Existence of traveling wave solutions for cholera model. (English) Zbl 1470.92376

Zhang, Tianran; Gou, Qingming; Wang, Xiaoli

Abstr. Appl. Anal. 2014, Article ID 201094, 11 p. (2014).

Summary: To investigate the spreading speed of cholera, C. T. Codeço’s cholera model [“Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir”, BMC Infect. Dis. 1, No. 1, 1–14 (2001; doi:10.1186/1471-2334-1-1)] is developed by a reaction-diffusion model that incorporates both indirect environment-to-human and direct human-to-human transmissions and the pathogen diffusion. The two transmission incidences are supposed to be saturated with infective density and pathogen density. The basic reproduction number \(R_0\) is defined and the formula for minimal wave speed \(c^*\) is given. It is proved by shooting method that there exists a traveling wave solution with speed \(c\) for cholera model if and only if \(c \geq c^*\).

MSC:

92D30

Epidemiology

References:

[2]	World Health Organization
[3]	Capasso, V.; Paveri-Fontana, S. L., Mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Revue d’Epidemiologie et de Sante Publique, 27, 2, 121-132 (1979)
[4]	Codeço, C. T., Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infectious Diseases, 1, 1, article 1 (2001)
[5]	Andrews, J. R.; Basu, S., Transmission dynamics and control of cholera in Haiti: an epidemic model, The Lancet, 377, 9773, 1248-1255 (2011)
[6]	Goh, K. T.; Teo, S. H.; Lam, S.; Ling, M. K., Person-to-person transmission of cholera in a psychiatric hospital, Journal of Infection, 20, 3, 193-200 (1990)
[7]	Weil, A. A.; Khan, A. I.; Chowdhury, F.; LaRocque, R. C.; Faruque, A. S. G.; Ryan, E. T.; Calderwood, S. B.; Qadri, F.; Harris, J. B., Clinical outcomes in household contacts of patients with cholera in Bangladesh, Clinical Infectious Diseases, 49, 10, 1473-1479 (2009)
[8]	Tien, J. H.; Earn, D. J. D., Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology, 72, 6, 1506-1533 (2010) · Zbl 1198.92030
[9]	Mukandavire, Z.; Liao, S.; Wang, J.; Gaff, H.; Smith, D. L.; Morris, J. G., Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proceedings of the National Academy of Sciences of the United States of America, 108, 21, 8767-8772 (2011)
[10]	Tian, J. P.; Wang, J., Global stability for cholera epidemic models, Mathematical Biosciences, 232, 1, 31-41 (2011) · Zbl 1217.92068
[11]	Hartley, D. M.; Morris, J. G.; Smith, D. L., Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics?, PLoS Medicine, 3, 1, 63-69 (2006)
[12]	Shuai, Z.; van den Driessche, P., Global dynamics of cholera models with differential infectivity, Mathematical Biosciences, 234, 2, 118-126 (2011) · Zbl 1244.92040
[13]	Shuai, Z.; Tien, J. H.; van den Driessche, P., Cholera models with hyperinfectivity and temporary immunity, Bulletin of Mathematical Biology, 74, 10, 2423-2445 (2012) · Zbl 1312.92041
[14]	Jensen, M. A.; Faruque, S. M.; Mekalanos, J. J.; Levin, B. R., Modeling the role of bacteriophage in the control of cholera outbreaks, Proceedings of the National Academy of Sciences of the United States of America, 103, 12, 4652-4657 (2006) · Zbl 1355.92115
[15]	Sanches, R. P.; Ferreira, C. P.; Kraenkel, R. A., The role of immunity and seasonality in cholera epidemics, Bulletin of Mathematical Biology, 73, 12, 2916-2931 (2011) · Zbl 1251.92031
[16]	Mutreja, A.; Kim, D. W.; Thomson, N. R.; Connor, T. R.; Lee, J. H.; Kariuki, S.; Croucher, N. J.; Choi, S. Y.; Harris, S. R.; Lebens, M.; Niyogi, S. K.; Kim, E. J.; Ramamurthy, T.; Chun, J.; Wood, J. L. N.; Clemens, J. D.; Czerkinsky, C.; Nair, G. B.; Holmgren, J.; Parkhill, J.; Dougan, G., Evidence for several waves of global transmission in the seventh cholera pandemic, Nature, 477, 7365, 462-465 (2011)
[17]	Colwell, R. R., Global climate and infectious disease: the cholera paradigm, Science, 274, 5295, 2025-2031 (1996)
[18]	Faruque, S. M.; Islam, M. J.; Ahmad, Q. S.; Faruque, A. S. G.; Sack, D. A.; Nair, G. B.; Mekalanos, J. J., Self-limiting nature of seasonal cholera epidemics: role of host-mediated amplification of phage, Proceedings of the National Academy of Sciences of the United States of America, 102, 17, 6119-6124 (2005)
[19]	Aniţa, S.; Capasso, V., Stabilization of a reaction-diffusion system modelling a class of spatially structured epidemic systems via feedback control, Nonlinear Analysis. Real World Applications, 13, 2, 725-735 (2012) · Zbl 1238.93081
[20]	Capasso, V., Asymptotic stability for an integro-differential reaction-diffusion system, Journal of Mathematical Analysis and Applications, 103, 2, 575-588 (1984) · Zbl 0595.45020
[21]	Capasso, V.; Kunisch, K., A reaction-diffusion system arising in modelling man-environment diseases, Quarterly of Applied Mathematics, 46, 3, 431-450 (1988) · Zbl 0704.35069
[22]	Capasso, V.; Maddalena, L., Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, Journal of Mathematical Biology, 13, 2, 173-184 (1981-1982) · Zbl 0468.92016
[23]	Capasso, V.; Wilson, R. E., Analysis of a reaction-diffusion system modeling man-environment-man epidemics, SIAM Journal on Applied Mathematics, 57, 2, 327-346 (1997) · Zbl 0872.35053
[24]	Bertuzzo, E.; Azaele, S.; Maritan, A.; Gatto, M.; Rodriguez-Iturbe, I.; Rinaldo, A., On the space-time evolution of a cholera epidemic, Water Resources Research, 44, 1 (2008)
[25]	Bertuzzo, E.; Casagrandi, R.; Gatto, M.; Rodriguez-Iturbe, I.; Rinaldo, A., On spatially explicit models of cholera epidemics, Journal of the Royal Society Interface, 7, 43, 321-333 (2010)
[26]	Mari, L.; Bertuzzo, E.; Righetto, L.; Casagrandi, R.; Gatto, M.; Rodriguez-Iturbe, I.; Rinaldo, A., Modelling cholera epidemics: the role of waterways, human mobility and sanitation, Journal of the Royal Society Interface, 9, 67, 376-388 (2012)
[27]	Diekmann, O., Thresholds and travelling waves for the geographical spread of infection, Journal of Mathematical Biology, 6, 2, 109-130 (1978) · Zbl 0415.92020
[28]	Lewis, M.; Rencławowicz, J.; van den Driessche, P., Traveling waves and spread rates for a West Nile virus model, Bulletin of Mathematical Biology, 68, 1, 3-23 (2006) · Zbl 1334.92414
[29]	Radcliffe, J.; Rass, L., The asymptotic speed of propagation of the deterministic nonreducible \(n\)-type epidemic, Journal of Mathematical Biology, 23, 3, 341-359 (1986) · Zbl 0606.92019
[30]	Zhao, X.-Q.; Wang, W., Fisher waves in an epidemic model, Discrete and Continuous Dynamical Systems B, 4, 4, 1117-1128 (2004) · Zbl 1097.34022
[31]	Xu, D.; Zhao, X.-Q., Bistable waves in an epidemic model, Journal of Dynamics and Differential Equations, 16, 3, 679-707 (2004) · Zbl 1073.34027
[32]	Jin, Y.; Zhao, X.-Q., Bistable waves for a class of cooperative reaction-diffusion systems, Journal of Biological Dynamics, 2, 2, 196-207 (2008) · Zbl 1140.92028
[33]	Hsu, C.-H.; Yang, T.-S., Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26, 1, 121-139 (2013) · Zbl 1264.35255
[34]	Dunbar, S. R., Travelling wave solutions of diffusive Lotka-Volterra equations, Journal of Mathematical Biology, 17, 1, 11-32 (1983) · Zbl 0509.92024
[35]	Dunbar, S. R., Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in \(R^4\), Transactions of the American Mathematical Society, 286, 2, 557-594 (1984) · Zbl 0556.35078
[36]	Li, W.-T.; Wu, S.-L., Traveling waves in a diffusive predator-prey model with Holling type-III functional response, Chaos, Solitons & Fractals, 37, 2, 476-486 (2008) · Zbl 1155.37046
[37]	Hsu, C.-H.; Yang, C.-R.; Yang, T.-H.; Yang, T.-S., Existence of traveling wave solutions for diffusive predator-prey type systems, Journal of Differential Equations, 252, 4, 3040-3075 (2012) · Zbl 1252.34036
[38]	Huang, W., Traveling wave solutions for a class of predator-prey systems, Journal of Dynamics and Differential Equations, 24, 3, 633-644 (2012) · Zbl 1365.35056
[39]	Huang, J.; Lu, G.; Ruan, S., Existence of traveling wave solutions in a diffusive predator-prey model, Journal of Mathematical Biology, 46, 2, 132-152 (2003) · Zbl 1018.92026
[40]	Lin, X.; Weng, P.; Wu, C., Traveling wave solutions for a predator-prey system with sigmoidal response function, Journal of Dynamics and Differential Equations, 23, 4, 903-921 (2011) · Zbl 1252.34048
[41]	Irving, R. S., Integers, Polynomials, and Rings (2004), New York, NY, USA: Springer, New York, NY, USA
[42]	Ma, Z.; Zhou, Y., Qualitative and Stable Method of Ordinary Differential Equations (2005), Beijing, China: Science Press, Beijing, China