Clustering for epidemics on networks: A geometric approach

Abstract

Infectious diseases typically spread over a contact network with millions of individuals, whose sheer size is a tremendous challenge to analyzing and controlling an epidemic outbreak. For some contact networks, it is possible to group individuals into clusters. A high-level description of the epidemic between a few clusters is considerably simpler than on an individual level. However, to cluster individuals, most studies rely on equitable partitions, a rather restrictive structural property of the contact network. In this work, we focus on Susceptible–Infected–Susceptible (SIS) epidemics, and our contribution is threefold. First, we propose a geometric approach to specify all networks for which an epidemic outbreak simplifies to the interaction of only a few clusters. Second, for the complete graph and any initial viral state vectors, we derive the closed-form solution of the nonlinear differential equations of the N-intertwined mean-field approximation of the SIS process. Third, by relaxing the notion of equitable partitions, we derive low-complexity approximations and bounds for epidemics on arbitrary contact networks. Our results are an important step toward understanding and controlling epidemics on large networks.

Predicting network dynamics without requiring the knowledge of the interaction graph

“…First, provided that the network has an equitable partition, the POD [2] is exact, and the agitation modes y p follow from the cells of the partition for a plethora of dynamical models ¶ This is the case provided that the functions f i and g are in the SINDy library of candidate functions, which must be preconstructed by the user of the SINDy algorithm. (70)(71)(72)(73)(74)(75). # Second, under some assumptions (76), if the network has a negligible degree correlation (18), then the dynamics can be approximated by the POD with one agitation mode y 1 .…”

Competing control scenarios in probabilistic SIR epidemics on social-contact networks

“…In sum, the components of each node state ( p S k , p I k , p R k ) on the social graph satisfy the property of probabilities, which allows a consistent probabilistic interpretation of the graph-based SIR dynamics with vaccination and confinement, Eqs. (2, 3, 4, 7), (Lajmanovich & Yorke, 1976;Gerbeau & Lombardi, 2014;Prasse et al, 2021;Broekaert & La Torre, 2021).…”

“…In previous work on infectious disease propagation, a probabilistic approach to SIS dynamics on coupled social-contact and economic production networks was explored (Broekaert & La Torre, 2021). In particular we applied the vector logistic equation on a graph for the infectionrecovery dynamics to preserve the probabilistic interpretation of the S-and I-propensity of each individual node (see also Lajmanovich and Yorke 1976;Gerbeau and Lombardi 2014;Prasse et al 2021). The adoption of a network-based approach to modeling an epidemic allows the description of patterns of interaction among individuals -the heterogeneity of the social-contact network-or to designate subpopulations with specific group properties.…”

Competing control scenarios in probabilistic SIR epidemics on social-contact networks

“…In sum, the components of the state (S k , I k , R k ) of the nodes on the social graph satisfy the property of probabilities, allowing a consistent probabilistic interpretation of SIR dynamics, Eqs. (2,3,4), on a social contact graph [15,16,17,14].…”

“…In a previous work on infectious disease propagation, a probabilistic approach to SIS dynamics on coupled social-contact and economic production networks was explored [14]. In particular we applied the vector logistic equation on a graph for the infection-recovery dynamics to preserve the probabilistic interpretation of the S-and I-propensity of each individual node (see also [15,16,17]). The adoption of a network-based approach to modeling an epidemic allows the description of patterns of interaction among individuals -the variety of ego network -or households.…”