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Complex dynamic behavior in a viral model with delayed immune response. (English) Zbl 1117.34081

By incorporating time delay into the immune response, the authors proposed the following viral model

x ' (t)=λ-dx(t)-βx(t)y(t),
y ' (t)=βx(t)y(t)-ay(t)-py(t)z(t),z ' (t)=cy(t-τ)-bz(t),

where x(t),y(t),z(t) are the numbers of susceptible host cells, virus population and cytotoxic T lymphocytes at time t, respectively, and all the parameters are positive. Then the stability of the equilibria and the permanence of the system are proved. Moreover, some numerical simulations are given to illustrate the complex dynamic behavior of the system, including the periodic solution, chaos and stability switches. The results in this paper imply that the basic reproductive ration of the virus is an important index in understanding the long time behavior of the immune state of patients.

MSC:
34K60Qualitative investigation and simulation of models
92D30Epidemiology
34K20Stability theory of functional-differential equations
34K25Asymptotic theory of functional-differential equations
34K23Complex (chaotic) behavior of solutions of functional-differential equations