SIS model on homogeneous networks with threshold type delayed contact reduction

We study the dynamics of a delayed SIS epidemic model on homogeneous networks, where it is assumed that individuals modify their contact patterns upon realizing the risk of infection. This decision is made with some time delay, and it is threshold type: when the density of infected nodes reaches a critical value, the number of links is reduced by a given factor. Such assumptions lead to a delay differential equation with discontinuous right hand side. We show that if the basic reproduction number R0≤1R0≤1, then the disease will be eradicated, while it persists for R0>1R0>1. In the latter case, there is a globally asymptotically stable endemic equilibrium, except for a crucial interval of reproduction numbers, where the system shows oscillations. We construct explicitly the unique slowly oscillatory periodic solution, which has strong attractivity properties, and show the existence of rapidly oscillatory periodic solutions with any frequency. The amplitude of the oscillations is determined by the time delay. Our results indicate that with such information delays, the link density of a network has an important effect on the qualitative dynamics of infectious diseases.