American Institute of Mathematical Sciences

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September  2021, 26(9): 4867-4885. doi: 10.3934/dcdsb.2020316

Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model

Weiwei Liu 1, , Jinliang Wang 1, and  Yuming Chen 2,,

1. 

School of Mathematical Science, Heilongjiang University, Harbin 150080, China
2. 

Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5 Canada

* Corresponding author: Yuming Chen

Received  December 2019 Revised  July 2020 Published  September 2021 Early access  November 2020

Fund Project: This research was partially supported by the Graduate Students Innovation Research Program of Heilongjiang University (No. YJSCX2020-211HLJU) (WL); the National Natural Science Foundation of China (Nos. 12071115, 11871179), Natural Science Foundation of Heilongjiang Province (Nos. LC2018002, LH209A021), Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems (JW); and NSERC of Canada (No. RGPIN-2019-05892) (YC)

Taking account of spatial heterogeneity, latency in infected individuals, and time for shed bacteria to the aquatic environment, we build a delayed nonlocal reaction-diffusion cholera model. A feature of this model is that the incidences are of general nonlinear forms. By using the theories of monotone dynamical systems and uniform persistence, we obtain a threshold dynamics determined by the basic reproduction number $ \mathcal {R}_0 $. Roughly speaking, the cholera will die out if $ \mathcal{R}_0<1 $ while it persists if $ \mathcal{R}_0>1 $. Moreover, we derive the explicit formulae of $ \mathcal{R}_0 $ for two concrete situations.

Keywords: Spatial heterogeneity, basic reproduction number, uniform persistence, threshold dynamics.
Mathematics Subject Classification: Primary: 92D30; Secondary: 35B35, 35B40, 35Q92, 37C75.
Citation: Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4867-4885. doi: 10.3934/dcdsb.2020316
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]

T. Berge, S. Bowong and J. M.-S. Lubuma, Global stability of a two-patch cholera model with fast and slow transmissions, Math. Comput. Simul., 133 (2017), 142-164.  doi: 10.1016/j.matcom.2015.10.013.

[3]

F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math Biosci. Eng., 10 (2013), 1335-1349.  doi: 10.3934/mbe.2013.10.1335.

[4]

F. Capone, V. De Cataldis and R. De Luca, Influence of diffusion on the stability of equilibria in a reaction-diffusion system modeling cholera dynamic, J. Math. Biol., 71 (2015), 1107-1131.  doi: 10.1007/s00285-014-0849-9.

[5]

M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den driessche, A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013), 105-112.  doi: 10.1016/j.mbs.2013.08.003.

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S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations, Vol. 48, American Matehmatical Society, Province, 2006, pp. 137–200. doi: 10.1007/s00285-006-0050-x.

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Z. Guo, F.-B. 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.

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D. M. Hartley, J. G. Morris Jr and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidmeics?, PLOS Med., 3 (2006), 63-69.  doi: 10.1371/journal.pmed.0030007.

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H. Li, R. Peng and F.-B. Wang, Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913. doi: 10.1016/j.jde.2016.09.044.

[11]

J. Lin, R. Xu and X. Tian, Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence, Appl. Math. Modelling, 63 (2018), 688-708.  doi: 10.1016/j.apm.2018.07.013.

[12]

J. Lin, R. Xu and X. Tian, Global dynamics of an age-structured cholera model with multiple transmissions, saturation incidence and imperfect vaccination, J. Biol. Dyn., 13 (2019), 69-102.  doi: 10.1080/17513758.2019.1570362.

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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.

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R. H. Martin Jr. and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

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J. A. J. Metz and O. Diekmann, Age dependence, The dynamics of physiologically structured populations (Amsterdam, 1983), Lecture Notes in Biomath., 68, Springer, Berlin, 1986,136–184. doi: 10.1007/978-3-662-13159-6_4.

[16]

J. B. H. Njagarah and F. Nyabadza, A metapopulation model for cholera transmission dynamics between communities linked by migration, Appl. Math. Comput., 241 (2014), 317-331.  doi: 10.1016/j.amc.2014.05.036.

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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.

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

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H. L. Smith and X. -Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. TMA, 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[20]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.

[21]

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.

[22]

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.

[23]

J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol., 72 (2010), 1506-1533.  doi: 10.1007/s11538-010-9507-6.

[24]

F.-B. Wang, J. Shi and X. Zou, Dynamics of a host-pathogen system on a bounded spatial domain, Commun. Pure Appl. Anal., 14 (2015), 2535-2560.  doi: 10.3934/cpaa.2015.14.2535.

[25]

X. Wang and J. Wang, Analysis of cholera epidemics with bacterial growth and spatial movement, J. Biol. Dyn., 9 (2015), 233-261.  doi: 10.1080/17513758.2014.974696.

[26]

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.

[27]

X. Wang, X.-Q. Zhao and J. Wang, A cholera epidemic model in a spatiotemporally heterogeneous environemnt, J. Math. Aanal. Appl., 468 (2018), 893-912.  doi: 10.1016/j.jmaa.2018.08.039.

[28]

World Health Organization, Cholera fact shettes, January 2019, available from http://www.who.int

[29]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Science, vol. 119, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[30]

Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.

[31]

Y. Wu and X. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differential Equations, 264 (2018), 4989-5024.  doi: 10.1016/j.jde.2017.12.027.

[32]

K. Yamazaki and X. Wang, Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model, Discrete Cont. Dynam. Syst., 21 (2016), 1297-1316.  doi: 10.3934/dcdsb.2016.21.1297.

[33]

K. Yamazaki and X. Wang, Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model, Math. Biosci. Eng., 14 (2017), 559-579.  doi: 10.3934/mbe.2017033.

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]

T. Berge, S. Bowong and J. M.-S. Lubuma, Global stability of a two-patch cholera model with fast and slow transmissions, Math. Comput. Simul., 133 (2017), 142-164.  doi: 10.1016/j.matcom.2015.10.013.

[3]

F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math Biosci. Eng., 10 (2013), 1335-1349.  doi: 10.3934/mbe.2013.10.1335.

[4]

F. Capone, V. De Cataldis and R. De Luca, Influence of diffusion on the stability of equilibria in a reaction-diffusion system modeling cholera dynamic, J. Math. Biol., 71 (2015), 1107-1131.  doi: 10.1007/s00285-014-0849-9.

[5]

M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den driessche, A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013), 105-112.  doi: 10.1016/j.mbs.2013.08.003.

[6]

S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations, Vol. 48, American Matehmatical Society, Province, 2006, pp. 137–200. doi: 10.1007/s00285-006-0050-x.

[7]

Z. Guo, F.-B. 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.

[8]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.

[9]

D. M. Hartley, J. G. Morris Jr and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidmeics?, PLOS Med., 3 (2006), 63-69.  doi: 10.1371/journal.pmed.0030007.

[10]

H. Li, R. Peng and F.-B. Wang, Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913. doi: 10.1016/j.jde.2016.09.044.

[11]

J. Lin, R. Xu and X. Tian, Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence, Appl. Math. Modelling, 63 (2018), 688-708.  doi: 10.1016/j.apm.2018.07.013.

[12]

J. Lin, R. Xu and X. Tian, Global dynamics of an age-structured cholera model with multiple transmissions, saturation incidence and imperfect vaccination, J. Biol. Dyn., 13 (2019), 69-102.  doi: 10.1080/17513758.2019.1570362.

[13]

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.

[14]

R. H. Martin Jr. and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[15]

J. A. J. Metz and O. Diekmann, Age dependence, The dynamics of physiologically structured populations (Amsterdam, 1983), Lecture Notes in Biomath., 68, Springer, Berlin, 1986,136–184. doi: 10.1007/978-3-662-13159-6_4.

[16]

J. B. H. Njagarah and F. Nyabadza, A metapopulation model for cholera transmission dynamics between communities linked by migration, Appl. Math. Comput., 241 (2014), 317-331.  doi: 10.1016/j.amc.2014.05.036.

[17]

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.

[18]

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

[19]

H. L. Smith and X. -Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. TMA, 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[20]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.

[21]

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.

[22]

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.

[23]

J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol., 72 (2010), 1506-1533.  doi: 10.1007/s11538-010-9507-6.

[24]

F.-B. Wang, J. Shi and X. Zou, Dynamics of a host-pathogen system on a bounded spatial domain, Commun. Pure Appl. Anal., 14 (2015), 2535-2560.  doi: 10.3934/cpaa.2015.14.2535.

[25]

X. Wang and J. Wang, Analysis of cholera epidemics with bacterial growth and spatial movement, J. Biol. Dyn., 9 (2015), 233-261.  doi: 10.1080/17513758.2014.974696.

[26]

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.

[27]

X. Wang, X.-Q. Zhao and J. Wang, A cholera epidemic model in a spatiotemporally heterogeneous environemnt, J. Math. Aanal. Appl., 468 (2018), 893-912.  doi: 10.1016/j.jmaa.2018.08.039.

[28]

World Health Organization, Cholera fact shettes, January 2019, available from http://www.who.int

[29]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Science, vol. 119, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[30]

Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.

[31]

Y. Wu and X. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differential Equations, 264 (2018), 4989-5024.  doi: 10.1016/j.jde.2017.12.027.

[32]

K. Yamazaki and X. Wang, Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model, Discrete Cont. Dynam. Syst., 21 (2016), 1297-1316.  doi: 10.3934/dcdsb.2016.21.1297.

[33]

K. Yamazaki and X. Wang, Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model, Math. Biosci. Eng., 14 (2017), 559-579.  doi: 10.3934/mbe.2017033.

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