Mathematical modelling of infectious disease spreading on temporal networks has recently gained popularity in complex systems science to understand the intricate interplay between social dynamics and epidemic processes. As analytic solutions can usually not be obtained, one has to resort to exact stochastic simulation algorithms, yet these have remained infeasible for the large sizes of realistic systems. Here, we introduce a rejection-based stochastic sampling algorithm with high acceptance probability (‘high-acceptance sampling’; HAS), tailored to simulate disease spreading on adaptive networks. We prove that HAS is exact and can be multiple orders faster than Gillespie’s algorithm. While its computational efficacy is dependent on model parameterization, we show that HAS is applicable regardless on whether contact dynamics are faster, on the same time-scale, or slower than the concurrent disease spreading dynamics. The algorithm is particularly suitable for processes where the spreading- and contact processes are co-dependent (adaptive networks), or when assumptions regarding time-scale separation become violated as the process unfolds. To highlight potential applications, we study the impact of diagnosis- and incidence-driven behavioural changes on virtual Mpox- and COVID-like epidemic and examine the impact of adaptive behaviour on the spreading processes.
