Stability and Bifurcations in an Epidemic Model with Varying Immunity Period
- 418 Downloads
An epidemic model with distributed time delay is derived to describe the dynamics of infectious diseases with varying immunity. It is shown that solutions are always positive, and the model has at most two steady states: disease-free and endemic. It is proved that the disease-free equilibrium is locally and globally asymptotically stable. When an endemic equilibrium exists, it is possible to analytically prove its local and global stability using Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and traceDDE to investigate different dynamical regimes in the model using numerical continuation for different values of system parameters and different integral kernels.
KeywordsVarying temporary immunity Epidemic model Delay differential equations
Unable to display preview. Download preview PDF.
- Anderson, R.M., May, R.M., 1991. Infectious Diseases in Humans: Dynamics and Control. Oxford University Press, Oxford. Google Scholar
- Arino, J., Cooke, K.L., van den Driessche, P., Velasco-Hernández, J., 2004. An epidemiology model that includes a leaky vaccine with a general waining function. Discrete Contin. Dyn. Syst. B 2, 479–495. Google Scholar
- Engelborghs, K., Luzyanina, T., Samaey, G., 2001. DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations. Technical Report No. TW-330, Department of Computer Science K.U. Leuven, Belgium. Google Scholar
- Gao, S., Teng, Z., Nieto, J.J., Torres, A., 2007. Analysis of an SIR epidemic model with pulse vaccination and distributed time delay. J. Biomed. Biotechnol. 64870. Google Scholar
- Krauskopf, B., 2005. Bifurcation analysis of lasers with delay. In: Kane, D.M., Shore, K.A. (Eds.), Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, pp. 147–183. Wiley, New York. Google Scholar
- Kuang, Y., 1993. Delay Differential Equations with Applications in Population Biology. Academic Press, New York. Google Scholar
- Ruan, S., 2006. Delay differential equations in single species dynamics. In: Arino, O., Hbid, M., Ait Dads, E. (Eds.), Delay Differential Equations with Applications. NATO Science Series II, vol. 205, pp. 477–517. Springer, Berlin. Google Scholar