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In the present work, a non-linear stochastic model is presented to study theeffect of time variation of transmission rates on the co-evolution of epidemicsand its corresponding awareness over a two layered multiplex network. In thismodel, the infection transmission rate of a given node in the epidemic layerdepends upon its awareness probability in the awareness layer. Similarly, theinfection information transmission rate of a node in the awareness layerdepends upon its infection probability in the epidemic layer. The spread ofdisease resulting from physical contacts is described in terms of SIS(Susceptible Infected Susceptible) process over the epidemic layer and thespread of information about the disease outbreak is described in terms of UAU(Unaware Aware Unaware) process over the virtual interaction mediated awarenesslayer. The time variation of the transmission rates and the resultingco-evolution of these mutually competing processes is studied in terms of anetwork topology depend parameter({\alpha}). Using a second order linear theoryit has been shown that in the continuous time limit, the co-evolution of theseprocesses can be described in terms of damped and driven harmonic oscillatorequations. From the results of the Monte-Carlo simulation, it is shown that forthe suitable choice of parameter({\alpha}), the two process can either exhibitsustained oscillatory or damped dynamics. The damped dynamics corresponds tothe endemic state. Further, for the case of endemic state it is shown that theinclusion of awareness layer significantly lowers the disease transmission rateand reduces the size of epidemic. The endemic state infection probability of agiven node corresponding to the damped dynamics is found to have dependenceupon both the transmission rates as well as on both absolute intra-layer andrelative inter-layer degree of the individual nodes.

Effect of time varying transmission rates on coupled dynamics of epidemic and awareness over multiplex network