Abrupt phase transition of epidemic spreading in simplicial complexes

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 the 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 reveals an interesting phase transition that becomes abrupt with the infectivity parameter of the 2-simplices. Our results pave the way 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.

Figure 1

Phase diagram of the mean-field approximation detailed in Eq. (4). We consider a network with average degree $\langle k\rangle =12$, an average number of triangles that each node is part of $\langle k^{▵}\rangle =5$, a recovery probability $\mu =0.2$, and two different infection probabilities through ternary interactions: (a) ${\beta }^{▵}=0.0$; (b) ${\beta }^{▵}=0.1$. Solid lines represent stable solutions and dashed lines depict unstable solutions to Eq. (4). The dotted area represents the region of parameters that enables a convergence to an endemic stable state from below.

Figure 2

Schematic representation of the contribution of different interactions to the joint probability between the states of node $i$ and node $j$, through node $i$. Black solid lines account for direct interactions between node $i$ and its neighbors. The checked pattern area represents a 2-simplex interaction with node $i$ where node $j$ is not participating, and the dotted pattern areas represent 2-simplex interactions with node $i$ where node $j$ is participating. All these contributions participate in Eqs. (9) to (11), respectively.

Figure 3

Incidence of the epidemic $\rho$ as function of infection probability $\beta$ on a random network of $N=2000$ nodes. We consider a network with average degree $\langle k\rangle =12$, an average number of triangles that each node is part of $\langle k^{▵}\rangle =5$, a recovery probability $\mu =0.2$, and two different infection probabilities through ternary interactions: (a) ${\beta }^{▵}=0.0$ and (b) $0.1$. The incidence has been analytically computed using ELE (solid line), MMCA (dashed line) and the mean-field approximation (dotted line). Results obtained from Monte Carlo simulations, performed using the quasistationary approach [33], are depicted by solid dots.

Figure 4

Incidence of the epidemic $\rho$ as function of infection probability $\beta$ on a random network with $N=8000$ nodes, built according to [34]. The average degree of the network is $\langle k\rangle =4$ and the average number of triangles per node is $\langle k^{▵}\rangle =3$. The recovery probability is $\mu =0.2$, and we use two different infection probabilities through ternary interactions: (a) ${\beta }^{▵}=0.00$ and (b) $0.14$. The incidence has been analytically computed using ELE (solid line), MMCA (dashed line) and the mean-field approximation (dotted line). Results obtained from Monte Carlo simulations, performed using the quasistationary approach [33], are depicted by solid dots.