  Mathematical models in epidemiology are an indispensable tool to determine
the dynamics and important characteristics of infectious diseases. Apart from
their scientific merit, these models are often used to inform political
decisions and intervention measures during an ongoing outbreak. However,
reliably inferring the dynamics of ongoing outbreaks by connecting complex
models to real data is still hard and requires either laborious manual
parameter fitting or expensive optimization methods which have to be repeated
from scratch for every application of a given model. In this work, we address
this problem with a novel combination of epidemiological modeling with
specialized neural networks. Our approach entails two computational phases: In
an initial training phase, a mathematical model describing the epidemic is used
as a coach for a neural network, which acquires global knowledge about the full
range of possible disease dynamics. In the subsequent inference phase, the
trained neural network processes the observed data of an actual outbreak and
infers the parameters of the model in order to realistically reproduce the
observed dynamics and reliably predict future progression. With its flexible
framework, our simulation-based approach is applicable to a variety of
epidemiological models. Moreover, since our method is fully Bayesian, it is
designed to incorporate all available prior knowledge about plausible parameter
values and returns complete joint posterior distributions over these
parameters. Application of our method to the early Covid-19 outbreak phase in
Germany demonstrates that we are able to obtain reliable probabilistic
estimates for important disease characteristics, such as generation time,
fraction of undetected infections, likelihood of transmission before symptom
onset, and reporting delays using a very moderate amount of real-world
observations.
