Numerical modelling in biosciences using delay differential equations.

*(English)*Zbl 0969.65124Summary: Our principal purposes here are (i) to consider, from the perspective of applied mathematics, models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics, (ii) to review the application of numerical techniques to investigate these models. We show that there are prima facie reasons for using such models: (i) they have a richer mathematical framework (compared with ordinary differential equations) for the analysis of biosystem dynamics, (ii) they display better consistency with the nature of certain biological processes and predictive results. We analyze both the qualitative and quantitative role that delays play in basic time-lag models proposed in population dynamics, epidemiology, physiology, immunology, neural networks and cell kinetics. We then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences, comparing them with those implemented by the bio-modellers.

##### MSC:

65R20 | Numerical methods for integral equations |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |

92-08 | Computational methods for problems pertaining to biology |

92D25 | Population dynamics (general) |

45J05 | Integro-ordinary differential equations |

45G10 | Other nonlinear integral equations |

92D30 | Epidemiology |

##### Keywords:

nonlinear integro-differential equations; biological systems; numerical modelling; parameter estimation; retarded functional differential equations; delay differential equations; biosystem dynamics; population dynamics; epidemiology; physiology; immunology; neural networks; cell kinetics
PDF
BibTeX
XML
Cite

\textit{G. A. Bocharov} and \textit{F. A. Rihan}, J. Comput. Appl. Math. 125, No. 1--2, 183--199 (2000; Zbl 0969.65124)

Full Text:
DOI

##### References:

[1] | C.T.H. Baker, G.A. Bocharov, F.A. Rihan, A report on the use of delay differential equations in numerical modelling in the biosciences, MCCM Technical Report, Vol. 343, (1999), Manchester, ISSN 1360-1725. (Available over the World-Wide Web from URL: ). |

[2] | C.T.H. Baker, E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, MCCM Technical Report, Vol. 345, ISSN 1360-1725, University of Manchester, 1999. |

[3] | Banks, R.B., Growth and diffusion phenomena. mathematical frameworks and applications, Texts in applied mathematics, Vol. 14, (1994), Springer Berlin |

[4] | Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press New York · Zbl 0115.30102 |

[5] | Cushing, J.M., Integrodifferential equations and delay models in population dynamics, Lecture notes in biomathematics, Vol. 20, (1977), Springer Berlin · Zbl 0363.92014 |

[6] | Diekmann, O.; van Gils, S.; Verduyn Lunel, S.; Walter, H.-O., Delay equations, functional-, complex-, and nonlinear analysis, (1995), Springer New York |

[7] | Driver, R.D., Ordinary and delay differential equations, Applied mathematics series, Vol. 20, (1977), Springer Berlin · Zbl 0374.34001 |

[8] | Elsgolt’s, L.E.; Norkin, S.B., Introduction to the theory and application of differential equations with deviating arguments, (1973), Academic Press New York · Zbl 0287.34073 |

[9] | Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Dordrecht · Zbl 0752.34039 |

[10] | Györi, I.; Ladas, G., Oscillation theory of delay equations with applications, Oxford mathematical monographs, (1991), Clarendon Press Oxford · Zbl 0780.34048 |

[11] | Halanay, A., Differential equations, stability, oscillations, time lags, (1966), Academic Press New York · Zbl 0144.08701 |

[12] | Hale, J.K., Theory of functional differential equations, (1977), Springer New York · Zbl 0425.34048 |

[13] | Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer New York · Zbl 0787.34002 |

[14] | Kolmanovskii, V.B.; Myshkis, A., Applied theory of functional differential equations, MIA, vol. 85, (1992), Kluwer Dordrecht |

[15] | Kolmanovskii, V.B.; Nosov, V.R., Stability of functional differential equations, (1986), Academic Press New York · Zbl 0593.34070 |

[16] | Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · Zbl 0777.34002 |

[17] | MacDonald, N., Time-lags in biological models, Lecture notes in biomathematics, Vol. 27, (1978), Springer Berlin |

[18] | G.I. Marchuk, Mathematical Modelling of Immune Response in Infectious Diseases, Kluwer, Dordrecht, 1997. · Zbl 0876.92015 |

[19] | May, R., Stability and complexity in model ecosystems, (1974), Princeton University Press Princeton, NJ |

[20] | Maynard Smith, J., Models in ecology, (1974), Cambridge University Press Cambridge · Zbl 0312.92001 |

[21] | Murray, J.D., Mathematical biology, (1989), Springer Berlin · Zbl 0682.92001 |

[22] | Waltman, P., Deterministic threshold models in the theory of epidemics, Lecture notes in biomathematics, Vol. 1, (1974), Springer Berlin |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.