Modelling and Analysis of the Intrinsic Dynamics of Cholera

Z. Mukandavire, A. Tripathi, C. Chiyaka, G. Musuka, F. Nyabadza, H. G. Mwambi

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

A simple mathematical model for cholera is presented using a system of ordinary differential equations. Comprehensive analysis of the important mathematical features of the model is carried out. The disease-free and endemic equilibria are obtained and their local stability investigated. We use the centre manifold theory to show the stability of the endemic equilibrium and suitable Lyapunov function for its global stability. Qualitative analysis of the model including positivity and boundedness of solutions are also presented. The cholera model is numerically analysed using published data to explore the effects of the recovery rate, rate of exposure to contaminated water and contribution of infected individuals to the population of Vibrio cholerae in the aquatic environment on the cumulative number of cholera infected individuals. The results demonstrate that proper management of the diseases will reduce the burden of cholera in endemic areas.

Original languageEnglish
Pages (from-to)253-265
Number of pages13
JournalDifferential Equations and Dynamical Systems
Volume19
Issue number3
Early online date22 Apr 2011
DOIs
Publication statusPublished - 1 Jul 2011
Externally publishedYes

Fingerprint

Endemic Equilibrium
Modeling
Boundedness of Solutions
Center Manifold
Local Stability
Qualitative Analysis
Lyapunov functions
Global Stability
System of Ordinary Differential Equations
Ordinary differential equations
Positivity
Lyapunov Function
Recovery
Model
Mathematical Model
Mathematical models
Water
Demonstrate

Keywords

  • Cholera model
  • Equilibria
  • Reproduction number
  • Stability

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Modelling and Analysis of the Intrinsic Dynamics of Cholera. / Mukandavire, Z.; Tripathi, A.; Chiyaka, C.; Musuka, G.; Nyabadza, F.; Mwambi, H. G.

In: Differential Equations and Dynamical Systems, Vol. 19, No. 3, 01.07.2011, p. 253-265.

Research output: Contribution to journalArticle

Mukandavire, Z. ; Tripathi, A. ; Chiyaka, C. ; Musuka, G. ; Nyabadza, F. ; Mwambi, H. G. / Modelling and Analysis of the Intrinsic Dynamics of Cholera. In: Differential Equations and Dynamical Systems. 2011 ; Vol. 19, No. 3. pp. 253-265.
@article{993591e7f43c4b698f42ff7123d4b3a3,
title = "Modelling and Analysis of the Intrinsic Dynamics of Cholera",
abstract = "A simple mathematical model for cholera is presented using a system of ordinary differential equations. Comprehensive analysis of the important mathematical features of the model is carried out. The disease-free and endemic equilibria are obtained and their local stability investigated. We use the centre manifold theory to show the stability of the endemic equilibrium and suitable Lyapunov function for its global stability. Qualitative analysis of the model including positivity and boundedness of solutions are also presented. The cholera model is numerically analysed using published data to explore the effects of the recovery rate, rate of exposure to contaminated water and contribution of infected individuals to the population of Vibrio cholerae in the aquatic environment on the cumulative number of cholera infected individuals. The results demonstrate that proper management of the diseases will reduce the burden of cholera in endemic areas.",
keywords = "Cholera model, Equilibria, Reproduction number, Stability",
author = "Z. Mukandavire and A. Tripathi and C. Chiyaka and G. Musuka and F. Nyabadza and Mwambi, {H. G.}",
year = "2011",
month = "7",
day = "1",
doi = "10.1007/s12591-011-0087-1",
language = "English",
volume = "19",
pages = "253--265",
journal = "Differential Equations and Dynamical Systems",
issn = "0971-3514",
publisher = "Springer Verlag",
number = "3",

}

TY - JOUR

T1 - Modelling and Analysis of the Intrinsic Dynamics of Cholera

AU - Mukandavire, Z.

AU - Tripathi, A.

AU - Chiyaka, C.

AU - Musuka, G.

AU - Nyabadza, F.

AU - Mwambi, H. G.

PY - 2011/7/1

Y1 - 2011/7/1

N2 - A simple mathematical model for cholera is presented using a system of ordinary differential equations. Comprehensive analysis of the important mathematical features of the model is carried out. The disease-free and endemic equilibria are obtained and their local stability investigated. We use the centre manifold theory to show the stability of the endemic equilibrium and suitable Lyapunov function for its global stability. Qualitative analysis of the model including positivity and boundedness of solutions are also presented. The cholera model is numerically analysed using published data to explore the effects of the recovery rate, rate of exposure to contaminated water and contribution of infected individuals to the population of Vibrio cholerae in the aquatic environment on the cumulative number of cholera infected individuals. The results demonstrate that proper management of the diseases will reduce the burden of cholera in endemic areas.

AB - A simple mathematical model for cholera is presented using a system of ordinary differential equations. Comprehensive analysis of the important mathematical features of the model is carried out. The disease-free and endemic equilibria are obtained and their local stability investigated. We use the centre manifold theory to show the stability of the endemic equilibrium and suitable Lyapunov function for its global stability. Qualitative analysis of the model including positivity and boundedness of solutions are also presented. The cholera model is numerically analysed using published data to explore the effects of the recovery rate, rate of exposure to contaminated water and contribution of infected individuals to the population of Vibrio cholerae in the aquatic environment on the cumulative number of cholera infected individuals. The results demonstrate that proper management of the diseases will reduce the burden of cholera in endemic areas.

KW - Cholera model

KW - Equilibria

KW - Reproduction number

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=84859725228&partnerID=8YFLogxK

U2 - 10.1007/s12591-011-0087-1

DO - 10.1007/s12591-011-0087-1

M3 - Article

VL - 19

SP - 253

EP - 265

JO - Differential Equations and Dynamical Systems

JF - Differential Equations and Dynamical Systems

SN - 0971-3514

IS - 3

ER -