# American Institute of Mathematical Sciences

August  2016, 21(6): 1869-1893. doi: 10.3934/dcdsb.2016027

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

 1 Department of Mathematics, University of Florida, 1400 Stadium Rd, Gainesville, FL 32611, United States 2 Department of Mathematics, University of Florida, 1400 Stadium Road, Gainesville, FL 32611

Received  April 2015 Revised  February 2016 Published  June 2016

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.
Citation: Evan Milliken, Sergei S. Pilyugin. A model of infectious salmon anemia virus with viral diffusion between wild and farmed patches. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1869-1893. doi: 10.3934/dcdsb.2016027
##### References:
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##### References:
 [1] E. Beretta and Y. Kuang, Modeling and analysis of marine bacteriophage infection,, Math. Biosci., 149 (1998), 57. doi: 10.1016/S0025-5564(97)10015-3. Google Scholar [2] G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems,, Proc. Am. Math. Soc., 96 (1986), 425. doi: 10.1090/S0002-9939-1986-0822433-4. Google Scholar [3] G. Butler and P. Waltman, Persistence in dynamical systems,, J. Differ. Equ., 63 (1986), 255. doi: 10.1016/0022-0396(86)90049-5. Google Scholar [4] P. DeLeenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis,, Math. Med. Biol., 25 (2008), 285. Google Scholar [5] K. Falk, E. Namork, E. Rimstad, S. Mjaaland and B. H. Dannevig, Characterization of infectious salmon anemia virus, an orthomyxo-like virus isolated from Atlantic salmon (Salmo salar L.),, J. Virol., 71 (1997), 9016. Google Scholar [6] A. Fonda, Uniformly persistent semidynamical systems,, Proc. Am. Math. Soc., 104 (1988), 111. doi: 10.1090/S0002-9939-1988-0958053-2. Google Scholar [7] H. I. Freedman, S. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set,, J. Dyn. Differ. Equ., 6 (1994), 583. doi: 10.1007/BF02218848. Google Scholar [8] B. Garay, Uniform persistence and chain recurrence,, J. Math. Anal. Appl., 139 (1989), 372. doi: 10.1016/0022-247X(89)90114-5. Google Scholar [9] M. G. Godoy, et al., Infectious salmon anemia virus (ISAV) in Chilean Atlantic salmon (Salmo salar) aquaculture: emergence of low pathogenic ISAV-HPR0 and re-emergence of ISAV-HPR$\Delta$: HPR3 and HPR14,, Virol. J., 10 (2013). Google Scholar [10] J. K. Hale and H. Koçak, Dynamics and Bifurcations,, Volume 3, (1991). doi: 10.1007/978-1-4612-4426-4. Google Scholar [11] J. A. P. Heesterbeek, A brief history of $\mathcalR_0$ and a recipe for its calculation,, Acta Biotheor., 50 (2002), 189. Google Scholar [12] J. Hofbauer and J. W.-H. So., Uniform persistence and repellers for maps,, Proc. Am. Math. Soc., 107 (1989), 1137. doi: 10.1090/S0002-9939-1989-0984816-4. Google Scholar [13] V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems,, Math. Biosci., 111 (1992), 1. doi: 10.1016/0025-5564(92)90078-B. Google Scholar [14] M. Krkosek, M. A. Lewis and J. P. Volpe, Transmission dynamics of parasitic sea lice from farm to wild salmon,, Proc. R. Soc. B, 272 (2005), 689. doi: 10.1098/rspb.2004.3027. Google Scholar [15] F. O. Mardones, A. M. Perez and T. E. Carpenter, Epidemiological investigation of the re-emergence of infectious salmon anemia virus in Chile,, Dis. Aquat. Organ., 84 (2009), 105. Google Scholar [16] M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology,, Oxford University Press, (2000). Google Scholar [17] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I: Dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [18] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, AMS, (1995). Google Scholar [19] H. L. Smith and P. DeLeenheer, Virus dynamics: A global analysis,, SIAM J. Appl. Math., 63 (2003), 1313. doi: 10.1137/S0036139902406905. Google Scholar [20] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. Google Scholar [21] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an epidemic model),, SIAM J. Math. Anal., 24 (1993), 407. doi: 10.1137/0524026. Google Scholar [22] H. R. Thieme, Mathematics in Population Biology,, Princeton University Press, (2003). Google Scholar [23] S. Vike, S. Nylund and A. Nylund, ISA virus in Chile: Evidence of vertical transmission,, Arch. Virol., 154 (2009), 1. doi: 10.1007/s00705-008-0251-2. Google Scholar [24] P. Waltman, A brief history of persistence in dynamical systems,, in Delay differential equations and and dynamical systems (eds. S. Busenberg and M. Martelli), (1991), 31. doi: 10.1007/BFb0083477. Google Scholar
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