AI 中文总结
本文研究数学生物学中保守确定性马尔可夫模型，证明其不可约性保证全局稳定平衡点的存在唯一，微观可逆性保证非振荡行为，为高效模型拟合与模拟提供方法。
AI 中文摘要
常微分方程（ODE）在各学科中被广泛用于构建随时间变化过程的机制模型，这类模型常为描述不同相互关联“状态”随时间演化的马尔可夫模型。当这些模型无“源”或“汇”时，会自然保持系统总种群守恒；若每个状态都能直接或间接从其他任意状态到达，则称为不可约。对许多应用而言，确定性常微分方程系统是最合适的建模方法。本文总结了重要数学结果，表明不可约性可保证全局稳定平衡点的存在与唯一性，并讨论了这类基于ODE模型的高效计算实现；还探讨了微观可逆性条件，证明其可保证非振荡行为，从而实现计算效率的进一步提升。这些性质和方法通过生物现象示例模型得到验证，展现了其对高效模型拟合与模拟的重要性。
英文摘要
Ordinary differential equations are commonly used throughout the sciences to build mechanistic models of time-dependent processes. Often, such models are Markov models describing the time-evolution of different interconnected "states". When these models have no "sources" or "sinks", they naturally conserve the total population of the system. When each state is reachable (directly or indirectly) from any other state, these models are called irreducible. For many applications, a deterministic system of ordinary differential equations (ODE) is the most suitable modelling approach. We summarise important mathematical results which show that this irreducibility property guarantees the existence and uniqueness of global stable equilibria, and discuss the computationally-efficient implementation of such ODE-based models. We also discuss the condition of microscopic reversibility and show how it guarantees non-oscillatory behaviour, which enables additional efficiencies in computations. These properties and methods are demonstrated through example models of biological phenomena, where we demonstrate their importance for efficient model fitting and simulation.
Comments35 pages, 7 figures
