Qualitative Analysis About A Certain Class Of Immune System Depending On Time Delay

Differential equations, as a class of mathematical models, have been being applied to various phenomena in the study of the natural science and engineering technology. All of them are based upon an assumption that the rules of variation are only in relation with the present state, but have nothing to do with the previous state. However, the fact is that many changes depend not only on their present state but also on their previous state. In such case, differential equations are inadequate in the exact description of the phenomena. Thus, difference-differential equations, especially differential equations with time delay, are the right ones to be turned to. In recent years, the time-delayed dynamical system has come to be the unportant objects of study in a variety of fields, such as medicine. Marchuk's model is a class of particular immune system with time delay. Marchuk's model of an immune system is one of the pioneering models in mathematical immunology. It was proposed by Marchuk in 1980. It is presented in the language of humoral immune reaction.The model was intensively studied by many people (see e.g. [2, 3,4, 5, 6, 7, 8, 9, 10, 11, 12]). For example, in reference [12], the author studies the stability of system equilibrium with the method of latent root and the periodic solution of the Marchuk's model with the method which provided by reference [13]. In this paper, we still regard time lag r as a parameter and deeply study the properties of the solution hi Marchuk's model of immune system based on different method. In the first, we study the stability of system equilibrium and the bound of these equilibriums by means of the method of latent root and analytical method. We find out there is a change in the stability of system equilibrium when time delay r experiences certain variation, i.e., when it changes from asymptotical stability (unstability) to unsiability (stability), these T become the system's bifurcation value (thesystem has small-scale amplitude nontrivial periodic solution around these values), and then we obtain the bound of stability of equilibrium. In the second, we make using of the method of HKW[14] which based on the center manifold theorem and normal form theory, and list two calculation formulas which reflect the stability with Hopf bifurcating periodic solution, and bifurcating direction. Thus it provide a basis for numerical simulation calculation. At last, we simulate the result we have obtained with mathematical software matlab6.5.