A model of infectious salmon anemia virus with viral diffusion between wild and farmed patches

Abstract

As the practice of aquaculture has increased the interplay between large fish farms and wild fisheries in close proximity has become ever more pressing. Infectious salmon anemia virus (ISAv) is a flu-like virus affecting a variety of finfish. In this article, we adapt the standard deterministic within host model of a viral infection to each patch of a two patch system and couple the patches via linear diffusion of the virus. We determine the basic reproductive ratio $\mathcal{R}^0$ for the full system as well as invariant subsystems. We show the existence of unique positive equilibrium in the full system and subsystems and relate the existence of the equilibrium to the $\mathcal{R}^0$ values. In particular, we show that if $\mathcal{R}^0>1$, the virus persists in the environment and is enzootic in the host population; if $\mathcal{R}^0\leq 1$, the virus is cleared and the system asymptotically approaches the disease free equilibrium. We also show that, with positive diffusivity, it is possible for the virus to be excluded when there is a susceptible host population in only one patch, but to persist if there are susceptible host populations in both patches. We analyze the local stability of the equilibria and show the existence of Hopf bifurcations.