GLOBAL STABILITY AND HOPF BIFURCATION OF A DELAYED EPIDEMIOLOGICAL MODEL WITH LOGISTIC GROWTH AND DISEASE RELAPSE YUGUANG MU, RUI XU

Available online at http://scik.org Commun. Math. Biol. Neurosci. 2018, 2018:7 https://doi.org/10.28919/cmbn/3636 ISSN: 2052-2541 GLOBAL STABILITY AND HOPF BIFURCATION OF A DELAYED EPIDEMIOLOGICAL MODEL WITH LOGISTIC GROWTH AND DISEASE RELAPSE YUGUANG MU, RUI XU Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, P.R. China Communicated by A. Elaiw Copyright c 2018 Mu and Xu. 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. Abstract. In this paper, an SIRI epidemiological model with relapse and a time delay describing the latent period of the disease is investigated. In the model, it is assumed that the susceptible population is subject to logistic growth in the absence of the disease. We show that the dynamic of the model are determined by the basic reproduction number. If the basic reproduction number is less than unity, then the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, Hopf bifurcation occurs as the time delay passes through a critical value. Numerical simulations are carried out to support our theoretical conclusion. Keywords: SIRI epidemic model; disease relapse; latent period; time delay; stability; Hopf bifurcation. 2010 AMS Subject Classification: 00A71. 1. Introduction Mathematical models describing the population dynamics of infectious disease have played an important role in better understanding epidemiological patterns and disease control for a long Corresponding author E-mail address: muyuguang2016mec@163.com Received January 2, 2018 1

2 YUGUANG MU, RUI XU time. For some diseases, recovered individuals may relapse with reactivation of latent infection and revert back to the infective class. This recurrence of disease is an important feature of some animal and human diseases, for example, tuberculosis, including human and bovine [1, 7], and herpes [1, 8]. For human tuberculosis, incomplete treatment can lead to relapse, but relapse can also occur in patients who took a full course of treatment and were declared cured. Most tuberculosis in human adults caused by Mycobacterium tuberculosis in the USA results from reactivation of latent infection [1]. Let St, It and Rt be the numbers of individuals in the susceptible, infective and the recovered classes, respectively, with the total population Nt = St + It + Rt, in [2], van den Driessche and Zou considered the following infectious disease model with relapse: Ṡt =d λstit dst, [ t ] İt = d + γit + λit 1 It γiξ e dt ξ Pt ξ dξ 0 t γiξ e dt ξ d t Pt ξ dξ, 0 t Rt = γiξ e dt ξ Pt ξ dξ, 0 1.1 where d > 0 is the birth rate and death rate constants, and λ > 0 is the average number of effective contacts of an infectious individual per unit time a fraction S / N are with susceptible. Denote by Pt the fraction of recovered individuals remaining in the recovered class t time units after recovery. By the meaning of Pt, it is reasonable to assume the following properties: A P : [0, [0, is differentiable hence continuous except at possibly finitely many points where it may have jump discontinuities, non-increasing and satisfies P0 = 1, lim t Pt = 0 and 0 Pudu is positive and finite. For Pt = e ατ, system 1.1 reduces to the following ODE system: Ṡt =d λstit dst, İt = d + γit + λit[1 It Rt] + αrt, 1.2 Ṙt = d + αrt + γit.

GLOBAL STABILITY AND HOPF BIFURCATION OF A DELAYED EPIDEMIOLOGICAL MODEL 3 This model is given in Diekmann and Heesterbeek [9]. For system 1.2, the basic reproduction number was identified and its threshold property was discussed in Tudor [11]. The epidemiological model with disease relapse above has been discussed by some authorssee, for example, [3-6] and the references cited therein. An important aspect of the mathematical study of epidemiology is the time delay describing a latent period. In order to study the effect of the latent period on the transmission dynamics of a disease, in [10], Liu et al. considered the following SIR epidemic model described by delay differential equations: Ṡt =Λ µst βe µτ StGIt τ, İt =βe µτ StGIt τ µ + γ + αit, 1.3 Ṙt =γit µrt, where Λ is the recruitment rate of the population, µ represents the natural death rate of the population, β is the transmission rate between compartments S and I, γ denotes the recovery rate of infectives, α stands for the disease-caused death rate of infectious individuals, the constant τ > 0 is the incubation time. The authors assumed that the force of infection at any time t is given by βe µτ StGIt τ, since those infected at time t τ become infectious at time τ latter. The term 0 < e µτ 1 denotes the survival of vector population in which the time taken to become infectious is τ. The parameters Λ, µ, β, γ, α are all positive constants. For more details about epidemiological model with latent period, we refer to [12-14] going in this direction. Motivated by the works of Liu et al. [10], van den Driessche and Zou [2], in this paper, we are concerned with the joint effects of disease relapse, saturation incidence and time delay describing latent period on the global dynamics of infectious diseases. To this end, we consider the following delayed differential equations: Ṡt = rst 1 St βstit K St + It, İt = βe µτ St τit τ St τ + It τ µ + γ + αit + δrt, 1.4 Ṙt = γit Rt.

4 YUGUANG MU, RUI XU where the parameters β, γ, µ and α are the same as that defined in model 1.3. τ 0 represents a time delay describing the latent period of the disease, the term βe µτ St τ It τ / St τ + It τ represents the individuals surviving in the latent period τ and becoming infective at time t. The parameter r > 0 is the intrinsic growth rate of the susceptible population, K > 0 is the carrying capacity of the environment of the area. δ > 0 is a constant representing the rate at which an individual in the recovered class reverts to the infective class. The initial conditions for system 1.4 are of the form St = φ 1 θ, It = φ 2 θ, Rt = φ 3 θ, φ i θ 0, θ [ τ,0], φ i 0 > 0 i = 1,2,3, 1.5 where φ 1 θ, φ 2 θ, φ 3 θ C[ τ,0], R 3 +0, the space of continuous functions mapping the interval [ τ,0] into R 3 +0, where R3 +0 = {x 1,x 2,x 3 : x i 0, i = 1,2,3}. It is well known by the fundamental theory of functional differential equations [15], system 1.4 has a unique solution St, It, Rt satisfying the initial condition 1.5. It is easy to show that all solutions of system 1.4 with initial condition 1.5 are defined on [0,+ and remain positive for all t 0. The organization of this paper is as follows. In the next section, the basic reproduction number is calculated, and the existence of equilibrium is discussed. In Section 3, we discuss the local stability of a disease-free equilibrium and an endemic equilibrium respectively and establish the existence of Hopf bifurcations at the endemic equilibrium of system 1.4. In Section 4, by means of Lyapunov functionals and LaSalle s invariance principle, we prove that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable; if the basic reproduction number is greater than unity, sufficient conditions are defined for the global asymptotic stability of the endemic equilibrium. In Section 5, numerical simulations are given to support our results. A brief remark is given in Section 6 to conclude this work. 2. Local stability and Hopf bifurcation

GLOBAL STABILITY AND HOPF BIFURCATION OF A DELAYED EPIDEMIOLOGICAL MODEL 5 In this section, we investigate local asymptotic stability of equilibrium and the existence of Hopf bifurcation for system 1.4. System 1.4 always has an disease-free equilibrium E 0 = S 0,0,0, where S 0 = K. Let R 0 = βe µτ µ + α + µγ. 2.1 R 0 represents the average number of secondary transmissions of a single infectious individual in a fully susceptible population. A direct calculation shows that if R 0 > 1, in addition to the disease-free equilibrium E 0 K,0,0, system 1.4 has a unique endemic equilibrium E S,I,R, where S = K r [r β + µ + α + µγ eµτ ], I = R 0 1S, R = γ I. 2.2 Theorem 2.1. If R 0 < 1, the disease-free equilibrium E 0 of system 1.4 is locally asymptotically stable; If R 0 > 1, E 0 becomes unstable. Proof. The characteristic equation at E 0 is λ + r[λ βe µτ λτ + µ + γ + αλ + δγ] = 0. 2.3 Obviously, λ = r is a negative real root of equation 2.3, and other roots are determined by the following equation: Let If R 0 > 1, then we have λ βe µτ λτ + µ + γ + αλ + δγ = 0. 2.4 f λ = λ βe µτ λτ + µ + γ + αλ + δγ. f 0 = [µ + α + µγ]1 R 0 < 0, lim f λ = +. λ + So there must be a λ 0 > 0 such that f λ 0 = 0, which implies that Equation 2.4 has at least one positive root. Therefore, the disease-free equilibrium E 0 S 0,0,0 is unstable when R 0 > 1. If R 0 < 1, then we have f 0 = [µ + α + µγ]1 R 0 > 0

6 YUGUANG MU, RUI XU and f λ = µ + α + µγ 1 R 0e λτ + δγ + 2λ + δ + + λτβe µτ λτ > 0 for all λ 0 and τ 0. Hence, Equation 2.4 has no positive real root. Suppose there is a complex root λ = x + iy with y 0. Filling in to Equation 2.4, and separating the real and imaginary parts, we obtain x + µ + γ + αx + y 2 = x + βe µτ xτ cosyτ + yβe µτ xτ sinyτ, y2x + 2µ + γ + α + δ = yβe µτ xτ cosyτ x + βe µτ xτ sinyτ. 2.5 Squaring and adding the two equations of Equation 2.5, we have y 2 = [µ + α + µγ R 0e xτ 1 γδ x]βe µτ xτ + x + µ + γ + α < 0. Therefore, all roots of Equation 2.4 have negative real parts. Hence, the disease-free equilibrium E 0 is locally asymptotically stable if R 0 < 1. This completes the proof. Next, we investigate stability and Hopf bifurcation of the endemic equilibrium for system 1.4. The characteristic equation at E is Pλ,τ + Qλ,τe λτ = 0, 2.6 where Pλ,τ =λ 3 + A 2 τλ 2 + A 1 τλ + A 0 τ, Qλ,τ =B 2 τλ 2 + B 1 τλ + B 0 τ, and A 0 τ =[ r + 2rS / K + β1 1 / R 0 2 ][µ + α + µγ], A 1 τ =2µ + γ + α + δ[ r + 2rS / K + β1 1 / R 0 2 ] + µ + α + µγ, A 2 τ =2µ + γ + α + δ r + 2rS / K + β1 1 / R 0 2, B 0 τ =[r 2rS / K][µ + α + µγ] / R 0, B 1 τ =[r 2rS / K ]µ + α + µγ / / R 0, B 2 τ = µ + α + µγ / / R 0.

GLOBAL STABILITY AND HOPF BIFURCATION OF A DELAYED EPIDEMIOLOGICAL MODEL 7 when τ = 0, Equation 2.6 becomes λ 3 + A 2 + B 2 λ 2 + A 1 + B 1 λ + A 0 + B 0 = 0. 2.7 Let H 1 = A 2 + B 2, H 2 = A 2 + B 2 A 1 + B 1 A 0 + B 0. After some tedious computations, it is clear that H 1 > 0, H 2 > 0. By the Routh-Hurwitz criterion, all roots of Equation 2.7 have negative real parts. Hence, the interior equilibrium E is locally asymptotically stable. Now, for τ > 0, let τ max is the maximum of τ under the condition that the equilibrium point E 1 exists. In order to apply the methods proposed by Kuang and Beretta in [16], we need to verify that the following conditions are established for τ [0,τ max : i P0,τ + Q0,τ 0 τ R +0, i.e., λ = 0 is not a characteristic root of 2.6; ii if λ = iω, ω R, then Piω,τ + Qiω,τ 0, for τ > 0; iii limsup { / } Qλ,τ Pλ,τ : λ,reλ 0 < 1 for any τ; iv Fω,τ = Piω,τ 2 Qiω,τ 2 for each τ has at most a finite number of real zeroes; v Each positive root ωτ of Fω,τ = 0 is continuous and differentiable in τ whenever it exists. Here, Pλ,τ and Qλ,τ are the same as those in 2.6. Because of P0,τ+Q0,τ = A 0 τ+ B 0 τ 0, the condition i is established clearly. Note that Piω,τ + Qiω,τ = iω 3 [A 2 τ + B 2 τ]ω 2 + i[a 1 τ + B 1 τ]ω + A 0 τ + B 0 τ =[A 0 τ + B 0 τ A 2 τ + B 2 τω 2 ] 2.8 + iω[a 1 τ + B 1 τ ω 2 ] 0. Therefore, the condition ii is established. From Equation 2.6, we obtain lim Qλ,τ Pλ,τ = lim B 2 τλ 2 + B 1 τλ + B 0 τ s λ 3 + A 2 τλ 2 + A 1 τλ + A 0 τ = 0, 2.9 s

8 YUGUANG MU, RUI XU This completes the proof of iii. To justify iv, we have Piω,τ 2 =ω 6 + [A 2 2τ 2A 1 τ]ω 4 + [A 2 1τ 2A 0 τa 2 τ]ω 2 + A 2 0τ, Qiω,τ 2 =B 2 2τω 4 + [B 2 1τ 2B 0 τb 2 τ]ω 2 + B 2 0τ, 2.10 Thus, we obtain that Fω,τ = ω 6 + a 1 τω 4 + a 2 τω 2 + a 3 τ, 2.11 where a 1 τ =A 2 2τ 2A 1 τ B 2 2τ, a 2 τ =A 2 1τ + 2B 0 τb 2 τ 2A 0 τa 2 τ B 2 1τ, a 3 τ =A 2 0τ B 2 0τ. Obviously, the condition iv is established. According to the Implicit Function Theorem, the condition v is also established. Let λ = iω 1 ω 1 > 0 be a root of Equation 2.6. Substituting it into Equation 2.6 and separating real and imaginary parts, we obtain B 2 τω1 2 B 0 τcosω 1 τ B 1 τω 1 sinω 1 τ = A 0 τ A 2 τω1, 2 B 2 τω1 2 B 0 τsinω 1 τ + B 1 τω 1 cosω 1 τ = ω1 3 A 1τω 1, 2.12 which gives sinω 1 τ = B 2τω 5 1 + [A 2τB 1 τ B 0 τ A 1 τb 2 τ]ω 3 1 B 2 2 τω4 1 + [B2 1 τ 2B 0τB 2 τ]ω 2 1 + B2 0 τ + [A 1 τb 0 τ A 0 τb 1 τ]ω 1 B 2 2 τω4 1 + [B2 1 τ 2B 0τB 2 τ]ω 2 1 + B2 0 τ, cosω 1 τ = [B 1τ A 2 τb 2 τ]ω1 4 A 0τB 0 τ B 2 2 τω4 1 + [B2 1 τ 2B 0τB 2 τ]ω1 2 + B2 0 τ 2.13 + [A 0τB 2 τ + B 0 τa 2 τ A 1 τb 1 τ]ω 2 1 B 2 2 τω4 1 + [B2 1 τ 2B 0τB 2 τ]ω 2 1 + B2 0 τ. Assume that I R +0 is the set where ω 1 τ is a positive root of Fω 1,τ = Piω 1,τ 2 Qiω 1,τ 2 and for τ / I, ω 1 τ is not definite. According to [16], we can define the angle

GLOBAL STABILITY AND HOPF BIFURCATION OF A DELAYED EPIDEMIOLOGICAL MODEL 9 θτ [0,2π], as the solution of 2.13 sinθτ =Im Piω 1,τ Qiω 1,τ, cosθτ = Re Piω 1,τ Qiω 1,τ, 2.14 and introduce the functions I R S n τ := τ Using Theorem 2.1 in [16], we have the following theorem. θτ + 2nπ, τ I, n N 0. 2.15 ω 1 τ Theorem 2.2. Assume that ω 1 τ is a positive real root of Fω 1,τ = 0 defined for τ I, I R +0 and at some τ I, S n τ = 0, f or some n N 0, 2.16 Then a pair of simple conjugate pure imaginary roots λ = ±iωτ of 2.6 exists at τ = τ which crosses the imaginary axis from left to right if δτ > 0 and crosses the imaginary axis from right to left if δτ < 0, where { dreλτ δτ = sign dτ τ=τ } = sign { F ωω 1 τ,τ } { dsn τ sign dτ τ=τ }. 2.17 3. Global stability Theorem 3.1. If R 0 < 1, then the disease-free equilibrium E 0 is globally asymptotically stable. Proof. Define where V 1 t = It + U t = δ Rt + e µτ U t, 3.1 t t τ βsθiθ dθ. 3.2 Sθ + Iθ

10 YUGUANG MU, RUI XU Then the time derivative of V 1 t along system 1.4 satisfies d dt V 1t = βe µτ St τit τ µ + γ + αit + δrt St τ + It τ + δ γit Rt + βe µτ StIt St + It = βe µτ StIt St + It µ + α + µγ = µ + α + µγ = βe µτ St τit τ St τ + It τ µ + α + µγ It StIt R 0 St + It It R 0 1 StIt St + It I2 t. St + It 3.3 It follows that V 1 t 0 if R 0 < 1. By Theorem 5.3.1 in [15], solutions limit to M 0, the largest invariant subset of { V 1 t = 0}. Clearly, it follows from 3.3 that V 1 t = 0 iff I = 0. Noting that M 0 is invariant, for each element in M 0, we have I = 0, İt = 0. We therefore obtain from the second equation of system 1.4 that 0 = İt = δrt, which yields R = 0. It follows from the first equation and V 1 t = 0 that Ṡt = rst1 St / K, which yields lim t St = K. Therefore, V 1 t = 0 iff S,I,R = K,0,0. Note that if R 0 < 1, the equilibrium E 0 is locally asymptotically stable. Hence, the global asymptotic stability of E 0 K,0,0 follows from LaSalle s invariance principle for delay differential systems. This completes the proof. Lemma 3.1. There is a constant H 0 > 0 such that It H 0 for any positive solution St,It,Rt of system 1.4. Proof. It is clear that all solutions of system 1.4 with initial condition 1.5 remain positive for all t 0. Let V t = e µτ St τ + It + Rt. Then the time derivative of V t along system 1.4 satisfies d V t =e µτ 2rSt τ r S2 t τ re µτ St τ dt K µ + αit µrt hv t + e µτ 2rSt τ r S2 t τ, K

GLOBAL STABILITY AND HOPF BIFURCATION OF A DELAYED EPIDEMIOLOGICAL MODEL 11 where h = min{r, µ + α, µ}. Let Gs = 2rs rs2 K, obviously, Gs gets the maximum G maxs = rk at s = K. Therefore It follow that d dt V t hv t + e µτ rk. 3.4 rk lim supv t t + he µτ : = N 0. Thus, there is a constant T 1 > 0 such that It N 0 for any ε > 0 when t > T 1. Lemma 3.2. There is an H 1 > 0 so that for any positive solution St,It,Rt of the system 1.4, when t is sufficiently large, there are St H 1 holds. Proof. By the proof of Lemma 3.1, when t > T 1, there are It N 0. From the first equation of the system 1.4 d St = rs 1 S βsi dt K S + I rs2 K + r βh 0 S. 3.5 S max + H 0 Let Hs = rs2 K + r βh 0 s, S max + H 0 Clearly, Hs = 0 has the only positive root s = K r βh 0, r S max + H 0 From the equation of 3.5 lim infst t + s, Thus, there is a constant T 2 > T 1 such that St s + ε 1 : = H 1 for any ε 1 > 0 when t > T 2. Theorem 3.2. If R 0 > 1 and H 1 K / 2, then the endemic equilibrium E is globally asymptotically stable. Proof. Let St, It, Rt be any positive solution of system 1.4 with initial conditions 1.5. Define St Ut =e St µτ S θ + I S S S + I θ dθ + It I I ln It I + δ Rt R R ln Rt R. 3.6

12 YUGUANG MU, RUI XU Calculating the derivative of Ut along positive solutions of system 1.4, we have d St + I Ut =e µτ 1 dt S + I =e µτ St + I 1 + 1 I + δ S St Ṡt + 1 I It İt + δ 1 R Rt S S + I rst 1 St βstit St K St + It βe µτ St τit τ µ + γ + αit + δrt It St τ + It τ 1 R γit Rt. Rt Ṙt 3.7 Substituting rs 1 S / K βs I / S + I = 0, βe µτ S I / S + I µ + γ + αi + δr = 0, and γi R = 0 into 3.7 yields d St + I Ut =e µτ 1 dt S + I S rs 1 S St K βstit + µ + γ + α 1 R St + It γ Rt + 1 I βe µτ St τit τ It St τ + It τ = rst S 2 e µτ S + I 1 S + St βe µτ S I St K S + I + rst 1 St + βs I K S + I γit Rt µ + γ + αit + δrt St + I S + I S St + βe µτ StIt St + I S St + It S + I St + βe µτ S I S + I βe µτ StIt St + It + βe µτ St τit τ St τ + It τ + µ + γ + αi δ I It Rt + δγ = rst S 2 e µτ S + I 1 S + St St K + + µ + α + µγ I St + I 1 S + I µ + α + µγ I 1 It + δr 2 It R I Rt I It µ + γ + αit βe µτ St τit τ St τ + It τ Rt R δγ R It It Rt + δr I It βe µτ StIt + βe µτ St τit τ St + It St τ + It τ S St + StIt St + It St + I StI I S + I St τit τ I S I St τ + It τ It. 3.8

GLOBAL STABILITY AND HOPF BIFURCATION OF A DELAYED EPIDEMIOLOGICAL MODEL 13 Let V 2 t = Ut + µ+αµ+δ+µγ µ+δ I U + t, where t [ U + t = t τ ln e µτ βsθiθ µ + α + µγi Sθ + Iθ 1 e µτ βsθiθ µ + α + µγi Sθ + Iθ ] dθ. 3.9 Calculating the derivative of U + t, d e µτ βstit dt U+ t = µ + α + µγi St + It + ln e µτ βst τit τ µ + α + µγi St τ + It τ St τit τst + It St τ + It τstit. 3.10 Using equality St τit τst + It S ln St τ + It τstit =ln + I St τit τ I S I St τ + It τ It + ln St + I S St + It S + I + ln St St + I, 3.11 we obtain d dt V 2t = d dt µ + α + µγ Ut + I d dt U+ t 1 S + St K = rst S 2 e µτ S + I St + + µ + α + µγ I St τit τst + It ln St τ + It τstit µ + α + µγ I St + I 1 S + I S St + StIt St + It St + I StI µ + α + µγ + I 1 It I S + I St τit τ I S I St τ + It τ It + δr 2 It R I Rt I Rt It R

14 YUGUANG MU, RUI XU = rst S 2 e µτ S + I 1 S + St St K + µ + α + µγ I St + I 1 S + I µ + α + µγ I S St + ln St + I S S + I St + 1 S + I St τit τ S I St τ + It τ It S + I St τit τ I +ln S I St τ + It τ It µ + α + µγ + I St + It 1 St + I + ln St + It St + I µ + α + µγ + δr 2 It R I Rt I StIt I 2 St + ItSt + I. It Rt R I 3.12 Since the function Ht = 1 f t + ln f t is always non-positive for any function f t > 0, and Ht = 0 iff f t = 1. From Lemma 3.2, when t is sufficiently large, there are St H 1 holds. Therefore, when H 1 K / 2, 1 S + St K 0. Thus, when R 0 > 1 and H 1 K / 2, there are dt d V 2t 0 holds. And dt d V 2t = 0 iff It = I,Rt = R,St = S. Then, the largest invariant set M 1 M = { S,I,R d dt V 2 t = 0 } is the singleton {E }. By LaSalle s Invariance Principle, the endemic equilibrium E is globally asymptotically stable. 4. Numerical simulations In this section, we perform numerical simulations to illustrate the existence of Hopf bifurcation for system 1.4. In the following, we choose a set of parameters r = 1, K = 5, β = 0.8, µ = 0.3, γ = 0.2, α = 0.1, δ = 0.1, τ = 0.4, where day is used as the unit of time. By calculation, we obtain that R 0 1.29 > 1. At this point, system1.4 has Hopf bifurcation at endemic equilibrium E.

GLOBAL STABILITY AND HOPF BIFURCATION OF A DELAYED EPIDEMIOLOGICAL MODEL 15 4.108 1.1902 4.106 4.104 1.19 1.1898 St 4.102 4.1 It 1.1896 1.1894 1.1892 4.098 4.096 1.189 1.1888 4.094 0 100 200 300 400 500 600 700 800 900 1000 t-time 1.1886 0 100 200 300 400 500 600 700 800 900 1000 t-time 0.5954 0.5952 0.595 Rt 0.5948 0.5946 0.5944 0.5942 0 100 200 300 400 500 600 700 800 900 1000 t-time FIGURE 1. There are periodic solutions bifurcated from the endemic equilibrium E of system 1.4 when τ = 0.4. 5. Conclusions In this paper, we have considered an epidemiological model with saturation incidence, a time delay describing the latent period and relapse. We have calculated the basic reproduction number R 0. It has shown that if R 0 < 1, then the disease-free equilibrium E 0 is globally asymptotically stable. If R 0 > 1, stability switch is shown to occur, and periodic solutions bifurcated from the endemic equilibrium. The establishment of global stability is crucial to our study, since local stability cannot rule out the possibility of periodic solutions far away from equilibria. From the expression of R 0, we see that the natural death rate µ, the disease-induced death rate α, the recovery rate γ, the constant rate δ at which an individual in the recovered class

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