×

Dynamical behavior of a stochastic epidemic model for cholera. (English) Zbl 1418.92179

Summary: In this paper, a stochastic epidemic model for cholera is proposed and investigated. Firstly, we establish sufficient conditions for extinction of the disease. Then we establish sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. The existence of an ergodic stationary distribution implies that all the individuals can be coexistent in the long run. Finally, some examples together with numerical simulations are introduced to illustrate our theoretical results.

MSC:

92D30 Epidemiology
93E03 Stochastic systems in control theory (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] ao, A. P.L.-P.; Silva, C. J.; Torres, D. F.M., An epidemic model for cholera with optimal control treatment, J. Comput. Appl. Math., 318, 168-180 (2017) · Zbl 1362.34078
[3] Shuai, Z.; Tien, J. H.; van den Driessche, P., Cholera models with hyperinfectivity and temporary immunity, Bull. Math. Biol., 74, 2423-2445 (2012) · Zbl 1312.92041
[4] Neilan, R. L.M.; Schaefer, E.; Gaff, H.; Fister, K. R.; Lenhart, S., Modeling optimal intervention strategies for cholera, Bull. Math. Biol., 72, 2004-2018 (2010) · Zbl 1201.92045
[5] Capasso, V.; Paveri-Fontana, S. L., A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. Epidemiol. Sante Publ., 27, 1979, 121-132 (1973)
[6] Mwasa, A.; Tchuenche, J. M., Mathematical analysis of a cholera model with public health interventions, Bull. Math. Biol., 105, 190-200 (2011)
[7] Codeço, C. T., Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infect. Dis., 1, 1 (2001)
[8] 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, 63-69 (2006)
[9] Capone, F.; De Cataldis, V.; De Luca, R., Influence of diffusion on the stability of equilibria in a reaction diffusion system modeling cholera dynamic, J. Math. Biol., 71, 1107-1131 (2015) · Zbl 1355.92107
[10] Hove-Musekwa, S. D.; Nyabadza, F.; Chiyaka, C.; Das, P.; Tripathi, A.; Mukandavire, Z., Modelling and analysis of the effects of malnutrition in the spread of cholera, Math. Comput. Model., 53, 1583-1595 (2010) · Zbl 1219.92053
[11] Joh, R. I.; Wang, H.; Weiss, H.; Weitza, J. S., Dynamics of indirectly transmitted infectious diseases with immunological threshold, Bull. Math. Biol., 71, 845-862 (2009) · Zbl 1163.92026
[12] Mukandavire, Z.; Mutasa, F. K.; Hove-Musekwa, S. D.; Dube, S.; Tchuenche, J. M., Mathematical analysis of a cholera model with carriers and assessing the effects of treatment, (Wilson, L. B., Mathematical Biology Research Trends (2008), Nova Science Publishers), 1-37
[13] Pascual, M.; Chaves, L. F.; Cash, B.; Rodo, X.; Yunus, M. D., Predicting endemic cholera: the role of climate variability and disease dynamics, Clim. Res., 36, 131-140 (2008)
[15] Matovinovic, J., A short history of quarantine (Victor c. Vaughan), Univ. Mich. Med. Cent. J., 35, 224-228 (1969)
[16] Dang, N. H.; Du, N. H.; Yin, G., Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differ. Equ., 257, 2078-2101 (2014) · Zbl 1329.60176
[17] Allen, L. J.S., A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis, Infect. Dis. Model., 2, 128-142 (2017)
[18] Altizer, S.; Ostfeld, R.; Johnston, P. T.J.; Kutz, S.; Harvell, C. D., Climate change and infectious diseases: from evidence to a predictive framework, Science, 341, 6145, 514-519 (2013)
[19] Jutla, A.; Whitcombe, E.; Hasan, N.; Haley, B.; Akanda, A.; Huq, A., Environmental factors influencing epidemic cholera, Am. J. Trop. Med. Hyg., 89, 3, 597-607 (2013)
[20] Wu, X.; Lu, Y.; Zhou, S.; Chen, L.; Xu, B., Impact of climate change on human infectious diseases: Empirical evidence and human adaptation, Environ. Int., 86, 14-23 (2016)
[21] Cai, Y.; Kang, Y.; Banerjee, M.; Wang, W., A stochastic epidemic model incorporating media coverage, Commun. Math. Sci., 14, 893-910 (2015) · Zbl 1344.92155
[22] Lahrouz, A.; Omari, L., Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Stat. Probab. Lett., 83, 960-968 (2013) · Zbl 1402.92396
[23] Hieu, N. T.; Du, N. H.; Auger, P.; Dang, N. H., Dynamical behavior of a stochastic SIRS epidemic model, Math. Model. Nat. Phenom., 10, 56-73 (2015) · Zbl 1337.34046
[24] Zhang, X.; Jiang, D.; Alsaedi, A.; Hayat, T., Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching, Appl. Math. Lett., 59, 87-93 (2016) · Zbl 1343.60095
[25] Cai, Y.; Kang, Y.; Banerjee, M.; Wang, W., A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differ. Equ., 259, 7463-7502 (2015) · Zbl 1330.35464
[26] Imhof, L.; Walcher, S., Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equ., 217, 26-53 (2005) · Zbl 1089.34041
[27] Mao, X., Stochastic Differential Equations and Applications (1997), Horwood Publishing: Horwood Publishing Chichester · Zbl 0874.60050
[28] Liu, Q.; Jiang, D.; Shi, N.; Hayat, T.; Alsaedi, A., Stationary distribution and extinction of a stochastic SIRS epidemic model with standard incidence, Phys. A, 469, 510-517 (2017) · Zbl 1400.92521
[29] Kutoyants, A. Y., Statistical Inference for Ergodic Diffusion Processes (2003), Springer: Springer London
[30] Peng, S.; Zhu, X., Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stoch. Process. Appl., 116, 370-380 (2006) · Zbl 1096.60026
[31] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), Academic Press: Academic Press New York · Zbl 0484.15016
[32] R.Z. Has’minskii, Stochastic Stability of Differential Equations, 1980, Sijthoff Noordhoff, Alphen aan den Rijn, The Netherlands.; R.Z. Has’minskii, Stochastic Stability of Differential Equations, 1980, Sijthoff Noordhoff, Alphen aan den Rijn, The Netherlands.
[33] Higham, D. J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43, 525-546 (2001) · Zbl 0979.65007
[34] Index Mundi, 2015, http://www.indexmundi.com/g/g.aspx?c=haandv=25; Index Mundi, 2015, http://www.indexmundi.com/g/g.aspx?c=haandv=25
[35] Sanches, R. P.; Ferreira, C. P.; Kraenkel, R. A., The role of immunity and seasonality in cholera epidemics, Bull. Math. Biol., 73, 2916-2931 (2011) · Zbl 1251.92031
[36] I. Mundi, 2015, http://www.indexmundi.com/g/g.aspx?c=haandv=26; I. Mundi, 2015, http://www.indexmundi.com/g/g.aspx?c=haandv=26
[37] World health organization, global task force on cholera control, cholera country profile: Haiti, 2011, http://www.who.int/cholera/countries/; World health organization, global task force on cholera control, cholera country profile: Haiti, 2011, http://www.who.int/cholera/countries/
[38] Bainov, D.; Simeonov, P., Impulsive Differential Equations: Periodic Solutions and Applications (1993), Longman, Harlow · Zbl 0815.34001
[39] Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of Impulsive Differential Equations (1989), World Scientific Press: World Scientific Press Singapore · Zbl 0719.34002
[40] Wang, Y.; Cao, J., Global dynamics of a network epidemic model for waterborne diseases spread, Appl. Math. Comput., 237, 474-488 (2014) · Zbl 1334.92433
[41] Wang, Y.; Cao, J., Global stability of general cholera models with nonlinear incidence and removal rates, J. Frankl. Inst., 352, 2464-2485 (2015) · Zbl 1395.92172
[42] Durrett, R., Stochastic Calculus (1996), CRC Press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.