American Institute of Mathematical Sciences

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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

Naveen K. Vaidya 1, , Feng-Bin Wang 1, and  Xingfu Zou 2,

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.
Keywords: basic reproductive number, threshold dynamics., diffusion, principal eigenvalue, heterogeneous environment, Avian influenza, spectral radius.
Mathematics Subject Classification: 35K57, 37N25, 92D3.
Citation: Naveen K. Vaidya, Feng-Bin Wang, Xingfu Zou. Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2829-2848. doi: 10.3934/dcdsb.2012.17.2829
References:
[1]

Best Country Report, Temperature map of Canada., Available from: \url{http://www.bestcountryreports.com/Temperature_Map_Canada.html} [Accessed date: 24 February, (2011). Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1637/8736-032509-ResNote.1. Google Scholar

[7]

K. Deimling, "Nonlinear Functional Analysis,", Springer-Verlag, (1985). Google Scholar

[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. doi: 10.1007/BF00178324. Google Scholar

[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. Google Scholar

[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). Google Scholar

[11]

P. Hess, "Periodic-parabolic Boundary Value Problem and Positivity,", Pitman Res. Notes Math., 247 (1991). Google Scholar

[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. doi: 10.1139/m80-108. Google Scholar

[13]

V. S. Hinshaw, R. G. Webster and B. Turner, Water-bone transmission of influenza A viruses?,, Intervirology, 11 (1979), 66. doi: 10.1159/000149014. Google Scholar

[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. doi: 10.1016/j.jde.2009.12.014. Google Scholar

[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., (). Google Scholar

[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. doi: 10.1016/j.jde.2004.05.002. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00285-010-0346-8. Google Scholar

[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. doi: 10.1637/7064. Google Scholar

[20]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. of A. M. S., 321 (1990), 1. Google Scholar

[21]

P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM. J. Math. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1073/pnas.0809026106. Google Scholar

[25]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Corrected reprint of the 1967 original, (1967). Google Scholar

[26]

H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", Math. Surveys Monogr., 41 (1995). Google Scholar

[27]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Anal., 47 (2001), 6169. doi: 10.1016/S0362-546X(01)00678-2. Google Scholar

[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. doi: 10.2307/1591428. Google Scholar

[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. doi: 10.1007/s11538-010-9507-6. Google Scholar

[30]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755. doi: 10.1007/BF00173267. Google Scholar

[31]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188. Google Scholar

[32]

N. K. Vaidya and L. M. Wahl, The sensitivity of avian influenza dynamics in wild birds to time-varying environmental temperature,, (2011), (2011). Google Scholar

[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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[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. doi: 10.1016/j.jde.2010.07.031. Google Scholar

[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. doi: 10.1137/090775890. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/0042-6822(78)90247-7. Google Scholar

[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. doi: 10.1098/rspa.2006.1806. Google Scholar

show all references

References:
[1]

Best Country Report, Temperature map of Canada., Available from: \url{http://www.bestcountryreports.com/Temperature_Map_Canada.html} [Accessed date: 24 February, (2011). Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1637/8736-032509-ResNote.1. Google Scholar

[7]

K. Deimling, "Nonlinear Functional Analysis,", Springer-Verlag, (1985). Google Scholar

[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. doi: 10.1007/BF00178324. Google Scholar

[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. Google Scholar

[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). Google Scholar

[11]

P. Hess, "Periodic-parabolic Boundary Value Problem and Positivity,", Pitman Res. Notes Math., 247 (1991). Google Scholar

[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. doi: 10.1139/m80-108. Google Scholar

[13]

V. S. Hinshaw, R. G. Webster and B. Turner, Water-bone transmission of influenza A viruses?,, Intervirology, 11 (1979), 66. doi: 10.1159/000149014. Google Scholar

[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. doi: 10.1016/j.jde.2009.12.014. Google Scholar

[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., (). Google Scholar

[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. doi: 10.1016/j.jde.2004.05.002. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00285-010-0346-8. Google Scholar

[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. doi: 10.1637/7064. Google Scholar

[20]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. of A. M. S., 321 (1990), 1. Google Scholar

[21]

P. Magal and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM. J. Math. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1073/pnas.0809026106. Google Scholar

[25]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Corrected reprint of the 1967 original, (1967). Google Scholar

[26]

H. L. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,", Math. Surveys Monogr., 41 (1995). Google Scholar

[27]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Anal., 47 (2001), 6169. doi: 10.1016/S0362-546X(01)00678-2. Google Scholar

[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. doi: 10.2307/1591428. Google Scholar

[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. doi: 10.1007/s11538-010-9507-6. Google Scholar

[30]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755. doi: 10.1007/BF00173267. Google Scholar

[31]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188. Google Scholar

[32]

N. K. Vaidya and L. M. Wahl, The sensitivity of avian influenza dynamics in wild birds to time-varying environmental temperature,, (2011), (2011). Google Scholar

[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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[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. doi: 10.1016/j.jde.2010.07.031. Google Scholar

[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. doi: 10.1137/090775890. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/0042-6822(78)90247-7. Google Scholar

[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. doi: 10.1098/rspa.2006.1806. Google Scholar

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