Wang Z, Wang Q, Klinke D J
Department of Computer Science, Mathematics and Engineering, Shepherd University, Shepherdstown, WV 25443, USA.
Department of Chemical Engineering, Mary Babb Randolph Cancer Center, West Virginia University, Morgantown, WV 25606, USA; Department of Microbiology, Immunology and Cell Biology, West Virginia University, Morgantown, WV 25606, USA.
J Comput Sci Syst Biol. 2016 Sep;9(5):163-172. doi: 10.4172/jcsb.1000234. Epub 2016 Sep 30.
Biological processes such as contagious disease spread patterns and tumor growth dynamics are modelled using a set of coupled differential equations. Experimental data is usually used to calibrate models so they can be used to make future predictions. In this study, numerical methods were implemented to approximate solutions to mathematical models that were not solvable analytically, such as a SARS model. More complex models such as a tumor growth model involve high-dimensional parameter spaces; efficient numerical simulation techniques were used to search for optimal or close-to-optimal parameter values in the equations. Runge-Kutta methods are a group of explicit and implicit numerical methods that effectively solve the ordinary differential equations in these models. Effects of the order and the step size of Runge-Kutta methods were studied in order to maximize the search accuracy and efficiency in parameter spaces of the models. Numerical simulation results showed that an order of four gave the best balance between truncation errors and the simulation speed for SIR, SARS, and tumormodels studied in the project. The optimal step size for differential equation solvers was found to be model-dependent.
诸如传染病传播模式和肿瘤生长动力学等生物过程是使用一组耦合微分方程进行建模的。实验数据通常用于校准模型，以便可以用它们来进行未来预测。在本研究中，采用数值方法来近似求解无法通过解析方法求解的数学模型，例如一个非典模型。诸如肿瘤生长模型等更复杂的模型涉及高维参数空间；使用高效的数值模拟技术在方程中搜索最优或接近最优的参数值。龙格 - 库塔方法是一组显式和隐式数值方法，可有效求解这些模型中的常微分方程。研究了龙格 - 库塔方法的阶数和步长的影响，以便在模型的参数空间中最大化搜索精度和效率。数值模拟结果表明，对于该项目研究的SIR、非典和肿瘤模型，四阶能够在截断误差和模拟速度之间实现最佳平衡。发现微分方程求解器的最优步长取决于模型。
