Abstract
The paper investigated an avian influenza virus propagation model with nonlinear incidence rate and delay based on SIR epidemic model. We regard delay as bifurcating parameter to study the dynamical behaviors. At first, local asymptotical stability and existence of Hopf bifurcation are studied; Hopf bifurcation occurs when time delay passes through a sequence of critical values. An explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcation periodic solutions is derived by applying the normal form theory and center manifold theorem. What is more, the global existence of periodic solutions is established by using a global Hopf bifurcation result.
Highlights
In March 2013, new avian-origin influenza A(H7N9) virus (A − OIV) broke out in Shanghai and the surrounding provinces of China [1]
(τ0)}), where μ2 determines the directions of the Hopf bifurcations, β2 determines the stability of the bifurcation periodic solutions, and T2 determines the odic solutions [9]
It is shown that all periodic solutions of system (2) are uniformly bounded
Summary
In March 2013, new avian-origin influenza A(H7N9) virus (A − OIV) broke out in Shanghai and the surrounding provinces of China [1]. The avian influenza virus propagation model based on SIR model has the following form:. Since R does not appear in the first two equations, and avoid excessive use of parentheses in some of the latter calculations, the avian influenza virus propagation model is transformed into the following form. Zhang and Li [6] studied the global stability of an SIR epidemic model with constant infectious periods. We investigated the Hopf bifurcation and the global existence of periodic solutions of model (2), which have not been reported yet.
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