The effects of an imperfect vaccine on cholera control

Jing’an Cui, Zhanmin Wu, Guohua Song

School of Science, Beijing University of Civil Engineering and Architecture, Beijing, China.

DOI: 10.4236/abb.2013.43052                

Abstract

In this paper, we consider a SVIR-B cholera model with imperfect vaccination. By analyzing the corresponding characteristic equations, the local asymptotically stability of a disease-free equilibrium and an endemic equilibrium is established. We calculate the certain threshold known as the basic reproduction number R_v. If R_v < 1, we obtain sufficient conditions for the global asymptotically stability of the disease- free equilibrium, the diseases will be eliminated from the community. By comparison arguments, it is proved that if R_v > 1, the unique endemic equilibrium is local asymptotically stable. We perform sensitivity analysis of R_v on the parameters in order to determine their relative importance to disease control and show that an imperfect vaccine is always beneficial in reducing disease spread within the community.

Keywords

Cholera Model; Stability; The Basic Reproduction Number; Sensitivity Analysis

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Kang, Y. and Zhang, H.Z. (2005) The study status of oral cholera vaccine. Chinese Frontier Health Quarantine, 28, 91-92.
[2]	Liao, S. and Wang, J. (2007) Global stability analysis of epidemiological models based on Volterra-Lyapunov stable matrices. Chaos, Solitons & Fractals, 45, 966-977. doi:10.1016/j.chaos.2012.03.009
[3]	Singh, S., Chandra, P. and Shukla, J.B. (2003) Modeling and analysis of the spread of carrier dependent infectious diseases with environmental effects. International Journal of Biomathematics, 11, 325-335.
[4]	World Health Organization (2012). http://www.who.int/gho/epidemic\_diseases/cholera/en/index.html
[5]	Yang, X. and Chen, L. (1996) Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models. Computers & Mathematics with Applications, 32, 109-116. doi:10.1016/0898-1221(96)00129-0
[6]	Van den Driessche, P. and Watmough, J. (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180, 29-48. doi:10.1016/S0025-5564(02)00108-6
[7]	Castillo-Chavez, C., Feng, Z.L. and Huang, W.H. (2003) On the computation of R0 and its role on global stability. Mathematical Approaches for Emerging and Reemerging Infectious Disease: An introduction, 65, 229-250.
[8]	Harley, D.M., Morris, J.G. and Smith, D.L. (2006) Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics? PLoS Medicine, 3, 63-69.
[9]	Codeco, C.T. (2001) Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir. BMC Infectious Diseases, 1, 1471-2334. doi:10.1186/1471-2334-1-1
[10]	Chen, M. (2011) The opinion of WHO about cholera vac- cine. International Journal of Biologicals, 34, 214-218.