  Self-adaptive dynamics occurs in many physical systems such as
socio-economics, neuroscience, or biophysics. We formalize a self-adaptive
modeling approach, where adaptation takes place within a set of strategies
based on the history of the state of the system. This leads to piecewise
deterministic Markovian dynamics coupled to a non-Markovian adaptive mechanism.
We apply this framework to basic epidemic models (SIS, SIR) on random networks.
We consider a co-evolutionary dynamical network where node-states change
through the epidemics and network topology changes through creation and
deletion of edges. For a simple threshold base application of lockdown measures
we observe large regions in parameter space with oscillatory behavior. For the
SIS epidemic model, we derive analytic expressions for the oscillation period
from a pairwise closed model. Furthermore, we show that there is a link to
self-organized criticality as the basic reproduction number fluctuates around
one. We also study the dependence of our results on the underlying network
structure.
