October  2022, 21(10): 3263-3282. doi: 10.3934/cpaa.2022099

Threshold dynamics of a reaction-diffusion cholera model with seasonality and nonlocal delay

1. 

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

2. 

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

*Corresponding author

Received  November 2021 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: W. Wu was supported by Graduate Students Innovation Research Program of Heilongjiang University, P. R. China (no. YJSCX2021-213HLJU). J. Wang was supported by National Natural Science Foundation of China (nos. 12071115, 11871179), Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, and Fundamental Research Funds for the Heilongjiang Education Department (no. 2021-KYYWF-0034)

In this paper, we investigate the threshold results for a nonlocal and time-delayed reaction-diffusion system involving the spatial heterogeneity and the seasonality. Due to the complexity of the model, we rigorously analyze the well-posedness of the model. The basic reproduction number $ \Re_0 $ is characterized with the next generation operator method. We show that the disease-free $ \omega $-periodic solution is globally attractive when $ \Re_0 < 1 $; while the system is uniformly persistent and a positive $ \omega $-periodic solution exists when $ \Re_0 > 1 $. In a special case that the parameters are all independent of the spatial heterogeneity and the seasonality, the global attractivity of the constant equilibria of the model is investigated by the technique of Lyapunov functionals.

Citation: Wenjing Wu, Tianli Jiang, Weiwei Liu, Jinliang Wang. Threshold dynamics of a reaction-diffusion cholera model with seasonality and nonlocal delay. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3263-3282. doi: 10.3934/cpaa.2022099
References:
[1]

L.J.S. AllenB.M. BolkerY. 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]

A. RoobthaisongK. Okada and N. Htun, Molecular epidemiology of Cholera outbreaks during the rainy season in Mandalay, Myanmar, Am. J. Trop. Med. Hyg., 97 (2017), 1323-1328. 

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N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.

[4]

Z. BaiR. Peng and X.-Q. Zhao, A reaction-diffusion malaria model with seasonality and incubation period, J. Math. Biol., 77 (2018), 201-228.  doi: 10.1007/s00285-017-1193-7.

[5]

F. BrauerS. 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.

[6]

F. CaponeV. De CataldisR. De Luca and P. van den Driessche, 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.

[7]

D. Danners and P. K. Medina, Abstract Evolution Equations, Peeriodic Problems and Applications, Pitman Research Notes in Mathematics Series, vol. 279. Chapman and Hall/CRC, London, 1992.

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M.C. EisenbergZ. ShuaiD.L. 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.

[9]

D.M. HartleyJ.G. Morris and D.L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidmeics?, PLoS Med., 3 (2006), 63-69. 

[10]

S. Itô, Diffusion Equations, Translations of Mathematical Monographs, vol. 114. American Mathematical Society, Providence, 1992. doi: 10.1090/mmono/114.

[11]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spreal and traveling waves for monotone semiflows with applications, Commun. Pure. Appl. Math., 60 (2008), 1-40.  doi: 10.1002/cpa.20154.

[12]

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.

[13]

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

[14]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68. Springer, New York, 1986. doi: 10.1007/978-3-642-93287-8_2.

[15]

Z. MukandavireS. Liao and J. Wang, Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe, Proc. Natl. Acad. Sci. USA., 108 (2011), 8767-8772. 

[16]

R. L. M. NeilanE. Schaefer and H. Gaff, Modeling optimal intervention strategies for cholera, Bull. Math. Biol., 72 (2010), 2004-2018.  doi: 10.1007/s11538-010-9521-8.

[17]

D. Posny and J. Wang, Modelling cholera in periodic environments, J. Biol. Dyn., 8 (2014), 1-19.  doi: 10.1080/17513758.2014.896482.

[18]

D. PosnyJ. WangZ. Mukandavire and C. Modnak, Analyzing transmission dynamics of cholera with public health interventions, Math. Biosci., 264 (2015), 38-53.  doi: 10.1016/j.mbs.2015.03.006.

[19]

Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2011), 118-126.  doi: 10.1016/j.mbs.2011.09.003.

[20]

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.

[21]

H. R. ThiemeC. Castillo-Chavez and T. Kuniya, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM. J. Appl. Math., 53 (1993), 1447-1479.  doi: 10.1137/0153068.

[22]

J. WangR. Zhang and T. Kuniya, A note on dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 13 (2016), 227-247.  doi: 10.3934/mbe.2016.13.227.

[23]

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.

[24]

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.

[25]

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

[26]

World Health Organization, Cholera, 2022. Available from: https://www.who.int/news-room/fact-sheets/detail/cholera.

[27]

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

[28]

L. ZhangZ. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reactin-diffusion epidemic model with latent period, J. Differ. Equ., 258 (2015), 3011-3036.  doi: 10.1016/j.jde.2014.12.032.

[29]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Differ. Equ., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.

show all references

References:
[1]

L.J.S. AllenB.M. BolkerY. 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]

A. RoobthaisongK. Okada and N. Htun, Molecular epidemiology of Cholera outbreaks during the rainy season in Mandalay, Myanmar, Am. J. Trop. Med. Hyg., 97 (2017), 1323-1328. 

[3]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.

[4]

Z. BaiR. Peng and X.-Q. Zhao, A reaction-diffusion malaria model with seasonality and incubation period, J. Math. Biol., 77 (2018), 201-228.  doi: 10.1007/s00285-017-1193-7.

[5]

F. BrauerS. 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.

[6]

F. CaponeV. De CataldisR. De Luca and P. van den Driessche, 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.

[7]

D. Danners and P. K. Medina, Abstract Evolution Equations, Peeriodic Problems and Applications, Pitman Research Notes in Mathematics Series, vol. 279. Chapman and Hall/CRC, London, 1992.

[8]

M.C. EisenbergZ. ShuaiD.L. 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.

[9]

D.M. HartleyJ.G. Morris and D.L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidmeics?, PLoS Med., 3 (2006), 63-69. 

[10]

S. Itô, Diffusion Equations, Translations of Mathematical Monographs, vol. 114. American Mathematical Society, Providence, 1992. doi: 10.1090/mmono/114.

[11]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spreal and traveling waves for monotone semiflows with applications, Commun. Pure. Appl. Math., 60 (2008), 1-40.  doi: 10.1002/cpa.20154.

[12]

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.

[13]

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

[14]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68. Springer, New York, 1986. doi: 10.1007/978-3-642-93287-8_2.

[15]

Z. MukandavireS. Liao and J. Wang, Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe, Proc. Natl. Acad. Sci. USA., 108 (2011), 8767-8772. 

[16]

R. L. M. NeilanE. Schaefer and H. Gaff, Modeling optimal intervention strategies for cholera, Bull. Math. Biol., 72 (2010), 2004-2018.  doi: 10.1007/s11538-010-9521-8.

[17]

D. Posny and J. Wang, Modelling cholera in periodic environments, J. Biol. Dyn., 8 (2014), 1-19.  doi: 10.1080/17513758.2014.896482.

[18]

D. PosnyJ. WangZ. Mukandavire and C. Modnak, Analyzing transmission dynamics of cholera with public health interventions, Math. Biosci., 264 (2015), 38-53.  doi: 10.1016/j.mbs.2015.03.006.

[19]

Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2011), 118-126.  doi: 10.1016/j.mbs.2011.09.003.

[20]

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.

[21]

H. R. ThiemeC. Castillo-Chavez and T. Kuniya, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM. J. Appl. Math., 53 (1993), 1447-1479.  doi: 10.1137/0153068.

[22]

J. WangR. Zhang and T. Kuniya, A note on dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 13 (2016), 227-247.  doi: 10.3934/mbe.2016.13.227.

[23]

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.

[24]

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.

[25]

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

[26]

World Health Organization, Cholera, 2022. Available from: https://www.who.int/news-room/fact-sheets/detail/cholera.

[27]

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

[28]

L. ZhangZ. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reactin-diffusion epidemic model with latent period, J. Differ. Equ., 258 (2015), 3011-3036.  doi: 10.1016/j.jde.2014.12.032.

[29]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Differ. Equ., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.

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