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Emergent Behavior in Multipartite Large Networks: Multi-virus Epidemics

Epidemics in large complete networks is well established. In contrast, we consider epidemics in non-complete networks. We establish the fluid limit macroscopic dynamics of a multi-virus spread over a multipartite network as the number of nodes at each partite or island grows large. The virus spread follows a peer-to-peer random rule of infection in line with the Harris contact process. The model conforms to an SIS (susceptible-infected-susceptible) type, where a node is either infected or it is healthy and prone to be infected. The local (at node level) random infection model induces the emergence of structured dynamics at the macroscale. Namely, we prove that, as the multipartite network grows large, the normalized Markov jump vector process $\left(\bar{\mathbf{Y}}^\mathbf{N}(t)\right) = \left(\bar{Y}_1^\mathbf{N}(t),\ldots, \bar{Y}_M^\mathbf{N}(t)\right)$ collecting the fraction of infected nodes at each island $i=1,\ldots,M$, converges weakly (with respect to the Skorokhod topology on the space of \emph{càdlàg} sample paths) to the solution of an $M$-dimensional vector nonlinear coupled ordinary differential equation. In the case of multi-virus diffusion with $K\in\mathbb{N}$ distinct strains of virus, the Markov jurmp matrix process $\left(\bar{\mathbf{Y}}^\mathbf{N}(t)\right)$, stacking the fraction of nodes infected with virus type $j$, $j=1,\ldots,K$, at each island $i=1,\ldots,M$, converges weakly as well to the solution of a $\left(K\times M\right)$-dimensional vector differential equation that is also characterized.