October  2021, 26(10): 5519-5549. doi: 10.3934/dcdsb.2020357

The spatially heterogeneous diffusive rabies model and its shadow system

College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang 150001, China

Received  August 2020 Revised  October 2020 Published  October 2021 Early access  December 2020

Fund Project: The author was partially supported by the Fundamental Research Funds for the Central Universities (GK2240260048), and the National Natural Science Foundation of China (11971088)

In this paper, we consider a class of spatially heterogeneous reaction diffusion rabies model which was used to describe population dynamics of the rabies epidemic disease observed in Europe. The dynamics of both the original non-degenerate reaction-diffusion system and its corresponding shadow system are investigated in great details. Firstly, we prove that under certain conditions, the in-time solutions of both the original non-degenerate reaction-diffusion system and its shadow system exist globally and remain uniformly bounded. Secondly, we are capable of showing that the shadow system is the nice approximations for the original non-degenerate reaction-diffusion system when the diffusion rate $ d_R $ of the infectious rabid individuals (R) is sufficiently large. This implies that the dynamics of the shadow system can say as much as possible about the dynamics of the original system when $ d_R $ is sufficiently large. Finally, we characterize the basic reproduction number for the shadow system, and study the stability/instability of the disease-free steady state.

Citation: Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5519-5549. doi: 10.3934/dcdsb.2020357
References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Academic Press, Amsterdam, 2003. 
[2]

S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, 1965. doi: 10.1090/chel/369.

[3]

K. M. AlanaziZ. Jackiewicz and H. R. Thieme, Spreading speeds of rabies with territorial and diffusing rabid foxes, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2143-2183.  doi: 10.3934/dcdsb.2019222.

[4]

R. M. AndersonH. C JacksonR. M. May and A. M. Smith, Population dynamics of fox rabies in Europe, Nature, 289 (1981), 765-771.  doi: 10.1038/289765a0.

[5]

E. ConwayD. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16.  doi: 10.1137/0135001.

[6]

R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Masson, Paris, 1988.

[7]

W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups, in Differential Equations in Banach Spaces, Lecture Notes in Mathematics, 1223 (eds. A. Favini and E. Obrecht), Springer-Verlag, (1986), 61–73. doi: 10.1007/BFb0099183.

[8]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[9]

A. DucrotJ.-S. Guo and M. Shimojo, Behaviors of solutions for a singular prey-predator model and its shadow system, J. Dynam. Differential Equations, 30 (2018), 1063-1079.  doi: 10.1007/s10884-017-9587-1.

[10]

S.-I. EiK. Ikeda and E. Yanagida, Instability of multi-spot patterns in shadow systems of reaction-diffusion equations, Commun. Pure Appl. Anal., 14 (2015), 717-736.  doi: 10.3934/cpaa.2015.14.717.

[11]

A. Fooks, F. Cliquet and S. Finke et al., Rabies, Nature Rev. Dis. Primers, 3 (2017), 17091.

[12]

J. K. Hale and K. Sakamoto, Shadow system and attractors in reaction-diffusion equations, Appl. Anal., 32 (1989), 287-303.  doi: 10.1080/00036818908839855.

[13]

K. Hampson, L. Coudeville and T. Lembo et al., Estimating the global burden of endemic canine rabies, PLoS Negl. Trop. Dis., 9 (2015), e0003709. doi: 10.1371/journal.pntd.0003709.

[14]

H. IkedaM. Mimura and T. Scotti, Shadow system approach to a plankton model generating harmful algal bloom, Discrete Contin. Dyn. Syst., 37 (2017), 829-858.  doi: 10.3934/dcds.2017034.

[15]

J. JangW.-M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2004), 297-320.  doi: 10.1007/s10884-004-2782-x.

[16]

H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751.  doi: 10.3934/dcds.2006.14.737.

[17]

J. P. Keener, Activators and inhibitors in pattern formation, Stud. Appl. Math., 59 (1978), 1-23.  doi: 10.1002/sapm19785911.

[18]

H. Kokubu, K. Mischaikow, Y. Nishiura, H. Oka and T. Takaishi, Connecting orbit structure of monotone solutions in the shadow system, J. Differential Equations, 140 (1997), 309–364. doi: 10.1006/jdeq.1997.3317.

[19]

S. Kondo and M. Mimura, A reaction-diffusion system and its shadow system describing harmful algal blooms, Tamkang J. Math., 47 (2016), 71-92.  doi: 10.5556/j.tkjm.47.2016.1916.

[20]

F. LiJ. Coville and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., 37 (2017), 879-903.  doi: 10.3934/dcds.2017036.

[21]

F. Li and W.-M. Ni, On the global existence and finite time blow-up of shadow systems, J. Differential Equations, 247 (2009), 1762-1776.  doi: 10.1016/j.jde.2009.04.009.

[22]

Q. Li and Y. Wu, Stability analysis on a type of steady state for the SKT competition model with large cross diffusion, J. Math. Anal. Appl., 462 (2018), 1048-1072.  doi: 10.1016/j.jmaa.2018.01.023.

[23]

Y. LouW.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435.

[24]

Y. LouW.-M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Contin. Dyn. Syst., 35 (2015), 1589-1607.  doi: 10.3934/dcds.2015.35.1589.

[25]

K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation, Japan J. Appl. Math., 4 (1987), 47-58.  doi: 10.1007/BF03167754.

[26]

J. D. Murray and W. L. Seward, On the spatial spread of rabies among foxes with immunity, J. theor. Biol., 156 (1992), 327-348.  doi: 10.1016/S0022-5193(05)80679-4.

[27]

J. D. MurrayE. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes, Proc. R. Soc. Lond. B., 229 (1986), 111-150.  doi: 10.1098/rspb.1986.0078.

[28]

W.-M. NiI. Takagi and E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J. Indust. Appl. Math., 18 (2001), 259-272.  doi: 10.1007/BF03168574.

[29]

W.-M. NiY. Wu and Q. Xu, The existence and stability of nontrivial steady states for S-K-T competition model with cross-diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 5271-5298.  doi: 10.3934/dcds.2014.34.5271.

[30]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13 (1959), 115-162. 

[31]

Y. Nishiura, Coexistence of infinite many stable solutions to reaction-diffusion systems in the singular limit, in Dynamics Reported: Expositions in Dynamical Systems 3, (eds. C. K. R. T. Jones, U. Kirchgraber and H. O. Walther), Springer-Verlag, (1994), 25–103.

[32]

Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593.  doi: 10.1137/0513037.

[33]

C. Ou and J. Wu, Spatial spread of rabies revisited: Influence of age-dependent diffusion on nonlinear dynamics, SIAM J. Appl. Math., 67 (2006), 138-163.  doi: 10.1137/060651318.

[34]

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.

[35]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.

[36]

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.

[37]

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.

[38]

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.

[39]

M. J. Ward and J. Wei, Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt model, European J. Appl. Math., 14 (2003), 677-711.  doi: 10.1017/S0956792503005278.

[40]

J. Wei and M. Winter, On the Gierer-Meinhardt system with saturation, Commun. Contemp. Math., 6 (2004), 259-277.  doi: 10.1142/S021919970400132X.

[41]

M. WinterL. XuJ. Zhai and T. Zhang, The dynamics of the stochastic shadow Gierer-Meinhardt system, J. Differential Equations, 260 (2016), 84-114.  doi: 10.1016/j.jde.2015.08.047.

[42]

F. Yi, Y. Tang and N. Tuncer, A coupled PDE-ODEs SIR model describing population dynamics of fox rabies-Ⅰ: Steady States, Stability and Bifurcations, preprint.

show all references

References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Academic Press, Amsterdam, 2003. 
[2]

S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, 1965. doi: 10.1090/chel/369.

[3]

K. M. AlanaziZ. Jackiewicz and H. R. Thieme, Spreading speeds of rabies with territorial and diffusing rabid foxes, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2143-2183.  doi: 10.3934/dcdsb.2019222.

[4]

R. M. AndersonH. C JacksonR. M. May and A. M. Smith, Population dynamics of fox rabies in Europe, Nature, 289 (1981), 765-771.  doi: 10.1038/289765a0.

[5]

E. ConwayD. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16.  doi: 10.1137/0135001.

[6]

R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Masson, Paris, 1988.

[7]

W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups, in Differential Equations in Banach Spaces, Lecture Notes in Mathematics, 1223 (eds. A. Favini and E. Obrecht), Springer-Verlag, (1986), 61–73. doi: 10.1007/BFb0099183.

[8]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[9]

A. DucrotJ.-S. Guo and M. Shimojo, Behaviors of solutions for a singular prey-predator model and its shadow system, J. Dynam. Differential Equations, 30 (2018), 1063-1079.  doi: 10.1007/s10884-017-9587-1.

[10]

S.-I. EiK. Ikeda and E. Yanagida, Instability of multi-spot patterns in shadow systems of reaction-diffusion equations, Commun. Pure Appl. Anal., 14 (2015), 717-736.  doi: 10.3934/cpaa.2015.14.717.

[11]

A. Fooks, F. Cliquet and S. Finke et al., Rabies, Nature Rev. Dis. Primers, 3 (2017), 17091.

[12]

J. K. Hale and K. Sakamoto, Shadow system and attractors in reaction-diffusion equations, Appl. Anal., 32 (1989), 287-303.  doi: 10.1080/00036818908839855.

[13]

K. Hampson, L. Coudeville and T. Lembo et al., Estimating the global burden of endemic canine rabies, PLoS Negl. Trop. Dis., 9 (2015), e0003709. doi: 10.1371/journal.pntd.0003709.

[14]

H. IkedaM. Mimura and T. Scotti, Shadow system approach to a plankton model generating harmful algal bloom, Discrete Contin. Dyn. Syst., 37 (2017), 829-858.  doi: 10.3934/dcds.2017034.

[15]

J. JangW.-M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations, 16 (2004), 297-320.  doi: 10.1007/s10884-004-2782-x.

[16]

H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751.  doi: 10.3934/dcds.2006.14.737.

[17]

J. P. Keener, Activators and inhibitors in pattern formation, Stud. Appl. Math., 59 (1978), 1-23.  doi: 10.1002/sapm19785911.

[18]

H. Kokubu, K. Mischaikow, Y. Nishiura, H. Oka and T. Takaishi, Connecting orbit structure of monotone solutions in the shadow system, J. Differential Equations, 140 (1997), 309–364. doi: 10.1006/jdeq.1997.3317.

[19]

S. Kondo and M. Mimura, A reaction-diffusion system and its shadow system describing harmful algal blooms, Tamkang J. Math., 47 (2016), 71-92.  doi: 10.5556/j.tkjm.47.2016.1916.

[20]

F. LiJ. Coville and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., 37 (2017), 879-903.  doi: 10.3934/dcds.2017036.

[21]

F. Li and W.-M. Ni, On the global existence and finite time blow-up of shadow systems, J. Differential Equations, 247 (2009), 1762-1776.  doi: 10.1016/j.jde.2009.04.009.

[22]

Q. Li and Y. Wu, Stability analysis on a type of steady state for the SKT competition model with large cross diffusion, J. Math. Anal. Appl., 462 (2018), 1048-1072.  doi: 10.1016/j.jmaa.2018.01.023.

[23]

Y. LouW.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435.

[24]

Y. LouW.-M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Contin. Dyn. Syst., 35 (2015), 1589-1607.  doi: 10.3934/dcds.2015.35.1589.

[25]

K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation, Japan J. Appl. Math., 4 (1987), 47-58.  doi: 10.1007/BF03167754.

[26]

J. D. Murray and W. L. Seward, On the spatial spread of rabies among foxes with immunity, J. theor. Biol., 156 (1992), 327-348.  doi: 10.1016/S0022-5193(05)80679-4.

[27]

J. D. MurrayE. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes, Proc. R. Soc. Lond. B., 229 (1986), 111-150.  doi: 10.1098/rspb.1986.0078.

[28]

W.-M. NiI. Takagi and E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J. Indust. Appl. Math., 18 (2001), 259-272.  doi: 10.1007/BF03168574.

[29]

W.-M. NiY. Wu and Q. Xu, The existence and stability of nontrivial steady states for S-K-T competition model with cross-diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 5271-5298.  doi: 10.3934/dcds.2014.34.5271.

[30]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13 (1959), 115-162. 

[31]

Y. Nishiura, Coexistence of infinite many stable solutions to reaction-diffusion systems in the singular limit, in Dynamics Reported: Expositions in Dynamical Systems 3, (eds. C. K. R. T. Jones, U. Kirchgraber and H. O. Walther), Springer-Verlag, (1994), 25–103.

[32]

Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593.  doi: 10.1137/0513037.

[33]

C. Ou and J. Wu, Spatial spread of rabies revisited: Influence of age-dependent diffusion on nonlinear dynamics, SIAM J. Appl. Math., 67 (2006), 138-163.  doi: 10.1137/060651318.

[34]

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.

[35]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.

[36]

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.

[37]

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.

[38]

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.

[39]

M. J. Ward and J. Wei, Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt model, European J. Appl. Math., 14 (2003), 677-711.  doi: 10.1017/S0956792503005278.

[40]

J. Wei and M. Winter, On the Gierer-Meinhardt system with saturation, Commun. Contemp. Math., 6 (2004), 259-277.  doi: 10.1142/S021919970400132X.

[41]

M. WinterL. XuJ. Zhai and T. Zhang, The dynamics of the stochastic shadow Gierer-Meinhardt system, J. Differential Equations, 260 (2016), 84-114.  doi: 10.1016/j.jde.2015.08.047.

[42]

F. Yi, Y. Tang and N. Tuncer, A coupled PDE-ODEs SIR model describing population dynamics of fox rabies-Ⅰ: Steady States, Stability and Bifurcations, preprint.

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