# American Institute of Mathematical Sciences

November  2012, 17(8): 2829-2848. doi: 10.3934/dcdsb.2012.17.2829

## Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment

 1 Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7, Canada, Canada 2 Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

Received  April 2011 Revised  August 2011 Published  July 2012

In this paper, we propose a mathematical model to describe the avian influenza dynamics in wild birds with bird mobility and heterogeneous environment incorporated. In addition to establishing the basic properties of solutions to the model, we also prove the threshold dynamics which can be expressed either by the basic reproductive number or by the principal eigenvalue of the linearization at the disease free equilibrium. When the environment factor in the model becomes a constant (homogeneous environment), we are able to find explicit formulas for the basic reproductive number and the principal eigenvalue. We also perform numerical simulation to explore the impact of the heterogeneous environment on the disease dynamics. Our analytical and numerical results reveal that the avian influenza dynamics in wild birds is highly affected by both bird mobility and environmental heterogeneity.
Citation: Naveen K. Vaidya, Feng-Bin Wang, Xingfu Zou. Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2829-2848. doi: 10.3934/dcdsb.2012.17.2829
##### References:
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Golender, et al., Avian influenza virus H9N2 survival at different temperatures and pHs, Avian. Dis., 54 (2010), 725-728. doi: 10.1637/8736-032509-ResNote.1. [7] K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985. [8] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproductionratio $R_0$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [9] R. A. Fouchier, V. Munster, A. Wallensten, T. M. Bestebroer and S. Herfst, et al., Characterization of a novel influenza A virus hemagglutinin subtype (H16) obtained from black-headed gulls, J. Virol., 79 (2005), 2814-2822. [10] V. Henaux, M. D. Samuel and C. M. Bunck, Model-based evaluation of highly and low pathogenic avian influenza dynamics in wild birds, PLoS One, 5 (2010), e10997. [11] P. Hess, "Periodic-parabolic Boundary Value Problem and Positivity," Pitman Res. Notes Math., 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [12] V. S. Hinshaw, R. G. Webster and B. Turner, The perpetuation of orthomyxoviruses and paramyxoviruses in Canadian waterfowl, Can.J. Microbiol., 26 (1980), 622-629. doi: 10.1139/m80-108. [13] V. S. Hinshaw, R. G. Webster and B. Turner, Water-bone transmission of influenza A viruses?, Intervirology, 11 (1979), 66-68. doi: 10.1159/000149014. [14] S.-B. Hsu, J. Jiang, and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Diff. Eqns., 248 (2010), 2470-2496. doi: 10.1016/j.jde.2009.12.014. [15] S.-B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, to appear in Journal of Dynamics and Differential Equations. [16] J. Jiang, X. Liang and X.-Q. Zhao, Saddle point behavior for monotone semiflows and reaction-diffusion models, J. Diff. Eqns., 203 (2004), 313-330. doi: 10.1016/j.jde.2004.05.002. [17] S. Krauss, D. Walker, S. P. Pryor, L. Niles and L. Chenghong, et al., Influenza A viruses of migrating wild aquatic birds in North America, Vector Borne Zoonotic Dis., 4 (2004), 177-189. [18] Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8. [19] H. Lu and A. E. Castro, Evaluation of the infectivity, length of infection, and immune response of a low-pathogenicity H7N2 avian influenza virus in specific-pathogen-free chickens, Avian. Dis., 48 (2004), 263-270. doi: 10.1637/7064. [20] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. of A. M. S., 321 (1990), 1-44. [21] P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. [22] B. Olsen, V. J. Munster, A. Wallensten, J. Waldenstrom and A. D. Osterhaus, et al., Global patterns of influenza a virus in wild birds, Science, 312, 384-388. [23] B. Roche, C. Lebarbenchon, M. Gauthier-Clerc, C. M. Chang and F. Thomas, et al., Water-borne transmission drives avian influenza dynamics in wild birds: The case of the 2005-2006 epidemics in the Camargue area, Infect. Genet. Evol., 9 (2009), 800-805. [24] P. Rohani, R. Breban, D. E. Stallknecht and J. M. Drake, Environmental transmission of low pathogenicity avian influenza viruses and its implications for pathogen invasion, Proc. Natl. Acad. Sci. USA, 106 (2009), 10365-10369. doi: 10.1073/pnas.0809026106. [25] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. [26] H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Math. Surveys Monogr., 41, American Mathematical Society, Providence, RI, 1995. [27] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2. [28] D. E. Stallknecht, S. M. Shane, M. T. Kearney and P. J. Zwank, Persistence of avian influenza viruses in water, Avian. Dis., 34 (1990), 406-411. doi: 10.2307/1591428. [29] J. H. Tien and D. J. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol., 72 (2010), 1506-1533. doi: 10.1007/s11538-010-9507-6. [30] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [31] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. [32] N. K. Vaidya and L. M. Wahl, The sensitivity of avian influenza dynamics in wild birds to time-varying environmental temperature, (2011), submitted. [33] P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [34] F.-B. Wang, A system of partial differential equations modeling the competition for two complementary resources in flowing habitats, J. Diff. Eqns., 249 (2010), 2866-2888. doi: 10.1016/j.jde.2010.07.031. [35] W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890. [36] R. G. Webster, W. J. Bean, O. T. Gorman, T. M. Chambers and Y. Kawaoka, Evolution and ecology of influenza A viruses, Microbiol. Rev., 56 (1992), 152-179. [37] R. G. Webster, M. Yakhno, V. S. Hinshaw, W. J. Bean and K. G. Murti, Intestinal influenza: Replication and characterization of influenza viruses in ducks, Virology, 84 (1978), 268-278. doi: 10.1016/0042-6822(78)90247-7. [38] K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusionmodel with a quiescent stage, Proc. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci., 463 (2007), 1029-1043. doi: 10.1098/rspa.2006.1806.

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##### References:
 [1] Best Country Report, Temperature map of Canada. Available from: http://www.bestcountryreports.com/Temperature_Map_Canada.html [Accessed date: 24 February, 2011]. [2] L. Bourouiba, J. Wu, S. Newman, J. Takekawa and T. Natdorj, et al., Spatial dynamics of bar-headed geese migration in the context of H5N1, J. R. Soc. Interface, 7 (2010), 1627-1639. [3] R. Breban, J. M. Drake, D. E. Stallknecht and P. Rohani, The role of environmental transmission in recurrent avian influenza epidemics, PLoS Comput. Biol., 5 (2009), e1000346. [4] J. D. Brown, G. Goekjian, R. Poulson, S. Valeika and D. E. Stallknecht, Avian influenza virus in water: Infectivity is dependent on pH, salinity and temperature, Vet. Microbiol., 136 (2009), 20-26. [5] J. D. Brown, D. E. Swayne, R. J. Cooper, R. E. Burns and D. E. Stallknecht, Persistence of H5 and H7 avian influenza viruses in water, Avian. Dis., 51 (2007), 285-289. [6] I. Davidson, S. Nagar, R. Haddas, M. Ben-Shabat and N. Golender, et al., Avian influenza virus H9N2 survival at different temperatures and pHs, Avian. Dis., 54 (2010), 725-728. doi: 10.1637/8736-032509-ResNote.1. [7] K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985. [8] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproductionratio $R_0$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [9] R. A. Fouchier, V. Munster, A. Wallensten, T. M. Bestebroer and S. Herfst, et al., Characterization of a novel influenza A virus hemagglutinin subtype (H16) obtained from black-headed gulls, J. Virol., 79 (2005), 2814-2822. [10] V. Henaux, M. D. Samuel and C. M. Bunck, Model-based evaluation of highly and low pathogenic avian influenza dynamics in wild birds, PLoS One, 5 (2010), e10997. [11] P. Hess, "Periodic-parabolic Boundary Value Problem and Positivity," Pitman Res. Notes Math., 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [12] V. S. Hinshaw, R. G. Webster and B. Turner, The perpetuation of orthomyxoviruses and paramyxoviruses in Canadian waterfowl, Can.J. Microbiol., 26 (1980), 622-629. doi: 10.1139/m80-108. [13] V. S. Hinshaw, R. G. Webster and B. Turner, Water-bone transmission of influenza A viruses?, Intervirology, 11 (1979), 66-68. doi: 10.1159/000149014. [14] S.-B. Hsu, J. Jiang, and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Diff. Eqns., 248 (2010), 2470-2496. doi: 10.1016/j.jde.2009.12.014. [15] S.-B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, to appear in Journal of Dynamics and Differential Equations. [16] J. Jiang, X. Liang and X.-Q. Zhao, Saddle point behavior for monotone semiflows and reaction-diffusion models, J. Diff. Eqns., 203 (2004), 313-330. doi: 10.1016/j.jde.2004.05.002. [17] S. Krauss, D. Walker, S. P. Pryor, L. Niles and L. Chenghong, et al., Influenza A viruses of migrating wild aquatic birds in North America, Vector Borne Zoonotic Dis., 4 (2004), 177-189. [18] Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8. [19] H. Lu and A. E. Castro, Evaluation of the infectivity, length of infection, and immune response of a low-pathogenicity H7N2 avian influenza virus in specific-pathogen-free chickens, Avian. Dis., 48 (2004), 263-270. doi: 10.1637/7064. [20] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. of A. M. S., 321 (1990), 1-44. [21] P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. [22] B. Olsen, V. J. Munster, A. Wallensten, J. Waldenstrom and A. D. Osterhaus, et al., Global patterns of influenza a virus in wild birds, Science, 312, 384-388. [23] B. Roche, C. Lebarbenchon, M. Gauthier-Clerc, C. M. Chang and F. Thomas, et al., Water-borne transmission drives avian influenza dynamics in wild birds: The case of the 2005-2006 epidemics in the Camargue area, Infect. Genet. Evol., 9 (2009), 800-805. [24] P. Rohani, R. Breban, D. E. Stallknecht and J. M. Drake, Environmental transmission of low pathogenicity avian influenza viruses and its implications for pathogen invasion, Proc. Natl. Acad. Sci. USA, 106 (2009), 10365-10369. doi: 10.1073/pnas.0809026106. [25] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. [26] H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Math. Surveys Monogr., 41, American Mathematical Society, Providence, RI, 1995. [27] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2. [28] D. E. Stallknecht, S. M. Shane, M. T. Kearney and P. J. Zwank, Persistence of avian influenza viruses in water, Avian. Dis., 34 (1990), 406-411. doi: 10.2307/1591428. [29] J. H. Tien and D. J. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol., 72 (2010), 1506-1533. doi: 10.1007/s11538-010-9507-6. [30] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [31] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. [32] N. K. Vaidya and L. M. Wahl, The sensitivity of avian influenza dynamics in wild birds to time-varying environmental temperature, (2011), submitted. [33] P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [34] F.-B. Wang, A system of partial differential equations modeling the competition for two complementary resources in flowing habitats, J. Diff. Eqns., 249 (2010), 2866-2888. doi: 10.1016/j.jde.2010.07.031. [35] W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890. [36] R. G. Webster, W. J. Bean, O. T. Gorman, T. M. Chambers and Y. Kawaoka, Evolution and ecology of influenza A viruses, Microbiol. Rev., 56 (1992), 152-179. [37] R. G. Webster, M. Yakhno, V. S. Hinshaw, W. J. Bean and K. G. Murti, Intestinal influenza: Replication and characterization of influenza viruses in ducks, Virology, 84 (1978), 268-278. doi: 10.1016/0042-6822(78)90247-7. [38] K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusionmodel with a quiescent stage, Proc. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci., 463 (2007), 1029-1043. doi: 10.1098/rspa.2006.1806.
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