  In our more and more interconnected world, a specific risk is that of a
cyber-epidemic (or cyber-pandemic), produced either accidentally or
intentionally, where a cyber virus propagates from device to device up to
undermining the global Internet system with devastating consequences in terms
of economic costs and societal harms related to the shutdown of essential
services. We introduce a compartmental model for studying the spreading of a
malware and of the awareness of its incidence through different waves which are
evolving on top of the same graph structure (the global network of connected
devices). This is realized by considering vectorial compartments made of two
components, the first being descriptive of the state of the device with respect
to the new malware's propagation, and the second accounting for the awareness
of the device's user about the presence of the cyber threat. By introducing
suitable transition rates between such compartments, one can then follow the
evolution of a cyber-epidemic from the moment at which a new virus is seeded in
the network, up to when a given user realizes that his/her device has suffered
a damage and consequently starts a wave of awareness which eventually ends up
with the development of a proper antivirus software. We then compare the
overall damage that a malware is able to produce in Erd\H{o}s-R\'enyi and
scale-free network architectures for both the case in which the virus is
causing a fixed damage on each device and the case where, instead, the virus is
engineered to mutate while replicating from device to device. Our result
constitute actually the attempt to build a specific compartmental model whose
variables and parameters are entirely customized for describing
cyber-epidemics.
