2018 3rd International Conference on Computational Modeling, Simulation and Applied Mathematics (CMSAM 2018) ISBN: 978-1-60595-035-8

**Stability Analysis and Simulation of a Fractional-order HBV Infection **

**Model Based on Saturation Incidence **

### Yong-mei SU

1### , Lan LIU

1### , Yong-an YE

2,*### and Xiao-ke LI

21_{Mathematics and Physics School, University of Science and technology Beijing, Beijing, China }
2_{Dongzhimen Hospital, Beijing University of Chinese Medicine, Beijing, China }

*Corresponding author

**Keyword: **Fractional calculus, Hepatitis B virus, Equilibrium points, Local stability.

**Abstract.** Fractional order model has the memory, while the characteristic of the immune response
contains the memory. In this paper, we set up a fractional-order HBV immune model based on
saturation incidence for the first time. We derive the basic reproductive number R0, the cytotoxic T

lymphocytes immune response reproductive number R1. There are three equilibrium points of our

model, the local stability of each equilibrium point was given with corresponding hypothesis about R0 or R1. Finally we also give some numerical simulation, the simulation shows the individual

difference in clinical may be reflected by fractional-order model.

**Introduction **

In recent years, fractional calculus has become a hotspot and undergone a huge development in many fields. By now, fractional differential equations are widely used in the fields of optics, fluid mechanics, signal processing and other natural sciences [1-3]. Many mathematicians and researchers in the application field are trying to model the differential equations of fractional order in biology, because the researchers found that the biological cell membranes have electron conductivity, which can be classified as a fractional order model [4-5]. In addition, some biological models established by fractional differential equations have proved to be more advantageous than integers [5]. In particular, The biggest difference between the fractional order model and the integer order model is that the fractional order model has the memory, while the characteristic of the immune response contains the memory [1, 5].

So when we discuss virus immune model, fractional mathematical models have become important tools. Paper [5] also proposed a fractional order HIV infection model, Paper [6] proposed a fractional order HIV infection model, considering the logistic growth of the healthy CTL cells, paper [3] had further proposed the following HIV model:

{

*xα _{=λ-μx+ρx(1-}*

_{(}

_{x+y}_{)}

_{x}*max*

⁄ )-βxv

*yα _{=βxv- δy}*

*vα*

_{=δy-γv}*, * (1)

It should be pointed that, when describe the infection between uninfected cells and virus, paper [1,5,6] all use the bilinear incidences βxv, so we will consider the model include the immune cell, while the cytotoxic T lymphocytes (CTL) immune response after viral infection is universal and necessary to eliminate or control the disease, as follows:

{

*xα _{=λ-dx- βxv}*

_{⁄}

_{(}

_{x+v}_{)}

_{+δy}*yα*

_{= βxv}_{⁄}

_{(x+v)}

_{-ay-pyz-δy}*vα*

_{=ky-μv}*zα _{=cyz-bz}*

* , *(2)

exponentially at a rate* bz*, which is proportional to their current concentration, the parameter p
expresses the efficacy of nonlytic component.

This paper is organized as follows. In section 2 and sections 3, we mainly discussed the existence
and uniqueness of positive solutions and the stability of the equilibrium point respectively, we give
the global stability of 𝐸_{0} when *δ=0*. This paper ended with a conclusion in section 4.

**The Existence and Uniqueness of Positive Solutions **

For the proof of the existence and uniqueness about the positive solution, we firstly prove that there exist a positively invariant region for system Eq.2.

Let

N(x)=*x(t)*+*y(t)*+*a _{k}v(t)*+

*p*, (3) We have

_{c}z(t)*Nα*(x)=λ-dx-ay-*aμ*
*k*

*v-pb*

*c* *z≤ λ- h *(*x+ y+*
*a*
*kv+*

*p*

*cz*) , (4)

which h=min {d,a,μ,b}, N(x)≤(-*λ*

*h+N(0))Eα*(-ht
*α*_{)+}*λ*

*h* , Let D={x+y+
*a*
*kv+*

*p*
*cz≤*

*λ*

*h,x,y,v,z≥0}*, it is easy

to see that 𝐷 is a positively invariant region for model Eq.2.

Theorem2.1. The system Eq.2 has an unique solution X(t)=(x(t),y(t),v(t),z(t))*T*, and the solution
will remain nonnegative for all *t≥0*.

Proof. Firstly we prove the existence uniqueness of solution. We denoting

f(t, X)=

(

*λ-dx- βxv*⁄(x+v)+δy
*βxv*⁄(x+v)*- ay- pyz- δy*

*ky-μv*

*cyz-bz* _{)}

, (5)

Obviously* f(t,X)* satisfies conditions (1)-(3) of the Unique Solution Lemma [7], we only prove
system Eq.2 satisfies the last condition (4).Let

*η*=(
*λ*
*0*
*0*
*0*

) , *X(t)*=(
*x(t)*
*y(t)*
*v(t)*
*z(t)*

) , *A1*=

(

*-d* *0* *0* *0*

*0* *-a* *0* *0*

*0* *0* *-μ* *0*

*0* *0* *0* *-b*_{)}

, *A2*=(

*-β* *0 0 0*

*β* *0 0 0*

*0* *0 0 0*

*0* *0 0 0*

), (6)

* A3*=(

*0* *0* *0 0*

*0 -p 0 0*

*0* *0* *0 0*

*0* *0* *0 0*

)* , A _{4}*=(

*0 0 0 0*
*0 0 0 0*
*0 0 0 0*
*0 0 0 c*

), *A5 *=(

*0* *δ* *0 0*

*0 -δ 0 0*

*0* *k* *0 0*

*0* *0* *0 0*

), (7)

‖f(t,X)‖=||A_{1}X(t)+ v(t)

x(t)+v(t)A2X(t)+z(t)A3X(t )+y(t) A4X(t)+ A5X(t)+η||

≤ ‖A_{1}X(t)‖+‖A_{2}X(t)‖+m‖A_{3}X(t)‖+m‖A_{4}X(t)‖+‖A_{5}X(t)‖+‖η‖

≤ω+λ||X|| , (8)
Where *m= λ h*⁄ *,ω=||η|| , * λ= ‖A*1*‖+‖A*2*‖+m‖A*3*‖+m‖A*4*‖+‖A*5*‖*. *By [7], system Eq.2 has a

unique solution. Next we prove the solution is nonnegative for all *t≥0*. For model Eq.2, we know

*xα*_{(t)|}

**Stable Analysis **

In this section, we will discuss the stability of the model Eq.2. This system always has an
infection-free equilibrium *E0=*(*x0,0,0,0*), where *x0= λ d*⁄ . The basic reproduction number is*R0= βk μ*⁄ (*a+δ*),

when 𝑅0 > 1, the system Eq.2 will have immune-absence equilibrium *E1=(x1,y _{1},v10),* where

* x1= _{kd+ }*

_{(}

_{a+μ }λk_{)}

_{( R}*0-1) , y _{1}=*

*λμ(R0-1)*

*kd+*(*a+μ*)*(R _{0}-1) , v1=*

*λk(R0-1)*

*kd+*(*a+μ*)*(R _{0}-1) . *We can see

*R0>1*,

*x1>0,y1>0 and v1>0*

and *z _{1}=0*,which means the infected cells and virus coexist but the immune response is not activated
yet, that is

*cy*. Further, we will give the immune response reproductive number

_{1}<b*R*⁄

_{1}= cy_{1}*b*, when 𝑅

_{1}> 1, that is

*cy*which means immune response is activated. So when 𝑅

_{1}>b,_{1}> 1, there is another immune-response equilibrium

*E*(

_{2}=*x*), where

_{2},y_{2},v_{2},z_{2} x*2=*

*Δ+*√*Δ2+4d(λ+δy _{2})v_{2}*

*2d* * ,y _{2}=*

*b*

*c , v2=*
*kb*

*μc , z _{2}=*

*λ-ay _{2}-dx2*

*py _{2}*

*.*(10)

Which *Δ= λ+δy _{2}-(β+d)v2*.

Now, we introduce the main theorem. Theorem 3.1 for the model Eq.2,

(1) If *R0<1, *the equilibrium* E0* is local asymptotically stable.

(2) If *R0>1,* the equilibrium *E0* is unstable.

Proof. The characteristic equation for the infection-free equilibrium 𝐸0 is given as follows:

(λ+d)(λ+b)(*λ2+(a+δ+μ)λ*+*μ(a+δ)-βk*)*=0*. (11)
We can see that the characteristic roots *λ _{1}=-d<0, λ_{2}=-b<0,* which satisfied |argλ

*|=π>απ/2. For another two characteristic roots, we will consider the equation*

_{1,2}*λ2+(a+δ+μ)λ+μ(a+δ)-βk=0.*Let

*B =a+δ+μ>0*,

*C=μ(a+δ)*-

*βk*. Obviously

*R*can ensure C>0 , so we have

_{0}<1*λ*, which satisfied |argλ

_{3,4}<0*3,4*|=π>απ/2. So if

*R0<1*, the equilibrium

*E0*is local asymptotically stable, if

*R0>1, E0*is unstable.

Theorem 3.2. For system Eq.2. when *R0>1*,

(1) when *R1<1*, if *D(P)>0*, 𝐸1 is locally asymptotically stable for *0<α<1*; if D(P)<0, then the

equilibrium *E1* is locally asymptotically stable for * 0<α< 2 3*⁄ .

(2) When *R _{1}>1*, the equilibrium

*E*is unstable.

_{1}Proof. The characteristic equation for the *E _{1}* is given as follows:

(*λ-cy _{1}+b*)(

*λ3+a1λ2+a2λ+a3*)=0. (12)

Where

*a1=μ+d+ βv12*⁄(*x1+v1*)*2>0*, * * (13)

*a2=μ(d+ βv12*⁄(x*1+v1*)*2*)+d(a+δ)+ βav*12*⁄*(x1+v1)2+μ(a+δ) (1- _{R}1*

*0*)*>0*,* * (14)

* a3=dμ(a+δ) (1- _{R}1*

*0*)*+*

*βav12*

*(x _{1}+v_{1})2>0* (15)

We can see the *λ _{1}=cy_{1}-b*, when

*R*

_{1}<1, 𝜆_{1}is negative and |argλ

*|=π>α( π 2⁄*

_{1}*)*hold.

*a1a2-a3=*(a+δ+d+
*βv12*

(x* _{1}+v1*)

*2*) (μ(

*d+*

*βv12*(x

*)*

_{1}+v1*2*)+

*d(a+δ)*+ *βv1*

*2*

(x* _{1}+v1*)2+

*μ(a+δ)*(*1-* *1*
*R0*))*+*

*βkx12*
(x* _{1}+v1*)2

*+μ2*_{(d+} *βv12*

Hence according to [8], we know when *R _{1}<1*, if

*D(P)>0*,

*E*is locally asymptotically stable for

_{1}*0<α<1*; if* D(P)<0*, then the equilibrium *E1* is locally asymptotically stable for* 0<α< 2 3*⁄ . When
*R1>1*, the equilibrium *E1* is unstable.

For the immune-response equilibrium E*2*=(*x2,y _{2},v2,z_{2}*) . We assume that the characteristic

equation for the *E _{2}* is given as follows: P(λ)=

*λ4+a*=0, By [2], when

_{1}λ3+a_{2}λ2+a_{3}λ+a_{4}*n=4*, the characteristic equation has negative real roots only if

*an>0,n=0,1,3*and

*a3a2a1>a12+a32a0*.

In the following part, we will give the global stability of 𝐸_{0} when *δ=0*.
Theorem 3.3 For system Eq.2,

(1) If *R0<1*, the equilibrium* E0* is global asymptotically stable.

(2) If *R0>1,* the equilibrium 𝐸0 is unstable.

Proof.

Let *V(x)=y(t)+p _{c}z(t)+a_{k}v(t).* We have

D*α _{V=D}α_{y(t)+}p*

*cD*

*α _{z(t)+}a*

*kD*

*α _{v(t) }*

=*βxv*⁄(*x+v*)*-ay- pyz+p*

*c*(cyz-bz)+
*a*

*k*(*ky-μv*)

= (x⁄(*x+v*))**βv- pbz c*⁄ *-μa*
*k* *v*

≤*βv- μav k*⁄ =(R*0-1)v k*⁄ . (17)

Since *R _{0}<1*, we have

*DαV≤0*. Let M={(x,y,v,z)∈

*D,DαV=0 },*obviously

*M⊂{(x,y,v,z)*∈

*D,v=0}*. Let 𝐸 is the largest positively invariant subset of 𝑀, by the third equation of system Eq.2, we can know

*y=0*. So in

*E*, the first and last equation will be as follows:

{*xα=λ-dx*

*zα _{=-bz}* . (18)

Its solution is

{*x(t)=*(- λ d⁄ *+x(0))Eα*(-dt

*α*_{)+ λ d}_{⁄}

*z*(*t*)=*z*(0)*Eα*(-*btα*)

. (19)

We have *lim*

* t⟶∞x(t)=*
*λ*

*d ,* _{ t⟶∞}lim*z(t)*= 0*.* Thus, by the Lvapunov-Lasalle Theorem [7], all solutions

in the set D approach the infection-free equilibrium 𝐸_{0}. Noting that 𝐸_{0} is locally asymptotically
stable, so 𝐸0 is global asymptotically stable.

**Discussion and Conclusion **

In this paper, we discussed a fractional order HBV model with saturation incidence. For the model
Eq.2, We obtain the basic reproductive numbers *R*0 and the cytotoxic T lymphocytes immune

response reproductive number *R*1. When *R*0 < 1, we have proved that *E*0 is global asymptotically

stable with different order α . When *R*0 >1, *R*1 < 1, *E*1 is locally asymptotically stable different

order α. When *R*1 > 1, we also give the local stable condition of *E*2. The simulation shows that the

**Acknowledgments **

This work is jointly supported by 2015 National traditional Medicine Clinical Research Base Business Construction Special Topics (JDZX2015299) and the Fundamental Research Funds for the Central Universities FRF-BR-16-019A.

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