June  2016, 21(4): 1297-1316. doi: 10.3934/dcdsb.2016.21.1297

Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model

1. 

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

2. 

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

Received  May 2015 Revised  December 2015 Published  March 2016

In this paper, we study the initial boundary value problem of a reaction-convection-diffusion epidemic model for cholera dynamics, which was developed in [38], named susceptible-infected-recovered-susceptible-bacteria (SIRS-B) epidemic PDE model. First, a local well-posedness result relying on the theory of cooperative dynamics systems is obtained. Via a priori estimates making use of the special structure of the system and continuation of local theory argument, we show that in fact this problem is globally well-posed. Secondly, we analyze the local asymptotic stability of the solutions based on the basic reproduction number associated with this model.
Citation: Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297
References:
[1]

J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: an epidemic model,, Lancet, 377 (2011), 1248. doi: 10.1016/S0140-6736(11)60273-0.

[2]

M. Bani-Yaghoub, R. Gautam, Z. Shuai, P. v. d. Driessche and R. Ivanek, Reproduction numbers for infections with free-living pathogens growing in the environment,, J. Biol. Dyn., 6 (2012), 923. doi: 10.1080/17513758.2012.693206.

[3]

O. Bratteli and P. E. T. Jorgensen, Positive Semigroups of Operators, and Applications,, Springer, (1984). doi: 10.1007/978-94-009-6484-6.

[4]

E. Bertuzzo, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, On spatially explicit models of cholera epidemics,, Journal of the Royal Society Interface, 7 (2010), 321. doi: 10.1098/rsif.2009.0204.

[5]

D. L. Chao, M. 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. doi: 10.1073/pnas.1102149108.

[6]

C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir,, BMC Infectious Diseases, 1 (2001). doi: 10.1186/1471-2334-1-1.

[7]

W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups,, in Differential Equations in Banach Spaces, 1223 (1986), 61. doi: 10.1007/BFb0099183.

[8]

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

[9]

P. V. D. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

[10]

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. doi: 10.1016/j.mbs.2013.08.003.

[11]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-state Problems,, Second edition, (2011). doi: 10.1007/978-0-387-09620-9.

[12]

J. P. Grover, S.-B. Hsu and F.-B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone,, Math. Biosci., 222 (2009), 42. doi: 10.1016/j.mbs.2009.08.006.

[13]

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

[14]

S.-B. Hsu, J. Jiang and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat,, J. Differential Equations, 248 (2010), 2470. doi: 10.1016/j.jde.2009.12.014.

[15]

S.-B. Hsu, F.-B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone,, J. Dynam. Differential Equations, 23 (2011), 817. doi: 10.1007/s10884-011-9224-3.

[16]

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

[17]

R. I. Joh, H. Wang, H. Weiss and J. S. Weitz, Dynamics of indirectly transmitted infectious diseases with immunological threshold,, Bull. Math. Biol., 71 (2009), 845. doi: 10.1007/s11538-008-9384-4.

[18]

S. Kakutani, Concrete representation of abstract (M)-spaces (a characterization of the space of continuous functions),, Ann. Math., 42 (1941), 994. doi: 10.2307/1968778.

[19]

C. Kapp, Zimbabwe's humanitarian crisis worsens,, Lancet, 373 (2009). doi: 10.1016/S0140-6736(09)60151-3.

[20]

T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1995).

[21]

S. Liao and J. Wang, Stability analysis and application of a mathematical cholera model,, Math. Biosci. Eng., 8 (2011), 733. doi: 10.3934/mbe.2011.8.733.

[22]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1. doi: 10.1090/S0002-9947-1990-0967316-X.

[23]

C. Mugero and A. Hoque, Review of Cholera Epidemic in South Africa with Focus on KwaZulu-Natal Province,, Technical Report. KwaZulu-Natal Department of Health, (2001).

[24]

Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. 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. doi: 10.1073/pnas.1019712108.

[25]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1.

[26]

R. Piarroux, R. Barrais, B. Faucher, R. Haus, M. Piarroux, J. Gaudart, R. Magloire and D. Raoult, Understanding the cholera epidemic, Haiti,, Emerg. Infect. Dis., 17 (2011), 1161. doi: 10.3201/eid1707.110059.

[27]

L. Righetto, E. Bertuzzo, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, Modeling human movement in a cholera spreading along fluvial systems,, Ecohydrol., 4 (2011), 49. doi: 10.1002/eco.122.

[28]

A. Rinaldo, E. Bertuzzo, L. Mari, L. Righetto, M. Blokesch, M. Gatto, R. Casagrandi, M. Murray, S. 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. doi: 10.1073/pnas.1203333109.

[29]

D. A. Sack, R. B. Sack and C.-L. Chaignat, Getting serious about cholera,, New Engl. J. Med., 355 (2006), 649. doi: 10.1056/NEJMp068144.

[30]

Z. Shuai, J. H. Tien and P. van den Driessche, Cholera models with hyperinfectivity and temporary immunity,, Bull. Math. Biol., 74 (2012), 2423. doi: 10.1007/s11538-012-9759-4.

[31]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions,, SIAM J. Appl. Math., 73 (2013), 1513. doi: 10.1137/120876642.

[32]

Z. Shuai, J. A. P. Heesterbeek and P. van den Driessche, Extending the type reproduction number to infectious disease control targeting contacts between types,, J. Math. Biol., 67 (2013), 1067. doi: 10.1007/s00285-012-0579-9.

[33]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Math. Surveys Monogr. 41, 41 (1995).

[34]

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

[35]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188. doi: 10.1137/080732870.

[36]

A. R. Tuite, J. H. Tien, M. C. Eisenberg, D. J. D. Earn, J. Ma and D. N. Fisman, Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions,, Annals of Internal Medicine, 154 (2011), 593. doi: 10.7326/0003-4819-154-9-201105030-00334.

[37]

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

[38]

X. Wang, D. Posny and J. Wang, A Reaction-Convection-Diffusion Model for Cholera Spatial Dynamics, submitted., ().

[39]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models,, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652. doi: 10.1137/120872942.

[40]

, World Health Organization (WHO),, Available from: , ().

[41]

, WHO web page,, Available from: , ().

show all references

References:
[1]

J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: an epidemic model,, Lancet, 377 (2011), 1248. doi: 10.1016/S0140-6736(11)60273-0.

[2]

M. Bani-Yaghoub, R. Gautam, Z. Shuai, P. v. d. Driessche and R. Ivanek, Reproduction numbers for infections with free-living pathogens growing in the environment,, J. Biol. Dyn., 6 (2012), 923. doi: 10.1080/17513758.2012.693206.

[3]

O. Bratteli and P. E. T. Jorgensen, Positive Semigroups of Operators, and Applications,, Springer, (1984). doi: 10.1007/978-94-009-6484-6.

[4]

E. Bertuzzo, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, On spatially explicit models of cholera epidemics,, Journal of the Royal Society Interface, 7 (2010), 321. doi: 10.1098/rsif.2009.0204.

[5]

D. L. Chao, M. 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. doi: 10.1073/pnas.1102149108.

[6]

C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir,, BMC Infectious Diseases, 1 (2001). doi: 10.1186/1471-2334-1-1.

[7]

W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups,, in Differential Equations in Banach Spaces, 1223 (1986), 61. doi: 10.1007/BFb0099183.

[8]

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

[9]

P. V. D. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

[10]

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. doi: 10.1016/j.mbs.2013.08.003.

[11]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-state Problems,, Second edition, (2011). doi: 10.1007/978-0-387-09620-9.

[12]

J. P. Grover, S.-B. Hsu and F.-B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone,, Math. Biosci., 222 (2009), 42. doi: 10.1016/j.mbs.2009.08.006.

[13]

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

[14]

S.-B. Hsu, J. Jiang and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat,, J. Differential Equations, 248 (2010), 2470. doi: 10.1016/j.jde.2009.12.014.

[15]

S.-B. Hsu, F.-B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone,, J. Dynam. Differential Equations, 23 (2011), 817. doi: 10.1007/s10884-011-9224-3.

[16]

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

[17]

R. I. Joh, H. Wang, H. Weiss and J. S. Weitz, Dynamics of indirectly transmitted infectious diseases with immunological threshold,, Bull. Math. Biol., 71 (2009), 845. doi: 10.1007/s11538-008-9384-4.

[18]

S. Kakutani, Concrete representation of abstract (M)-spaces (a characterization of the space of continuous functions),, Ann. Math., 42 (1941), 994. doi: 10.2307/1968778.

[19]

C. Kapp, Zimbabwe's humanitarian crisis worsens,, Lancet, 373 (2009). doi: 10.1016/S0140-6736(09)60151-3.

[20]

T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1995).

[21]

S. Liao and J. Wang, Stability analysis and application of a mathematical cholera model,, Math. Biosci. Eng., 8 (2011), 733. doi: 10.3934/mbe.2011.8.733.

[22]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1. doi: 10.1090/S0002-9947-1990-0967316-X.

[23]

C. Mugero and A. Hoque, Review of Cholera Epidemic in South Africa with Focus on KwaZulu-Natal Province,, Technical Report. KwaZulu-Natal Department of Health, (2001).

[24]

Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. 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. doi: 10.1073/pnas.1019712108.

[25]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1.

[26]

R. Piarroux, R. Barrais, B. Faucher, R. Haus, M. Piarroux, J. Gaudart, R. Magloire and D. Raoult, Understanding the cholera epidemic, Haiti,, Emerg. Infect. Dis., 17 (2011), 1161. doi: 10.3201/eid1707.110059.

[27]

L. Righetto, E. Bertuzzo, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, Modeling human movement in a cholera spreading along fluvial systems,, Ecohydrol., 4 (2011), 49. doi: 10.1002/eco.122.

[28]

A. Rinaldo, E. Bertuzzo, L. Mari, L. Righetto, M. Blokesch, M. Gatto, R. Casagrandi, M. Murray, S. 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. doi: 10.1073/pnas.1203333109.

[29]

D. A. Sack, R. B. Sack and C.-L. Chaignat, Getting serious about cholera,, New Engl. J. Med., 355 (2006), 649. doi: 10.1056/NEJMp068144.

[30]

Z. Shuai, J. H. Tien and P. van den Driessche, Cholera models with hyperinfectivity and temporary immunity,, Bull. Math. Biol., 74 (2012), 2423. doi: 10.1007/s11538-012-9759-4.

[31]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions,, SIAM J. Appl. Math., 73 (2013), 1513. doi: 10.1137/120876642.

[32]

Z. Shuai, J. A. P. Heesterbeek and P. van den Driessche, Extending the type reproduction number to infectious disease control targeting contacts between types,, J. Math. Biol., 67 (2013), 1067. doi: 10.1007/s00285-012-0579-9.

[33]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Math. Surveys Monogr. 41, 41 (1995).

[34]

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

[35]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188. doi: 10.1137/080732870.

[36]

A. R. Tuite, J. H. Tien, M. C. Eisenberg, D. J. D. Earn, J. Ma and D. N. Fisman, Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions,, Annals of Internal Medicine, 154 (2011), 593. doi: 10.7326/0003-4819-154-9-201105030-00334.

[37]

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

[38]

X. Wang, D. Posny and J. Wang, A Reaction-Convection-Diffusion Model for Cholera Spatial Dynamics, submitted., ().

[39]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models,, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652. doi: 10.1137/120872942.

[40]

, World Health Organization (WHO),, Available from: , ().

[41]

, WHO web page,, Available from: , ().

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