Download CALCULATION OF THE FLOCK REPRODUCTION NUMBER r

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Impact of the implementation of rest days in live bird markets on the dynamics of
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H5N1 Highly Pathogenic Avian Influenza
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G. Fournié, F.J. Guitian, P. Mangtani, A.C. Ghani
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ELECTRONIC SUPPLEMENTARY MATERIAL – TEXT ESM1
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CALCULATION OF THE FLOCK REPRODUCTION NUMBER r
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Content
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10
S1.1 – From farm to farms within cluster
11
S1.2 – From farm to farms in other clusters
12
S1.3 – From farm to market within cluster
13
S1.3A) Via indirect contacts
14
S1.3B) Via poultry movements
15
S1.4 – From farm to markets in other clusters
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S1.4A) Via indirect contacts
17
S1.4B) Via poultry movements
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S1.5 – From market to farms within cluster
19
S1.6 – From market to farms in other clusters
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S1.7 – From market to markets in the other clusters
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S1.8 – Validation of the flock reproduction number calculation
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S1.9 – References
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1
1
The flock reproduction number r is defined as the mean number of secondary cases
2
(farm or market) per single infected case in a population initially free of the disease.
3
Here the unit for a “case” is defined as a farm or a market.
4
To calculate r we used the method outlined by Diekmann & Heesterbeek [1]. We have
5
two categories of cases: farms, F and markets, M. These are further stratified by their
6
cluster. Between these categories there are here seven different routes of transmission
7
(see also Figure S1.1):
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1. From an initial infected farm to another farm in the same cluster
9
2. From an initial infected farm to another farm in a different cluster
10
3. From an initial infected farm to a market in the same cluster
11
4. From an initial infected farm to a market in a different cluster
12
5. From an initial infected market to a farm in the same cluster
13
6. From an initial infected market to a farm in a different cluster
14
7. From an initial infected market to a market in a different cluster.
15
Regional market
Regional market
5
3
4
Farm
2
1
6
Farm
Cluster
16
17
18
19
20
21
22
7
Cluster
Figure S1.1: Transmission routes
1: From farm to farms within the same cluster; 2: From farm to farms in other clusters; 3: From farm to
market within the same cluster; 4: From farm to markets in other clusters; 5: From market to farms
within the same cluster; 6: From market to farms in other clusters; 7: From market to markets in other
clusters.
23
To calculate the overall reproduction number the expected number of new “cases”
24
(farms and/or markets) produced by an initial infected “case” (farm or market)
25
through each transmission route is calculated to populate the next generation matrix.
26
The dominant eigenvalue is used as an estimate of r.
27
2
1
For illustration, firstly consider a two-cluster model with two farms and a single
2
market in each cluster. Let Fc(1) and Fc(2) denote the farms in cluster c and Fc’(1) and
3
Fc’(2) denote the two farms in cluster c’. Similarly let Mc denote the market in cluster
4
c and Mc’ the market in cluster c’. The next generation matrix then has the following
5
elements:
6
Fc (1) Fc (2) Fc ' (1) Fc ' (2) M C
r3
r2
r2
Fc (1)
0
r1
2
2
2
r3
r2
r2
Fc (2)
r1
0
2
2
2
r2
r2
r4
Fc ' (1)
0
r1
2
2
2
r2
r2
r4
Fc ' (2)
r1
0
2
2
2
MC
r5
r5
r6
r6
0
MC'
r6
r6
r5
r5
r7
7
8
9
10
11
MC'
r4
2
r4
2
r3
2
r3
2
r7
0
This can be generalised to a setting with Nr clusters and with n farms and 1 market in
each cluster to give a matrix of the form:
FC1 (1)
..
FC1 (n)
FC2 (1)
..
..
FCN ( n)
r
M C1
FC1 (1)
0
r1
n 1
r1
n 1
r2
n  N r  1
r2
n  N r  1
FC2 (n)
r2
r2
..
n  N r  1
n  N r  1
..
FCN (1)
r2
n  N r  1
r2
n  N r  1
r3
n
r4
r4
..
n  N r  1
n  N r  1
..
r1
n 1
0
r1
n 1
r2
n  N r  1
r2
n  N r  1
r2
r2
..
n  N r  1
n  N r  1
r2
n  N r  1
r2
n  N r  1
r3
n
r4
r4
..
n  N r  1
n  N r  1
FC1 (n)
r1
n 1
r1
n 1
0
r2
n  N r  1
r2
n  N r  1
r2
r2
..
n  N r  1
n  N r  1
r2
n  N r  1
r2
n  N r  1
r3
n
r4
r4
..
n  N r  1
n  N r  1
FC2 (1)
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
0
r1
n 1
r1
n 1
..
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r4
n  N r  1
r3
n
..
r4
n  N r  1
..
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r1
n 1
0
r1
n 1
..
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r4
n  N r  1
r3
n
..
r4
n  N r  1
FC2 (n)
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r1
n 1
r1
n 1
0
..
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r4
n  N r  1
r3
n
..
r4
n  N r  1
..
..
r
M C2
..
M CN
r
..
..
..
..
..
..
..
..
..
..
..
..
..
FCN (1)
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r2
..
n  N r  1
0
r1
n 1
r1
n 1
r4
n  N r  1
r4
..
n  N r  1
r3
n
..
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r2
..
n  N r  1
r1
n 1
0
r1
n 1
r4
n  N r  1
r4
..
n  N r  1
r3
n
FCN (n)
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r2
n  N r  1
r2
..
n  N r  1
r1
n 1
r1
n 1
0
r4
n  N r  1
r4
..
n  N r  1
r3
n
M C1
r5
r5
r5
r6
Nr 1
r6
Nr 1
r6
Nr 1
..
r6
Nr 1
r6
Nr 1
r6
Nr 1
0
r7
Nr 1
..
r7
Nr 1
M C2
r6
Nr 1
r6
Nr 1
r6
Nr 1
r5
r5
r5
..
r6
Nr 1
r6
Nr 1
r6
Nr 1
r7
Nr 1
0
..
r7
Nr 1
..
r6
Nr 1
..
r6
Nr 1
..
r6
Nr 1
..
r6
Nr 1
..
r6
Nr 1
..
r6
Nr 1
..
..
..
..
..
r5
r5
r5
..
r7
Nr 1
..
..
..
r7
Nr 1
..
0
r
r
..
M CN
r
12
13
In the following sections we detail the calculations for each of the seven elements,
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r1..r7 in this next generation matrix.
3
1
S1.1 – From farm to farms within cluster (r1)
2
Consider a farm infected at time t=0. For this farm, either the course of infection will
3
go to its end and no infectious birds will pass into the market chain (i.e. all susceptible
4
birds die due to the infection or the infection is detected and remaining birds are
5
culled out) with the farm infectious for V1 days, or birds are transferred to the market
6
chain before the infection is detected and the farm would be infectious for V2 days.
7
8
The total onward infectivity of a farm will therefore be

9
t  min V 1,V 2 
 F  t  dt
t 0
(1)
10
11
where  F (t ) is the environmental load.
12
The farm susceptibility is unchanged during the production cycle, thus the infection
13
can occur at any point during a farm production cycle of length Tcycle. We assume that
14
birds will be culled due to infection x days after the infection event. The probability
15
that infection occurs at time u, with 0<u<Tcycle, is therefore 1/ Tcycle. Thus, V1=x,
16
V2=Tcycle-u, and the total onward infectivity of a farm is therefore:
K 
17
u Tcycle
u 0
1
Tcycle


t  min x ,Tcycle u
t 0

 F  t  dt du
(2)
18
19
The total environmental load for the farm F (t ) was calculated numerically.
20
The expected number of birds infected by a farm via this route, b, is therefore given
21
by
22
23
b   ff KS
24
where γff is the rate of transmission between farms within a cluster. S is the number of
25
susceptible birds given by the expression:
26
S  N b N f  1
T
(3)
Tcycle
cycle
 Trepop 
(4)
27
28
where N f is the number of farms in a cluster, N b is the number of birds per farm and
29
Trepop is the length of time that a farm remains empty between two successive cycles.
30
31
To infer the expected number of infected farms from the number of infected birds b,
32
we need to take into account the probability that several infected birds can belong to
4
N f 1
1
the same farm. At each time step
cycles start or end in the cluster,
2
therefore the population at risk (i.e. populated farms in the same cluster) changes
3
during that period. Moreover the environmental load of the primary case,  F t  ,
4
varies as a function of time, and the infectious period can be shortened due to the end
5
of the production cycle. We therefore calculate the number of farms at risk weighted
6
by the proportion of the index farm’s infectious period that they are at risk in order to
7
calculate the number of farms infected by b birds. We term this the “weighted number
8
of at risk farms”.
Tcycle  Trepop
9
10
Let y1 be the weighted number of populated at risk farms which are already populated
11
when the primary case gets infected. This is given by the expression:
y1 
12
Tcycle  N f  1
Tcycle  Trepop

v Tcycle
v 0
u Tcycle
1
Tcycle u 0

1
t  min  u ,min( v , x ) 
 F  t  dt
t 0

Tcycle
l  min( v , x )
 F  l  dl
l 0
du dv
(5)
13
Tcycle  N f  1
14
Here, reading from the left, the first expression,
15
farms populated in the same cluster when the primary case gets infected. The first
16
integral is over v which is the length of time between the infection of the primary case
17
and the end of its production cycle. The second integral is over u which is the time
18
during which a susceptible farm remains populated once the primary case gets
19
infected. The last component,

Tcycle  Trepop
, is the number of other
t  min  u ,min( v , x ) 
 F  t  dt
t 0

l  min( v , x )
 F  l  dl
l 0
, is the proportion of the total
20
onward infectivity of the primary case that the secondary susceptible farm is exposed
21
to.
22
Similarly, let y2 denote the weighted number of populated at risk farms which start a
23
cycle during the infectious period of the primary case. This is given by the expression:
24
y2 
N f 1
Tcycle  Trepop
t  min( v , x )

v Tcycle
v 0
1
u  min( v , x )
Tcycle u 0


t  min( v , x ) u
l  min( v , x )
l 0
 F  t  dt
 F  l  dl
du dv
(6)
25
5
N f 1
1
Here the first term,
2
cluster. As before, the first integral is over v which is the length of time between the
3
infection of the primary case and the end of its production cycle whilst the second
4
integral is over u, the time at which a susceptible farm starts a new cycle during the
Tcycle  Trepop
, is the rate of new population of farms in the same
t  min( v , x )
5
infectious period of the primary case. The last component,


t  min( v , x ) u
l  min( v , x )
l 0
 F  t  dt
 F  l  dl
6
proportion of the total onward infectivity a susceptible farm is exposed to.
7
Thus the total weighted number of populated at risk farms is then given by
, is the
8
9
y  y1  y2
10
The probability of getting k infected farms when b birds are infected can be expressed
11
as a conditional probability:
12
13
14
15
(7)
P X  k b  P X  k b 1P X  k b  X  k b 1  P X  k 1 b 1P X  k b  X  k 1 b 1
(8)
Therefore, the expected number of infected farms is:
y
r1   kP X  k / b 
16
(9)
k 0
17
18
S1.2 – From farm to farms in other clusters (r2)
19
We can use the same method detailed in section S1.1 (from equation (1) to (9)), to
20
calculate the expected number of farms r2 infected in other clusters. In equation (3)
21
 ff is replaced with  ff , the rate of transmission between farms in other clusters.
22
Equation (4), giving the number of susceptible birds S, equation (5), giving the
23
weighted number of populated at risk farms which are already populated when the
24
primary case gets infected, y1, and equation (6), giving the weighted number of
25
populated at risk farms which start a cycle during the infectious period of the primary
26
case y2, are replaced by
27
S  N b N f N r  1
T
Tcycle
cycle
 Trepop 
(10)
28
29
where Nr is the number of clusters.
6
y1 
1
Tcycle N f  N r  1
Tcycle  Trepop

v Tcycle
v 0
u Tcycle
1
Tcycle u 0
1

t  min  u ,min( v , x ) 
 F  t  dt
t 0

Tcycle
l  min( v , x )
 F  l  dl
l 0
du dv
(11)
2
Tcycle N f  N r  1
3
Here, reading from the left, the first expression,
4
other farms populated in other clusters when the primary case gets infected. The first
5
integral is over v which is the length of time between the infection of the primary case
6
and the end of its production cycle. The second integral is over u which is the time
7
during which a susceptible farm remains populated once the primary case gets
8
infected. The last component,

10
11
, is the number of
t  min  u ,min( v , x ) 
 F  t  dt
t 0

l  min( v , x )
 F  l  dl
l 0
9
Tcycle  Trepop
, is the proportion of the total
onward infectivity of the primary case that the secondary susceptible farm is exposed
to.
y2 
N f  N r  1
Tcycle  Trepop
t  min( v , x )

v Tcycle
v 0
1
u  min( v , x )
Tcycle u 0


t  min( v , x ) u
l  min( v , x )
l 0
 F  t  dt
 F  l  dl
du dv
(12)
12
N f  N r  1
13
Here the first term,
14
clusters. As before, the first integral is over v which is the length of time between the
15
infection of the primary case and the end of its production cycle whilst the second
16
integral is over u , the time at which a susceptible farm starts a new cycle during the
Tcycle  Trepop
, is the rate of new population of farms in other
t  min( v , x )
17
infectious period of the primary case. The last component,


t  min( v , x ) u
l  min( v , x )
l 0
18
 F  t  dt
 F  l  dl
, is the
proportion of the total onward infectivity a susceptible farm is exposed to.
19
20
S1.3 – From farm to market within cluster (r3)
21
There are two routes by which a farm can infect a market:
22
23
A) A farm could infect a market via indirect contacts which can occur at any
point during the production cycle.
7
1
B) If the infection occurs less than x days (x being the length of time a farm is
2
infectious) before the end of the cycle, then the farm could infect a market via
3
animal movements (i.e. transfer of the flock at the end of a cycle into the
4
market chain).
5
6
S1.3A) Via indirect contacts
7
Infection may occur at any point during the production cycle and so the infectivity of
8
the farm, K, is given by equation (2) in section S1.1.
9
10
A market is considered to be infected when at least one bird is infectious. However, at
11
market, an infected bird can be sold and slaughtered before becoming infectious. Let
12
p(Tr,Tlatent) be the probability that a bird becomes infectious when infected at the
13
market. p depends on the latent period Tlatent and the turn-over period Tr. The
14
probability that a market is infected by a farm within the same cluster is given by the
15
expression:
r3A   fm KSp(Tr , Tlatent )
16
17
(13)
18
where  fm is the rate of transmission between farms and markets within a cluster and S
19
is the number of susceptible birds at a market given by
S
20
T
Nb N f Nr
cycle
1
Tm
 Trepop  N r
(14)
21
22
Where the first expression,
T
Nb N f N r
cycle
 Trepop 
, is the number of chickens entering into the
23
market chain each day, and Tm is the average time spent by poultry in the regional
24
market.
25
26
S1.3B) Via poultry movements
27
In this case infection starts in a farm at time u, such that Tcycle-x<u<Tcycle. Let H(u)
28
denote the probability that if an infectious bird is introduced into a farm at time u, an
29
infectious bird will be introduced into the regional market. This will depend on the
30
within-farm disease dynamics as well as virus amplification in the wholesale market.
8
1
We calculate this quantity numerically. Then the probability that a regional market
2
becomes infected by animal movement is given by
r3B  (1  r3A( B ) )
3
1 u Tcycle
H (u )du
x u Tcycle  x
(15)
4
5
where (1  r3A( B ) ) is the probability that the market has not already been infected via
6
indirect contact from the farm. The expression for r3 A ( B ) closely resembles equation
7
(13) but here we need to restrict the calculation of the probability of infection via
8
indirect contact to times u such that Tcycle-x<u<Tcycle:
r3A( B)   fm K 1Sp(Tr , Tlatent )
9
10
11
(16)
Similarly K1 is the infectivity of the farm restricted to these times:
1 t Tcycle F
  t  dt du
u Tcycle  x x t u
K1  
12
u Tcycle
(17)
13
14
where the first integral is over u, the time at which the farm gets infected. The second
15
integral is over t, the time during which the farm remains infected. S is as defined as
16
in equation (14).
17
18
As before, infection can occur in the farm at any point during the production cycle, so
19
the probability that it occurs less than x days before the end of the cycle is
20
Therefore, the overall probability that a market becomes infected is
r3  r3A 
x
Tcycle
.
x
22
(18)
r3B
Tcycle
As there is only one market per cluster this is also the expected number of markets
23
that are infected by a single infected farm in the cluster.
21
24
25
S1.4 – From farm to markets in other clusters (r4)
26
S1.4A) Via indirect contacts
27
The calculation for the probability of infection occurring from a farm to markets in
28
other clusters closely follows that in the previous section. From equation (13), we
29
derive b as the number of infected birds at markets in other clusters
9
b   fm KSp(Tr , Tlatent )
1
(19)
2
3
where Γfm is the rate of transmission between farms and markets in different clusters,
4
and the number of susceptible birds S is defined as
S
5
T
b N f Nr
cycle
 Trepop 
Nr 1
Tm
Nr
(20)
6
7
where the first expression,
T
b N f Nr
cycle
 Trepop 
, is the number of chickens entering into the
8
market chain each day. As detailed in section 1, equations (8) and (9) can be used to
9
infer the number of infected markets from the number of infected birds b. We applied
10
these equations to markets, y being now the number of populated markets in other
11
clusters
y  Nr 1
12
(21)
13
14
S1.4B) Via animal movements
15
The number of regional markets infected by animal movements is given by:
16
17
r4B  ( N r  1)r3B '
18
where r3B ' is the probability that a regional market becomes infected by animal
19
movement as defined in equation (15) to (17). In equation (16)  fm is replaced by Γfm.
(22)
20
21
Therefore, the expected number of infected markets is
r4  r4A 
22
x
Tcycle
r4B
(23)
23
24
S1.5 – From market to farms within cluster (r5)
25
The infectivity of a market depends on two factors:
26
-
27
28
29
the number of birds initially infected at the market (due to indirect contact
with an infected farm or via animal movements from an infected farm),
-
when the infection starts at market (i.e. how long before the next scheduled
rest day).
30
10
1
A market is equally likely to be infected at any time point between two successive rest
2
days, so that the probability that some infected birds are introduced at time v, with
3
0<v<Trestd, is 1/Trestd. Trestd is defined as the period between two successive rest days.
4
When a market is infected via indirect contact, its infectivity is then:
LA  
5
v Trestd
v 0
1
Trestd

t Trestd
t v
 AM (t )dt dv
(24)
6
7
The total environmental load for the market  AM (t ) was calculated numerically. The
8
first integral is over v, the time at which the market gets infected. The second integral
9
is over t, the length of time the market remains infected. When a market is infected
10
via animal movement, its infectivity is then:
LB  
11
ux
u 0
1 v Trestd 1 t Trestd M
 B (t , u )dt dv du
x v 0 Trestd t v
(25)
12
13
with BM (t , u ) the environmental load for the market is again calculated numerically.
14
The first integral is over u, the time at which the infection occurs in a farm before the
15
end of the production cycle. The second integral is over v, the time at which the
16
market gets infected. The third integral is over t, the length of time the market remains
17
infected.
18
19
Thus the market infectivity is given by the expression:
L
20
r3A A r3B B
L 
L
r3
r3
(26)
21
22
The expected number of birds infected in farms in the same cluster is then given by:
23
24
b   fm LS
25
where  fm is the rate of transmission between farms and markets in the same cluster
26
and S is the number of susceptible birds in farms in the same cluster, given by the
27
expression:
28
S  Nb N f
T
Tcycle
cycle
 Trepop 
(27)
(28)
29
11
1
To infer the number of infected farms, we again need to allow for several birds
2
becoming infected in the same farm using equations (8) and (9). Let y1 be the number
3
of populated farms at risk weighted by the infectious period in the market which are
4
already populated when the market gets infected
5
y1 
Tcycle N f
Tcycle  Trepop

v Trestd
v 0
u Tcycle
1
Trestd u 0

1
t  min  u ,v 
 M  t  dt
t 0

Tcycle
l v
l 0

M
 l  dl
du dv
(29)
6
Tcycle N f
7
Here, reading from the left, the first expression,
8
farms populated in the same cluster when the primary market case gets infected. The
9
first integral is over v which is the length of time between the infection of the primary
10
case and the next rest day. The second integral is over u which is the time during
11
which a susceptible farm remains populated once the primary case gets infected. The
12

last component,
Tcycle  Trepop
, is the number of other
t  min  u ,v 
 M  t  dt
t 0

l v
l 0
 M  l  dl
, is the proportion of the total onward infectivity of
13
the primary case that the secondary susceptible farm is exposed to. Similarly let y2
14
denote the number of populated farms at risk weighted by the infectious period in the
15
market which start a cycle during the infectious period of the market
16
y2 
Nf
Tcycle  Trepop

v Trestd
v 0
1
u v
Trestd u 0

t  min( u Tcycle ,v )
 M  t  dt
t u

l v
l 0
 M  l  dl
du dv
(30)
17
Nf
18
Here the first term,
19
cluster. As before, the first integral is over v which is the length of time between the
20
infection of the primary market case and the next rest day whilst the second integral is
21
over u, the time at which a susceptible farm starts a new cycle during the infectious
22
Tcycle  Trepop
, is the rate of new population of farms in the same
period of the primary case. The last

component,
t  min( u Tcycle ,v )
 M  t  dt
t u

l v
l 0
23

M
 l  dl
, is the
proportion of the total onward infectivity a susceptible farm is exposed to.
24
25
12
1
The total weighted number of populated at risk farms is then given by
y  y1  y2
2
3
(31)
4
S1.6 – From market to farms in other clusters (r6)
5
The infectivity LA of a market infected by direct contact and the infectivity LB of a
6
market infected via animal movement are as defined in equations (24) and (25) in
7
section 5. The global infectivity L of the market is:
L
8
r4A A r4B B
L 
L
r4
r4
(32)
9
10
The number of infected birds is expressed as
b   fm LS
11
(33)
12
13
where Γfm is the rate of transmission between farms and markets in other clusters. The
14
number of susceptible birds S is
S  N b N f N r  1
15
T
Tcycle
cycle
 Trepop 
(34)
16
17
The number of infected farms is inferred from b using equation (8) and (9) as detailed
18
in section S1.1. The number of populated farms at risk weighted by the infectious
19
period in the market which are already populated when the market gets infected, y1,
20
and the number of populated farms at risk weighted by the infectious period in the
21
market which start a cycle during the infectious period of the market, y2, are here
22
given by the expressions:
23
y1 
Tcycle N f  N r  1
Tcycle  Trepop

v Trestd
v 0
1
u Tcycle
Trestd u 0
1
Tcycle

t  min( u ,v )
 M  t  dt
t 0

l v
l 0
 M  l  dl
du dv
(35)
24
Tcycle N f  N r  1
25
Here, reading from the left, the first expression,
26
farms populated in other clusters when the primary market case gets infected. The
27
first integral is over v which is the length of time between the infection of the primary
28
case and the next rest day. The second integral is over u which is the time during
29
which a susceptible farm remains populated once the primary case gets infected. The
Tcycle  Trepop
, is the number of
13
1

last component,
t  min  u ,v 
 M  t  dt
t 0

l v
l 0
2
 M  l  dl
, is the proportion of the total onward infectivity of
the primary case that the secondary susceptible farm is exposed to.
3
4
y2 
N f  N r  1
Tcycle  Trepop

v Trestd
v 0
1
u v
Trestd u 0

t  min( u Tcycle ,v )
 M  t  dt
t u

l v
l 0

M
 l  dl
du dv
(36)
5
N f  N r  1
6
Here the first term,
7
clusters. As before, the first integral is over v which is the length of time between the
8
infection of the primary market case and the next rest day whilst the second integral is
9
over u, the time at which a susceptible farm starts a new cycle during the infectious
10
Tcycle  Trepop
, is the rate of new population of farms in other
period of the primary case. The last

component,
t  min( u Tcycle ,v )
 M  t  dt
t u

l v
l 0
11
 M  l  dl
, is the
proportion of the total onward infectivity a susceptible farm is exposed to.
12
13
S1.7 – From market to markets in the other clusters (r7)
14
The infectivity L of the market is defined in equation (32) in section S1.6. The
15
expected number of infected birds in markets is given by
16
b  mm LSp(Tr , Tlatent )
(37)
17
18
where mm is the rate of transmission between markets. The number of susceptible
19
birds S is defined as in equation (20) in section S1.4.
20
21
The number of infected markets is inferred from b using equations (8) and (9) as
22
detailed in section 1. y is defined as in equation (21) in section S1.4.
23
24
S1.8 – Validation of the flock reproduction number calculation
25
To numerically check our expression for r the stochastic metapopulation model was
26
run for different values of the flock reproduction number r. For each simulation, the
27
disease is introduced randomly at any point during the first cycle of a farm. In order to
14
1
reduce the computing time, the length of the farm production cycle is 45 days, and
2
rest days occur every 14 days. The disease is considered as endemic if at least one
3
farm or market is infected at t=365 days. 1000 simulations were run for each value of
4
r. Figure S1.3 presents the proportion of simulations in which the disease becomes
5
endemic. As expected, the slope of the curve increases sharply when r gets higher
6
than one, and then decreases for higher values of r (r>2).
7
8
9
10
11
Figure S1.3: Proportion of simulations the disease gets endemic as function of the flock
reproduction number
Full red line: proportion of simulations; Dotted line: values for 1 – 1/r.
12
S1.9 – References
13
14
15
16
17
1. Diekmann O, Heesterbeek JA (2000) Mathematical Epidemiology of Infectious
Diseases : Model Building, Analysis and Interpretation; Series W, editor.
Chichester.
15