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Analysis of all time-delay stability for biological systems using symbolic computation methods
LIU Jun, NIU Wei
Sino-French Engineer School, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract: All time-delay stability for biological systems shows that the time-delay system possess good reliability, so this issue has always been the highlight of the scholars research. However, researchers usually adopt the traditional mathematical methods or numerical calculation methods. Based on Hurwitz criterion and polynomial complete discriminant system, a sufficient and necessary algebraic criterion of all time-delay stability for nonlinear biological systems with parameters was introduced. By using symbolic computation methods, such as the methods of Grbner basis,triangular decomposition and real solution classification, a systematic and algorithmic approach for automatically analyzing all time-delay stability of biological systems with parameters was proposed. All the computations in our approach are all exact, which may help biologists and engineers to perform algebraic analysis for certain biological models. The successful experiments on the all time-delay stability analysis of several biological models, such as time delayed Lotka-Volterra systems and SIR epidemic models with time delay, showed the feasibility of our algebraic approach and also the superiority of symbolic computation methods compared with traditional mathematic methods.
Key words: symbolic computation     all time-delay stability     algebraic approach     biological systems     nonlinear

1 时滞微分系统正平衡点的稳定性

2 时滞系统全时滞稳定的代数判据

1) det[λI－A－$\sum\limits_{k=1}^{n}{{{A}_{k}}}$]=0的根具有负实部.

2) 对∀τ>0及任意实数y,都有

3 代数方法分析

 图 1 代数方法分析算法步骤 Fig. 1 Algorithmic steps using algebraic approaches
4 应用实例 4.1 实例验证

τ=0,那么平衡点问题可化为

4.2 无选择性的捕食-食饵系统

f(y,z)g(y,z)都是含有参数和变量的多项式.表 1中是f(y,z)g(y,z)的项数和最高次数,该问题使用传统的数学计算方法很难得出结果,而利用符号计算方法在几秒之内,可得

 多项式 项数 最高次数 f(y,z) 62 9 g(y,z) 44 8

4.3 SIR传染病模型

SIR传染病模型是一个重要的生物模型,Cooke在文献[18]中提出了时滞SIR传染病模型,并指出t时刻的传染能力为βS(t)I(t－τ′),其中,β>0为每天每个感染者接触的人数,τ′≥0为病毒在被感染者体内的作用时间.文献[19]中时滞SIR传染病系统为

5 结 论

1) 经实验验证表明该算法可实现较为优异的计算性能,例如计算含有62项的多项式所用时间仅仅为几秒,这是传统的数学计算所达不到的.

2) 此外,仍在进行任意多时滞微分系统的全时滞稳定性分析的算法研究及实验.

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#### 文章信息

LIU Jun, NIU Wei

Analysis of all time-delay stability for biological systems using symbolic computation methods

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(12): 2363-2369.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0794