Global Hopf Bifurcation Analysis for an Avian Influenza Virus Propagation Model with Nonlinear Incidence Rate and Delay

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 242410, 7 pages http://dx.doi.org/10.1155/2014/242410 Research Article Global Hopf Bifurcation Analysis for an Avian Influenza Virus Propagation Model with Nonlinear Incidence Rate and Delay Yanhui Zhai, Ying Xiong, Xiaona Ma, and Haiyun Bai School of Science, Tianjin Polytechnic University, Tianjin 300387, China Correspondence should be addressed to Ying Xiong; Received 1 January 2014; Accepted 14 June 2014; Published 14 July 2014 Academic Editor: Zhichun Yang Copyright © 2014 Yanhui Zhai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. 1. Introduction In March 2013, new avian-origin influenza 𝐴(𝐻7𝑁9) virus (𝐴 − 𝑂𝐼𝑉) broke out in Shanghai and the surrounding provinces of China [1]. During the first week of April, this virus had been detected in six provinces and municipal cities; this virus has caused global concern as a potential pandemic threat [2]. The virus fast took people’s life without timely treatment. Therefore, strong measures should be taken to control the spread of H7N9 viruses. 𝐻7𝑁9 is an infectious disease caused by influenza A virus. Moreover, it is essential to study and to dominate the spread of 𝐻7𝑁9. Mathematical models become important instruments in the analysis and control of infectious diseases. The present study evaluates the possible application of SIR model for 𝐻7𝑁9 spreading. Let 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡) be the population densities of susceptible, infective, and recovered, respectively. Recruitment of new individuals is into the susceptible class at a constant rate 𝐵 [3]. Parameters 𝜇1 , 𝜇2 , and 𝜇3 are positive constants which represent the death rate of the classes, respectively. 𝜏 is the length of the infectious period; 1/𝛾 is the average time spent in class 𝐼 before recovery [3]. In 1979, Cooke [4] used mass action incidence 𝛽𝑆(𝑡)𝐼(𝑡 − 𝜏). In 2009, Xu and Ma [5] developed the model with the force of infection given by 𝛽𝑆(𝑡)(𝐼(𝑡 − 𝜏)/(1 + 𝛼𝐼(𝑡 − 𝜏))), where 𝛼 determines the level at which the force of infection saturates and 𝛽 is a contract [5]. Then, the avian influenza virus propagation model based on SIR model has the following form: 𝑆 ̇ (𝑡) = 𝐵 − 𝜇1 𝑆 (𝑡) − 𝐼 ̇ (𝑡) = 𝛽𝑆 (𝑡) 𝐼 (𝑡 − 𝜏) , 1 + 𝛼𝐼 (𝑡 − 𝜏) 𝛽𝑆 (𝑡) 𝐼 (𝑡 − 𝜏) − (𝜇2 + 𝛾) 𝐼 (𝑡) , 1 + 𝛼𝐼 (𝑡 − 𝜏) (1) 𝑅̇ (𝑡) = 𝛾𝐼 (𝑡) − 𝜇3 𝑅 (𝑡) . 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 𝑆 ̇ (𝑡) = 𝐵 − 𝜇1 𝑆 (𝑡) − 𝐼 ̇ (𝑡) = 𝛽𝑆 (𝑡) 𝐼 (𝑡 − 𝜏) , 1 + 𝛼𝐼 (𝑡 − 𝜏) 𝛽𝑆 (𝑡) 𝐼 (𝑡 − 𝜏) − (𝜇2 + 𝛾) 𝐼 (𝑡) , 1 + 𝛼𝐼 (𝑡 − 𝜏) 𝑅̇ (𝑡) = 𝛾𝐼 (𝑡) − 𝜇3 𝑅 (𝑡) , (2) (3) 2 Abstract and Applied Analysis with the following initial condition: 𝑆 (0) ∈ 𝑅+ , 𝐼 (𝜃) = 𝜙 (𝜃) Then, we get for 𝜃 ∈ [−𝜏, 0] , where 𝜙 ∈ 𝐶 ([−𝜏, 0] , 𝑅+ ) , cos 𝜔𝜏 = (4) sin 𝜔𝜏 = which was presented and studied in [3]. The steady state of the model and the stability of epidemic models have been studied in many papers. Zhang and Li [6] studied the global stability of an SIR epidemic model with constant infectious periods. Xu and Ma [5] showed the global stability of the endemic equilibrium for the case of the reproduction number 𝑅0 > 1. McCluskey [3] shown that the endemic equilibrium is globally asymptotically stable whenever it exists. In this paper, we investigated the Hopf bifurcation and the global existence of periodic solutions of model (2), which have not been reported yet. The organization of this paper is as follows. In Section 2, we will investigate the local asymptotical stability and existence of Hopf bifurcation by analyzing the associated characteristic equation. In Section 3, an explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcation periodic solutions will be derived by applying the normal form theory and center manifold theorem. In Section 4, existence of global periodic solutions will be established by using a global Hopf bifurcation result. In Section 5, a brief discussion is offered to conclude this work. (5) where 𝑝0 = (𝜇2 + 𝛾)(𝜇1 + 𝛽𝐼∗ /(1 + 𝛼𝐼∗ )), 𝑝1 = 𝜇1 + 𝜇2 + 2 𝛾 + 𝛽𝐼∗ /(1 + 𝛼𝐼∗ ), 𝑞0 = −𝛽𝜇1 𝑆∗ /(1 + 𝛼𝐼∗ ) , and 𝑞1 = −𝛽𝑆∗ / 2 (1 + 𝛼𝐼∗ ) . If 𝑅0 > 1 (𝑃1 ) ∗ hold, when 𝜏 = 0, the endemic equilibrium 𝐸 of system (2) is locally stable [5]. If 𝑖𝜔 (𝜔 > 0) is a solution of system (2), separating real and imaginary parts, we obtain the following: 𝑝1 𝜔 = 𝑞0 sin 𝜔𝜏 − 𝑞1 𝜔 cos 𝜔𝜏, 𝜔2 − 𝑝0 = 𝑞0 cos 𝜔𝜏 + 𝑞1 𝜔 sin 𝜔𝜏. (6) 𝑞02 + 𝑞12 𝜔2 (7) . 𝜔4 + (𝑝12 − 2𝑝0 − 𝑞12 ) 𝜔2 + 𝑝02 − 𝑞02 = 0. (8) Letting 𝑧 = 𝜔2 , we get 𝑧2 + (𝑝12 − 2𝑝0 − 𝑞12 ) 𝑧 + 𝑝02 − 𝑞02 = 0. (9) It is easy to show that 𝑝12 − 2𝑝0 − 𝑞12 = (𝜇1 + 2 𝛽𝐼∗ ) 1 + 𝛼𝐼∗ 2 2 + (𝜇2 + 𝛾) − (𝜇2 + 𝛾) (1 + 𝛼𝐼∗ )2 > 0, 𝑝02 − 𝑞02 (10) = (𝜇2 + 𝛾) [(𝜇2 + 𝛾) (𝜇1 + 2. Local Stability and Hopf Bifurcation 𝜆2 + 𝑝1 𝜆 + 𝑝0 + (𝑞1 𝜆 + 𝑞0 ) 𝑒−𝜆𝜏 = 0, 𝑝1 𝑞0 𝜔 + (𝜔2 − 𝑝0 ) 𝑞1 𝜔 It follows that × (𝜇1 − Some results can be directly obtained from [3, 5]. The basic reproduction number for the model is 𝑅0 = 𝐵𝛽/𝜇1 (𝜇2 + 𝛾). System (2) always has a disease-free equilibrium 𝐸1 = (𝐵/𝜇1 , 0). If 𝐵𝛽 > 𝜇1 (𝜇2 + 𝛾), system (2) has a unique endemic equilibrium 𝐸∗ = (𝑆∗ , 𝐼∗ ) = ((𝐵𝛼 + 𝜇2 + 𝛾)/(𝛽 + 𝛼𝜇1 ), (𝐵𝛽 − 𝜇1 (𝜇2 + 𝛾))/(𝜇2 + 𝛾)(𝛽 + 𝛼𝜇1 )) [3]. The characteristic equation of system (2) at the endemic equilibrium 𝐸∗ is (𝑞0 − 𝑝1 𝑞1 ) 𝜔2 − 𝑝0 𝑞0 , 𝑞02 + 𝑞12 𝜔2 𝛽𝐼∗ 𝛽𝜇1 𝑆∗ ) + ] 1 + 𝛼𝐼∗ (1 + 𝛼𝐼∗ )2 𝜇1 𝛽𝐼∗ + ). 1 + 𝛼𝐼∗ 1 + 𝛼𝐼∗ The case of 𝛽 ≥ 𝜇1 𝛼 (𝐻1 ) has been discussed in [5]. We obtain global asymptotic stability of the endemic equilibrium when 𝑅0 > 1. If 𝛽 < 𝜇1 𝛼 (𝐻2 ) hold, that is, (𝛽 − 𝜇1 𝛼)𝐼∗ /(1 + 𝛼𝐼∗ ) < 0, we have 𝑝02 − 𝑞02 < 0. Following the theorem given by Ruan [7], there exists critical value (𝑗) 𝜏𝑘 = 2 2 (𝜔𝑘 − 𝑝0 ) 𝑞0 − 𝑝1 𝑞1 𝜔𝑘 2𝑗𝜋 1 arccos + , 𝜔𝑘 𝜔𝑘 𝑞02 + 𝑞12 𝜔𝑘2 (11) with 𝜔𝑗 2 2 2 2 1/2 [ 2𝑝0 + 𝑝1 − 𝑞1 + √(2𝑝0 + 𝑝1 − 𝑞1 ) − 4 (𝑝0 − 𝑞0 ) ] =[ ] , 2 [ ] (12) 2 2 2 where 𝑘 = 1, 2, . . ., 𝑗 = 0, 1, 2, . . .. If (𝑃1 ) and (𝐻2 ) are satisfied, (6) has a pair of purely imaginary roots ±𝜔0 𝑖 when Abs (...truncated)