(287d) Coarse-Grained Dynamics for Epidemics on Adaptive Networks

Modeling of epidemic dynamics is crucial in understanding and mitigating the spreading of diseases in the real world, and it has attracted increased attention from mathematicians again due to the ongoing coronavirus break-out since last year. The more well-known mathematical models for epidemic dynamics are the family of compartmental models, which can be studied in both a mean-field setting and with complex, possibly adaptive, network coupling [1]. Adaptive networks can successfully incorporate the change of status of individuals and their connections in a larger community. The so-called Adaptive Susceptible-Infected-Susceptible (SIS) network is actually a dynamic graph in which every person appears as a node with social connections as edges [2].

Such models can accurately account for the interacting individual level; yet, the most valuable information comes at the system-level. For example, healthcare policies must be made or revised based on a few collective variable predictions, rather than the time-series of individuals. Here, we use Diffusion Maps (DMAP), a popular dimension-reduction approach, to parameterize and learn coarse-grained dynamics. However, as with other unsupervised learning methods, DMAP is typically applied to data points that are vectors in Euclidean space with an appropriate distance metric; we thus need to transform our original data, where each point is a large labelled graph. To achieve this we consider the different types of graph nodes separately, and for each type, we obtain information about corresponding subgraph densities (e.g. number of edges, 2-connected edges and triangles, etc, see Figure for illustration); we concatenate these features, and normalize the resulting vector [3]. These vectors will be fed into the DMAP algorithm to generate our data-driven low-dimensional representations.

We then use black-box models (e.g. deep neural networks) to learn, and subsequently predict temporal evolution of these coarse variables, circumventing the time-consuming computation involved in long-time runs of the adaptive SIS network stochastic simulations. Additionally, after these tasks have been performed, we use bifurcation analysis on the surrogate macroscopic dynamics to explore how predictions will change with operating/policy parameters. Finally, we suggest such bifurcation/stability analysis and the resulting predictions could enable closed-loop control of epidemic processes, and therefore possibly faster epidemic containment.

[1] Eubank S, Guclu H, Anil Kumar VS, Marathe MV, Srinivasan A, Toroczkai Z, Wang N. Modelling disease outbreaks in realistic urban social networks. Nature 2004; 429:180-4; PMID:15141212; http://dx.doi.org/10.1038/nature02541

[2] Gross T, Kevrekidis IG. Robust oscillations in SIS epidemics on adaptive networks: Coarse graining by automated moment closure. Epl-Europhys Lett 2008; 82:38004; http://dx.doi.org/10.1209/0295-5075/82/38004

[3] Kattis1 AA, Holiday A, Stoica A-A, Kevrekidis IG. Modeling epidemics on adaptively evolving networks: A data-mining perspective. Virulence 2016; 7(2): 153; http://dx.doi.org/10.1080/21505594.2015.1121357

Figure: Converting a labelled graph to a vector containing information of 2-connect edge densities. Here, black and white nodes respectively stand for susceptible and infected individuals, and edges reflect their social connections.