In 1927 Kermack and McKendrick formulated a model to describe an epidemic outbreak in a demographically closed population. A misconception
maintains that their paper is only about the SIR model. In fact it allows for a general expected contribution $A(\tau)$ to the prevailing force
of infection by individuals that were infected tau units of time ago. The mathematical formulation takes the form of a nonlinear renewal
equation with kernel A. This equation easily provides an expression for the basic reproduction number $R_0$ and equations for the Malthusian
parameter $r$ and the final epidemic size.
Erik Volz and Joel Miller extended the SIR model to the configuration network in the infinite population limit. The first aim of the talk
is to similarly extend the general K-McK model. This leads once again to a nonlinear renewal equation and next to useful characterizations
of $R_0$, $r$ and the final size.
The second aim is to introduce some dynamic variants of the configuration network. Concerning epidemic spread, we now limit ourselves to
the SIR model. We find that the situation is much easier when only edges come and go than when there is turnover of vertices as well. The
intuitive explanation involves the 'mean field at distance one' of the title.
The third aim is to admit a serious weakness: the considered networks lack (short) cycles! This confession mainly serves to raise a
question: is it possible to constructively define an infinite population network that admits both cycles and a statistical description
of its spatio-temporal properties that sustains an analytical treatment of epidemic spread across the network?
The lecture is based on joint work with KaYin Leung (now at Stockholm University) and Mirjam Kretzschmar (Julius Center, Utrecht,
and RIVM, Bilthoven).