Space-Time Modelling of the Spread of Salmon Lice between and within Norwegian Marine Salmon Farms

Parasitic salmon lice are potentially harmful to salmonid hosts and farm produced lice pose a threat to wild salmonids. To control salmon lice infections in Norwegian salmonid farming, numbers of lice are regularly counted and lice abundance is reported from all salmonid farms every month. We have developed a stochastic space-time model where monthly lice abundance is modelled simultaneously for all farms. The set of farms is regarded as a network where the degree of contact between farms depends on their seaway distance. The expected lice abundance at each farm is modelled as a function of i) lice abundance in previous months at the same farm, ii) at neighbourhood farms, and iii) other, unspecified sources. In addition, the model includes explanatory variables such as seawater temperature and farm-numbers of fish. The model gives insight into factors that affect salmon lice abundance and contributing sources of infection. New findings in this study were that 66% of the expected salmon lice abundance was attributed to infection within farms, 28% was attributed to infection from neighbourhood farms and 6% to non-specified sources of infection. Furthermore, we present the relative risk of infection between neighbourhood farms as a function of seaway distance, which can be viewed as a between farm transmission kernel for salmon lice. The present modelling framework lays the foundation for development of future scenario simulation tools for examining the spread and abundance of salmon lice on farmed salmonids under different control regimes.

Here, y 0 i(t−1) is an indicator variable that is 1 if y i(t−1) = 0, i.e. if there was a zero lice count at farm i at month t − 1, and 0 elsewhere. Furthermore, the β z -s are coefficients as usual. The interpretation of the first and the last two terms in (1) is that the sum of these constitute three different intercepts.
The intercept is β z 0 if the current farm was active in the previous month (S i(t−1) = 1) with positive lice counts (y 0 i(t−1) = 0). If zero lice was counted in the previous month (y 0 Finally, if the current farm was in-active in the previous month (S i(t−1) = 0), the intercept is β z 0 + β z 4 .
The Box-Cox transformation allows for a non-linear dependency of the expected lice abundance µ it on the logit scale, and β z 2 controls this non-linear dependency.

Modelling the negative binomial distribution
If X is negative binomially distributed with parameters R and P , its probability distribution is [1] P The mean is RP/(1 − P ) and the variance is RP/(1 − P ) 2 . If we introduce the i and t indexes, in our case with a zero-inflated negative binomial distribution the mean in the negative binomial part is where n = 20 is the number if fish in the sample, and the expected salmon lice abundance µ it and the excess zero probability p z it are modelled as described above.
In addition, we model the parameter R it as a function of a Box-Cox transformation of µ it as where the β R -s are coefficients that are to be estimated from the data. A simpler alternative would be to model R it as a constant, i.e. with β R 1 = 0, but this would give a significantly poorer fit and provide a less precise description of the probability distribution for the counts. On the other hand, R it could also have been modelled as a separate function of the explanatory variables as in Jansen et al. ( [2], but our alternative model in Eq. (3) is a parsimonious compromise between these two extremes.
The parameter P it is then implicitly given as The likelihood We drop the farm and month indexes for a moment and let p 0 denote the probability for the number of counted lice, y, being 0. This is the sum of the probability for excess zeroes and of the probability for an "ordinary" zero from the negative binomial distribution, given by The probability for the zero-inflated negative binomial distribution is then where, as before, y 0 = 1 if y = 0 and y 0 = 0 if y > 0.
The log likelihood of our data is therefore dropping terms that do not depend on the parameters.
Results for p x it and R it Table A shows the estimated values for the parameters in the sub-models for p z it and R it . The probability for excess zeroes increased if the observed lice abundance the previous month was 0 (beta z 3 > 0) and when the current farm was in-active the previous month β z 4 > 0. The probability for excess zeroes increased also by increasing expected lice abundance, which at first glance may seen counter-intuitive. But this does not mean that the total probability for observing 0 lice abundance increased, since the probability for "ordinary" zeroes in the negative binomial distribution at the same time increased. The R parameter in the negative binomial distribution decreased by increasing expected lice abundance, since β R 1 was positive when β R 2 was negative.  Table B corresponds to Table 1 in the main text and shows the estimated parameters in the expected abundance when medical treatment the previous month is included in the model.  Other, non-optimal variants of the model We also investigated several variants of the model, but none of these gave improved BIC values compared to the selected model. These include:

Results including medical treatment the previous month
• Models with either a third order polynomial of seawater temperature, a cross product of latitude and squared temperature, or a second order polynomial of the logarithm of the fish weight.
• A model with seawater temperatures from month t − 1 instead of month t. • The lice counts show a strong seasonal variation, which potentially can be accounted for by the seawater temperature. To investigate if there was further seasonal variation, we fitted a model with a pair of sine and cosine functions with a period of 12 months in Eq. (3) in the main text, and a model that in addition included a pair with a period of six months.