www.nr.no
Space-time modeling of the spread of a
disease between marine fish farms
Magne Aldrin, Norwegian Computing Center and the
University of Oslo
Ragnar Bang Huseby, Norwegian Computing Center
in collaboration with the National Veterinary Institute
and with A. Frigessi and others at the University of Oslo
INFER March 2011
Similar ideas used in work on
• Foot-and-mouth disease
• Swine fever virus
• Hospital infections
1
The aquaculture industry
• In Norway:
Export value of farmed fish > export value of wild fish!
• 1200 fish farms in Norway, spread along the coast,
mostly producing salmon
2
Industry threatened by infectious diseases
• Infectious Salmon Anaemia (ISA) epidemics:
? Scotland 1999: cost 100 million £
? Shetland 2009: cost 40 million £
? Chile 2008: 7000 jobs lost
3
Norway: 70 ISA outbreaks 2003-2009
0°0'
10°0'E
20°0'E
30°0'E
a
Probability of infection from
neighbourhood versus other sources
70°0'N
Neighbour
Other
65°0'N
±
60°0'N
100
Kilometers
10°0'E
20°0'E
4
ISA outbreaks in Troms 2007-2009
17°0'E
Most likely
infection pathway
a
Probability
0 - 0,25
0,25 - 0,5
0,5 - 0,75
69°10'N
0,75 - 1
69°0'N
68°50'N
68°40'N
±
5
Kilometers
17°0'E
5
Data
• Monthly data 2003-2009
• Information on all farms active in this period
? Biomass of fish, tells when a farm is active
? Sea distances between all farms
? Local contact networks between all farms
(same ownership, may share staff and equipment)
6
Three diseases
Marine Environment and Health Series No. 22, 2005
_______________________________________________________________________________________________
2 PANCREAS DISEASE IN IRELAND: EPIDEMIOLOGICAL SURVEY RESULTS FOR
2003 AND 2004
Hamish Rodger & Susie Mitchell
• Information on outbreaks of three diseases
? When and where
? 500 HSMI
? 300 PD
? 70 ISA
• No information on infection time, except
infection time < outbreak time
Vet-Aqua International, Oranmore Business Park, Oranmore, Co. Galway
2.1.Introduction
Pancreas disease aetiology & clinical disease
Pancreas disease (PD) is an infectious viral disease of marine stage farmed Atlantic salmon
(Salmo salar) that was first described by Munro et al. (1984) in Scotland. The disease has been
reported in Ireland (1984), Norway (1985), France (1986), USA (1987) and Spain (1989) (Kent
& Elston, 1987, Raynard et al., 1992) with the most serious impacts from the disease reported in
Ireland, Norway and Scotland.
The causal agent of the disease is an atypical alphavirus, the salmon pancreas disease virus
(SPDV), of which there appear to be at least two subtypes (Hodneland et al., 2005, Weston et al.,
2005); one subtype is associated with PD in Scotland and Ireland and another is associated with
PD in Norway. A third salmonid alphavirus, very closely related to SPDV, and known as the
sleeping disease (SD) virus, causes a similar condition in rainbow trout (Oncorhynchus mykiss)
in freshwater and is reported in France, Italy, Germany and the UK (Castric et al., 1997,
Bergmann et al., 2005). The name salmonid alphavirus (SAV) has been proposed for these
closely related viral isolates (Weston et al., 2002).
Infection with these viruses can lead to clinical disease due to acinar pancreatic necrosis and
fibrosis as well as a range of myopathies (cardiac, skeletal, oesophageal) (Ferguson et al., 1986,
Rodger et al., 1994, McLoughlin et al., 2002). Clinically affected fish present with lethargy,
anorexia and increased mortality. Mortalities can reach over 40% in affected pens and failure to
thrive is a further consequence of the disease resulting in poor condition, thin fish that are
susceptible to parasitism and secondary bacterial disease (Figure 1).
Figure 1: Three salmon from a pen affected by pancreas disease (PD) in Ireland with the top
two fish exhibiting low condition factor, typical of those affected by chronic PD.
In a farm affected by PD, the majority of fish will exhibit pancreatic regeneration and recovery
and appetite then returns. However, in fish affected by the later developing and longer lasting
myopathies, further mortalities can then ensue when the fish exhibit a strong feeding response
(Rodger et al., 1991), probably due to either cardiac muscle exhaustion, further exertional
7
Basic data unit in our modeling
• Fish cohort: A group of fish at a fish farm
• Stocked at some time and slaugthered/removed 2-3
years later
• Only one cohort at a fish farm at a time
• During some years a farm have several
subsequent cohorts
• Note: we do not consider individual fish
8
Susceptible and infectious cohorts
susceptible
a)
stocked
removed
susceptible
time
infectious
b)
stocked
infected
susceptible
visible
outbreak
removed
time
infectious
c)
stocked
infected
removed time
9
Two sub-models
• Infection process - but infection time unobserved
? Infection rate λ
• Outbreak process - outbreak time observed
? Outbreak rate γ
10
Model overwiev
Fixed and known time
Susceptible
λ
Infected
and
infectious
Fixed and known time
Removed
γ
Visible
outbreak
of disease
Fixed and known time
11
Infection model
Additive-multiplicative model for
the total infection rate (intensity)
for fish cohort i at time t
λi(t) = Si(t) · κsi(t) · [λdi(t) + λci(t) + λpi(t) + λoi(t)]
12
Multiplicative terms
• Si(t) is an at-risk indicator, which
is 1 when fish cohort i is susceptible and 0 otherwise
• κsi(t) is a factor proportional to
the susceptibility of fish cohort i, dependent on
? size of the fish cohort (number of fish)
? sea temperature
? season
13
Four transmission pathways
• λdi(t) - Transmission from
infected fish cohorts at neighbouring farms,
dependent on seaway distance to infected fish cohorts
• λci(t) - Transmission from infected fish cohorts at farms
in the same local contact network (e.q. same workers)
•
p
λi (t)
- Transmission from previous infected fish cohorts
at the same fish farm i (e.g. remaining disease agent)
• λoi(t) - Transmission via other, non-specified, pathways
14
Neighbourhood transmission λdi(t)
The sum over the individual contributions from all other
fish cohorts
X
λdi(t) =
λdij (t)
j6=i
15
Contribution from fish cohort j
λdij (t) = Ij (t) · exp(−φ · dij ) · κij (t)
• Ij (t): an indicator variable that is 1 if salmon farm j is
infectious at time t, and 0 otherwise
• dij : the seaway distance between fish cohorts i and j,
i.e. the distance between their farms
• φ: parameter - the effect of the seaway distance
• κij (t): a factor proportional to
the infectiousness of fish cohort j, dependent on
? the size of fish cohort j
? the time since it was infected, based on a SIR model
16
Outbreak model
The outbreak rate for fish cohort i at time t
γi(t) = Ii(t) · γi∗(t)
• Ii(t) is an at-risk indicator, which is 1 when fish farm i
is infected and yet not slaugthered and 0 otherwise
• γi∗(t) a smooth function of time since cohort i was
infected, based on a SIR model
17
A Bayesian data augmentation approach
• Outbreak times are observed data, Dobs
• Infection times are missing data, Dmis
• Parameters θ
Posterior
p(Dmis, θ|Dobs) ∝ p(Dobs, Dmis|θ) · π(θ)
18
Probability density p(Dobs, Dmis|θ)
• Contributions from each infection process:
Z t∗
λi(t∗) · exp(−
λ(t) dt) if infection occurs at t = t∗
tstart
Z
tend
exp(−
λ(t) dt)
if cohort i is never infected
tstart
• Similar contributions from the outbreak processes
19
Priors
• Mostly vague priors for the parameters,
but some restricted to be non-negative
• For some parameters, the possibility of zero effect is of
special interest
? E.g. prior for the distance parameter φ:
∗ P (φ = 0) = 0.5 - i.e. no effect of seaway distance
between fish cohorts (farms)
∗ P (φ > 0) = 0.5 and then
φ ∼ Uniform(0,large number)
20
Estimation
• 15-30 parameters with priors
• More than 3000 cohorts with unknown infection times
• Markov Chain Monte Carlo techniques
21
Results - Posteriors for HSMI
Posterior 95 % cred.int.
mean
low
high
Mean time from infection to outbreak
2.7 2.1
3.4
- given visible outbreak
months
% of cohorts infected
20 18
22
% of infected cohorts with outbreak
76 70
82
22
Effect of sea temperature and season
0 2 4 6 8
12
A farm in Southern Norway 2006
2
4
6
8
10
12
months in year
23
Effect of distance - φ
40
Phi
Relative infection rate
0
0.00
10
0.04
20
30
0.08
Prior
Posterior
0.00
0.02
0.04
0.06
0.08
0.10
0
20
40
60
80
100
seaway distance (km) to infectious farm
24
Effect of time since infection
0.0
0.4
0.8
on infectiousness in the infection process
and on the outbreak rate
0
5
10
15
20
months since infection
25
Relative importance of transmission pathways
Proportion of infections related to neighbourhood
transmission:
average the following expression over all infections
p
d
d
c
λi (t)/(λi (t) + λi (t) + λi (t) + λoi(t))
Similar definitions for the other pathways
Pathway
Neigbourhood
54
Relative importance in % Contact network 18
Previous cohort
9
Other
19
26
Potential use of models for spread of diseases
• Better understanding of infection processes
• Tracing transmission routes
• Estimating the protective effect of vaccination,
given data on vaccination
• Investigating the effects of potential interventions by
scenario simulations - what-if
? the accumulated effects of vaccination on a network
of fish farms for a specified effect of the vaccine
? the effect of re-locations of fish farms
27
References
• Aldrin, Lyngstad, Kristoffersen, Storvik, Borgan, Jansen (2011).
Modelling the spread of infectious salmon anaemia (ISA) among salmon farms
based on seaway distances between farms and genetic relationships between ISA
virus isolates.
To appear in the Journal of the Royal Society Interface.
• Aldrin, Storvik, Frigessi, Viljugrein, Jansen (2010).
A stochastic model for the assessment of the transmission pathways of heart and
skeleton muscle inflammation, pancreas disease and infectious salmon anaemia
in marine fish farms in Norway.
Preventive Veterinary Medicine, Vol. 93, p. 51-61.
• Scheel, Aldrin, Frigessi, Jansen (2007).
A stochastic model for infectious salmon anemia (ISA) in Atlantic salmon
farming.
Journal of the Royal Society Interface, Vol. 4, p. 699-706.
28
Salmon Farming Industry
ThanksHandbook
for your 2009
attention!
The Marine Harvest Salmon Industry Handbook
The purpose of this document is to give financial analysts and
investors a better insight into the salmon farming industry and in
Marine Harvest’s view the most important value drivers.
Our intention is to update and publish this document annually.
Enjoy your lunch today!
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