# Nonlinear Dynamics and Chaos in a Fractional-Order HIV Model

Authors: Haiping Ye; Yongsheng Ding
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

We introduce fractional order into an HIV model. We consider
the effect of viral diversity on the human immune system with frequency dependent rate of proliferation of cytotoxic T-lymphocytes (CTLs) and rate of
elimination of infected cells by CTLs, based on a fractional-order differential
equation model. For the one-virus model, our analysis shows that the interior
equilibrium which is unstable in the classical integer-order model can become
asymptotically stable in our fractional-order model and numerical simulations
confirm this. We also present simulation results of the chaotic behaviors produced from the fractional-order HIV model with viral diversity by using an
Adams-type predictor-corrector method.


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## Body

## 1. Introduction

An important part of the human immune response against viral infections is cytotoxic T lymphocytes (CTLs) [1]. They recognize and kill cells which are infected by virus. There are many immune models describing the virus dynamics with CTL immune response. Nowak and Bangham [2, 3] proposed an ODE model which explores the relation among CTL immune responses, virus load, and virus diversity. In [2], a rate of specific CTL (Zj) proliferation in response to the corresponding specific infected cells (Ij) depends on the mass action law cIjZj. This model has been important in the field of mathematical modelling of HIV infection. In their model, there is no interaction among different types of CTL (Zj). Iwami et al. [4] assumed that the correlation is incorporated as a function of the frequency that the specific CTLs (Zj) encounter in the specific infected cells (Ij). In a similar manner, they considered the rate of elimination of specific infected cells (Ij) by the specific CTLs (Zj) to be proportional to this frequency. However, these models do not take into account the fractional order derivatives that have been extensively applied in many fields (e.g., [5–17] and the reference cited therein). Recently many mathematicians and applied researchers have tried to model real processes using the fractional order differential equations (FODE) [16]. In biology, it has been deduced that the membranes of cells of biological organism have fractional order electrical conductance [13] and then, they are classified into group of noninteger order models. Also, it has been shown that modelling the behavior of brainstem vestibule-oculumotor neurons by FODE has more advantages than classical integer order modelling [8].Particular emphasis is that a major difference between fractional order models and integer order models is that fractional order models possess memory [5, 12], while the main features of immune response involve memory [18]. Hence, we attempt to model HIV infection with immune response using a fractional order system. Our presentation is based on the immune model of HIV infection which is developed by Iwami et al. [4]. For the one-virus model, we carry out a detailed analysis on stability of equilibrium. Our analysis shows that the interior equilibrium which is unstable in the classical integer order model can become asymptotically stable in our fractional order model. We also find that chaos does exist in the fractional order HIV model with viral diversity.


## 2. Model Derivation

We first give the definition of fractional order integration and fractional order differentiation [14, 16]. For the concept of fractional derivative we will adopt Caputo's definition which is a modification of the Riemann-Liouville definition and has the advantage of dealing properly with initial value problems.Definition 2.1.
The fractional integral of orderα>0 of a function f:R+→R is given by
(2.1)Iαf(x)=1Γ(α)∫0x(x-t)α-1f(t)dt
provided the right side is pointwise defined on R+.Definition 2.2.
The Caputo fractional derivative of orderα∈(n-1,n) of a continuous function f:R+→R is given by
(2.2)Dαf(x)=In-αDnf(x),D=ddt.Now we introduce fractional order into the ODE model by Iwami et al. [4]. The new system is described by the following set of FODE:
(2.3)Dq1T=λ-dT-∑l=1nβl′TVl,Dq2jIj=βj′TVj-aIj-bZjIjT+∑l=1nIl,Dq3jZj=cZjIjT+∑l=1nIl-δZj,Dq4jVj=kaIj-uVj(j=1,2,…,n),
where T(t) represents the concentration of uninfected cells at time t, Ij(t) represents the concentration of infected cells with a virus particle of type j, Vj(t) the concentration of free virus particle of type j, and Zj(t) denotes the magnitude of the specific CTL response against variant j. Here, 0.95≤q1,q2j,q3j,q4j≤1(j=1,2,…,n) are restricted such that fractional derivative can be approximately described the rate of change in number.Following [4], uninfected cells are assumed to be generated at a constant rate λ. Uninfected cells, infected cells, free viruses, and CTLs decline at rates d,a,u, and δ, respectively. The total number of virus particles produced from one cell is k. The rate of CTL proliferation in response to antigen is given by cZjIj/(T+∑l=1nIl) and the specific infected cells are killed by specific CTLs at rate bZjIj/(T+∑l=1nIl), while infected cells are produced from uninfected cells and free virus at rate βl′TVl. That is, a rate of specific CTL(Zj) proliferation in response to the corresponding specific infected cells (Ij) depends on the frequency, instead of the mass action law.To simplify the model, it is reasonable to assume that the decay rate of free virus,u, is much larger than that of the infected cells, a, and this system describes the qualitative dynamics of the asymptomatic phase of HIV infection. Thus, we may introduce as a good approximation that the virus is in steady state (i.e., Dq4jVj=0) and hence Vj=kaIj/u (see [4, 19]). This leads to the following simplified system of FODE:
(2.4)Dq1T=λ-dT-∑l=1nβlTIl,Dq2jIj=βjTIj-aIj-bZjIjT+∑l=1nIl,Dq3jZj=cZjIjT+∑l=1nIl-δZj(j=1,2,…,n),
where βj=kaβj′/u.


## 3. One-Virus Model

In this section, we discuss in detail an important special case of model (2.4) and perform an equilibrium and stability analysis for this special case. We consider the one-virus model (n=1) and assume that q1=q21=q31=α(0.95≤α≤1). This one-virus model is described by the following system of FODE:
(3.1)DαT=λ-dT-β1TI1,DαI1=β1TI1-aI1-bZ1I1T+I1,DαZ1=cZ1I1T+I1-δZ1.To evaluate the equilibria, let(3.2)DαT=0,DαI1=0,DαZ1=0.
Then system (3.1) has three equilibria: the uninfected equilibrium EH=(λ/d,0,0), the boundary equilibrium EI=(T*,I1*,0), where
(3.3)T*=aβ1,I1*=λa-dβ1,
and the interior equilibrium Ec=(T̂,Î1,Ẑ1), where
(3.4)T̂=-d+d2+4λβ̂2β̂,Î1=δc-δT̂,Ẑ1=cT̂b(c-δ)(β1T̂-a),β̂=δβ1c-δ.
Following the analysis in [4], we introduce a basic reproduction number which is defined by
(3.5)R0=λβ1ad.DenoteR+3={(T,I1,Z1)∈R3∣T≥0,I1≥0,Z1≥0} and we always assume that c>δ.  Note that DαZ1<0 always holds true if c≤δ. By generalized mean value theorem [15], we get Z1(t) is decreasing if c≤δ.Next we will discuss the existence and stability of the equilibria of the model (3.1).Theorem 3.1.
(a) The uninfected equilibriumEH is locally asymptotically stable (LAS) if R0∈(0,1) and unstable if R0>1.
(b)
IfR0>1, then the boundary equilibrium EI exists. This equilibrium is LAS if 1<R0<(aδ/d(c-δ))+1 and unstable if R0>(aδ/d(c-δ))+1.(c)
IfR0>(aδ/d(c-δ))+1, then Ec exists in IntR+3, where IntR+3 is the interior of R+3.Proof.
(a) The Jacobian matrixJ(EH) for system (3.1) evaluated at EH is given by
(3.6)J(EH)=(-d-β1λd00β1λd-a000-δ).EH is locally asymptotically stable if all of the eigenvalues p of the Jacobian matrix J(EH) satisfy the following condition [6, 17]:
(3.7)|arg(p)|>απ2.
The eigenvalues of J(EH) are -d,β1λ/d-a,-δ. It is clear that EH is LAS if R0<1 and is unstable if R0>1.
(b) IfR0>1, then the existence of EI is obvious.
The Jacobian matrixJ(EI) for system (3.1) evaluated at EI is given by
(3.8)J(EI)=(-d-β1I1*-β1T*0β1I1*0-bI1*T*+I1*00cI1*T*+I1*-δ).
For J(EI) given by (3.8), the characteristic equation becomes
(3.9)[p2+(d+β1I1*)p+β12I1*T*](p-cI1*T*+I1*+δ)=0.
and hence all the eigenvalues are
(3.10)p1,2=-(d+β1I1*)±(d+β1I1*)2-4β12I1*T*2,p3=cI1*T*+I1*-d.
If R0>1, then T*>0,I1*>0, and p1,2 have negative real parts. Furthermore, if 1<R0<(aδ/d(c-δ))+1, thenp3<0 and EI is LAS. If R0>(aδ/d(c-δ))+1, then p3>0 and EI is unstable.
(c) IfR0>(aδ/d(c-δ))+1, then we obtain Ẑ1>0. Thus, Ec exists in IntR+3. Therefore, the proof is complete.To discuss the local stability of the interior equilibriumEc, we consider the linearized system of (3.1) at Ec. The Jacobian matrix at Ec is given by
(3.11)J(Ec)=(-d-β1Î1-β1T̂0β1Î1+bẐ1Î1(T̂+Î1)2β1T̂-a-bẐ1T̂(T̂+Î1)2-bÎ1T̂+Î1-cẐ1Î1(T̂+Î1)2cẐ1T̂(T̂+Î1)2cÎ1T̂+Î1-δ).
For convenience, we denote T̂=T,Î1=I,Ẑ1=Z, and β1=β. In view of the above assumptions and using
(3.12)βT-a-bZT+I=0,λ-dT-βTI=0,cIT+I=δ,TT+I=c-δc,J(Ec) can now be written as follows:
(3.13)J(Ec)=(-d-βI-βT0βI+δc(βT-a)δc(βT-a)-bδc-δb(βT-a)c-δb(βT-a)0).
Then the characteristic equation of the linearized system of (3.1) is
(3.14)Φ(p)=p3+a1p2+a2p+a3=0,
where
(3.15)a1=d+δac+δ2βTc(c-δ),a2=βT(βI+δc(βT-a))+δc(c-δ)(βT-a)-δc(d+βI)(βT-a),a3=δc(c-δ)(βT-a)(d+βI)+βδ2cT(βT-a).Proposition 3.2.
The interior equilibriumEc is LAS if all of the eigenvalues p of J(Ec) satisfy |arg(p)|>απ/2.
Denote(3.16)D(Φ)=-|1a1a2a3001a1a2a332a1a200032a1a200032a1a2|=18a1a2a3+(a1a2)2-4a3a13-4a23-27a32.Using the results of [5, 20], we have the following proposition.Proposition 3.3.
One assumes thatEc exists in IntR+3.
(i)
If the discriminant ofΦ(p),D(Φ) is positive and Routh-Hurwitz conditions are satisfied, that is,
(3.17)D(Φ)>0,a1>0,a3>0,a1a2>a3,
then the interior equilibrium Ec is LAS.(ii)
IfD(Φ)<0,a1>0,a2>0,a1a2=a3,α∈[0,1), then the interior equilibrium Ec is LAS.(iii)
IfD(Φ)<0,a1<0,a2<0,α>2/3, then the interior equilibrium Ec is unstable.In our first example we setλ=10,d=0.02 which are chosen according to [21] and set a=δ=0.04,c=b=0.8,β1=4×10-4 which come from [4]. With these parameter values, R0=5>(aδ/d(c-δ))+1,D(Φ)=-9.4073×10-7<0,a1a2<a3. By Proposition 3.2., we obtain the interior equilibrium Ec=(362.0335,19.0544,49.9289) is LAS when α<0.9916. Numerical simulations show that trajectories of system (3.1) approach to the interior equilibrium (see Figures 1(a) and 1(b)). However, when α=1 (that is the case of classical integer order), Ec is unstable by the Routh-Hurwitz criterion(see Figures 2(a) and 2(b)).Numerical solutions of system (3.1). The plots show that trajectories of system (3.1) approach to the interior equilibrium for α=0.95.
(a)(b)Numerical solutions of system (3.1). The plots show that the interior equilibrium is unstable for α=1.
(a)(b)


## 4. Two-Virus Model

In this section, we consider viral diversity. We examine the two-virus model using numerical simulations. By examining the behavior of this simpler model we hope to get an idea as to how the more general models in system (2.4) may behave. The two-virus model is given by the following system of FODE:
(4.1)Dq1T=λ-dT-β1TI1-β2TI2,Dq21I1=β1TI1-aI1-bZ1I1T+I1+I2,Dq22I2=β2TI2-aI2-bZ2I2T+I1+I2,Dq31Z1=cZ1I1T+I1+I2-δZ1,Dq32Z2=cZ2I2T+I1+I2-δZ2,
with initial value condition
(4.2)T(0)=T0,Ii(0)=Ii0,Zi(0)=Zi0,i=1,2,
where 0.95≤q1,q2j,q3j(j=1,2)≤1.To find numerical solution to (4.1) and (4.2) in the interval [0,T], we reduce the systems (4.1) and (4.2) to a set of fractional integral equations, by using an equivalence (see [16, Theorem 3.24])
(4.3)DαX=f(X)⟺X(t)=X(0)+Iαf(X).
Then we apply the generalized Adams-type predictor-corrector method or, more precisely, Predict, Evaluate, Correct, Evaluate (PECE) methods (see [22, 23]).For notational convenience, we denoteα=(q1,q21,q22,q31,q32). We carry out numerical simulations for system, (4.1) and (4.2) with parameters λ=10,b=c=0.8,a=0.031,δ=0.03,d=0.02,β1=4×10-4, and β2=2.8×10-4 for the step size 0.07. Numerical solutions of systems (4.1) and (4.2) support that the system exhibits a chaotic behavior and systems (4.1) and (4.2) have a strange attractor in IntR+5 for α=[0.95,0.95,1,1,1] (see Figures 3(a)–3(c)). It is clear that chaos does exist in our fractional order model with viral diversity as in the case of integer order model. The effect of viral diversity and the frequency dependence results in collapse of the immune system and make the behavior of the system dynamics complex [4]. However, as the value of some component or more components of the order α further decreases, for example, α=[0.95,0.95,0.95,1,1], the chaotic motion disappears and the systems (4.1) and (4.2) stabilize to a fixed point (see Figures 4(a)–4(c)).Numerical solutions of system (4.1) for α=[0.95,0.95,1,1,1]. (a) A strange attractor in the Z1-Z2-I1  phase. (b) Infected cell 1. (c) CTL 2.
(a)(b)(c)Numerical solutions of system (4.1) for α=[0.95,0.95,0.95,1,1]. (a) Z1-Z2-I1 phase. (b) Infected cell 1. (c) CTL 2.
(a)(b)(c)


## 5. Conclusions

In this paper, we have proposed a fractional order HIV model, as a generalization of an integer order model, developed by Iwami et al. [4]. The premise of the proposed model is the fact that fractional order models possess memory while the main features of immune response involve memory. It is an attempt to incorporate fractional order into the mathematical model of HIV-immune system dynamics and it is still an interesting exercise to determine, mathematically, how the order of a fractional differential system affects the dynamics of system.In the case of one-virus model, the fractional order system has an interior equilibrium under some restriction. By using stability analysis on fractional order system, we obtain sufficient condition on the parameters for the stability of the interior equilibrium. Our analysis shows that the interior equilibrium which is unstable in the classical integer order model can become asymptotically stable in our fractional order model. Note that the interior equilibrium is globally asymptotically stable (GAS) (see [24]) if the terms associated with immune reactions are given by cZ1I1 and bZ1I1 instead of cZ1I1/(T+I1) and bZ1I1/(T+I1) in (3.1). That is, the interior equilibrium of the one-virus model can become unstable because of the frequency dependence (see [4]). However, in our fractional order model with the frequency dependence, the interior equilibrium can also become asymptotically stable if the order α<0.9916.We then consider viral diversity. If the terms associated with immune reaction depend on the mass action law instead of frequency, an interior equilibrium in [24] is GAS. Similar to the integer order model in [4], we find that strange chaotic attractors can be obtained under fractional order model with frequency dependence. That is, the effect of viral diversity and the frequency dependence results in collapse of the immune system and make the behavior of the system dynamics complex. However the chaotic motion may disappear and the fractional order system stabilizes to a fixed point if the value of the order α decreases. The specific biological meaning is deserved to further study.



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*Source: 378614-2009-06-24.xml*