In biochemically reactive systems with small copy numbers of one or more
reactant molecules, the dynamics is dominated by stochastic effects. To
approximate those systems, discrete state-space and stochastic
simulation approaches have been shown to be more relevant than
continuous state-space and deterministic ones. In systems characterized
by having simultaneously fast and slow timescales, existing discrete
space-state stochastic path simulation methods, such as the stochastic
simulation algorithm (SSA) and the explicit tau-leap (explicit-TL)
method, can be very slow. Implicit approximations have been developed to
improve numerical stability and provide efficient simulation algorithms
for those systems. Here, we propose an efficient Multilevel Monte Carlo
(MLMC) method in the spirit of the work by Anderson and Higham (SIAM
Multiscal Model. Simul. 10(1), 2012).
This method uses split-step implicit tau-leap (SSI-TL) at levels where
the explicit-TL method is not applicable due to numerical stability
issues. We present numerical examples that illustrate the performance of
the proposed method.