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 and Continuous Dynamical Systems - B, 2016, 21 (6) : 1869-1893. doi: 10.3934/dcdsb.2016027
References:
[1]

E. Beretta and Y. Kuang, Modeling and analysis of marine bacteriophage infection, Math. Biosci., 149 (1998), 57-76. doi: 10.1016/S0025-5564(97)10015-3.

[2]

G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Am. Math. Soc., 96 (1986), 425-430. doi: 10.1090/S0002-9939-1986-0822433-4.

[3]

G. Butler and P. Waltman, Persistence in dynamical systems, J. Differ. Equ., 63 (1986), 255-263. doi: 10.1016/0022-0396(86)90049-5.

[4]

P. DeLeenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis, Math. Med. Biol., 25 (2008), 285-322.

[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), 9016-9023.

[6]

A. Fonda, Uniformly persistent semidynamical systems, Proc. Am. Math. Soc., 104 (1988), 111-116. doi: 10.1090/S0002-9939-1988-0958053-2.

[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), 583-600. doi: 10.1007/BF02218848.

[8]

B. Garay, Uniform persistence and chain recurrence, J. Math. Anal. Appl., 139 (1989), 372-381. doi: 10.1016/0022-247X(89)90114-5.

[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), p344.

[10]

J. K. Hale and H. Koçak, Dynamics and Bifurcations, Volume 3, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4.

[11]

J. A. P. Heesterbeek, A brief history of $\mathcalR_0$ and a recipe for its calculation, Acta Biotheor., 50 (2002), 189-204.

[12]

J. Hofbauer and J. W.-H. So., Uniform persistence and repellers for maps, Proc. Am. Math. Soc., 107 (1989), 1137-1142. doi: 10.1090/S0002-9939-1989-0984816-4.

[13]

V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci., 111 (1992), 1-71. doi: 10.1016/0025-5564(92)90078-B.

[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), 689-696. doi: 10.1098/rspb.2004.3027.

[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), 105-114.

[16]

M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, UK, 2000.

[17]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I: Dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[18]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, RI, 1995.

[19]

H. L. Smith and P. DeLeenheer, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.

[20]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, New York, 1995. doi: 10.1017/CBO9780511530043.

[21]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an epidemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026.

[22]

H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003.

[23]

S. Vike, S. Nylund and A. Nylund, ISA virus in Chile: Evidence of vertical transmission, Arch. Virol., 154 (2009), 1-8. doi: 10.1007/s00705-008-0251-2.

[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), Lecture Notes in Mathematics, Volume 1475, Springer, (1991), 31-40. doi: 10.1007/BFb0083477.

show all references

References:
[1]

E. Beretta and Y. Kuang, Modeling and analysis of marine bacteriophage infection, Math. Biosci., 149 (1998), 57-76. doi: 10.1016/S0025-5564(97)10015-3.

[2]

G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Am. Math. Soc., 96 (1986), 425-430. doi: 10.1090/S0002-9939-1986-0822433-4.

[3]

G. Butler and P. Waltman, Persistence in dynamical systems, J. Differ. Equ., 63 (1986), 255-263. doi: 10.1016/0022-0396(86)90049-5.

[4]

P. DeLeenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis, Math. Med. Biol., 25 (2008), 285-322.

[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), 9016-9023.

[6]

A. Fonda, Uniformly persistent semidynamical systems, Proc. Am. Math. Soc., 104 (1988), 111-116. doi: 10.1090/S0002-9939-1988-0958053-2.

[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), 583-600. doi: 10.1007/BF02218848.

[8]

B. Garay, Uniform persistence and chain recurrence, J. Math. Anal. Appl., 139 (1989), 372-381. doi: 10.1016/0022-247X(89)90114-5.

[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), p344.

[10]

J. K. Hale and H. Koçak, Dynamics and Bifurcations, Volume 3, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4.

[11]

J. A. P. Heesterbeek, A brief history of $\mathcalR_0$ and a recipe for its calculation, Acta Biotheor., 50 (2002), 189-204.

[12]

J. Hofbauer and J. W.-H. So., Uniform persistence and repellers for maps, Proc. Am. Math. Soc., 107 (1989), 1137-1142. doi: 10.1090/S0002-9939-1989-0984816-4.

[13]

V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci., 111 (1992), 1-71. doi: 10.1016/0025-5564(92)90078-B.

[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), 689-696. doi: 10.1098/rspb.2004.3027.

[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), 105-114.

[16]

M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, UK, 2000.

[17]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I: Dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[18]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, RI, 1995.

[19]

H. L. Smith and P. DeLeenheer, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.

[20]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, New York, 1995. doi: 10.1017/CBO9780511530043.

[21]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an epidemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026.

[22]

H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003.

[23]

S. Vike, S. Nylund and A. Nylund, ISA virus in Chile: Evidence of vertical transmission, Arch. Virol., 154 (2009), 1-8. doi: 10.1007/s00705-008-0251-2.

[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), Lecture Notes in Mathematics, Volume 1475, Springer, (1991), 31-40. doi: 10.1007/BFb0083477.

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