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A reaction-convection-diffusion model for cholera spatial dynamics
1. | Washington State University, Department of Mathematics and Statistics, Pullman, WA 99164-3113 |
2. | NSF Center for Integrated Pest Management, North Carolina State University, Raleigh, NC 27606, United States |
3. | Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403 |
References:
[1] |
D. Aronson, A comparison method for stability analysis of nonlinear parabolic problems, SIAM Review, 20 (1978), 245-264.
doi: 10.1137/1020038. |
[2] |
J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: an epidemic model, Lancet, 377 (2011), 1248-1255.
doi: 10.1016/S0140-6736(11)60273-0. |
[3] |
D. Butler, Cholera tightens grip on Haiti, Nature, 468 (2010), 483-484.
doi: 10.1038/468483a. |
[4] |
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. |
[5] |
E. I. Jury and M. Mansour, Positivity and nonnegativity conditions of a quartic equation and related problems, IEEE Trans. Automat. Control, AC, 26 (1981), 444-451.
doi: 10.1109/TAC.1981.1102589. |
[6] |
M. Bani-Yaghoub, R. Gautam, Z. Shuai, P. van den Driessche and R. Ivanek, Reproduction numbers for infections with free-living pathogens growing in the environment, J Biol. Dyn., 6 (2012), 923-940.
doi: 10.1080/17513758.2012.693206. |
[7] |
E. Bertuzzo, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, On spatially explicit models of cholera epidemics, J R. Soc. Interface, 7 (2010), 321-333.
doi: 10.1098/rsif.2009.0204. |
[8] |
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., 108 (2011), 7081-7085.
doi: 10.1073/pnas.1102149108. |
[9] |
C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect. Dis., 1 (2001), p1. |
[10] |
L. Ellwein, H. Tran, C. Zapata, V. Novak and M. Olufsen, Sensitivity analysis and model assessment: Mathematical models for arterial blood flow and blood pressure, J. Cardiovasc. Eng., 8 (2008), 94-108.
doi: 10.1007/s10558-007-9047-3. |
[11] |
M. Ghosh, P. Chandra, P. Sinha and J. B. Shukla, Modeling the spread of carrier-dependent infectious diseases with environmental effect, Appl. Math. Comput, 152 (2004), 385-402.
doi: 10.1016/S0096-3003(03)00564-2. |
[12] |
Y. H. Grad, J. C. Miller and M. Lipsitch, Cholera modeling: Challenges to quantitative analysis and predicting the impacts of interventions, Epidemiology, 23 (2012), 523-530.
doi: 10.1097/EDE.0b013e3182572581. |
[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 Medicine, 3 (2006), e7.
doi: 10.1371/journal.pmed.0030007. |
[14] |
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), 265-297.
doi: 10.1016/j.jde.2013.04.006. |
[15] |
M. Jensen, S. M. Faruque, J. J. Mekalanos and B. Levin, Modeling the role of bacteriophage in the control of cholera outbreaks, Proc. Nat. Acad. Sci. 103 (2006), 4652-4657.
doi: 10.1073/pnas.0600166103. |
[16] |
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), 845-862.
doi: 10.1007/s11538-008-9384-4. |
[17] |
C. Kapp, Zimbabwe's humanitarian crisis worsens, Lancet, 373 (2009), p447.
doi: 10.1016/S0140-6736(09)60151-3. |
[18] |
S. Kabir, Cholera vaccines: The current status and problems, Rev. Med. Microbiol., 16 (2005), 101-116.
doi: 10.1097/01.revmedmi.0000174307.33651.81. |
[19] |
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. |
[20] |
S. Liao and J. Wang, Stability analysis and application of a mathematical cholera model, Math. Biosci. Eng., 8 (2011), 733-752.
doi: 10.3934/mbe.2011.8.733. |
[21] |
J. Lin, V. Andreasen, R. Casagrandi and S. A. Levin, Traveling waves in a model of influenza A drift, J. Theor. Biol., 222 (2003), 437-445.
doi: 10.1016/S0022-5193(03)00056-0. |
[22] |
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, Pietermaritzburg, South America. 2001. |
[23] |
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., 108 (2011), 8767-8772. |
[24] |
J. D. Murray, Mathematical Biology, Springer, Berlin, 2003.
doi: 10.1007/b98869. |
[25] |
R. L. M. Neilan, E. Schaefer, H. Gaff, K. R. Fister and S. Lenhart, Modeling optimal intervention strategies for cholera, Bull. Math. Biol., 72 (2010), 2004-2018.
doi: 10.1007/s11538-010-9521-8. |
[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), 1161-1168,
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 speading along fluvial systems, Ecohydrology, 4 (2011), 49-55. |
[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., 109 (2012), 6602-6607.
doi: 10.1073/pnas.1203333109. |
[29] |
J. Reidl and K. E. Klose, Vibrio cholerae and cholera: out of the water and into the host, FEMS Microbiol. Rev, 26 (2002), 125-139. |
[30] |
S. L. Robertson, M. C. Eisenberg and J. H. Tiend, Heterogeneity in multiple transmission pathways: Modelling the spread of cholera and other waterborne disease in networks with a common water source, J. Biol. Dyn., 7 (2013), 254-275.
doi: 10.1080/17513758.2013.853844. |
[31] |
D. A. Sack, R. Sack and C.-L. Chaignat, Getting serious about cholera, New Engl. J. Med., 355 (2006), 649-651.
doi: 10.1056/NEJMp068144. |
[32] |
Z. Shuai, J. H. Tien and P. van den Driessche, Cholera models with hyperinfectivity and temporary immunity, Bull. Math. Biol., 74 (2012), 2423-2445.
doi: 10.1007/s11538-012-9759-4. |
[33] |
Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.
doi: 10.1137/120876642. |
[34] |
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), 1067-1082.
doi: 10.1007/s00285-012-0579-9. |
[35] |
J. P. Tian, S. Liao and J. Wang, Analyzing the infection dynamics and control strategies of cholera, Discret. Contin. Dyn. S., (2013), 747-757.
doi: 10.3934/proc.2013.2013.747. |
[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] |
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. |
[38] |
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. |
[39] |
J. H. Tien, H. N. Poinar, D. N. Fisman and D. J. Earn, Herald Waves of Cholera in Nineteenth Century London, J. R. Soc., 8 (2011), 756-760.
doi: 10.1098/rsif.2010.0494. |
[40] |
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, Ann. Intern. Med., 154 (2011), 593-601.
doi: 10.7326/0003-4819-154-9-201105030-00334. |
[41] |
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. |
[42] |
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. |
[43] |
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. |
[44] |
W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. S., 11 (2012), 1652-1673.
doi: 10.1137/120872942. |
[45] |
, World Health Organization (WHO) web page: http://www.who.org. |
[46] |
, WHO web page: http://www.who.int/csr/don/2014_05_30/en/. |
[47] |
, WHO web page: http://www.who.int/mediacentre/factsheets/fs107/en/index.html. |
show all references
References:
[1] |
D. Aronson, A comparison method for stability analysis of nonlinear parabolic problems, SIAM Review, 20 (1978), 245-264.
doi: 10.1137/1020038. |
[2] |
J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: an epidemic model, Lancet, 377 (2011), 1248-1255.
doi: 10.1016/S0140-6736(11)60273-0. |
[3] |
D. Butler, Cholera tightens grip on Haiti, Nature, 468 (2010), 483-484.
doi: 10.1038/468483a. |
[4] |
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. |
[5] |
E. I. Jury and M. Mansour, Positivity and nonnegativity conditions of a quartic equation and related problems, IEEE Trans. Automat. Control, AC, 26 (1981), 444-451.
doi: 10.1109/TAC.1981.1102589. |
[6] |
M. Bani-Yaghoub, R. Gautam, Z. Shuai, P. van den Driessche and R. Ivanek, Reproduction numbers for infections with free-living pathogens growing in the environment, J Biol. Dyn., 6 (2012), 923-940.
doi: 10.1080/17513758.2012.693206. |
[7] |
E. Bertuzzo, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo, On spatially explicit models of cholera epidemics, J R. Soc. Interface, 7 (2010), 321-333.
doi: 10.1098/rsif.2009.0204. |
[8] |
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., 108 (2011), 7081-7085.
doi: 10.1073/pnas.1102149108. |
[9] |
C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect. Dis., 1 (2001), p1. |
[10] |
L. Ellwein, H. Tran, C. Zapata, V. Novak and M. Olufsen, Sensitivity analysis and model assessment: Mathematical models for arterial blood flow and blood pressure, J. Cardiovasc. Eng., 8 (2008), 94-108.
doi: 10.1007/s10558-007-9047-3. |
[11] |
M. Ghosh, P. Chandra, P. Sinha and J. B. Shukla, Modeling the spread of carrier-dependent infectious diseases with environmental effect, Appl. Math. Comput, 152 (2004), 385-402.
doi: 10.1016/S0096-3003(03)00564-2. |
[12] |
Y. H. Grad, J. C. Miller and M. Lipsitch, Cholera modeling: Challenges to quantitative analysis and predicting the impacts of interventions, Epidemiology, 23 (2012), 523-530.
doi: 10.1097/EDE.0b013e3182572581. |
[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 Medicine, 3 (2006), e7.
doi: 10.1371/journal.pmed.0030007. |
[14] |
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), 265-297.
doi: 10.1016/j.jde.2013.04.006. |
[15] |
M. Jensen, S. M. Faruque, J. J. Mekalanos and B. Levin, Modeling the role of bacteriophage in the control of cholera outbreaks, Proc. Nat. Acad. Sci. 103 (2006), 4652-4657.
doi: 10.1073/pnas.0600166103. |
[16] |
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), 845-862.
doi: 10.1007/s11538-008-9384-4. |
[17] |
C. Kapp, Zimbabwe's humanitarian crisis worsens, Lancet, 373 (2009), p447.
doi: 10.1016/S0140-6736(09)60151-3. |
[18] |
S. Kabir, Cholera vaccines: The current status and problems, Rev. Med. Microbiol., 16 (2005), 101-116.
doi: 10.1097/01.revmedmi.0000174307.33651.81. |
[19] |
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. |
[20] |
S. Liao and J. Wang, Stability analysis and application of a mathematical cholera model, Math. Biosci. Eng., 8 (2011), 733-752.
doi: 10.3934/mbe.2011.8.733. |
[21] |
J. Lin, V. Andreasen, R. Casagrandi and S. A. Levin, Traveling waves in a model of influenza A drift, J. Theor. Biol., 222 (2003), 437-445.
doi: 10.1016/S0022-5193(03)00056-0. |
[22] |
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, Pietermaritzburg, South America. 2001. |
[23] |
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., 108 (2011), 8767-8772. |
[24] |
J. D. Murray, Mathematical Biology, Springer, Berlin, 2003.
doi: 10.1007/b98869. |
[25] |
R. L. M. Neilan, E. Schaefer, H. Gaff, K. R. Fister and S. Lenhart, Modeling optimal intervention strategies for cholera, Bull. Math. Biol., 72 (2010), 2004-2018.
doi: 10.1007/s11538-010-9521-8. |
[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), 1161-1168,
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 speading along fluvial systems, Ecohydrology, 4 (2011), 49-55. |
[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., 109 (2012), 6602-6607.
doi: 10.1073/pnas.1203333109. |
[29] |
J. Reidl and K. E. Klose, Vibrio cholerae and cholera: out of the water and into the host, FEMS Microbiol. Rev, 26 (2002), 125-139. |
[30] |
S. L. Robertson, M. C. Eisenberg and J. H. Tiend, Heterogeneity in multiple transmission pathways: Modelling the spread of cholera and other waterborne disease in networks with a common water source, J. Biol. Dyn., 7 (2013), 254-275.
doi: 10.1080/17513758.2013.853844. |
[31] |
D. A. Sack, R. Sack and C.-L. Chaignat, Getting serious about cholera, New Engl. J. Med., 355 (2006), 649-651.
doi: 10.1056/NEJMp068144. |
[32] |
Z. Shuai, J. H. Tien and P. van den Driessche, Cholera models with hyperinfectivity and temporary immunity, Bull. Math. Biol., 74 (2012), 2423-2445.
doi: 10.1007/s11538-012-9759-4. |
[33] |
Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.
doi: 10.1137/120876642. |
[34] |
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), 1067-1082.
doi: 10.1007/s00285-012-0579-9. |
[35] |
J. P. Tian, S. Liao and J. Wang, Analyzing the infection dynamics and control strategies of cholera, Discret. Contin. Dyn. S., (2013), 747-757.
doi: 10.3934/proc.2013.2013.747. |
[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] |
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. |
[38] |
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. |
[39] |
J. H. Tien, H. N. Poinar, D. N. Fisman and D. J. Earn, Herald Waves of Cholera in Nineteenth Century London, J. R. Soc., 8 (2011), 756-760.
doi: 10.1098/rsif.2010.0494. |
[40] |
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, Ann. Intern. Med., 154 (2011), 593-601.
doi: 10.7326/0003-4819-154-9-201105030-00334. |
[41] |
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. |
[42] |
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. |
[43] |
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. |
[44] |
W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. S., 11 (2012), 1652-1673.
doi: 10.1137/120872942. |
[45] |
, World Health Organization (WHO) web page: http://www.who.org. |
[46] |
, WHO web page: http://www.who.int/csr/don/2014_05_30/en/. |
[47] |
, WHO web page: http://www.who.int/mediacentre/factsheets/fs107/en/index.html. |
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