• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Stochastic responses of tumor immune system with periodic treatment?

    2017-08-30 08:25:00DongXiLi李東喜andYingLi李穎
    Chinese Physics B 2017年9期
    關(guān)鍵詞:李穎

    Dong-Xi Li(李東喜)and Ying Li(李穎)

    1 College of Data Science,Taiyuan University of Technology,Taiyuan 030024,China

    2 College of Mathematics,Taiyuan University of Technology,Taiyuan 030024,China

    Stochastic responses of tumor immune system with periodic treatment?

    Dong-Xi Li(李東喜)1,?and Ying Li(李穎)2

    1 College of Data Science,Taiyuan University of Technology,Taiyuan 030024,China

    2 College of Mathematics,Taiyuan University of Technology,Taiyuan 030024,China

    We investigate the stochastic responses of a tumor–immune system competition model with environmental noise and periodic treatment.Firstly,a mathematical model describing the interaction between tumor cells and immune system under external fluctuations and periodic treatment is established based on the stochastic differential equation.Then,sufficient conditions for extinction and persistence of the tumor cells are derived by constructing Lyapunov functions and Ito’s formula.Finally,numerical simulations are introduced to illustrate and verify the results.The results of this work provide the theoretical basis for designing more effective and precise therapeutic strategies to eliminate cancer cells,especially for combining the immunotherapy and the traditional tools.

    stochastic responses,environmental noise,tumor–immune system,extinction

    1.Introduction

    Cancer is becoming the leading cause of death around the world.Traditional cancer treatments include surgery,radiation therapy,and chemotherapy.Cancer immunotherapy has recently gained exciting progress.Studies of tumor and immune system have largely been inspired by the works in Refs.[1] and[2],the authors showed that the immune system can recognize and eliminate malignant tumors.So immunotherapy, such as the cellular immunotherapy,[3]has been studied by researchers.And a number of tumor–immune system competition models have been proposed,such as Kuznetsov–Taylor model[4]and Kirschner–Panetta model.[5]In fact,tumor mi-croenvironment is inevitably affected by environmental noise in realism.Nowadays,noise dynamics have been widely studied in different fields such as metapopulation system[6]and Van der Pol oscillator.[7]In the last years,researchers have studied stochastic growth models of cancer cells,[8–11]using the Lyapunov exponent method and the Fokker–Planck equation method to investigate the stability of the stochastic model. Moreover,from a biological or a clinical point of view,investigations including treatments such as periodic ones are important for a successful treatment,e.g.,Thibodeaux and Schlittenhard[12]investigated the effect of a periodic treatment in the within-hostdynamics of malaria infection and suggested that synchronization with the intrinsic oscillation of infected erythrocytes takes place,leading to an optimal treatment.Sotolongo et al.[13]investigated the effect of immunotherapy under periodic treatment on a deterministic model of tumor– immune system and considered the possibility of suppression of tumor growth.Ideta et al.[14]considered the intermittent hormonal therapy in a model of prostate cancer and they suggested the existence of an optimal protocol to the intermittent therapy.Up to now,the effect of noise and cyclic treatment in the tumor dynamics has been widely studied.And fluctuations induced extinction and stochastic resonance in a model of tumor growth with periodic treatment have been studied.[15]Aisu and Horita[16]numerically investigated the stochastic extinction of tumor cells due to the synchronization effect through a time periodic treatment in a tumor–immune interaction model.

    The aim of this paper is to explore the dynamics of a simplified Kuznetsov–Taylor model[17]with both environmental noise and periodic treatment,especially the extinction and persistence.One of the advantages of our study is that we make use of the methods of It?o’s stochastic integral and Lyapunov function to derive and analyze the properties of the stochastic tumor–immune system competition model,which is different from the approaches of Fokker–Planck equation and effective potential function used in the existing literature.The other advantage is that the conditions for extinction and strong persistence in the mean of tumor cells are obtained by the strict mathematical proofs.The sufficient conditions for extinction and persistence could provide us a more effective and precise therapeutic schedule to eliminate tumor cells and improve the treatment of cancer.

    This paper is organized as follows.In Section 2,thestochastic tumor–immune model with periodic treatment is derived.In Section 3,we establish the sufficient conditions for extinction and strong persistence in the mean of tumor cells. Numerical simulations are presented in Section 4,which are used to verify and illustrate the theorems of Section 3.In Section 5,we present the conclusion and discuss future directions of this research.

    2.Stochastic tumor–immune system with periodic treatment

    In this section,the Kuznetsov–Taylor model[4]and its modified version by Galach[17]are introduced.The Kuznetsov–Taylor model describes the response of effector cells to the growth of tumor cells and takes into account the penetration of tumor cells by effector cells,which simultaneously causes the inactivation of effector cells.The Kuznetsov–Taylor model reads

    where s is the normal(i.e.,not increased by the presence of the tumor)rate of the flow of adult effector cells into the tumor site in units of cells per day,p and g are positive constants in the function F(E,T)=pE T/(g+T)that describes the accumulation of effector cells in the tumor site,p is in units of day?1and g is in units of cells.m denotes the coefficient of inactivation of effector cells during the formation and decomposition of EC-TC compounds and is in units of day?1·cells?1. d is the coefficient of the destruction and migration of effector cells and is in units of day?1.a is the coefficient of the maximal growth of tumor and is in units of day?1.b?1is the environment capacity,and b is in units of cells?1.n represents the inactivation rate of tumor cells due to the immune system response and is in units of day?1·cells?1.The dimensionless form of the model is

    where x=E/E0,y=T/T0,ε=s/(nE0T0),ρ=p/(nT0), η=g/T0,μ=m/n,δ=d/(nT0),α=a/(nT0),β=bT0,and E0=T0=106cells.

    In 2003,Galach proposed the modified version of model (1),which reads

    where x denotes the dimensionless density of effector cells;y stands for the dimensionless density of the population of tumor cells;ε,δ,α,1/β have the same meanings as those in Eq.(1),and ω represents the immune response to the appearance of the tumor cells(i.e.,immune coefficient).In this paper, we consider the case of ω>0,which means that the immune response is positive.

    System(2)always has the equilibrium

    If ω>0 and αδ<ε,then P0is the unique equilibrium of model(2)and it is globally stable.If ω>0 and αδ>ε,then P0is unstable and there is an equilibrium

    which is globally stable.Here Δ=α2(βδ?ω)2+4αβεω.

    In fact,the growth of tumor cells is influenced by many environmental factors,[18]e.g.,the supply of oxygen and nutrients,the degree of vascularization of tissues,the immunological state of the host,chemical agents,temperature,etc.So, it is inevitable to consider the tumor–immune system competition model with environmental noises.In this paper,taking into account the effect of randomly fluctuating environment, we assume that the fluctuations in the environment mainly affect the immune coefficient ω,

    where B(t)is the standard Brownian motion with B(0)=0, and the intensity of white noise σ2>0.We are interested in the stochastic responses of the tumor immune system driven by a controllable therapy.Here,the influence of the therapeutic factors is studied by considering a periodic treatment (chemo-or radiation-therapy).The treatment scheme[19]can be expressed as

    Here Φ stands for the Heaviside function reflecting the on-off switch of the cyclic treatment performed with the intensity A and frequency f.Now the tumor–immune system competition model with environmental noise and periodic treatment can be rewritten as

    where all the parameters are positive and bounded.For convenience,we define the following notions:

    3.Theoretical analysis of extinction and persistence under periodic treatment

    Our primary interests in tumor dynamics are the extinction and survival of tumors.In order to study the extinction and survival,we need some appropriate definitions about extinction and persistence.Here we adopt the concepts of extinction and strong persistence in the mean.[20]In addition,some of our proofs are motivated by the works of Liu,[20]Mao,[21]and Jiang.[22]Some useful definitions are as follows:

    1)The tumor cells y(t)will go to extinction a.s.if limt→+∞y(t)=0.

    2)The tumor cells y(t)will be strongly persistent in the mean a.s.if〈y(t)〉?>0.

    Next we establish the sufficient conditions of extinction and persistence for our model.

    Lemma 1 For any positive initial value(x0,y0),if 0<x0<1/β,the solution of Eq.(4)obeys

    Proof According to the second equation of model(4),we have

    Firstly,we discuss y for x in different value ranges.

    Consequently,we have proved y(t)≤max{y0,1/β}.Then we will show that x(t)is bounded.Applying the Ito’s formula[23]to the first equation of model(4),

    Integrating both sides from 0 to t,we obtain

    where

    Therefore,

    That is to say if x0<1/β,then x(t)<1/β for all t>0.

    Lemma 2 Let f∈C[[0,∞)×?,(0,∞)]and F(t)∈C([0,∞)×?,R).If there exist positive constants λ0,λ,and T such that

    for all t≥T,and limt→∞F(t)/t=0 a.s.,then

    Proof The proof is similar to that of lemma in Ref.[22]. Note that limt→∞F(t)/t=0 a.s.;then for arbitrary k>0,there exists a T0=T0(ω)>0 and a set ?ksuch that P(?k)≥1?k and F(t)/t≤k for all t≥T0,ω∈?k.LetˉT=max{T,T0} and

    Since f∈C[[0,∞)×?,(0,∞)],then ?(t)is differentiable ona.s.,and

    Taking the limit inferior of both sides and applying L’Hospital’s rule on the right-hand side of this inequality,we obtain

    Letting k→0 yields

    Theorem 1 For any positive initial value(x0,y0),particularly,when x0<1/β,equation(4)has a positive unique global solution(x(t),y(t))on t≥0 a.s.

    Proof To obtain a unique global solution for any given initial value,the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition.[24]However,the coefficients of model(4)do not satisfy the linear growth condition,so the solution may explode in a finite time.Since the coefficients of Eq.(4)are locally Lipschitz continuous for any given initial value(x0,y0)∈,there is a unique maximal local solution(x(t),y(t))on t∈[0,τe],where τeis the explosion time.[24]To show that this solution is global,we only need to show τe=∞.To this end,let k0>0 be sufficiently large so that x0,y0all lie within the interval[1/k0,k0].For each integer k≥k0,we define the stopping time τk=inf{t∈[0,τe]:min{x(t),y(t)}≤1/k or max{x(t),y(t)}≥k}.Clearly,τkis increasing as k→∞.Set τ∞=limk→+∞τk,thus τ∞≤τea.s.In other words,we only need to prove τ∞=∞.If this statement is false,there exists constants T>0 and ε∈(0,1)such that P{τ∞<∞}>ε. Thus there is an integer k1>k0such that

    Define a C2-function V:→R+by V(x,y)=(x?1?ln x)+ (y?1?ln y).The nonnegativity of this function can be seen from u?1?ln u≥0,?u>0.Let k≥k0and T>0 be arbitrary.Applying the It?o’s formula,we have

    Here,L is a positive constant and in the proof of the last inequality,we have used Lemma 3(i.e.,for?t≥0,x(t)and y(t) are bounded).The inequality(9)implies

    Taking expectation on both sides of the above inequality,we can obtain

    Let ?k={τk∧T},then by inequality(8),we have P(?k)≥ε. Note that for any ω∈?k,x(τk,ω),y(τk,ω)equals either k or 1/k,hence V(x(τk,ω),y(τk,ω))is no less than min{2(k?1?ln k),2(1/k?1+ln k),k+1/k?2)}.By formula(10)we have

    where 1?kis the indicator function of ?k.Let k→∞,there exists the contradiction∞>V(x0,y0)+LT=∞,which completes the proof.

    Remark 1 In order to guarantee the existence and uniqueness of the solution of model(4),we discuss the extinction and persistence of y(t)under the condition x0<1/β below.

    Theorem 2 Let(x(t),y(t))be the solution of system(4) with positive initial value(x0,y0),if αδ?ε<A/2,then

    Proof An integration of the first equation of model(4) yields

    We compute

    where

    which is a local continuous martingale and N1(0)=0.Moreover

    By strong law of large numbers for local martingales,[24]we obtain

    Taking the limit inferior of both sides of inequality(12),we have

    Applying the It?o’s formula to the second equation of model(4) yields

    Integrating this from 0 to t and dividing by t on both sides,we have

    Taking the limit superior of both sides of inequality(15)and substituting inequality(14)into inequality(15)yield

    If the condition αδ?ε<A/2 is satisfied,then

    which implies

    Applying the It?o’s formula to the first equation of model(4) leads to

    Integrating this from 0 to t,we have

    By virtue of the exponential martingale inequality,[24]for any positive constants T,a,and b,we have

    Choosing T=n,a=1,and b=2ln n,we obtain

    An application of Borel–Cantelli lemma[22]yields that for almost all ω∈?,there is a random integer n0=n0(ω)such that for n≥n0,

    That is to say,

    for all 0≤t≤n,n≥n0a.s.Substituting the above inequality into inequality(17)leads to

    Theorem 3 If δα?ε>A/2,then the tumor cells y(t) will be almost surely strong persistent in the mean.

    Proof An integration of system(4)is

    Substituting Eq.(20)to Eq.(21)yields

    Consequently,we can derive that if δα?ε>A/2,then〈y(t)〉?>0 a.s.

    4.Simulations and discussion

    In this section,we use the Euler–Maruyama numerical algorithm mentioned by Higham[25]to support our results.The parameters in model(4)are chosen as α=1.636,β=0.002, δ=0.3743,ε=0.5181,and ω=0.0115,which are approximated to the experimental values.[4,17]

    Figure 1 shows the simulation results of Theorem 2. Clearly,the parameters satisfy the condition αδ?ε<A/2.In view of Theorem 2,the tumor cells y(t)will go to extinction, and the effector cells x(t)have the property limt→+∞〈x(t)〉= (ε+A/2)/δ=1.785.Figure 1 confirms the results of Theorem 2.

    Fig.1.(color online)Solutions of model(4)with periodic treatment s(t)=A[1?Φ(cos(2π ft))]for A=0.3,σ=0.03,f=0.05,step size Δt=0.01,and initial value(x(0),y(0))=(1.5,25).

    Figure 2 shows the simulation results of Theorem 2.In Fig.2,it is clear that the parameters of the example meet the condition δα?ε>A/2.According to Theorem 2,the tumor cells y(t)will be almost surely strongly persistent in the mean. It can be seen from Fig.2 that the tumor cells will decrease firstly and then exhibit a period-like evolution at a relative low concentration under periodic treatment,but do not tend to zero.This phenomenon implies that the tumor cells could be suppressed by the periodic treatment but not be completely eliminated when the intensity of the treatment is not enough to cure the tumor,i.e.,the tumor cells could be controlled and will not deteriorate in this case.Moreover,it can be seen that as long as the conditions of persistence δα?ε>A/2 are satisfied,the tumor cell will be strongly persistent in the mean almost surely when the model is with noise.And,the tumor cells will also be persistent or survival when the model is without noises.The difference is that the persistence of the former is in the sense of the mean,which is random;while the persistence of the latter is expressed as persistence or survival, which is deterministic.

    Fig.2.(color online)Solutions of model(4)with periodic treatment s(t)=A[1?Φ(cos(2π ft))]for A=0.07,σ=0.03,f=0.05,step size Δt=0.01,and initial value(x(0),y(0))=(1.5,25).

    Figure 3 shows the evolution of tumor cells y(t)as a function of time t for three different values of A=0.07,0.3,3. Clearly,when A is 0.07,it satisfies the condition δα?ε>A/2.According to Theorem 2,the tumor cells y(t)will be strongly persistent in the mean.With increasing strength of the treatment,A is taken as 0.3 or 3,they satisfy the condition αδ?ε<A/2.According to Theorem 2,the tumor cells y(t) will tend to be extinction.Moreover,by comparing curves(ii) and(iii),we find that the tumor cells will be extinct faster with the increase of the treatment intensity A.This behavior indicates that increasing the intensity of the treatment is beneficial to accelerate the extinction of the tumor cells.

    Fig.3.(color online)Solutions of tumor cells for σ=0.03,f=0.05,step size Δt=0.01,and initial value(x(0),y(0))=(2.6,25).

    5.Conclusion

    We study stochastic responses of a tumor–immune system competition model with environmental noise and periodic treatment.Firstly,the environmental noise(Gaussian white noise)is taken into account and the periodic treatment is regarded as a Heaviside function.Then,sufficient conditions for extinction and strong persistence in the mean of tumor cells are derived by constructing Lyapunov functions.The detail results and biological significance are as follows:

    (A)If αδ?ε<A/2,then the effector cells x(t)have property limt→+∞〈x(t)〉=(ε+A/2)/δ,and the tumor cells y(t)will go to extinction a.s.

    (B)If αδ?ε>A/2,then the tumor cells y(t)will be strongly persistent in the mean a.s.

    According to the theorems and figures,the extinction and survival of the tumor cells rely on the strength of the periodic treatment.With the increasing intensity of the periodic treatment,the tumor cells will experience the process from strongly persistence in the mean to extinction.In addition,the synchronization effect between the environmental noises and the periodic treatment on the tumor–immune system competition model is obtained by strict proof and simulation.Our theoretical results will be beneficial to design more effective and feasible treatment therapies.

    Some interesting questions deserve further investigations. For example,in our model,we assume that fluctuations in the environment mainly affect the immune coefficient ω.It is interesting to study what happens if it affects other parameters of the tumor–immune system.Another question of interest is to consider the stability in distribution(e.g.,Refs.[26]and[27]) and time delay(e.g.Ref.[28])of the tumor–immune system.

    [1]Parish C R 2003 Immunol.Cell.Biol 81 106

    [2]Smyth M J,Godfrey D I and Trapani J A 2001 Nat.Immunol.2 293

    [3]Rosenberg S A,Spiess P and Lafreniere R 1986 Science 233 1318

    [4]Kuznetsoz V A,Makalkin I A,Taylor M A and Perelson A S 1994 Bull. Math.Biol 56 295

    [5]Kirschner D and Panetta J C 1998 J.Math.Biol 37 235

    [6]Wang K K and Liu X B 2013 Chin.Phys.Lett 30 070504

    [7]Yang Y G,Xu W,Sun Y H and Gu X D 2016 Chin.Phys.B 25 020201

    [8]Zhong W R,Shao Y Z and He Z H 2006 Phys.Rev.E 73 060902

    [9]Albano G and Giorno V 2006 J.Theor Biol 242 329

    [10]Lenbury Y,Triampo Wannapong,Tang IMand Picha P 2006 J.Korean. Phys.Soc 49 1652

    [11]Ferrante L,Bompadre S,Possati L and Leone L 2000 Biometrics 56 1076

    [12]Thibodeaux J J and Schlittenhardt T P 2011 Bull.Math.Biol.73 2791

    [13]Sotolongo-Costam O,Molina L M,Perez D R,Antranz J C and Reys M C 2003 Physica D 178 242

    [14]Ideta A M,Tanaka G,Takeuchi T and Aihara K 2008 J.Nonlinear Sci. 18 593

    [15]Li D X,Xu W,Guo Y and Xu Y 2011 Phys.Lett.A 375 886

    [16]Aisu R and Horita T 2012 Nonlinear Theory and Its Applications,IEICE 3 191

    [17]Galach M 2003 Int.J.Appl.Math.Comput.Sci.13 395

    [18]Fiasconaro A,Spagnolo B,Ochabmarcinek A and Gudowskanowak E 2006 Phys.Rev.E 74 041904

    [19]Fiasconaro A,Ochab-Marcinek A,Spagnolo B and Gudowska-Nowak E 2008 Eur.Phys.J.B 65 435

    [20]Liu M and Wang K 2011 J.Math.Anal.Appl.375 443

    [21]Mao X,Marion G and Renshaw E 2002 Stoch.Proc.Appl.97 95

    [22]Zhao Y,Jiang D and O’Regan D 2013 Physica A 392 4916

    [23]Evans L C 2013 An Introduction to Stochastic Differential Equations (New York:Amer Mathematical Society)pp.77–79

    [24]Mao X 1997 Stochastic Differential Equations and Applications (Chichester:Horwood)pp.31–84

    [25]Higham D J 2001 SIAM Rev.43 525

    [26]Liu M and Bai C 2016 Appl.Math.Comput.284 308

    [27]Liu M and Bai C 2016 Appl.Math.Comput.276 301

    [28]Jin Y F and Xie W X 2015 Chin.Phys.B 24 110501

    26 February 2017;revised manuscript

    9 May 2017;published online 24 July 2017)

    10.1088/1674-1056/26/9/090203

    ?Project supported by the National Natural Science Foundation of China(Grant Nos.11402157 and 11571009),Shanxi Scholarship Council of China(Grant No.2015-032),Technological Innovation Programs of Higher Education Institutions in Shanxi,China(Grant No.2015121),and Applied Basic Research Programs of Shanxi Province,China(Grant No.2016021013).

    ?Corresponding author.E-mail:dxli0426@126.com

    ?2017 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn

    猜你喜歡
    李穎
    An overview of quantum error mitigation formulas
    Effect of astrocyte on synchronization of thermosensitive neuron–astrocyte minimum system
    《二次根式》拓展精練
    Assessment of cortical bone fatigue using coded nonlinear ultrasound?
    完形填空專練(三)
    Ultrasonic backscatter characterization of cancellous bone using a general Nakagami statistical model?
    李穎、李鳳華作品
    Human body
    一雉雞翎的傷痛
    小說月刊(2017年8期)2017-08-16 22:34:39
    李穎、李鳳華作品
    一边亲一边摸免费视频| 亚洲精品国产av蜜桃| 欧美一区二区亚洲| 边亲边吃奶的免费视频| 久久久久久久久久久丰满| 超碰av人人做人人爽久久| 国产成人精品久久久久久| 又爽又黄a免费视频| 综合色av麻豆| 纵有疾风起免费观看全集完整版 | 日本免费在线观看一区| 三级国产精品片| 日本免费a在线| 中文欧美无线码| 天天一区二区日本电影三级| 狠狠精品人妻久久久久久综合| 18禁在线无遮挡免费观看视频| 在线观看一区二区三区| 免费观看av网站的网址| 97超碰精品成人国产| 又爽又黄a免费视频| 91精品国产九色| 人妻一区二区av| 国产精品久久久久久精品电影| 欧美不卡视频在线免费观看| 午夜精品一区二区三区免费看| 亚洲精品国产av蜜桃| 性色avwww在线观看| 丰满乱子伦码专区| 一个人看视频在线观看www免费| 直男gayav资源| 高清欧美精品videossex| 国产国拍精品亚洲av在线观看| 日韩中字成人| 1000部很黄的大片| 麻豆精品久久久久久蜜桃| 国产不卡一卡二| 中文天堂在线官网| 欧美人与善性xxx| 乱码一卡2卡4卡精品| 亚洲国产色片| 欧美一区二区亚洲| 最新中文字幕久久久久| 成人毛片60女人毛片免费| 国产成人freesex在线| videos熟女内射| 欧美丝袜亚洲另类| 日韩av在线大香蕉| 亚洲内射少妇av| 日本色播在线视频| 在线a可以看的网站| 免费少妇av软件| 秋霞在线观看毛片| 亚洲婷婷狠狠爱综合网| 国产黄a三级三级三级人| 精品久久久久久久久亚洲| 天堂中文最新版在线下载 | 嫩草影院新地址| 高清日韩中文字幕在线| 男的添女的下面高潮视频| 亚洲av中文av极速乱| 2022亚洲国产成人精品| 亚洲国产欧美人成| 国产又色又爽无遮挡免| 亚洲美女搞黄在线观看| 成人性生交大片免费视频hd| 一个人观看的视频www高清免费观看| 99久久中文字幕三级久久日本| 久久久久久九九精品二区国产| 大香蕉久久网| 亚洲丝袜综合中文字幕| 亚洲不卡免费看| 国产精品久久久久久精品电影小说 | a级一级毛片免费在线观看| 九九爱精品视频在线观看| 中文乱码字字幕精品一区二区三区 | 床上黄色一级片| 国产av不卡久久| 在线免费观看的www视频| 国产 一区精品| 婷婷色麻豆天堂久久| 国内少妇人妻偷人精品xxx网站| 国产男女超爽视频在线观看| 国产又色又爽无遮挡免| 亚洲av成人精品一区久久| 精品亚洲乱码少妇综合久久| 亚洲精品国产av蜜桃| 秋霞在线观看毛片| 乱系列少妇在线播放| 久久国内精品自在自线图片| 好男人视频免费观看在线| 国产精品三级大全| 国产男人的电影天堂91| 日本午夜av视频| 成人毛片a级毛片在线播放| 亚洲性久久影院| 免费人成在线观看视频色| 内地一区二区视频在线| 日韩一区二区三区影片| 亚洲综合精品二区| 毛片一级片免费看久久久久| 六月丁香七月| 亚洲国产欧美在线一区| 草草在线视频免费看| 91av网一区二区| 精品人妻视频免费看| 亚洲美女视频黄频| 国产一区亚洲一区在线观看| 成年女人看的毛片在线观看| 国产伦在线观看视频一区| 欧美成人精品欧美一级黄| 午夜激情欧美在线| 午夜精品在线福利| 亚洲精品成人av观看孕妇| 日韩欧美三级三区| 亚洲av日韩在线播放| 成年女人看的毛片在线观看| 久久久久国产网址| 亚洲无线观看免费| 欧美潮喷喷水| 成人一区二区视频在线观看| 日韩视频在线欧美| 国产精品麻豆人妻色哟哟久久 | 成人高潮视频无遮挡免费网站| 国产在线男女| 国产又色又爽无遮挡免| 高清视频免费观看一区二区 | xxx大片免费视频| 菩萨蛮人人尽说江南好唐韦庄| 美女大奶头视频| 91aial.com中文字幕在线观看| 久久99蜜桃精品久久| 国产黄色免费在线视频| 成人毛片a级毛片在线播放| 你懂的网址亚洲精品在线观看| 一个人看的www免费观看视频| 在线播放无遮挡| 国产一区二区三区综合在线观看 | 亚洲图色成人| 97精品久久久久久久久久精品| 久久久久九九精品影院| 亚洲最大成人手机在线| 精品久久久精品久久久| 99久久精品国产国产毛片| 国内精品宾馆在线| 亚洲国产精品成人久久小说| 午夜激情欧美在线| 人妻少妇偷人精品九色| 日韩 亚洲 欧美在线| 欧美丝袜亚洲另类| 男人狂女人下面高潮的视频| 超碰97精品在线观看| 18禁动态无遮挡网站| 久久久久久久国产电影| 免费黄网站久久成人精品| 国产麻豆成人av免费视频| 一本一本综合久久| 免费观看的影片在线观看| 精品欧美国产一区二区三| 亚洲国产色片| 能在线免费观看的黄片| 三级国产精品片| 亚洲国产成人一精品久久久| 国产v大片淫在线免费观看| 亚洲人成网站在线播| av黄色大香蕉| 亚洲人成网站高清观看| 黄色欧美视频在线观看| 天堂av国产一区二区熟女人妻| 亚洲av电影不卡..在线观看| 国产91av在线免费观看| av线在线观看网站| 精品人妻一区二区三区麻豆| 十八禁国产超污无遮挡网站| 天堂网av新在线| 天堂av国产一区二区熟女人妻| 欧美日韩一区二区视频在线观看视频在线 | 日本与韩国留学比较| 亚洲自拍偷在线| 亚洲av.av天堂| 一级二级三级毛片免费看| 91精品伊人久久大香线蕉| 精品久久久精品久久久| 老司机影院成人| 亚洲精品第二区| 国产乱来视频区| 久久99热6这里只有精品| 国产单亲对白刺激| 国产精品久久久久久久久免| 久久久久久久久中文| 91狼人影院| 搞女人的毛片| 免费大片黄手机在线观看| 大又大粗又爽又黄少妇毛片口| 精品人妻熟女av久视频| 国产亚洲最大av| 日韩中字成人| 高清午夜精品一区二区三区| 少妇的逼水好多| 久久精品久久久久久久性| 久久午夜福利片| 国国产精品蜜臀av免费| 街头女战士在线观看网站| 看免费成人av毛片| 不卡视频在线观看欧美| 亚洲在线观看片| 亚洲无线观看免费| 岛国毛片在线播放| 免费av不卡在线播放| 午夜福利网站1000一区二区三区| 1000部很黄的大片| 久久久午夜欧美精品| 久久99热这里只有精品18| 少妇的逼好多水| 欧美一区二区亚洲| 97热精品久久久久久| 久99久视频精品免费| 亚洲av不卡在线观看| 国产日韩欧美在线精品| 熟妇人妻久久中文字幕3abv| 三级男女做爰猛烈吃奶摸视频| 在线观看av片永久免费下载| 亚洲av.av天堂| 国产一区二区亚洲精品在线观看| 亚洲欧美清纯卡通| 禁无遮挡网站| 男人狂女人下面高潮的视频| 亚洲av成人精品一区久久| 国产毛片a区久久久久| 亚洲精品亚洲一区二区| 国产av不卡久久| 国产成人a∨麻豆精品| 亚洲精品中文字幕在线视频 | 偷拍熟女少妇极品色| 欧美zozozo另类| 免费观看的影片在线观看| 久99久视频精品免费| 极品教师在线视频| 丝袜喷水一区| 国产成人午夜福利电影在线观看| 国产伦理片在线播放av一区| 女人久久www免费人成看片| 成年女人看的毛片在线观看| 天天躁夜夜躁狠狠久久av| 国产真实伦视频高清在线观看| 久久精品久久久久久噜噜老黄| 国产成人免费观看mmmm| 精品久久国产蜜桃| 久久97久久精品| 亚洲av免费高清在线观看| 亚洲精品成人av观看孕妇| 成年av动漫网址| 黄色欧美视频在线观看| 伦精品一区二区三区| 夜夜看夜夜爽夜夜摸| 国产黄频视频在线观看| 美女黄网站色视频| 亚洲av一区综合| 18禁裸乳无遮挡免费网站照片| 亚洲国产精品成人久久小说| 欧美成人精品欧美一级黄| 国产探花在线观看一区二区| 国产伦在线观看视频一区| 国产精品久久久久久久电影| 日日啪夜夜撸| 日韩三级伦理在线观看| 日本免费a在线| 免费在线观看成人毛片| 欧美潮喷喷水| 国产精品1区2区在线观看.| 免费高清在线观看视频在线观看| 97超视频在线观看视频| 两个人视频免费观看高清| 春色校园在线视频观看| 91精品国产九色| 中文字幕av在线有码专区| 亚洲欧美精品自产自拍| 亚洲怡红院男人天堂| 国产成人福利小说| 在线观看免费高清a一片| 日韩 亚洲 欧美在线| 乱码一卡2卡4卡精品| 十八禁网站网址无遮挡 | 成年女人看的毛片在线观看| 汤姆久久久久久久影院中文字幕 | 亚洲va在线va天堂va国产| 女人被狂操c到高潮| 国产伦精品一区二区三区四那| 人人妻人人澡人人爽人人夜夜 | 丰满人妻一区二区三区视频av| 国产综合精华液| 免费观看的影片在线观看| 精品久久久久久久久久久久久| 伦精品一区二区三区| 少妇人妻一区二区三区视频| 国产欧美另类精品又又久久亚洲欧美| 亚洲va在线va天堂va国产| 秋霞在线观看毛片| 大又大粗又爽又黄少妇毛片口| 色综合色国产| 26uuu在线亚洲综合色| 啦啦啦中文免费视频观看日本| 噜噜噜噜噜久久久久久91| 国模一区二区三区四区视频| 日韩av在线大香蕉| 亚洲国产色片| 国产在线一区二区三区精| 国产精品99久久久久久久久| 男的添女的下面高潮视频| 亚洲精品乱久久久久久| 一个人观看的视频www高清免费观看| 午夜激情欧美在线| 国产一区有黄有色的免费视频 | 精品一区二区三区视频在线| 亚洲av电影不卡..在线观看| 亚洲最大成人中文| 亚洲自偷自拍三级| 春色校园在线视频观看| 亚洲av在线观看美女高潮| 亚洲最大成人中文| 国产精品国产三级专区第一集| 精品久久久久久久久av| 成人无遮挡网站| 高清毛片免费看| 国产白丝娇喘喷水9色精品| 只有这里有精品99| 有码 亚洲区| 精品一区二区三区视频在线| 国产 一区 欧美 日韩| 日本熟妇午夜| 国产精品国产三级专区第一集| 亚洲国产欧美在线一区| 成人无遮挡网站| 18禁动态无遮挡网站| 国产老妇女一区| 看非洲黑人一级黄片| 国产精品久久久久久av不卡| 精品久久久噜噜| 国产精品福利在线免费观看| 卡戴珊不雅视频在线播放| 乱码一卡2卡4卡精品| 亚洲第一区二区三区不卡| 国产视频首页在线观看| 国产在视频线在精品| 22中文网久久字幕| 听说在线观看完整版免费高清| 国产久久久一区二区三区| 欧美最新免费一区二区三区| 夫妻午夜视频| 免费看av在线观看网站| 亚洲成人av在线免费| 人人妻人人澡欧美一区二区| 男女视频在线观看网站免费| 亚洲在线观看片| 精华霜和精华液先用哪个| 国产精品熟女久久久久浪| 久久鲁丝午夜福利片| 免费不卡的大黄色大毛片视频在线观看 | 亚洲最大成人av| 99久国产av精品| 寂寞人妻少妇视频99o| 亚洲最大成人av| 内地一区二区视频在线| 日本免费在线观看一区| 一级a做视频免费观看| 熟妇人妻不卡中文字幕| 91午夜精品亚洲一区二区三区| 18禁裸乳无遮挡免费网站照片| 免费看不卡的av| 色综合亚洲欧美另类图片| 国产精品一二三区在线看| 亚洲欧美精品自产自拍| 亚洲aⅴ乱码一区二区在线播放| 国产综合懂色| 黄片无遮挡物在线观看| 18禁裸乳无遮挡免费网站照片| 18+在线观看网站| 亚洲av成人精品一二三区| 简卡轻食公司| 久久久久久久久中文| 久久精品夜夜夜夜夜久久蜜豆| 婷婷色综合www| 老女人水多毛片| 久久精品人妻少妇| 亚洲婷婷狠狠爱综合网| 天堂av国产一区二区熟女人妻| 精品午夜福利在线看| 观看免费一级毛片| 日本猛色少妇xxxxx猛交久久| 联通29元200g的流量卡| 一级片'在线观看视频| 成人欧美大片| 亚洲国产欧美人成| 日韩在线高清观看一区二区三区| 亚洲av一区综合| 搞女人的毛片| 91精品伊人久久大香线蕉| 免费av不卡在线播放| 性插视频无遮挡在线免费观看| 免费av不卡在线播放| 97热精品久久久久久| 精品国产一区二区三区久久久樱花 | 一区二区三区免费毛片| www.av在线官网国产| 成人午夜高清在线视频| 人人妻人人澡人人爽人人夜夜 | 国产午夜精品论理片| 日本免费a在线| 天天一区二区日本电影三级| 欧美成人a在线观看| 精品国产露脸久久av麻豆 | 伊人久久精品亚洲午夜| 亚洲国产精品成人综合色| 亚洲av二区三区四区| 精品久久久久久久末码| 精品一区二区三区人妻视频| 久久精品夜色国产| 精华霜和精华液先用哪个| 精品99又大又爽又粗少妇毛片| 亚洲欧美成人综合另类久久久| av黄色大香蕉| 高清av免费在线| 免费av观看视频| 国产综合精华液| 久久鲁丝午夜福利片| 欧美xxxx黑人xx丫x性爽| 夜夜看夜夜爽夜夜摸| 日韩亚洲欧美综合| 亚洲真实伦在线观看| 婷婷色av中文字幕| 免费av毛片视频| 国产成人a区在线观看| 亚洲内射少妇av| 久久亚洲国产成人精品v| 听说在线观看完整版免费高清| 色5月婷婷丁香| 69av精品久久久久久| 在线a可以看的网站| 18禁裸乳无遮挡免费网站照片| 国内精品美女久久久久久| 亚洲成人久久爱视频| 亚洲乱码一区二区免费版| 国产老妇伦熟女老妇高清| 老司机影院毛片| 91久久精品国产一区二区成人| 久久精品熟女亚洲av麻豆精品 | 丰满乱子伦码专区| 亚洲国产精品国产精品| 欧美最新免费一区二区三区| 日韩强制内射视频| 欧美日韩亚洲高清精品| 午夜日本视频在线| 欧美激情在线99| 国产在线男女| 国产一区二区三区av在线| 久久韩国三级中文字幕| 久久这里有精品视频免费| 久久热精品热| 视频中文字幕在线观看| 国产淫语在线视频| 日韩精品有码人妻一区| 能在线免费观看的黄片| 岛国毛片在线播放| 超碰av人人做人人爽久久| 中文字幕久久专区| 亚洲国产精品专区欧美| 精品久久久久久久久av| 欧美成人一区二区免费高清观看| 在线播放无遮挡| 日韩强制内射视频| 国内精品一区二区在线观看| 一本久久精品| 亚洲av国产av综合av卡| 欧美日韩亚洲高清精品| 免费不卡的大黄色大毛片视频在线观看 | h日本视频在线播放| 亚洲综合精品二区| 亚洲精品亚洲一区二区| 内地一区二区视频在线| 亚洲熟妇中文字幕五十中出| 九九爱精品视频在线观看| 少妇被粗大猛烈的视频| 欧美3d第一页| 亚洲精品乱码久久久久久按摩| 一区二区三区乱码不卡18| 亚洲精品一二三| 日韩三级伦理在线观看| 中文字幕免费在线视频6| 日产精品乱码卡一卡2卡三| 亚洲欧美中文字幕日韩二区| 免费观看在线日韩| 一级毛片我不卡| 精品酒店卫生间| 91久久精品国产一区二区成人| 麻豆成人av视频| 听说在线观看完整版免费高清| 精品国内亚洲2022精品成人| 99久久中文字幕三级久久日本| 蜜桃久久精品国产亚洲av| 草草在线视频免费看| 一级av片app| 免费看a级黄色片| 最近2019中文字幕mv第一页| 免费无遮挡裸体视频| 99九九线精品视频在线观看视频| 久久久久久久亚洲中文字幕| 久久久久久久久大av| 亚洲图色成人| 又大又黄又爽视频免费| 中文字幕制服av| 日韩中字成人| 精品久久久久久成人av| 最新中文字幕久久久久| 亚洲国产成人一精品久久久| 美女国产视频在线观看| 国产午夜精品一二区理论片| 久久韩国三级中文字幕| 国产精品综合久久久久久久免费| 欧美 日韩 精品 国产| 97超视频在线观看视频| 久久久久免费精品人妻一区二区| 丰满少妇做爰视频| 日日啪夜夜撸| 高清在线视频一区二区三区| 色综合站精品国产| 99久久九九国产精品国产免费| 在线 av 中文字幕| 国产 一区精品| 真实男女啪啪啪动态图| 免费大片18禁| 亚洲电影在线观看av| 高清欧美精品videossex| 永久网站在线| 日本欧美国产在线视频| 亚洲av日韩在线播放| 国产老妇伦熟女老妇高清| 中文天堂在线官网| 国产人妻一区二区三区在| 欧美精品一区二区大全| 在现免费观看毛片| 永久网站在线| 麻豆成人午夜福利视频| 国产一级毛片七仙女欲春2| 婷婷色综合www| 欧美xxⅹ黑人| 亚洲最大成人手机在线| 永久免费av网站大全| 在线免费十八禁| 22中文网久久字幕| 午夜激情福利司机影院| 日本与韩国留学比较| 啦啦啦啦在线视频资源| 国产 一区精品| 伊人久久国产一区二区| 国产精品爽爽va在线观看网站| 伦理电影大哥的女人| 国产人妻一区二区三区在| 久久精品久久久久久噜噜老黄| 午夜免费观看性视频| 亚洲国产精品成人综合色| 1000部很黄的大片| 亚洲美女搞黄在线观看| 色综合站精品国产| 三级男女做爰猛烈吃奶摸视频| 少妇熟女欧美另类| 色吧在线观看| 日本熟妇午夜| 国产精品嫩草影院av在线观看| 国产 一区精品| 成人一区二区视频在线观看| 亚洲aⅴ乱码一区二区在线播放| 午夜福利高清视频| 别揉我奶头 嗯啊视频| or卡值多少钱| 亚洲第一区二区三区不卡| 国产精品伦人一区二区| 有码 亚洲区| 亚洲经典国产精华液单| 99热全是精品| 爱豆传媒免费全集在线观看| 国产成人freesex在线| 精品人妻熟女av久视频| 精品久久久久久电影网| 中国美白少妇内射xxxbb| 国产精品熟女久久久久浪| eeuss影院久久| 看黄色毛片网站| 性色avwww在线观看| 哪个播放器可以免费观看大片| 免费少妇av软件| 日日摸夜夜添夜夜爱| 汤姆久久久久久久影院中文字幕 | 久久热精品热| 亚洲国产精品国产精品| 一级二级三级毛片免费看| 亚洲欧美精品自产自拍| 免费黄网站久久成人精品| 狂野欧美激情性xxxx在线观看| 日本免费在线观看一区| 欧美 日韩 精品 国产| 亚洲精品国产成人久久av| 国产有黄有色有爽视频| 久久草成人影院| 成人特级av手机在线观看| 插逼视频在线观看| 91精品国产九色| 久久久久久国产a免费观看| 插逼视频在线观看| 99久国产av精品| 久久久久久久午夜电影| 免费观看av网站的网址| 成人亚洲精品av一区二区| 久久久精品欧美日韩精品| 国产成人精品久久久久久| av线在线观看网站|