Stability of SIS Spreading Processes in Networks with Non-Markovian Transmission and Recovery

Although viral spreading processes taking place in networks are often analyzed using Markovian models in which both the transmission and the recovery times follow exponential distributions, empirical studies show that, in many real scenarios, the distribution of these times are not necessarily exponential. To overcome this limitation, we first introduce a generalized susceptible-infected-susceptible (SIS) spreading model that allows transmission and recovery times to follow phase-type distributions. In this context, we derive a lower bound on the exponential decay rate towards the infection-free equilibrium of the spreading model without relying on mean-field approximations. Based on our results, we illustrate how the particular shape of the transmission/recovery distribution influences the exponential rate of convergence towards the equilibrium.

💡 Research Summary

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The paper addresses a fundamental limitation of the classic susceptible‑infected‑susceptible (SIS) epidemic model, namely the assumption that both transmission and recovery times are exponentially distributed. Empirical evidence from social media, human contact networks, and disease progression shows that these times often follow non‑exponential laws such as log‑normal, Erlang, or Coxian distributions. To capture this realism without resorting to mean‑field approximations, the authors introduce the Generalized Networked SIS (GeNeSIS) model, in which transmission and recovery intervals are modeled as phase‑type (PH) distributions.

Phase‑type distributions are defined as the absorption time of a continuous‑time Markov chain with a single absorbing state. By choosing an appropriate number of transient phases p and a Metzler transition matrix T together with an initial probability vector φ, any positive‑valued distribution can be approximated arbitrarily closely. This property makes PH distributions a versatile tool for representing realistic waiting‑time statistics while retaining a Markovian structure amenable to analytical treatment.

The authors first develop a vectorial representation of a generic PH distribution using the canonical basis vectors e_m and the elementary matrices E_{mm’}. They then embed each node’s infection state into a higher‑dimensional vector that tracks the phase of its ongoing transmission attempts and its recovery process. The network dynamics are expressed as a system of stochastic differential equations with Poisson jumps: each possible infection attempt from node i to neighbor j is driven by an independent Poisson counter, and each recovery event is driven by another independent Poisson counter. Lemma II.1 provides an Itô‑type formula for functions of jump processes, while Lemma II.2 yields a closed‑form expression for the time derivative of the expectation when the jump functions are affine in the stochastic intensity.

Using these tools, the authors derive a linear differential equation for the expected state vector: \