DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi 10.1155/2017/2340549 2340549 Research Article Bifurcation of a Delayed SEIS Epidemic Model with a Changing Delitescence and Nonlinear Incidence Rate http://orcid.org/0000-0003-3209-143X Liu Juan 1 Yang Lu-Xing Department of Mathematics and Physics Bengbu University Bengbu 233030 China bbc.edu.cn 2017 952017 2017 16 02 2017 28 03 2017 952017 2017 Copyright © 2017 Juan Liu. 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.

This paper is concerned with a delayed SEIS (Susceptible-Exposed-Infectious-Susceptible) epidemic model with a changing delitescence and nonlinear incidence rate. First of all, local stability of the endemic equilibrium and the existence of a Hopf bifurcation are studied by choosing the time delay as the bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are determined based on the normal form theory and the center manifold theorem. At last, numerical simulations are carried out to illustrate the obtained theoretical results.

Natural Science Foundation of the Higher Education Institutions of Anhui Province KJ2015A144
1. Introduction

The outbreak of infectious diseases had not only caused the loss of billions of lives but also badly damaged the social economy in a short time, which brought much pain to human society . Thus, it has been an increasingly urgent issue to understand how to prevent or slow down the transmission of infectious diseases. To this end, many mathematical models have been proposed for describing the spread process of infectious diseases . However, all the epidemic models above do not consider the change of delitescence of the infectious diseases. Considering that the diversity of the delitescence period in each infected individual who is infected with disease virus is mainly due to the variation of the virus and the distinct constitution of different people for some disease, such as H1N1 disease, Wang proposed the following SEIS epidemic model with a changing delitescence and a nonlinear incidence rate :(1)dStdt=A-dSt-βStIt1+αIt+γIt,dEtdt=μβStIt1+αIt-d+εEt,dItdt=1-μβStIt1+αIt+εEt-d+γ+δIt,where S(t), E(t), and I(t) denote the numbers of the susceptible, exposed, and infectious populations at time t, respectively. A is the recruitment rate of the susceptible population; d is the natural death rate of the population; δ is the death rate due to the disease of the infected population; ε is the rate at which the exposed population becomes infectious; γ is the rate at which the infected population returns to the susceptible population because of the treatment; μ is the rate at which the infected population becomes the exposed one; and 1-μ is the rate at which the infected population becomes infectious directly. βSI/(1+αI) is the nonlinear incidence rate, where β measures the infection force of the disease and α measures the inhibition effect from the behavioral change of the susceptible population. Wang investigated global stability of system (1).

In fact, many infectious diseases have different kinds of delays during their spreading process in the population, such as latent period delay [9, 1216], immunity period delay [17, 18], and infection period delay . The time delay may induce Hopf bifurcation and periodic solutions. The occurrence of a Hopf bifurcation means that the state of the epidemic disease prevalence changes from an equilibrium to a limit cycle. Therefore, the time delay can influence the dynamics of infectious diseases. So it is necessary and useful to investigate system (1) with time delay. Based on this fact and taking the period used to cure the infectious population, we consider the following delayed epidemic system:(2)dStdt=A-dSt-βStIt1+αIt+γIt-τ,dEtdt=μβStIt1+αIt-d+εEt,dItdt=1-μβStIt1+αIt+εEt-d+δIt-γIt-τ,where τ is the time delay due to the period that is used to cure the infectious population. That is, we assume that all the infectious populations will survive after time τ. The initial conditions for system (2) are (3)ϕ1θ,ϕ2θ,ϕ3θC=C-τ,0,R+3,ϕ1θ>0,ϕ2θ>0,ϕ3θ>0,where R+3=(S,E,I)R+3.

The outline of this paper is as follows. In the next section, stability of the endemic equilibrium is analyzed and the critical value of the time delay at which a Hopf bifurcation occurs is obtained. In Section 3, direction and stability of the Hopf bifurcation are investigated. In Section 4, the obtained theoretical results are verified by some numerical simulations. Finally, this work is summarized in Section 5.

2. Stability of the Endemic Equilibrium and Existence of Hopf Bifurcation

By a direct computation, we know that if (I) b2=0 and b0/b1<0, (II) b12-4b0b2>0 and b0/b2<0, (III) b12-4b0b2>0, b0=0 and b1/b2<0, or (IV) b12-4b0b2=0 and b1/b2<0, then system (2) has a unique endemic equilibrium P(S,E,I), where(4)S=d+εd+γ+δ1+αIβ1-μd+ε+μβε,E=μβSId+ε1+αI,and I is the unique positive root of the following equation:(5)b2I2+b1I+b0=0,where(6)b0=dd+εd+γ+ε-Aβ1-μd+ε+με,b1=d+ε2dα+βd+γ+ε-β1-μd+ε+μεAα+γ,b2=αd+εdα+βd+γ+ε.

Let u1(t)=S(t)-S, u2(t)=E(t)-E, u3(t)=I(t)-I. We can rewrite system (2) as the following form:(7)u˙1t=a11u1t+a13u3t+b13u3t-τ+i+j21i!j!fij1u1itu3jt,u˙2t=a21u1t+a22u2t+a23u3t+i+j21i!j!fij2u1itu3jt,u˙3t=a31u1t+a32u2t+a33u3t+b33u3t-τ+i+j21i!j!fij3u1itu3jt,where(8)a11=-d+βI1+αI,a13=-βS1+αI,b13=γ,a21=μβI1+αI,a22=-d+ε,a23=μβS1+αI,a31=1-μβI1+αI,a32=ε,a33=1-μβI1+αI-d+δ,b33=-γ,fijk=i+jfkS,E,Iu1itu3jt,f1=A-du1t-βu1tu3t1+αu3t+γu3t-τ,f2=μβu1tu3t1+αu3t-d+εu2t,f3=1-μβu1tu3t1+αu3t+εu2t-d+δu3t-γu3t-τ.Then we obtain the linearized system of system (2)(9)u˙1t=a11u1t+a13u3t+b13u3t-τ,u˙2t=a21u1t+a22u2t+a23u3t,u˙3t=a31u1t+a32u2t+a33u3t+b33u3t-τ.The characteristic equation is(10)λ3+A2λ2+A1λ+A0+B2λ2+B1λ+B0e-λτ=0,where(11)A0=a13a22a31-a21a32+a11a23a32-a22a33,A1=a11a22+a22a33+a11a33A2=-a11+a22+a33,B0=b13a22a31-a21a32-a11a22b33,B1=b33a11+a22-a13b13,B2=-b33.When τ=0, (10) reduces to(12)λ3+A2+B2λ2+A1+B1λ+A0+B0=0.

Routh-Hurwitz criterion implies that P is locally asymptotically stable without delay if condition (H1) holds.

(H1) A2+B2>0, (A2+B2)(A1+B1)>A0+B0>0.

For τ>0, substituting λ=iω (ω>0) into (10), we obtain(13)B1ωsinτω+B0-B2ω2cosτω=A2ω2-A0,B1ωcosτω-B0-B2ω2sinτω=ω3-A1ω.Then(14)ω6+a2ω4+a1ω2+a0=0,where(15)a0=A02-B02,a1=A12-2A0A2-B12+2B0B2,A2=A22-2A1-B22.Let ω2=v; then(16)v3+a2v2+a1v+a0=0,where f(v)=v3+a2v2+a1v+a0. According to the analysis about the distribution of roots of (16) in Song et al. , we have the following result.

Lemma 1.

For the polynomial equation (16),

if a0<0, then (16) has at least one positive root;

if a00 and =a22-3a10, then (16) has no positive roots;

if a00 and =a22-3a1>0, then (16) has positive roots if and only if v1=(-a2+)/3>0 and f(v1)0.

Next, we assume that the coefficients in (16) satisfy the following condition.

( H 2 ) (i) a0<0 or (ii) a00, =a22-3a1>0, v1=(-a2+)/3>0, and f(v1)0.

Thus, (14) has at least one positive root such that (10) has a pair of purely imaginary roots ±iω0. The corresponding critical value τ0 can be obtained from (13)(17)τ0=1ω0arccosB1-A2B2ω04+A2B0+A0B2-A1B1ω02-A0B0B12ω02+B0-B2ω022.Taking derivative with respect to τ on both sides of (10), we obtain(18)dλdτ-1=-3λ2+2A2λ+A1λλ3+A2λ2+A1λ+A0+2B2λ+B1λB2λ2+B1λ+B0-τλ.Further, we have(19)Redλdττ=τ0-1=fω02B12ω02+B0-B2ω022.

Thus, if the condition (H3): f(ω02)0 holds, then Redλ/dττ=τ0-10. Then, based on the Hopf bifurcation theorem in , we have the following.

Theorem 2.

For system (2), if the conditions (H1)(H3) hold, then the endemic equilibrium E(S,E,I) of system (2) is asymptotically stable for τ[0,τ0) and system (2) undergoes a Hopf bifurcation at the endemic equilibrium E(S,E,I) when τ=τ0, where τ0 is defined in (17).

3. Direction and Stability of the Hopf Bifurcation

Let τ=τ0+μ, μR; then μ=0 is the Hopf bifurcation value of system (2). Rescaling the time delay t(t/τ), then system (2) can be transformed into an FDE in C=C([-1,0],R3) as follows:(20)u˙t=Lμut+Fμ,ut,where(21)Lμϕ=τ0+μa110a13a21a22a23a31a32a33ϕ0+τ0+μ00b1300000b33ϕ-1,Fμ,ϕ=τ0+μF1,F2,F3T,where F1, F2, and F3 are defined by Appendix A.

By the Riesz representation theorem, there exists a 3×3 matrix function η(θ,μ), θ[-1,0], whose components are of bounded variation, such that(22)Lμϕ=-10dηθ,μϕθ,ϕC-1,0,R3.In fact, we choose(23)ηθ,μ=τ0+μa110a13a21a22a23a31a32a33ϕ0+τ0+μ00b1300000b33ϕ-1.For ϕC([-1,0],R3), we define(24)Aμϕ=dϕθdθ,-1θ<0,-10dηθ,μϕθ,θ=0,Rμϕ=0,-1θ<0,Fμ,ϕ,θ=0.Then system (20) is equivalent to(25)u˙t=Aμut+Rμut.For φC1([0,1]),(R3), the adjoint operator A of A is defined as(26)Aφ=-dφsds,0<s1,-10dηTs,0φ-s,s=0,and a bilinear inner product is defined by(27)φs,ϕθ=φ¯0ϕ0-θ=-10ξ=0θφ¯ξ-θdηθϕξdξ,where η(θ)=η(θ,0).

Let q(θ)=(1,q2,q3)Teiω0τ0θ be the eigenvector of A(0) belonging to +iω0τ0 and q(s)=D(1,q2,q3)eiω0τ0s be the eigenvector of A(0) belonging to -iω0τ0. By a direct computation, we can get(28)q2=a21+a23q3iω0-a22,q3=iω0-a11a13+b13e-iτ0ω0,q2=-a32q3iω0+a22,q3=a13+b13eiτ0ω0iω0+a22a23a32-iω0+a22iω0+a33+b33eiτ0ω0.From (27), we can get(29)qs,qθ=D¯1+q2q¯2+q3q¯3+τ0e-iτ0ω0q3b13+b33q¯3.Then we choose(30)D¯=1+q2q¯2+q3q¯3+τ0e-iτ0ω0q3b13+b33q¯3-1.such that q,q=1.

Next, we can obtain the coefficients g20, g11, g02, and g21 by using the method introduced in  and a computation process similar to that in . The expressions of g20, g11, g02, and g21 are defined by Appendix B.

Then, we can get the following coefficients which determine the properties of the Hopf bifurcation:(31)C10=i2τ0ω0g11g20-2g112-g0223+g212,μ2=-ReC10Reλτ0,β2=2ReC10,T2=-ImC10+μ2Imλτ0τ0ω0.In conclusion, we have the following results.

Theorem 3.

For system (2), if μ2>0  (μ2<0), then the Hopf bifurcation is supercritical (subcritical). If β2<0  (β2>0), then the bifurcating periodic solutions are stable (unstable). If T2>0  (T2<0), then the bifurcating periodic solutions increase (decrease).

4. Numerical Simulations

In order to verify the efficiency of the obtained results in the paper, we carry out some numerical simulations in this section. By extracting some values from  and considering the conditions for the existence of the Hopf bifurcation, we consider the special case of system (2) with the parameters A=5, d=0.01, β=0.5, α=0.4, γ=0.5, μ=0.65, ε=0.1, and δ=0.02, Then, system (2) becomes the following form:(32)dStdt=5-0.01St-0.5StIt1+0.4It+0.5It-τ,dEtdt=0.325StIt1+0.4It-0.11Et,dItdt=0.175StIt1+0.4It+0.1Et-0.03It-0.5It-τ,(33)0.0014I2-0.0999I-0.2584=0,from which we can obtain the unique positive root I=73.8562 and then we get the unique endemic equilibrium P(34.3750,245.5930,73.8562). Then, we can obtain ω0=0.3950, τ0=5.1686, and λ(τ0)=0.0012-0.0759i. Thus, based on Theorem 2, we know that the endemic equilibrium P(34.3750,245.5930,73.8562) is locally asymptotically stable when τ<τ0=5.1686, which can be illustrated by Figures 1 and 2. In this case, the disease can be controlled easily. Once the value of the delay passes through the critical value τ0=5.1686, then the endemic equilibrium P(34.3750,245.5930,73.8562) loses its stability and a Hopf bifurcation occurs, and a family of periodic solutions bifurcate from the endemic equilibrium P(34.3750,245.5930,73.8562). This property can be shown as in Figures 3 and 4. In this case, the disease will be out of control.

P is locally asymptotically stable for τ=4.475<τ0=5.1686 with initial values “27.5, 260.5, 65.”

P is locally asymptotically stable for τ=4.475<τ0=5.1686 with initial values “22, 200, 75.”

P becomes unstable and a Hopf bifurcation occurs when τ=5.485>τ0=5.1686 with initial values “27.5, 260.5, 65.”

P becomes unstable and a Hopf bifurcation occurs when τ=5.485>τ0=5.1686 with initial values “22, 200, 75.”

In addition, according to (31), we get C1(0)=-1.0027-0.9244i, μ2=835.5833>0, β2=-2.0054<0, and T2=31.5171>0. Therefore, we can conclude that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable and increase. Since the bifurcating periodic solutions are stable, it can be concluded that the populations in system (32) can coexist from the view of ecology. Based on this fact, we can conclude that the time delay is harmful for system (32).

5. Conclusions

We generalize a delayed SEIS (Susceptible-Exposed-Infectious-Susceptible) epidemic model with a changing delitescence and nonlinear incidence rate in this paper by introducing the time delay due to the period that is used to cure the infectious population into the SEIS model considered in the literature . Compared with the literature , we mainly consider the effect of the time delay on the model.

The main results are given in terms of local stability and Hopf bifurcation. Stability of the endemic equilibrium is investigated by analyzing the corresponding characteristic equation. By choosing the time delay as a bifurcation parameter, sufficient conditions have been established for local existence of Hopf bifurcation at the endemic equilibrium. Then, with the help of the normal form theory and the center manifold theorem due to Hassard et al. , direction and stability of the Hopf bifurcation are determined. Finally, through numerical simulations, it can be concluded that the period used to cure the infectious population plays an important role in the disease spreading and the disease may be controlled by shortening the period used to cure the infectious population.

Appendix A. The Expressions of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M167"><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M168"><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M169"><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

(A.1) F 1 = g 1 ϕ 3 2 0 + g 2 ϕ 1 0 ϕ 3 0 + g 3 ϕ 1 0 ϕ 3 2 0 + g 4 ϕ 3 3 0 + , F 2 = h 1 ϕ 3 2 0 + h 2 ϕ 1 0 ϕ 3 0 + h 3 ϕ 1 0 ϕ 3 2 0 + h 4 ϕ 3 3 0 + , F 3 = k 1 ϕ 3 2 0 + k 2 ϕ 1 0 ϕ 3 0 + k 3 ϕ 1 0 ϕ 3 2 0 + k 4 ϕ 3 3 0 + , g 1 = α β S 1 + α I 3 , g 2 = - β 1 + α I 2 , g 3 = α β 1 + α I 3 , g 4 = - α 2 β S 1 + α I 4 , h 1 = - μ α β S 1 + α I 3 , h 2 = μ β 1 + α I 2 , h 3 = - μ α β 1 + α I 3 , h 4 = μ α 2 β S 1 + α I 4 , k 1 = - 1 - μ α β S 1 + α I 3 , k 2 = 1 - μ β 1 + α I 2 , k 3 = - 1 - μ α β 1 + α I 3 , k 4 = 1 - μ α 2 β S 1 + α I 4 .

B. The Expressions of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M171"><mml:mrow><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mn>20</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M172"><mml:mrow><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M173"><mml:mrow><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mn>02</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M174"><mml:mrow><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

(B.1) g 20 = 2 τ 0 D ¯ g 1 q 3 0 2 + g 2 q 1 0 q 3 0 + q ¯ 2 h 1 q 3 0 2 + h 2 q 1 0 q 3 0 + q ¯ 3 k 1 q 3 0 2 + k 2 q 1 0 q 3 0 , g 11 = τ 0 D ¯ 2 g 1 q 3 0 q ¯ 3 0 + g 2 q 1 0 q ¯ 3 0 + q ¯ 1 0 q 3 0 + q ¯ 2 2 h 1 q 3 0 q ¯ 3 0 + h 2 q 1 0 q ¯ 3 0 + q ¯ 1 0 q 3 0 + q ¯ 3 2 k 1 q 3 0 q ¯ 3 0 + k 2 q 1 0 q ¯ 3 0 + q ¯ 1 0 q 3 0 , g 02 = 2 τ 0 D ¯ g 1 q ¯ 3 0 2 + g 2 q ¯ 1 0 q ¯ 3 0 + q ¯ 2 h 1 q ¯ 3 0 2 + h 2 q ¯ 1 0 q ¯ 3 0 + q ¯ 3 k 1 q ¯ 3 0 2 + k 2 q ¯ 1 0 q ¯ 3 0 , g 21 = 2 τ 0 D ¯ g 1 2 W 11 3 0 q 3 0 + W 20 3 0 q ¯ 3 0 + g 2 W 11 1 0 q 3 0 + 1 2 W 20 1 0 q ¯ 3 0 + W 11 3 0 q 1 0 + 1 2 W 20 3 0 q ¯ 1 0 + g 3 q ¯ 1 0 q 3 0 2 + 2 q 1 0 q 3 0 q ¯ 3 0 + 3 g 4 q 3 0 2 q ¯ 3 0 + q ¯ 2 h 1 2 W 11 3 0 q 3 0 + W 20 3 0 q ¯ 3 0 + h 2 W 11 1 0 q 3 0 + 1 2 W 20 1 0 q ¯ 3 0 + W 11 3 0 q 1 0 + 1 2 W 20 3 0 q ¯ 1 0 + h 3 q ¯ 1 0 q 3 0 2 + 2 q 1 0 q 3 0 q ¯ 3 0 + 3 h 4 q 3 0 2 q ¯ 3 0 + q ¯ 3 k 1 2 W 11 3 0 q 3 0 + W 20 3 0 q ¯ 3 0 + k 2 W 11 1 0 q 3 0 + 1 2 W 20 1 0 q ¯ 3 0 + W 11 3 0 q 1 0 + 1 2 W 20 3 0 q ¯ 1 0 + k 3 q ¯ 1 0 q 3 0 2 + 2 q 1 0 q 3 0 q ¯ 3 0 + 3 k 4 q 3 0 2 q ¯ 3 0 , W 20 θ = i g 20 q 0 τ 0 ω 0 e i τ 0 ω 0 θ + i g ¯ 02 q ¯ 0 3 τ 0 ω 0 e - i τ 0 ω 0 θ + E 1 e 2 i τ 0 ω 0 θ , W 11 θ = - i g 11 q 0 τ 0 ω 0 e i τ 0 ω 0 θ + i g ¯ 11 q ¯ 0 τ 0 ω 0 e - i τ 0 ω 0 θ + E 2 , E 1 = 2 i ω 0 - a 11 0 - a 13 - b 13 e - 2 i τ 0 ω 0 - a 21 2 i ω 0 - a 22 - a 23 - a 31 - a 32 2 i ω 0 - a 33 - b 33 e - 2 i τ 0 ω 0 - 1 × E 1 1 E 1 2 E 1 3 , E 2 = - a 11 a 12 a 13 + b 13 a 21 a 22 a 23 a 31 a 32 a 33 + b 33 - 1 × E 2 1 E 2 2 E 2 3 , E 1 1 = g 1 q 3 0 2 + g 2 q 1 0 q 3 0 , E 1 2 = h 1 q 3 0 2 + h 2 q 1 0 q 3 0 , E 1 3 = k 1 q 3 0 2 + k 2 q 1 0 q 3 0 , E 2 1 = 2 g 1 q 3 0 q ¯ 3 0 + g 2 q 1 0 q ¯ 3 0 + q ¯ 1 0 q 3 0 , E 2 2 = 2 h 1 q 3 0 q ¯ 3 0 + h 2 q 1 0 q ¯ 3 0 + q ¯ 1 0 q 3 0 , E 2 3 = 2 k 1 q 3 0 q ¯ 3 0 + k 2 q 1 0 q ¯ 3 0 + q ¯ 1 0 q 3 0 .

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2015A144).

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