Universality of small initial conditions in PDE models of epidemics

In this talk we give an introduction to deterministic and stochastic models of epidemics. We are interested in epidemics evolving on social networks, especially in the limit behaviour when the size of the underlying graph goes to infinity. For this we describe some notions of graph limits and a recent result describing the PDE limit of the stochastic process on the "not too sparse" regime where the average degrees are diverging but might be much smaller compared to the number of vertices.

In the second part, we are investigating a property of the PDE limit dubbed the Universality of Small Initial Conditions (USIC). We show that if initially the ratio of infected individuals is small than any initial configuration will lead to approximately the same epidemic curve up to time translation. The limit curve starting from the infinite past from infinitesimally small infections is identified as the "nontrivial eternal solution" of the PDE, well defined for both all positive and negative time values and connecting the infection less constant zero solution at minus infinity with the stationary endemic solution at plus infinity.