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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: { x=λ-μx+ρx(1(x+y) xmax ⁄ )-βxv y=βxvδy v=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: { x=λ-dxβxv (x+v) ⁄ +δy y= βxv (x+v) ⁄ -ay-pyz-δy v=ky-μv z=cyz-bz , (2) with 0<α<1, where the meaning of x, y and v are the same as model Eq.1, z represents the number of CTL. The immune response is assumed to get stronger at a rate cyz, the immune response decays 28 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 E0 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 c z(t) , (3) We have N(x)=λ-dx-ayaμ k vpb c z≤ λh (x+ y+ a k v+ p c z) , (4) which h=min {d,a,μ,b}, N(x)≤(λ h +N(0))Eα(-ht )+ λ h , Let D= {x+y+ a k v+ p c z≤ 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

Stability Analysis and Simulation of a Fractional-order HBV Infection Model Based on Saturation Incidence