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Modelling and analysis of the effects of malnutrition in the spread of cholera. (English) Zbl 1219.92053

Summary: Although cholera has existed for ages, it has continued to plague many parts of the world. In this study, a deterministic model for cholera in a community is presented and rigorously analysed in order to determine the effects of malnutrition in the spread of the disease. The important mathematical features of the cholera model are thoroughly investigated. The epidemic threshold known as the basic reproductive number and equilibria for the model are determined, and stabilities are investigated. The disease-free equilibrium is shown to be globally asymptotically stable. Local stability of the endemic equilibrium is determined using centre manifold theory and conditions for its global stability are derived using a suitable Lyapunov function. Numerical simulations suggest that an increase in susceptibility to cholera due to malnutrition results in an increase in the number of cholera infected individuals in a community. The results suggest that nutritional issues should be addressed in impoverished communities affected by cholera in order to reduce the burden of the disease.

MSC:

92C60 Medical epidemiology
92C50 Medical applications (general)
37N25 Dynamical systems in biology
65C20 Probabilistic models, generic numerical methods in probability and statistics
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[1] Cholera 2005, Weekly epidemiological record, No. 31, 2006.; Cholera 2005, Weekly epidemiological record, No. 31, 2006.
[2] Jensen, M. A.; Faruque, S. M.; Mekalanos, J. J.; Levin, B. R., Modeling the role of bacteriophage in the control of cholera outbreaks, PNAS, 103, 4652-4657 (2006) · Zbl 1355.92115
[3] de Magny, G. C., Environmental signatures associated with cholera epidemics, PNAS, 105, 17676-17681 (2008)
[4] Mugoya, I.; Kariuki, S.; Galgalo, T., Rapid spread of vibrio cholera O1 throughout Kenya, Am. J. Trop. Med. Hyg., 77, 527-533 (2008)
[5] Cholera: risk factors. http://www.mayoclinic.com/health/cholera/ds00579/dsection=risk-factors; Cholera: risk factors. http://www.mayoclinic.com/health/cholera/ds00579/dsection=risk-factors
[6] E.G. Piwoza, E.A. Preble, HIV/AIDS and nutrition: a review of the literature and recommendation for nutritional care and support in sub-Saharan Africa, Support for Analysis and Research in Africa (SARA) Project Bureau for Africa, Office of Sustainable Development US Agency for International Development, 2000.; E.G. Piwoza, E.A. Preble, HIV/AIDS and nutrition: a review of the literature and recommendation for nutritional care and support in sub-Saharan Africa, Support for Analysis and Research in Africa (SARA) Project Bureau for Africa, Office of Sustainable Development US Agency for International Development, 2000.
[7] Gaffga, N. H.; Tauxe, R. V.; Mintz, E. D., Cholera: a new homeland in Africa?, Am. J. Trop. Med. Hyg., 77, 705-713 (2007)
[8] Somer, A.; Katz, J.; Tawrwotjo, I., Increased risk of respiratory disease and diarrhea in children with pre-existing mild vitamin A deficiency, Am. J. Clin. Nutr., 40, 1090-1095 (1984)
[9] Malnutrition, cholera bite DRC’s war-ravaged community, http://www.afrol.com/articles/26984; Malnutrition, cholera bite DRC’s war-ravaged community, http://www.afrol.com/articles/26984
[10] Malnutrition aids cholera epidemic in Malawi, WHO 2002, http://www.highbeam.com/doc/1P2-13299869.html; Malnutrition aids cholera epidemic in Malawi, WHO 2002, http://www.highbeam.com/doc/1P2-13299869.html
[11] Global News: Agencies: Zimbabwe malnutrition. http://www.philstar.com/Article.aspx?articleid=427501.html; Global News: Agencies: Zimbabwe malnutrition. http://www.philstar.com/Article.aspx?articleid=427501.html
[12] W.R. Beisel, Infection-induced malnutrition—from cholera to cytokines, Hermn Award Lecture, 1995.; W.R. Beisel, Infection-induced malnutrition—from cholera to cytokines, Hermn Award Lecture, 1995.
[13] Mahatanabis, D., Nitrogen balance during recovery from secretory diarrhea of cholera in children, Am. J. Clin. Nutr., 34, 1548-1551 (1981)
[14] Codeço, C. T., Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMJ Infect. Dis., 1 (2001)
[15] Hartley, D. M.; Morris, J. G.; Smith, D. L., Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics?, PLoS Med., 3, e7 (2006)
[16] Hyman, J. M.; Li, J., Disease transmission models with biased partnership selection, Appl. Numer. Math., 24, 379-392 (1997) · Zbl 0878.92024
[17] Hyman, J. M.; Li, J.; Stanley, E. A., The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155, 77-109 (1999) · Zbl 0942.92030
[18] Hyman, J. M.; Li, J., Differential susceptibility in epidemic models, J. Math. Biol., 50, 626-644 (2005) · Zbl 1066.92044
[19] Hyman, J. M.; Li, J., Differential susceptibility and infectivity epidemic models, Math. Biosci. Eng., 3, 89-100 (2006) · Zbl 1101.34034
[20] Nelson, E. J.; Harris, J. B.; Morris, J. G.; Calderwood, S. B.; Camilli, A., Cholera transmission: the host pathogen and bacteriophage dynamic, Nature Rev. Microbiol., 7, 693 (2009)
[21] Glass, R. I., Endemic cholera in rural Bangladesh, 1966-1980, Am. J. Epidemiol., 116, 959-970 (1982)
[22] Clemens, J. D., Biotype as determinant of natural immunising effect of cholera, Lancet, 337, 883-884 (1991)
[23] Koelle, K.; Pascual, M., Disentangling extrinsic from intrinsic factors in disease dynamics: a nonlinear time series approach with an application to cholera, Am. Nat., 163, 901-913 (2004)
[24] Levine, M. M., (Holme, T.; Holmgren, J.; Merson, M. H.; Mollby., R., Acute Enteric Infections in Children: New Prospects for Treatment and Prevention (1981), Elsevier/North-Holland Biomedical Press: Elsevier/North-Holland Biomedical Press Amsterdam), 443-459
[25] Cash, R. A., Response of man to infection with Vibrio cholerae. II. Protection from illness afforded by previous disease and vaccine, J. Infect. Dis., 130, 325-333 (1974)
[26] King, A. A.; Lonides, E. L.; Pascual, M.; Bouma, M. J., Inapparent infections and cholera dynamics, Nature, 454 (2008)
[27] Koelle, K.; Rodo, X.; Pascual, M.; Yunus, M.; Mostafa, G., Refractory periods and climate forcing in cholera dynamics, Nature, 436, 696-700 (2005)
[28] Islam, M. S.; Drasar, B.; Bradley, D. J., Survival of toxigenic Vibrio cholerae O1 with a common duckweed, Lemna minor, in artificial aquatic ecosystems, Trans. R. Soc. Trop. Med. Hyg., 84, 422-424 (1990)
[29] Anderson, R. M.; May, R. M., Infectious Diseases of Humans (1991), Oxford University Press: Oxford University Press London, New York
[30] Brauer, F.; Castillo-Chavez, C., Mathematical models in population biology and epidemiology, (Texts in Applied Mathematics Series, vol. 40 (2001), Springer-Verlag: Springer-Verlag New York) · Zbl 1302.92001
[31] C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of \(\mathcal{R}_0\) www.math.la.asu.edu/chavez/2002/JB276.pdf; C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of \(\mathcal{R}_0\) www.math.la.asu.edu/chavez/2002/JB276.pdf · Zbl 1021.92032
[32] Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 599-653 (2000) · Zbl 0993.92033
[33] Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., On the definition and computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28, 365-382 (1990) · Zbl 0726.92018
[34] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29-48 (2002) · Zbl 1015.92036
[35] Lakshmikantham, V.; Leela, S.; Martynyuk, A. A., Stability Analysis of Nonlinear Systems (1989), Marcel Dekker Inc.: Marcel Dekker Inc. New York · Zbl 0676.34003
[36] Carr, J., Applications of Centre Manifold Theory (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0464.58001
[37] Castillo-Chavez, C.; Song, B., Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1, 361-404 (2004) · Zbl 1060.92041
[38] Naresh, R.; Tripathi, A.; Sharma, D., Modelling and analysis of the spread of AIDS epidemic with immigration of HIV infectives, Math. Comput. Modelling, 49, 880-892 (2009) · Zbl 1165.34377
[39] Naresh, R.; Tripathi, A.; Biazar, J.; Sharma, D., Analysis of the effect of vaccination on the spread of AIDS epidemic using Adomian Decomposition Method, J. Nat. Sci. Sust. Tech., 2, 183-213 (2008)
[40] Tripathi, A.; Naresh, R.; Sharma, D., Modelling the effect of screening of unaware infectives on the spread of HIV infection, Appl. Math. Comput., 184, 1053-1968 (2007) · Zbl 1111.92051
[41] Mukandavire, Z.; Garira, W., HIV/AIDS model for assessing the effects of prophylactic sterilizing vaccines, condoms and treatment with amelioration, J. Biol. Syst., 14, 323-355 (2006) · Zbl 1116.92042
[42] Sepulveda, J.; Gomez-Dantes, H.; Bronfman, M., Cholera in the Americas: an overview, Infection, 20, 243-248 (1992)
[43] Chitnis, N.; Hyman, J. M.; Cushing, J. M., Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70, 1272-1296 (2008) · Zbl 1142.92025
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