6.2. In Black-throated Blue Warblers,
120
density dependence has been demonstrated to act primarily through fecundity rather than
121
adult survivorship. Under the assumption that a breeding site can hold 1000 individuals,
122
and that adult survivorship is very high, the density-independent component of the birth
123
rate,
b1 » b0 /1000 = 0.0062 .
124
125
Migration strategy (typical)
126
127
The northbound spring migration for this species typically occurs from Mar 15-Apr 30 (6
128
weeks = 0.125yr); the species stays at the breeding site from May 1 to August 31 (4 months
129
= Ts = 0.33yr); fall migration occurs from Sep 1-Oct 15 (6 weeks = 0.125yr); the rest of the
130
time is spent on the wintering grounds (Oct 15-Mar 15 =5 months = 0.42yr).
131
132
Since Sillett and Holmes (2002) report roughly equal survival probabilities at breeding and
133
wintering sites, the typical migration distance is assumed to be representative of the model
134
parameter db. The species migrates overland from its breeding grounds in the northeastern
7
135
US to Florida, before embarking on an overseas crossing to wintering sites in the
136
Caribbean. Therefore we crudely estimate the migration distance as the straight-line
137
distance from New Hampshire to Orlando, FL (= 2000km) plus the straight-line distance
138
from Orlando, FL to Havana, Cuba (= 600km), yielding db= 2600km. Since the migration
139
typically takes 6 weeks, the average migration speed, v = 2600/0.125 = 20,800km/yr.
140
141
The shape parameter governing how winter mortality declines with distance from the
142
breeding site, n, is unknown. However, there are very few winter records of this species
143
from the US with the exception of Florida, where a handful of overwintering birds are
144
detected most years (www.eBird.org). Choosing a shape parameter of n=5 yields a monthly
145
overwinter survival rate of 0.8 at d=2000km (i.e. the Orlando area).
146
147
References:
148
149
Holmes, R. T., N. L. Rodenhouse and T. S. Sillett. 2005. Black-throated Blue Warbler
150
(Setophaga caerulescens), The Birds of North America Online (A. Poole, Ed.). Ithaca: Cornell
151
Lab of Ornithology; Retrieved from the Birds of North America Online:
152
http://bna.birds.cornell.edu/bna/species/087
153
154
Sillett, T. S., and R. T. Holmes. 2002. Variation in survivorship of a migratory
155
songbird throughout its annual cycle. Journal of Animal Ecology 71: 296–308.
156
8
157
Data S4: Calculation of the the disease-free equilibrium population size and basic
158
reproductive number (R0)
159
160
In the breeding season in year Y, the population grows logistically to a maximum
161
population size NY (Tmax ) where Tmax = min (Tb,Ts ) , and then a proportion s =
Õ
si
i=nb,m,w,m
162
survives until the start of the next breeding season, where the s i are the proportions
163
surviving each migratory stage (expressions 3-5 in the manuscript). The population size at
164
the start of the breeding season in the following year is therefore
165
NY +1 ( 0) = s NY (Tmax ) .
(S4.1)
166
We can write down the population size at the end of the breeding period in year Y+1 by
167
solving the logistic equation directly and substituting expression (1):
NY +1 (Tmax ) =
168
K
æ
ö
K
1+ ç
-1÷ exp ( -rTmax )
è s NY (Tmax ) ø
.
(S4.2)
169
Note that in our model, r = b0 - mb and K = ( b0 - mb ) b1 . If we define F = exp ( rTmax ) as the
170
maximum (density-independent) per-capita growth rate over the breeding period, we can
171
write this as
172
NY +1 (Tmax ) =
KF
F -1+
K
s NY (Tmax )
(S4.3)
173
9
174
The system reaches an annual equilibrium profile when NY +1 ( t ) = NY ( t ) = N(t) for all
175
within-year times t Î[ 0,1]. Denoting the maximum equilibrium population size as N max , we
176
can rearrange expression (2) to obtain
N max =
177
K ( Fs -1)
( F -1)s
(S4.4)
178
and the minimum equilibrium population size is found by multiplying (3) by the proportion
179
surviving the nonbreeding period, s :
N min =
180
181
K ( Fs -1)
.
F -1
(S4.5)
Finally, for 0 £ t £ Tmax , we can calculate the equilibrium population size at time t,
N (t ) =
182
K
æ K
ö
1+ ç
-1÷ exp ( -rt )
è N min ø
(S4.6)
.
183
184
For our calculation of R0 below, we will need the integral of this expression over Tmax.
185
æ K
ö
Making the substitution u = ç
-1÷ exp ( -rt ) , we can solve this integral directly:
è N min ø
u2 =
Tmax
ò
N(t)dt
=
(
) exp( -rT
K
N min -1
max
ò
u1 = N K -1
t=0
min
)
Tmax
K é æ1 öù
K é æ exp ( rt ) ö ù
ln ç +1÷ ú = ê ln ç K
+1 ú
ê
r ë è u ø û u1 r êë è N min -1 ÷ø úû
t=0
u2
186
=
=
u2
u
2
1
1
é K æ 1+ u ö ù
æ K ö du K
=
- du = ê ln ç
çè
÷ø
÷ú
ò
1+ u -ru r u1 1+ u u
ë r è u ø û u1
K æ exp ( rTmax ) +
ln
K
r çè
N min
K
N min
æ exp ( rTmax ) ö
+1÷
K ç NKmin -1
÷
= ln ç
1
r ç
+1 ÷
çè NKmin -1
÷ø
(S4.7)
-1ö K æ N min
ö
÷ = r ln çè 1+ K ( exp ( rTmax ) -1)÷ø
ø
10
187
Note that since F = exp ( rTmax ) , and using expression (A4.5) to substitute for Nmin, this
188
integral can be written in the simple form
Tmax
K æ Fs -1
ö K
ò N (t ) dt = r ln çè1+ F -1 ( F -1)÷ø = r ln ( Fs ).
189
(S4.8)
0
190
For an SI model where a species exhibits logistic growth in a constant environment, the
191
basic reproductive number of a pathogen is simply the product of the disease-free
192
equilibrium host population size (i.e. the host carrying capacity), the pathogen
193
transmission rate and infectious period (the inverse of the per capita mortality rate of
194
infected individuals); a derivation of this result may be found in Otto and Day 2007a. For a
195
migratory species, the transmission rate will vary as the population size and contact
196
probability varies over the migratory cycle, and the infectious period will be determined by
197
differential host mortality and costs-of-infection throughout the migratory cycle. The
198
seasonal, discontinuous variation in host and pathogen parameters, and resultant lack of a
199
constant disease-free equilibrium host population, means that the standard analytical
200
techniques for deriving R0 (e.g. evaluating the condition for the disease-free equilibrium to
201
become unstable in the Jacobean matrix, or using next generation matrices) are not
202
applicable.Instead, Floquet theory can be applied (Shulgin et al 1998) to determine the
203
expression forR0in a migratory species that is analogous to the expression for R0 in the non-
204
seasonal, constant environment. We demonstrate its utility as an invasion threshold by
205
remarking that the region of (Tb,d) parameter space for which R0>1 corresponds to the
206
region in which the equilibrium pathogen prevalence is zero from numerical solution of the
207
model.
208
11
209
The basic reproductive number of a migratory species can therefore be expressed as the
210
product of the infectious period, transmission rate and equilibrium population size
211
averaged over the annual cycle,
1
212
R0 =
1
b (t)N(t)dt
m ò0
(S4.9)
213
where 1/m is the infectious period. It is a standard assumption in SI models that the
214
infectious period follows an exponential distribution (e.g., P27, Keeling and Rohani 2008),
215
and that its expected value is the inverse of the per capita mortality rate of infected
216
individuals. Under this assumption, the probability than an infected individual survives an
217
annual cycle is given by the product of the proportions surviving the breeding season, 2
218
way migration, and overwintering periods:
2
s b,Is m,I
s w,I
p(1) =
219
æ -mbTb ö
æ -2mmTm ö
æ -mwTw ö
= exp ç
exp ç
exp ç
÷
÷
è 1 - cb ø
è 1- cm ø
è 1- cw ÷ø
(S4.10)
exp ( -m)
=
220
221
222
wheremisthe mortality rate of an infected individual averaged over the annual cycle,
m=
Ti mi
i=b,nb,m,w,m 1- ci
å
(S4.11)
223
and the Tiand mican be expressed as functions of Tband d as defined in the manuscript. In
224
our case, the transmission rate is constant on the breeding site and zero elsewhere, so that
225
R0 =
b
Tb
m ò0
N(t)dt
(S4.12)
12
226
The time at which the maximum population size is attained (Tmax ) , and therefore the
227
maximum per capita growth rate during the breeding season (F) and nonbreeding survival
228
probability ( s ) and the will depend on the migratory strategy deployed:
229
230
(i) Migrant departs before onset of unfavourable season (Tb £ TS )
231
232
233
234
235
236
237
238
239
240
241
242
In this case, Tmax = Tb , so F = exp ( rTb ) and
é
s = s m2 s w = exp ê -2mm
ë
d
döù
æ
- mw ( d ) ç1- Tb - 2 ÷ ú
è
v
v ø û.
(S4.13)
The annual mortality rate is then
m=
(
)
d
Tb mb 2 d v mm 1- Tb - 2 v mw ( d )
+
+
1- cb
1- cm
1- cw
(S4.14)
and from (A4.7) and (A4.12), the basic reproductive number is
R0 =
bK
æ N
ö
ln ç1+ min ( exp ( rTb ) -1)÷ .
mr è
K
ø
(S4.15)
Substituting for Nminfrom expression (A4.5) and noting thatF=exp(rTb) yields
R0 =
bK
æ Fs -1
ö bK
ln ç1+
F -1)÷ =
(
( ln F + ln s ) .
ø mr
mr è
F -1
(S4.16)
and noting that rTb = lnF, we can multiply this expression by (Tb/Tb) to give
R0 =
b Tb K
m ln F
( ln F + ln s ) =
b Tb K æ
ln s ö
çè1+
÷
m
ln F ø
(S4.17)
Finally, can can substitute for F and sigma from (A4.13) to give
13
R0 =
243
(
æ 2m d + m ( d ) 1- T - 2 d
m
w
b
v
v
ç1m ç
rTb
è
b Tb K
) ö÷
÷ø
(S4.18)
244
This expression is conceptually useful as it allows us to see more clearly the analogy to the
245
constant environment R0; the expression outside the parentheses represents the product of
246
the constant environment carrying capacity (K), time-averaged transmission rate ( b Tb )
247
and infectious period (1/m), while the second term in parentheses can be interpreted as
248
the proportional reduction in R0 due to migration.
249
250
(ii) Migrant departing after onset of unfavourable season (Tb > TS )
251
In this case, Tmax = TS , F = exp ( rTS ), and the proportion of susceptibles surviving (necessary
252
to calculate the disease-free equilibrium population) is
253
254
255
256
é
s = s nbs m2 s w = exp ê -mnb (Tb - TS ) - 2mm
ë
d
döù
æ
- mw ( d ) ç1- Tb - 2 ÷ ú .
è
v
vøû
(S4.19)
The average mortality rate is then
(
)
d
TS mb (Tb - TS ) mnb 2 d v mm 1- Tb - 2 v mw ( d )
m=
+
+
+
1- cb
1- cnb
1- cm
1- cw
(S4.20)
and the basic reproductive number is
257
R0 =
258
R0 =
259
bæ
TS
ö
N(t)dt
+
N(t)dt
ç
÷
ò
m è ò0
ø
TS
bæ
TS
Tb
N(t)dt + N max
m çè ò0
Tb -Ts
ò
0
ö
s nb dt ÷
(S4.21)
ø
which ‘simplifies’ to
14
bK
æ N
ö b N max
ln ç1+ min ( exp ( rTS ) -1)÷ +
1- exp ( -mnb (Tb - TS ))
mr è
K
ø mmnb
(
)
260
R0 =
261
and again by substituting expression (A4.5) for Nmin and noting thatF=exp(rTs), this can be
262
written more compactly as
R0 =
263
ln s
ö
+ f÷
çè1+
ø
m
ln F
(S4.22)
b TS K æ
(S4.23)
264
where f is the additional transmission opportunity by remaining at the breeding site
265
beyond the end of the breeding season:
f=
266
( Fs -1) (1- s nb ) .
( F -1)s Ts
(S4.24)
267
References
268
Keeling, M. J. and Rohani, P. 2008. Modeling infectious diseases in humans and animals.
269
Princeton University Press.
270
271
Data S5: calculation of the optimal migratory strategy.
272
273
Our procedure for finding the migratory strategy (i.e. the combination of unique values of
274
Tb and d) that maximises population host population size following arrival of a pathogen
275
was as follows. First, we discretized (Tb, d) space (within-year time Tb took the full range of
276
values from 0 to 1 in steps of 0.01; the distance migrated, d was varied from 2000 to
277
4000km in steps of 20km). For each parameter combination, we calculated the disease free
278
host population size from expression (A4.5); if this number was less than 1 we assumed the
279
host population could not persist. For all combinations of Tb and d for which host
280
populations could persist, we set the initial number of susceptibles at the start of the
15
281
breeding season to the disease-free equilibrium, and introduced a small number of infected
282
individuals. The model was then run until one of the following occurred:
283
(i)
The number of infected individuals dropped below one, at which point we
284
assumed that the pathogen could not persist (I=0), and the equilibrium
285
susceptible population size was recorded as the disease-free equilibrium.
286
287
288
(ii)
The difference between the susceptible and infected population size at the start
of the breeding season in subsequent years was less than some tolerance:
SY +1 (0) - SY ( 0) + IY +1 (0) - IY ( 0) <1
289
If this occurred, the population was assumed to have reached equilibrium, and
290
the model outputs for Sy+1(0 ) and Sy+1(0 ) were recorded as the equilibrium
291
values.
292
The plots of the raw data from these simulations are shown below for a given pathogen (i.e.
293
fixed transmission rate and costs of infection). To verify that our proposed expression for
294
R0 was indeed acting as an invasion threshold, we also calculated R0 for each of our
295
migration strategies, and noted an excellent convergence between those strategies for
296
which R0>1 and for which the equilibrium prevalence calculated from solving the model
297
numerically was non-zero.
298
16
(a) population at start of breeding season, no pathogen
(b) population at start of breeding season, with pathogen
400
350
3.0
300
2.8
250
2.6
200
150
2.4
100
2.2
2.0
0.1
3.2
distance migrated, d, 103 km
distance migrated, d, 103 km
3.2
50
0.2
0.3
Time spent at breeding site, T , yr
0.4
300
2.8
100
2.2
3.0
1.25
3.0
2.8
1.2
2.6
1.15
2.4
1.1
2.2
1.05
0.4
1
distance migrated, d, 103 km
distance migrated, d, 103 km
3.2
b
50
0.2
0.3
Time spent at breeding site, T , yr
0.4
0
(d) pathogen prevalence at start of breeding season
1.3
0.3
150
b
(c) basic reproductive number, R0
0.2
200
2.4
3.2
Time spent at breeding site, T , yr
250
2.6
b
2.0
0.1
350
3.0
2.0
0.1
0
400
0.12
0.1
2.8
0.08
2.6
0.06
2.4
0.04
2.2
2.0
0.1
0.02
0.2
0.3
Time spent at breeding site, T , yr
0.4
b
299
300
SupplementaryFig. 2. Effect of migratory strategy on host population size (a) before and (b)
301
after introduction of a pathogen, and on pathogen invasion success as characterized by (c)
302
the basic reproductive number, R0, and (d) equilibrium pathogen prevalence. This is the
303
raw data used to produce Fig. 2 in the manuscript; each rectangle represents a combination
304
of Tb and d used to define the migratory strategy, with the value of the dependent variable
305
denoted by the colour bar.
17
306
The optimal strategy was simply the combination of Tb and d which resulted in the
307
maximum equilibrium population size over the simulated range. Extensive simulation
308
confirmed that the optimum was unique.
309
310
Finding the optimal strategy as a function of pathogen traits
311
312
We investigated the effects of two pathogen properties (the transmission rate, b , and the
313
cost of infection to migratory survival, cm) by calculating the optimal migration strategy
314
using the above protocol for each combination of cmand b . We chose three values of b
315
(0.01, 0.02 and 0.03) and varied cm between 0.2 and 0.8 in steps of 0.025. We noted that
316
error in numerical solution for the optimal migration strategy could arise from multiple
317
sources. First, our discretization of migratory strategy (Tb,d) space could have meant that
318
the ‘true’ optimum lay between two of our grid points. Second, our imposition of a
319
minimum persistence threshold (i.e. we assumed pathogen extinction if the infected host
320
population size dropped below one) could also cause our numerically-derived optimum to
321
differ from the true optimum of the deterministic model, where host population sizes can
322
become arbitrarily small. Third, we note that for our chosen parameterization of the
323
environmental gradient (see Supplementary Fig. 1), small increases in the distance
324
migrated could result in large increases in overwintering survival. Hence the absolute size
325
of changes tod(opt)as pathogen parameters vary may be small relative to the 20km step
326
size used to discretised.
327
Our solution to detecting numerical errors was to record the ‘top 10’ migration strategies
328
that resulted in the 10 highest equilibrium population sizes for each combination of
18
329
pathogen traits, assuming that a consistent pattern in the resulting 10 values of Tb(opt) and
330
d(opt) would confirm that the simulations were converging on the true optimum
331
(Supplementary Fig. 3, below). The tight bounds on the range of values for the optimal time
332
at the breeding site, Tb(opt), and the associated equilibrium population size and prevalence
333
(Supp. Figs 3a-c) , suggest that the numerical optimization is indeed converging on the
334
‘true’ optimal strategy. As expected, there is more ‘noise’ in the calculations of the optimal
335
distance migrated (d(opt), Supp. Figs e-f); nonetheless, there is a clear trend that further-
336
migrating strategies perform better than the disease-free optimum if costs of infection are
337
sufficiently high, and that further-migrating strategies (the maximum value of d(opt)) can
338
be more advantageous for more highly-transmissible pathogens.
339
340
Supplementary Fig. 3. The migratory strategy that maximizes population size following
341
pathogen introduction, as a function of two key pathogen traits: the cost-of-infection to
342
migratory survival (cm), and three values of transmission rate, b : low ( b =0.01, dashed
343
line), intermediate ( b =0.02, thin line) and high ( b =0.03, thick line). The response
344
variables for the optimal migratory strategy are (a) time spent at breeding site (Tb), (b)
345
equilibrium host population size (N), (c) equilibrium pathogen prevalence (I/N) and (d-f)
346
distance migrated, for each of the three transmission rates. All quantities are measured at
347
the beginning of the breeding season. To account for numerical error in the calculation of
348
the optimum, the range of each output variable was shown for the top 10 strategies
349
maximizing population size (grey shading). The dashed lines represent the mean values of
350
each output variable; additionally, the loess regression line is drawn in red in (d)-(f).
351
19
(d) low transmission rate ( b = 0.01)
(a)
0.35
optimal distance migrated, d(opt)
3.0
b
optimal time spent at breeding site, T (opt)
352
0.30
0.25
0.20
0.15
0.2
0.3
0.4
0.5
0.6
cost to migratory survival, c
0.7
0.3
0.4
0.5
0.6
cost to migratory survival, c
0.7
0.8
3.0
400
optimal distance migrated, d(opt)
optimal population size at start of breeding, N(opt)
2.6
(e) medium transmission rate ( b = 0.02)
350
300
250
200
150
0.3
0.4
0.5
0.6
0.7
cost to migratory survival, cm
2.9
2.8
2.7
2.6
2.5
0.2
0.8
0.3
0.4
0.5
0.6
cost to migratory survival, c
0.7
0.8
m
(f) high transmission rate ( b = 0.03)
(c)
1.0
3.0
optimal distance migrated, d(opt)
pathogen prevalence at optimum, I/N(opt)
2.7
m
(b)
0.8
0.6
0.4
0.2
0.0
0.2
2.8
2.5
0.2
0.8
m
100
0.2
2.9
0.3
0.4
0.5
0.6
cost to migratory survival, c
m
0.7
0.8
2.9
2.8
2.7
2.6
2.5
0.2
0.3
0.4
0.5
0.6
cost to migratory survival, c
0.7
0.8
m
353
20