**Přednášející: **Gergely Rost

**Pracoviště: **Mathematical Institute, University of Oxford

**Datum a čas konání: **24.11.2017 - 14:00

**Podrobnosti:**

A large number of models (nearly 200 in the past six years) have been proposed in the mathematical epidemiology literature to capture the phenomenon that individuals modify their behavior during an epidemic outbreak. This can be due to directly experiencing the rising number of infections, media coverage, or intervention policies. In this talk we show that a delayed activation of such a response can lead to some interesting dynamics. For an SIS type process, if the delayed response occurs with a jump in the contact rate when the density of infection reaches some threshold, we show that for some interval of reproduction numbers, the system is oscillatory. The oscillation frequency is a discrete Lyapunov functional, and there exists a unique slowly oscillatory periodic solution with strong attractivity properties. We also construct rapidly oscillatory periodic solutions of any frequency. In the case of continuously decreasing transmission rate, if the response is not too sharp, the system preserves global stability. However, for sharp delayed response, we can observe stability switches as the basic reproduction number is increasing. First, the stability is passed from the disease free equilibrium to an endemic equilibrium via transcritical bifurcation as usual, but a further increase of the reproduction number causes oscillations, which later disappear, forming a structure in the bifurcation diagram what we call endemic bubble.
(Joint work with Maoxing Liu, Eduardo Liz, Gabriella Vas.)