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Simulation implementation details (TODO: cleanup)

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Enrico Trombetta edited this page Nov 22, 2021 · 2 revisions

Salmon

Salmon fish live in a sea cage in a farm. Over time it grows and has a chance of dying.

Growth rate

  • 10000 / (1 + exp(-0.01 * (t - 475))) based on a fitted logistic curve to data from FAO

Death rate

Sea lice

The life cycle of sea lice is modelled as 5 stages:

L1: Nauplius

The sea lice nauplii hatch from the eggs and drift in the water with some very limited ability to move. This stage is also called planctonic.

L2: Copepodid

Nauplii develop further into copepodids. They also drift in water and have limited movement but can perform initial attachment to the fish (infection). Note that a sea lice needs to be attached to a fish to develop further into the chalimus.

L3 & L4: Chalimus & Pre-adult

Chalimus performs further attachement to the fish. They develop into pre-adult sea lice and are now mobile and can move on the host and swim in water column. They cannot yet reproduce.

L5: Adult

Adult sea lice can reproduce. They are further split into adult male (L5m) and adult female (L5f).

See here for more details and here for a summary graph (page 6, Figure I).

Additionally:

  • Adult male and adult female attached to the same fish can reproduce.
  • The sea lice development depends on the water temperature (warmer -> faster)
  • The sea lice can develop resistance to a chemical treatment and pass it on to their offspring
  • Sea lice have a chance of dying depending on their development stage and gender (background death) and the treatment (treatment death)

Sea Lice Events

Development

Progression in the life cycle of the sea lice. Based on Aldrin et al 2017.

Infection

Attachement to the host fish. Based on Aldrin et al 2017.

Mating

Originally devised: sea lice grows on a given fish obtained by this formula: adult_males * adult_females * (1/4) * (2.5 / avg_adult_female_lifetime)

TODO: source.

New approach: use a sigmoid function on the current adult female population. Based on Aldrin et al. 2017.

Advantages:

  • Simple and straightforward

Disadvantages/Opinionated?

  • Simplistic: does not take into account the available lice. In particular, it does not consider that lice are usually unavailable for 3 days after mating (see assumptions below). TODO: Does this matter?
  • Assumes a bias-free sex distribution. Some papers discuss that sex biases may be plausible as lice appear to have promiscuous behaviour and propose better models. See Cox et al., 2017.

Egg production

Based on Aldrin, but may change according to the assumptions below.

Old notes

6-11 broods of 500 ova on average extrude a pair every 3 days (alternatively a constant production of 70 nauplii/day for 20 days post-mating), based on Costello 2006.

Assumptions
  • lice is unavailable for 3 days after mating
  • a female will mate as soon as a male is available
  • only adult (5F and 5M) sea lice can reproduce
  • the offspring's inherited resistence is calculated as linear function of the parents' resistances

Background death

Constant rates based on Stien et al 2005.

Treatment death

The treatment death in a stage depends on three factors:

  • the type of treatment
  • the genotype distribution of the lice
  • the temperature

The first and second factor are heavily intertwined: different genetic mechanisms influence resistance in different ways that are not fully understood to this day. For example, Deltamethrin resistance is mainly affected by mithocondrial haplotype, while Emabectine resistance is modelled by heterozygous monogenic allele. For more information:

The third factor is mainly taken into account to determine the length for which the treatment has noticeable effects. For example, it is known that once EMB is applied the effect can range between 10 and 28 days according to the temperature (the higher, the shorter). These are discussed by Aldrin et al. (2017) (see §2.2.3, equations 19-20).

Farms, Sea cages & Reservoir

Farm consists of multiple sea cages. Reservoir is the external environment that is also modelled as a constant inflow of sea lice into farms.

Reservoir is modelled after the following hierarchical model (ours, not based on other models):

a = constant (we estimate 0.003)
b = const (we estimate 150)
D(t) = a*N_offspring(t) + b = a*sum((offspring(t, farm) for farm in farms))
alpha(t) = alpha(t-1) + D(t)/sum(D(t)
Ext(t, cage) = Multinomial(sum(D(t)), Dirichlet(alpha(t))

In other worlds, we describe a linear relationship between the number of offspring adsorbed by the reservoir and the actual number of L1-L2 lice that will thus re-enter the cages. The multinomial distribution naturally models the generation of discrete events without replacement, and the Dirichlet distribution is the conjugated prior of the multinomial distribution, allowing for a natural application of Bayes's rule to infer the new population genotype resistance.

We assume lice can enter the cages only once they have hatched, and assume the choice between the stages they can assume (L1, L2) is chosen with a simple Bernoullian trial for each genotype in Ext(t, cage).

Model outline

Given sea cages populated with fish (based on farm data) at each development stage, at each considered timestep perform the following updates:

  • get dead sea lice from background death
  • get dead sea lice from treatment death
  • progress the development of the sea lice
  • progress fish growth
  • get dead fish (natural death or from sea lice)
    • remove lice associated with dead fish
  • perform infection
  • get new offspring from sea lice mating
  • update the population deltas (changes in population of fish and sea lice)

Also:

Treatment

EMB Treatment Model

TODO: source of data (f_meanEMB, f_sigEMB, EMBmort, env_meanEMB, env_sigEMB)

We follow Jensen et al (2017) and model resistance according to genotype population. Unfortunately the paper does not provide "survival rates per genotype" but rather expected lifespan under treatment. We currently approximate with a simple geometric distribution (every day only a fixed fraction of lice belonging to a certain stage and genetic group can die).

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