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.
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.
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 delay describes the time necessary for the newly produced virions to become mature and infectious and the second delay represents the time needed to activate the humoral immune response. Then, the model is presented as follows:where the general incidence functions and assumed to be continuously differentiable satisfy the following hypotheses [14, 15]:(i)(): (0, I) = 0, for all ; (or is a monotone increasing function with respect to when ), and , for all and (ii)(): , for all and (iii)(): is a monotone increasing function with respect to (or when is a strictly monotone increasing function with respect to ), for any fixed and (iv)(): is a monotone decreasing function with respect to and
In biological terms, , , , and indicate the densities of susceptible cells, infected cells, CHIKV particles, and antibodies at time , respectively. The parameter is the recruitment rate of uninfected cells, and is the production rate of free CHIKV particles by infected cells. The CHIKV particles are attacked by the antibodies at rate . The antibodies are created at rate and multiplicated at rate . The parameters , , , and 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 or by direct contact with an infected cell at rate . In addition, particular cases of the incidence function and 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 and the model proposed in [8] when and .
The 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. 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:where and . is the Banach space of continuous functions mapping the interval into 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 .
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 .
Proof. We first demonstrate that for all . By contradiction, we assume that there exists a first time such that and . From the first equation of system (1), we have , which leads a contradiction. Then, for all . Since , and similar to the above, we deduce that for all . According to (1), we havewhich implies that and are nonnegative for all .
We consider the following function:Then,where . Hence,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 . Thus, we characterize the basic reproduction number of our model as follows:
To locate different equilibria of (1), we solve the accompanying system:
From (8)–(11), we obtain , , , and
leads to . Hence, there is no biological equilibrium when . Accordingly, we consider the function defined on by
We have andwith . Then, the equation admits a unique solution . Thus, . Since if , we deduce that there exists such that .
Substitute and in (8), and define a function as . Due to the fact that , , and is a strictly decreasing function of , we deduce that there exists a unique such that . Therefore, model (1) has a unique chronic infection equilibrium when .
The precedent conversations can be summed up in the accompanying outcome.
Theorem 2. (i)For , model (1) has one infection-free equilibrium (ii)For , model (1) has a unique chronic infection equilibrium with , , , and
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
First, we have the following result.
Theorem 3. For any and , the infection-free equilibrium is locally asymptotically stable if and becomes unstable if .
Proof. Examining (15) at , we obtainWhen , from equation (16), we obtainTherefore, the roots of this equation areof which . Obviously, , , and are negative. Furthermore, is negative if and positive if . Consequently, is locally asymptotically stable if and unstable if .
The following theorem characterizes the global stability of the infection-free equilibrium when .
Theorem 4. For any and , the infection-free equilibrium is globally asymptotically stable if .
Proof. We establish a Lyapunov function as follows:Computing the time derivative of along the solutions of (1), we findGiven that , we have . Likewise, it is not difficult to show that the largest invariant set in is . By the LaSalle’s invariance principle [19], is globally asymptotically stable for .
Next, we focus on the global stability of the chronic infection equilibrium by assuming that , and for all , , , we consider the following hypothesis:
Theorem 5. Assume that (21) holds. For any , if and , then the chronic infection equilibrium is globally asymptotically stable.
Proof. Consider the following Lyapunov function:where , . Thus, the time derivative of along the positive solutions of (1) satisfieswhere . Therefore, we haveSubstituting , , and , we obtainThus,By (), we find thatBy (21), we obtainSince , we have with equality if and only if , , , and . From LaSalle’s invariance principle, we deduce that the chronic infection equilibrium is globally asymptotically stable when .
4. Hopf Bifurcation Analysis
In this section, we investigate the bifurcation at the infection equilibrium . By computing the characteristic equation for system (1) at , we findwherewith , , and .
However, when , equation (29) is too complicated. Therefore, in the following discussions, we assume that and . Then, equation (29) is diminished towhere
Let be a purely imaginary root of (31). Separating real and imaginary parts, it follows that
Squaring and adding the two equations of (33), it follows thatwhere
Let , then equation (34) becomes
It is clear that when , equation (36) has at least one positive root because and . Moreover, we obtain
Denote that
Applying the Cardano formula, the cubic equation (37) has the following roots:
When , the first root is a real number and the other two, and , are conjugate complex numbers. In this situation,
We assume that is a decreasing function on the interval and increasing function on . Since for all , it attains its strict global minimum at .
When , all roots are real with and . Then,
Hence, is a decreasing function on and increasing function on . Also, it attains its strict global minimum at . Consequently, if and , then equation (36) has a positive root if and only if and .
When , all three roots are real and distinct. In this case, can be changed as
Similarly, we obtain that if and , then equation (36) has a positive root if and only if there exists at least one such that and .
Outlining the above discussions, we obtain the following lemma.
Lemma 1. For the polynomial equation (36), the following results are true:(i)If , then equation (36) has at least one positive root(ii)If and , then equation (36) has positive root if and only if and (iii)If and , then equation (36) has a positive root if and only if there exists at least one such that and In light of this lemma, we acknowledge the following conditions:(a)(b), , , and (c), , and there exists at least one such that and
In the case that the conditions are not fulfilled, then equation (14) has no positive roots. Consequently, the infection equilibrium is locally asymptotically stable for all delay . As a result, the existence of Hopf bifurcation is preposterous.
Presently, we expect that one of the conditions, , is fulfilled. We suppose that equation (36) has k0 positive roots, where . Denote the positive root of (36) by . Then, equation (34) has positive roots . In accordance with (33), we obtain
Definewhere and . Hence,
Let be the root of equation (31) at satisfying and . Then, we obtain the following result.
Lemma 2. If , then and have the same sign.
Proof. Differentiating both sides of equation (31) with respect to and noticing that is a function of yieldThe fact thatleads toThus, .
Summarizing the above and the Hopf bifurcation theorem [20] allows us to state the following results.
Theorem 6. If and (21) are satisfied, then the following results hold:(i)If the conditions are all not satisfied, then the infection equilibrium is locally asymptotically stable for all delay , which is called the delay-independent stability(ii)If one of the conditions is satisfied, then the infection equilibrium is locally asymptotically stable for all (iii)If one of the conditions is satisfied and , then the transversality condition holds and model (1) undergoes a Hopf bifurcation at infection equilibrium when
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].
Lemma 3. Let be a polynomial of degree 4 with real coefficients:(i)If the quartic equation has only a single positive and simple root , then (ii)If the quartic equation has only two positive and simple roots and (setting ), then and (iii)If the quartic equation has only three positive and simple roots , then , , and (iv)If the quartic equation has only four positive roots , then , , , and
Theorem 7. Assume that and (21) holds. Define by (45).(i)If equation (36) has only a single positive and simple root , then is locally asymptotically stable for and unstable for . Besides, a Hopf bifurcation happens when , .(ii)If equation (36) has only two positive and simple roots and with , then there exists a finite number of intervals such that if the delay is fixed in these intervals, the equilibrium is locally asymptotically stable, while unstable if does not belong to the ones. In this case, 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 Theorem 5, is locally asymptotically stable for . Then, equation (31) has complex roots with negative real parts for . If equation (36) has only one positive and simple root , then is a pair of purely imaginary roots of equation (31) with .
By applying Lemmas 2 and 3, we obtainThen, all roots of (31) have negative real parts for and it has at least one root with positive real part for . Therefore, we obtain .
For (ii), we have . From (44), we find that there exists such that . From Lemma 2, we obtain that and . Hence, and . We deduce that switches from stability to instability when the parameters ; is locally asymptotically stable when and unstable when . 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 and :where and are positive constants that measure the saturation effect, is the virus-to-cell infection rate, and is the cell-to-cell transmission rate. Apparently, the hypotheses ()–() hold and we have
Therefore, hypothesis (21) is verified. From Theorems 4 and 5, we have the following result.
Corollary 1. (i)If , then the infection-free equilibrium of system (50) is globally asymptotically stable(ii)If , then the infection-free equilibrium becomes unstable and the chronic infection equilibrium of system (50) is globally asymptotically stable for Furthermore, theorem 7 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) when , the infection-free equilibrium is globally asymptotically stable for any time delays and , which naturally implies that the virus is cleared and the infection vanishes; (ii) when and (H4) holds, the chronic infection equilibrium is globally asymptotically stable for anytime delay and , meaning that the infection perseveres in the host; (iii) when and , we obtain the sufficient conditions on the existence of Hopf bifurcation at .
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.