Abrupt phase transition of epidemic spreading in simplicial complexes

Abstract

Recent studies on network geometry, a way of describing network structures as geometrical objects, are revolutionizing our way to understand dynamical processes on networked systems. Here, we cope with the problem of epidemic spreading, using the Susceptible-Infected-Susceptible (SIS) model, in simplicial complexes. In particular, we analyze the dynamics of SIS in complex networks characterized by pairwise interactions (links), and three-body interactions (filled triangles, also known as 2-simplices). This higher-order description of the epidemic spreading is analytically formulated using a microscopic Markov chain approximation. The analysis of the fixed point solutions of the model, reveal an interesting phase transition that becomes abrupt with the infectivity parameter of the 2-simplices. Our results are of outmost importance for network theorists to advance in our physical understanding of epidemic spreading in real scenarios where diseases are transmitted among groups as well as among pairs, and to better understand the behavior of dynamical processes in simplicial complexes.The collective behavior of dynamical systems on networks, has been a major subject of research in the physics community during the last decades [1][2][3][4][5]. In particular, our understanding of both natural and man-made systems has significantly improved by studying how network structures and dynamical processes combined shape the overall systems' behavior. Recently, the network science community has turned its attention to network geometry [6-9] to better represent the kinds of interactions that one can find beyond typical pairwise interactions.These higher-order interactions are encoded in geometrical structures that describe the different kinds of simplex structure present in the network: a filled clique of m + 1 nodes is known as an m-simplex, and together a set of 1-simplexes (links), 2-simplexes (filled triangles), etc., comprise the simplicial complex. While simplicial complexes have been proven to be very useful for the analysis and computation in high dimensional data sets, e.g., using persistent homologies [10][11][12][13][14], little is understood about their role in shaping dynamical processes, save for a handful of examples [15][16][17][18].A more accurate description of dynamical processes on complex systems necessarily requires a new paradigm where the network structure representation helps to include higher-order interactions [19]. Simplicial geometry of complex networks is a natural way to extend manybody interactions in complex systems. The standard approach so far consists in understanding the coexistence of two-body (link) interactions and three-body interactions (filled triangles). Note that this approach is different from considering pairwise interactions among three elements of a triangle, it refers to the interaction of the three elements, in the filled triangle, at unison.Here we present a probabilistic formalization of the higher-order interactions of an epidemic process, represented by the well-known Suscep...

Anticipating the novel coronavirus disease (COVID-19) pandemic

“…Since human to human contact is a leading transmitter of the disease, therefore, by-passing a certain threshold of infected cases, the growth curve shows an increasing slope and finally depicts a sudden large shift/transition in the fraction of affected populations ( Fig. 1) [25]. It is important to note that the growth in number of cases for China and South Korea, the countries which initiated public monitoring/social distancing actions relatively earlier as compared to the other countries, saturates after nearly 3-4 weeks from the initiation of the lockdown.…”

A mathematical model for the spatiotemporal epidemic spreading of COVID19

“…To this aim, we rely on previous metapopulation models by the authors [18][19][20][21] including the spatial demographical distribution and recurrent mobility patterns, and develop a more refined epidemic model that incorporates the stratification of population by age in order to consider the different epidemiological and clinical features associated to each group age that have been reported so far. The mathematical formulation of these models rely on the Microscopic Markov Chain Approach formulation for epidemic spreading in complex networks [22][23][24][25][26][27].…”

Random walks on hypergraphs

“…Interestingly, more complicated nonlinear dynamics have been also recently studied on simplicial complexes [16,20,49,50] or in a pure multi-body frame [51]. Once again, however, the focus is placed on lowdimensional simplicial complexes (triangles).…”

“…Once again, however, the focus is placed on lowdimensional simplicial complexes (triangles). Recently, several dynamics, including epidemic spreading [16,49,52] and synchronisation [38], have been shown to produce new collective behaviours when higher-order interactions are assumed to shape the networked arrangement. In the projected network each hyperedge Eα becomes a complete clique of size |Eα|, with thus |Eα|(|Eα| − 1)/2 pairwise interactions.…”