School of Mathematics and Physics, University of Queensland, Brisbane, Australia.
Department of Mathematical Sciences, RMIT University, Melbourne, Australia.
J Math Biol. 2022 Jun 25;84(7):63. doi: 10.1007/s00285-022-01769-5.
In mathematical biology, there is a great deal of interest in producing continuum models by scaling discrete agent-based models governed by local stochastic rules. We discuss a particular example of this approach: a model for the proliferation of neural crest cells that can help us understand the development of Hirschprung's disease, a potentially-fatal condition in which the enteric nervous system of a new-born child does not extend all the way through the intestine and colon. Our starting point is a discrete-state, continuous-time Markov chain model proposed by Hywood et al. (2013a) for the location of the neural crest cells that make up the enteric nervous system. Hywood et al. (2013a) scaled their model to derive an approximate second order partial differential equation describing how the limiting expected number of neural crest cells evolve in space and time. In contrast, we exploit the relationship between the above-mentioned Markov chain model and the well-known Yule-Furry process to derive the exact form of the scaled version of the process. Furthermore, we provide expressions for other features of the domain agent occupancy process, such as the variance of the marginal occupancy at a particular site, the distribution of the number of agents that are yet to reach a given site and a stochastic description of the process itself.
在数学生物学中，人们对通过缩放受局部随机规则控制的离散基于代理的模型来生成连续模型产生了浓厚的兴趣。我们讨论了这种方法的一个特殊例子：一种用于神经嵴细胞增殖的模型，它可以帮助我们理解先天性巨结肠的发展，这是一种潜在致命的疾病，新生婴儿的肠神经系统无法延伸到整个肠道和结肠。我们的起点是 Hywood 等人（2013a）提出的用于构成肠神经系统的神经嵴细胞位置的离散状态、连续时间马尔可夫链模型。Hywood 等人（2013a）对他们的模型进行了缩放，以推导出一个描述神经嵴细胞的极限预期数量如何在空间和时间中演变的近似二阶偏微分方程。相比之下，我们利用上述马尔可夫链模型与著名的 Yule-Furry 过程之间的关系，推导出了该过程缩放版本的精确形式。此外，我们还提供了关于域代理占用过程的其他特征的表达式，例如特定位置的边际占用的方差、尚未到达给定位置的代理数量的分布以及过程本身的随机描述。
