Reaction-diffusion processes and metapopulation models on duplex networks

Reaction-diffusion processes, used to model various spatially distributed dynamics such as epidemics, have been studied mostly on regular lattices or complex networks with simplex links that are identical and invariant in transferring different kinds of particles. However, in many self-organized systems, different particles may have their own private channels to keep their purities. Such division of links often significantly influences the underlying reaction-diffusion dynamics and thus needs to be carefully investigated. This article studies a special reaction-diffusion process, named susceptible-infected-susceptible (SIS) dynamics, given by the reaction steps $\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\alpha}$ and $\ensuremath{\alpha}+\ensuremath{\beta}\ensuremath{\rightarrow}2\ensuremath{\beta}$, on duplex networks where links are classified into two groups: $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ links used to transfer $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ particles, which, along with the corresponding nodes, consist of an $\ensuremath{\alpha}$ subnetwork and a $\ensuremath{\beta}$ subnetwork, respectively. It is found that the critical point of particle density to sustain reaction activity is independent of the network topology if there is no correlation between the degree sequences of the two subnetworks, and this critical value is suppressed or extended if the two degree sequences are positively or negatively correlated, respectively. Based on the obtained results, it is predicted that epidemic spreading may be promoted on positive correlated traffic networks but may be suppressed on networks with modules composed of different types of diffusion links.