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December  2017, 22(10): 3797-3820. doi: 10.3934/dcdsb.2017191

Threshold dynamics of a reaction-diffusion epidemic model with stage structure

Liang Zhang and  Zhi-Cheng Wang ,

School of Mathematics and Statistics, Lanzhou University, and Key Laboratory of Applied Mathematics and Complex Systems of Gansu province, Lanzhou, Gansu 730000, China

* Corresponding author

Received  December 2016 Revised  May 2017 Published  July 2017

A time-delayed reaction-diffusion epidemic model with stage structure and spatial heterogeneity is investigated, which describes the dynamics of disease spread only proceeding in the adult population. We establish the basic reproduction number $\mathcal{R}_0$ for the model system, which gives the threshold dynamics in the sense that the disease will die out if $\mathcal{R}_0＜1$ and the disease will be uniformly persistent if $\mathcal{R}_0>1.$ Furthermore, it is shown that there is at least one positive steady state when $\mathcal{R}_0>1.$ Finally, in terms of general birth function for adult individuals, through introducing two numbers $\check{\mathcal{R}}_0$ and $\hat{\mathcal{R}}_0$, we establish sufficient conditions for the persistence and global extinction of the disease, respectively.

Keywords: Stage structure, spatial heterogeneity, basic reproduction number, persistence and extinction.
Mathematics Subject Classification: 35K57, 35B35, 35B40, 92D25, 92D30.
Citation: Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191
References:
[1]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

[2]

B. M. Bolker and B. T. Grenfen, Space, persistence, and dynamics of measles epidemics, Phil. Trans. R. Soc. Lond., B237 (1995), 309-320.   Google Scholar

[3]

L. Burlando, Monotonicity of spectral radius for positive operators on ordered Banach spaces, Arch. Math., 56 (1991), 49-57.  doi: 10.1007/BF01190081.  Google Scholar

[4]

R. S. Cantrell and C. Cosner Spatial Ecology via Reaction-Diffusion Equations, UK: John Wiley and Sons Ltd. , 2003. doi: 10.1002/0470871296.  Google Scholar

[5]

V. Capasso, Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35 (1978), 274-284.  doi: 10.1137/0135022.  Google Scholar

[6]

D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Mathematics Series, 279. Longman, Harlow, UK, 1992.  Google Scholar

[7]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the deinition and the computation of the basic production ratio $\mathcal{R}_0$ in the models for infectious disease in heterogeneous population, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[8]

W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.  Google Scholar

[9]

W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model for indirectly transmitted diseases, Math. Biosci., 206 (2007), 233-248.  doi: 10.1016/j.mbs.2005.07.005.  Google Scholar

[10]

I. Gudelj and K. A. J. White, The effects of spatial movement and group interactions on disease dynamics of social animals, Bull. Math. Biol., 66 (2004), 91-108.  doi: 10.1016/S0092-8240(03)00075-2.  Google Scholar

[11]

Z. Guo, F. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410.  doi: 10.1007/s00285-011-0500-y.  Google Scholar

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs 25), American Mathematical Society, Providence, RI, 1988.  Google Scholar

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[14]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[15]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.  doi: 10.3934/mbe.2004.1.57.  Google Scholar

[16]

D. Le, Dissipativity and global attractors for a class of quasilinear parabolic systems, Commun. Partial. Diff. Eqns., 22 (1997), 413-433.  doi: 10.1080/03605309708821269.  Google Scholar

[17]

J. Li and X. Zou, Generalization of the Kermack-McKendrick SIR model to a patchy environment for a disease with latency, Math. Model. Nat. Phenom., 4 (2009), 92-118.  doi: 10.1051/mmnp/20094205.  Google Scholar

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J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079.  doi: 10.1007/s11538-009-9457-z.  Google Scholar

[19]

J. Li and X. Zou, Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment, J. Math. Biol., 60 (2010), 645-686.  doi: 10.1007/s00285-009-0280-9.  Google Scholar

[20]

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.  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), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[22]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.1090/S0002-9947-1990-0967316-X.  Google Scholar

[23]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.  Google Scholar

[24]

R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.  Google Scholar

[25]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[26]

S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, Mathematics for Life Science and Medicine (eds. Y. Iwasa, K. Sato and Y. Takeuchi), Biol. Med. Phys, (2007), 97-122.   Google Scholar

[27]

S. Ruan and J. Wu, Modeling Spatial Spread of Communicable Diseases Involving Animal Hosts, Chapman & Hall/CRC, Boca Raton, (2009), 293-316.   Google Scholar

[28]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems, Math. Surveys and Monographs, 41, Amer. Math. Soc. , Providence, RI, 1995.  Google Scholar

[29]

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.  Google Scholar

[30]

H. R. Thieme, Book review. Linda Rass, John Radcliffe, spatial deterministic epidemics, AMS, 2003, ISBN:0821804995, Math. Biosci., 202 (2006), 218-225.  doi: 10.1016/j.mbs.2006.03.015.  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), 188-211.  doi: 10.1137/080732870.  Google Scholar

[32]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.  Google Scholar

[33]

P. van den Driessche and J. 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.  Google Scholar

[34]

W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112.  doi: 10.1016/j.mbs.2002.11.001.  Google Scholar

[35]

W. Wang and X.-Q. Zhao, An age-structured epidemic model in a patchy environment, SIAM J. Appl. Math., 65 (2005), 1597-1614.  doi: 10.1137/S0036139903431245.  Google Scholar

[36]

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.  Google Scholar

[37]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.  Google Scholar

[38]

Z.-C. Wang and W.-T. Li, Dynamics of a non-local delayed reaction-diffusion equation without quasi-monotonicity, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1081-1109.  doi: 10.1017/S0308210509000262.  Google Scholar

[39]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied mathematical sciences, 119. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[40]

J. Wu, Spatial structure: Partial differential equations models, Math. Biosci. Subser, (2008).  doi: 10.1007/978-3-540-78911-6_8.  Google Scholar

[41]

J. Wu and X.-Q. Zhao, Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations, J. Differential Equations, 186 (2002), 470-484.  doi: 10.1016/S0022-0396(02)00012-8.  Google Scholar

[42]

L. Zhang and Z.-C. Wang, Spatial dynamics of a diffusive predator-prey model with stage structure, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1831-1853.  doi: 10.3934/dcdsb.2015.20.1831.  Google Scholar

[43]

X. -Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[44]

X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Q., 17 (2009), 271-281.   Google Scholar

show all references

References:
[1]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

[2]

B. M. Bolker and B. T. Grenfen, Space, persistence, and dynamics of measles epidemics, Phil. Trans. R. Soc. Lond., B237 (1995), 309-320.   Google Scholar

[3]

L. Burlando, Monotonicity of spectral radius for positive operators on ordered Banach spaces, Arch. Math., 56 (1991), 49-57.  doi: 10.1007/BF01190081.  Google Scholar

[4]

R. S. Cantrell and C. Cosner Spatial Ecology via Reaction-Diffusion Equations, UK: John Wiley and Sons Ltd. , 2003. doi: 10.1002/0470871296.  Google Scholar

[5]

V. Capasso, Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35 (1978), 274-284.  doi: 10.1137/0135022.  Google Scholar

[6]

D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Mathematics Series, 279. Longman, Harlow, UK, 1992.  Google Scholar

[7]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the deinition and the computation of the basic production ratio $\mathcal{R}_0$ in the models for infectious disease in heterogeneous population, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[8]

W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.  Google Scholar

[9]

W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model for indirectly transmitted diseases, Math. Biosci., 206 (2007), 233-248.  doi: 10.1016/j.mbs.2005.07.005.  Google Scholar

[10]

I. Gudelj and K. A. J. White, The effects of spatial movement and group interactions on disease dynamics of social animals, Bull. Math. Biol., 66 (2004), 91-108.  doi: 10.1016/S0092-8240(03)00075-2.  Google Scholar

[11]

Z. Guo, F. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410.  doi: 10.1007/s00285-011-0500-y.  Google Scholar

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs 25), American Mathematical Society, Providence, RI, 1988.  Google Scholar

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[14]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[15]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.  doi: 10.3934/mbe.2004.1.57.  Google Scholar

[16]

D. Le, Dissipativity and global attractors for a class of quasilinear parabolic systems, Commun. Partial. Diff. Eqns., 22 (1997), 413-433.  doi: 10.1080/03605309708821269.  Google Scholar

[17]

J. Li and X. Zou, Generalization of the Kermack-McKendrick SIR model to a patchy environment for a disease with latency, Math. Model. Nat. Phenom., 4 (2009), 92-118.  doi: 10.1051/mmnp/20094205.  Google Scholar

[18]

J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079.  doi: 10.1007/s11538-009-9457-z.  Google Scholar

[19]

J. Li and X. Zou, Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment, J. Math. Biol., 60 (2010), 645-686.  doi: 10.1007/s00285-009-0280-9.  Google Scholar

[20]

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.  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), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[22]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.1090/S0002-9947-1990-0967316-X.  Google Scholar

[23]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.  Google Scholar

[24]

R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.  Google Scholar

[25]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[26]

S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, Mathematics for Life Science and Medicine (eds. Y. Iwasa, K. Sato and Y. Takeuchi), Biol. Med. Phys, (2007), 97-122.   Google Scholar

[27]

S. Ruan and J. Wu, Modeling Spatial Spread of Communicable Diseases Involving Animal Hosts, Chapman & Hall/CRC, Boca Raton, (2009), 293-316.   Google Scholar

[28]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems, Math. Surveys and Monographs, 41, Amer. Math. Soc. , Providence, RI, 1995.  Google Scholar

[29]

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.  Google Scholar

[30]

H. R. Thieme, Book review. Linda Rass, John Radcliffe, spatial deterministic epidemics, AMS, 2003, ISBN:0821804995, Math. Biosci., 202 (2006), 218-225.  doi: 10.1016/j.mbs.2006.03.015.  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), 188-211.  doi: 10.1137/080732870.  Google Scholar

[32]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.  Google Scholar

[33]

P. van den Driessche and J. 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.  Google Scholar

[34]

W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112.  doi: 10.1016/j.mbs.2002.11.001.  Google Scholar

[35]

W. Wang and X.-Q. Zhao, An age-structured epidemic model in a patchy environment, SIAM J. Appl. Math., 65 (2005), 1597-1614.  doi: 10.1137/S0036139903431245.  Google Scholar

[36]

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.  Google Scholar

[37]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.  Google Scholar

[38]

Z.-C. Wang and W.-T. Li, Dynamics of a non-local delayed reaction-diffusion equation without quasi-monotonicity, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1081-1109.  doi: 10.1017/S0308210509000262.  Google Scholar

[39]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied mathematical sciences, 119. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[40]

J. Wu, Spatial structure: Partial differential equations models, Math. Biosci. Subser, (2008).  doi: 10.1007/978-3-540-78911-6_8.  Google Scholar

[41]

J. Wu and X.-Q. Zhao, Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations, J. Differential Equations, 186 (2002), 470-484.  doi: 10.1016/S0022-0396(02)00012-8.  Google Scholar

[42]

L. Zhang and Z.-C. Wang, Spatial dynamics of a diffusive predator-prey model with stage structure, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1831-1853.  doi: 10.3934/dcdsb.2015.20.1831.  Google Scholar

[43]

X. -Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[44]

X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Q., 17 (2009), 271-281.   Google Scholar

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