Bifurcation and Stability Analysis of the Dynamics of Cholera Model with Controls
Cholera is a disease that is predominately common in
developing countries due to poor sanitation and overcrowding
population. In this paper, a deterministic model for the dynamics of
cholera is developed and control measures such as health educational
message, therapeutic treatment, and vaccination are incorporated in
the model. The effective reproduction number is computed in terms
of the model parameters. The existence and stability of the
equilibrium states, disease free and endemic equilibrium states are
established and showed to be locally and globally asymptotically
stable when R0 < 1 and R0 > 1 respectively. The existence of
backward bifurcation of the model is investigated. Furthermore,
numerical simulation of the model developed is carried out to show
the impact of the control measures and the result indicates that
combined control measures will help to reduce the spread of cholera
in the population.
[1] O.A. Adagbada, S.A. Adesida, F.O. Nwaokorie, M. Niemogha, and A.O.
Coker, “Cholera epidemiology in Nigeria: An overview”, Pan African
Medical Journal, vol. 12, no. 59, 2012.
[2] C. Chin, J. Sorenson, J.B. Harris, W.R. Robins, R.C. Charles, R.R. Jean-
Charles, et al, “The origin of the Haitian cholera outbreak strain”, The
New England Journal of Medicine, vol. 364, pp. 33 – 42, 2011.
[3] F.J. Luquero, L. Grout, I. Ciglenecki, K. Sakoba, B. Traore, M. Helie, et
al, “Use of vibrio cholerae vaccine in an outbreak in Guinea”, The New
England Journal of Medicine, vol. 370, pp. 2111 – 2120, 2014.
[4] H.D.N. Nyamogoba, A. A. Obala, and R. Kakai, “Combating cholera
epidemics by targeting reservoirs of infection and transmission routes: A
review”, East African Medical Journal, vol. 79, no. 3, pp. 150 – 155,
March 2002.
[5] World Health Organisation, “Cholera 2013”, Weekly epidemiological
record, vol. 89, no. 31, pp. 345 -356, 2014.
[6] S.M. Farugue, M.J. Albert, and J.J. Mekalanos, “Epidemiology,
genetics, and ecology of toxigenic vibrio cholerae”, Microbiology and
Molecular Biology Reviews, vol. 62, no. 4, pp. 1301 – 1314, 1998.
[7] J. Einardóttir, A. Passa, and G. Gunnlaugsson, “Health education and
cholera in rural Guinea-bissau”, Int. J. Infect. Dis., vol. 5, no. 3, pp. 133
– 138, 2001.
[8] H. W. Hethcote, “The Mathematics of Infectious Diseases”, SIAM
REVIEW, vol. 42, no. 4, pp. 599-653, 2000.
[9] M.J. Ochoche, C.E. Madubueze, and T.B. Akaabo, “A mathematical
model on the control of cholera: hygiene consciousness as a strategy”, J.
Math. Comput. Sci., vol. 5, no. 2, pp. 172 – 187.
[10] M. Al-arydah, A. Mwasa, J.M. Tchuenche, and R.J. Smith, “Modelling
cholera disease with education and chlorination”, Journal of Biological
System, Vol. 21, no. 4, 1340007 (20 pp.), 2013.
[11] J. Cui, Z. Wu, and X. Zhou, “Mathematical analysis of a cholera model
with vaccination”, Journal of Applied Mathematics, vol. 2014, article
ID324767, 16 pp., 2014.
[12] J. Wang and C. “Modnak, Modeling cholera dynamics with controls,”
Canadian Applied Mathematics Quarterly, vol. 19, no. 3, pp. 255 – 273,
2011.
[13] P. Van Den Driessche, and J. Watmough, “Reproduction numbers and
sub-thresholds endemic equilibrium for compartmental models of
disease transmission”, Mathematical Bioscience, vol. 180, no. 2002, pp.
29 – 48, 2002.
[14] C. Castillo-Chavez, Z. Feng, and W. Huang, “On the computation of
[1] O.A. Adagbada, S.A. Adesida, F.O. Nwaokorie, M. Niemogha, and A.O.
Coker, “Cholera epidemiology in Nigeria: An overview”, Pan African
Medical Journal, vol. 12, no. 59, 2012.
[2] C. Chin, J. Sorenson, J.B. Harris, W.R. Robins, R.C. Charles, R.R. Jean-
Charles, et al, “The origin of the Haitian cholera outbreak strain”, The
New England Journal of Medicine, vol. 364, pp. 33 – 42, 2011.
[3] F.J. Luquero, L. Grout, I. Ciglenecki, K. Sakoba, B. Traore, M. Helie, et
al, “Use of vibrio cholerae vaccine in an outbreak in Guinea”, The New
England Journal of Medicine, vol. 370, pp. 2111 – 2120, 2014.
[4] H.D.N. Nyamogoba, A. A. Obala, and R. Kakai, “Combating cholera
epidemics by targeting reservoirs of infection and transmission routes: A
review”, East African Medical Journal, vol. 79, no. 3, pp. 150 – 155,
March 2002.
[5] World Health Organisation, “Cholera 2013”, Weekly epidemiological
record, vol. 89, no. 31, pp. 345 -356, 2014.
[6] S.M. Farugue, M.J. Albert, and J.J. Mekalanos, “Epidemiology,
genetics, and ecology of toxigenic vibrio cholerae”, Microbiology and
Molecular Biology Reviews, vol. 62, no. 4, pp. 1301 – 1314, 1998.
[7] J. Einardóttir, A. Passa, and G. Gunnlaugsson, “Health education and
cholera in rural Guinea-bissau”, Int. J. Infect. Dis., vol. 5, no. 3, pp. 133
– 138, 2001.
[8] H. W. Hethcote, “The Mathematics of Infectious Diseases”, SIAM
REVIEW, vol. 42, no. 4, pp. 599-653, 2000.
[9] M.J. Ochoche, C.E. Madubueze, and T.B. Akaabo, “A mathematical
model on the control of cholera: hygiene consciousness as a strategy”, J.
Math. Comput. Sci., vol. 5, no. 2, pp. 172 – 187.
[10] M. Al-arydah, A. Mwasa, J.M. Tchuenche, and R.J. Smith, “Modelling
cholera disease with education and chlorination”, Journal of Biological
System, Vol. 21, no. 4, 1340007 (20 pp.), 2013.
[11] J. Cui, Z. Wu, and X. Zhou, “Mathematical analysis of a cholera model
with vaccination”, Journal of Applied Mathematics, vol. 2014, article
ID324767, 16 pp., 2014.
[12] J. Wang and C. “Modnak, Modeling cholera dynamics with controls,”
Canadian Applied Mathematics Quarterly, vol. 19, no. 3, pp. 255 – 273,
2011.
[13] P. Van Den Driessche, and J. Watmough, “Reproduction numbers and
sub-thresholds endemic equilibrium for compartmental models of
disease transmission”, Mathematical Bioscience, vol. 180, no. 2002, pp.
29 – 48, 2002.
[14] C. Castillo-Chavez, Z. Feng, and W. Huang, “On the computation of
@article{"International Journal of Engineering, Mathematical and Physical Sciences:71147", author = "C. E. Madubueze and S. C. Madubueze and S. Ajama", title = "Bifurcation and Stability Analysis of the Dynamics of Cholera Model with Controls", abstract = "Cholera is a disease that is predominately common in
developing countries due to poor sanitation and overcrowding
population. In this paper, a deterministic model for the dynamics of
cholera is developed and control measures such as health educational
message, therapeutic treatment, and vaccination are incorporated in
the model. The effective reproduction number is computed in terms
of the model parameters. The existence and stability of the
equilibrium states, disease free and endemic equilibrium states are
established and showed to be locally and globally asymptotically
stable when R0 < 1 and R0 > 1 respectively. The existence of
backward bifurcation of the model is investigated. Furthermore,
numerical simulation of the model developed is carried out to show
the impact of the control measures and the result indicates that
combined control measures will help to reduce the spread of cholera
in the population.", keywords = "Backward bifurcation, cholera, equilibrium,
dynamics, stability.", volume = "9", number = "11", pages = "640-7", }