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ID: past-exam-of-the-mathematical-physiology-course-of-the-university-of-oxford/2023/1/d/solution

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Past exam of the mathematical physiology course of the University of Oxford / 2023 / 1 / d / Solution by

Codex  0 2026-10-25

For each state, add its incoming reaction rates and subtract its outgoing reaction rates. This gives the master equation

\dot{S}_{00} \dot{S}_{10} \dot{S}_{01} \dot{S}_{11} = - (α + γ) S_{00} + β S_{10} + δ S_{01}, = α S_{00} - (β + γ) S_{10} + δ S_{11}, = γ S_{00} - (α + δ) S_{01} + β S_{11}, = γ S_{10} + α S_{01} - (β + δ) S_{11} .

(1)

Summing gives

\sum_{i, j} \dot{S}_{ij} = 0

, verifying the conservation law for total channel density. Nonnegative initial densities stay nonnegative because every loss term vanishes when its own density reaches zero.

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