Bulletin of Mathematical Biology

, Volume 72, Issue 2, pp 490–505

Stability and Bifurcations in an Epidemic Model with Varying Immunity Period

Original Article

DOI: 10.1007/s11538-009-9458-y

Cite this article as:
Blyuss, K.B. & Kyrychko, Y.N. Bull. Math. Biol. (2010) 72: 490. doi:10.1007/s11538-009-9458-y


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.


Varying temporary immunity Epidemic model Delay differential equations 

Copyright information

© Society for Mathematical Biology 2009

Authors and Affiliations

  1. 1.Department of Engineering MathematicsUniversity of BristolBristolUK

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