  In the present work, a non-linear stochastic model is presented to study the
effect of time variation of transmission rates on the co-evolution of epidemics
and its corresponding awareness over a two layered multiplex network. In this
model, the infection transmission rate of a given node in the epidemic layer
depends upon its awareness probability in the awareness layer. Similarly, the
infection information transmission rate of a node in the awareness layer
depends upon its infection probability in the epidemic layer. The spread of
disease resulting from physical contacts is described in terms of SIS
(Susceptible Infected Susceptible) process over the epidemic layer and the
spread of information about the disease outbreak is described in terms of UAU
(Unaware Aware Unaware) process over the virtual interaction mediated awareness
layer. The time variation of the transmission rates and the resulting
co-evolution of these mutually competing processes is studied in terms of a
network topology depend parameter({\alpha}). Using a second order linear theory
it has been shown that in the continuous time limit, the co-evolution of these
processes can be described in terms of damped and driven harmonic oscillator
equations. From the results of the Monte-Carlo simulation, it is shown that for
the suitable choice of parameter({\alpha}), the two process can either exhibit
sustained oscillatory or damped dynamics. The damped dynamics corresponds to
the endemic state. Further, for the case of endemic state it is shown that the
inclusion of awareness layer significantly lowers the disease transmission rate
and reduces the size of epidemic. The endemic state infection probability of a
given node corresponding to the damped dynamics is found to have dependence
upon both the transmission rates as well as on both absolute intra-layer and
relative inter-layer degree of the individual nodes.
