| In this talk I will present the derivation of an epidemic model with distributed time delay that in order to describe the dynamics of infectious diseases with varying immunity. I will show that solutions of the model 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. In the case of a constant period of temporary immunity, numerical simulations are performed to illustrate how the dynamics changes depending on the duration of immunity. |
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