SIRS epidemics on complex networks: Concurrence of exact Markov chain and approximated models

Abstract

Abstract-We study the SIRS (Susceptible-InfectedRecovered-Susceptible) spreading processes over complex networks, by considering its exact 3 n -state Markov chain model. The Markov chain model exhibits an interesting connection with its 2n-state nonlinear "mean-field" approximation and the latter's corresponding linear approximation. We show that under the specific threshold where the disease-free state is a globally stable fixed point of both the linear and nonlinear models, the exact underlying Markov chain has an O(log n) mixing time, which means the epidemic dies out quickly. In fact, the epidemic eradication condition coincides for all the three models. Furthermore, when the threshold condition is violated, which indicates that the linear model is not stable, we show that there exists a unique second fixed point for the nonlinear model, which corresponds to the endemic state. We also investigate the effect of adding immunization to the SIRS epidemics by introducing two different models, depending on the efficacy of the vaccine. Our results indicate that immunization improves the threshold of epidemic eradication. Furthermore, the common threshold for fast-mixing of the Markov chain and global stability of the disease-free fixed point improves by the same factor for the vaccination-dominant model.

Epidemic spread over networks with agent awareness and social distancing

“…Agents take preventative distancing actions which are linear functions of the epidemic state. We proved the standard persistence ratio ([5] - [6]) is a sufficient condition for the existence of a nontrivial fixed point in this approximation and that no combination of awareness parameters can restore stability of the disease-free equilibrium. Such a fixed point obeys a strict partial order in relation to the fixed point of the benchmark MFA model.…”

“…The benchmark model (no awareness) is defined by the same transition probabilities with a i (s(t)) = 1, ∀i. When λ max (βA C + (1 − δ)I n ) < 1, the mixing time of the benchmark model is O(log n) ( [14], [15], [6]), i.e the epidemic dies out quickly from any initial condition. We are interested in the case when λ max (βA C + (1 − δ)I n ) > 1, the regime where the epidemic persists for a long period of time before eventually becoming extinct.…”

“…There are two regimes of dynamics -one in which any initial epidemic dies out exponentially fast, and the other where the epidemic persists for a long period of time before dying out. Analysis on the latter in [5], [6], and [7] show existence and uniqueness of a metastable equilibrium, which is of particular interest in this paper as well. Specifically, we are interested in the metastable state in the setting of agent awareness and social distancing -i.e when nodes in the network take preventative actions to reduce the chance of becoming infected based on available information.…”

On the dynamics of deterministic epidemic propagation over networks

“…This work confirms the existence of an epidemic threshold, as a function of the spectral radius of the contact network. Further recent results on the discrete-time model are obtained by Ahn et al [1] and by Azizan Ruhi et al [3].…”

Inference, prediction and control of networked epidemics

“…In this section, we develop an observation model for the SIS epidemic process (1). As our goal is to develop a tractable method which controls the process without approximating the epidemic dynamics, we can neither use the commonly studied mean field approximations, nor can we propagate the exact process dynamics forward for every node, as the former would introduce unacceptable approximation errors, and the latter would necessitate the use of 2 n equations to propagate the joint distribution [16]. The approach we develop in this paper uses observations of the realized compartmental memberships of a subset of nodes as the process evolves in order to tractably make inferences and predictions.…”