Numerical modelling of biological systems with memory using delay differential equations

This is a review article, to show the consistency of delay differential equations with biological systems with memory, in which we present a class of mathematical models with time-lags in immunology, physiology, epidemiology and cell growth. We also incorporate optimal control parameters into a delay model to describe the interactions of the tumour cells and immune response cells with external therapy. We then study parameter estimations and sensitivity analysis with delay differential equations. Sensitivity analysis is an important tool for understanding a particular model, which is considered as an issue of stability with respect to structural perturbations in the model. We introduce a variational method to evaluate sensitivity of the state variables to small perturbations in the initial conditions and parameters appear in the model. The presented numerical simulations show the consistency of delay differential equations with biological systems with memory. The displayed results may bridge the gap between the mathematics reserach and its applications in biology and medicine.