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Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions
Threshold dynamics of a reaction-diffusion epidemic model with stage structure
School of Mathematics and Statistics, Lanzhou University, and Key Laboratory of Applied Mathematics and Complex Systems of Gansu province, Lanzhou, Gansu 730000, China |
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.
References:
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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. |
[2] |
B. M. Bolker and B. T. Grenfen,
Space, persistence, and dynamics of measles epidemics, Phil. Trans. R. Soc. Lond., B237 (1995), 309-320.
|
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L. Burlando,
Monotonicity of spectral radius for positive operators on ordered Banach spaces, Arch. Math., 56 (1991), 49-57.
doi: 10.1007/BF01190081. |
[4] |
R. S. Cantrell and C. Cosner Spatial Ecology via Reaction-Diffusion Equations, UK: John Wiley and Sons Ltd. , 2003.
doi: 10.1002/0470871296. |
[5] |
V. Capasso,
Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35 (1978), 274-284.
doi: 10.1137/0135022. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs 25), American Mathematical Society, Providence, RI, 1988. |
[13] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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. |
[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. |
[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.
|
[27] |
S. Ruan and J. Wu,
Modeling Spatial Spread of Communicable Diseases Involving Animal Hosts, Chapman & Hall/CRC, Boca Raton, (2009), 293-316.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[40] |
J. Wu,
Spatial structure: Partial differential equations models, Math. Biosci. Subser, (2008).
doi: 10.1007/978-3-540-78911-6_8. |
[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. |
[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. |
[43] |
X. -Q. Zhao,
Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[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.
|
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. |
[2] |
B. M. Bolker and B. T. Grenfen,
Space, persistence, and dynamics of measles epidemics, Phil. Trans. R. Soc. Lond., B237 (1995), 309-320.
|
[3] |
L. Burlando,
Monotonicity of spectral radius for positive operators on ordered Banach spaces, Arch. Math., 56 (1991), 49-57.
doi: 10.1007/BF01190081. |
[4] |
R. S. Cantrell and C. Cosner Spatial Ecology via Reaction-Diffusion Equations, UK: John Wiley and Sons Ltd. , 2003.
doi: 10.1002/0470871296. |
[5] |
V. Capasso,
Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35 (1978), 274-284.
doi: 10.1137/0135022. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs 25), American Mathematical Society, Providence, RI, 1988. |
[13] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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. |
[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. |
[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.
|
[27] |
S. Ruan and J. Wu,
Modeling Spatial Spread of Communicable Diseases Involving Animal Hosts, Chapman & Hall/CRC, Boca Raton, (2009), 293-316.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[40] |
J. Wu,
Spatial structure: Partial differential equations models, Math. Biosci. Subser, (2008).
doi: 10.1007/978-3-540-78911-6_8. |
[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. |
[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. |
[43] |
X. -Q. Zhao,
Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[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.
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