Efficient Low-Order Approximation of First-Passage Time Distributions

We consider the problem of computing first-passage time distributions for reaction processes modeled by master equations. We show that this generally intractable class of problems is equivalent to a sequential Bayesian inference problem for an auxiliary observation process. The solution can be approximated efficiently by solving a closed set of coupled ordinary differential equations (for the low-order moments of the process) whose size scales with the number of species. We apply it to an epidemic model and a trimerization process and show good agreement with stochastic simulations.

Figure 1

Results for the epidemic system (12). (a) Simulated path of the process. (b) Mean, variance, and mode of the FPT distribution of species $I$ becoming extinct as a function of the initial populations $x_0$ of species $S$, from the stochastic simulation algorithm (SSA, dots, $10^6$ samples per point) and from BFPT (lines). The rate constants are set to $k_1=0.5$ and $k_2=1$, and the initial value for species $I$ is set to $y_0=2x_0$. (c),(d) FPT distributions as obtained from the SSA (dots, $10^7$ samples per parameter set) and BFPT (lines) for the parameter set $(x_0,y_0,k_1,k_2)$ chosen as (6,1,0.25,1) [blue, (c)], (20,10,0.5,1) [red, (c)], (20,1,0.5,2) [blue, (d)], and (40,10,0.25,1) [red, (d)]. The parameter $a$ in (11) was chosen as $a=-3$ for the blue curve in (c) and $a=-1.5$ for all other figures. (e),(f) The same results as (c),(d) but logarithmic scale.

Figure 2

Results for the polymerization system in (13). (a) FPT distribution for the parameters $k_1=k_2={10}^{-3}$ obtained from the SSA (dots, $10^4$ samples). (b),(c) Heat plots of the mean and the coefficient of variation (defined as the standard deviation divided by the mean) of the FPT to produce 200 trimers starting with $10^3$ monomers, as a function of $k_1$ and $k_2$ on the logarithmic scale. (d) Corresponding 3D plot for the normalization of the FPT distribution, that is, the probability with which at least 200 trimers are being produced. The white areas in (b) and (c) indicate that either the value is larger than the plotted range or that the target state is reached with such small probability that an estimation of moments is not sensible. The parameter $a$ in (11) was fixed to $a=0.2$ for all figures.