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

Rakkiyappan, R.; Latha, V. Preethi; Rihan, Fathalla A.

doi:https://doi.org/10.1155/2019/4178073

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Abstract Introduction Model Formulation Conclusion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Recent Advances in Modeling, Analysis, and Synchronization of Chaotic Systems

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Research Article | Open Access

Volume 2019 | Article ID 4178073 | https://doi.org/10.1155/2019/4178073

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

R. Rakkiyappan,¹V. Preethi Latha,¹and Fathalla A. Rihan²

Guest Editor: Serdar Çiçek

Received19 Aug 2019

Revised21 Oct 2019

Accepted31 Oct 2019

Published14 Nov 2019

Abstract

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 , , and . The model undergoes a Hopf bifurcation at the threshold parameters , , and . 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.

1. Introduction

Zika infection is a mosquito-borne disease, transmitted to humans through the bite of an infected Aedes mosquito. It was first discovered in Uganda in 1947 in rhesus monkey. The first 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]. There 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 differential 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 real-life phenomena. Time delay has influence on dynamical behaviours of biological systems in various aspects. Therefore, considering time delays in the investigation of biological systems is significant 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. Therefore, 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, 7–12]). Dengue fever is analyzed in [13], using a system of four nonlinear differential equations with two time delays. In [12], the authors considered the vector-borne epidemic model with time delay. The authors intensively discussed the impact of time delay in the host-to-vector 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 [15–20]. 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 fractional-order models with systems of memory [22–26]. 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. The 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]. The authors in [31] have compared the Zika infection model with dengue to show effect of the virus on population. The dynamical analysis of the SIS model is studied by considering bifurcation parameters in [32]. The authors [33] have discussed absence and presence of diffusion in the Zika virus disease model. The stability analysis and Hopf bifurcation point for various generalized epidemic models have been discussed in the literature [33–35]. 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. The 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 [38–40].

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.

1.1. Preliminaries

Herein, we provide some basic definitions and properties of integration and differentiation with fractional-order (free order) α (see [41]).

Definition 1. Let , the operator on is defined bywhich is called the fractional integral (or Riemann–Liouville integral) of order , where is the identity operator.

Definition 2. Let and , where , and the operator is defined for bywhich is called the Riemann–Liouville fractional derivative of order α.

Definition 3. Let and f is such that exists, where (the set of all function provided that be absolutely continuous), then we define the operator bywhich exists for almost everywhere . The operator is called the Caputo fractional derivative of order α. In particular, when , we have

Remark 1. Let and . Then,(i)If and if , then (ii) uniformly on , , where (iii) weakly(iv)If is absolutely continuous on , then (v)Thus, (Riemann–Liouville sense) and (Caputo sense)

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.

2. Model Formulation

The 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.

The underlying model is governed by a system of fractional-order differential equations with multiple time delays for Zika virus infection. The model includes the dynamics of susceptible individuals, , with Zika symptoms and infected portion, , and recovered portion, , individuals recovered from Zika, the susceptible mosquitoes, , in infected mosquitoes, . Thus, the total human population . The overall vector (mosquito) population, at time t, is Assume that is the transmission rate from humans to mosquitoes. is the transmission rate of Zika from the vector (mosquitoes) to humans. Natural death rate of host is denoted by . The recruitment rate into susceptible population is denoted by . Natural death rate of vector is denoted by . η is the recovery rate from treatment. is the recruitment rate into susceptible mosquito population. Also, γ 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 to represent the transferring of the infection from infected mosquitoes into suspected humans. The incubation period (time delay) is incorporated to represent the time required for an individual/susceptible to become infectious, after becoming infected. is the incubation period of susceptible mosquitoes to become infectious (see Figure 1). The 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. The model then takes the following form:

The initial conditions for system (5) should be provided so that , , , , and , when for , time lag, .

Remark 3. The fractional derivative 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. The 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). Therefore, as , the influence of memory decreases.

2.1. 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 (see [42]).

Lemma 1. The closed set is positively invariant with respect to model (5).

Proof. In order to prove the nonnegativity of system (5), it is assumed that there exists a such that and for where is sufficiently close to . If ,Thus, one obtains for all and where . Hence, one derivesSince , one gets , which contradicts the assumption. Hence, for any . In the same manner, we have are nonnegative.
To show that the system is bounded, we add the first three equation of System (5), and we getWe know that all parameters value is positive, and one can obtainwhere , and solving this equation, we haveThe solution is given by , where is the Mittag-Leffler function. Considering the fact that Mittag-Leffler function has an asymptotic behaviour,One can observe that as . The proof of the mosquitoes (vector) population is similar to human (host) population, and we obtain . Therefore, all solutions of the model with initial conditions in remain bounded in the positively invariant region for all . The region is positively invariant with respect to model (5).

The equilibrium points (steady states) are obtained by setting , in model (5). The model has two equilibrium points: (i) disease-free equilibrium point and (ii) endemic steady state , which is

Here, is the positive root of the following equation:

3. Stability and Bifurcation Analysis

To study the stability of model (5), suppose is the steady state of the linearized system:

Taking Laplace transform [43] on both sides of the linearized system (14), we obtainwhere , , , , and are Laplace transforms of , , , and , respectively, with , , , , and . Then, (15) can be written in the following matrix form asin whichwhere , , , , , , , , and and is considered as the characteristic matrix of system (5) The characteristic polynomial is then

The coefficients , , are estimated by Mathematica and given in Appendix.

Case 1.
When the characteristic equation (18) becomeswhereNow, we prove that the characteristic equation (19) has no pure imaginary roots for any . Assume that characteristic equation (19) has pure imaginary root, and let it be . If we multiply on both sides of equation (19), we getNow, we substitute the expression of s into (21) to haveThe coefficients and are real and imaginary parts of , and , respectively, so thatSeparating real and imaginary parts yieldsIt follows from (14) thatUsing the fact that , we have , and then (25) can be written in the following form:It can be concluded from (26) thatwhereThe quadratic equation (27) has roots, so we can obtain the expression of and denote , where is a function of .
Substituting the expression of into (27), we can get expression of . Assume that . Moreover, we have . Thus, it follows from thatWe suppose that has at least one positive root, and thus, the bifurcation point is defined asNow, differentiating equation (21) with respect to , we obtainwhere , and are derivatives of , and , respectively. It follows thatFrom (32), by some computation, we deduce thatwhere and are the real and imaginary parts of . Also, stands for the critical value and denotes the bifurcation point. Here,

Case 2.
When the characteristic equation (18) becomeswhereNow, we prove that the characteristic equation (35) has no pure imaginary roots for any . Assume that characteristic equation (35) has pure imaginary root, and let it be . Now, multiplying on both sides of equation (35), we getSubstitute the expression of s into (37) to havewhere and are real and imaginary parts of , and . Here,Separating real and imaginary parts yieldsIt follows from (40) thatWe know that ; by using it, we have , and then (41) can be written in the following form:It can be concluded from (42) thatwhereAs we know, the quadratic equation (43) has roots, we can obtain the expression of and denote , where is a function of .
Substituting the expression of into (43), we can get expression of . Let us denote , where is a function with respect to . Moreover, . Thus, it follows from thatClearly, has at least one positive root. The bifurcation point is defined asWe obtain the transversality condition of the occurrence for Hopf bifurcation at .
Differentiating equation (37) with respect to yieldswhere , and are derivatives of , and , respectively. It follows thatFrom (48), by some computation, we deduce thatwhere and are the real and imaginary parts of . Also stands for the critical value and denotes bifurcation point. Here,

Case 3.
When the characteristic equation (18) becomeswhereAgain, we prove that the characteristic equation (51) has no pure imaginary roots for any . Here, we assume that characteristic equation (51) has pure imaginary root, let it be Now, we substitute the expression of s into (51), and we havewhere and are real and imaginary parts of and , respectively. Here,Separation of real and imaginary parts yieldsFrom (14), we haveIt is clear that from (56),Hence, it follows from thatWe suppose that (57) have at least one positive root. The bifurcation point is defined asWe obtain the transversality condition of the occurrence for Hopf bifurcation at .
Now, differentiating equation (51) with respect to , we obtainwhere and are derivatives of and , respectively. It follows thatFrom (61), by some computation, we deduce thatwhere and are the real and imaginary parts of . Also stands for the critical value and denotes bifurcation point. Here,

Case 4.
When the characteristic equation (18) becomesHere,From the Routh–Hurwitz criteria, if we choose , and , then the five eigenvalues of the characteristic equation (64) have negative real parts. Hence, the steady state fractional-order system (5) is asymptotically stable when (without time delays).
We arrive at the following theorem.

Theorem 1. If and an endemic equilibrium point exists for system (5), then the following results hold:(i)When , the endemic steady state is asymptotically stable for and the system undergoes a Hopf bifurcation at the origin at and the transversality condition holds, (ii)When , the endemic steady state is asymptotically stable for and system undergoes a Hopf bifurcation at the origin when and the transversality condition holds, (iii)When , of is asymptotically stable for and system (5) undergoes a Hopf bifurcation at the origin at and transversality condition holds, (iv)When and holds, the endemic steady state is asymptotically stable for

Remark 4. Theorem 1 reports the asymptotic stability of the endemic equilibrium point . The analysis can be extended to investigate the stability of infection-free equilibrium points for the fractional-order model.

4. Numerical Simulations and Observations

In this section, we provide some numerical simulations for system (5) to demonstrate the effectiveness of our main results. The simulations have been done by using stable implicit Euler approximation scheme, discussed in [44]. Of course, many other methods have been used for fractional-order delay differential equations such as the Adams–Bashforth–Moulton scheme [45]. The parameter values of system (5) are taken as follows:

Case 1. . In this case, time-delay is chosen as the bifurcation parameter. We then discuss the dynamic effect of system (5) with the above parameter values. We calculate the critical frequency and bifurcation point of various fractional-order α. Figure 2 shows the numerical simulations of model (5) when , and , with different fractional orders and estimated bifurcation point (see Table 1). Here, for the fractional-order and whereas which satisfies the condition (i) in Theorem 1. The equilibrium of the model (5) is asymptotically stable for . When , which does not satisfies the condition (i) of Theorem 1, the system undergoes a Hopf bifurcation for the functional-order .

Case 2. . We choose time-delay as a bifurcation parameter of system (5) with parameter values:We then calculate the critical frequency and bifurcation point of various fractional-order α (see Table 2). Figure 3 shows the dynamics of system (5) for , and , with values of , and 0.8. The corresponding bifurcation point is . satisfies the condition (ii) of Theorem 1. Therefore, the equilibrium of the model (5) is asymptotically stable for , which is shown in Figure 3. However, for , a Hopf bifurcation occurs for the functional-order .

Case 3. . We consider time-delay as a bifurcation parameter of system (5) with parameter values:We calculate the critical frequency and bifurcation point of various fractional-order α. When , and , the dynamics of system (5) is shown in Figure 4 with different fractional-order , and 0.8, its corresponding bifurcation points (see Table 1). Here, and a Hopf bifurcation occurs for the factional-order which not satisfies the condition (iii) in Theorem 1. Therefore, the equilibrium point is of model (5) is unstable, which is shown in Figure 4.

Case 4. , without time delays. Assume the parameter values:Hence, system (5) is asymptotically stable, which is shown in Figure 5.

5. Conclusion

Fractional derivatives have the unique property of capturing the history of the variable; that is, they have short and long memory. This cannot be easily done by means of the integer-order derivatives. In this paper, we proposed a fractional-order model for Zika virus infection with multiple time delays , , and . 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: , , and . The 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. These 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.

Appendix

The coefficients of equation (18) are as follows:

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This manuscript was funded from the project no. 31S265 (UAE University) and DST-SERB Project # SB/FTP/MS-045/2013 (Bharathiar University).

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Copyright © 2019 R. Rakkiyappan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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