# Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease Model with Two Delays and Reinfection

Authors: Yanxia Zhang; Long Li; Junjian Huang; Yanjun Liu
Journal: Computational and Mathematical Methods in Medicine
 (2021)

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## Abstract

In this paper, a vector-borne disease model with two delays and reinfection is established and considered. First of all, the existence of the equilibrium of the system, under different cases of two delays, is discussed through analyzing the corresponding characteristic equation of the linear system. Some conditions that the system undergoes Hopf bifurcation at the endemic equilibrium are obtained. Furthermore, by employing the normal form method and the center manifold theorem for delay differential equations, some explicit formulas used to describe the properties of bifurcating periodic solutions are derived. Finally, the numerical examples and simulations are presented to verify our theoretical conclusions. Meanwhile, the influences of the degree of partial protection for recovered people acquired by a primary infection on the endemic equilibrium and the critical values of the two delays are analyzed.


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## Body

## 1. Introduction

Vector-borne diseases (VBDs) are one of the complex infectious diseases that endanger human beings. Vectors are living biological agents, such as ticks, mosquitoes, and fleas, with the ability to transmit parasites, bacteria, or viruses between people or from animals to people. It is reported that VBDs cause more than 1 billion infections and 1 million deaths worldwide every year [1]. Malaria is the most prevalent parasitic vector-borne disease caused by plasmodium parasitizing the human body [2–5]. Plasmodium enters stem cells through the blood to parasitize and reproduce. It then invades red blood cells to reproduce after maturity, causing red blood cells to burst in batches and attack. The source of malaria infection is malaria patients and those with Plasmodium. The natural transmission medium of malaria is female mosquitoes of the genus Anopheles, which is transmitted by biting the human body, and a few by blood transfusion and vertical transmission, with Anopheles gambiae being a major carrier of the disease [2, 5, 6]. Different types of malaria have different incubation periods, some of which are about 7-12 days, and some of which are more than 6 months. The morbidity and mortality of malaria are high [7]. According to the latest WHO reports, in 2018, there were estimated 228 million malaria cases worldwide, of which the death toll was about 405,000. It has led to great global economic and social losses, especially in the tropical and subtropical regions on five continents [3].The population is generally susceptible to malaria. Although there is a certain degree of immunity after several infections, its acquisition process is so slow that it may take years or decades to develop, and it gradually weakens over time [5, 7, 8]. The remaining plasmodium may escape from the immune function due to the antigenic variation and reproduce again (recrudescence); even if the parasite has been eliminated by human immunity or drugs after the initial onset of malaria, the possibility of relapse is not excluded with time. That is, reinfection likely occurs [6, 7]. It is shown that time delay is of great significance in many biological modelling, and its change may affect the dynamic behavior of the system [4–6, 9–19], such as stability, uniqueness, and oscillation of solution. So, it is instructive to consider a mathematical model with time delays to research the influence of immunity on disease control of malaria transmission.Considering the incubation period of virus transmission in the vector population, some vector-borne epidemic models with a delay were considered [4–6]. In [5], Xu and Zhou proposed a delayed vector-borne epidemic model and reinfection, investigated its existence and stability of equilibrium, and analyzed its dynamical behavior. It suggested that there are two effective preventive measures to reduce infections: one is to minimize vector to human contacts and the other is to use insecticides to control vector. According to [4, 5], a vector-borne disease model with delay-saturated infection rate and cure rate was given by [6]. The existence and local stability of the epidemic equilibrium were discussed, and the length of the delay of the system preserving stability was estimated. This paper develops an improved vector-borne disease model with two delays and reinfection to consider the time required for the malaria virus to spread to the host population and vector population.The remainder of this paper is organized as follows. In Section2, an improved vector-borne disease model is formulated. In Section 3, the stability of the equilibrium and the existence of local Hopf bifurcation are discussed. In Section 4, some explicit formulas determining properties bifurcating periodic solutions are obtained by employing the normal form method and the center manifold theorem for the delay differential equations developed by Hassard et al. [20]. Some numerical examples and simulations are performed in Section 5 to demonstrate the main theoretical results, and the conclusions of this paper are summarized in Section 6.


## 2. Model Description

Generally, in the mathematical model of a vector-transmitted disease, the host population size at timet, denoted by N1t, is divided into three subclasses: susceptible, infected, and recovered, with numbers denoted by St, It, and Rt, respectively. The vector population size at time t, given by N2t, is partitioned into two subclasses: susceptible vectors Mt and infectious vectors Vt. Based on the models [4–6], we consider an improved vector-borne disease model with two delays and reinfection as follows:
(1)dSdt=Λ1−bβ1St−τ1Vt−τ1−μ1St,dIdt=bβ1St−τ1Vt−τ1+σbβ1RtVt−μ1+γIt,dRdt=γIt−σbβ1RtVt−μ1Rt,dMdt=Λ2−bβ2Mt−τ2It−τ2−μ2Mt,dVdt=bβ2Mt−τ2It−τ2−μ2Vt,where Λ1 and Λ2 are the recruitment rates of the hosts and vectors, respectively. b is the average number of bites per mosquito per day. β1 and β2 represent the infection rates from vector to human and human to vector, respectively. μ1 and μ2 are the natural death rates of the hosts and vectors, respectively. σ0≤σ≤1 is the degree of partial protection of individuals that recovered from primary infection. γ is the recovery rate per capita of the infective host population. τ1 and τ2 are two delays, in which τ1 represents the incubation period of the host population, and τ2 denotes the incubation period of the vector population. The term bβ1St−τ1Vt−τ1 represents the incidence number of the susceptible host infections caused by the infective vector at time t−τ1. bβ2Mt−τ2It−τ2 denotes the number of infections of the susceptible vectors that bite the infected host at time t−τ2 and become infective at time t.According to [21], system (1) is equivalent to the following model:
(2)dSdt=Λ1−bβ1St−τ1Vt−τ1−μ1St,dIdt=bβ1St−τ1Vt−τ1+σbβ1Λ1μ1−St−ItVt−μ1+γIt,dVdt=bβ2Λ2μ2−Vt−τ2It−τ2−μ2Vt.Notice thatNi′t=Λi−μiNit, i=1,2, and its solution is Nit=Ni0e−μit+Λi/μi1−e−μit.Obviously,limt→∞Nit=Λi/μi,i=1,2. So, for t→∞ and any ε>0, 0≤S+I≤N1t<Λ1/μ1+ε holds by limit theorem, which follows that 0≤S+I≤Λ1/μ1. Similarly, 0≤V≤Λ2/μ2 also holds. Thus, all the solutions of system (2) enter into the region:
(3)Ω=S,I,V∈R+30≤S+I≤Λ1μ1,0≤V≤Λ2μ2,S≥0,I≥0.For the existence of the equilibrium of system (2), according to [5], one can have the following result.Lemma 1.
For system (2), E0=Λ1/μ1,0,0 is a disease-free equilibrium if R0≤1, and E∗=S∗,I∗,V∗ is an endemic equilibrium if R0>1, where R0=b2Λ1Λ2β1β2/μ1μ22μ1+γ is the basic reproductive number, S∗=Λ1μ2bβ2I∗+μ2/b2Λ2β1β2I∗+μ1μ2bβ2I∗+μ2, V∗=bΛ2β2I∗/μ2bβ2I∗+μ2, and I∗ satisfy the following equation:
(4)P2I∗2+P1I∗+P0=0,where
(5)P2=μ1b2β22bσβ1Λ2+μ1μ2bΛ2β1+μ1μ2+γb2β22μ1μ2bΛ2β1+μ1μ2,P1=bβ2μ1μ22μ1+γbΛ2β1+μ1μ2+bβ2μ12μ22bσβ1Λ2+μ1μ2+bγβ2μ12μ23−b3Λ1Λ2β1β22bσβ1Λ2+μ1μ2,P0=μ12μ24μ1+γ1−R0.The disease-free equilibriumE0 denotes no infection, and the endemic equilibrium E∗ represents that the disease will exist and persist. The basic reproductive number R0 describes the expected number of secondary infections, which is mainly sensitive to parameters b and β1 but not affected by parameter σ [5]. The process of deriving R0 using the next-generation method [22–24] is presented in Appendix A.


## 3. Stability of Endemic Equilibrium and Hopf Bifurcation

The Jacobian matrix of the linear system (2) at E∗=S∗,I∗,V∗ is
(6)J=−μ1−bβ1V∗e−λτ10−bβ1S∗e−λτ1−σ−e−λτ1bβ1V∗−μ1−γ−bσβ1V∗bβ1S∗e−λτ1+bσβ1Λ1μ1−I∗−S∗0bβ2Λ2μ2−V∗e−λτ2−μ2−bβ2I∗e−λτ2,and the characteristic equation detλI−J=0 is
(7)λ3+m2λ2+m1λ+m0+n2λ2+n1λ+n0e−λτ1+q2λ2+q1λ+q0e−λτ2+r1λ+r0e−λτ1+τ2=0,where
(8)m2=μ2+2μ1+γ+bσβ1V∗,m1=μ1+γ+bσβ1V∗μ1+μ2+μ1μ2,m0=μ1μ2μ1+γ+bσβ1V∗,n2=bβ1V∗,n1=bβ1V∗μ2+μ1+γ+bσβ1V∗,n0=bβ1V∗μ1+γ+bσβ1V∗μ2,q2=bβ2I∗,q1=bβ2I∗2μ1+γ+bσβ1V∗−bσβ1Λ1μ1−I∗−S∗bβ2Λ2μ2−V∗,q0=bβ2I∗μ1μ1+γ+bσβ1V∗−bσβ1Λ1μ1−I∗−S∗bβ2Λ2μ2−V∗μ1,r1=b2β1β2V∗I∗−bβ2Λ2μ2−V∗bβ1S∗,r0=b2β1β2V∗I∗μ1+γ+bσβ1V∗−b3β12β2S∗σV∗Λ2μ2−V∗−b3β12β2σV∗Λ1μ1−I∗−S∗Λ2μ2−V∗+μ1b2β1β2S∗Λ2μ2−V∗.Case (1).
τ1=τ2=0.Characteristic equation (7) becomes
(9)λ3+A12λ2+A11λ+A10=0,where A12=m2+n2+q2, A11=m1+n1+q1+r1, and A10=m0+n0+q0+r0.When the following condition (H1) holds, all roots of equation (9) have negative real parts.
(10)H1:A12>0,A11>0,A10>0,A12A11−A10>0.Hence, according to the Routh-Hurwitz criterion, the following conclusion can be drawn.Theorem 2.
If (H1) holds, the endemic equilibriumE∗ is locally asymptotically stable when τ1=τ2=0.Case (2).
τ1>0,τ2=0.The characteristic equation (7) becomes
(11)λ3+A22λ2+A21λ+A20+B22λ2+B21λ+B20e−λτ1=0,where A22=m2+q2,A21=m1+q1,A20=m0+q0,B22=n2,B21=n1+r1,andB20=n0+r0.Supposeλ=iω1ω1>0 is a root of equation (11), replacing it into equation (11) and separating the real and imaginary parts, then, we can obtain
(12)A22ω12−A20=B20−B22ω12cosω1τ1+B21ω1sinω1τ1,ω13−A21ω1=−B20−B22ω12sinω1τ1+B21ω1cosω1τ1.It follows that(13)ω16+A222−2A21−B222ω14+A212−B212−2A20A22+2B20B22ω12+A202−B202=0.Letz=ω12, p2=A222−2A21−B222, p1=A212−B212−2A20A22+2B20B22, and p0=A202−B202, then, equation (13) is equivalent to
(14)fz=z3+p2z2+p1z+p0=0.It is easy to get that the two roots off′z=3z2+2p2z+p1=0 are z1=1/3−p2+p22−3p1 and z2=1/3−p2−p22−3p1.According to [25, 26], we can get the conditions that equation (14) exists positive roots.Lemma 3.
For equation (14),
(i)
Ifp0<0, then equation (14) has at least one positive root(ii)
Ifp0≥0 and Δ1=p22−3p1≤0, then equation (14) has no positive roots(iii)
Ifp0≥0 and Δ1=p22−3p1>0, then equation (14) has positive roots if and only if z1=1/3−p2+Δ1>0 and fz1≤0By Lemma3, it is easy to see that the stability of E∗ will not change when τ changes if (ii) is set up. However, the stability may change when τ crosses through some critical values if equation (14) has a positive root.Assume thatz1, z2, and z3 are three positive roots of equation (14), then, there are three positive numbers ω11=z1, ω12=z2, and ω13=z3. It follows from equation (12) that
(15)cosω1τ1=A22ω12−A20B20−B22ω12+B21ω1ω13−A21ω1B20−B22ω122+B212ω12.Denote(16)τ1kj=1ω1karccosA22ω12−A20B20−B22ω12+B21ω1ω13−A21ω1B20−B22ω122+B212ω12+2jπω1k,where j=0,1,2,⋯, k=1,2,3, then, ±iω1k is a pair of purely imaginary roots of equation (11) for τ1=τ1kj. Let τ10=mink∈1,2,3τ1k0, ω10 be the ω1k that corresponds to the minimum τ1k0, and λτ1=ατ1+iωτ1 be the root of equation (11) near τ1=τ1kj satisfying ατ1kj=0 and ωτ1kj=ω1k.Furthermore, we consider the transversality condition. By differentiating both sides of equation (11) with respect to τ1, we get
(17)dλdτ1−1=3λ2+2A22λ+A21−λλ3+A22λ2+A21λ+A20+2B22λ+B21λB22λ2+B21λ+B20−τ1λ.It follows that(18)Redλdτ1τ1=τ10−1=Re3λ2+2A22λ+A21eλτ1+2B22λ+B21λB22λ2+B21λ+B20−τ1λτ1=τ10=3ω14+2A222−4A21−2B222ω12+A212−2A20A22−B212+2B20B22ω12B212ω14+B20−B22ω122ω12=ω12f′ω12B212ω14+B20−B22ω122ω12.Thus,(19)signdReλdτ1τ1=τ10=signRedλdτ1τ1=τ10−1=signf′ω12.The transversality condition is satisfied whenf′ω12≠0.According to [27], and applying Lemma 3 and the above transversal condition to system (2), the following theorem is obtained.Theorem 4.
For system (2), when τ2=0,
(i)
ifp0≥0 and Δ1=p22−3p1≤0 hold, then, the endemic equilibrium E∗ is locally asymptotically stable for all τ1>0(ii)
ifp0<0 or p0≥0, Δ1=p22−3p1>0, z1>0, and fz1≤0, then, the endemic equilibrium E∗ is locally asymptotically stable for τ1∈0,τ10(iii)
if the conditions of (ii) andf′z1≠0 hold, then, it undergoes a Hopf bifurcation at E∗ when τ1=τ10Case (3).
τ1>0,τ2>0.Fixingτ1 in its stable interval and regarding τ2 as a parameter, we analyze the roots of characteristic equation (7). Let λ=iωω>0 be a characteristic root of equation (7). By performing some calculations as those in case (2), we can obtain
(20)λ3+m2λ2+m1λ+m0eλτ2+n2λ2+n1λ+n0e−λτ1eλτ2+q2λ2+q1λ+q0+r1λ+r0e−λτ1=0,(21)A31cosωτ2+A32sinωτ2=A34,A31sinωτ2−A32cosωτ2=A35,(22)ω6+k5ω5+k4ω4+k3ω3+k2ω2+k1ω+k0=0,where
(23)A31=n0−n2ω2cosωτ1+n1ωsinωτ1+m0−m2ω2,A32=n0−n2ω2sinωτ1−n1ωcosωτ1+ω3−m1ω,A34=−r1ωsinωτ1−r0cosωτ1+q2ω2−q0,A35=−r1ωcosωτ1+r0sinωτ1−q1ω,k0=n02+m02−r02−q02+2n0m0cosωτ1−2r0q0cosωτ1,k1=2n1m0−n0m1−r1q0+r0q1sinωτ1,k2=n12+m12−2n0n2−2n0m2cosωτ1−2n2m0cosωτ1−2m0m2+2n1m1cosωτ1−r12+2r0q2cosωτ1+2q0q2−2r1q1cosωτ1−q12,k3=2n0+m1n2−n1m2+r1q2sinωτ1,k4=n22+m22+2n2m2cosωτ1−2n1cosωτ1−2m1−q22,k5=−2n2sinωτ1.Furthermore, we suppose that(H2): equation (22) has finite positive roots ω2i,i=1,2,⋯,k.Then, for every fixedω2i, there exists a sequence of critical values τ2ij such that equation (22) holds, where
(24)τ2ij=1ω2iarccosA31A34−A32A35A312+A322+2jπω2i,i=1,2,⋯,k;j∈ℕ0.Letτ2∗=minτ2i0i=1,2,⋯,k, when τ2=τ2∗, then equation (7) has a pair of purely imaginary roots ±iω2∗.Next, we also check the transversality condition. By taking the differentiation of (20) with respect to τ2 and further calculating, we can obtain
(25)dλdτ2−1=−f31λf32λ+τ2λ,where
(26)f31λ=3λ2+2m2λ+m1eλτ2+2n2λ+n1eλτ2−τ1−n2λ2+n1λ+n0τ1eλτ2−τ1+2q2λ+q1+r1e−λτ1−r1λ+r0τ1e−λτ1,f32λ=λ4+m2λ3+m1λ2+m0λeλτ2+n2λ3+n1λ2+n0λeλτ2−τ1.Thus, we can get(27)Redλdτ2τ2=τ2∗−1=−P31Q31+P32Q32Q312+Q322,where
(28)Q31=n0ω2∗−n2ω2∗3sinω2∗τ1−n1ω2∗2cosω2∗τ1+ω2∗4−m1ω2∗2cosω2∗τ2∗−n0ω2∗−n2ω2∗3cosω2∗τ1+n1ω2∗2sinω2∗τ1+m0ω2∗−m2ω2∗3sinω2∗τ2∗,Q32=n0ω2∗−n2ω2∗3sinω2∗τ1−n1ω2∗2cosω2∗τ1+ω2∗4−m1ω2∗2sinω2∗τ2∗+n0ω2∗−n2ω2∗3cosω2∗τ1+n1ω2∗2sinω2∗τ1−m2ω2∗3+m0ω2∗cosω2∗τ2∗,P31=τ1n2ω2∗2−τ1n0+n1cosω2∗τ1+2n2−τ1n1ω2∗sinω2∗τ1−3ω2∗2+m1cosω2∗τ2∗+τ1n2ω2∗2−τ1n0+n1sinω2∗τ1−2n2−τ1n1ω2∗cosω2∗τ1−2m2ω2∗sinω2∗τ2∗+r1−r0τ1cosω2∗τ1−τ1r1ω2∗sinω2∗τ1+q1,P32=τ1n2ω2∗2−τ1n0+n1cosω2∗τ1+2n2−τ1n1ω2∗sinω2∗τ1−3ω2∗2+m1sinω2∗τ2∗−τ1n2ω2∗2−τ1n0+n1sinω2∗τ1−2n2−τ1n1ω2∗cosω2∗τ1−2m2ω2∗cosω2∗τ2∗−r1−r0τ1sinω2∗τ1−τ1r1ω2∗cosω2∗τ1+2q2ω2∗.Therefore, when(H3):P31Q31+P32Q32≠0 holds, then Redλ/dτ2τ2=τ2∗−1≠0, i.e., the transversality condition is satisfied.According to the above discussions and based on [27], we can get the following result.Theorem 5.
Letτ1∈0,τ10, if (H2) and (H3) hold, then, the endemic equilibrium E∗ is locally asymptotically stable for τ2∈0,τ2∗ and is unstable for τ2∈τ2∗,+∞. System (2) undergoes Hopf bifurcation at the endemic equilibrium E∗ for τ2=τ2∗.


## 4. Properties of Hopf Bifurcation

From the analysis in the last section, we can see that for some critical values of delays, system (2) can occur in a series of periodic solutions at the equilibrium. On the basis of Theorem 5, we will employ the normal form method and the center manifold theorem introduced by Hassard et al. [20] to provide the properties of bifurcating periodic solutions in this section. Without loss of generality, we assume that τ1∗<τ2∗, τ1∗∈0,τ10, τ2=τ2∗+μ, u1t=Sτt−S∗, u2t=Iτt−I∗, and u3t=Vτt−V∗. Then, equation (2) can be rewritten into the following functional differential equation in the Banach Space C=C−1,0,ℝ3.
(29)u̇t=Lμut+Fμ,ut,where ut=u1t,u2t,u3tT∈ℝ3, utθ=ut+θ, Lμ:C→ℝ3, and Fμ,⋅:ℝ×C→ℝ3 are defined as follows:
(30)Lμϕ=τ2∗+μA′ϕ0+B′ϕ−τ1∗τ2∗+C′ϕ−1,Fμ,ϕ=τ2∗+μF1,F2,F3T,where
(31)ϕθ=ϕ1θ,ϕ2θ,ϕ3θT∈C,A′=−μ100−σbβ1V∗−μ1+γ+bσβ1V∗bσβ1Λ1μ1−I∗−S∗00−μ2,B′=−bβ1V∗0−bβ1S∗bβ1V∗0bβ1S∗000,C′=0000000bβ2Λ2μ2−V∗−bβ2I∗,F1=−bβ1ϕ1−τ1∗τ2∗ϕ3−τ1∗τ2∗,F2=bβ1ϕ1−τ1∗τ2∗ϕ3−τ1∗τ2∗−bσβ1ϕ10ϕ30−bσβ1ϕ20ϕ30,F3=−bβ2ϕ2−1ϕ3−1.Based on the Riesz representation theorem, we know that there exists a function of bounded variation componentsηθ,μ, θ∈−1,0, such that
(32)Lμϕ=∫−10dηθ,μϕθ,ϕθ∈C−1,0,ℝ3.In fact, we can take(33)ηθ,μ=τ2∗+μA′+B′+C′,θ=0,τ2∗+μB′+C′,θ∈−τ1∗τ2∗,0,τ2∗+μC′,θ∈−1,−τ1∗τ2∗,0,θ=−1.DefineAμ and Rμ by
(34)Aμϕ=dϕθdθ,θ∈−1,0,∫−10dηs,μϕs,θ=0,(35)Rμϕ=0,θ∈−1,0,Fμ,ϕ,θ=0.Then, system (29) can be further represented as
(36)u̇t=Aμut+Rμut,where utθ=ut+θ=u1t+θ,u2t+θ,u3t+θ,θ∈−1,0.Forφ∈C0,1,ℝ3∗, we define
(37)A∗φs=−dψsds,s∈0,1,∫−10dηTt,0φ−t,s=0,and the bilinear inner product
(38)φs,ϕθ=φ¯0ϕ0−∫−10∫ξ=0θφ¯ξ−θdηθϕξdξ,where ηθ=ηθ,0. Let A=A0, then A and A∗ are a pair of adjoint operators. From the discussions in Section 3, we know that ±iω2∗τ2∗ are a pair of eigenvalues of A0, it follows that they are also a pair of eigenvalues of A∗. Next, we calculate the eigenvectors of A0 and A∗ with respect to iω2∗τ2∗ and −iω2∗τ2∗, respectively.Suppose thatqθ is an eigenfunction of A0 corresponding to iω2∗τ2∗. Then, by the definition of A, we can take
(39)qθ=1,q2,q3Teiω2∗τ2∗θ,where
(40)q2=iω2∗+μ2+bβ2I∗e−iω2∗τ2∗iω2∗+μ1+bβ1V∗e−iω2∗τ1∗−bβ2Λ2/μ2−V∗e−iω2∗τ2∗,q3=iω2∗+μ1+bβ1V∗e−iω2∗τ1∗−bβ1S∗e−iω2∗τ1∗.Similarly,q∗s=D1,q2∗,q3∗Teiω2∗τ2∗s,s∈0,1 is the eigenfunction of A∗0 corresponding to −iω2∗τ2∗, where
(41)q2∗=−−iω2∗+μ1+bβ1V∗eiω2∗τ1∗σ−eiω2∗τ1∗bβ1V∗,q3∗=−iω2∗+μ1+γ+bσβ1V∗−iω2∗+μ1+bβ1V∗eiω2∗τ1∗bβ2Λ2/μ2−V∗eiω2∗τ2∗σ−eiω2∗τ1∗bβ1V∗.From equation (38), we can get
(42)q∗s,qθ=q¯∗0q0−∫−10∫ξ=0θq¯∗ξ−θdηθqξdξ=D¯1,q¯2∗,q¯3∗1,q2,q3T−D¯∫−10∫ξ=0θ1,q¯2∗,q¯3∗eiω2∗τ2∗θ−ξdηθ1,q2,q3Teiω2∗τ2∗ξdξ=D¯1+q2q¯2∗+q3q¯3∗−1,q¯2∗,q¯3∗∫−10θeiω2∗τ2∗θdηθ1,q2,q3T=D¯1+q2q¯2∗+q3q¯3∗+τ1∗bβ1e−iω2∗τ1∗−V∗+q¯2∗−S∗q3+q¯2∗q3+τ2∗bβ2e−iω2∗τ2∗Λ2μ2−V∗q2−I∗q3q¯3∗.Thus, one can chooseD as
(43)D=1+q¯2q2∗+q¯3q3∗+τ1∗bβ1eiω2∗τ1∗−V∗+q¯2∗−S∗q3+q¯2∗q3+τ2∗bβ2eiω2∗τ2∗Λ2μ2−V∗q2−I∗q3q¯3∗−1,which satisfiesq∗s,qθ=1.Next, using the methods given in [20], we can calculate some explicit expressions as follows that are used to determine the qualities of bifurcating periodic solution.
(44)g20=2D¯τ2∗−bβ1q3e−2iω2∗τ1∗+q¯2∗bβ1q3e−2iω2∗τ1∗−σbβ1q3−σbβ1q2q3−q¯3∗bβ2q2q3e−2iω2∗τ2∗,(45)g11=D¯τ2∗−bβ1q¯3+q3+q¯2∗1−σbβ1q¯3+q3−bσβ1q2q¯3+q¯2q3−q¯3∗bβ2q2q¯3+q¯2q3,(46)g02=2D¯τ2∗−bβ1q¯3e2iω2∗τ1∗+q¯2∗bβ1q¯3e2iω2∗τ1∗−σbβ1q¯3−σbβ1q¯2q¯3−q¯3∗bβ2q¯2q¯3e2iω2∗τ2∗,(47)g21=D¯τ2∗−bβ12W113−τ1∗τ2∗e−iω2∗τ1∗+W203−τ1∗τ2∗eiω2∗τ1∗+W201−τ1∗τ2∗q¯3eiω2∗τ1∗+2W111−τ1∗τ2∗q3e−iω2∗τ1∗+q¯2∗bβ12W113−τ1∗τ2∗e−iω2∗τ1∗+W203−τ1∗τ2∗eiω2∗τ1∗+W201−τ1∗τ2∗q¯3eiω2∗τ1∗+2W111−τ1∗τ2∗q3e−iω2∗τ1∗−σbβ12W1130+W2030+W2010q¯3+2W1110q3−bσβ12W1130q2+W2030q¯2+W2020q¯32W1120q3−q¯3∗bβ22W113−1q2e−iω2∗τ2∗+W203−1q¯2eiω2∗τ2∗+W202−1q¯3eiω2∗τ2∗+2W112−1q3e−iω2∗τ2∗,where
(48)W20θ=ig20ω2∗τ2∗q0eiω2∗τ2∗θ+ig¯023ω2∗τ2∗q¯0e−iω2∗τ2∗θ+E1e2iω2∗τ2∗θ,W11θ=−ig11ω2∗τ2∗q0eiω2∗τ2∗θ+ig¯11ω2∗τ2∗q¯0e−iω2∗τ2∗θ+E2,E1=2Δ11Δ1,Δ12Δ1,Δ13Δ1,E2=Δ21Δ2,Δ22Δ2,Δ23Δ2,Δ1=2iω2∗+μ1+bβ1V∗e−2iω2∗τ1∗0bβ1S∗e−2iω2∗τ1∗σ−e−2iω2∗τ1∗bβ1V∗2iω2∗+μ1+γ+bσβ1V∗−bβ1S∗e−2iω2∗τ1∗−bσβ1Λ1μ1−I∗−S∗0−bβ2Λ2μ2−V∗e−2iω2∗τ2∗2iω2∗+μ2+bβ2I∗e−2iω2∗τ2∗,Δ2=μ1+bβ1V∗0bβ1S∗−1−σbβ1V∗μ1+λ+bσβ1V∗−μ1+γI∗V∗0−bβ2Λ2μ2−V∗μ2+bβ2I∗.Δ11, Δ12, and Δ13 are the third-order determinants obtained by substituting the first, second, and third columns of Δ1 by vector P1,P2,P3T, respectively. Δ21, Δ22, and Δ23 are also third-order determinants obtained by replacing the first, second, and third columns of Δ2 by vector Q1,Q2,Q3T, respectively, where
(49)P1=−bβ1q3e2iω2∗τ1∗,P2=bβ1q3e−2iω2∗τ1∗−σbβ1q3−σbβ1q2q3,P3=−bβ2q2q3e−2iω2∗τ2∗,Q1=−bβ1Req3,Q2=1−σbβ1Req3−bσβ1Req2q¯3,Q3=−bβ2Req2q¯3.Thus, the following quantities can be calculated:(50)c10=i2ω2∗τ2∗g20g11−2g112−13g022+12g21,μ2=−Rec10Reλ′τ2∗,β2=2Rec10,T2=−1ω2∗τ2∗Imc10+μ2Imλ′τ2∗.Therefore, based on [20], we can obtain the main results in the present section as follows.Theorem 6.
For system (2), the periodic solution is supercritical (resp., subcritical) if μ2>0 (resp., μ2<0). The bifurcating periodic solutions are orbitally asymptotically stable with an asymptotical phase (resp., unstable) if β2<0 (resp., β2>0). The period of the bifurcating periodic solutions increases (resp., decreases) if T2>0 (resp. T2<0).


## 5. Numerical Examples and Simulations

We select the parameters to simulate the above theoretical results as follows:(51)Λ1=9,b=0.3,β1=0.002,μ1=0.01,σ=0.8,γ=0.6,Λ2=9,β2=0.005,μ2=0.05.By calculations, we can obtain that the endemic equilibrium of system (2) is E∗=S∗,I∗,V∗=102.5734,85.6437,129.5701. Under the above parameters, the condition (H1) is satisfied. According to Theorem 2, we know that the endemic equilibrium E∗ of system (2) is asymptotically stable when τ1=τ2=0 (see Figure 1). When τ1>0, τ2=0, we get p0=−4.7847×10−5<0, τ10=23.3291, z1=0.0053, and f′z1=0.017≠0; from Theorem 4, the endemic equilibrium E∗ is locally asymptotically stable for τ1∈0,τ10 and is unstable for τ1>τ10, and a Hopf bifurcation occurs at τ1=τ10 (see Figures 2–4). When τ2>0, fixing τ1=23 that belongs to its stable interval 0,τ10, then, by calculation, we obtain that a positive root of equation (22) is ω=0.0933 and τ2∗=21.4739, which follow from Theorem 5 that the endemic equilibrium E∗ is locally asymptotically stable for τ2∈0,τ2∗ and is unstable for τ2∈τ2∗,+∞, a Hopf bifurcation exits at τ2=τ2∗ (see Figures 5–7).Figure 1
The phase graph of system (2) with τ1=τ2=0.Figure 2
The endemic equilibriumE∗ is asymptotically stable with τ1=23<τ10.Figure 3
The endemic equilibriumE∗ is unstable with τ1=24>τ10.Figure 4
The bifurcation diagrams withτ1 as parameter and τ2=0. A Hopf bifurcation exists at τ1=τ10.Figure 5
The endemic equilibriumE∗ is asymptotically stable with τ1=23, τ2=21<τ2∗.Figure 6
The endemic equilibriumE∗ is unstable with τ1=23 and τ2=22>τ2∗.Figure 7
The bifurcation diagrams withτ2 as parameter and τ1=23. A Hopf bifurcation exists at τ2=τ2∗.The numerical results show thatc10=1.0266×10−5−4.7626×10−6i, μ2=−0.0024<0, β2=2.0531×10−5>0, and T2=2.3115×10−6>0, which imply that, for system (2), its periodic solution is subcritical at τ2∗ and unstable, and the period of the bifurcating periodic solutions increases (see Figure 8). Also, when the two delays are greater than their critical values, the system will produce irregular periodic behaviors, which is depicted in Figure 9.Figure 8
The bifurcation periodic solutions ofIt with τ1=23 and τ2=22,28,38 (from the top to the first of the bottom). The bifurcation periodic solution is unstable with τ1=23 and τ2=38.Figure 9
The system appears irregular periodic behaviors whenτ1=31 and τ2=31.Besides, the influences of the parameterσ on the endemic equilibrium E∗ and the critical values τ10 and τ2∗ of the two delays τ1 and τ2 are shown in Table 1. When σ decreases, I∗ and V∗ also decrease, but S∗, τ10, and τ2∗ increase, which suggests that the degree of partial protection for recovered people acquired by a primary infection can change the densities of distributions of susceptible and infective people and can also affect on the stability of positive equilibrium. It is noteworthy that it can not change the direction of Hopf bifurcation and the stability of bifurcating periodic solutions.Table 1
The relationships among the partial immunityσ, the endemic equilibrium E∗, and the critical values τ10 and τ2∗.
σ10.80.60.40.2S∗97.4478102.5734111.3948129.1222170.4044I∗107.056685.643763.424641.203021.8956V∗137.2619129.5701117.989699.502371.3613τ1021.589523.329126.611234.364756.7849τ2∗17.812021.473925.592530.626744.0800


## 6. Conclusions

In this paper, we considered a vector-borne disease model with two delays and reinfection based on the models proposed by [4–6]. Some theoretical analyses of this model were implemented. From the characteristic equations, firstly, we discussed the conditions of the endemic equilibrium E∗, which is locally asymptotically stable in the model without delays by the Routh-Hurwitz criterion. Then, by setting τ2=0, regarding τ1 as a bifurcating parameter, and setting τ1 in its stable interval, regarding τ2 as a parameter, respectively, we employed the Hopf bifurcation theorem to analyze the conditions for the stability of equilibrium and existence of Hopf bifurcation. Furthermore, when τ1 is in its stable interval, for parameter τ2, the directions and stability of Hopf bifurcation were investigated, and some explicit formulas were derived by using the normal form theory and center manifold theorem. In the end, our numerical examples and simulations verified the correctness of theoretical conclusions. As the facts show, the disease will be persistent under the set of parameters, and the number of the host population and vector population will fluctuate. Also, when both delays exceed their critical values, the stability of the system and the properties and behaviors of the bifurcating periodic solutions may lead to change. The oscillatory behavior and phenomenon that an increased partial immunity may give rise to instability will prompt medical workers engaged in infectious diseases to remain vigilant against the virus even if they have noticed that the number of infected populations decreases. The qualitative analysis and numerical simulation used in the research of dynamic behavior of system can provide theoretical basis for revealing the development process, transmission pattern, and change trend of diseases and give important reference to the analysis, prediction, and control of human infectious diseases.



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