Two-Scale Epidemic Spread Model: Comparison to Single-Layer Gillespie Algorithm

This article is one in a series by the authors on epidemic outbreak modeling and presents a practical implementation and validation of a two-scale (micro–macro) scheme for simulating epidemic spread on networks with explicit community structure. At the micro level, each community is modeled by a stochastic SI process with a fixed transmission coefficient on a complete internal graph; at the macro level, communities are connected by rare import events whose intensities depend on the donor’s current ‘infectiousness,’ the recipient’s ‘susceptibility,’ and the weights of inter-community links. Layer synchronization is performed step-by-step with no more than one import per global step; a residual waiting-time budget is used to ensure correct timing of internal events.
The goal is to assess how well the two-scale scheme reproduces the temporal profiles of a full Gillespie simulator on identical networks and parameters, and to examine how the inter-community scale parameter T behaves across different community sizes. We consider eight scenarios, varying the number of communities, community size, and inter-community edge weights. For each scenario we run 100 replications for both approaches; within each group a representative trajectory is selected using RMSE strictly as a curve-selection criterion.
The experiments show qualitative agreement between the two-scale model and the full simulator in timing and curve shape, yet reveal a key sensitivity: using raw counts of infected and susceptible individuals in the inter-community term makes T non-portable across scales (early overshoot in small networks, under-import in medium ones, pronounced over-import in large ones). We put forward an idea to stabilize the behavior of T, narrow its calibration range, and improve curve alignment across all configurations.
The results also indicate that full Gillespie simulation on a static network exhibits noticeable delays in cross-community spread, because import can occur only through nodes that actually bridge communities. In addition, the two-scale approach reduces the number of operations, making it suitable for rapid scenario analysis and decision support.