Simulating epidemic peak dynamics on complex networks using efficient Gillespie algorithms

We present an integrated study of epidemic spreading on complex networks that (i) reveals how network structure and targeted interventions shape the peak count of infected nodes (PCIN) and its timing, (ii) supplies an open-source dashboard that lets researchers explore these effects with realistic model extensions, and (iii) delivers a high-performance simulation engine which improves the time complexity of existing sparse network implementations of Gillespie's algorithm by multiplicative factors. Continuous-time SI/SIS/SIR dynamics are analyzed with respect to two intervention knobs: edge-specific infection rate reduction and node-level recovery acceleration, yielding explicit bounds on peak height and delay across heterogeneous topologies. To test scenarios interactively, we extend the dashboard simulator to include non-exponential recovery times, temporal rewiring, weighted and multi-type contacts, simulation of antigenically equivalent mutant strains, and a novel visual aggregation of likely infection routes. Moreover, we redesign Gillespie's algorithm for sparse graphs at its core by succinctly and incrementally updating the (sums of the) transition rate and maintaining a sorted infected node list, achieving speed increases with a multiplicative factor compared to the state-of-the-art implementation of the adjacency list. Additional speed gains can be achieved by our new algorithm when infection is in its early stage and only a few nodes of a large network are infected; a complementary dense matrix variant covers non-sparse cases. Benchmarks on Barabási–Albert networks confirm up-to-order-of-magnitude gains over the standard adjacency-list implementation of Gillespie's algorithm.