Appendix S4
The model
The probability that a tick in life stage Z feeds is given by


 
thZ sZ
P

1

exp
H
H
, where  
, Z = L, N, A, h = 1, 2
ZU cZ
Z
f
Z
1 1
Z
2 2
Z
h
(See Table 1 for parameter definitions). L, N, or A corresponds to the larva, nymph, or adult life
stage of the tick, and 1 and 2 refer to the host species 1 (H1) and host species 2 (H2), respectively.
Z U is the number of unfed ticks in life stage Z. The probability that a tick is infected upon feeding
on a host is modelled from simple probability rules:


PinfZ  PHZ1sr  PctZ  PHZ1i  PctZ  PstZ  1  PctZ
.
PHZ1sr is the proportion of susceptible or recovered H1 hosts (i.e., non-viraemic), and PHZ1i is the
Z
Z
proportion of infected H1 hosts (i.e., viraemic) encountered by ticks of stage class Z. Pct and Pst
are the probabilities of cofeeding and systemic transmission in ticks of stage class Z, respectively.
Here we assume that cofeeding transmission is the only way in which ticks can be infected by
feeding on susceptible or recovered hosts, and that cofeeding and systemic transmission are
mutually exclusive but either is possible on infected hosts. For the cofeeding transmission rate,
we adopt and modify the formulation used by Rosà and Pugliese [1] to include the immunity
condition (i.e., recovered or not) of the competent host. We set the parameters such that
cofeeding transmission on recovered hosts is more difficult than on susceptible or infected hosts
as Labuda et al. [2] found in their experiments. The probability of cofeeding transmission to a
larva from a larva becomes,



i L
r L si
L


L
P
P

P
r
si
Q
,
t
f LL
LL
H
1
H
1


P

1

exp

LL


H
1


40
where


r
r
si
si
LL
LL





1




1

and
LL
LL
LL
LL


L L
L L





kk 

kk 
The λ’s are the probabilities of cofeeding transmission enhanced by aggregation of ticks
Z
(negative binomial). The parameter k is the aggregation (or dispersion) parameter of the
negative binomial distribution for ticks in life stage Z [3,4]. The parameter  ZZ  is the correlation
between burdens of ticks in stage Z and those in stage Z’ on a host [5]. For example, when the
same host individuals tend to harbour a large number of larvae and nymphs, the correlation
LNNL is close to 1. If Z = Z’, correlation is identically 1. The probability of cofeeding
transmission to a larva from a nymph is similarly,




i N
r N si
N

N
P
P

P
r
s i
Q
,
t
f LN
LN
H
1
H
1

H
1


P

1

exp

LN
.


H
1


Then the total probability of a larva to be infected via cofeeding transmission is
L

. The total probability of a nymph to be infected via cofeeding
P
P

P
1

P
ct
LL
LN
LL
transmission is modeled similarly. For adult ticks, both the probabilities of systemic and
cofeeding transmission are zero since they are assumed to feed only on the incompetent host
which supports neither transmission route. The probability of not being infected upon feeding is
the complement of that of being infected:
 

Z
ZZ
ZZ
Z
ZZ
P

1

P

P

1

P

P

1

P

1

P

P
sr
i
non

inf
inf
ct
st
ct
H
2
H
1
H
1
The infection rate of the competent host is also modelled as mass action [1]:



 
H
1
L
L
i
N
N
i
A
A
i
P

1

exp

q
L

q
N

q
A
inf
H
1
Q
,
t
H
1
Q
,
t
H
1
Q
,
t.
The host mortality rate was chosen to reflect longevity of about one year, which is typical in
small rodents in the upper midwest to northeastern United States [6]. The birth rate was set so
that the total host density remains constant.
41
The model components and equations

The rate of feeding on any host = PfZ  1  exp  1Z H1   2Z H 2

t hZ  que _ d Z
, Z = L, N, A, h = 1, 2
ZU  atch _ d Z
L, N, or A corresponds to the larva, nymph, or adult life stage of the tick, and 1 and 2 refers to the
host species 1 (H1) and host species 2 (H2), respectively.
where  hZ 
Let Pst , Pct the probability of systemic transmission and cofeeding transmission, respectively;
The rate of feeding but not being infected (FEED_NOT_INFECTEDZ)
=






PfZ  PHZ1sr  1  PctZ  PHZ1i  1  PstZ  1  PctZ  PHZ2
The rate of feeding and being infected (FEED_AND_INFECTEDZ)
=



PfZ  PHZ1sr  PctZ  PHZ1i  PctZ  PstZ  1  PctZ

The rate of cofeeding transmission from a larva to a larva

=

 LiQ ,t PfL rLL PHL1r  siLL PHL1s  H 1i
1  exp  
PLL =
H1


 
The rate of cofeeding transmission from a nymph to a larva

=

 N Qi ,t PfN rLN PHN1r  siLN PHN1s  H 1i
1  exp  
PLN = 

H1




 
The rate of cofeeding transmission from a nymph to a nymph

=

 N Qi ,t PfN rNN PHN1r  siNN PHN1s  H 1i
1  exp  
PNN = 

H1


The rate of cofeeding transmission from a larva to a nymph

=

 LiQ ,t PfL rNL PHL1r  siNL PHL1s  H 1i
1  exp  
PNL =
H1


where,

r
1 
rLL   LL


 LL 

 LL
si
 ,  siLL   LL
1 

L L 
k k 
k Lk L





42


 


 




 si

 LN 
si
 ,  LN   LN
1 


L N 
L N 
k
k
k
k






 NN  si
 NN 
r
si
1 
 ,  NN   NN
1 

rNN   NN


N N 
N N 
k
k
k
k






 NL  si
 NL 
r
si
1 
 ,  NL   NL
1 

rNL   NL


N L 
N L 
k
k
k
k




represent the aggregation-corrected cofeeding transmission rates on recovered (r) or
susceptible/infected (si) hosts. LL = from a larva to a larva, LN = from a nymph to a larva, NN =
from a nymph to a nymph, NL = from a larva to a nymph.
r
1 
rLN   LN

 LN
and,
1Z H1
P  Proportion of H1 hosts encountered = Z
1 H1   2Z H 2
Z
H1
 2Z H 2
P  Proportion of H2 hosts encountered = Z
1 H1   2Z H 2
Z
r
PHZ1r  Proportion of H1r hosts encountered = Z 1 H 1 Z
1 H 1   2 H 2
Z
s
i
PHZ1s  H 1i  Proportion of H1s or H 1i hosts encountered = Z1 H 1  ZH 1 
1 H 1   2 H 2
Z
i
PHZ1i  Proportion of H 1i hosts encountered = Z 1 H 1 Z
1 H 1   2 H 2
Z
s
r
1 
PHZ1s H 1r  Proportion of H1s or H1r hosts encountered = Z1 H 1  H
1 H 1   2Z H 2
Z
H2
Then, the rate of cofeeding transmission of larvae
 PLL  PLN 1  PLL   PLN  PLL 1  PLN 


 LiQ ,t PfL rLL PHL1r  siLL PHL1s  H 1i
 1  exp  

H1


 


 N Qi ,t PfN rLN PHN1r  siLN PHN1s  H 1i
 1  exp  

H1



 LiQ ,t PfL rLL PHL1r  siLL PHL1s  H 1i
 exp  

H1

 


That of nymphs
43


 


 PNN  PNL 1  PNN   PNL  PNN 1  PNL 


 N Qi ,t PfN rNN PHN1r  siNN PHN1s  H 1i
 1  exp  

H1




 LiQ ,t PfL rNL PHL1r  siNL PHL1s  H 1i
 1  exp  

H1



 N Qi ,t PfN rNN PHN1r  siNN PHN1s  H 1i
 exp  

H1

 


 
 




These rates are zero for adult ticks.
The monthly equations of the ticks (with cohort overlap)
July


LUs ,t 1  aT  viab  AFs ,t  exp  d F
LsF ,t 1  0

A
  1   a
0
T

LiU ,t 1  aT  viab   0 AFi ,t  exp  d F
LiF ,t 1  0



NUs ,t 1  NUs ,t  1  PfN  exp  dU

N Fi ,t 1
 N Fi ,t

A



 FEED _ NOT _ INFECTED  N  exp  d 
 N  1  P  exp  d 
 N  P  exp  d   N  FEED _ AND _ INFECTED  exp  d 
 exp  d 
N Fs ,t 1  NUs ,t  exp  dU
NUi ,t 1
N
A

 viab  AFi ,t  exp  d F
i
U ,t
N
N
F
N
N
f
i
U ,t
N
s
F ,t
U
N
N
f
s
U ,t
U
N
N
U
N
F
At 1  0
August



LUs ,t 1  LUs ,t  1  PfL  exp  dU

LiF ,t 1  LiU ,t
N
s
U ,t 1
N

 FEED _ NOT _ INFECTED
 1  P  exp  d 
 P  exp  d   L  FEED _ AND _ INFECTED
 1  P  exp  d 
LsF ,t 1  LUs ,t  exp  dU
LiU ,t 1  LiU ,t
L
s
U ,t
L
L
L
L
f
U
L
L
f
s
U ,t
U
N
f
N
U
44
L

 exp  dU
L


 FEED _ NOT _ INFECTED  N  exp  d 
N
 N  1  P  exp  d 
N
 N  P  exp  d   N  FEED _ AND _ INFECTED
 exp  d   N  exp  d 
N Fs ,t 1  NUs ,t  exp  dU
i
U ,t 1
i
U ,t
i
F ,t 1
i
U ,t
N
N
U
N
N
f
N
F
N
N
f
s
U ,t
U
N
N
i
F ,t
U
N
s
F ,t
F
At 1  0
September



LUs ,t 1  LUs ,t  1  PfL  exp  dU

 FEED _ NOT _ INFECTED  L  exp  d 
 L  1  P  exp  d 
 L  P  exp  d   L  FEED _ AND _ INFECTED  exp  d 
 exp  d 
LsF ,t 1  LUs ,t  exp  dU
LiU ,t 1
LiF ,t 1
 LiF ,t
i
U ,t
L
i
U ,t
L
s
F ,t
F
L
U
L
L
f
s
U ,t
U
L
L
U
L
F

N Fs ,t 1  NUs ,t  exp  dU
N
L
L
f
NQs ,t 1  0
i
Q, t 1

L
0
N
 FEED _ NOT _ INFECTED

N Fi ,t 1  NUi ,t  PfN  exp  dU

 N Fi ,t  exp  d F
N

N
 N
s
U ,t
N

 N Fs ,t  exp  d F
N


 FEED _ AND _ INFECTED N  exp  dU
At 1  0
October



LUs ,t 1  LUs ,t  1  PfL  exp  dU

 FEED _ NOT _ INFECTED  L  exp  d 
 L  1  P  exp  d 
 L  P  exp  d   L  FEED _ AND _ INFECTED  exp  d 
 exp  d 
LsF ,t 1  LUs ,t  exp  dU
LiU ,t 1
LiF ,t 1
 LiF ,t

L
i
U ,t
L
F
U
L
L
f
s
U ,t
U
U
L




N
  1    m  N
AUi ,t 1   N  m N  N Fi ,t  exp  d F
N
N
L
L
F
AUs ,t 1  m N  N Fs ,t  exp  d F
AFs ,t 1  0
L
s
F ,t
L
L
f
i
U ,t
N t 1  0
L
N

AFi ,t 1  0
45
i
F ,t

 exp  d F
N

N

November
LUs ,t 1  0

LsF ,t 1  LUs ,t  exp  dU
LiU ,t 1  0
 FEED _ NOT _ INFECTED
L

LiF ,t 1  LiU ,t  PfL  exp  dU

 LiF ,t  exp  d F
N t 1  0

L

L
 L
s
U ,t


AUs ,t 1  AUs ,t  1  PfA  exp  dU

AFi ,t 1  AUi ,t

 LsF ,t  exp  d F
L


 FEED _ AND _ INFECTED L  exp  dU
L


 FEED _ NOT _ INFECTED
 1  P  exp  d 
 P  exp  d   A  FEED _ AND _ INFECTED
AFs ,t 1  AUs ,t  exp  dU
AUi ,t 1  AUi ,t
A
L
A
A
A
A
f
U
A
A
f
s
U ,t
U
A

 exp  dU
A

December
Lt 1  0

NUs ,t 1  m L  LsF ,t  exp  d F
N Fs ,t 1  0
L
  1    m
L

L


A

NUi ,t 1   L  m L  LiF ,t  exp  dU
N
i
F ,t 1
0


AUs ,t 1  AUs ,t  1  PfA  exp  dU

AFi ,t 1
 AFi ,t

 LiF ,t  exp  d F
i
U ,t
A
A
i
U ,t
A
s
U ,t
U

N


N

NUs ,t 1  NUs ,t  exp  dU
NUi ,t 1  NUi ,t  exp  dU



AUs ,t 1  AUs ,t  1  PfA  exp  dU
A
A
U
A
0
F
A
F
N Fs ,t 1  0
A
s
F ,t
U
A
f
Lt 1  0
N

A
A
f
January
i
F ,t 1
L
 FEED _ NOT _ INFECTED  A  exp  d 
 A  1  P  exp  d 
 A  P  exp  d   A  FEED _ AND _ INFECTED  exp  d 
 exp  d 
AFs ,t 1  AUs ,t  exp  dU
AUi ,t 1
L

46

 FEED _ NOT _ INFECTED  A  exp  d 
 A  1  P  exp  d 
 A  P  exp  d   A  FEED _ AND _ INFECTED  exp  d 
 exp  d 
AFs ,t 1  AUs ,t  exp  dU
AUi ,t 1
AFi ,t 1
 AFi ,t
i
U ,t
i
U ,t
A
A
U
A
A
f
U
s
U ,t
U
February

N


N

NUs ,t 1  NUs ,t  exp  dU
NUi ,t 1  NUi ,t  exp  dU
N
i
F ,t 1
0

 exp  d
 exp  d
 exp  d
AUs ,t 1  AUs ,t  exp  dU
A
AFs ,t 1  AFs ,t
A
AUi ,t 1  AUi ,t
AFi ,t 1  AFi ,t
F
A
U
A
F




March
Lt 1  0

N


N

NUs ,t 1  NUs ,t  exp  dU
N Fs ,t 1  0
NUi ,t 1  NUi ,t  exp  dU
N
i
F ,t 1
0

 exp  d
 exp  d
 exp  d
AUs ,t 1  AUs ,t  exp  dU
A
AFs ,t 1  AFs ,t
A
AUi ,t 1  AUi ,t
AFi ,t 1  AFi ,t
F
A
U
A
F




April
Lt 1  0

NUs ,t 1  NUs ,t  exp  dU
N
A
A
A
N Fs ,t 1  0
F
A
A
f
F
Lt 1  0
A
s
F ,t

N Fs ,t 1  0
47

NUi ,t 1  NUi ,t  exp  dU
N Fi ,t 1  0

N



AUs ,t 1  AUs ,t  1  PfA  exp  dU
s
F ,t 1
A
AUi ,t 1
AFi ,t 1
 AFi ,t

 FEED _ NOT _ INFECTED  A  exp  d 
 A  1  P  exp  d 
 A  P  exp  d   A  FEED _ AND _ INFECTED  exp  d 
 exp  d 
A
s
U ,t

A
 exp  dU
i
U ,t
A
A
F
A
A
f
i
U ,t
A
s
F ,t
U
A
A
f
s
U ,t
U
A
A
U
A
F
May
Lt 1  0

N


N

NUs ,t 1  NUs ,t  exp  dU
N Fs ,t 1  0
NUi ,t 1  NUi ,t  exp  dU
N Fi ,t 1  0



AUs ,t 1  AUs ,t  1  PfA  exp  dU

AFi ,t 1
 AFi ,t

 FEED _ NOT _ INFECTED  A  exp  d 
 A  1  P  exp  d 
 A  P  exp  d   A  FEED _ AND _ INFECTED  exp  d 
 exp  d 
AFs ,t 1  AUs ,t  exp  dU
AUi ,t 1
A
i
U ,t
i
U ,t
A
A
A
s
F ,t
F
A
A
f
U
A
A
f
s
U ,t
U
A
A
U
A
F
June
Lt 1  0



NUs ,t 1  NUs ,t  1  PfN  exp  dU

N Fi ,t 1  NUi ,t
AUs ,t 1  0
N
N
N
N
f
U
N
N
f

AUi ,t 1  0

A
 FEED _ NOT _ INFECTED
AFi ,t 1  AUi ,t  PfA  exp  dU
 AFi ,t  exp  d F
A
s
U ,t
U
AFs ,t 1  AUs ,t  exp  dU


 FEED _ NOT _ INFECTED
 1  P  exp  d 
 P  exp  d   N  FEED _ AND _ INFECTED
N Fs ,t 1  NUs ,t  exp  dU
NUi ,t 1  NUi ,t
N

A
 A
s
U ,t
A
N

 exp  dU

 AFs ,t  exp  d F

A

 FEED _ AND _ INFECTED A  exp  dU
48
N
A


The host equations
To accommodate the time scales of the host infectious period (2 – 3 days), we approximate the
monthly host dynamics using discrete-time equations derived from the integration of differential
equations that describe the within-month infection, recovery, and mortality processes in the host
population.


 H
exp     exp   exp  d 
exp    d  
 
H s1  H s exp    d H
H i 1  H i
H

s

H






H r1  H r exp  d H  H i 1  exp   exp  d H  H s 1 
exp    
exp    exp  d H
  
   





References
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persistence of tick-borne infections. Mathematical BioSciences 208: 216–240.
2. Labuda M, Kozuch O, Zuffova E, Eleckova E, Hails R et al. (1997) Tick-borne encephalitis
virus transmission between ticks cofeeding on specific immune natural rodent hosts. Virology
235: 138-143.
3. Randolph SE, Miklisova D, Lysy J, Rogers DJ, Labuda M. (1999) Incidence from
coincidence: patterns of tick infestations on rodents facilitate transmission of tick-borne
encephalitis virus. Parasitology 118: 177-186.
4. Brunner JL, Ostfeld RS. (2008) Multiple causes of variable tick burdens on small-mammal
hosts. Ecology 89: 2259-2272.
5. Rosà R, Pugliese A, Norman R, Hudson PJ. (2003) Thresholds for disease persistence in
models for tick-borne infections including non-viraemic transmission, extended feeding and
tick aggregation. Journal of Theoretical Biology 224: 359-376.
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vol. 247, pp. 1-10.
49

