April  2017, 14(2): 559-579. doi: 10.3934/mbe.2017033

Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model

1. 

Department of Mathematics, University of Rochester, Rochester, NY 14627, USA

2. 

Department of Mathematics and Statistics, Washington State University, Pullman, WA 99164-3113, USA

Received  May 10, 2016 Accepted  September 19, 2016 Published  October 2016

Fund Project: This work was partially supported by a grant from the Simons Foundation (#317047 to Xueying Wang)

We study the global stability issue of the reaction-convection-diffusion cholera epidemic PDE model and show that the basic reproduction number serves as a threshold parameter that predicts whether cholera will persist or become globally extinct. Specifically, when the basic reproduction number is beneath one, we show that the disease-free-equilibrium is globally attractive. On the other hand, when the basic reproduction number exceeds one, if the infectious hosts or the concentration of bacteria in the contaminated water are not initially identically zero, we prove the uniform persistence result and that there exists at least one positive steady state.

Citation: Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033
References:
[1]

E. BertuzzoR. CasagrandiM. GattoI. Rodriguez-Iturbe and A. Rinaldo, On spatially explicit models of cholera epidemics, Journal of the Royal Society Interface, 7 (2010), 321-333.  doi: 10.1098/rsif.2009.0204.  Google Scholar

[2]

E. BertuzzoL. MariL. RighettoM. GattoR. CasagrandiM. BlokeschI. Rodriguez-Iturbe and A. Rinaldo, Prediction of the spatial evolution and effects of control measures for the unfolding Haiti cholera outbreak, Geophys. Res. Lett., 38 (2011), 1-5.  doi: 10.1029/2011GL046823.  Google Scholar

[3]

V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. Epidemiol. Sante, 27 (1979), 121-132.   Google Scholar

[4]

A. Carpenter, Behavior in the time of cholera: Evidence from the 2008-2009 cholera outbreak in Zimbabwe, in Social Computing, Behavioral-Cultural Modeling and Prediction, Springer, 8393 (2014), 237-244. doi: 10.1007/978-3-319-05579-4_29.  Google Scholar

[5]

D. L. ChaoM. E. Halloran and I. M. Longini Jr., Vaccination strategies for epidemic cholera in Haiti with implications for the developing world, Proc. Natl. Acad. Sci. USA, 108 (2011), 7081-7085.  doi: 10.1073/pnas.1102149108.  Google Scholar

[6]

S. F. Dowell and C. R. Braden, Implications of the introduction of cholera to Haiti, Emerg. Infect. Dis., 17 (2011), 1299-1300.  doi: 10.3201/eid1707.110625.  Google Scholar

[7]

M. C. EisenbergZ. ShuaiJ. 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.  Google Scholar

[8]

L. Evans, Partial Differential Equations American Mathematics Society, Providence, Rhode Island, 1998.  Google Scholar

[9]

H. I. Freedman and X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Differential Equations, 137 (1997), 340-362.  doi: 10.1006/jdeq.1997.3264.  Google Scholar

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems Mathematical surveys and monographs, American Mathematics Society, Providence, Rhode Island, 1988.  Google Scholar

[11]

D. M. HartleyJ. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Med., 3 (2006), e7.  doi: 10.1371/journal.pmed.0030007.  Google Scholar

[12]

S.-B. HsuF.-B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297.  doi: 10.1016/j.jde.2013.04.006.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

Z. MukandavireS. LiaoJ. WangH. GaffD. L. Smith and J. G. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Natl. Acad. Sci. USA, 108 (2011), 8767-8772.   Google Scholar

[17]

R. L. M. NeilanE. SchaeferH. GaffK. R. Fister and S. Lenhart, Modeling optimal intervention strategies for cholera, B. Math. Biol., 72 (2010), 2004-2018.  doi: 10.1007/s11538-010-9521-8.  Google Scholar

[18]

R. PiarrouxR. BarraisB. FaucherR. HausM. PiarrouxJ. GaudartR. Magloire and D. Raoult, Understanding the cholera epidemic, Haiti, Emerg. Infect. Dis., 17 (2011), 1161-1168.  doi: 10.3201/eid1707.110059.  Google Scholar

[19]

A. RinaldoE. BertuzzoL. MariL. RighettoM. BlokeschM. GattoR. CasagrandiM. MurrayS. M. Vesenbeckh and I. Rodriguez-Iturbe, Reassessment of the 2010-2011 Haiti cholera outbreak and rainfall-driven multiseason projections, Proc. Natl. Acad. Sci. USA, 109 (2012), 6602-6607.  doi: 10.1073/pnas.1203333109.  Google Scholar

[20]

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.  Google Scholar

[21]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Math. Surveys Monogr. 41 American Mathematical Society, Providence, Rhode Island, 1995.  Google Scholar

[22]

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

[23]

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.  Google Scholar

[24]

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

[25]

J. P. Tian and J. Wang, Global stability for cholera epidemic models, Math. Biosci., 232 (2011), 31-41.  doi: 10.1016/j.mbs.2011.04.001.  Google Scholar

[26]

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

[27]

J. H. TienZ. ShuaiM. C. Eisenberg and P. van den Driessche, Disease invasion on community net-works with environmental pathogen movement, J. Math. Biology, 70 (2015), 1065-1092.  doi: 10.1007/s00285-014-0791-x.  Google Scholar

[28]

N. K. VaidyaF.-B. Wang and X. Zou, Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2829-2848.  doi: 10.3934/dcdsb.2012.17.2829.  Google Scholar

[29]

J. Wang and S. Liao, A generalized cholera model and epidemic-endemic analysis, J. Biol. Dyn., 6 (2012), 568-589.  doi: 10.1080/17513758.2012.658089.  Google Scholar

[30]

J. Wang and C. Modnak, Modeling cholera dynamics with controls, Canad. Appl. Math. Quart., 19 (2011), 255-273.   Google Scholar

[31]

X. WangD. Posny and J. Wang, A reaction-convection-diffusion model for cholera spatial dynamics, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2785-2809.  doi: 10.3934/dcdsb.2016073.  Google Scholar

[32]

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.  Google Scholar

[33]

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

[34]

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

[35]

WHO Cholera outbreak, South Sudan Disease Outbreak News, 2014. Available from: http://www.who.int/csr/don/2014_05_30/en/. Google Scholar

[36] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996.  doi: 10.1007/978-1-4612-4050-1.  Google Scholar
[37]

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

[38]

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

show all references

References:
[1]

E. BertuzzoR. CasagrandiM. GattoI. Rodriguez-Iturbe and A. Rinaldo, On spatially explicit models of cholera epidemics, Journal of the Royal Society Interface, 7 (2010), 321-333.  doi: 10.1098/rsif.2009.0204.  Google Scholar

[2]

E. BertuzzoL. MariL. RighettoM. GattoR. CasagrandiM. BlokeschI. Rodriguez-Iturbe and A. Rinaldo, Prediction of the spatial evolution and effects of control measures for the unfolding Haiti cholera outbreak, Geophys. Res. Lett., 38 (2011), 1-5.  doi: 10.1029/2011GL046823.  Google Scholar

[3]

V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. Epidemiol. Sante, 27 (1979), 121-132.   Google Scholar

[4]

A. Carpenter, Behavior in the time of cholera: Evidence from the 2008-2009 cholera outbreak in Zimbabwe, in Social Computing, Behavioral-Cultural Modeling and Prediction, Springer, 8393 (2014), 237-244. doi: 10.1007/978-3-319-05579-4_29.  Google Scholar

[5]

D. L. ChaoM. E. Halloran and I. M. Longini Jr., Vaccination strategies for epidemic cholera in Haiti with implications for the developing world, Proc. Natl. Acad. Sci. USA, 108 (2011), 7081-7085.  doi: 10.1073/pnas.1102149108.  Google Scholar

[6]

S. F. Dowell and C. R. Braden, Implications of the introduction of cholera to Haiti, Emerg. Infect. Dis., 17 (2011), 1299-1300.  doi: 10.3201/eid1707.110625.  Google Scholar

[7]

M. C. EisenbergZ. ShuaiJ. 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.  Google Scholar

[8]

L. Evans, Partial Differential Equations American Mathematics Society, Providence, Rhode Island, 1998.  Google Scholar

[9]

H. I. Freedman and X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Differential Equations, 137 (1997), 340-362.  doi: 10.1006/jdeq.1997.3264.  Google Scholar

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems Mathematical surveys and monographs, American Mathematics Society, Providence, Rhode Island, 1988.  Google Scholar

[11]

D. M. HartleyJ. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Med., 3 (2006), e7.  doi: 10.1371/journal.pmed.0030007.  Google Scholar

[12]

S.-B. HsuF.-B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297.  doi: 10.1016/j.jde.2013.04.006.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

Z. MukandavireS. LiaoJ. WangH. GaffD. L. Smith and J. G. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Natl. Acad. Sci. USA, 108 (2011), 8767-8772.   Google Scholar

[17]

R. L. M. NeilanE. SchaeferH. GaffK. R. Fister and S. Lenhart, Modeling optimal intervention strategies for cholera, B. Math. Biol., 72 (2010), 2004-2018.  doi: 10.1007/s11538-010-9521-8.  Google Scholar

[18]

R. PiarrouxR. BarraisB. FaucherR. HausM. PiarrouxJ. GaudartR. Magloire and D. Raoult, Understanding the cholera epidemic, Haiti, Emerg. Infect. Dis., 17 (2011), 1161-1168.  doi: 10.3201/eid1707.110059.  Google Scholar

[19]

A. RinaldoE. BertuzzoL. MariL. RighettoM. BlokeschM. GattoR. CasagrandiM. MurrayS. M. Vesenbeckh and I. Rodriguez-Iturbe, Reassessment of the 2010-2011 Haiti cholera outbreak and rainfall-driven multiseason projections, Proc. Natl. Acad. Sci. USA, 109 (2012), 6602-6607.  doi: 10.1073/pnas.1203333109.  Google Scholar

[20]

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.  Google Scholar

[21]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Math. Surveys Monogr. 41 American Mathematical Society, Providence, Rhode Island, 1995.  Google Scholar

[22]

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

[23]

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.  Google Scholar

[24]

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

[25]

J. P. Tian and J. Wang, Global stability for cholera epidemic models, Math. Biosci., 232 (2011), 31-41.  doi: 10.1016/j.mbs.2011.04.001.  Google Scholar

[26]

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

[27]

J. H. TienZ. ShuaiM. C. Eisenberg and P. van den Driessche, Disease invasion on community net-works with environmental pathogen movement, J. Math. Biology, 70 (2015), 1065-1092.  doi: 10.1007/s00285-014-0791-x.  Google Scholar

[28]

N. K. VaidyaF.-B. Wang and X. Zou, Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2829-2848.  doi: 10.3934/dcdsb.2012.17.2829.  Google Scholar

[29]

J. Wang and S. Liao, A generalized cholera model and epidemic-endemic analysis, J. Biol. Dyn., 6 (2012), 568-589.  doi: 10.1080/17513758.2012.658089.  Google Scholar

[30]

J. Wang and C. Modnak, Modeling cholera dynamics with controls, Canad. Appl. Math. Quart., 19 (2011), 255-273.   Google Scholar

[31]

X. WangD. Posny and J. Wang, A reaction-convection-diffusion model for cholera spatial dynamics, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2785-2809.  doi: 10.3934/dcdsb.2016073.  Google Scholar

[32]

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.  Google Scholar

[33]

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

[34]

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

[35]

WHO Cholera outbreak, South Sudan Disease Outbreak News, 2014. Available from: http://www.who.int/csr/don/2014_05_30/en/. Google Scholar

[36] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996.  doi: 10.1007/978-1-4612-4050-1.  Google Scholar
[37]

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

[38]

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

Table 1.  Definition of parameters in model (1)
ParameterDefinition
$b$Recruitment rate of susceptible hosts
$d$Natural death rate of human hosts
$\gamma$Recovery rate of infectious hosts
$\sigma$Rate of host immunity loss
$\delta$Natural death rate of bacteria
$\xi$Shedding rate of bacteria by infectious hosts
$\beta_{1}$Direct transmission parameter
$\beta_{2}$Indirect transmission parameter
$K$Half saturation rate of bacteria
$U$Bacterial convection coefficient
$K_{B}$Maximal carrying capacity of bacteria in the environment
ParameterDefinition
$b$Recruitment rate of susceptible hosts
$d$Natural death rate of human hosts
$\gamma$Recovery rate of infectious hosts
$\sigma$Rate of host immunity loss
$\delta$Natural death rate of bacteria
$\xi$Shedding rate of bacteria by infectious hosts
$\beta_{1}$Direct transmission parameter
$\beta_{2}$Indirect transmission parameter
$K$Half saturation rate of bacteria
$U$Bacterial convection coefficient
$K_{B}$Maximal carrying capacity of bacteria in the environment
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