HIV is a retrovirus that infects and impairs the cells and functions of the immune system. It has caused a great challenge to global public health systems and leads to Acquired Immunodeficiency Syndrome (AIDS), if not...HIV is a retrovirus that infects and impairs the cells and functions of the immune system. It has caused a great challenge to global public health systems and leads to Acquired Immunodeficiency Syndrome (AIDS), if not attended to in good time. Antiretroviral therapy is used for managing the virus in a patient’s lifetime. Some of the symptoms of the disease include lean body mass and many opportunistic infections. This study has developed a SIAT mathematical model to investigate the impact of inconsistency in treatment of the disease. The arising non-linear differential equations have been obtained and analyzed. The DFE and its stability have been obtained and the study found that it is locally asymptotically stable when the basic reproduction number is less than unity. The endemic equilibrium has been obtained and found to be globally asymptotically stable when the basic reproduction number is greater than unity. Numerical solutions have been obtained and analyzed to give the trends in the spread dynamics. The inconsistency in treatment uptake has been analyzed through the numerical solutions. The study found that when the treatment rate of those infected increases, it leads to an increase in treatment population, which slows down the spread of HIV and vice versa. An increase in the rate of treatment of those with AIDS leads to a decrease in the AIDS population, the reverse happens when this rate decreases. The study recommends that the community involvement in advocating for consistent treatment of HIV to curb the spread of the disease.展开更多
This study directs the discussion of HIV disease with a novel kind of complex dynamical generalized and piecewise operator in the sense of classical and Atangana Baleanu(AB)derivatives having arbitrary order.The HIV i...This study directs the discussion of HIV disease with a novel kind of complex dynamical generalized and piecewise operator in the sense of classical and Atangana Baleanu(AB)derivatives having arbitrary order.The HIV infection model has a susceptible class,a recovered class,along with a case of infection divided into three sub-different levels or categories and the recovered class.The total time interval is converted into two,which are further investigated for ordinary and fractional order operators of the AB derivative,respectively.The proposed model is tested separately for unique solutions and existence on bi intervals.The numerical solution of the proposed model is treated by the piece-wise numerical iterative scheme of Newtons Polynomial.The proposed method is established for piece-wise derivatives under natural order and non-singular Mittag-Leffler Law.The cross-over or bending characteristics in the dynamical system of HIV are easily examined by the aspect of this research having a memory effect for controlling the said disease.This study uses the neural network(NN)technique to obtain a better set of weights with low residual errors,and the epochs number is considered 1000.The obtained figures represent the approximate solution and absolute error which are tested with NN to train the data accurately.展开更多
Biologically,because of the impact of reproduction period and nonlocal dispersal of HIV-infected cells,time delay and spatial heterogeneity should be considered.In this paper,we establish an HIV infection model with n...Biologically,because of the impact of reproduction period and nonlocal dispersal of HIV-infected cells,time delay and spatial heterogeneity should be considered.In this paper,we establish an HIV infection model with nonlocal dispersal and infection age.Moreover,applying the theory of Fourier transformation and von Foerster rule,we transform the model to an integrodifferential equation with nonlocal time delay and dispersal.The well-posedness,positivity,and boundedness of the solution for the model are studied.展开更多
In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatmen...In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.展开更多
The HIV problem is studied by version of delay mathematical models which consider the apoptosis of uninfected CD4<sup>+</sup> T cells which cultured with infected T cells in big volume. The opportunistic i...The HIV problem is studied by version of delay mathematical models which consider the apoptosis of uninfected CD4<sup>+</sup> T cells which cultured with infected T cells in big volume. The opportunistic infection and the apoptosis of uninfected CD4<sup>+</sup> T cells are caused directly or indirectly by a toxic substance produced from HIV genes. Ubiquitously, the nonlinear incidence rate brings forth the increasing number of infected CD4<sup>+</sup> T cells with introduction of small time delay, and in addition, there also exists a natural time delay factor during the process of virus replication. With state feedback control of time delay, the bifurcating periodical oscillating phenomena is induced via Hopf bifurcation. Mathematically, with the geometrical criterion applied in the stability analysis of delay model, the critical threshold of Hopf bifurcation in multiple delay differential equations which satisfy the transversal condition is derived. By applying reduction dimensional method combined with the center manifold theory, the stability of the bifurcating periodical solution is analyzed by the perturbation near Hopf point.展开更多
In this paper, we provide a new approach to solve approximately a system of fractional differential equations (FDEs). We extend this approach for approximately solving a fractional-order differential equation model of...In this paper, we provide a new approach to solve approximately a system of fractional differential equations (FDEs). We extend this approach for approximately solving a fractional-order differential equation model of HIV infection of CD4<sup>+</sup>T cells with therapy effect. The fractional derivative in our approach is in the sense of Riemann-Liouville. To solve the problem, we reduce the system of FDE to a discrete optimization problem. By obtaining the optimal solutions of new problem by minimization the total errors, we obtain the approximate solution of the original problem. The numerical solutions obtained from the proposed approach indicate that our approximation is easy to implement and accurate when it is applied to a systems of FDEs.展开更多
Considering the antiviral drugs can act on the fusion,reverse transcription,and budding stages of HIV infected cells,in this paper,we formulate a two-periodic delay heterogeneous space diffusion HIV model with three-s...Considering the antiviral drugs can act on the fusion,reverse transcription,and budding stages of HIV infected cells,in this paper,we formulate a two-periodic delay heterogeneous space diffusion HIV model with three-stage infection process to study the effects of periodic antiviral treatment and spatial heterogeneity on HIV infection process.We first study the well-posedness of the full system and then derive the basic reproduction number R_(0),which is defined as the spectral radius of the next generation operator.We further prove that R_(0) is a threshold for the elimination and persistence of HIV infection by comparison principle and persistence theory for non-autonomous system.In the spatial homogeneous case,the explicit expression of R_(0) is derived and the global attractivity of the positive steady state is proved by using the fluctuation method.Some numerical simulations are conducted to illustrate the theoretical results and our works suggest that both spatial heterogeneity and periodic delays caused by periodic antiviral therapy have a remarkable impact on the progression of HIV infection and should not be overlooked in clinical treatment process.展开更多
This paper mainly investigates dynamics behavior of HIV （human immunodeficiency virus） infectious disease model with switching parameters, and combined bounded noise and Gaussian white noise. This model is different...This paper mainly investigates dynamics behavior of HIV （human immunodeficiency virus） infectious disease model with switching parameters, and combined bounded noise and Gaussian white noise. This model is different from existing HIV models. Based on stochastic Ito lemma and Razumikhin-type approach, some threshold conditions are established to guarantee the disease eradication or persistence. Results show that the smaller amplitude of bounded noise and R0 〈 1 can cause the disease to die out; the disease becomes persistent if R0 〉 1. Moreover, it is found that larger noise intensity suppresses the prevalence of the disease even if R0 〉 1. Some numerical examples are given to verify the obtained results.展开更多
United Nations Political Declaration 2011 on HIV and AIDS calls to reduce the sexual transmission and the transmission of HIV among people, who inject drugs by 50% by 2015, through different control strategies and pre...United Nations Political Declaration 2011 on HIV and AIDS calls to reduce the sexual transmission and the transmission of HIV among people, who inject drugs by 50% by 2015, through different control strategies and precautionary measures. In this paper, we propose and study a simple SI type model that considers the effect of various precaution- ary measures to control HIV epidemic. We show, unlike conventional epidemic models, that the basic reproduction number which essentially considered as the disease eradica- tion condition is no longer sufficient to eliminate HIV infection. In particular, we show that even when the basic reproduction number is made less than unity, the disease may persist if the initial outbreak is not low. Eradication of disease is however guaranteed if the ensemble control measure exceeds some upper critical value. It is also shown that an epidemic model with mass action incidence may exhibit backward bifurcation and bistability if density-dependent demography is considered. Our theoretical study thus indicates that extra attention should be given in controlling HIV epidemic to achieve the desired result.展开更多
In this paper, the human immunodeficiency virus （HIV） infection model of CD4＋ T-cells is considered. In order to numerically solve the model problem, an operational method is proposed. For that purpose, we construc...In this paper, the human immunodeficiency virus （HIV） infection model of CD4＋ T-cells is considered. In order to numerically solve the model problem, an operational method is proposed. For that purpose, we construct the operational matrix of integration for bases of Taylor polynomials. Then, by using this matrix operation and approximation by polynomials, the HIV infection problem is transformed into a system of algebraic equations, whose roots are used to determine the approximate solutions. An important feature of the method is that it does not require collocation points. In addition, an error estimation technique is presented. We apply the present method to two numerical examples and compare our results with other methods.展开更多
The dynamics of a single strain HIV model is studied. The basic reproduction number R0 used as a bifurcation parameter shows that the system undergoes transcritical and saddle-node bifurcations. The usual threshold un...The dynamics of a single strain HIV model is studied. The basic reproduction number R0 used as a bifurcation parameter shows that the system undergoes transcritical and saddle-node bifurcations. The usual threshold unit value of R0 does not completely determine the eradication of the disease in an HIV infected person. In particular, a sub-threshold value Rc is established which determines the system＇s number of endemic states： multiple if Rc 〈 Ro 〈 1, only one if Rc=Ro = 1, and none if R0 〈 Rc 〈 1.展开更多
In this paper,the authors develop and study an HIV infection system with two distinct cell subsets and nonlinear stochastic perturbation.Firstly,the authors obtain that the solution of the system is positive and globa...In this paper,the authors develop and study an HIV infection system with two distinct cell subsets and nonlinear stochastic perturbation.Firstly,the authors obtain that the solution of the system is positive and global.Secondly,for the corresponding linear case,the authors derive a critical condition R0S similar to deterministic system.When R0S>1,the authors establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution to the stochastic system,respectively.Finally,the authors give sufficient criterions for extinction of the diseases.The proposed work provides a new method in overcoming difficulty conduced by nonlinear stochastic perturbation.展开更多
In this paper, an HIV dynamics model with two distributed intracellular delays incorporating Crowley-Martin functional response infection rate is investigated. The authors take into account multiple stage disease tran...In this paper, an HIV dynamics model with two distributed intracellular delays incorporating Crowley-Martin functional response infection rate is investigated. The authors take into account multiple stage disease transmission and the latently infected cells（not yet producing virus） in our system. The authors consider nonnegativity, boundedness of solutions, and global asymptotic stability of the system. By constructing suitable Lyapunov functionals and using the Lyapunov-La Salle invariance principle, the authors prove the global stability of the infected（endemic） equilibrium and the diseasefree equilibrium for time delays. The authors have proven that if the basic reproduction number R_0 is less than unity, then the disease-free equilibrium is globally asymptotically stable, and if R_0 is greater than unity, then the infected equilibrium is globally asymptotically stable. The results obtained show that the global dynamic behaviors of the model are completely determined by the basic reproduction number R_0 and that the time delay does not affect the global asymptotic properties of the model.展开更多
In this research work,we study the Human Immunodeficiency Virus(HIV)infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations.The objecti...In this research work,we study the Human Immunodeficiency Virus(HIV)infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations.The objective of the analysis is to present an approximate mathematical solution to the model that gives the count of the numbers of uninfected and infected helper T cells and the number of free virus particles present at a given instant of time.The system of nonlinear ODEs is converted into a system of nonlinear algebraic equations using spectral collocation method with three different basis functions such as Chebyshev,Legendre and Jacobi polynomials.Some factors such as the production of helper T cells and infection of these cells play a vital role in infected and uninfected cell counts.Detailed error analysis is done to compare our results with the existing methods.It is shown that the spectral collocation method is a very reliable,efficient and robust method of solution compared to many other solution procedures available in the literature.All these results are presented in the form of tables and figures.展开更多
In this work, biologically-inspired computing framework is developed for HIV infection of CD4＋ T-cell model using feed-forward artificial neural networks （ANNs）, genetic algorithms （GAs）, sequential quadratic pro...In this work, biologically-inspired computing framework is developed for HIV infection of CD4＋ T-cell model using feed-forward artificial neural networks （ANNs）, genetic algorithms （GAs）, sequential quadratic programming （SQP） and hybrid approach based on GA-SQP. The mathematical model for HIV infection of CD4＋ T-cells is represented with the help of initial value problems （IVPs） based on the system of ordinary differential equations （ODEs）. The ANN model for the system is constructed by exploiting its strength of universal approximation. An objective function is developed for the system through unsupervised error using ANNs in the mean square sense. Training with weights of ANNs is carried out with GAs for effective global search supported with SQP for efficient local search. The proposed scheme is evaluated on a number of scenarios for the HIV infection model by taking the different levels for infected cells, natural substitution rates of uninfected cells, and virus particles. Comparisons of the approximate solutions are made with results of Adams numerical solver to establish the correctness of the proposed scheme. Accuracy and convergence of the approach are validated through the results of statistical analysis based on the sufficient large number of independent runs.展开更多
In this study, we consider two target-cell limited models with saturation type infec- tion rate and intracellular delay： one without self-proliferation and the other with self- proliferation of activated CD4＋T cells...In this study, we consider two target-cell limited models with saturation type infec- tion rate and intracellular delay： one without self-proliferation and the other with self- proliferation of activated CD4＋T cells. We discuss about the local and global behavior of both the systems in presence and absence of intracellular delay. It is shown that the endemic equilibrium of a target-cell limited model would be unstable in presence and absence of intraeellular delay only when self-proliferation of activated CD4＋T cell is considered. Otherwise, all positive solutions converge to the endemic equilibrium or disease-free equilibrium depending on whether the basic reproduction ratio is greater than or less than unity. Our study suggests that amplitude of oscillation is negatively correlated with the constant input rate of CD4＋T cell when intracellular delay is absent or low. However, they are positively correlated if the delay is too high. Amplitude of oscillation, on the other hand, is always positively correlated with the proliferation rate of CD4＋T cell for all delay. Our mathematical and simulation analysis also suggest that there are many potential contributors who are responsible for the variation of CD4＋T cells and virus particles in the blood plasma of HIV patients.展开更多
In this paper, a fractional-order model which describes the human immunodeficiency type-1 virus （HIV-1） infection is presented. Numerical solutions are obtained using a generalized Euler method （GEM） to handle the...In this paper, a fractional-order model which describes the human immunodeficiency type-1 virus （HIV-1） infection is presented. Numerical solutions are obtained using a generalized Euler method （GEM） to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. We show that the model established in this paper possesses non-negative solutions. Comparisons between the results of the fractional-order model, the results of the integer model and the measured real data obtained from 10 patients during primary HIV-1 infection are presented. These compar- isons show that the results of the fractional-order model give predictions to the plasma virus load of the patients better than those of the integer model.展开更多
In this paper, an exponential method is presented for the approximate solutions of the HIV infection model of CD4＋T. The method is based on exponential polynomi- als and collocation points. This model problem corresp...In this paper, an exponential method is presented for the approximate solutions of the HIV infection model of CD4＋T. The method is based on exponential polynomi- als and collocation points. This model problem corresponds to a system of nonlinear ordinary differential equations. Matrix relations are constructed for the exponential functions. By aid of these matrix relations and the collocation points, the proposed technique transforms the model problem into a system of nonlinear algebraic equations. By solving the system of the algebraic equations, the unknown coefficients are com- puted and thus the approximate solutions are obtained. The applications of the method for the considered problem are given and the comparisons are made with the other methods.展开更多
Mathematical models and computer simulations are useful experimental tools for building and testing theories. Many mathematical models in biology can be formulated by a nonlinear system of ordinary differential equati...Mathematical models and computer simulations are useful experimental tools for building and testing theories. Many mathematical models in biology can be formulated by a nonlinear system of ordinary differential equations. This work deals with the numerical solution of the hantavirus infection model, the human immunodeficiency virus （HIV） infection model of CD4^＋T cells and the susceptible-infected-removed （SIR） epidemic model using a new reliable algorithm based on shifted Boubaker Lagrangian （SBL） method. This method reduces the solution of such system to a system of linear or non- linear algebraic equations which are solved using the Newton iteration method. The obtained results of the proposed method show highly accurate and valid for an arbitrary finite interval. Also, those are compared with fourth-order Runge-Kutta （RK4） method and with the solutions obtained by some other methods in the literature.展开更多
文摘HIV is a retrovirus that infects and impairs the cells and functions of the immune system. It has caused a great challenge to global public health systems and leads to Acquired Immunodeficiency Syndrome (AIDS), if not attended to in good time. Antiretroviral therapy is used for managing the virus in a patient’s lifetime. Some of the symptoms of the disease include lean body mass and many opportunistic infections. This study has developed a SIAT mathematical model to investigate the impact of inconsistency in treatment of the disease. The arising non-linear differential equations have been obtained and analyzed. The DFE and its stability have been obtained and the study found that it is locally asymptotically stable when the basic reproduction number is less than unity. The endemic equilibrium has been obtained and found to be globally asymptotically stable when the basic reproduction number is greater than unity. Numerical solutions have been obtained and analyzed to give the trends in the spread dynamics. The inconsistency in treatment uptake has been analyzed through the numerical solutions. The study found that when the treatment rate of those infected increases, it leads to an increase in treatment population, which slows down the spread of HIV and vice versa. An increase in the rate of treatment of those with AIDS leads to a decrease in the AIDS population, the reverse happens when this rate decreases. The study recommends that the community involvement in advocating for consistent treatment of HIV to curb the spread of the disease.
基金supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University(IMSIU)(grant number IMSIU-RP23066).
文摘This study directs the discussion of HIV disease with a novel kind of complex dynamical generalized and piecewise operator in the sense of classical and Atangana Baleanu(AB)derivatives having arbitrary order.The HIV infection model has a susceptible class,a recovered class,along with a case of infection divided into three sub-different levels or categories and the recovered class.The total time interval is converted into two,which are further investigated for ordinary and fractional order operators of the AB derivative,respectively.The proposed model is tested separately for unique solutions and existence on bi intervals.The numerical solution of the proposed model is treated by the piece-wise numerical iterative scheme of Newtons Polynomial.The proposed method is established for piece-wise derivatives under natural order and non-singular Mittag-Leffler Law.The cross-over or bending characteristics in the dynamical system of HIV are easily examined by the aspect of this research having a memory effect for controlling the said disease.This study uses the neural network(NN)technique to obtain a better set of weights with low residual errors,and the epochs number is considered 1000.The obtained figures represent the approximate solution and absolute error which are tested with NN to train the data accurately.
基金Supported by Funding for the National Natural Science Foundation of China(12201557,12001483,61807006)。
文摘Biologically,because of the impact of reproduction period and nonlocal dispersal of HIV-infected cells,time delay and spatial heterogeneity should be considered.In this paper,we establish an HIV infection model with nonlocal dispersal and infection age.Moreover,applying the theory of Fourier transformation and von Foerster rule,we transform the model to an integrodifferential equation with nonlocal time delay and dispersal.The well-posedness,positivity,and boundedness of the solution for the model are studied.
文摘In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.
文摘The HIV problem is studied by version of delay mathematical models which consider the apoptosis of uninfected CD4<sup>+</sup> T cells which cultured with infected T cells in big volume. The opportunistic infection and the apoptosis of uninfected CD4<sup>+</sup> T cells are caused directly or indirectly by a toxic substance produced from HIV genes. Ubiquitously, the nonlinear incidence rate brings forth the increasing number of infected CD4<sup>+</sup> T cells with introduction of small time delay, and in addition, there also exists a natural time delay factor during the process of virus replication. With state feedback control of time delay, the bifurcating periodical oscillating phenomena is induced via Hopf bifurcation. Mathematically, with the geometrical criterion applied in the stability analysis of delay model, the critical threshold of Hopf bifurcation in multiple delay differential equations which satisfy the transversal condition is derived. By applying reduction dimensional method combined with the center manifold theory, the stability of the bifurcating periodical solution is analyzed by the perturbation near Hopf point.
文摘In this paper, we provide a new approach to solve approximately a system of fractional differential equations (FDEs). We extend this approach for approximately solving a fractional-order differential equation model of HIV infection of CD4<sup>+</sup>T cells with therapy effect. The fractional derivative in our approach is in the sense of Riemann-Liouville. To solve the problem, we reduce the system of FDE to a discrete optimization problem. By obtaining the optimal solutions of new problem by minimization the total errors, we obtain the approximate solution of the original problem. The numerical solutions obtained from the proposed approach indicate that our approximation is easy to implement and accurate when it is applied to a systems of FDEs.
基金supported by the National Natural Science Foundation of China(No.12201557)the Foundation of Zhejiang Provincial Education Department(No.Y202249921).
文摘Considering the antiviral drugs can act on the fusion,reverse transcription,and budding stages of HIV infected cells,in this paper,we formulate a two-periodic delay heterogeneous space diffusion HIV model with three-stage infection process to study the effects of periodic antiviral treatment and spatial heterogeneity on HIV infection process.We first study the well-posedness of the full system and then derive the basic reproduction number R_(0),which is defined as the spectral radius of the next generation operator.We further prove that R_(0) is a threshold for the elimination and persistence of HIV infection by comparison principle and persistence theory for non-autonomous system.In the spatial homogeneous case,the explicit expression of R_(0) is derived and the global attractivity of the positive steady state is proved by using the fluctuation method.Some numerical simulations are conducted to illustrate the theoretical results and our works suggest that both spatial heterogeneity and periodic delays caused by periodic antiviral therapy have a remarkable impact on the progression of HIV infection and should not be overlooked in clinical treatment process.
基金supported by the National Natural Science Foundation of China(Grant Nos.11172233,11472212,11272258,and 11302170)the Natural Science and Engineering Research Council of Canada(NSERC)
文摘This paper mainly investigates dynamics behavior of HIV （human immunodeficiency virus） infectious disease model with switching parameters, and combined bounded noise and Gaussian white noise. This model is different from existing HIV models. Based on stochastic Ito lemma and Razumikhin-type approach, some threshold conditions are established to guarantee the disease eradication or persistence. Results show that the smaller amplitude of bounded noise and R0 〈 1 can cause the disease to die out; the disease becomes persistent if R0 〉 1. Moreover, it is found that larger noise intensity suppresses the prevalence of the disease even if R0 〉 1. Some numerical examples are given to verify the obtained results.
文摘United Nations Political Declaration 2011 on HIV and AIDS calls to reduce the sexual transmission and the transmission of HIV among people, who inject drugs by 50% by 2015, through different control strategies and precautionary measures. In this paper, we propose and study a simple SI type model that considers the effect of various precaution- ary measures to control HIV epidemic. We show, unlike conventional epidemic models, that the basic reproduction number which essentially considered as the disease eradica- tion condition is no longer sufficient to eliminate HIV infection. In particular, we show that even when the basic reproduction number is made less than unity, the disease may persist if the initial outbreak is not low. Eradication of disease is however guaranteed if the ensemble control measure exceeds some upper critical value. It is also shown that an epidemic model with mass action incidence may exhibit backward bifurcation and bistability if density-dependent demography is considered. Our theoretical study thus indicates that extra attention should be given in controlling HIV epidemic to achieve the desired result.
文摘In this paper, the human immunodeficiency virus （HIV） infection model of CD4＋ T-cells is considered. In order to numerically solve the model problem, an operational method is proposed. For that purpose, we construct the operational matrix of integration for bases of Taylor polynomials. Then, by using this matrix operation and approximation by polynomials, the HIV infection problem is transformed into a system of algebraic equations, whose roots are used to determine the approximate solutions. An important feature of the method is that it does not require collocation points. In addition, an error estimation technique is presented. We apply the present method to two numerical examples and compare our results with other methods.
文摘The dynamics of a single strain HIV model is studied. The basic reproduction number R0 used as a bifurcation parameter shows that the system undergoes transcritical and saddle-node bifurcations. The usual threshold unit value of R0 does not completely determine the eradication of the disease in an HIV infected person. In particular, a sub-threshold value Rc is established which determines the system＇s number of endemic states： multiple if Rc 〈 Ro 〈 1, only one if Rc=Ro = 1, and none if R0 〈 Rc 〈 1.
基金supported by the Natural Science Foundation of Shandong Province of China under Grant Nos.ZR2022MA008,ZR2018MA023,ZR2020QA008,ZR2019BA022the National Natural Science Foundation of China under Grant No.11901329the Project of Shandong Province Higher Educational Science and Technology Program of China under Grant No.J16LI09。
文摘In this paper,the authors develop and study an HIV infection system with two distinct cell subsets and nonlinear stochastic perturbation.Firstly,the authors obtain that the solution of the system is positive and global.Secondly,for the corresponding linear case,the authors derive a critical condition R0S similar to deterministic system.When R0S>1,the authors establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution to the stochastic system,respectively.Finally,the authors give sufficient criterions for extinction of the diseases.The proposed work provides a new method in overcoming difficulty conduced by nonlinear stochastic perturbation.
基金supported partially by Scientific Research Staring Foundation,Henan Normal University(qd13045)
文摘In this paper, an HIV dynamics model with two distributed intracellular delays incorporating Crowley-Martin functional response infection rate is investigated. The authors take into account multiple stage disease transmission and the latently infected cells（not yet producing virus） in our system. The authors consider nonnegativity, boundedness of solutions, and global asymptotic stability of the system. By constructing suitable Lyapunov functionals and using the Lyapunov-La Salle invariance principle, the authors prove the global stability of the infected（endemic） equilibrium and the diseasefree equilibrium for time delays. The authors have proven that if the basic reproduction number R_0 is less than unity, then the disease-free equilibrium is globally asymptotically stable, and if R_0 is greater than unity, then the infected equilibrium is globally asymptotically stable. The results obtained show that the global dynamic behaviors of the model are completely determined by the basic reproduction number R_0 and that the time delay does not affect the global asymptotic properties of the model.
文摘In this research work,we study the Human Immunodeficiency Virus(HIV)infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations.The objective of the analysis is to present an approximate mathematical solution to the model that gives the count of the numbers of uninfected and infected helper T cells and the number of free virus particles present at a given instant of time.The system of nonlinear ODEs is converted into a system of nonlinear algebraic equations using spectral collocation method with three different basis functions such as Chebyshev,Legendre and Jacobi polynomials.Some factors such as the production of helper T cells and infection of these cells play a vital role in infected and uninfected cell counts.Detailed error analysis is done to compare our results with the existing methods.It is shown that the spectral collocation method is a very reliable,efficient and robust method of solution compared to many other solution procedures available in the literature.All these results are presented in the form of tables and figures.
文摘In this work, biologically-inspired computing framework is developed for HIV infection of CD4＋ T-cell model using feed-forward artificial neural networks （ANNs）, genetic algorithms （GAs）, sequential quadratic programming （SQP） and hybrid approach based on GA-SQP. The mathematical model for HIV infection of CD4＋ T-cells is represented with the help of initial value problems （IVPs） based on the system of ordinary differential equations （ODEs）. The ANN model for the system is constructed by exploiting its strength of universal approximation. An objective function is developed for the system through unsupervised error using ANNs in the mean square sense. Training with weights of ANNs is carried out with GAs for effective global search supported with SQP for efficient local search. The proposed scheme is evaluated on a number of scenarios for the HIV infection model by taking the different levels for infected cells, natural substitution rates of uninfected cells, and virus particles. Comparisons of the approximate solutions are made with results of Adams numerical solver to establish the correctness of the proposed scheme. Accuracy and convergence of the approach are validated through the results of statistical analysis based on the sufficient large number of independent runs.
文摘In this study, we consider two target-cell limited models with saturation type infec- tion rate and intracellular delay： one without self-proliferation and the other with self- proliferation of activated CD4＋T cells. We discuss about the local and global behavior of both the systems in presence and absence of intracellular delay. It is shown that the endemic equilibrium of a target-cell limited model would be unstable in presence and absence of intraeellular delay only when self-proliferation of activated CD4＋T cell is considered. Otherwise, all positive solutions converge to the endemic equilibrium or disease-free equilibrium depending on whether the basic reproduction ratio is greater than or less than unity. Our study suggests that amplitude of oscillation is negatively correlated with the constant input rate of CD4＋T cell when intracellular delay is absent or low. However, they are positively correlated if the delay is too high. Amplitude of oscillation, on the other hand, is always positively correlated with the proliferation rate of CD4＋T cell for all delay. Our mathematical and simulation analysis also suggest that there are many potential contributors who are responsible for the variation of CD4＋T cells and virus particles in the blood plasma of HIV patients.
文摘In this paper, a fractional-order model which describes the human immunodeficiency type-1 virus （HIV-1） infection is presented. Numerical solutions are obtained using a generalized Euler method （GEM） to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. We show that the model established in this paper possesses non-negative solutions. Comparisons between the results of the fractional-order model, the results of the integer model and the measured real data obtained from 10 patients during primary HIV-1 infection are presented. These compar- isons show that the results of the fractional-order model give predictions to the plasma virus load of the patients better than those of the integer model.
文摘In this paper, an exponential method is presented for the approximate solutions of the HIV infection model of CD4＋T. The method is based on exponential polynomi- als and collocation points. This model problem corresponds to a system of nonlinear ordinary differential equations. Matrix relations are constructed for the exponential functions. By aid of these matrix relations and the collocation points, the proposed technique transforms the model problem into a system of nonlinear algebraic equations. By solving the system of the algebraic equations, the unknown coefficients are com- puted and thus the approximate solutions are obtained. The applications of the method for the considered problem are given and the comparisons are made with the other methods.
文摘Mathematical models and computer simulations are useful experimental tools for building and testing theories. Many mathematical models in biology can be formulated by a nonlinear system of ordinary differential equations. This work deals with the numerical solution of the hantavirus infection model, the human immunodeficiency virus （HIV） infection model of CD4^＋T cells and the susceptible-infected-removed （SIR） epidemic model using a new reliable algorithm based on shifted Boubaker Lagrangian （SBL） method. This method reduces the solution of such system to a system of linear or non- linear algebraic equations which are solved using the Newton iteration method. The obtained results of the proposed method show highly accurate and valid for an arbitrary finite interval. Also, those are compared with fourth-order Runge-Kutta （RK4） method and with the solutions obtained by some other methods in the literature.
