2022 Annual Meeting

(688g) Tipping Point Dynamics for Epidemiological Networks. Constructing Reduced Dynamical Data-Driven Models for Evolving Graphs

Authors

Evangelou, N. - Presenter, Johns Hopkins University
Cui, T., Johns Hopkins University
Bello-Rivas, J., Princeton University
Makeev, A., Faculty CMC Lomonosov Moscow State University
Bertalan, T., Johns Hopkins University
Kevrekidis, I. G., Princeton University
The construction of mathematical models of epidemic dynamics is instrumental to formulating healthcare policies [1]. In our work we use one of the most prevalent models of epidemics, the susceptible-infected-susceptible (SIS) model, which considers the epidemic dynamics as an evolving graph of connected Susceptible or Infected individuals [1-3]. The complexity and computational cost of large-scale network dynamics impedes utilization and understanding of these models and their predictions. To circumvent this issue, coarse/macroscopic variables that aim to summarize the important dynamic features can be used [1-2]. When such coarse observables are not a priori available, tools from machine learning and data science may be used to discover latent observables that capture the intrinsic information of the full models [4].

In our work we study a full complex epidemics network model in a set of coarse mean field variables, and we also use the manifold learning technique Diffusion Maps [4] to discover corresponding sets of data-driven variables that also capture the intrinsic structure of the sampled data. We show that the obtained data-driven variables can be physically interpretable. We further construct reduced dynamical models in terms of both sets of coarse variables (physical and data-driven) in a data-assisted manner. We used novel deep learning algorithms to learn a state as well as parameter dependent stochastic differential equation [5] in terms of both our coarse variable sets. We check and verify that the identified dynamics of the coarse models agree with the expected full system behavior. We construct the effective bifurcation diagram based on the deterministic part of the identified stochastic differential equation both in the physical coarse variables and in the data-driven ones. Furthermore, we compare the escape time distribution of our data-driven reduced dynamical models with the full epidemic network. The computation of the exit times was performed both with kinetic Monte Carlo simulations and by solving a partial differential equation boundary value problem arising from the Feynman-Kac formula[6].

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[2] 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
[3] Gross T, Kevrekidis IG. Robust oscillations in SIS epidemics on adaptive networks: Coarse graining by automated
moment closure. Epl-Europhys Lett 2008; 82:38004;

[4] Coifman, R. R., & Lafon, S. (2006). Diffusion maps. Applied and computational harmonic analysis, 21(1), 5-30.

[5] Dietrich, F., Makeev, A., Kevrekidis, G., Evangelou, N., Bertalan, T., Reich, S., & Kevrekidis, I. G. (2021). Learning effective stochastic differential equations from microscopic simulations: combining stochastic numerics and deep learning. arXiv preprint arXiv:2106.09004.

[6] B. Øksendal, Stochastic Differential Equations, Sixth. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. doi: 10.1007/978-3-642-14394-6.