Impact of temporary lockdown on disease extinction in assortative networks

Changing environmental conditions can significantly affect the dynamics of disease spread. These changes may arise naturally or result from human interventions; in the latter case, lockdown measures that lead to abrupt but temporary reductions in transmission rates are used to combat disease spread. However, the impact of these measures on rare events in realistic populations has not been studied so far. Here, we analyze the susceptible-infected-susceptible (SIS) model in a stochastic setting where disease extinction – a sudden clearance of the infection – occurs via a rare, large fluctuation. We use a semiclassical approximation and extensive numerical simulations to show how the extinction risk of the disease depends on both the duration and magnitude of the lockdown, in heterogeneous assortative networks, with degree-degree correlations between neighboring nodes.

💡 Research Summary

This paper investigates how a temporary lockdown influences the extinction probability of an infectious disease modeled by the susceptible‑infected‑susceptible (SIS) process on heterogeneous, assortatively mixed networks. The authors begin by formulating the stochastic SIS dynamics on a network whose degree distribution (P(k)) and degree‑degree correlation function (P(k’|k)) are parameterized by an assortativity coefficient (\alpha) (0 ≤ α ≤ 1). When α = 0 the network is uncorrelated; α → 1 forces links to connect nodes of the same degree, creating a highly assortative structure. Using the annealed‑network approximation, the infection and recovery rates for nodes of degree k are expressed as
(W^+k = \beta \sum{k’} k P(k’|k) x_{k’}) and (W^-k = \gamma x_k),
where (x_k) is the fraction of infected degree‑k nodes. The epidemic threshold is determined by the largest eigenvalue (\Upsilon^{(1)}) of the connectivity matrix (C{kk’} = k P(k’|k)); the critical infection rate is (\beta_c = 1/\Upsilon^{(1)}) and the basic reproduction number is (R_0 = (\beta/\gamma)\Upsilon^{(1)}).

In a finite population (N ≫ 1) the endemic fixed point becomes metastable: demographic noise eventually drives the system to the absorbing disease‑free state. The extinction probability (P(t)) grows with time and is related to the mean time to extinction (MTE) τ. For large N, the authors employ a semiclassical (WKB) approximation to estimate the action governing rare large fluctuations, but because assortativity and degree heterogeneity make the system high‑dimensional, they rely on extensive kinetic Monte‑Carlo (Gillespie) simulations to obtain quantitative results.

A lockdown is introduced as a temporary reduction of the infection rate: \