SIS Epidemic Spreading with Heterogeneous Infection Rates

In this work, we aim to understand the influence of the heterogeneity of infection rates on the Susceptible-Infected-Susceptible (SIS) epidemic spreading. Employing the classic SIS model as the benchmark, we study the influence of the independently identically distributed infection rates on the average fraction of infected nodes in the metastable state. The log-normal, gamma and a newly designed distributions are considered for infection rates. We find that, when the recovery rate is small, i.e.\ the epidemic spreads out in both homogeneous and heterogeneous cases: 1) the heterogeneity of infection rates on average retards the virus spreading, and 2) a larger even-order moment of the infection rates leads to a smaller average fraction of infected nodes, but the odd-order moments contribute in the opposite way; when the recovery rate is large, i.e.\ the epidemic may die out or infect a small fraction of the population, the heterogeneity of infection rates may enhance the probability that the epidemic spreads out. Finally, we verify our conclusions via real-world networks with their heterogeneous infection rates. Our results suggest that, in reality the epidemic spread may not be so severe as the classic SIS model indicates, but to eliminate the epidemic is probably more difficult.

💡 Research Summary

This paper investigates how heterogeneity in infection rates influences the dynamics of the Susceptible‑Infected‑Susceptible (SIS) epidemic model on networks. While the classic SIS framework assumes a uniform infection rate β for all edges and a uniform recovery rate δ for all nodes, the authors replace β with independent and identically distributed (i.i.d.) random variables βij, keeping the mean infection rate fixed at 1. Three families of distributions are examined: log‑normal, gamma, and a newly introduced symmetric polynomial (SP) distribution, each allowing the variance (and higher moments) to be tuned while preserving the mean.

The study employs continuous‑time Gillespie‑type simulations on two canonical network topologies: Erdős‑Rényi (ER) random graphs and scale‑free (SF) networks, both with N = 10⁴ nodes and average degree ⟨k⟩ = 4. For each configuration, 1,000 independent realizations are run, and the steady‑state infected fraction y∞ (the average proportion of infected nodes in the metastable state) is recorded. The recovery rate δ is varied to explore two regimes: (i) small δ, where the effective infection rate τ = β/δ is well above the epidemic threshold τc and the disease persists; (ii) large δ, where τ is close to τc and the disease may die out in the homogeneous case.

Key findings are:

1. Variance suppresses prevalence in the persistent regime. When δ is small, increasing the normalized variance v of the infection‑rate distribution consistently reduces y∞, regardless of the specific distribution. This effect is driven primarily by even‑order moments (ν₂, ν₄, …): larger even moments lead to a stronger decline in prevalence. Odd‑order moments (ν₃, ν₅, …) have the opposite, albeit weaker, effect, slightly raising y∞. Intuitively, a broader spread of β values creates many low‑β edges that hinder overall transmission, while a few high‑β edges cannot compensate.

2. Heavy tails can rescue the epidemic near the threshold. In the regime where δ is large (τ≈τc), the homogeneous SIS model predicts extinction, but heterogeneous models with high variance can sustain a non‑zero y∞. The presence of a small fraction of edges with exceptionally large β (the “super‑spreader” links) acts as bridges that keep the infection alive even when the average transmissibility is subcritical. Log‑normal distributions, with their heavier tails, are more effective at this than gamma distributions.

3. Analytical insight via moment expansion. The infection probability over a time interval T is ρ(T)=∫fB(β)