A Fractional-Order Model for Zika Virus Infection with Multiple Delays

Time delays and fractional order play a vital role in biological systems with memory. In this paper, we propose an epidemic model for Zika virus infection using delay differential equations with fractional order. Multiple time delays are incorporated in the model to consider the latency of the infection in a vector and the latency of the infection in the infected host. We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to three time delays τ 1 , τ 2 , and τ 3 . The model undergoes a Hopf bifurcation at the threshold parameters τ ∗ 1 , τ ∗ 2 , and τ ∗ 3 . Some numerical simulations are given to show the effectiveness of obtained results. The numerical simulations confirm that combination of fractional order and time delays in the epidemic model effectively enriches the dynamics and strengthens the stability condition of the model.


Introduction
Zika infection is a mosquito-borne disease, transmitted to humans through the bite of an infected Aedes mosquito. It was rst discovered in Uganda in 1947 in rhesus monkey. e rst human cases were reported in Nigeria in 1954. Zika was thought to cause mild symptoms in humans, including mild fever, skin rashes, conjunctivitis, muscle and joint pain, and headache, which lasts for three to twelve days normally. However, the World Health Organization (WHO) has concluded that Zika virus infection during pregnancy is also a cause of congenital brain abnormalities, including microcephaly [1]. Moreover, Zika virus is a trigger of Guillain-Barre syndrome [2]. ere is no doubt that mathematical modeling of Zika infection plays an important role in gaining understanding of transmission of disease and to predict the behaviour of any outbreak [3,4].
Recently, mathematical modeling of dynamics of infectious diseases, using di erential equations with memory (time-delay terms or fractional orders), has attracted much attention of many researchers (see, e.g., [5] and references therein). Time delay in models of population dynamics and in particular in macroscopic models of the immune response are natural and common [6]. Naturally, time delay or memory is an unavoidable factor in dynamics of most reallife phenomena. Time delay has in uence on dynamical behaviours of biological systems in various aspects. erefore, considering time delays in the investigation of biological systems is signi cant in both theoretical and practical point of views. In fact, when immune system works against the non-self-cells, it may take some time (time lag) to interact with the pathogen. erefore, time delays cannot be ignored in models for immune response. Accordingly, the analysis of dynamical properties of system with time delays is important (see [5,[7][8][9][10][11][12]). Dengue fever is analyzed in [13], using a system of four nonlinear di erential equations with two time delays. In [12], the authors considered the vectorborne epidemic model with time delay. e authors intensively discussed the impact of time delay in the host-tovector transmission term that can destabilize the system. Periodic solutions can also be raised through Hopf bifurcation.
In the existing literature, most of the biological problems are studied through the integer-order mathematical modeling by using ordinary, partial, and delay differential equations [9,10,14]. In the last few decades, fractional-order models have been incorporated in several areas of science, engineering, applied mathematics, economics, and bioengineering [15][16][17][18][19][20]. One advantage of the fractional-order differential equation is that they provide a powerful instrument for incorporation of memory and hereditary properties of the systems as opposed to the integer-order models, where such effects are neglected or difficult to incorporate. In addition, when fitting data, the fractional models have one more degree of freedom than the integer-order model (see [21]). Based on these advantages, some authors have developed interesting applications to investigate the dynamics of such fractionalorder models with systems of memory [22][23][24][25][26]. In [5,22], the authors studied fractional-order cancer immune systems. In [25], a fractional-order model for HIV with nonlinear incidence has been considered and stability for various equilibrium points has also been discussed. e authors in [27] investigated the dynamics of Ebola virus with time delay and fractional order and reported that combination of time delay and fractional order can effectively enrich the dynamics and strengthen the stability condition of the infection model. Analysis and dynamics of Zika transmission have been examined by many researchers (see, e.g., [3,28,29]. In [3], a mathematical model for transmission of Zika virus has been proposed with control measures of Zika virus. Stability properties of the Zika infection model have been investigated in [30]. e authors in [31] have compared the Zika infection model with dengue to show effect of the virus on population. e dynamical analysis of the SIS model is studied by considering bifurcation parameters in [32]. e authors [33] have discussed absence and presence of diffusion in the Zika virus disease model. e stability analysis and Hopf bifurcation point for various generalized epidemic models have been discussed in the literature [33][34][35]. However, the dynamics of fractional order with multiple time-delay models for Zika virus infection has not been yet studied in mathematical epidemiology.
Herein, we demonstrate that a nonlinear fractional-order differential equations model, with multiple time delays, can simulate the dynamics of Zika virus infection much more than the classical epidemic models. e application of fractional derivatives is in several cases justified because they provide a better model than integer-order derivative models do [36,37]. One important feature of fractional derivatives is that they are nonlocal opposed to the local behaviour of integer derivatives. In this way, the next state of a fractional system depends not only upon its current state but also upon all of its historical states [38][39][40].
Motivated by the above discussion, in this paper, we investigate the dynamics of Zika virus infection with fractional order and time delays. In Section 2, we formulate the model and study the nonnegativity of the solutions. In Section 3, we investigate the asymptotic stability analysis and Hopf bifurcation properties by taking time-delay parameters as bifurcation parameters. Sufficient conditions are derived to ensure the asymptotic stability and Hopf bifurcation behaviours of the addressed model. Finally, some numerical simulations are provided with various fractional orders and time delays to demonstrate the effectiveness of our theoretical findings in Section 4. We then conclude in Section 5.
Before we start analysis, we provide some useful preliminaries.

Preliminaries.
Herein, we provide some basic definitions and properties of integration and differentiation with fractional-order (free order) α (see [41]).
which is called the fractional integral (or Riemann-Liouville integral) of order α, where I 0 a � Id is the identity operator. (2) which is called the Riemann-Liouville fractional derivative of order α.
⟶ R provided that f (n− 1) be absolutely continuous), then we define the operator C D α a by which exists for almost everywhere x ∈ [a, b]. e operator C D α a f(t) is called the Caputo fractional derivative of order α. In particular, when 0 < α ≤ 1, we have Remark 1. Let β, c ∈ R + and α ∈ (0, 1). en, Remark 2. We notice that the fractional derivatives involve an integration and are nonlocal operators, which can be used for modeling systems with memory.
We should mention here that Caputo's definition of fractional derivative is a modification of the Riemann-Liouville definition and has the advantage of dealing with initial value problems in a proper way.

Model Formulation
e literature reveals that most mathematical modeling of biological systems with memory is based either on delay differential equations (DDEs) with integer-order or fractional-order differential equations without a delay. However, fractional-order calculus is more suitable, than integer-order ones, in modeling biological systems with intrinsic memory and long-range interactions such as epidemic evolution systems [42]. Modeling of such systems by fractional-order differential equations has more advantages than classical integer-order mathematical modeling, in which the effects of memory or long-range interactions are neglected. Indeed, memory effects play an essential role in the spreading of diseases. Including memory effects in the susceptible-infected-recovered (SIR) epidemic models seems very appropriate for such an investigation (see Remark 2). Herein, we investigate the impact of combining both time delays and fractional order in an epidemic model for Zika virus infection. e underlying model is governed by a system of fractional-order differential equations with multiple time delays for Zika virus infection. e model includes the dynamics of susceptible individuals, H S (t), with Zika symptoms and infected portion, H I (t), and recovered portion, H R (t), individuals recovered from Zika, the susceptible mosquitoes, M S (t), in infected mosquitoes, M I (t).
us, the total human population N H (t) � H s (t) + H I (t) + H R (t). e overall vector (mosquito) population, at time t, is N m (t) � M s (t) + M I (t). Assume that β h is the transmission rate from humans to mosquitoes. β m is the transmission rate of Zika from the vector (mosquitoes) to humans. Natural death rate of host is denoted by d h . e recruitment rate into susceptible population is denoted by λ h . Natural death rate of vector is denoted by d m . η is the recovery rate from treatment. λ m is the recruitment rate into susceptible mosquito population. Also, c is the average infectious period for humans. We use time delays in the model to consider the latency of the infection in a vector and the latency of the infection in an infected host. In our model, we consider time-delay τ 1 to represent the transferring of the infection from infected mosquitoes into suspected humans. e incubation period (time delay) τ 2 is incorporated to represent the time required for an individual/susceptible to become infectious, after becoming infected. τ 3 is the incubation period of susceptible mosquitoes to become infectious (see Figure 1). e memory of the earlier times, which are represented by time lags, could have less effect on the present situation, as compared to more recent times. However, it is expected that long-range memory, represented by fractional order, effects decay in time more slowly than an exponential decay but can typically behave like a power-law damping function. e model then takes the following form: e initial conditions for system (5) should be provided

Remark 3.
e fractional derivative α ∈ (0, 1] is defined by Caputo sense (4), so that introducing a convolution integral with a power-law memory kernel is useful to describe memory effects in dynamical systems. e decaying rate of the memory kernel (a time correlation function) depends on α. A lower value of α corresponds to more slowly decaying time-correlation functions (long memory). erefore, as α ⟶ 1, the influence of memory decreases.

Nonnegative Solution.
Since model (5) monitors the dynamics of human populations, therefore, all the parameters are assumed to be nonnegative. Furthermore, it can be shown that all state variables of the model are nonnegative and bounded for all time t ≥ 0 (see [42]).

Lemma 1.
e closed set Ω � (H s , H I , H R , M S , M I ) ∈ R 5 + : Proof. In order to prove the nonnegativity of system (5), it is assumed that there exists a t * > t 0 such that H S (t * ) � 0 and Since H S (t * ) � 0, one gets H S (t) > 0, t ∈ [t * , t 1 ], which contradicts the assumption. Hence, H S (t) > 0 for any t > t 0 . In the same manner, we have H I (t), H R (t), M S (t), and M I (t) are nonnegative.
To show that the system is bounded, we add the first three equation of System (5), and we get We know that all parameters value is positive, and one can obtain where N H � H S (t) + H I (t) + H R (t), and solving this equation, we have e solution is given by , where E α,β is the Mittag-Leffler function. Considering the fact that Mittag-Leffler function has an asymptotic behaviour, e model has two equilibrium points: Here, H * I is the positive root of the following equation:

Separation of real and imaginary parts yields
From (14), we have It is clear that cos 2 θ + sin 2 θ � 1; from (56), Hence, it follows from cos ξτ 3 � f 1 (ξ) that We suppose that (57) have at least one positive root. e bifurcation point is defined as We obtain the transversality condition of the occurrence for Hopf bifurcation at τ 3 � τ * 3 . Now, differentiating equation (51) with respect to τ 3 , we obtain where P 7 ′ (s) and P 8 ′ (s) are derivatives of P 7 (s) and P 8 (s), respectively. It follows that From (61), by some computation, we deduce that where M 1 , N 1 and M 2 , N 2 are the real and imaginary parts of M(s), N(s). Also ξ 0 stands for the critical value and τ * 3 denotes bifurcation point. Here, (63)

Conclusion
Fractional derivatives have the unique property of capturing the history of the variable; that is, they have short and long memory. is cannot be easily done by means of the integerorder derivatives. In this paper, we proposed a fractionalorder model for Zika virus infection with multiple time delays τ 1 , τ 2 , and τ 3 . We studied the asymptotic stability and Hopf bifurcation properties for the model. Time delays and fractional order play a vital role in the stability and complexity of the model. By evaluating the characteristics, some sufficient conditions have derived to ensure the asymptotic stability in terms of the fractional order and time delays. Moreover, we estimated the thresholds bifurcation parameters: τ * 1 , τ * 2 , and τ * 3 . e transversality conditions have been obtained to confirm the existence of Hopf bifurcations for different values at the threshold parameters and particular values of fractional orders. Our findings illustrate that using the time delays as bifurcation points, one can conclude that when time delay increases, the equilibrium loses its stability and Hopf bifurcation occurs. ese models can be used to understand key aspects of the viral life cycle and to predict antiviral efficacy. Finally, numerical simulations show that a combination of fractional order and time delays in the model effectively enriches the dynamics and strengthens the stability condition of the model.
Including control variables in the model is desirable to determine the best strategy of treatment and control and eliminate the infection, which will be considered in future work.

Data Availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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