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Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period

Introduction

Since the pioneering work of Kermack and McKendrick on compartment modeling, mathematical modeling has become an important tool in analyzing the spread and control of infectious diseases. Recently, 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.

Delays play an important role in the dynamics of populations. In many real-world processes, especially, in a lot of biological phenomena, the present dynamics of the state variables depends not only on the present state of the processes but also on the history of the phenomenon, that is, on the past values of state variables. The time delay may influence the dynamics of infectious diseases. In fact, many diseases have different kinds of delays when they spread, such as immunity period delay [7, 8], infection period delay, and incubation period delay [10–14]. It is well known that the dynamical behaviors (including stability, attractivity, persistence, periodic oscillation, bifurcation, and chaos) of population models with time delay have become a subject of intense research activities. In particular, the properties of periodic solutions arising from the Hopf bifurcation are of great interest. A number of epidemic models with time delay have been developed in the literature to gain insights into the effect of time delay on the dynamic behavior of the model (see, e.g., [15–27]). Li et al. investigated the existence of a positive solution and local stability for the steady state of an age-structured SEIR epidemic model. Röst and Wu analyzed the global stability of an SEIR model with distributed infinite time delay when the infectivity depends on the age of infection. Gao et al. 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 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.

In addition, the course of an epidemic depends on the contact rate between susceptible and infected individuals and on the assumption that the net rate at which infections are acquired is proportional to the number of encounters between susceptible and infected individuals denoted by S and I, respectively. The constant of proportionality β is sometimes called the transmission coefficient. This transmission coefficient may well depend on the population size. If the total population size N is not too large, then the bilinear incidence, denoted by \documentclass[12pt]{minimal}

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\begin{document}$\beta S I$\end{document}βSI, is proper for the model because the number of adequate contacts by an individual per unit time should increase as the total population size N increases. On the other hand, if the population size N is quite large, then the standard incidence, denoted by \documentclass[12pt]{minimal}

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\begin{document}$\beta S I/N$\end{document}βSI/N, is more realistic. These two incidences are widely used in modeling the transmission dynamics of the human diseases [13, 28, 30]. Thus, the formulation of the incidence function is an important aspect of the mathematical study of epidemiology.

In view of the above, the aim of this paper is to formulate and analyze a delayed SEIR epidemic model, in which the latent and infected states are infective, for the occurrence of Hopf bifurcation. The paper is organized as follows. In Sect. 2, we present a delayed SEIR epidemic model with infectious force in latent and infected periods and give the basic properties of the model. The local and global asymptotic stabilities of disease-free equilibrium are established in Sect. 3. The local stability of the endemic equilibrium and sufficient and necessary conditions for the existence of the Hopf bifurcation are analyzed in Sect. 4. In Sect. 5, when the model exhibits the Hopf bifurcation, we employ the normal form theory and center manifold approach to derive formulas for determining the direction and stability of bifurcating periodic solutions. Numerical simulations are carried out in Sect. 6 to illustrate the main theoretical results, and a brief discussion is given in Sect. 7 to conclude this work.

Definition 2.1

([27])

The positive equilibrium is absolutely stable if it is asymptotically stable for every delay \documentclass[12pt]{minimal}

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\begin{document}$\tau\geq0$\end{document}τ≥0 and is conditionally stable if it is asymptotically stable for τ in some finite interval.

Theorem 3.1

The disease-free equilibrium

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\begin{document}$\mathcal{E}_{0}$\end{document}E0

of model (2.1) is

(i)absolutely stable if

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\begin{document}$R_{0}<1$\end{document}R0<1,(ii)linearly neutrally stable if

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\begin{document}$R_{0}=1$\end{document}R0=1, and(iii)unstable if

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\begin{document}$R_{0} > 1$\end{document}R0>1.

Proof

By linearization the Jacobian of system (2.1) evaluated at \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}_{0}$\end{document}E0 is given by

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\begin{document}$\lambda_{1} = \lambda_{2} = -\mu$\end{document}λ1=λ2=−μ and the roots of the transcendental polynomial

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\begin{document}$$ f(\lambda)= \lambda^{2}+ (k_{1}+k_{2} )\lambda +k_{1}k_{2}- (\beta_{1} \lambda+k_{1} k_{2} R_{0} )e^{-\lambda\tau}=0. $$\end{document}f(λ)=λ2+(k1+k2)λ+k1k2−(β1λ+k1k2R0)e−λτ=0.

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\begin{document}$\tau=0$\end{document}τ=0, (3.1) reduces to

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\begin{document}$$ f(\lambda)= \lambda^{2}+ \biggl(k_{2}+\frac{\sigma\beta _{2}}{k_{2}}+k_{1}(1-R_{0}) \biggr)\lambda +k_{1}k_{2}(1-R_{0})=0. $$\end{document}f(λ)=λ2+(k2+σβ2k2+k1(1−R0))λ+k1k2(1−R0)=0. It is easy to see that if \documentclass[12pt]{minimal}

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\begin{document}$R_{0} <1$\end{document}R0<1, then the roots of (3.2) have negative real parts. Thus the disease-free equilibrium is locally asymptotically stable when \documentclass[12pt]{minimal}

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\begin{document}$\tau=0$\end{document}τ=0.

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\begin{document}$\lambda=i\omega(\omega>0)$\end{document}λ=iω(ω>0) be the root of (3.1). After substituting and separating into real and imaginary parts, we have

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\begin{document}$$ \begin{aligned}& \beta_{1}\omega\sin\omega \tau+k_{1} k_{2} R_{0} \cos\omega\tau = - \omega^{2} +k_{1}k_{2}, \\ &{-}k_{1} k_{2} R_{0}\omega\sin\omega\tau+ \beta_{1}\omega\cos\omega\tau = (k_{1} + k_{2}) \omega, \end{aligned} $$\end{document}β1ωsinωτ+k1k2R0cosωτ=−ω2+k1k2,−k1k2R0ωsinωτ+β1ωcosωτ=(k1+k2)ω, which implies

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\begin{document}$$ \omega^{4}+a_{1}\omega^{2}+a_{0}=0, $$\end{document}ω4+a1ω2+a0=0, where

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\begin{document} $$\begin{aligned} &a_{0} = k_{1}^{2}k_{2}^{2} \bigl(1-R_{0}^{2}\bigr), \\ &a_{1} = k_{2}^{2}+(k_{1}+ \beta_{1}) \biggl(\frac{\beta_{2}\sigma }{k_{2}}+k_{1}(1-R_{0}) \biggr). \end{aligned}$$ \end{document}a0=k12k22(1−R02),a1=k22+(k1+β1)(β2σk2+k1(1−R0)). Since \documentclass[12pt]{minimal}

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\begin{document}$\omega^{2}$\end{document}ω2 is negative. The contradiction shows that (3.1) has no purely imaginary root for \documentclass[12pt]{minimal}

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\begin{document}$\tau>0$\end{document}τ>0. Hence, by Definition 2.1 and Lemma 3.5(i), the disease-free equilibrium \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}_{0}$\end{document}E0 is absolutely stable for \documentclass[12pt]{minimal}

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\begin{document}$\tau\geq0$\end{document}τ≥0.

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\begin{document}$R_{0}=1$\end{document}R0=1, then the transcendental polynomial (3.1) becomes

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\begin{document}$$ \lambda^{2}+ (k_{1}+k_{2} )\lambda +k_{1}k_{2}- (\beta_{1} \lambda+k_{1}k_{2} )e^{-\lambda\tau}=0. $$\end{document}λ2+(k1+k2)λ+k1k2−(β1λ+k1k2)e−λτ=0. It is clear that \documentclass[12pt]{minimal}

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\begin{document}$\lambda= 0$\end{document}λ=0 is a simple root of (3.5). We will further show that any root of (3.5) must have a negative real part except \documentclass[12pt]{minimal}

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\begin{document} $$\begin{aligned} &{u}^{2}-{\omega}^{2}+(k_{{2}}+k_{{1}})u+k_{{1}}k_{{2}} = \bigl( ( \beta_{{1}}u+k_{{1}}k_{{2}} ) \cos ( \omega \tau ) - \beta_{{1}}\omega \sin ( \omega\tau ) \bigr) {e}^{- u\tau} , \\ &\omega ( k_{{2}}+k_{{1}}+2 u ) = \bigl( ( \beta_{{ 1}}u+k_{{1}}k_{{2}} ) \sin ( \omega\tau ) + \beta_{{ 1}}\omega \cos ( \omega\tau ) \bigr) {e}^{- u\tau}, \end{aligned}$$ \end{document}u2−ω2+(k2+k1)u+k1k2=((β1u+k1k2)cos(ωτ)−β1ωsin(ωτ))e−uτ,ω(k2+k1+2u)=((β1u+k1k2)sin(ωτ)+β1ωcos(ωτ))e−uτ, which, together with \documentclass[12pt]{minimal}

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\begin{document}$u \geq0$\end{document}u≥0, implies that

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\begin{document} $$\begin{aligned} &\bigl( {u}^{2}+2 k_{{2}}u+{\omega}^{2}+{k_{{2}}}^{2} \bigr) \bigl( {u}^{2}+2 k_{{1}}u+{\omega}^{2}+{k_{{1}}}^{2} \bigr) \\ &\quad= \bigl[ ( \beta_{{1}}u+k_{{1}}k_{{2}} ) ^{2}+{\beta_{{1}}}^{2}{ \omega}^{2} \bigr]{e}^{-2 u\tau} \\ &\quad\leq ( \beta_{{1}}u+k_{{1}}k_{{2}} ) ^{2}+{\beta_{{1}}}^{2}{ \omega}^{2}. \end{aligned}$$ \end{document}(u2+2k2u+ω2+k22)(u2+2k1u+ω2+k12)=[(β1u+k1k2)2+β12ω2]e−2uτ≤(β1u+k1k2)2+β12ω2. 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 \documentclass[12pt]{minimal}

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\begin{document}$R_{0} = 1$\end{document}R0=1.

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\begin{document}$(-\infty,\infty)$\end{document}(−∞,∞) it follows that the transcendental equation (3.1) has at least one positive real root. Hence \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}_{0}$\end{document}E0 is unstable. Therefore the theorem is proved.

Define the region

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\begin{document} $$\begin{aligned} \Omega=\bigl\{ (S,E,I,R) \in\mathbb{R}^{4}_{+}: S+E+I+R \leq \Pi/\mu\bigr\} . \end{aligned}$$ \end{document}Ω={(S,E,I,R)∈R+4:S+E+I+R≤Π/μ}. Adding all equations in (2.1) gives \documentclass[12pt]{minimal}

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\begin{document}$dN/dt = \Pi/\mu- \alpha I$\end{document}dN/dt=Π/μ−αI. Consequently, in the absence of infection, \documentclass[12pt]{minimal}

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\begin{document}$N(0)\leq\Pi/\mu$\end{document}N(0)≤Π/μ. Also, if \documentclass[12pt]{minimal}

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\begin{document}$N(0) > \Pi/\mu$\end{document}N(0)>Π/μ, then N will decrease to this level. Thus Ω is positively invariant with respect to system (2.1). The global stability of the disease-free equilibrium is therefore established in the following theorem.

Theorem 3.2

The disease-free equilibrium

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\begin{document}$\mathcal{E}_{0}$\end{document}E0

of system (2.1) is globally asymptotically stable in Ω if

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\begin{document}$R_{0}\leq1$\end{document}R0≤1.

Proof

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\begin{document}$x_{t}$\end{document}xt represent the translation of the solution of system (2.1) with initial conditions (2.2), that is, \documentclass[12pt]{minimal}

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\begin{document}$N(t+\theta)=S(t+\theta)+E(t+\theta)+I(t+\theta)+R(t+\theta)$\end{document}N(t+θ)=S(t+θ)+E(t+θ)+I(t+θ)+R(t+θ) for \documentclass[12pt]{minimal}

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\begin{document}$\theta\in[0,\infty)$\end{document}θ∈[0,∞). We introduce the Lyapunov function

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\begin{document}$$ V(x_{t})=\frac{\beta_{1}k_{2}+\beta_{2}\sigma}{k_{1}k_{2}}E+\frac {\beta_{2}}{k_{2}}I + \frac{\beta_{1}k_{2}+\beta_{2}\sigma}{k_{1}k_{2}} \int_{t-\tau }^{t} \biggl(\frac{\beta_{1}E(\theta)+\beta_{2}I(\theta)}{N(\theta )}S(\theta) \biggr) \,d\theta. $$\end{document}V(xt)=β1k2+β2σk1k2E+β2k2I+β1k2+β2σk1k2∫t−τt(β1E(θ)+β2I(θ)N(θ)S(θ))dθ. Note that \documentclass[12pt]{minimal}

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\begin{document}$V\geq0$\end{document}V≥0 along the solutions of system (2.1). In addition, \documentclass[12pt]{minimal}

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\begin{document}$V=0$\end{document}V=0 if and only if both E and I are zero. The derivative of V along the solutions of system (2.1) is given by

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\begin{document} $$\begin{aligned} \frac{dV}{dt}={}&\frac{\beta_{1}k_{2}+\beta_{2}\sigma }{k_{1}k_{2}} \biggl( \frac{\beta_{1}E(t-\tau)+\beta_{2}I(t-\tau )}{N(t-\tau)}S(t-\tau)-k_{1}E(t) \biggr) +\frac{\beta_{2}}{k_{2}} \bigl(\sigma E(t)-k_{2}I(t) \bigr) \\ &{}+\frac{\beta_{1}k_{2}+\beta_{2}\sigma}{k_{1}k_{2}} \biggl(\frac {\beta_{1}E(t)+\beta_{2}I(t)}{N(t)}S(t) \biggr) \\ &{}-\frac{\beta_{1}k_{2}+\beta_{2}\sigma}{k_{1}k_{2}} \biggl(\frac {\beta_{1}E(t-\tau)+\beta_{2}I(t-\tau)}{N(t-\tau)}S(t-\tau) \biggr) \\ = {}&\biggl(\frac{\beta_{1}k_{2}+\beta_{2}\sigma}{k_{1}k_{2}} \biggr) \frac{(\beta_{1}E(t)+\beta_{2}I(t))S(t)}{N(t)}-(\beta_{1}E+ \beta_{2} I) \\ ={}&(\beta_{1}E+\beta_{2} I) \biggl(R_{0}\frac{S(t)}{N(t)}-1 \biggr) \\ \leq{}&{-}(\beta_{1}E+ \beta_{2}I) (1-R_{0}). \end{aligned}$$ \end{document}dVdt=β1k2+β2σk1k2(β1E(t−τ)+β2I(t−τ)N(t−τ)S(t−τ)−k1E(t))+β2k2(σE(t)−k2I(t))+β1k2+β2σk1k2(β1E(t)+β2I(t)N(t)S(t))−β1k2+β2σk1k2(β1E(t−τ)+β2I(t−τ)N(t−τ)S(t−τ))=(β1k2+β2σk1k2)(β1E(t)+β2I(t))S(t)N(t)−(β1E+β2I)=(β1E+β2I)(R0S(t)N(t)−1)≤−(β1E+β2I)(1−R0). Thus \documentclass[12pt]{minimal}

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\begin{document}$dV/dt=0$\end{document}dV/dt=0 if and only if \documentclass[12pt]{minimal}

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\begin{document}$E=I=0$\end{document}E=I=0. Consequently, the maximum invariance set in \documentclass[12pt]{minimal}

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\begin{document}$\{(S,E,I,R)\in\Omega:dV/dt=0\}$\end{document}{(S,E,I,R)∈Ω:dV/dt=0} when \documentclass[12pt]{minimal}

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\begin{document}${\mathcal{E}_{0}}$\end{document}E0}. Therefore, the LaSalle’s invariance principle implies that \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}_{0}$\end{document}E0 is globally asymptotically stable in Ω. This proves the theorem.

Theorem 4.1

If

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\begin{document}$R_{0} > 1$\end{document}R0>1

and condition (4.8) holds, then the endemic equilibrium

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\begin{document}$\mathcal{E}^{*}$\end{document}E∗

of model (2.1) is absolutely stable for

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\begin{document}$\tau\geq0$\end{document}τ≥0.

Case 2. Rewriting r as

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\begin{document} $$\begin{aligned} r=-R_{{0}} \bigl[ k_{{1}}k_{{2}}R_{{0}}-(3k_{{1}}k_{{2}}- \sigma \alpha) \bigr] -\sigma\alpha, \end{aligned}$$ \end{document}r=−R0[k1k2R0−(3k1k2−σα)]−σα, we see that \documentclass[12pt]{minimal}

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\begin{document}$r<0$\end{document}r<0 if the following condition holds:

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\begin{document} $$\begin{aligned} R_{{0}} > 2+\frac{k_{1} k_{2}-\sigma\alpha}{k_{1} k_{2}}. \end{aligned}$$ \end{document}R0>2+k1k2−σαk1k2. This gives the following condition for the contact rate β:

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\begin{document}$$ \beta> \frac{3k_{1} k_{2}-\sigma \alpha}{\beta_{E}k_{2}+\beta_{I}\sigma}. $$\end{document}β>3k1k2−σαβEk2+βIσ. By Lemma 3.3(c), (4.3) has positive real roots, that is, the characteristic equation (4.6) has a pair of purely imaginary roots of the form \documentclass[12pt]{minimal}

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\begin{document}$\lambda=\pm i\omega_{0}$\end{document}λ=±iω0.

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\begin{document}$\omega=\omega_{0}$\end{document}ω=ω0 into (4.4)–(4.5) and solving for τ, we get the corresponding \documentclass[12pt]{minimal}

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\begin{document}$\tau_{n}>0$\end{document}τn>0, \documentclass[12pt]{minimal}

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\begin{document}$n=0,1,2,\dots$\end{document}n=0,1,2,…:

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\begin{document}$$ \tau_{n} = \frac{1}{\omega_{0}}\cos^{-1} \biggl\{ \frac{(\omega _{0}^{3}-a_{1}\omega_{0})b_{1}\omega_{0}-(a_{2}\omega_{0}^{2}-a_{0})(b_{2}\omega _{0}^{2}-b_{0})}{(b_{0}-b_{2}\omega_{0}^{2})^{2}+(b_{1}\omega_{0})^{2}} \biggr\} +\frac {2\pi n}{\omega_{0}}. $$\end{document}τn=1ω0cos−1{(ω03−a1ω0)b1ω0−(a2ω02−a0)(b2ω02−b0)(b0−b2ω02)2+(b1ω0)2}+2πnω0. By Lemma 3.4(c) in all roots of (4.3) have negative real parts for \documentclass[12pt]{minimal}

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\begin{document}$\tau\in[0,\tau_{0})$\end{document}τ∈[0,τ0). Therefore, by Lemma 3.5(ii), we obtain the following theorem.

Theorem 4.2

If

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\begin{document}$R_{0} > 1$\end{document}R0>1

and condition (4.9) holds, then the endemic equilibrium

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\begin{document}$\mathcal{E}^{*}$\end{document}E∗

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\begin{document}$\tau\in [0,\tau_{0})$\end{document}τ∈[0,τ0).

For the bifurcation analysis, the time delay τ is chosen as the bifurcation parameter, and we will show that there exists at least one eigenvalue with positive real part for \documentclass[12pt]{minimal}

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\begin{document}$\frac{d(\operatorname{Re}\lambda)}{d\tau} \vert _{\tau=\tau_{0}}>0$\end{document}d(Reλ)dτ|τ=τ0>0.

The derivative of (4.3) with respect to τ is given by

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\begin{document} $$\begin{aligned} &(\lambda+\mu) \biggl\{ \bigl(3\lambda^{2}+2a_{2} \lambda+a_{1}\bigr)\frac{d\lambda }{d\tau}- \bigl(b_{2} \lambda^{2}+b_{1}\lambda+b_{0}\bigr)e^{-\lambda\tau} \biggl(\lambda+\tau \frac{d\lambda}{d\tau} \biggr) \\ &\quad{}+e^{-\lambda\tau}(2b_{2} \lambda+b_{1})\frac{d\lambda}{d\tau} \biggr\} \\ &\quad{}+ \bigl(\lambda^{3}+a_{2}\lambda^{2}+a_{1} \lambda+a_{0}+\bigl(b_{2}\lambda^{2} +b_{1}\lambda+b_{0}\bigr)e^{-\lambda\tau} \bigr) \frac{d\lambda}{d\tau}=0. \end{aligned}$$ \end{document}(λ+μ){(3λ2+2a2λ+a1)dλdτ−(b2λ2+b1λ+b0)e−λτ(λ+τdλdτ)+e−λτ(2b2λ+b1)dλdτ}+(λ3+a2λ2+a1λ+a0+(b2λ2+b1λ+b0)e−λτ)dλdτ=0. After rearranging this equation, we get

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\begin{document} $$\begin{aligned} \biggl(\frac{d\lambda}{d\tau} \biggr)^{-1} = \frac{2\lambda^{3}+a_{2}\lambda^{2}-a_{0}}{-\lambda^{2}(\lambda ^{3}+a_{2}\lambda^{2}+a_{1}\lambda+a_{0})}+\frac{b_{2}\lambda^{2}-b_{0}}{\lambda ^{2}(b_{2}\lambda^{2}+b_{1}\lambda+b_{0})}-\frac{\tau}{\lambda}. \end{aligned}$$ \end{document}(dλdτ)−1=2λ3+a2λ2−a0−λ2(λ3+a2λ2+a1λ+a0)+b2λ2−b0λ2(b2λ2+b1λ+b0)−τλ. Therefore

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\begin{document} $$\begin{aligned} &\operatorname{sign} \biggl\{ \frac{d(\operatorname{Re}\lambda)}{d\tau} \biggr\} _{\tau=\tau_{0}} \\ &\quad= \operatorname{sign} \biggl\{ \operatorname{Re} \biggl(\frac{d\lambda }{d\tau} \biggr)^{-1}_{\lambda=\text{i}\omega_{0}} \biggr\} \\ &\quad= \operatorname{sign} \biggl\{ \frac{(a_{2}\omega^{2}+a_{0})(a_{2}\omega ^{2}-a_{0})-2\omega^{3}(a_{1}\omega-\omega^{3})+(b_{0}+b_{2}\omega ^{2})(b_{0}-b_{2}\omega^{2})}{\omega^{2}[(b_{0}-b_{2}\omega^{2})^{2}+(b_{1}\omega )^{2}]} \biggr\} \\ &\quad= \operatorname{sign} \biggl\{ \frac{-(a_{0}^{2}-a_{2}^{2}\omega^{4})-2a_{1}\omega ^{4}+2\omega^{6}+b_{0}^{2}-b_{2}^{2}\omega^{4}}{\omega^{2}[(b_{0}-b_{2}\omega ^{2})^{2}+(b_{1}\omega)^{2}]} \biggr\} \\ &\quad = \operatorname{sign} \biggl\{ \frac{2\omega^{6}+p\omega^{4}-r}{\omega ^{2}[(b_{0}-b_{2}\omega^{2})^{2}+(b_{1}\omega)^{2}]} \biggr\} . \end{aligned}$$ \end{document}sign{d(Reλ)dτ}τ=τ0=sign{Re(dλdτ)λ=iω0−1}=sign{(a2ω2+a0)(a2ω2−a0)−2ω3(a1ω−ω3)+(b0+b2ω2)(b0−b2ω2)ω2[(b0−b2ω2)2+(b1ω)2]}=sign{−(a02−a22ω4)−2a1ω4+2ω6+b02−b22ω4ω2[(b0−b2ω2)2+(b1ω)2]}=sign{2ω6+pω4−rω2[(b0−b2ω2)2+(b1ω)2]}. Here, \documentclass[12pt]{minimal}

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\begin{document}$p>0$\end{document}p>0 and \documentclass[12pt]{minimal}

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\begin{document}$r>0$\end{document}r>0 under condition (4.9). Thus \documentclass[12pt]{minimal}

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\begin{document}$\frac{d(\operatorname{Re}\lambda)}{d\tau} \vert _{\tau=\tau_{0}}>0$\end{document}d(Reλ)dτ|τ=τ0>0. This result shows that the root of characteristic (4.2) crosses the imaginary axis from left to right as τ continuously varies from a number less than \documentclass[12pt]{minimal}

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\begin{document}$\tau_{0}$\end{document}τ0. Therefore, the conditions for Hopf bifurcation are satisfied at \documentclass[12pt]{minimal}

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\begin{document}$\tau=\tau_{0}$\end{document}τ=τ0. From Theorem 4.2 and our analysis we obtain the following theorem.

Theorem 4.3

Suppose that

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\begin{document}$\mathcal{R}_{0}>1$\end{document}R0>1. Then the endemic equilibrium

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\begin{document}$\mathcal{E}^{*}$\end{document}E∗

of model (2.1) is

(i)

absolutely stable for

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\begin{document}$\tau\geq0$\end{document}τ≥0

whenever

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\begin{document}$\frac{k_{1} k_{2}}{\beta_{E} k_{2} + \beta_{I} \sigma}<\beta< \frac{2 k_{1} k_{2} -\sigma\alpha}{\beta_{E} k_{2} + \beta_{I} \sigma}$\end{document}k1k2βEk2+βIσ<β<2k1k2−σαβEk2+βIσ

and

(ii)conditionally stable for

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\begin{document}$\tau\in[0,\tau_{0})$\end{document}τ∈[0,τ0)

whenever

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\begin{document}$\beta> \frac{3k_{1} k_{2}-\sigma \alpha}{\beta_{E}k_{2}+\beta_{I}\sigma}$\end{document}β>3k1k2−σαβEk2+βIσ. System (2.1) with

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\begin{document}$\tau=\tau_{0}$\end{document}τ=τ0

given in (4.10) undergoes a Hopf bifurcation.

Theorem 5.1

For delayed model (2.1), when

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\begin{document}$\tau=\tau_{0}$\end{document}τ=τ0, the direction and stability of a periodic solution of Hopf bifurcation are determined by considering the signs of

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\begin{document}$\tilde{\beta}_{2}$\end{document}β˜2, and

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\begin{document}$T_{2}$\end{document}T2, respectively, given in (5.26). Then

(i)if

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\begin{document}$\mu_{2}<0$\end{document}μ2<0

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\begin{document}$(\mu_{2}>0)$\end{document}(μ2>0), then the Hopf bifurcation is subcritical (supercritical) and the bifurcation periodic solutions exist for

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\begin{document}$\tau<\tau_{0}$\end{document}τ<τ0

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\begin{document}$(\tau>\tau_{0})$\end{document}(τ>τ0);(ii)if

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\begin{document}$\tilde{\beta}_{2}>0$\end{document}β˜2>0

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\begin{document}$(\tilde{\beta}_{2}<0)$\end{document}(β˜2<0), then the bifurcation periodic solutions are unstable (stable);(iii)if

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\begin{document}$T_{2}<0$\end{document}T2<0

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\begin{document}$(T_{2}>0)$\end{document}(T2>0), then the period of the bifurcating periodic solutions decreases (increases).

Numerical simulations

To illustrate the dynamic behavior and the phenomenon of Hopf bifurcation of a delayed SEIR epidemic model, we integrate system (2.1) numerically by using the standard MATLAB algorithm with the parameter values/ranges in Table 1.

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\begin{document}$R_{0} = 0.085<1$\end{document}R0=0.085<1. As is evident from Fig. 1, whenever \documentclass[12pt]{minimal}

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\begin{document}$R_{0} < 1$\end{document}R0<1, the solution profiles converge to a disease-free equilibrium \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}_{0}$\end{document}E0 for any chosen time delay τ, as in Theorems 3.1(i) and 3.2. By comparing with \documentclass[12pt]{minimal}

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\begin{document}$\tau=0$\end{document}τ=0, time delay has effect to the profiles of exposed and infectious individuals, making them oscillately converge as shown in Fig. 1(a), (d). On the other hand, the time delay has no impact on the profiles of susceptible and recovered individuals as τ increases; see Fig. 1(b,c). These results can be interpreted so that the disease is delayed and eventually extinct, that is, the disease disappears in the population.

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\begin{document}$R_{0} > 1$\end{document}R0>1, the dynamics behavior of model (2.1) is explored with various contact rates and time delays. The contact rate β is chosen to be \documentclass[12pt]{minimal}

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\begin{document}$\beta= 3.6$\end{document}β=3.6 and \documentclass[12pt]{minimal}

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\begin{document}$\beta= 7.2875$\end{document}β=7.2875. It is found that, when \documentclass[12pt]{minimal}

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\begin{document}$\beta= 3.6$\end{document}β=3.6, the condition in Theorem 4.3(i) holds. It is seen that all solutions of model (2.1) converge to an endemic equilibrium \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}^{*}$\end{document}E∗ for all chosen τ; see Fig. 2. This verifies that the endemic equilibrium of (2.1) is absolutely stable, as guaranteed by Theorem 4.3(i). The results also show that the qualitative behavior of the model does not change as time delay increases.

Biologically, we observe that, as time delay increases, the numbers of exposed and infectious individuals decrease (see Figs. 2(b), (c)), whereas the numbers of susceptible and recovered individuals increase (see Figs. 2 (a,d)) due reduction in the chance of infection of susceptible individuals, and infectious population recovers from the disease (then they become members of the recovered group).

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\begin{document}$\beta= 7.2875$\end{document}β=7.2875 with the other parameters in Table 1, the condition in Theorem 4.3(ii) holds. Further, we have \documentclass[12pt]{minimal}

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\begin{document}$R_{0}=3.1$\end{document}R0=3.1, an endemic equilibrium \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}^{*}=(1.9243\times10^{5},30.2160,47.8420,4.0403 \times 10^{5})$\end{document}E∗=(1.9243×105,30.2160,47.8420,4.0403×105), and the critical time delay \documentclass[12pt]{minimal}

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\begin{document}$\tau_{0}=25.86$\end{document}τ0=25.86. The solutions of model (2.1) as τ increases are illustrated in Figs. 3–5. We have found that \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}^{*}$\end{document}E∗ is asymptotically stable when \documentclass[12pt]{minimal}

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\begin{document}$\tau= 25 <\tau_{0}$\end{document}τ=25<τ0 (see Fig. 3), limit circle when \documentclass[12pt]{minimal}

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\begin{document}$\tau\approx \tau_{0}$\end{document}τ≈τ0 (see Fig. 4), and asymptotically unstable when (see Fig. 5), respectively. Furthermore, we can calculate the following values: \documentclass[12pt]{minimal}

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\begin{document}$c_{1}(0)=1.6548\times 10^{-13}-3.5212\times10^{-11}\text{i}, \mu_{2}=-4.6952\times10^{-7}, \tilde{\beta}_{2}=3.3096\times10^{-13}$\end{document}c1(0)=1.6548×10−13−3.5212×10−11i,μ2=−4.6952×10−7,β˜2=3.3096×10−13, which verify that the endemic equilibrium \documentclass[12pt]{minimal}

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\begin{document}$0< \tau<\tau_{0}$\end{document}0<τ<τ0 (see Fig. 3); when \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}^{*}$\end{document}E∗ loses its stability (see Fig. 5), and a Hopf bifurcation occurs at \documentclass[12pt]{minimal}

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\begin{document}$\tau\approx \tau_{0}$\end{document}τ≈τ0 (see Fig. 4), that is, a family of periodic solutions bifurcate from \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}^{*}$\end{document}E∗ (see Fig. 5), as guaranteed by Theorem 5.1.

In addition, we see that the critical time delay for Hopf bifurcation is a large number (\documentclass[12pt]{minimal}

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\begin{document}$\tau_{0}=25.86$\end{document}τ0=25.86), which is realistic in the study of the effect of time delay in an epidemic model because adding a time delay in the model destabilizes the system and periodic solutions can arise through Hopf bifurcation, which impacts the effectiveness of disease control. If Hopf bifurcation, therefore, occurs at a large time delay, then the authorities involved with disease control may have enough time to act before the exposed individuals can become infective and infect other members of the population.

Conclusion

This paper presents a delayed SEIR epidemic model with infectious force in latent and infected periods for studying the existence of Hopf bifurcation. The model is rigorously analyzed to gain insight into its dynamical features. The study results are summarized as follows. By using the Lyapunov functional method and the LaSalle invariance principle, the disease-free equilibrium is globally asymptotically stable if a certain threshold quantity, known as the reproductive number and denoted by \documentclass[12pt]{minimal}

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\begin{document}$R_{0}> 1$\end{document}R0>1, the contact rate β and time delay τ are regraded as bifurcated parameters. The study results show that if the contact rate β satisfies condition (4.8), then the endemic equilibrium \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}^{*}$\end{document}E∗ of model (2.1) is absolutely stable, that is, \documentclass[12pt]{minimal}

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\begin{document}$\tau\geq0$\end{document}τ≥0. Meanwhile, if the contact rate β satisfies condition (4.9), then the endemic equilibrium, \documentclass[12pt]{minimal}

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\begin{document}$\mathcal{E}^{*}$\end{document}E∗ of model (2.1) is conditionally stable, that is, \documentclass[12pt]{minimal}

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\begin{document}$\tau\in[0,\tau_{0})$\end{document}τ∈[0,τ0), and the Hopf bifurcation occurs at \documentclass[12pt]{minimal}

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\begin{document}$\tau= \tau_{0}$\end{document}τ=τ0. It is observed that the delayed SEIR epidemic model with infectious force in latent and infected period (2.1) exhibits a Hopf bifurcation, called subcritical, which is a different result from the epidemic models with bilinear incidence rate and nonlinear incidence rate that exhibit supercritical Hopf bifurcation; see [27, 41–44]. This gives the new result that the type of Hopf bifurcation depends on the type of incidence function used in the epidemic model. In addition, the phenomenon of Hopf bifurcation in the delayed SEIR epidemic model with infectious force in latent and infected period depends on contact rate in the sense that the contact rate is a crucial condition to ensure the Hopf bifurcation and time delay can cause the loss of stability via subcritical Hopf bifurcation at the critical time delay \documentclass[12pt]{minimal}

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\begin{document}$\tau= \tau_{0}$\end{document}τ=τ0.

In terms of disease control campaigns, this study result shows that the infection rate can be effectively controlled in a community if some public health measures are initiated that can reduce the contact rate. There exists an endemic equilibrium state, which is asymptotically stable, and the transmission of the disease seems to happen immediately (without any delay). Besides public health education, there is another way to destabilize this state and make the education more effective: by the management and care of exposed individuals in a timely fashion at the supervision or the direction of a legally qualified medical practitioner. So, delay in diagnosis and treatment of disease is one of the reasons for the failure in the control of the disease.