Abstraction-Guided Truncations for Stationary Distributions of Markov Population Models

Abstract

To understand the long-run behavior of Markov population models, the computation of the stationary distribution is often a crucial part. We propose a truncation-based approximation that employs a state-space lumping scheme, aggregating states in a grid structure. The resulting approximate stationary distribution is used to iteratively refine relevant and truncate irrelevant parts of the state-space. This way, the algorithm learns a well-justified finite-state projection tailored to the stationary behavior. We demonstrate the method’s applicability to a wide range of non-linear problems with complex stationary behaviors.

Appendices

A Detailed Results

See Tables 2, 3 and Fig. 6.

Table 2. Detailed results for Model 2. The errors are computed wrt. the reference Poissonian product. The total absolute error and the maximum absolute errors are given.

Fig. 6.

The error over the truncation wrt. the analytical solution

Table 3. Detailed results for Model 3. Upper bounds on the total absolute error and the maximum absolute error are given. The worst-case errors are computed wrt. the reference Geobound solution with $\epsilon _{\ell }=1e-2$.

B Lyapunov Analysis of the p53 Oscillator

We now derive Lyapunov-sets for the p53 oscillator case study (Model 4). Let the Lyapunov function

$$\begin{aligned} g(x) = 120 x_{\mathrm {p53}} + 0.2 x_{\mathrm {pMdm2}} + 0.1 x_{\mathrm {Mdm2}}\,. \end{aligned}$$

(16)

Then the drift

$$\begin{aligned} d(x) =&- \frac{k_3 x_{\mathrm {Mdm2}} x_{\mathrm {p53}}}{x_{\mathrm {p53}} + k_7} - 0.1 k_6 x_{\mathrm {Mdm2}} + 120 k_1 \nonumber \\&- 120 k_2 x_{\mathrm {p53}} + 0.2 k_4 x_{\mathrm {p53}} - 0.1 k_5 x_{\mathrm {pMdm2}} \nonumber \\ =&- \frac{204 x_{\mathrm {Mdm2}} x_{\mathrm {p53}}}{x_{\mathrm {p53}} + 0.01} - 0.096 x_{\mathrm {Mdm2}} - 0.02 x_{\mathrm {p53}} \nonumber \\&- 0.0093 x_{\mathrm {pMdm2}} + 10800\,. \end{aligned}$$

(17)

Clearly, $c = \sup _{x\in {S}} d(x) = 10800$. In particular, the supremum c is at the origin since all non-constant terms are negative. The slowest rate of decrease for (17) is $x_{\mathrm {p53}}$ with $x_{\mathrm {Mdm2}} = x_{\mathrm {pMdm2}} = 0$. We are content with a superset of a Lyapunov set (9) for some threshold $\epsilon _{\ell }$. Therefore taking (9), we can solve the inequality

$$ \frac{\epsilon _{\ell }}{c}(c - 0.02 x_{\mathrm {p53}}) > \epsilon _{\ell } - 1 $$

for $x_{\mathrm {p53}}$ and

$$\begin{aligned} \frac{c}{0.02 \epsilon _{\ell }} < x_{\mathrm {p53}}\,. \end{aligned}$$

(18)

Therefore

$$\begin{aligned} \pi _{\infty }\left( \left\{ x\in \mathcal {S} \mid \frac{c}{0.2\epsilon _{\ell }} < \Vert x \Vert \right\} \right) > 1 - \epsilon _{\ell }\,. \end{aligned}$$

(19)