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The Covid-19 outbreak hit us as a serious health crisis with vaccination to be seen as the only way out of it. And media can play the role of an advocate to fight against this epidemic by spreading important awareness regarding various protocols and vaccination strategies. Since breakthrough infections are becoming highly common, a two dose vaccine regime may be the need of the hour even for individuals with a pre-existing illness to build better immunity. Thus in this paper, our aim is to analyse a mathematical model with two dose vaccination strategy in the presence of media and breakthrough infections. An SIV1V2R model is formulated and the dynamical analysis is done. The basic reproduction number is obtained and the local stability analysis of both the disease-free and endemic equilibrium point is discussed based on reproduction number. The global stability of the endemic equilibrium point is done by graph theoretic approach. Finally, the numerical validation of the analytic solution is done using MATLAB using the real data of India for some important parameters to address a few vital questions which involves the role of media on the vaccination strategy. And sensitive model parameters effecting the basic reproduction number and endemic equilibrium points are identified using sensitivity analysis techniques like PRCC(partial rank correlation coefficient). Thus, the outcomes demonstrated the trend a two-dose vaccine model can follow and how the effect of media can help bring down the infections. This model provides support that two dose vaccination against COVID-19 with media presence for awareness is highly effective in controlling this epidemic. [ABSTRACT FROM AUTHOR]

There are relatively few theoretical papers to consider the positivity of solutions of discrete time stochastic difference equations (DSDEs), compared to many publications on theoretical analysis of solutions of deterministic difference equations and stochastic differential equations. Additionally, no papers theoretically investigate the global stability of nontrivial solutions of n-dimensional DSDEs. In this paper, we consider the Euler-Maruyama scheme for n-dimensional stochastic difference equations that are a generalization of a two-dimensional model of stochastic predator-prey interactions, and show the positivity and the global stability of nontrivial solutions of the scheme by applying a new discretized version of the Itô formula. Numerical simulations are introduced to support the results. [ABSTRACT FROM AUTHOR]

The Covid-19 outbreak hit us as a serious health crisis with vaccination to be seen as the only way out of it. And media can play the role of an advocate to fight against this epidemic by spreading important awareness regarding various protocols and vaccination strategies. Since breakthrough infections are becoming highly common, a two dose vaccine regime may be the need of the hour even for individuals with a pre-existing illness to build better immunity. Thus in this paper, our aim is to analyse a mathematical model with two dose vaccination strategy in the presence of media and breakthrough infections. An SIV1V2R model is formulated and the dynamical analysis is done. The basic reproduction number is obtained and the local stability analysis of both the disease-free and endemic equilibrium point is discussed based on reproduction number. The global stability of the endemic equilibrium point is done by graph theoretic approach. Finally, the numerical validation of the analytic solution is done using MATLAB using the real data of India for some important parameters to address a few vital questions which involves the role of media on the vaccination strategy. And sensitive model parameters effecting the basic reproduction number and endemic equilibrium points are identified using sensitivity analysis techniques like PRCC(partial rank correlation coefficient). Thus, the outcomes demonstrated the trend a two-dose vaccine model can follow and how the effect of media can help bring down the infections. This model provides support that two dose vaccination against COVID-19 with media presence for awareness is highly effective in controlling this epidemic. [ABSTRACT FROM AUTHOR]

There are many publications on theoretical analysis of deterministic difference equationsand stochastic differential equations. However, relatively few theoretical papers are pub-lished to consider the positivity of solutions of discrete-time stochastic difference equations(DSDEs), and no theoretical papers investigate the global stability of nontrivial solutionsof DSDEs with nonlinear terms. In this paper, we consider a DSDE model that is ageneralization of two-dimensional nonlinear models of stochastic predator-prey interac-tions, and show the positivity and global stability of the nontrivial solutions by using ournew discretized version of the Itô formula. In addition, our results are compared withthose of continuous-time stochastic differential equations and discrete-time deterministicdifference equations. Numerical simulations are introduced to support the results. [ABSTRACT FROM AUTHOR]

There are many publications on theoretical analysis of deterministic difference equations and stochastic differential equations. However, relatively few theoretical papers are published to consider the positivity of solutions of discrete-time stochastic difference equations (DSDEs), and no theoretical papers investigate the global stability of nontrivial solutions of DSDEs with nonlinear terms. In this paper, we consider a DSDE model that is a generalization of two-dimensional nonlinear models of stochastic predator-prey interactions, and show the positivity and global stability of the nontrivial solutions by using our new discretized version of the It^o formula. In addition, our results are compared with those of continuous-time stochastic differential equations and discrete-time deterministic difference equations. Numerical simulations are introduced to support the results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the global stability of the solutions, the boundedness and the periodic character of the nonlinear di¤erence equation... where the parameters α,β,γ, a; b; c and d are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1, x0 are positive real numbers where s = max{l,k}. Some numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonstandard finite difference (NSFD) scheme to solve numerically a cholera epidemic model with spatial diffusion. Through constructing discrete Lyapunov functions, we prove the globally asymptotical stabilities of the diseasefree equilibrium and the chronic infection equilibrium, which coincide with the corresponding continuous model. Finally, numerical simulations are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonstandard finite difference (NSFD) scheme to solve numerically a cholera epidemic model with spatial diffusion. Through constructing discrete Lyapunov functions, we prove the globally asymptotical stabilities of the diseasefree equilibrium and the chronic infection equilibrium, which coincide with the corresponding continuous model. Finally, numerical simulations are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the global stability of the solutions, the boundedness and the periodic character of the nonlinear difference equation xn+1 = αxn+βxn-l+γxn-k + axnxn-k/bxn + cxn-l + dxn-k, n = 0, 1, ..., where the parameters α, β, γ, a, b, c and d are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1, x0 are positive real numbers where s = max{l, k}. Some numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

This paper proposes a latent Chikungunya viral infection model with saturated incidence rate. To take into account the time lag between the initial viral contacts uninfected monocytes and the production of new active CHIKV particles the model is incorporated by intracellular discrete or distributed time delays. We study the qualitative behavior of the model. Using the method of Lyapunov function, we established the global stability of the steady states of the model. The effect of the time delay on the stability of the steady states has also been shown by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper studies the stability of a mathematical model for within-host Chikungunya virus (CHIKV) infection. The model incorporates (i) two types of infected monocytes, latently infected monocytes which do not generate CHIKV until they have been activated and actively infected monocytes, (ii) antibody immune response, and (iii) saturated incidence rate. We derive a biological threshold number R0. Using the method of Lyapunov function, we established the global stability of the steady states of the model. We have proven that, when R0 ≤ 1, then Q0 is globally asymptotically stable and when R0 > 1, the endemic equilibrium Q1 is globally asymptotically stable. The theoretical results have been supported by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, some qualitative properties are discussed such as the boundedness, the periodicity and the global stability of the positive solutions of the nonlinear difference equation... where the coefficients A; αi; βi 2 (0,1), i = 1,...., 5; while the initial conditions y-5,y-4,y-3,y-2; y-1; y0 are arbitrary positive real numbers. Some numerical examples will be given to illustrate our results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the global stability of the solutions, the boundedness and the periodic character of the nonlinear difference equationxn+1 = βxn+βxn-l+γxn-k + axnxn-k/bxn + cxn-l + dxn-k, n = 0, 1, ..., where the parameters α, β, γ, a, b, c and d are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1, x0 are positive real numbers where s = max{l, k}. Some numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

This paper proposes a latent Chikungunya viral infection model with saturated incidence rate. To take into account the time lag between the initial viral contacts uninfected monocytes and the production of new active CHIKV particles the model is incorporated by intracellular discrete or distributed time delays. We study the qualitative behavior of the model. Using the method of Lyapunov function, we established the global stability of the steady states of the model. The effect of the time delay on the stability of the steady states has also been shown by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper studies the stability of a mathematical model for within-host Chikungunya virus (CHIKV) infection. The model incorporates (i) two types of infected monocytes, latently infected monocytes which do not generate CHIKV until they have been activated and actively infected monocytes, (ii) antibody immune response, and (iii) saturated incidence rate. We derive a biological threshold number R0. Using the method of Lyapunov function, we established the global stability of the steady states of the model. We have proven that, when R0 ≤ 1, then Q0 is globally asymptotically stable and when R0 > 1, the endemic equilibrium Q1 is globally asymptotically stable. The theoretical results have been supported by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, some qualitative properties are discussed such as the boundedness, the periodicity and the global stability of the positive solutions of the nonlinear difference equation ..., where the coefficients A, αi, βi ∈ (0, ∞), i = 1, ..., 5, while the initial conditions y-5, y-4, y-3, y-2, y-1, y0 are arbitrary positive real numbers. Some numerical examples will be given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability analysis of a virus dynamics model with CTL immune response and with both cell-to-cell and virus-to-cell transmissions. The model contains three types of distributed time delays. The existence and global stability of all steady states of the model are determined by two parameters, the basic reproduction number (R0) and the CTL immune response activation number (R1). By using suitable Lyapunov functionals, we show that if R0 ≤ 1, then the infection-free steady state E0 is globally asymptotically stable; if R1 ≤ 1 < R0, then the CTL-inactivated infection steady state E1 is globally asymptotically stable; if R1 > 1, then the CTL-activated infection steady state E2 is globally asymptotically stable. Numerical simulations are conducted to support the theoretical results. [ABSTRACT FROM AUTHOR]

This paper study the dynamical behavior of HIV-1 infection model with saturated virus-target and infected-target incidences. The model is incorporated by two types of intracellular distributed time delays. The model generalizes all the existing HIV-1 infection models with cell-to-cell transmission presented in the literature by considering saturated incidence rate. The nonnegativity and boundedness of the solutions of the model as well as global stability of the steady states are studied. The global stability are established using Lyapunov method. Using MATLAB we conduct some numerical simulations to confirm our results. The effect of the saturated incidence of the HIV-1 dynamics is shown. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions, the boundedness and the periodic character of the difference equation xn+1 = Axn + Bxn ¡ l + Cxn ¡ k + γxn ¡ k/Dxn ¡ s + Exn ¡ t, n=0, 1, ..., where the parameters A, B, C, D, E,γ ∊ (0,∞) and the initial conditions x¡ σ, x¡ σ+1,..., x¡1, x0 are positive real numbers where σ= max{l, k, s, t}. Numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the effect of the cytotoxic T lymphocytes (CTLs) on an HIV-1 dynamics. The model considers that the virus infects the macrophages in addition to the CD4+ T cells. The role of the CTLs is to kill the infected macrophages and CD4+ T cells. The time delay which accounts the time of infection and the time of producing new active HIV-1 is modeled. The HIV-1 dynamics is modeled as a 6-dimensional nonlinear delay differential equations. The incidence rate of infection and killer rate of infected cells are given by general nonlinear functions. We study the qualitative behavior of the system. The global stability analysis has been established using Lyapunov method and LaSalle invariance principle. We present an example and perform numerical simulations to emphasize our theoretical results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation xn+1 = αxn + βxn-l + γxn-k + axn-l + bxn-k/cxn-l + dxn-k, n + 0, 1, ..., where the parameters α,β,γ,a,b,c,d ∊(0,∞) and the initial conditions x-s, x-s+1 ..., x-l and x0 are positive real numbers where s = max{l,k}. Examples to illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

This paper study the dynamical behavior of a general HIV-1 infection model under the effect of Highly active antiretroviral therapies (HAART). The model includes three types of infected cells (i) long-lived productively infected cells which live for long time and creat small amount of HIV-1 particles, (ii) latently infected cells which do not creat HIV-1 until they have been activated (iii) short-lived productively infected cells which live for long time and creat large amount of HIV-1. The model incorporates humoral immune response and general nonlinear forms for the incidence rate of infection, the generation and removal rates of all compartments. The nonnegativity and boundedness of the solutions of the model as well as global stability of the steady states are studied. The global stability are established using Lyapunov method. Using MATLAB we conduct some numerical simulations to confirm our results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation xn+1 = A kΣi=0 xn-i/B+C kπi=0xn-i, n=0, 1, ..., where the parameters A, B and C are positive real numbers and the initial conditions x-k, x-k+1, ..., x-, x0 are nonnegative real numbers. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation xn+1 = axn + . . ., n=0,1, ..., where the parameters α, β, γ and a are positive real numbers and the initial conditions x-t, x-t+1 ..., x-1 and x0 are positive real numbers where t = max{l, k}. Numerical examples to the difference equation are given to explain our results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the qualitative behaviors of three viral infection models with two types of cocirculating target cells. The models take into account both antibodies and latently infected cells. The incidence rate is represented by bilinear, saturation and general function. For the first two models, we have derived two threshold parameters, R0 and R1 which completely determined the global properties of the models. Lyapunov functions are constructed and LaSalle's invariance principle is applied to prove the global asymptotic stability of all equilibria of the models. For the third model, we have established a set of conditions on the general incidence rate function which are sufficient for the global stability of the equilibria of the model. Theoretical results have been checked by numerical simulations. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation xn+1 = axn + αxnxn-l/β+γn-k,n = 0, 1, ..., where the parameters αβγ and a are positive real numbers and the initial conditions x-t,x-t+1 ..., x-1 and x0 are positive real numbers where t = {l,k}. Numerical examples to the difference equation are given to explain our results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the qualitative behaviors of three viral infection models with two types of cocirculating target cells. The models take into account both antibodies and latently infected cells. The incidence rate is represented by bilinear, saturation and general function. For the first two models, we have derived two threshold parameters, R0 and R1 which completely determined the global properties of the models. Lyapunov functions are constructed and LaSalle's invariance principle is applied to prove the global asymptotic stability of all equilibria of the models. For the third model, we have established a set of conditions on the general incidence rate function which are sufficient for the global stability of the equilibria of the model. Theoretical results have been checked by numerical simulations. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the periodic character and the global stability of the positive solutions of the difference equation xn+1 = axn + bxn-k + cxn-l - dxn-s/exn-s - αxn-t, n = 0, 1, ..., where the parameters a, b, c, d, e and α are positive real numbers and the initial conditions x-σ, x-σ+1, ..., x-1, x0 are positive real numbers where σ = max{s, t, k,}. Some numerical examples were given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior of a nonlinear model for viral infection with Cytotoxic T Lymphocyte (CTL) immune response. The model is a generalization of several models presented in the literature by considering more general functions for the: (i) intrinsic growth rate of uninfected cells; (ii) incidence rate of infection; (iii) natural death rate of infected cells; (iv) rate at which the infected cells are killed by CTL cells; (v) production and removal rates of viruses; (vi) activation and natural death rates of CTLs. We derive two threshold parameters R0 (the basic infection reproduction number) and R1 (the CTL immune response activation number) and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the model. By using suitable Lyapunov functions and LaSalle's invariance principle, we prove the global asymptotic stability of all equilibria of the model. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the periodic character and the global stability of the positive solutions of the difference equation ... where the parameters a, b, c, d, e and a are positive real numbers and the initial conditions x --&#963, x-σ+1, ···, x-1, xo, xo are positive real numbers where σ = max {s, t, l, k} Some numerical examples were given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior of a nonlinear model for viral infection with Cytotoxic T Lymphocyte (CTL) immune response. The model is a generalization of several models presented in the literature by considering more general functions for the: (i) intrinsic growth rate of uninfected cells; (ii) incidence rate of infection; (iii) natural death rate of infected cells; (iv) rate at which the infected cells are killed by CTL cells; (v) production and removal rates of viruses; (vi) activation and natural death rates of CTLs. We derive two threshold parameters R0 (the basic infection reproduction number) and R1 (the CTL immune response activation number) and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the model. By using suitable Lyapunov functions and LaSalle's invariance principle, we prove the global asymptotic stability of all equilibria of the model. [ABSTRACT FROM AUTHOR]

In this paper, we study the global stability of a mathematical model that describes the virus dynamics under the e¤ect of antibody immune response. The model is a modi?cation of some of the existing virus dynamics models by considering the latently infected cells and nonlinear incidence rate for virus infections. We show that the global dynamics of the model is completely determined by two threshold values R0, the corresponding reproductive number of viral infection and R1, the corresponding reproductive number of antibody immune response, respectively. Using Lyapunov method, we have proven that, if R0 ≤ 1, then the uninfected steady state is globally asymptotically stable (GAS), if R1 ≤ 1 < R0, then the infected steady state without antibody immune response is GAS, and if R1 > 1, then the infected steady state with antibody immune response is GAS. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a viral infection model with humoral immunity. The incidence rate is given by Hill type infection rate. We have derived two threshold parameters, R0 and R1 which completely determined the global properties of the model. By constructing suitable Lyapunov functions and applying LaSalle's invariance principle we have established the global asymptotic stability of all steady states of the model. We have proven that, if R0 ⩽ 1, then the infection-free steady state is globally asymptotically stable (GAS), if R1 ⩽ 1 < R0, then the chronic-infection steady state without humoral immune response is GAS, and if R1 > 1, then the chronic-infection steady state with humoral immune response is GAS. [ABSTRACT FROM AUTHOR]

In this paper, we study the global dynamics of a viral infection model with antibody immune response. The incidence rate is given by a general function of the population of the uninfected target cells, infected cells and free viruses. We have established a set of conditions on the general incidence rate function and determined two threshold parameters R0 (the basic infection reproduction number) and R1 (the antibody immune response activation number) which are su¢ cient to determine the global behavior of the model. The global asymptotic stability of the equilibria of the model has been proven by using direct Lyapunov method and applying LaSalle's invariance principle. [ABSTRACT FROM AUTHOR]

In this paper, we study the global properties of a viral infection model with antibody immune response. The incidence rate is given by a general function of the populations of the uninfected target cells, infected cells and free viruses. The model contains two types of intracellular discrete time delays to describe the time required for viral contacting an uninfected target cell and viral emission. We have established a set of conditions on the general incidence rate function and determined two threshold parameters R0 (the basic infection reproduction number) and R1 (the antibody immune response activation number) which are sufficient to determine the global behavior of the model. The global asymptotic stability of the equilibria of the model has been proven by using direct Lyapunov method and applying LaSalle's invariance principle. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a viral infection model with humoral immunity. The incidence rate is given by Hill type infection rate. We have derived two threshold parameters, R0 and R1 which completely determined the global properties of the model. By constructing suitable Lyapunov functions and applying LaSalle's invariance principle we have established the global asymptotic stability of all steady states of the model. We have proven that, if R0 ≤ 1, then the infection-free steady state is globally asymptotically stable (GAS), if R1 ≤ 1 < R0, then the chronic-infection steady state without humoral immune response is GAS, and if R1 > 1, then the chronic-infection steady state with humoral immune response is GAS. [ABSTRACT FROM AUTHOR]

In this paper, we study the global properties of a viral infection model with antibody immune response. The incidence rate is given by a general function of the populations of the uninfected target cells, infected cells and free viruses. The model contains two types of intracellular discrete time delays to describe the time required for viral contacting an uninfected target cell and viral emission. We have established a set of conditions on the general incidence rate function and determined two threshold parameters R0 (the basic infection reproduction number) and R1 (the antibody immune response activation number) which are sufficient to determine the global behavior of the model. The global asymptotic stability of the equilibria of the model has been proven by using direct Lyapunov method and applying LaSalle's invariance principle. [ABSTRACT FROM AUTHOR]

In this paper, we study the global dynamics of a viral infection model with antibody immune response. The incidence rate is given by a general function of the population of the uninfected target cells, infected cells and free viruses. We have established a set of conditions on the general incidence rate function and determined two threshold parameters R0 (the basic infection reproduction number) and R1 (the antibody immune response activation number) which are sufficient to determine the global behavior of the model. The global asymptotic stability of the equilibria of the model has been proven by using direct Lyapunov method and applying LaSalle's invariance principle. [ABSTRACT FROM AUTHOR]

In this paper we study the boundedness, investigate the global convergence and the periodicity of the solutions to the following recursive sequence xn+1 = axn + bxn-1² + cxn-2xn-3/dxn-1² + exn-2xn-3, n = 0, 1, ..., where the parameters a, b, c, d and e are positive real numbers and the initial conditions x-3, x-2, x-1 and x0 ∈ (0, ∞). Also we give some numerical examples of some special cases of considered equation and presented some related graphs and figures using Matlab. [ABSTRACT FROM AUTHOR]

In this paper we study the boundedness, investigate the global convergence and the periodicity of the solutions to the following recursive sequence ... , where the parameters a, b, c, d and e are positive real numbers and the initial conditions x-3, x-2, x-1 and .... Also we give some numerical examples of some special cases of considered equation and presented some rleated graphs and figures using Matlab. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to investigate the qualitative behavior of the solutions for the following rational difference equation: ... where α, β, ai,bi ∈ (0,∞),i =0, 1, ..., k; with the initial conditions x0,x-1, ..., x-2k,x-2k-1 ∈ (0,∞). We determine the equilibrium points of the considered equation and then study their local stability. Also we study the boundedness and the permanence of the solutions. Finally we investigate the global asymptotically stable of the equilibrium points. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to investigate the qualitative behavior of the solutions for the following rational difference equation: ... where α, β, ai , bi ∊ (0, ∊), i = 0,1 ,...,k; with the initial conditions xo , x-i , ..., x-2k , x-2k- 1 ∊ (0, ∊). We determine the equilibrium points of the considered equation and then study their local stability. Also we study the boundedness and the permanence of the solutions. Finally we investigate the global asymptotically stable of the equilibrium points. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the basic and global properties of a virus infection model with humoral immunity and distributed intracellular delays. The incidence rate of the infection is given by Crowley-Martin functional response. Two types of distributed time delays have been incorporated into the model to describe the time needed for infection of uninfected cell and virus replication. Using the method of Lyapunov functional, we have established that the global stability of the model is completely determined by two threshold numbers, the basic reproduction number R0 and the humoral immunity reproduction number R1. We have proven that if R0 ≤ 1 then the uninfected steady state is globally asymptotically stable (GAS), if R1 ≤ 1 R0, then the infected steady state without immune response is GAS, and if R1 > 1, then the infected steady state with humoral immunity is GAS. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the basic and global properties of a virus infection model with humoral immunity and distributed intracellular delays. The incidence rate of the infection is given by Crowley- Martin functional response. Two types of distributed time delays have been incorporated into the model to describe the time needed for infection of uninfected cell and virus replication. Using the method of Lyapunov functional, we have established that the global stability of the model is completely determined by two threshold numbers, the basic reproduction number R0 and the humoral immunity reproduction number R1. We have proven that if R0 ≤ 1 then the uninfected steady state is globally asymptotically stable (GAS), if R1 ≤ 1 ≤ Ro, then the infected steady state without immune response is GAS, and if R1 ≥ 1, then the infected steady state with humoral immunity is GAS. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global dynamics of virus infection model with humoral immune response and distributed intracellular delays. The model is 4-dimensional nonlinear delay differential equations that describe the interaction of the virus with target cells, taking into account the humoral immune system response. The model has two types of distributed time delays which describe the time needed for infection of target cell and virus replication. Lyapunov functionals are constructed to establish the global asymptotic stability of the steady states of the model. We have proven that if the basic reproduction number R0 is less than or equal unity then the uninfected steady state is globally asymptotically stable (GAS), and if the antibody immune response reproduction number R1 is less than or equal unity and R0 > 1, then the infected steady state without immune response exists and it is GAS; if R1 > 1 then the infected steady state with immune response exists and it is GAS. [ABSTRACT FROM AUTHOR]

In this paper, we study the global properties of two mathematical models which describe the interaction of the human immunodeficiency virus (HIV) with two classes of target cells, CD4+ T cells and macrophages. The incidence rate of virus infection is given by the Crowley-Martin functional response. The first model has two types of discrete delays while the second one incorporates two types of distributed delays to describe the time needed for infection of cell and virus replication. The basic reproduction number R0 is identified which completely determines the global dynamics of the models. By constructing suitable Lyapunov functionals, we have proven that if R0 ≤ 1 then the uninfected steady state is globally asymptotically stable (GAS), and if R0 > 1 then the infected steady state exists and it is GAS. [ABSTRACT FROM AUTHOR]

In this paper, we study the global analysis of virus dynamics models with discrete delay and with distributed delay. The models describe the interaction of the HIV with two classes of target cells, CD4+ T cells and macrophages. The incidence rate of virus infection is given by the Beddington-DeAngelis functional response. The models have two types of discrete time delay or distributed delay describing the time needed for infection of cell and virus replication. The basic reproduction number R0 is identified which completely determines the global dynamics of the models. By constructing suitable Lyapunov functionals, we have proven that if R0 ≤ 1 then the uninfected steady state is globally asymptotically stable (GAS), and if R0 > 1 then the infected steady state exists and it is GAS. [ABSTRACT FROM AUTHOR]

This paper investigates the qualitative behavior of viral infection model with multitarget cells in vivo. The infection rate is given by Crowley-Martin functional response. By assuming that the virus attack n classes of uninfected target cells, we study a viral infection model of dimension 2n + 1 with distributed delay. To describe the latent period for the contacted target cells with viruses to begin producing viruses, two types of distributed delay are incorporated into the model. The basic reproduction number R0 of the model is defined which determines the dynamical behavior of the model. Utilizing Lyapunov functionals and LaSalle's invariance principle, we have proven that if R0 ≤ 1 then the uninfected steady state is globally asymptotically stable, and if R0 > 1 then the infected steady state is globally asymptotically stable. [ABSTRACT FROM AUTHOR]