Sea-lice infection models for fishes

As free-living sea-lice larvae are difficult to sample directly, lice abundances on fish have recently been used to study larvae in the water. In the KLV problem, juvenile wild salmon migrate past a salmon farm, and the change of infection with distance along the migration route is used to estimate larvae production from the farm. In the farm problem, time-varying infection of sea-cage fish is used to estimate the time-variation of free-living larvae in waters near the farm. Both inverse problems require good forward models for infection. In the farm problem, hosts are relatively large and lice pathogenesis is seldom mortal, whereas in the KLV problem hosts are small and lice-induced host mortality can affect lice abundance; thus, infection models for the farm problem are special cases of models for the KLV problem. Here I give an infection model for the KLV problem that explicitly includes lice clumping and host mortality, showing that Krkosek et al. (Proc R Soc B 272:689-696, 2005) (KLV) probably underestimated larvae production by the salmon farm, and further, that if lice development rates were known from laboratory data, lice abundance field data could be directly inverted for lice-induced host mortality during migration. If lice-induced host mortality is negligible, or if lice are Poisson distributed, infection models of arbitrary complexity reduce to Erlang models. I give two useful Erlang models with their solutions for non-zero initial conditions.