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2018 | OriginalPaper | Buchkapitel

Possibility and Causes of Backward Bifurcation in a Cholera Model

verfasst von : Sandeep Sharma, Nitu Kumari

Erschienen in: Applications of Fluid Dynamics

Verlag: Springer Singapore

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Abstract

Backward bifurcation in an epidemiological model is a phenomenon in which the model possesses stable endemic equilibria together with a stable disease-free equilibrium. Till now, this phenomenon has been observed in a number of epidemic models. In this work, we investigate the possibility of backward bifurcation in a cholera model. We also explore the role of various factors, which induce backward bifurcation in other epidemic models. We believe the present work provides an insight of the dynamics of a cholera model and possible causes of backward bifurcation in the same.

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Literatur
Zurück zum Zitat Alexander ME, Moghadas SM (2004) Periodicity in an epidemic model with a generalized non-linear incidence. Math Biosci 189(1):75–96CrossRefMATHMathSciNet Alexander ME, Moghadas SM (2004) Periodicity in an epidemic model with a generalized non-linear incidence. Math Biosci 189(1):75–96CrossRefMATHMathSciNet
Zurück zum Zitat Arino J, McCluskey CC, Van den Driessche P (2003) Global results for an epidemic model with vaccination that exhibits backward bifurcation. SIAM J Appl Math 64(1):260–276 Arino J, McCluskey CC, Van den Driessche P (2003) Global results for an epidemic model with vaccination that exhibits backward bifurcation. SIAM J Appl Math 64(1):260–276
Zurück zum Zitat Buonomo B, Lacitignola D (2011) On the backward bifurcation of a vaccination model with nonlinear incidence. Nonlinear Anal: Model Control 16(1):30–46MATHMathSciNet Buonomo B, Lacitignola D (2011) On the backward bifurcation of a vaccination model with nonlinear incidence. Nonlinear Anal: Model Control 16(1):30–46MATHMathSciNet
Zurück zum Zitat Buonomo B, Lacitignola D (2010) Analysis of a tuberculosis model with a case study in Uganda. J Biol Dyn 4(6):571–593CrossRefMathSciNet Buonomo B, Lacitignola D (2010) Analysis of a tuberculosis model with a case study in Uganda. J Biol Dyn 4(6):571–593CrossRefMathSciNet
Zurück zum Zitat Buonomo B, Lacitignola D (2012) Forces of infection allowing for backward bifurcation in an epidemic model with vaccination and treatment. Acta Applicandae Mathematicae 122(1):283–293MATHMathSciNet Buonomo B, Lacitignola D (2012) Forces of infection allowing for backward bifurcation in an epidemic model with vaccination and treatment. Acta Applicandae Mathematicae 122(1):283–293MATHMathSciNet
Zurück zum Zitat Buonomo B, De-León CV (2013) Stability and bifurcation analysis of a vector-bias model of malaria transmission. Math Biosci 242(1):59–67 Buonomo B, De-León CV (2013) Stability and bifurcation analysis of a vector-bias model of malaria transmission. Math Biosci 242(1):59–67
Zurück zum Zitat Capasso V, Fontana PSL (1979) A mathematical model for the 1973 cholera epidemic in the European mediterranean region. Revued’épid émiologie et de Santé Publiqué 27(2):121–132 Capasso V, Fontana PSL (1979) A mathematical model for the 1973 cholera epidemic in the European mediterranean region. Revued’épid émiologie et de Santé Publiqué 27(2):121–132
Zurück zum Zitat Chitnis N, Cushing JM, Hyman JM (2006) Bifurcation analysis of a mathematical model for malaria transmission. SIAM J Appl Math 67(1):24–45CrossRefMATHMathSciNet Chitnis N, Cushing JM, Hyman JM (2006) Bifurcation analysis of a mathematical model for malaria transmission. SIAM J Appl Math 67(1):24–45CrossRefMATHMathSciNet
Zurück zum Zitat Codeço CT (2001) Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. BMC Infect Dis 1(1):1CrossRef Codeço CT (2001) Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. BMC Infect Dis 1(1):1CrossRef
Zurück zum Zitat Cui J, Wu Z, Zhou X (2014) Mathematical analysis of a cholera model with vaccination. J Appl Math article ID324767, 16 pp Cui J, Wu Z, Zhou X (2014) Mathematical analysis of a cholera model with vaccination. J Appl Math article ID324767, 16 pp
Zurück zum Zitat Dushoff J, Huang W, Chavez CC (1998) Backwards bifurcations and catastrophe in simple models of fatal diseases. J Math Biol 36(3):227–248CrossRefMATHMathSciNet Dushoff J, Huang W, Chavez CC (1998) Backwards bifurcations and catastrophe in simple models of fatal diseases. J Math Biol 36(3):227–248CrossRefMATHMathSciNet
Zurück zum Zitat Gerberry DJ (2016) Practical aspects of backward bifurcation in a mathematical model for tuberculosis. J Theor Biol 388:15–36CrossRefMATHMathSciNet Gerberry DJ (2016) Practical aspects of backward bifurcation in a mathematical model for tuberculosis. J Theor Biol 388:15–36CrossRefMATHMathSciNet
Zurück zum Zitat Hartley DM, Morris JG Jr, Smith DL (2005) Hyperinfectivity: a critical element in the ability of v. cholerae to cause epidemics? PLoS Med 3(1):e7CrossRef Hartley DM, Morris JG Jr, Smith DL (2005) Hyperinfectivity: a critical element in the ability of v. cholerae to cause epidemics? PLoS Med 3(1):e7CrossRef
Zurück zum Zitat Kribs-Zaleta CM, Velasco-Hernandez JX (2000) A simple vaccination model with multiple endemic states. Math Biosci 164(2):183–201CrossRefMATH Kribs-Zaleta CM, Velasco-Hernandez JX (2000) A simple vaccination model with multiple endemic states. Math Biosci 164(2):183–201CrossRefMATH
Zurück zum Zitat Misra AK, Mishra SN, Pathak AL, Misra P, Naresh R (2012) Modeling the effect of time delay in controlling the carrier dependent infectious disease–cholera. Appl Math Comput 218(23):11547–11557MATHMathSciNet Misra AK, Mishra SN, Pathak AL, Misra P, Naresh R (2012) Modeling the effect of time delay in controlling the carrier dependent infectious disease–cholera. Appl Math Comput 218(23):11547–11557MATHMathSciNet
Zurück zum Zitat Misra AK, Singh V (2012) A delay mathematical model for the spread and control of water borne diseases. J Theor Biol 301:49–56CrossRefMathSciNet Misra AK, Singh V (2012) A delay mathematical model for the spread and control of water borne diseases. J Theor Biol 301:49–56CrossRefMathSciNet
Zurück zum Zitat Mukandavire Z, Gumel AB, Garira W, Tchuenche JM (2009) Mathematical analysis of a model for HIV-malaria co-infection. Math Biosci Eng 6(2):333–362CrossRefMATHMathSciNet Mukandavire Z, Gumel AB, Garira W, Tchuenche JM (2009) Mathematical analysis of a model for HIV-malaria co-infection. Math Biosci Eng 6(2):333–362CrossRefMATHMathSciNet
Zurück zum Zitat Mukandavire Z, Liao S, Wang J, Gaff H, Smith DL, Morris JG (2011) Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe. Proc Natl Acad Sci 108(21):8767–8772CrossRef Mukandavire Z, Liao S, Wang J, Gaff H, Smith DL, Morris JG (2011) Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe. Proc Natl Acad Sci 108(21):8767–8772CrossRef
Zurück zum Zitat Mwasa A, Tchuenche JM (2011) Mathematical analysis of a cholera model with public health interventions. Biosystems 105(3):190–200CrossRef Mwasa A, Tchuenche JM (2011) Mathematical analysis of a cholera model with public health interventions. Biosystems 105(3):190–200CrossRef
Zurück zum Zitat Posny D, Wang J, Mukandavire Z, Modnak C (2015) Analyzing transmission dynamics of cholera with public health interventions. Math Biosci 264:38–53CrossRefMATHMathSciNet Posny D, Wang J, Mukandavire Z, Modnak C (2015) Analyzing transmission dynamics of cholera with public health interventions. Math Biosci 264:38–53CrossRefMATHMathSciNet
Zurück zum Zitat Safi MA, Melesse DY, Gumel AB (2013) Dynamics analysis of a multi-strain cholera model with an imperfect vaccine. Bull Math Biol 75(7):1104–1137CrossRefMATHMathSciNet Safi MA, Melesse DY, Gumel AB (2013) Dynamics analysis of a multi-strain cholera model with an imperfect vaccine. Bull Math Biol 75(7):1104–1137CrossRefMATHMathSciNet
Zurück zum Zitat Sanchez JL, Vasquez B, Begue RE, Meza R, Castellares G, Cabezas C, Watts DM, Svennerholm AM, Sadoff JC, Taylor DN (1994) Protective efficacy of oral whole-cell/recombinant-b-subunit cholera vaccine in Peruvian military recruits. Lancet 344(8932):1273–1276CrossRef Sanchez JL, Vasquez B, Begue RE, Meza R, Castellares G, Cabezas C, Watts DM, Svennerholm AM, Sadoff JC, Taylor DN (1994) Protective efficacy of oral whole-cell/recombinant-b-subunit cholera vaccine in Peruvian military recruits. Lancet 344(8932):1273–1276CrossRef
Zurück zum Zitat Sharomi O, Gumel AB (2009) Re-infection-induced backward bifurcation in the transmission dynamics of chlamydia trachomatis. J Math Anal Appl 356(1):96–118CrossRefMATHMathSciNet Sharomi O, Gumel AB (2009) Re-infection-induced backward bifurcation in the transmission dynamics of chlamydia trachomatis. J Math Anal Appl 356(1):96–118CrossRefMATHMathSciNet
Zurück zum Zitat Sharomi O, Gumel AB (2011a) Dynamical analysis of a sex-structured chlamydia trachomatis transmission model with time delay. Nonlinear Anal: Real World Appl 12(2):837–866CrossRefMATHMathSciNet Sharomi O, Gumel AB (2011a) Dynamical analysis of a sex-structured chlamydia trachomatis transmission model with time delay. Nonlinear Anal: Real World Appl 12(2):837–866CrossRefMATHMathSciNet
Zurück zum Zitat Sharomi O, Gumel AB (2011b) Mathematical study of a risk-structured two group model for chlamydia transmission dynamics. Appl Math Model 35(8):3653–3673CrossRefMATHMathSciNet Sharomi O, Gumel AB (2011b) Mathematical study of a risk-structured two group model for chlamydia transmission dynamics. Appl Math Model 35(8):3653–3673CrossRefMATHMathSciNet
Zurück zum Zitat Sharomi O, Podder CN, Gumel AB, Elbasha EH, Watmough J (2007) Role of incidence function in vaccine-induced backward bifurcation in some HIV models. Math Biosci 210(2):436–463CrossRefMATHMathSciNet Sharomi O, Podder CN, Gumel AB, Elbasha EH, Watmough J (2007) Role of incidence function in vaccine-induced backward bifurcation in some HIV models. Math Biosci 210(2):436–463CrossRefMATHMathSciNet
Zurück zum Zitat Van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1):29–48CrossRefMATHMathSciNet Van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1):29–48CrossRefMATHMathSciNet
Zurück zum Zitat Zhang X, Liu X (2008) Backward bifurcation of an epidemic model with saturated treatment function. J Math Anal Appl 348(1):433–443CrossRefMATHMathSciNet Zhang X, Liu X (2008) Backward bifurcation of an epidemic model with saturated treatment function. J Math Anal Appl 348(1):433–443CrossRefMATHMathSciNet
Zurück zum Zitat Zhou XY, Cui J, Zhang ZH (2012) Global results for a cholera model with imperfect vaccination. J Franklin Inst 349(3):770–791CrossRefMATHMathSciNet Zhou XY, Cui J, Zhang ZH (2012) Global results for a cholera model with imperfect vaccination. J Franklin Inst 349(3):770–791CrossRefMATHMathSciNet
Zurück zum Zitat Zhou X, Cui J (2011) Modeling and stability analysis for a cholera model with vaccination. Math Methods Appl Sci 34(14):1711–1724MATHMathSciNet Zhou X, Cui J (2011) Modeling and stability analysis for a cholera model with vaccination. Math Methods Appl Sci 34(14):1711–1724MATHMathSciNet
Zurück zum Zitat Zhou X, Shi X, Cui J (2017) Stability and backward bifurcation on a cholera epidemic model with saturated recovery rate. Math Methods Appl Sci 40(4):1288–1306 Zhou X, Shi X, Cui J (2017) Stability and backward bifurcation on a cholera epidemic model with saturated recovery rate. Math Methods Appl Sci 40(4):1288–1306
Metadaten
Titel
Possibility and Causes of Backward Bifurcation in a Cholera Model
verfasst von
Sandeep Sharma
Nitu Kumari
Copyright-Jahr
2018
Verlag
Springer Singapore
DOI
https://doi.org/10.1007/978-981-10-5329-0_51

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