Abstract
In this paper, we analyze a delayed SEIR epidemic model in which the latent and infected states are infective. The model has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number R_{0}, is less than or equal to unity. We investigate the effect of the time delay on the stability of endemic equilibrium when R_{0}>1. We give criteria that ensure that endemic equilibrium is asymptotically stable for all time delays and a Hopf bifurcation occurs as time delay exceeds the critical value. We give formulae for the direction of Hopf bifurcations and the stability of bifurcated periodic solutions by applying the normal form theory and the center manifold reduction for functional differential equations. Numerical simulations are presented to illustrate the analytical results.
Highlights
Since the pioneering work of Kermack and McKendrick [1] on compartment modeling, mathematical modeling has become an important tool in analyzing the spread and control of infectious diseases
Great attention has been paid to developing realistic mathematical models for the transmission dynamics of infectious diseases, such as the severe acute respiratory syndromes (SARS) outbreak in 2003 [2, 3], the avian influenza A (H7N9) outbreak in China in 2013 [4, 5], and potential mechanisms behind the spread of AH1N1 influenza virus in different regions around the world [6]
It is well known that the dynamical behaviors of population models with time delay have become a subject of intense research activities
Summary
Since the pioneering work of Kermack and McKendrick [1] on compartment modeling, mathematical modeling has become an important tool in analyzing the spread and control of infectious diseases. Gao et al [17] formulated an SEIR epidemic model with two time delays and pulse vaccination for studying the control of spread and transmission of an infectious disease. Tipsri and Chinviriyasit [27] investigated the effect of time delay on the stability of bifurcating periodic solutions and direction of Hopf bifurcation of an SEIR model with nonlinear incidence. The local and global asymptotic stabilities of disease-free equilibrium are established in Sect. It is easy to check that this inequality is not true This shows that any root of (3.5) has a negative real part except λ = 0, which implies that E0 is linearly neutrally stable when R0 = 1. From the continuity of the function f (λ) on (–∞, ∞) it follows that the transcendental equation (3.1) has at least one positive real root.
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