A reaction-convection-diffusion model for cholera spatial dynamics

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October 2016, 21(8): 2785-2809. doi: 10.3934/dcdsb.2016073

A reaction-convection-diffusion model for cholera spatial dynamics

Xueying Wang ^1, , Drew Posny ^2, and Jin Wang ^3,

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

Received September 2015 Revised November 2015 Published September 2016

Abstract
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In this paper, we propose a general partial differential equation (PDE) model of cholera epidemics that extends previous mathematical cholera studies. Our new formation concerns the impact of the bacterial and human diffusion, bacterial convection, and their interaction with the intrinsic bacterial growth and multiple disease transmission pathways. A sensitivity analysis for a few key model parameters indicates the significance of diffusion and convection in shaping cholera epidemics. We then investigate the traveling wave solutions of our PDE model based on analytical derivation and numerical simulation, with a focus on the interplay of different biological, environmental and physical factors that determines the spatial spreading speeds of cholera. In addition, disease threshold dynamics are studied by computing the basic reproduction number associated with the PDE model, using both asymptotic analysis and numerical calculation.

Keywords: reaction-convection-diffusion model, traveling wave speeds, Cholera spatial dynamics, threshold dynamics..

Mathematics Subject Classification: 35K57, 35C07, 37N25, 92D3.

Citation: Xueying Wang, Drew Posny, Jin Wang. A reaction-convection-diffusion model for cholera spatial dynamics. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2785-2809. doi: 10.3934/dcdsb.2016073

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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