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Nonnegative compartment dynamical system modelling with stochastic differential equations

机译：具有随机微分方程的非负部分动力系统建模

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摘要

Compartment models are widely used in various physical sciences and adequately describe many biological phenomena. Elements such as blood, gut, liver and lean tissue are characterized as homogeneous compartments, within which the drug resides for a time, later to transit to another compartment, perhaps recycling or eventually vanishing. We address the issue of compartment dynamical system modelling using multidimensional stochastic differential equations, rather than the classical approach based on the continuous-time Markov chain. Pure-jump processes are employed as perturbation to the deterministic compartmental dynamical system. Unlike with the Brownian motion, noise can be incorporated into both outflows and inter-compartmental flows without violating nonnegativity of the compartmental system, under mild technical conditions. The proposed formulation is easy to simulate, shares various essential properties with the corresponding deterministic ordinary differential equation, such as asymptotic behaviors in mean, steady states and average residence times, and can be made as close to the corresponding diffusion approximation as desired. Asymptotic mean-square stability of the steady state is proved to hold under some assumptions on the model structure. Numerical results are provided to illustrate the effectiveness of our formulation.
机译：隔室模型被广泛用于各种物理科学中，并充分描述了许多生物现象。诸如血液，肠，肝和瘦肉组织之类的元素的特征是均质的隔室，药物在其中停留一段时间，然后转移到另一个隔室，也许是再循环或最终消失。我们使用多维随机微分方程，而不是基于连续时间马尔可夫链的经典方法，来解决车厢动力学系统建模的问题。纯跳跃过程被用作确定性隔室动力学系统的扰动。与布朗运动不同，在温和的技术条件下，噪声可以合并到流出物和隔室之间，而不会破坏隔室系统的非负性。所提出的公式易于模拟，并与相应的确定性常微分方程共享各种基本特性，例如平均，稳态和平均停留时间的渐近行为，并且可以使其接近所需的扩散近似。在模型结构的一些假设下，证明了稳态的渐近均方稳定性。提供了数值结果以说明我们配方的有效性。

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