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

A generalized chikungunya virus (CHIKV) infection model with nonlinear incidence functions and two time delays is proposed and investigated. )e 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.


Introduction
e 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 singlestranded, thermosensitive RNA, about 15,000 nucleotides long. e 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. e virus multiplies with great ease in vitro, but also in vivo in mosquito cells, which explains the high infective power of contaminated Aedes. e 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). e 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. e hips and shoulders are more rarely affected. ese 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.
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. is 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]. e 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]. is emergence has made it possible to highlight the need to strengthen the knowledge of health professionals with regard to arboviroses. erefore, 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][11][12][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 delay τ 1 describes the time necessary for the newly produced virions to become mature and infectious and the second delay τ 2 represents the time needed to activate the humoral immune response. en, the model is presented as follows: where the general incidence functions f(U, I, C) and g(U, I) assumed to be continuously differentiable satisfy the following hypotheses [14,15]: is a monotone increasing function with respect to T when f ≡ 0), and (zg/zI)(U, I) ≤ 0, for all U ≥ 0 and I ≥ 0 (ii) (H 1 ): f(0, I, C) � 0, for all I ≥ 0 and C ≥ 0 (iii) (H 2 ): f(U, I, C) is a monotone increasing function with respect to U (or (zf/zU)(U, I, C) ≥ 0 when g(U, I) is a strictly monotone increasing function with respect to U), for any fixed I ≥ 0 and C ≥ 0 (iv) (H 3 ): f(U, I, C) is a monotone decreasing function with respect to I and C In biological terms, U(t), I(t), C(t), and A(t) indicate the densities of susceptible cells, infected cells, CHIKV particles, and antibodies at time t, respectively. e parameter s is the recruitment rate of uninfected cells, and υ is the production rate of free CHIKV particles by infected cells.
e CHIKV particles are attacked by the antibodies at rate pAC. e antibodies are created at rate σ and multiplicated at rate qAC. e parameters d 1 , d 2 , d 3 , and d 4 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 f(U, I, C)C or by direct contact with an infected cell at rate g(U, I)I. 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 τ 1 � τ 2 � 0 and the model proposed in [8] when f(U, I, C) � β 1 U and g(U, I) � 0.  e rest of this paper is organized as follows. In Section 2, 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. is paper ends with a conclusion in Section 6.

Preliminary Results
In this section, we first prove the existence, positivity, and boundedness of solutions. After that, we discuss the existence of equilibria.
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 that U(t) > 0 for all t ≥ 0. By contradiction, we assume that there exists a first time t 1 > 0 such that U(t 1 ) � 0 and _ U(t 1 ) ≤ 0. From the first equation of system (1), we have _ U(t 1 ) � s > 0, which leads a contradiction. en, U(t) > 0 for all t ≥ 0. Since _ A(t) � η > 0, and similar to the above, we deduce that A(t) > 0 for all t ≥ 0. According to (1), we have which implies that I(t) and C(t) are nonnegative for all t ≥ 0. We consider the following function: en, which implies that all solutions of system (1) are bounded. is completes the proof.

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 Q f ((s/d 1 ), 0, 0, (σ/d 4 )). us, we characterize the basic reproduction number of our model as follows: To locate different equilibria of (1), we solve the accompanying system: Discrete Dynamics in Nature and Society

(12)
A � σ/d 4 − qC ≥ 0 leads to C < d 4 /q. Hence, there is no biological equilibrium when C ≥ d 4 /q. Accordingly, we consider the function ψ defined on [0, (d 4 /q)) by with From (10) and (11), we find Substitute C � C * and I � I * in (8), and define a function φ 3 as φ 3 e precedent conversations can be summed up in the accompanying outcome.

Stability Analysis of Equilibria
In this section, we concentrate on the stability of infectionfree equilibrium of system (1). e characteristic equation of system (1) is noted as First, we have the following result.

Theorem 3. For any τ 2 and τ
Proof. Examining (15) at Q f , we obtain When τ 1 � 0, from equation (16), we obtain erefore, the roots of this equation are e following theorem characterizes the global stability of the infection-free equilibrium Q f when R 0 ≤ 1.

Theorem 4. For any τ 1 and τ 2 , the infection-free equilibrium
Proof. We establish a Lyapunov function as follows: Computing the time derivative of L along the solutions of (1), we find Given that R 0 ≤ 1, we have (dL/dt) ≤ 0. Likewise, it is not difficult to show that the largest invariant set in } is Q f . By the LaSalle's invariance principle [19], Q f is globally asymptotically stable for R 0 ≤ 1.
Next, we focus on the global stability of the chronic infection equilibrium Q * by assuming that R 0 > 1, and for all U, I, C > 0, we consider the following hypothesis: Theorem 5. Assume that (21) holds. For any τ 1 , if τ 2 � 0 and R 0 > 1, then the chronic infection equilibrium Q * is globally asymptotically stable.
Proof. Consider the following Lyapunov function: where Φ(x) � x − 1 − ln x, x > 0. us, the time derivative of W along the positive solutions of (1) satisfies Discrete Dynamics in Nature and Society 5 where (25) us, 6 Discrete Dynamics in Nature and Society By (H 2 ), we find that By (21), we obtain Since Φ(x) ≥ 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 R 0 > 1.
Outlining the above discussions, we obtain the following lemma. In the case that the conditions (a) − (c) are not fulfilled, then equation (14) has no positive roots. Consequently, the infection equilibrium Q * is locally asymptotically stable for all delay τ 2 ≥ 0. As a result, the existence of Hopf bifurcation is preposterous.
Proof. Differentiating both sides of equation (31) with respect to τ 2 and noticing that ξ is a function of τ yield Discrete Dynamics in Nature and Society e fact that us, sign dReξ(τ 2 )/dτ 2 | τ 2 �τ 0 � sign Ψ ′ (z 0 ) . Summarizing the above and the Hopf bifurcation theorem [20] allows us to state the following results. From Lemma 2, 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].

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) when R 0 ≤ 1, the infection-free equilibrium Q f is globally asymptotically stable for any time delays τ 1 ≥ 0 and τ 2 ≥ 0, which naturally implies that the virus is cleared and the infection vanishes; (ii) when R 0 > 1 and (H 4 ) holds, the chronic infection equilibrium Q * is globally asymptotically stable for anytime delay τ 1 ≥ 0 and τ 2 � 0, meaning that the infection perseveres in the host; (iii) when τ 2 > 0 and τ 1 � 0, we obtain the sufficient conditions on the existence of Hopf bifurcation at Q * .

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.