### Epidemics on networks: From complicated structures to simple dynamics

## Abstract

The spread of an infectious disease crucially depends on the contact patterns of individuals, which range from superspreaders and clustered communities to isolated individuals with only a few regular contacts. The contact network specifies all contacts either between individuals in a population or, on a coarser scale, the contacts between groups of individuals, such as households, age groups or geographical regions. The structure of the contact network has a decisive impact on the viral dynamics. However, in most scenarios, the precise network structure is unknown, which constitutes a tremendous obstacle to understanding and predicting epidemic outbreaks.

This talk focusses on a stark contrast: network structures are complicated, but viral dynamics on networks are simple. Specifically, denote the N x 1 viral state vector by I(t) = (I_1(t), ..., I_N(t)), where N is the network size and I_i(t) is the infection probability of individual i at time t. The dynamics are “simple” in the way that the state I(t) evolves in a subspace X of R^N of astonishingly low dimension dim(X) << N. The low dimensionality of the viral dynamics has far-reaching consequences. First, it is possible to predict an epidemic outbreak, even without knowing the network structure. Second, provided that the basic reproduction number R_0 is close to one, the Susceptible-Infectious-Susceptible (SIS) epidemic model has a closed-form solution for arbitrarily large and heterogeneous contact networks.