A new mathematical model was proposed to study the effect of self-proliferation and delayed activation of immune cells in the process of virus infection. The global stability of the boundary equilibria was obtained by constructing appropriate Lyapunov functional. For positive equilibrium, the conditions of stability and Hopf bifurcation were obtained by taking the delay as the bifurcation parameter. Furthermore, the direction and stability of the Hopf bifurcation are derived by using the theory of normal form and center manifold. These results indicate that self-proliferation intensity can significantly affect the kinetics of viral infection, and the delayed activation of immune cells can induce periodic oscillation scenario. Along with the increase of delay time, numerical simulations give the corresponding bifurcation diagrams under different self-proliferation rates, and verify that there exists stability switch phenomenon under some conditions.

中文翻译：

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具有自我增殖和免疫细胞延迟激活的病毒模型的稳定性和Hopf分支

提出了一个新的数学模型来研究病毒感染过程中免疫细胞自我增殖和延迟激活的作用。边界平衡的全局稳定性是通过构建适当的Lyapunov函数获得的。对于正平衡，通过将时延作为分岔参数来获得稳定性和Hopf分岔的条件。此外，利用法线形式和中心流形理论推导了霍普夫分支的方向和稳定性。这些结果表明，自我增殖强度可以显着影响病毒感染的动力学，而免疫细胞的延迟激活可以诱导周期性振荡。随着延迟时间的增加，