# Stability and Hopf Bifurcation of a Generalized Chikungunya Virus Infection Model with Two Modes of Transmission and Delays

Authors: Hajar Besbassi; Khalid Hattaf; Noura Yousfi
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

A generalized chikungunya virus (CHIKV) infection model with nonlinear incidence functions and two time delays is proposed and investigated. The model takes into account both modes of transmission that are virus-to-cell infection and cell-to-cell transmission. Furthermore, the local and global stabilities of the disease-free equilibrium and the chronic infection equilibrium are established by using the linearization and Lyapunov functional methods. Moreover, the existence of Hopf bifurcation is also analyzed. Finally, an application is presented in order to support the analytical results.


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## Body

## 1. Introduction

The CHIKV belongs to the family Togaviridae, a term built from the Roman toga, to describe the draped appearance of their envelope [1]. Its genetic material consists of a single-stranded, thermosensitive RNA, about 15,000 nucleotides long. The multiplication of the viral genome in the cell is not strictly accurate, a common property of RNA viruses, which results in mutations that can affect not only the infective and pathogenic powers of the virus but also its passage from one kind of Aedes to another. Viral RNA of the infecting virion is included in a spherical particle made up of viral or nucleocapsid proteins assembled regularly and is of a size of around 70 nanometres. The virus multiplies with great ease in vitro, but also in vivo in mosquito cells, which explains the high infective power of contaminated Aedes. The female mosquito infects itself during a blood meal (necessary for laying) on a contaminated individual (man especially in the epidemic phase and also bats, monkeys, and other vertebrates). The virus proliferates in the insect. It is injected into a man or animal during a subsequent blood meal, during the initial phase of the bite, which includes the injection of “saliva” from the infected insect, before the blood meal itself [2].In [3], the authors described the CHIKV replication cycle (see Figure 1) and the results of chikungunya virus infection particularly intense joint and muscle pain that forces patients to lean forward. After about one week of incubation, the pain appears, especially in the wrists, fingers, knees, ankles, and feet. The hips and shoulders are more rarely affected. These pains are accompanied by severe headaches, fever (over 38.5°C), and rash in the chest and limbs, as well as lymph node swelling and conjunctivitis. Other symptoms sometimes appear, including bleeding of the gums or nose and neurological disorders.Figure 1
Schematic representation of CHIKV dissemination to different tissues and organs [3].Medical management is purely symptomatic, based on pain and anti-inflammatory treatments. However, these treatments have no preventive effect on the occurrence of a chronic evolution. First isolated in Uganda in 1953, CHIKV circulates mainly in the intertropical zone. This disease is particularly virulent in Africa and South Asia. However, cases were detected in the French territory as early as 2010 (in the south of France) and in 2013 and 2014 in the West Indies [4]. It can be responsible for important epidemics [5]. The risk of emergence in Europe is ever increasing due to the increase in “tiger mosquito,” Aedes albopictus [6]. First observed in 2004 in the Alpes-Maritimes, the vector was established and active in 33 metropolitan departments in May 2017 [7]. This emergence has made it possible to highlight the need to strengthen the knowledge of health professionals with regard to arboviroses. Therefore, a few mathematical models have been established to describe dynamics of CHIKV viral infection, mostly focusing on virus-to-cell transmission [8, 9]. However, CHIKV can be spread by cell-to-cell transmission mode [10–13].In view of this, we will formulate and analyse a generalized within-host CHIKV viral infection model taking into the account both modes of transmission and two discrete delays, in which the first delayτ1 describes the time necessary for the newly produced virions to become mature and infectious and the second delay τ2 represents the time needed to activate the humoral immune response. Then, the model is presented as follows:(1)U˙=s−d1Ut−fUt,It,CtCt−gUt,ItIt,I˙=fUt,It,CtCt+gUt,ItIt−d2It,C˙=υIt−τ1−d3Ct−pAtCt,A˙=σ+qAt−τ2Ct−τ2−d4At,where the general incidence functions fU,I,C and gU,I assumed to be continuously differentiable satisfy the following hypotheses [14, 15]:(i)
(H0): g (0, I) = 0, for all I≥0; ∂g/∂UU,I≥0 (or gU,I is a monotone increasing function with respect to T when f≡0), and ∂g/∂IU,I≤0, for all U≥0 and I≥0(ii)
(H1): f0,I,C=0, for all I≥0 and C≥0(iii)
(H2): fU,I,C is a monotone increasing function with respect to U (or ∂f/∂UU,I,C≥0 when gU,I is a strictly monotone increasing function with respect to U), for any fixed I≥0 and C≥0(iv)
(H3): fU,I,C is a monotone decreasing function with respect to I and CIn biological terms,Ut, It, Ct, and At indicate the densities of susceptible cells, infected cells, CHIKV particles, and antibodies at time t, respectively. The parameter s is the recruitment rate of uninfected cells, and υ is the production rate of free CHIKV particles by infected cells. The CHIKV particles are attacked by the antibodies at rate pAC. The antibodies are created at rate σ and multiplicated at rate qAC. The parameters d1, d2, d3, and d4 are, respectively, the death rates of susceptible cells, infected cells, free CHIKV virions, and antibodies. Moreover, susceptible cells become infected either by free virus at rate fU,I,CC or by direct contact with an infected cell at rate gU,II. In addition, particular cases of the incidence function f and g are used by Elaiw et al. [16] to model the dynamics of CHIKV with cellular infection and delays. On the other hand, system (1) extends the model presented in [17] when τ1=τ2=0 and the model proposed in [8] when fU,I,C=β1U and gU,I=0.The rest of this paper is organized as follows. In Section2, we provide some preliminary results concerning the existence, positivity, and boundedness of solutions. Also, we discuss the existence of equilibria. In Section 3, we analyse the stability for the equilibria. We investigate the existence of Hopf bifurcation in Section 4. An application is presented in Section 5. This paper ends with a conclusion in Section 6.


## 2. Preliminary Results

In this section, we first prove the existence, positivity, and boundedness of solutions. After that, we discuss the existence of equilibria.


### 2.1. Existence, Positivity, and Boundedness of Solutions

According to biological meanings, the initial condition of system (1) is given as follows:(2)Uθ=ϕ1θ≥0,Iθ=ϕ2θ≥0,Cθ=ϕ3θ≥0,Aθ=ϕ4θ≥0,θ∈−τ,0,where τ=maxτ1,τ2 and ϕ1θ,ϕ2θ,ϕ3θ,ϕ4θ∈C+=C−τ,0,IR+4. C+ is the Banach space of continuous functions mapping the interval −τ,0 into IR+4 with the topology of uniform convergence.It follows from the fundamental theory of functional differential equations [18] that there exists a unique solution of system (1) with initial condition ϕ1,ϕ2,ϕ3,ϕ4∈C+.Next, we investigate the positivity and boundedness of this solution under initial condition (2).Theorem 1.
Under the initial condition (2), the solution of system (1) remains bounded and positive for all t>0.Proof.
We first demonstrate thatUt>0 for all t≥0. By contradiction, we assume that there exists a first time t1>0 such that Ut1=0 and U˙t1≤0. From the first equation of system (1), we have U˙t1=s>0, which leads a contradiction. Then, Ut>0 for all t≥0. Since A˙t=η>0, and similar to the above, we deduce that At>0 for all t≥0. According to (1), we have(3)It=ϕ20e−d2t∫0tgUθ,Iθdθ+∫0tfUξ,Iξ,CξCξe−d2t−ξ+∫0tgUθ,Iθdθdξ,Ct=ϕ30e−d3t−p∫0tAθdθ+υ∫0tIξ−τ1e−d3t−ξ−∫0tpAθdθdξ,which implies that It and Ct are nonnegative for all t≥0.
We consider the following function:(4)Nt=Ut+It+d22υCt+τ1+pd22qυAt+τ1+τ2.
Then,(5)N˙t=s−d1Ut−d22It−d2d32υCt+τ1+pd2σ2qυ−pd2d42qυAt+τ1+τ2≤s+pd2σ2qυ−δNt,where δ=mind1,d2/2,d3,d4. Hence,(6)lim supt⟶∞Nt≤sδ+pd2σ2qυδ,which implies that all solutions of system (1) are bounded. This completes the proof.


### 2.2. Existence of the Equilibria

Presently, we examine the existence of equilibria. By a basic calculation, system (1) has constantly one infection-free equilibrium of the form Qfs/d1,0,0,σ/d4. Thus, we characterize the basic reproduction number of our model as follows:(7)R0=υfs/d1,0,0+d3+pσ/d4gs/d1,0d2d3+pσ/d4.To locate different equilibria of (1), we solve the accompanying system:(8)s−d1U−fU,I,CC−gU,II=0,(9)fU,I,CC+gU,II−d2I=0,(10)υI−d3C−pAC=0,(11)σ+qAC−d4A=0.From (8)–(11), we obtain A=σ/d4−qC, I=d3d4−qC+pσ/υd4−qCC=φ1C, U=s−d2φ1C/d1=φ2C, and(12)υd4−qCfφ2C,φ1C,C+d3d4−qC+pσgφ2C,φ1C=d2d3d4−qC+pσ.A=σ/d4−qC≥0 leads to C<d4/q. Hence, there is no biological equilibrium when C≥d4/q. Accordingly, we consider the function ψ defined on 0,d4/q by(13)ψC=υd4−qCfφ2C,φ1C,C+d3d4−qC+pσ⋅gφ2C,φ1C−d2d3d4−qC+pσ.We haveφ20=s/d1>0 and(14)limC⟶d4/q−φ2C=−∞andφ2′C=−d2d1φ1′C<0,with φ1′C=d3d4−qC2+pσd4/υd4−qC2>0. Then, the equation φ2C=0 admits a unique solution C˜∈0,d4/q. Thus, A˜=σ/d4−qC˜>0andψC˜=−d2d3d4−qC˜+pσ<0. Since ψ0=d2d3d4+pσR0−1>0 if R0>1, we deduce that there exists C∗∈0,C˜ such that ψC∗=0.From (10) and (11), we find A∗=σ/d4−qC∗>0andI∗=d3+pA∗/υC∗>0.SubstituteC=C∗ and I=I∗ in (8), and define a function φ3 as φ3U=s−d1U−fU,I∗,C∗C∗−gU,I∗I∗. Due to the fact that φ30=s>0, φ3s/d=−fs/d1,I∗,C∗C∗−gs/d,I∗I∗<0, and φ3 is a strictly decreasing function of U, we deduce that there exists a unique U∗∈0,s/d1 such that φ3U∗=0. Therefore, model (1) has a unique chronic infection equilibrium Q∗U∗,I∗,C∗,A∗ when R0>1.The precedent conversations can be summed up in the accompanying outcome.Theorem 2.
(i)
ForR0≤1, model (1) has one infection-free equilibrium Qfs/d1,0,0,σ/d4(ii)
ForR0>1, model (1) has a unique chronic infection equilibrium Q∗U∗,I∗,C∗,A∗ with U∗∈0,s/d1, I∗>0, C∗>0, and A∗>0


## 2.1. Existence, Positivity, and Boundedness of Solutions

According to biological meanings, the initial condition of system (1) is given as follows:(2)Uθ=ϕ1θ≥0,Iθ=ϕ2θ≥0,Cθ=ϕ3θ≥0,Aθ=ϕ4θ≥0,θ∈−τ,0,where τ=maxτ1,τ2 and ϕ1θ,ϕ2θ,ϕ3θ,ϕ4θ∈C+=C−τ,0,IR+4. C+ is the Banach space of continuous functions mapping the interval −τ,0 into IR+4 with the topology of uniform convergence.It follows from the fundamental theory of functional differential equations [18] that there exists a unique solution of system (1) with initial condition ϕ1,ϕ2,ϕ3,ϕ4∈C+.Next, we investigate the positivity and boundedness of this solution under initial condition (2).Theorem 1.
Under the initial condition (2), the solution of system (1) remains bounded and positive for all t>0.Proof.
We first demonstrate thatUt>0 for all t≥0. By contradiction, we assume that there exists a first time t1>0 such that Ut1=0 and U˙t1≤0. From the first equation of system (1), we have U˙t1=s>0, which leads a contradiction. Then, Ut>0 for all t≥0. Since A˙t=η>0, and similar to the above, we deduce that At>0 for all t≥0. According to (1), we have(3)It=ϕ20e−d2t∫0tgUθ,Iθdθ+∫0tfUξ,Iξ,CξCξe−d2t−ξ+∫0tgUθ,Iθdθdξ,Ct=ϕ30e−d3t−p∫0tAθdθ+υ∫0tIξ−τ1e−d3t−ξ−∫0tpAθdθdξ,which implies that It and Ct are nonnegative for all t≥0.
We consider the following function:(4)Nt=Ut+It+d22υCt+τ1+pd22qυAt+τ1+τ2.
Then,(5)N˙t=s−d1Ut−d22It−d2d32υCt+τ1+pd2σ2qυ−pd2d42qυAt+τ1+τ2≤s+pd2σ2qυ−δNt,where δ=mind1,d2/2,d3,d4. Hence,(6)lim supt⟶∞Nt≤sδ+pd2σ2qυδ,which implies that all solutions of system (1) are bounded. This completes the proof.


## 2.2. Existence of the Equilibria

Presently, we examine the existence of equilibria. By a basic calculation, system (1) has constantly one infection-free equilibrium of the form Qfs/d1,0,0,σ/d4. Thus, we characterize the basic reproduction number of our model as follows:(7)R0=υfs/d1,0,0+d3+pσ/d4gs/d1,0d2d3+pσ/d4.To locate different equilibria of (1), we solve the accompanying system:(8)s−d1U−fU,I,CC−gU,II=0,(9)fU,I,CC+gU,II−d2I=0,(10)υI−d3C−pAC=0,(11)σ+qAC−d4A=0.From (8)–(11), we obtain A=σ/d4−qC, I=d3d4−qC+pσ/υd4−qCC=φ1C, U=s−d2φ1C/d1=φ2C, and(12)υd4−qCfφ2C,φ1C,C+d3d4−qC+pσgφ2C,φ1C=d2d3d4−qC+pσ.A=σ/d4−qC≥0 leads to C<d4/q. Hence, there is no biological equilibrium when C≥d4/q. Accordingly, we consider the function ψ defined on 0,d4/q by(13)ψC=υd4−qCfφ2C,φ1C,C+d3d4−qC+pσ⋅gφ2C,φ1C−d2d3d4−qC+pσ.We haveφ20=s/d1>0 and(14)limC⟶d4/q−φ2C=−∞andφ2′C=−d2d1φ1′C<0,with φ1′C=d3d4−qC2+pσd4/υd4−qC2>0. Then, the equation φ2C=0 admits a unique solution C˜∈0,d4/q. Thus, A˜=σ/d4−qC˜>0andψC˜=−d2d3d4−qC˜+pσ<0. Since ψ0=d2d3d4+pσR0−1>0 if R0>1, we deduce that there exists C∗∈0,C˜ such that ψC∗=0.From (10) and (11), we find A∗=σ/d4−qC∗>0andI∗=d3+pA∗/υC∗>0.SubstituteC=C∗ and I=I∗ in (8), and define a function φ3 as φ3U=s−d1U−fU,I∗,C∗C∗−gU,I∗I∗. Due to the fact that φ30=s>0, φ3s/d=−fs/d1,I∗,C∗C∗−gs/d,I∗I∗<0, and φ3 is a strictly decreasing function of U, we deduce that there exists a unique U∗∈0,s/d1 such that φ3U∗=0. Therefore, model (1) has a unique chronic infection equilibrium Q∗U∗,I∗,C∗,A∗ when R0>1.The precedent conversations can be summed up in the accompanying outcome.Theorem 2.
(i)
ForR0≤1, model (1) has one infection-free equilibrium Qfs/d1,0,0,σ/d4(ii)
ForR0>1, model (1) has a unique chronic infection equilibrium Q∗U∗,I∗,C∗,A∗ with U∗∈0,s/d1, I∗>0, C∗>0, and A∗>0


## 3. Stability Analysis of Equilibria

In this section, we concentrate on the stability of infection-free equilibrium of system (1). The characteristic equation of system (1) is noted as(15)−d1−C∂f∂U−I∂g∂U−ξ−C∂f∂I−I∂g∂I−gU,I−C∂f∂C−fU,I,C0C∂f∂U+I∂g∂UC∂f∂I+I∂g∂I+gU,I−d2−ξC∂f∂C+fU,I,C00υe−ξτ1−d3−pA−ξ−pC00qAe−ξτ2qCe−ξτ2−d4−ξ=0.First, we have the following result.Theorem 3.
For anyτ2 and τ1=0, the infection-free equilibrium Qf is locally asymptotically stable if R0<1 and becomes unstable if R0>1.Proof.
Examining (15) at Qf, we obtain(16)ξ+d1ξ+d4ξ2+d3+d2+pσd4−gsd1,0ξ−υe−ξτ1fsd1,0,0−d3+pσd4gsd1,0−d2=0.
Whenτ1=0, from equation (16), we obtain(17)ξ+d1ξ+d4ξ2+d3+d2+pσd4−gsd1,0ξ+d2d3+pσd41−R0=0.
Therefore, the roots of this equation are(18)ξ1=−d1,ξ2=−d4,ξ3=−d2+d3+pσ/d4−gs/d1,0−Δ2,ξ4=−d2+d3+pσ/d4−gs/d1,0+Δ2,of which Δ=d2+d3+pσ/d4−gs/d1,02−4d2d3+pσ/d41−R0. Obviously, ξ1, ξ2, and ξ3 are negative. Furthermore, ξ4 is negative if R0<1 and positive if R0>1. Consequently, Qf is locally asymptotically stable if R0<1 and unstable if R0>1.
The following theorem characterizes the global stability of the infection-free equilibriumQf when R0≤1.Theorem 4.
For anyτ1 and τ2, the infection-free equilibrium Qf is globally asymptotically stable if R0≤1.Proof.
We establish a Lyapunov function as follows:(19)Lt=It+fs/d1,0,0d3+pσ/d4Ct+υfs/d1,0,0d3+pσ/d4∫t−τ1tIθdθ.
Computing the time derivative ofL along the solutions of (1), we find(20)dLdt=fU,I,C−d3+pAd3+pσ/d4fsd1,0,0C+d2υfs/d1,0,0+d2+pσ/d4gU,Id2d3+pσ/d4−1I≤fU,0,0−fsd1,0,0C+aR0−1I≤aR0−1I.
Given thatR0≤1, we have dL/dt≤0. Likewise, it is not difficult to show that the largest invariant set in U,I,C,A|dL/dt=0 is Qf. By the LaSalle’s invariance principle [19], Qf is globally asymptotically stable for R0≤1.
Next, we focus on the global stability of the chronic infection equilibriumQ∗ by assuming that R0>1, and for all U, I, C>0, we consider the following hypothesis:(21)1−fU,I,CfU,I∗,C∗fU,I∗,C∗fU,I,C−CC∗≤0,1−fU∗,I∗,C∗gU,IfU,I∗,C∗gU∗,I∗fU,I∗,C∗gU∗,I∗fU∗,I∗,C∗gU,I−II∗≤0.Theorem 5.
Assume that (21) holds. For any τ1, if τ2=0 and R0>1, then the chronic infection equilibrium Q∗ is globally asymptotically stable.Proof.
Consider the following Lyapunov function:(22)Wt=Ut−U∗−∫U∗UfU∗,I∗,C∗fX,I∗,C∗dX+I∗ΦItI∗+fU∗,I∗,C∗C∗υI∗C∗ΦCtC∗+qfU∗,I∗,C∗C∗qυI∗A∗ΦAtA∗+fU∗,I∗,C∗C∗∫t−τ1tΦIθI∗dθ,where Φx=x−1−lnx, x>0. Thus, the time derivative of W along the positive solutions of (1) satisfies(23)dWdt=1−fU∗,I∗,C∗fU,I∗,C∗U˙+1−I∗II˙+fU∗,I∗,C∗C∗υI∗1−C∗CC˙pfU∗,I∗,C∗C∗qυI∗1−A∗AA˙+fU∗,I∗,C∗C∗ddt∫t−τ1tΦIθI∗dθ,where d/dt∫t−τ1tΦIθ/I∗dθ=I−It−τ1/I∗+lnIt−τ1/I. Therefore, we have(24)dWdt=1−fU∗,I∗,C∗fU,I∗,C∗s−d1U−fU,I,CC−gU,II+1−I∗IfU,I,CC+gU,II−d2I+fU∗,I∗,C∗C∗υI∗1−C∗CυIt−τ1−d3C−pAC+pfU∗,I∗,C∗C∗qυI∗1−A∗Aσ+qAC−d4A+fU∗,I∗,C∗C∗I−It−τ1I∗+lnIt−τ1I.
Substitutings=d1U∗+fU∗,I∗,C∗C∗+gU∗,I∗I∗, υI∗=d3C∗+qA∗C∗, and σ=d4A∗−qA∗C∗, we obtain(25)dWdt=d1U∗1−UU∗1−fU∗,I∗,C∗fU,I∗,C∗+fU∗,I∗,C∗C∗−1−CC∗+fU,I,CCfU,I∗,C∗C∗+fU,I∗,C∗fU,I,C+fU∗,I∗,C∗C∗4−fU∗,I∗,C∗fU,I∗,C∗−fU,I∗,C∗fU,I,C−fU,I,CCI∗fU∗,I∗,C∗C∗I−C∗It−τ1CI∗+lnIt−τ1I+gU∗,I∗I∗−1−II∗−fU∗,I∗,C∗gU,IIfU,I∗,C∗gU∗,I∗I∗+fU,I∗,C∗gU∗,I∗fU∗,I∗,C∗gU,I+gU∗,I∗I∗3−fU∗,I∗,C∗fU,I∗,C∗−fU,I∗,C∗gU∗,I∗fU∗,I∗,C∗gU,I−gU,IgU∗,I∗−pσfU∗,I∗,C∗C∗qυI∗A∗AA−A∗2.
Thus,(26)dWdt=d1U∗1−UU∗1−fU∗,I∗,C∗fU,I∗,C∗+fU∗,I∗,C∗C∗−1−CC∗+fU,I,CCfU,I∗,C∗C∗+fU,I∗,C∗fU,I,C+gU∗,I∗I∗−1−II∗−fU∗,I∗,C∗gU,IIfU,I∗,C∗gU∗,I∗I∗+fU,I∗,C∗gU∗,I∗fU∗,I∗,C∗gU,I−fU∗,I∗,C∗C∗ΦfU∗,I∗,C∗fU,I∗,C∗+ΦfU,I∗,C∗fU,I,C+ΦC∗It−τ1CI∗+ΦfU,I,CCI∗fU∗,I∗,C∗C∗I−gU∗,I∗I∗ΦfU∗,I∗,C∗fU,I∗,C∗+ΦfU,I∗,C∗gU∗,I∗fU∗,I∗,C∗gU,I+ΦgU,IgU∗,I∗−pσfU∗,I∗,C∗C∗qυI∗A∗AA−A∗2.
By (H2), we find that(27)1−UU∗1−fU∗,I∗,C∗fU,I∗,C∗≤0.
By (21), we obtain(28)−1−CC∗+fU,I,CCfU,I∗,C∗C∗+fU,I∗,C∗fU,I,C=1−fU,I,CfU,I∗,C∗fU,I∗,C∗fU,I,C−CC∗≤0,−1−II∗−fU∗,I∗,C∗gU,IIfU,I∗,C∗gU∗,I∗I∗+fU,I∗,C∗gU∗,I∗fU∗,I∗,C∗gU,I=1−fU∗,I∗,C∗gU,IfU,I∗,C∗gU∗,I∗fU,I∗,C∗gU∗,I∗fU∗,I∗,C∗gU,I−II∗≤0.
SinceΦx≥0, we have dW/dt≤0 with equality if and only if U=U∗, I=I∗, C=C∗, and A=A∗. From LaSalle’s invariance principle, we deduce that the chronic infection equilibrium Q∗ is globally asymptotically stable when R0>1.


## 4. Hopf Bifurcation Analysis

In this section, we investigate the bifurcation at the infection equilibriumQ∗. By computing the characteristic equation for system (1) at Q∗, we find(29)ξ4+a1ξ3+a2ξ2+a3ξ+a4+b1ξ3+b2ξ2+b3ξ+b4e−ξτ2+c1ξ2+c2ξ+c3e−ξτ1+r1ξ+r2e−ξτ1+τ2=0,where(30)a1=d1+d3+d4+M+A∗−N,a2=d1+M−Nd3+d4+pA∗+d4d3+pA∗+d1d2−N+Md2,a3=d1d2−Nd3+d4+pA∗+d3+pA∗d4d1+M−N+Md2+Md2d4,a4=d4d3+pA∗d1d2−N+Md2,b1=−qC∗,b2=−qC∗d1+d3+M−N,b3=−qC∗d2M−pA∗+d1d2−N+d3d1+M−N,b4=−qC∗pA∗Nd1−Md2+d3+pA∗d1d2−N+Md2,c1=−υQ,c2=−υQd4+1,c3=−υQd4,r1=qC∗υ,r2=qC∗υd1,with M=C∗∂f/∂UU∗,I∗,C∗+I∗∂g/∂UU∗,I∗, Q=C∗∂f/∂CU∗,I∗,C∗+fU∗,I∗,C∗, and N=C∗∂f/∂IU∗,I∗,C∗+I∗∂g/∂IU∗,I∗+gU∗,I∗.However, whenτ1>0, equation (29) is too complicated. Therefore, in the following discussions, we assume that τ1=0 and τ2>0. Then, equation (29) is diminished to(31)ξ4+H1ξ3+H2ξ2+H3ξ+H4+I1ξ3+I2ξ2+I3ξ+I4e−ξτ2=0,where(32)H1=a1=d1+d3+d4+M+A∗−N,H2=a2+c1=d1+M−Nd3+d4+pA∗+d4d3+pA∗+d1d2−N+Md2−υQ,H3=a3+c2=d1d2−Nd3+d4+pA∗+d3+pA∗d4d1+M−N+Md2+Md2d4−υQd4+1,H4=a4+c3=d4d3+pA∗d1d2−N+Md2−υQd4,I1=b1=−qC∗,I2=b2=−qC∗d1+d3+M−N,I3=b3+r1=−qC∗d2M−pA∗+d1d2−N+d3d1+M−N−υ,I4=b4+r2=−qC∗pA∗Nd1−Md2+d3+pA∗d1d2−N+Md2−υd1.Letξ=iωω>0 be a purely imaginary root of (31). Separating real and imaginary parts, it follows that(33)ω4−H2ω2+H4=I2ω2−I4cosτ2ω+I1ω3−I3ωsinτ2ω,−H1ω3+H3ω=I1ω3−I3ωcosτ2ω−I2ω2−I4sinτ2ω.Squaring and adding the two equations of (33), it follows that(34)ω8+L1ω6+L2ω4+L3ω2+L4=0,where(35)L1=H12−2H2−I12,L2=H22−2H1H3+2H4+2I1I3−I22,L3=H32−2H2H4+2I2I4−I32,L4=H42−I42.Letz=ω2, then equation (34) becomes(36)Ψz≔z4+L1z3+L2z2+L3z+L4=0.It is clear that whenL4<0, equation (36) has at least one positive root because Ψ0=L4<0 and limz⟶+∞Ψz=+∞. Moreover, we obtain(37)Ψ′z=4z3+3L1z2+2L2z+L3.Denote that(38)γ1=8L2−3L1216,δ1=L13−4L1L2+8L232,Δ=γ123+δ123,j=ei2π/3=−12+i32.Applying the Cardano formula, the cubic equation (37) has the following roots:(39)ν1=−δ12+Δ3+−δ12+Δ3−L14,ν2=j−δ12+Δ3+j2−δ12+Δ3−L14,ν3=j2−δ12+Δ3+j−δ12+Δ3−L14.WhenΔ>0, the first root ν1 is a real number and the other two, ν2 and ν3, are conjugate complex numbers. In this situation,(40)Ψ′z=4z−ν1z2−2Reν2z+ν22.We assume thatΨz is a decreasing function on the interval −∞,ν1 and increasing function on ν1,+∞. Since z2−2Reν2z+ν22>0 for all z∈ℝ, it attains its strict global minimum at z=ν1.WhenΔ=0, all roots are real with ν1=3γ1/δ1−L1/4 and ν2=ν3=3γ1/2δ1−L1/4. Then,(41)Ψ′z=4z−ν1z−ν22.Hence,Ψz is a decreasing function on −∞,ν1 and increasing function on ν1,+∞. Also, it attains its strict global minimum at z=ν1. Consequently, if L4≥0 and Δ≥0, then equation (36) has a positive root if and only if ν1>0 and Ψν1≤0.WhenΔ<0, all three roots are real and distinct. In this case, Ψ′z can be changed as(42)Ψ′z=4z−ν1z−ν2z−ν3.Similarly, we obtain that ifL4≥0 and Δ<0, then equation (36) has a positive root if and only if there exists at least one ν∗∈ν1,ν2,ν3 such that ν∗>0 and Ψν∗≤0.Outlining the above discussions, we obtain the following lemma.Lemma 1.
For the polynomial equation (36), the following results are true:(i)
IfL4<0, then equation (36) has at least one positive root(ii)
IfL4≥0 and Δ≥0, then equation (36) has positive root if and only if ν1>0 and Ψν1≤0(iii)
IfL4≥0 and Δ<0, then equation (36) has a positive root if and only if there exists at least one ν∗∈ν1,ν2,ν3 such that ν∗>0 and Ψν∗≤0
In light of this lemma, we acknowledge the following conditions:(a)
L4<0(b)
L4≥0, Δ≥0, ν1>0, and Ψν1≤0(c)
L4≥0, Δ<0, and there exists at least one ν∗∈ν1,ν2,ν3 such that ν∗>0 and Ψν∗≤0In the case that the conditionsa−c are not fulfilled, then equation (14) has no positive roots. Consequently, the infection equilibrium Q∗ is locally asymptotically stable for all delay τ2≥0. As a result, the existence of Hopf bifurcation is preposterous.Presently, we expect that one of the conditions,a−c, is fulfilled. We suppose that equation (36) has k0 positive roots, where k0∈1,2,3,4. Denote the positive root of (36) by zk,k=1,2,…,k0. Then, equation (34) has positive roots ωk=zk. In accordance with (33), we obtain(43)cosωkτ2=ωk4−H2ωk2+H4I2ωk2−I4+−H1ωk3+H3ωkI1ωk3−I3ωkI2ωk2−I42+I1ωk3−I3ωk2=Υ1ωk,sinωkτ2=ωk4−H2ωk2+H4I1ωk3−I3ωk−−H1ωk3+H3ωkI2ωk2−I4I2ωk2−I42+I1ωk3−I3ωk2=Υ2ωk.Define(44)τ2,kn=1ωkarccosΥ1ωk+2nπ,IfY2ωk≥0,1ωk2π−arccosΥ1ωk+2nπ,IfΥ2ωk<0,where k=1,2,…,k0 and n∈ℕ. Hence,(45)τ0=τ2,k00=min1≤k≺k0τ2,k0,ω0=ωk0andz0=zk0.Letξτ=ατ+iωτ be the root of equation (31) at τ=τ0 satisfying ατ0=0 and ωτ0=ω0. Then, we obtain the following result.Lemma 2.
IfΨ′ν0≠0, then dReξτ2/dτ2|τ2=τ0 and Ψ′τ2 have the same sign.Proof.
Differentiating both sides of equation (31) with respect to τ2 and noticing that ξ is a function of τ yield(46)dξdτ2−1=4ξ3+3H1ξ2+2H2ξ+H3−ξe−ξτ2I1ξ3+I2ξ2+I3ξ+I4+3I1ξ2+2I2ξ+I3ξI1ξ3+I2ξ2+I3ξ+I4−τ2ξ=4ξ3+3H1ξ2+2H2ξ+H3−ξξ4+H1ξ3+H2ξ2+H3ξ+H4+3I1ξ2+2I2ξ+I3ξI1ξ3+I2ξ2+I3ξ+I4−τ2ξ4ξ3+3H1ξ2+2H2ξ+H3dξdτ2+e−ξτ23I1ξ2+2I2ξ+I3dξdτ2−e−ξτ2I1ξ3+I2ξ2+I3ξ+I4τ2dξdτ2+ξ=0.
The fact that(47)signdReξτ2dτ2|τ2=τ0=signRedξτ2dτ2−1|τ2=τ0,leads to(48)signdReξτ2dτ2|τ2=τ0=signRe4ξ3+3H1ξ2+2H2ξ+H3−ξξ4+H1ξ3+H2ξ2+H3ξ+H4|τ2=τ0+Re3I1ξ2+2I2ξ+I3ξI1ξ3+I2ξ2+I3ξ+I4|ξ=iω0=sign4ω06+3L1ω04+2L2ω02+L3I2ω02−I42+I1ω03−I3ω02=signΨ′ω02I2ω02−I42+I1ω03−I3ω02=signΨ′z0I2ω02−I42+I1ω03−I3ω02.
Thus,signdReξτ2/dτ2|τ2=τ0=signΨ′z0.
Summarizing the above and the Hopf bifurcation theorem [20] allows us to state the following results.Theorem 6.
IfR0>1 and (21) are satisfied, then the following results hold:(i)
If the conditionsa−c are all not satisfied, then the infection equilibrium Q∗ is locally asymptotically stable for all delay τ2≥0, which is called the delay-independent stability(ii)
If one of the conditionsa−c is satisfied, then the infection equilibrium Q∗ is locally asymptotically stable for all τ∈0,τ0(iii)
If one of the conditionsa−c is satisfied and Ψ′z0≠0, then the transversality condition holds and model (1) undergoes a Hopf bifurcation at infection equilibrium Q∗ when τ2=τ0From Lemma2, we see that, to ensure the condition of transversality, it is mandatory that the positive roots of equation (36) are simple. First, we need the following lemma given by Hattaf [21].Lemma 3.
LetPx be a polynomial of degree 4 with real coefficients:(i)
If the quartic equationPx=0 has only a single positive and simple root x1, then P′x1>0(ii)
If the quartic equationPx=0 has only two positive and simple roots x1 and x2 (setting x2<x1), then P′x1>0 and P′x2<0(iii)
If the quartic equationPx=0 has only three positive and simple roots x3<x2<x1, then P′x1>0, P′x2<0, and P′x3>0(iv)
If the quartic equationPx=0 has only four positive roots x4<x3<x2<x1, then P′x1>0, P′x2<0, P′x3>0, and P′x4<0Theorem 7.
Assume thatR0>1 and (21) holds. Define τ2,k0n by (45).(i)
If equation (36) has only a single positive and simple root z1, then Q∗ is locally asymptotically stable for τ2∈0,τ2,10 and unstable for τ2>τ2,10. Besides, a Hopf bifurcation happens when τ2>τ2,1n, n∈ℕ.(ii)
If equation (36) has only two positive and simple roots z1 and z2 with z2<z1, then there exists a finite number of intervals such that if the delay τ2 is fixed in these intervals, the equilibrium Q∗ is locally asymptotically stable, while unstable if τ2 does not belong to the ones. In this case, Q∗ changes from stability to instability.(iii)
If equation (36) has a minimum of three positive and simple roots, then there exists at least one stability change.Proof.
According to Theorem5, Q∗ is locally asymptotically stable for τ2=0. Then, equation (31) has complex roots with negative real parts for τ2=0. If equation (36) has only one positive and simple root z1, then ±iω1 is a pair of purely imaginary roots of equation (31) with τ2=τ2,1n.
By applying Lemmas2 and 3, we obtain(49)signdReξτ2dτ2|τ2=τ2,1n=signΨ′z0>0.
Then, all roots of (31) have negative real parts for τ2∈0,τ2,10 and it has at least one root with positive real part for τ2>τ2,10. Therefore, we obtain i.
For (ii), we havez2<z1. From (44), we find that there exists l∈ℕ such that τ2,1l−τ2,1l−1=2π/ω1<2π/ω2=τ2,2l−τ2,2l−2. From Lemma 2, we obtain that Ψ′z1>0 and Ψ′z2<0. Hence, dReξτ2/dτ2|τ2=τ2,1n>0 and dReξτ2/dτ2|τ2=τ2,2n<0. We deduce that l switches from stability to instability when the parameters τ2,10<τ2,20<τ2,11<⋯<τ2,1l−1<τ2,2l−1<τ2,1l; Q∗ is locally asymptotically stable when τ2∈0,τ2,10∪τ2,20,τ2,11∪,…,∪τ2,2l−1,τ2,1l and unstable when τ2∈τ2,10,τ2,20∪τ2,11,τ2,21∪,…,∪τ2,1l−1,τ2,2l−1∪τ2,1l,+∞. This demonstrates (ii) and, additionally, we can undoubtedly obtain the outcome yielding (iii).


## 5. Application

The purpose of this section is to illustrate our theoretical results to the following model, which is a special case of system (1) by letting fU,I,C=β1U/1+α1C and gU,I=β2U/1+α2I:(50)U˙=s−d1Ut−β1UtCt1+α1Ct−β2UtIt1+α2It,I˙=β1UtCt1+α1Ct+β2UtIt1+α2It−d2It,C˙=υIt−τ1−d3Ct−pAtCt,A˙=σ+qAt−τ2Ct−τ2−d4At,where α1 and α2 are positive constants that measure the saturation effect, β1 is the virus-to-cell infection rate, and β2 is the cell-to-cell transmission rate. Apparently, the hypotheses (H0)–(H3) hold and we have(51)1−fU,I,CfU,I∗,C∗fU,I∗,C∗fU,I,C−CC∗=−α1C−C∗2C∗1+α1C1+α1C∗≤0,1−fU∗,I∗,C∗gU,IfU,I∗,C∗gU∗,I∗fU,I∗,C∗gU∗,I∗fU∗,I∗,C∗gU,I−II∗=−α2I−I∗2I∗1+α2I1+α2I∗≤0.Therefore, hypothesis (21) is verified. From Theorems 4 and 5, we have the following result.Corollary 1.
(i)
IfR0≤1, then the infection-free equilibrium Qf of system (50) is globally asymptotically stable(ii)
IfR0>1, then the infection-free equilibrium Qf becomes unstable and the chronic infection equilibrium Q∗ of system (50) is globally asymptotically stable for τ2=0
Furthermore, theorem7 holds true for system (50).


## 6. Conclusion

In this paper, we have presented a delayed CHIKV infection model with general incidence functions that include various forms existing in the literature. Initially, we have examined the nonnegativity, boundedness of the solutions, and the existence of equilibria. By building appropriate Lyapunov function, utilizing Lyapunov–LaSalle invariance principle and Hopf bifurcation theory, we have demonstrated the following outcomes: (i) whenR0≤1, the infection-free equilibrium Qf is globally asymptotically stable for any time delays τ1≥0 and τ2≥0, which naturally implies that the virus is cleared and the infection vanishes; (ii) when R0>1 and (H4) holds, the chronic infection equilibrium Q∗ is globally asymptotically stable for anytime delay τ1≥0 and τ2=0, meaning that the infection perseveres in the host; (iii) when τ2>0 and τ1=0, we obtain the sufficient conditions on the existence of Hopf bifurcation at Q∗.



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