  We investigate the evolution of epidemics over dynamical networks when nodes
choose to interact with others in a selfish and decentralized manner.
Specifically, we analyze the susceptible-asymptomatic-infected-recovered (SAIR)
epidemic in the framework of activity-driven networks with heterogeneous node
degrees and time-varying activation rates, and derive both individual and
degree-based mean-field approximations of the exact state evolution. We then
present a game-theoretic model where nodes choose their activation
probabilities in a strategic manner using current state information as
feedback, and characterize the quantal response equilibrium (QRE) of the
proposed setting. We then consider the activity-driven
susceptible-infected-susceptible (SIS) epidemic model, characterize equilibrium
activation probabilities and analyze epidemic evolution in closed-loop. Our
numerical results provide compelling insights into epidemic evolution under
game-theoretic activation. Specifically, for the SAIR epidemic, we show that
under suitable conditions, the epidemic can persist, as any decrease in
infected proportion is counteracted by an increase in activity rates by the
nodes. For the SIS epidemic, we show that in regimes where there is an endemic
state, the infected proportion could be significantly smaller under
game-theoretic activation if the loss upon infection is sufficiently high.
