• <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 ?

    Structure Preserving Algorithm for Fractional Order Mathematical Model of COVID-19

    2022-08-24 03:25:56ZafarIqbalMuhammadAzizurRehmanNaumanAhmedAliRazaMuhammadRafiqIlyasKhanandKottakkaranSooppyNisar
    Computers Materials&Continua 2022年5期

    Zafar Iqbal,Muhammad Aziz-ur Rehman,Nauman Ahmed,Ali Raza,Muhammad Rafiq Ilyas Khanand Kottakkaran Sooppy Nisar

    1Department of Mathematics,University of Management and Technology,Lahore,Pakistan

    2Department of Mathematics and Statistics,The University of Lahore,Lahore,Pakistan

    3Stochastic Analysis&Optimization Research Group,Department of Mathematics,Air University,Islamabad,44000,Pakistan

    4Department of Mathematics,National College of Business Administration and Economics,Lahore,Pakistan

    5Department of Mathematics,F(xiàn)aculty of Sciences,University of Central Punjab,Lahore,54500,Pakistan

    6Faculty of Mathematics and Statistics,Ton Duc Thang University,Ho Chi Minh City,72915,Vietnam

    7Department of Mathematics,College of Arts and Science at Wadi Aldawaser,Prince Sattam Bin Abdulaziz University,Alkharj,11991,Kingdom of Saudi Arabia

    Abstract:In this article,a brief biological structure and some basic properties of COVID-19 are described.A classical integer order model is modified and converted into a fractional order model with ξ as order of the fractional derivative.Moreover, a valued structure preserving the numerical design,coined as Grunwald–Letnikov non-standard finite difference scheme, is developed for the fractional COVID-19 model.Taking into account the importance of the positivity and boundedness of the state variables,some productive results have been proved to ensure these essential features.Stability of the model at a corona free and a corona existing equilibrium points is investigated on the basis of Eigen values.The Routh–Hurwitz criterion is applied for the local stability analysis.An appropriate example with fitted and estimated set of parametric values is presented for the simulations.Graphical solutions are displayed for the chosen values of ξ (fractional order of the derivatives).The role of quarantined policy is also determined gradually to highlight its significance and relevancy in controlling infectious diseases.In the end, outcomes of the study are presented.

    Keywords: Coronavirus pandemic model; deterministic ordinary differential equations; numerical methods; convergence analysis

    1 Introduction(COVID-19)

    Novel coronavirus is a spherical or pleomorphic shaped, particle having single stranded(positive sense) RNA (Ribonucleic Acid) linked to a nucleoprotein surrounded by a special type of protein.The outer surface of the coronavirus contains the projections of the club-shaped structure.The Classification of the coronaviruses depends upon the appearance of the outer surface (whether it is crown like or halo like), the replication mechanism and the distinct features related to the chemistry of the virus.In general, these viruses belong to OC43-like or 229Elike serotypes.Avian and mammalian species serve as hosts for them.Both types are similar with respect to morphology and chemical structure.Corona viruses present in human beings and animals are antigenically similar.These are capable of attacking on different types of tissues in animals.But, in human beings this family of viruses generally cause only the upper respiratory tract infection.This virus belons to the subclass Orthocoronavirinae, class Coronaviridae, order Nidovirales, realm Riboviria, kingdom Orthcornavirae and phylum Pisuviricota.The dimension of this virus varies from 26 to 32 kilobases, which is largest in the class of RNA viruses.They have distinct protruded club or clove shaped studs or spikes [1].Like other corona viruses, COVID-19 also contains protein in the form of spikes ejecting outside from the surface.These spikes cling with the host (human) cells then its genome bears a structural change and the viral membrane fuse with the host cell cytoplasm.After this step, the viral genes of the COVID-19 enter into the host cell for replication and multiplication of the viruses.Depending upon the protease of the host cell,cleavage reaction permits it to reach into the host cell by endocytosis or fusion.After entering into the host cell, the virus becomes uncovered and their genome attacks on the cell cytoplasm.The genome of the coronavirus works as a messenger and it is translated by the ribosomes of the host cells.These viruses are divided into four categories as alpha coronavirus, beta coronavirus, gamma coronavirus and delta coronavirus.The first two viruses infect the mammals while the last two viruses initially attack the birds.The genera and species of these viruses are described as follows:the species Alphacoronavirus 1, Human coronavirus 229E, Human coronavirus NL63, Miniopterus bat coronavirus 1, Miniopterus bat cor onavirus HKU8, Porcine epidemic diarrhea virus, Rhinolophus bat coronavirus HKU2 and Scotophilus bat coronavirus 512 belong to the Alpha coronavirus.While the species, Betacoronavirus 1(Bovine Coronavirus, Human coronavirus OC43),Hedgehog coronavirus 1, Human coronavirus HKU1, Middle East respiratory syndrome-related coronavirus, Murine coronavirus, Pipistrellus bat coronavirus HKU5, Rousetlus bat coronavirus HKU9, Severe acute respiratory syndrome- related coronavirus (SARS-Cov, SARS-Cov-2) and Tylonycteris bat coronavirus HKU4 belong to Beta coronavirus.Furthermore, the species Avian coronavirus and Beluga whale coronavirus SW1 are the members of the Gamma coronavirus.Lastly, the Bulbul coronavirus HKU11 and Porcine coronavirus HKU15 species are the family members of the Delta coronavirus.Coronaviruses are deleterious to health with high risk factor.Some of them have more than 30% mortality rate, for instance MERS-COV.But other are not so harmful like as common cold.All types of the coronaviruses can be the causative agent of cold with prime symptoms including fever, sore throat and swollen adenoids.Moreover,they can cause primary viral pneumonia or secondary bacterial pneumonia or bronchitis in the same way as that of pneumonia [2].The SARS-COV appeared in 2003, resulted in severe acute respiratory syndrome (SARS).It effected both the upper and lower respiratory tract due to an unmatched pathogenesis.There are six classes of human coronaviruses that are known so far,each specie is categorized into two types.There are seven types of human coronaviruses.Four coronaviruses which show mild symptoms are:Human coronavirus OC43 (HCOV-OC43),β-Cov,Human coronavirus HKU1 (HCOV-HKU1) andβ-Cov, Human coronavirus 229E (HCOV-229E),α-Cov, Human coronavirus NL63 (HCOV-NL63) andα-Cov.Three coronaviruses which show severe symptoms are middle east respiratory syndrome-related coronavirus (MERS-COV),β-Cov,Severe acute respiratory syndrome coronavirus (SARS-COV),β-Cov and Severe acute respiratory syndrome coronavirus-2 (SARS-COV-2) andβ-Cov.The HCOV-OC43, HCOV-HKU1, HCOV-229E and HCOV-NL63.They periodically produce the mild symptoms of the common cold in the population all around the year.The outburst of pneumonia in Wuhan, China declared a pandemic by WHO (World Health Organization).This is considered due to a novel type of coronavirus, provisionally named as 2019-nCov by WHO, later it was renamed as SARS-COV-2 by the international committee on Taxonomy of viruses.As of 24 June 2020, 476,911 deaths and more than 9,237691 confirmed cases of COVID-19 were recorded.The Wuhan breed is recognized as a new class of Beta coronavirus, which is genetically similar to SARS-COV.Since COVID-19 has a great resemblance with the bat coronavirus so it is suspected to be initiated from bats also [3].As there is no vaccination or treatment for this disease so it has become a challenge for the scientists, health workers and policy makers to control the spreading of the infection.However, the research community and scientists are making efforts to find the treatment, vaccine or factors that are helpful in slowing down the dynamics of the disease.As a matter of fact,the virus is new however the virus family is not new for the human beings.The humanity has already faced such types of viruses on different scales in the near past.Currently, this infection has frozen all types of academic, business, sports and many other routine activities which created many problems and difficulties for the human beings as well as for the society [4].The eradication of the COVID-19 is an uphill task for the relevant authorities.The modern world is fighting against the infectious diseases on the one hand and changing environmental conditions that are favorable for the emergence of the viral diseases on the other hand.The examination of dead bodies revealed that most of the patients were diagnosed with severe heart, lungs, diabetes and some other diseases.The Disease can be communicated easily through the nasal viral secretion that is transmitted directly or indirectly to the susceptible person.Generally, the symptoms of the infection are mild but in some cases painful death is also observed.The effective forecast for the disease dynamics is a prolific study matter regarding epidemiology, mathematical modeling and simulations.There exist many classes of mathematical models that depend upon the assumptions imposed on the process of dynamics.For instance, SIS, SIR, SEIR, SEIQR and many other compartmental models are used to formulate nonlinear incidence rates and double epidemic hypothesis [5].In these types of modelsS,E,I,QandRdescribe the susceptible, exposed,infected, quarantined and recovered individuals.These cellular or compartmental systems are used for adjusted incidence rate and imperfect vaccinations [6].In these models, it is assumed that the susceptible individuals are aware of the infection’s presence [7].It is worth mentioning that most of the existing deterministic models rest on the ordinary differential equations which imply the assumption of constant diffusion in the domain population.On the other hand, the use of partial differential equations model highlights the non-constant diffusion of the infection [8–10].Although, many studies have been carried out for the providing the deeper insight into the disease dynamics.Chen et al.[11] considered a model with four compartments to investigate the dynamics of the novel infection.Shim and co-authors [12] addressed the questions relating to the effect analysis of the disease.Naveed et al.[13] developed a mathematical model to analyze the virus communication among the population and calculated the basic reproductive number.Many other approaches on the infection propagation may be seen in the sequel.Fractional calculus (FC) is the extension of the integer order calculus.On the basis of FC, the researchers are trying to understand the real world phenomenon of the infectious diseases in a more comprehensive way.They are developing mathematical models with derivatives of non-integer order.By the usage of these types of fractional operators many fruitful studies have been made in the recent scenario.New features and properties of the FC have fascinated the researchers, engineers and scientists to model the problems in the frame work of fractional calculus.The development of new non integer order operators have brought a number of essential features of many physical problems in to the lime light.The history of non-integer order calculus starts with a question posed by Leibniz in 1965.There is a long list of existing fractional order differential operators depending upon the nature of Kernels.Caputo, Riemann–Liouvelle and Katugampola fractional differential operators are developed by using singular kernels.While the fractional operators without singular kernels are of two types.In first type, exponential kernel is used for instance, Caputo–Fabrizio fractional differential operator.Whereas Mittag-Leffler kernel is used in the second type of operators e.g.,Atangana–Baleanu fractional operator in Caputo sense.Due to the salient features of the FC, it is used to model a wide range of physical and dynamical problems in various fields of physical sciences, mathematics, life sciences and engineering [14].Positivity, boundedness and the stability of the equilibrium points for the fractional order physical problems is a challenging task for the scientists and mathematicians.Some researchers are working in this line, but a lot of work is still to be done.Memory effect and hereditary properties are among the celebrated features of the non-integer order derivatives that are helpful in describing the disease dynamics.Taking in to account the memory effect, the fractional order models provide all the important informations from the past that are helpful in forecasting the dynamics of the infection more accurately and comprehensively.Saeedian and co-authors [15] designed a fractional SIR infectious disease model with memory effect and investigated the infection spread in the population.Ucar et al.[16] studied the dynamics of a fractional order smoking model.In the current study, an integer order model is considered initially then we switched on the non-integer order model by using Caputo differential operator.The fractional order model of COVID-19 can describe the complex dynamics of the physical phenomena with a more realistic approach.

    2 Preliminaries

    In this section, we will present some fundamental definitions of non-integer order derivatives,their key properties and notations used in this article.

    2.1 Non Integer Order Derivatives

    Fractional order derivatives have been defined by many researchers in a number of ways according to the nature of the kernel used therein.Some basic fractional order operators are defined in this section.Firstly, the Riemann–Liouville non integer order derivative of order 0<ξ≤1 is defined as,

    wherek=[ξ]+1,k-1<ξ≤k,is thekthorder derivative and Γ (.) is the extensively used gamma function presented by Euler Fractional order derivative of a function is not defined in a unique way, as in the classical calculus an integer order derivative of a function is defined.Fractional derivative is a generic name given to a class of differential operators used to find the non-integer order derivative of a function.These operators are defined by a number of researchers in different ways.Some of them are Liouville, Riemann–Liouville, Caputo, Fabrizio,Grunwald–Letnikov and many more can be studied in the sequel.Among many other questions,it is important that which differential operator best suits for the underlying model.A more generic fractional operator should be selected for the purpose.The Caputo fractional operator is defined as,

    The importance of this operator, when applied to solve a system of fractional differential equations is that it can be associated with initial conditions of classical order, which results in an initial value problem in the desired form as,

    A very useful definition relating to this article by using the classical finite differences on a uniform mesh partionised in [0,t] is described as follows, consider that) observes the particular smoothness constraints in every interval (0,t),t≤Twith mesh points as

    0=τ0<τ1<...<τn+1=t=(n+1)h

    wherehis defined as the difference ofτn+1andτn.By using the classical finite difference symbols,we have

    This expression is derived from the famous Euler method.Consider the fractional differential equation

    Now, by applying the G-L scheme on a uniform mesh, we obtain the following expression

    whereeandγnsatisfy the following relationsand

    whereμ,k∈N ∪{0}.

    Furthermore,eiandγiobserve the relations as stated in the Lemma 1.

    Lemma 1 [17]:Let 0<ξ <1, then the coefficients expressed byare positive and obey the relation

    Also the following two relations are satisfied

    In this section, we present the GL-NSFD hybrid scheme is formulated by combining the GL scheme for numerical approximation of the fractional order derivatives and NSFD scheme constructed by using the standard rules designed by Mickens.More details can be seen in [18].The system of equations for COVID-19 is described as follows:

    2.2 GL-NSFD Scheme

    In this portion, we will construct the proposed scheme.The discretization of fractional derivativeC0) is given as,

    The above formula is used on the left hand side of Eq.(6) to get the following expression

    After some simplifications, we have the final form as,

    Similar procedure is adopted for the remaining compartments and we have the final forms as,

    2.3 Positivity of the Solution

    In this portion, positivity of the solution will be investigated.Positivity is an important feature of the compartmental models.Since, the state variables in these type of models describe the size of the population that cannot be negative.So, positivity is the basic requirement of the solutions at every moment of time.Following result is helpful in this regard.

    Theorem:Assume that all the unknowns and parameters arose in the model are non-negative i.e.,So,Eo,Io,QoandRoare positive.Alsoλξ,μξ,kξ,rξ,αξ,d1ξ,d2ξ,β1ξ,β2ξ,q1ξ,q2ξ,qξandφ(h)ξall are ≥0.ThenSn,En,In,QnandRnall are ≥0 ?n∈Z+.

    Proof:Taking in to account the Eqs.(11) to (15) forn=0, we have

    From the restrictions imposed on the state variables and parameters, it is evident thatS1≥0.Similarly,E1≥0,I1≥0,Q1≥0and R1≥0.Continuing in the same way and by straight forward calculations, it is easy to conclude thatSn+1≥0,En+1≥0,In+1≥0,Qn+1≥0and Rn+1≥0.i.e.,

    2.4 Boundedness

    Since the state variables in the model represent the subpopulation of a certain compartment.So the sum of values of all the state variables must be less than or equal to the total population or equivalently the sum of solutions at any time must be bounded.The following result is helpful in this regard.

    Theorem:LetS0,E0,I0,Q0andR0are all finite quantities andS0+E0+I0+Q0+R0=N0.Moreover, all the parameters involved in the model are positive.i.e.,λξ,μξ,qξ,qξ1,qξ2,αξ,γξ,kξandφ(h)ξare positive, then there is a constantBn+1such thatSn+1≤Bn+1,En+1≤Bn+1,In+1≤Bn+1,Qn+1≤Bn+1,Rn+1≤Bn+1.

    Proof:Considering the Eqs.(11) to (15), we have

    By applying the principle of mathematical induction,

    forn=0, we have

    In the same wayE1≤B1,I1≤B1,Q1≤B1andR1≤B1.

    Now, we calculate the expression (16) forn=1, and obtain the following relations,

    The above inequalities help us to reach at

    Now, let

    Sm≤Bm,Em≤Bm,Im≤Bm,Qm≤Bm and Rm≤Bm

    For somem∈Z+.

    where,

    Now forn∈Z+, we arrive at,

    In the same fashion a adopted before, we conclude that

    So, the given expression is true for all positive values ofn.

    Hence, the solutions are bounded ?n∈Z+.

    2.5 Stability of the Model

    In this portion, we will investigate the stability of the model at both the points of equilibria i.e., at a corona free equilibrium point and a corona existing equilibrium point.

    The corona free equilibrium state of the model is given as

    Theorem:The corona free equilibriumC1=(So,Eo,Io,Qo,Ro) =of the model is locally asymptotically stable if Ro<1, otherwise unstable for Ro>1.

    Proof:The corona-free equilibriumC1=(So,Eo,Io,Qo,Ro) =is locally asymptotically stable (LAS) if all the Eigenvaluesλi <0,i=1,2,3,4,5 with condition |arg(λi)|>For the Eigen values, the Jacobean matrix atis given as follows:

    Notice that the two Eigen values are repeated asλ1=-μξ <0,λ2=-μξ <0 and third Eigen value is

    By using the Routh–Hurwitz Criterion of 2ndorder polynomial as,

    Hence, all Eigenvalues are negative and by Routh-Hurwitz criteria the given equilibrium pointC1is locally asymptotically stable.

    Theorem:The corona existence equilibriumof the model is locally asymptotically stable if Ro>1, otherwise unstable for Ro<1.

    Proof:The corona existence equilibriumis locally asymptotically stable (LAS) if all the Eigenvaluesλi <0,i=1,2,3,4,5 with condition

    For the eigen values, the Jacobean matrix atis given as follows:

    Notice that, the Eigen values areλ1=-μξ <0 andλ2=-(qξ+μξ+0

    By using the Routh–Hurwitz Criterion of 3rdorder polynomial, we get the following expression:

    and

    Thus, we have concluded that all Eigenvalues are negative and by Routh Hurwitz criteria,the given equilibrium pointC2is locally asymptotically stable.Here, we will present a suitable numerical example and graphical solutions of the state variables involved in the model.This whole stuff is presented with the aid of computer simulations.

    3 Numerical Example and Simulations

    In this portion, an example of the fractional order COVID-19 model is provided.The parametric values are mentioned in Tab.1.Also, non-negative initial conditions are considered.

    Computer aided graphs are submitted to support our assertions.These sketches support the fact that proposed numerical device is a structure preserving tool for solving the nonlinear fractional systems.The device encounters the positivity, stability and boundedness of the solutions.All the graphs in Fig.1 reveals that all the subpopulations converge at the virus free equilibrium point (with different values ofξ).E(t)→0,I(t)→0,Q(t)→0 andR(t)→0, when the population is infection free and the values of the parameters are chosen suitably as listed in Tab.1.The graphs in Fig.1 part (a) illustrate that all the curved trajectories representing the susceptible populace growing with timetapproach towards the disease-free value of the susceptible individualswhich is one in this case.Each trajectory is drawn against a certain value ofξ(the order of the fractional derivative) as mentioned in the figure.Moreover,the rate of the convergence towards the VFE of each trajectory is different, depending upon the value ofξ.Similarly, the other sketches in Fig.1 part (b)–part (e) provide the strong evidence for our declaration about the proposed numerical design.The Fig.2 exhibits the simulation results of our proposed scheme endemic equilibrium.The graphs (a), (b), (c), (d) and (e) in Fig.2 provide the graphical solutions toS,E,I,QandRfor some selected values ofξ, whereξis the fractional order of the derivative.The trajectories in Fig.2 part (a) converge at the virus restricted state i.e., endemic state for different values ofξ, while the other parametric values are kept same as mentioned in Tab.1.Each curved line in the graph Fig.2 part (a) attains the equilibrium state ofS1(t) which is represented as,

    with a certain rate of convergence according to the value ofξ.Similarly, the other curved trajectories in Fig.2 part (b)–part (e) depict that they attain the virus endemic equilibrium state for various values ofξ.The endemic equilibrium is expressed as(S1(t),E1(t),I1(t),Q1(t),R1(t))and the value of each state variable is stated earlier in the section of the stability of the model.The Fig.3 describes the quarantine approach to control the infection in the populace.The values of the quarantine factorqconsidered in this figure are asq1=0.1,q2=0.3,q3=0.5 andq4=0.7 while theξ=0.9 is fixed.All the four curved representations in Fig.3 unveil the key fact that by increasing the quarantine or isolation approach, the flock of infected individuals can be minimized to a certain level.In this area, we will submit the fruitful conclusion about the current study and some future directions will be pointed out.

    Table 1:Fitted and estimated parameters values for coronavirus model

    Figure 1:The simulations results for all the subpopulations using proposed method at disease free equilibrium with the variation of fractional order ξ

    Figure 2:The simulations results for all the subpopulations using proposed method at endemic free equilibrium with the variation of fractional order ξ

    Figure 3:The effect of quarantined strategy on infected population by increasing the values of q1 for ξ=0.9

    4 Conclusion

    In this study, some biological and physical features of the novel corona virus-2019 are described.A classicalSIEQRmodel is converted to fractional order compartmental model withξas order of the fractional derivatives.The GL-NSFD scheme is proposed to study the propagation of the COVID-19 along with some leading properties of the system.Moreover, the numerical study is made to ensure the pre-assumed results about the numerical design.The equilibrium points of the system are also described to detect the local stability of the model.The decisive role ofR0(reproductive number) in describing the stability of the system is also discussed.Positivity and boundedness of the numerical design is also investigated to exhibit the productiveness of the scheme.The computer-aided graphs are presented via computer simulations.These solutions coincide with the exact equilibrium points for different values ofξ.As, the proposed scheme preserves the structure of the system.So, it can be used successfully to solve many other nonlinear physical systems.Moreover, this tool may be used to solve delay models, advection and diffusion reaction models in future.

    Funding Statement:The authors received no specific funding for this study.

    Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.

    国产一区二区激情短视频| 免费看a级黄色片| 久久婷婷人人爽人人干人人爱| 99在线人妻在线中文字幕| 男女视频在线观看网站免费| 校园春色视频在线观看| 人人妻人人澡人人爽人人夜夜 | 欧美色欧美亚洲另类二区| 亚洲国产精品久久男人天堂| 国产精品国产高清国产av| 亚洲激情五月婷婷啪啪| 丰满乱子伦码专区| 久久久久网色| 免费人成在线观看视频色| 欧美日韩综合久久久久久| 亚洲成人精品中文字幕电影| 我要搜黄色片| 国产单亲对白刺激| 两性午夜刺激爽爽歪歪视频在线观看| 欧美日韩在线观看h| 久久久久久国产a免费观看| 免费观看的影片在线观看| 老司机福利观看| 91精品一卡2卡3卡4卡| 国产高潮美女av| 免费电影在线观看免费观看| 春色校园在线视频观看| 久久久久久久亚洲中文字幕| 国产极品精品免费视频能看的| 久久精品夜夜夜夜夜久久蜜豆| 一级二级三级毛片免费看| 国产精品一区www在线观看| 国产在线精品亚洲第一网站| 精品一区二区三区人妻视频| 亚洲熟妇中文字幕五十中出| 精品一区二区三区视频在线| 日韩一区二区视频免费看| 亚洲第一区二区三区不卡| 日韩制服骚丝袜av| 日韩高清综合在线| 日本一本二区三区精品| 身体一侧抽搐| 亚洲欧洲日产国产| 在线免费观看不下载黄p国产| 亚洲av成人精品一区久久| 在线观看66精品国产| 美女大奶头视频| 99九九线精品视频在线观看视频| 免费看av在线观看网站| 亚洲在久久综合| av天堂在线播放| 久久久久久国产a免费观看| 成人av在线播放网站| 久久久精品大字幕| 国产精品综合久久久久久久免费| 亚洲精品国产成人久久av| 成人av在线播放网站| 免费人成在线观看视频色| 1000部很黄的大片| 丰满乱子伦码专区| 国产精品伦人一区二区| 国产精品1区2区在线观看.| 黄片wwwwww| 69人妻影院| 国产精品伦人一区二区| 欧美最新免费一区二区三区| 一本—道久久a久久精品蜜桃钙片 精品乱码久久久久久99久播 | a级一级毛片免费在线观看| 身体一侧抽搐| 婷婷精品国产亚洲av| 日本色播在线视频| 亚洲七黄色美女视频| 又爽又黄无遮挡网站| 国产成人一区二区在线| 久久99精品国语久久久| 国产黄片视频在线免费观看| 亚洲不卡免费看| 人妻制服诱惑在线中文字幕| 国产蜜桃级精品一区二区三区| 夜夜爽天天搞| 久久久a久久爽久久v久久| 国产亚洲精品久久久久久毛片| 男女做爰动态图高潮gif福利片| 免费人成视频x8x8入口观看| 又爽又黄a免费视频| 国产高清不卡午夜福利| 国产成人精品婷婷| 国产黄色视频一区二区在线观看 | 日韩一本色道免费dvd| 精品久久国产蜜桃| 色视频www国产| 久久久久免费精品人妻一区二区| 国产一区二区在线观看日韩| 99riav亚洲国产免费| 午夜久久久久精精品| 99视频精品全部免费 在线| 91麻豆精品激情在线观看国产| 97热精品久久久久久| 中文在线观看免费www的网站| 人体艺术视频欧美日本| 国产成人福利小说| 青春草国产在线视频 | 精品久久国产蜜桃| 国产v大片淫在线免费观看| 久久鲁丝午夜福利片| avwww免费| 嫩草影院入口| 欧美最黄视频在线播放免费| 自拍偷自拍亚洲精品老妇| 国产精品女同一区二区软件| 久久精品国产亚洲av天美| av天堂中文字幕网| 麻豆国产97在线/欧美| 国产成人aa在线观看| 久久久欧美国产精品| 最好的美女福利视频网| 少妇猛男粗大的猛烈进出视频 | 欧美一区二区国产精品久久精品| 免费看日本二区| 欧美激情久久久久久爽电影| 国产高清视频在线观看网站| 成年免费大片在线观看| 波野结衣二区三区在线| 亚洲内射少妇av| 少妇人妻精品综合一区二区 | 麻豆国产97在线/欧美| 少妇熟女欧美另类| 国语自产精品视频在线第100页| 日韩制服骚丝袜av| 午夜激情欧美在线| 亚洲色图av天堂| 日日撸夜夜添| 国产中年淑女户外野战色| 国产伦理片在线播放av一区 | 韩国av在线不卡| av在线亚洲专区| 少妇丰满av| 成人无遮挡网站| 亚洲欧美日韩高清在线视频| 久久99蜜桃精品久久| 日日干狠狠操夜夜爽| 欧洲精品卡2卡3卡4卡5卡区| 日韩欧美 国产精品| 淫秽高清视频在线观看| 精品人妻一区二区三区麻豆| 亚洲精品日韩在线中文字幕 | 在线天堂最新版资源| 九色成人免费人妻av| 中文字幕精品亚洲无线码一区| 亚洲无线观看免费| av免费在线看不卡| 美女 人体艺术 gogo| 久久久久久久久久成人| 亚洲18禁久久av| 精品无人区乱码1区二区| www.av在线官网国产| 国产精品一及| 最近的中文字幕免费完整| 亚洲在久久综合| 91麻豆精品激情在线观看国产| 国产人妻一区二区三区在| 色5月婷婷丁香| 国产高清激情床上av| 久久久久网色| .国产精品久久| 国产成人精品婷婷| 亚洲av电影不卡..在线观看| 精品人妻偷拍中文字幕| 六月丁香七月| 在线播放无遮挡| 亚洲欧洲国产日韩| 久久人人爽人人片av| h日本视频在线播放| 国产高清视频在线观看网站| 亚洲久久久久久中文字幕| 国产精品一区二区性色av| 欧美性猛交╳xxx乱大交人| 免费黄网站久久成人精品| 亚洲欧美精品自产自拍| 成人午夜高清在线视频| 国产色婷婷99| 一区福利在线观看| 1024手机看黄色片| 特级一级黄色大片| 一边亲一边摸免费视频| 久久精品国产亚洲av天美| 日本免费一区二区三区高清不卡| 中国美女看黄片| 国产在线男女| 亚洲欧洲国产日韩| 啦啦啦啦在线视频资源| 两个人的视频大全免费| 午夜亚洲福利在线播放| 真实男女啪啪啪动态图| 男女下面进入的视频免费午夜| 亚洲国产精品成人久久小说 | 日本一二三区视频观看| 大香蕉久久网| 狂野欧美激情性xxxx在线观看| 99热这里只有精品一区| 午夜福利在线在线| 99在线视频只有这里精品首页| 日韩成人伦理影院| 亚洲国产欧洲综合997久久,| 内地一区二区视频在线| 国产av在哪里看| 午夜福利在线观看吧| 国产激情偷乱视频一区二区| 少妇被粗大猛烈的视频| 一级黄片播放器| 国产精品1区2区在线观看.| 免费在线观看成人毛片| 人体艺术视频欧美日本| 18禁在线播放成人免费| 丝袜喷水一区| 亚洲国产精品sss在线观看| 国产亚洲av嫩草精品影院| 国产伦理片在线播放av一区 | 丝袜美腿在线中文| 青春草视频在线免费观看| 美女脱内裤让男人舔精品视频 | 又黄又爽又刺激的免费视频.| 久久99精品国语久久久| 精品一区二区三区视频在线| 伦理电影大哥的女人| 变态另类丝袜制服| 国产在视频线在精品| 国产三级在线视频| 精品一区二区三区视频在线| 伦理电影大哥的女人| a级毛片免费高清观看在线播放| 欧美高清性xxxxhd video| 国内少妇人妻偷人精品xxx网站| 欧美激情在线99| 国产精品电影一区二区三区| 99热只有精品国产| 成熟少妇高潮喷水视频| 免费看a级黄色片| 99久国产av精品国产电影| 熟妇人妻久久中文字幕3abv| 亚洲国产欧美人成| av黄色大香蕉| 五月伊人婷婷丁香| 日韩一区二区三区影片| 边亲边吃奶的免费视频| 天天躁日日操中文字幕| 不卡视频在线观看欧美| 狂野欧美激情性xxxx在线观看| 老师上课跳d突然被开到最大视频| 久久99蜜桃精品久久| 好男人在线观看高清免费视频| 成人高潮视频无遮挡免费网站| 哪里可以看免费的av片| 国产大屁股一区二区在线视频| 亚洲乱码一区二区免费版| 精品欧美国产一区二区三| 国产精品.久久久| 丰满乱子伦码专区| 国产精品电影一区二区三区| av又黄又爽大尺度在线免费看 | 国产在视频线在精品| 精品人妻视频免费看| 精品欧美国产一区二区三| 看十八女毛片水多多多| 一夜夜www| 美女被艹到高潮喷水动态| 婷婷六月久久综合丁香| 午夜精品国产一区二区电影 | 乱系列少妇在线播放| 亚洲av二区三区四区| 中国国产av一级| 国产日韩欧美在线精品| 亚洲欧美成人综合另类久久久 | 免费看光身美女| 久久久久久久午夜电影| 成人综合一区亚洲| 我的女老师完整版在线观看| 中出人妻视频一区二区| 欧美日韩在线观看h| 乱系列少妇在线播放| 亚洲中文字幕日韩| 国产成人影院久久av| 日本熟妇午夜| 嫩草影院新地址| 日韩欧美在线乱码| 亚洲国产欧美人成| 可以在线观看毛片的网站| 精品一区二区免费观看| 高清毛片免费看| 美女大奶头视频| 日韩一区二区三区影片| 神马国产精品三级电影在线观看| 午夜爱爱视频在线播放| 国产极品精品免费视频能看的| 亚洲精品影视一区二区三区av| 国产伦理片在线播放av一区 | 又黄又爽又刺激的免费视频.| 最新中文字幕久久久久| 亚洲欧美精品综合久久99| av天堂在线播放| 国产精品一区www在线观看| av.在线天堂| 日韩av在线大香蕉| 色综合色国产| 亚洲av熟女| 国产伦理片在线播放av一区 | 成人美女网站在线观看视频| 色吧在线观看| 中文字幕人妻熟人妻熟丝袜美| 久久精品久久久久久噜噜老黄 | 伊人久久精品亚洲午夜| 一级毛片久久久久久久久女| 日韩成人伦理影院| 久久综合国产亚洲精品| 亚洲国产日韩欧美精品在线观看| 黄色视频,在线免费观看| 国产av麻豆久久久久久久| av又黄又爽大尺度在线免费看 | 亚洲精华国产精华液的使用体验 | 不卡视频在线观看欧美| 狠狠狠狠99中文字幕| 国产伦在线观看视频一区| 亚洲国产欧美人成| 中文字幕av在线有码专区| 床上黄色一级片| 最近最新中文字幕大全电影3| 久久精品国产亚洲av香蕉五月| 一级av片app| 2022亚洲国产成人精品| 少妇熟女欧美另类| 日本与韩国留学比较| 国产成人aa在线观看| 国产免费一级a男人的天堂| 日韩高清综合在线| 欧美三级亚洲精品| 亚洲四区av| 99视频精品全部免费 在线| 亚洲精品影视一区二区三区av| 国产毛片a区久久久久| 亚洲av一区综合| 中文资源天堂在线| 99九九线精品视频在线观看视频| 国产伦在线观看视频一区| 国产精品精品国产色婷婷| 在线国产一区二区在线| 噜噜噜噜噜久久久久久91| 中文字幕av在线有码专区| 欧美一区二区国产精品久久精品| 日韩欧美在线乱码| 欧美性感艳星| 久久久久久久久久成人| 少妇裸体淫交视频免费看高清| 免费av观看视频| 国产精品日韩av在线免费观看| 中文字幕久久专区| 97在线视频观看| 天堂av国产一区二区熟女人妻| 欧美精品国产亚洲| 精品国内亚洲2022精品成人| 国产黄a三级三级三级人| 啦啦啦观看免费观看视频高清| 亚洲国产欧美在线一区| 免费观看精品视频网站| 国产黄a三级三级三级人| 99精品在免费线老司机午夜| 亚洲欧洲日产国产| 91精品一卡2卡3卡4卡| 青春草亚洲视频在线观看| 午夜免费男女啪啪视频观看| 最近视频中文字幕2019在线8| 欧美变态另类bdsm刘玥| 久久综合国产亚洲精品| 久久鲁丝午夜福利片| 一区二区三区四区激情视频 | 久久久久网色| 亚洲精品自拍成人| 性欧美人与动物交配| 亚洲国产精品sss在线观看| 蜜桃亚洲精品一区二区三区| 国产不卡一卡二| 老司机福利观看| 午夜a级毛片| 蜜桃久久精品国产亚洲av| 国产爱豆传媒在线观看| 国产麻豆成人av免费视频| 亚洲精品日韩在线中文字幕 | 国产蜜桃级精品一区二区三区| 国产精品永久免费网站| 久久中文看片网| 国产日韩欧美在线精品| 一级毛片电影观看 | 成人亚洲欧美一区二区av| 99riav亚洲国产免费| av.在线天堂| 久久午夜亚洲精品久久| 午夜福利高清视频| 国产麻豆成人av免费视频| 在线观看av片永久免费下载| 麻豆乱淫一区二区| 国产真实伦视频高清在线观看| 又爽又黄无遮挡网站| 欧美bdsm另类| av黄色大香蕉| 99在线人妻在线中文字幕| 午夜视频国产福利| 欧美一区二区亚洲| 国产黄片美女视频| 天天躁日日操中文字幕| 永久网站在线| 狂野欧美白嫩少妇大欣赏| 最好的美女福利视频网| 校园春色视频在线观看| 亚洲一级一片aⅴ在线观看| 国产精品麻豆人妻色哟哟久久 | 久久久久久久久久黄片| 欧美潮喷喷水| 听说在线观看完整版免费高清| 色5月婷婷丁香| 日日干狠狠操夜夜爽| h日本视频在线播放| 亚洲精品国产av成人精品| 午夜福利在线观看免费完整高清在 | 一进一出抽搐动态| 深夜a级毛片| 内地一区二区视频在线| 五月玫瑰六月丁香| 97超碰精品成人国产| 久久综合国产亚洲精品| 一个人看的www免费观看视频| 中国美女看黄片| 亚洲中文字幕一区二区三区有码在线看| 亚洲欧美中文字幕日韩二区| 国产av在哪里看| 亚洲欧美精品综合久久99| 久久婷婷人人爽人人干人人爱| 又黄又爽又刺激的免费视频.| 精品久久久久久久久av| 久久久久免费精品人妻一区二区| 一个人观看的视频www高清免费观看| 变态另类成人亚洲欧美熟女| 日韩亚洲欧美综合| 国产三级在线视频| 国产精品永久免费网站| 日本色播在线视频| 成年av动漫网址| 亚洲av男天堂| 麻豆一二三区av精品| 久久综合国产亚洲精品| 黄色日韩在线| 老司机影院成人| 91午夜精品亚洲一区二区三区| 欧美丝袜亚洲另类| 亚洲欧美日韩无卡精品| 国产乱人偷精品视频| 国产成年人精品一区二区| 美女被艹到高潮喷水动态| 欧美性感艳星| 在线a可以看的网站| 毛片女人毛片| 此物有八面人人有两片| 晚上一个人看的免费电影| 亚洲av男天堂| 国产在线男女| 日韩av在线大香蕉| 天堂中文最新版在线下载 | 成年女人看的毛片在线观看| 三级毛片av免费| 精华霜和精华液先用哪个| 一本精品99久久精品77| 国产一区二区亚洲精品在线观看| 精品少妇黑人巨大在线播放 | 亚洲天堂国产精品一区在线| 成年女人永久免费观看视频| 人妻制服诱惑在线中文字幕| 午夜免费激情av| 日本免费一区二区三区高清不卡| 亚洲欧美精品综合久久99| av免费在线看不卡| 97超碰精品成人国产| 免费人成视频x8x8入口观看| 我的老师免费观看完整版| 免费在线观看成人毛片| 久久99热这里只有精品18| 啦啦啦韩国在线观看视频| 少妇裸体淫交视频免费看高清| 在线a可以看的网站| 亚洲成人久久爱视频| 日本色播在线视频| 亚洲在线自拍视频| 免费观看人在逋| 美女脱内裤让男人舔精品视频 | 不卡一级毛片| 久久久国产成人精品二区| 美女国产视频在线观看| 久久这里有精品视频免费| 日本免费a在线| 成人亚洲欧美一区二区av| 欧美xxxx性猛交bbbb| av免费观看日本| 精品久久久久久久久久免费视频| 嫩草影院精品99| 国产精品一区二区三区四区免费观看| 毛片女人毛片| 精品人妻视频免费看| 国产 一区 欧美 日韩| 欧美一区二区国产精品久久精品| 白带黄色成豆腐渣| av女优亚洲男人天堂| 精华霜和精华液先用哪个| 国产成人精品一,二区 | 日本黄色视频三级网站网址| 一级黄片播放器| 久久99热这里只有精品18| 亚洲av中文字字幕乱码综合| av视频在线观看入口| 三级国产精品欧美在线观看| 嘟嘟电影网在线观看| 国产一区亚洲一区在线观看| 久久精品国产亚洲av涩爱 | 波多野结衣高清无吗| 欧美极品一区二区三区四区| 国产成人福利小说| 成年版毛片免费区| 欧美xxxx性猛交bbbb| 亚洲美女视频黄频| 中文字幕av在线有码专区| 男女视频在线观看网站免费| 男女啪啪激烈高潮av片| 成年av动漫网址| 久久精品综合一区二区三区| 波野结衣二区三区在线| 99热6这里只有精品| 久久久成人免费电影| 免费av观看视频| 最后的刺客免费高清国语| 亚洲精品日韩在线中文字幕 | 欧美性猛交黑人性爽| 伦理电影大哥的女人| 亚洲成人久久爱视频| 亚洲美女视频黄频| 日韩欧美在线乱码| 一本久久中文字幕| 国产精品福利在线免费观看| 最近最新中文字幕大全电影3| 深爱激情五月婷婷| 国内精品美女久久久久久| 特大巨黑吊av在线直播| 亚洲精品粉嫩美女一区| 色综合色国产| 国产中年淑女户外野战色| 久久久久久伊人网av| 日韩成人伦理影院| 给我免费播放毛片高清在线观看| 村上凉子中文字幕在线| 欧美一级a爱片免费观看看| 亚洲七黄色美女视频| 一进一出抽搐gif免费好疼| 国产精品久久电影中文字幕| 国国产精品蜜臀av免费| 小说图片视频综合网站| 欧美3d第一页| 99在线人妻在线中文字幕| 成年版毛片免费区| 久久人人爽人人片av| 久久中文看片网| 久久精品国产亚洲av涩爱 | 18禁黄网站禁片免费观看直播| 天堂网av新在线| 日日干狠狠操夜夜爽| 在线观看av片永久免费下载| 日本三级黄在线观看| 我要看日韩黄色一级片| 亚洲欧美精品自产自拍| 精品99又大又爽又粗少妇毛片| 久久久久久大精品| av.在线天堂| 中文字幕制服av| 国国产精品蜜臀av免费| 插逼视频在线观看| 99久国产av精品| 国产黄a三级三级三级人| 国产毛片a区久久久久| 欧美xxxx性猛交bbbb| 免费电影在线观看免费观看| 国产精品99久久久久久久久| 日本欧美国产在线视频| 婷婷色综合大香蕉| 午夜免费男女啪啪视频观看| 春色校园在线视频观看| 26uuu在线亚洲综合色| kizo精华| 国产精华一区二区三区| 欧美bdsm另类| 国产亚洲精品av在线| 亚洲欧美日韩高清专用| 草草在线视频免费看| 亚洲性久久影院| 欧美色视频一区免费| 日韩制服骚丝袜av| 熟妇人妻久久中文字幕3abv| 能在线免费观看的黄片| 亚洲av男天堂| 麻豆成人av视频| 久久午夜亚洲精品久久| 久久这里只有精品中国| 麻豆成人av视频| 亚洲成人久久爱视频| 一本—道久久a久久精品蜜桃钙片 精品乱码久久久久久99久播 | 一级黄色大片毛片| 好男人视频免费观看在线| 国产午夜福利久久久久久| 精品人妻熟女av久视频| 精品久久久久久久久亚洲| 亚洲在线观看片| 免费大片18禁|