October  2012, 17(7): 2615-2634. doi: 10.3934/dcdsb.2012.17.2615

A vector-bias malaria model with incubation period and diffusion

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China

2. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7

Received  June 2011 Revised  March 2012 Published  July 2012

This paper is devoted to the study of the global dynamics of a vector-bias malaria model with incubation period and diffusion. The global attractivity of the disease-free or endemic equilibrium is first proved for the spatially homogeneous system. Then the threshold dynamics is established for the spatially heterogeneous system in terms of the basic reproduction ratio. A set of sufficient conditions is further obtained for the global attractivity of the positive steady state.
Citation: Zhiting Xu, Xiao-Qiang Zhao. A vector-bias malaria model with incubation period and diffusion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2615-2634. doi: 10.3934/dcdsb.2012.17.2615
References:
[1]

F. Chamchod and N. F. Britton, Analysis of a vector-bias model on malaria transmisson,, Bull. Math. Biol., 73 (2011), 639.  doi: 10.1007/s11538-010-9545-0.  Google Scholar

[2]

C. A. Guerra, R. W. Snow and S. I. Hay, Mapping the global extent of malaria in 2005,, Trends Parasitol, 22 (2006), 353.  doi: 10.1016/j.pt.2006.06.006.  Google Scholar

[3]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Appl. Math. Sci., (1993).   Google Scholar

[4]

J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria,, Am. Nat., 130 (1987), 811.  doi: 10.1086/284749.  Google Scholar

[5]

R. Lacroix, W. R. Mukabana, L. C. Gouagna and J. C. Koella, Malaria infection increases attractiveness of humans to mosquitoes,, PLOS Biol., 3 (2005), 1590.   Google Scholar

[6]

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

[7]

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

[8]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1.  doi: 10.1090/S0002-9947-1990-0967316-X.  Google Scholar

[9]

K. Mischaikow, H. L. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions,, Trans. Amer. Math. Soc., 347 (1995), 1669.  doi: 10.1090/S0002-9947-1995-1290727-7.  Google Scholar

[10]

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

[11]

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

[12]

R. W. Snow, C. A. Guerra, A. M. Noor, H. Y. Myint and S. I. Hay, The global distribution of clinical episodes of Plasmodium falciparum malaria,, Nature, (2005), 214.  doi: 10.1038/nature03342.  Google Scholar

[13]

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

[14]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188.  doi: 10.1137/080732870.  Google Scholar

[15]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model,, Nonlinear Analysis: Real World Application, 2 (2001), 145.   Google Scholar

[16]

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

[17]

J. Wu, "Theory and Applications of Partial Functional Differential Equations,", Applied Mathematical Sciences, 119 (1996).   Google Scholar

[18]

X.-Q. Zhao, "Dynamical Systems in Population Biology,", CMS Books in Mathematics/Ouvrages de Math®¶matiques de la SMC, 16 (2003).   Google Scholar

[19]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations,, Canadian Appl. Math. Quarterly, 4 (1996), 421.   Google Scholar

show all references

References:
[1]

F. Chamchod and N. F. Britton, Analysis of a vector-bias model on malaria transmisson,, Bull. Math. Biol., 73 (2011), 639.  doi: 10.1007/s11538-010-9545-0.  Google Scholar

[2]

C. A. Guerra, R. W. Snow and S. I. Hay, Mapping the global extent of malaria in 2005,, Trends Parasitol, 22 (2006), 353.  doi: 10.1016/j.pt.2006.06.006.  Google Scholar

[3]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Appl. Math. Sci., (1993).   Google Scholar

[4]

J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria,, Am. Nat., 130 (1987), 811.  doi: 10.1086/284749.  Google Scholar

[5]

R. Lacroix, W. R. Mukabana, L. C. Gouagna and J. C. Koella, Malaria infection increases attractiveness of humans to mosquitoes,, PLOS Biol., 3 (2005), 1590.   Google Scholar

[6]

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

[7]

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

[8]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1.  doi: 10.1090/S0002-9947-1990-0967316-X.  Google Scholar

[9]

K. Mischaikow, H. L. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions,, Trans. Amer. Math. Soc., 347 (1995), 1669.  doi: 10.1090/S0002-9947-1995-1290727-7.  Google Scholar

[10]

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

[11]

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

[12]

R. W. Snow, C. A. Guerra, A. M. Noor, H. Y. Myint and S. I. Hay, The global distribution of clinical episodes of Plasmodium falciparum malaria,, Nature, (2005), 214.  doi: 10.1038/nature03342.  Google Scholar

[13]

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

[14]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188.  doi: 10.1137/080732870.  Google Scholar

[15]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model,, Nonlinear Analysis: Real World Application, 2 (2001), 145.   Google Scholar

[16]

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

[17]

J. Wu, "Theory and Applications of Partial Functional Differential Equations,", Applied Mathematical Sciences, 119 (1996).   Google Scholar

[18]

X.-Q. Zhao, "Dynamical Systems in Population Biology,", CMS Books in Mathematics/Ouvrages de Math®¶matiques de la SMC, 16 (2003).   Google Scholar

[19]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations,, Canadian Appl. Math. Quarterly, 4 (1996), 421.   Google Scholar

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