Fast computation method for comprehensive agent-level epidemic dissemination in networks

TL;DR

This paper introduces an algebraic approach using the squared norm of probability vectors to efficiently analyze stationary states in epidemic models on networks, leveraging symmetries to reduce computational complexity.

Contribution

It presents a novel algebraic method for evaluating epidemic spreading in networks, enabling analysis of larger populations through symmetry-based eigenvalue problems.

Findings

Reduces eigenvalue problem complexity to O(N) sparse matrix diagonalization.

Provides perturbative tools for assessing health policy impacts.

Enables analytical evaluation of stationary states in epidemic models.

Abstract

Two simple agent based models are often employed in epidemic studies: the susceptible-infected (SI) and the susceptible-infected-susceptible (SIS). Both models describe the time evolution of infectious diseases in networks in which vertices are either susceptible (S) or infected (I) agents. Predicting the effects of disease spreading is one of the major goals in epidemic studies, but often restricted to numerical simulations. Analytical methods using operatorial content are subjected to the asymmetric eigenvalue problem, restraining the usability of standard perturbative techniques, whereas numerical methods are limited to small populations since the vector space increases exponentially with population size $N$. Here, we propose the use of the squared norm of probability vector, $\vert P(t)\vert ^2$, to obtain an algebraic equation which allows the evaluation of stationary states, in time independent Markov processes. The equation requires eigenvalues of symmetrized time generators, which take full advantage of system symmetries, reducing the problem to an $O(N)$ sparse matrix diagonalization. Standard perturbative methods are introduced, creating precise tools to evaluate the effects of health policies.