Mathematical Model Of HIV-1 Infection Research Articles

Dynamics of an HIV infection model with two time delays

In this paper, a mathematical model for HIV-1 infection and immune response is considered, involving two discrete time delays in the intracellular as well as in activation of immune response. Using a recently developed geometric method for studying a class of transcendental equation with two time delays and delay dependent coefficients, we obtain the stability and bifurcation results at the non-trivial equilibrium. In particular, the crossing curves on the two-delays parameter plane can be completely characterized, on which Hopf and double-Hopf bifurcation will take place. In the case of Hopf bifurcation, there exist stability switches, and the direction and stability of delay induced Hopf-bifurcation can be determined using normal form theory and center manifold theorem. These results imply the model will exhibit complex temporal dynamics, such as period oscillations, quasi-periodic solutions, etc. Numerical examples are also carried out to verify these results.

An analytic study of the fractional order model of HIV-1 virus and CD4+ T-cells using adomian method

In this article, we study the fractional mathematical model of HIV-1 infection of CD4+ T-cells, by studying a system of fractional differential equations of first order with some initial conditions, we study the changing effect of many parameters. The fractional derivative is described in the caputo sense. The adomian decomposition method (Shortly, ADM) method was used to calculate an approximate solution for the system under study. The nonlinear term is dealt with the help of adomian polynomials. Numerical results are presented with graphical justifications to show the accuracy of the proposed methods.

Numerical Solution of Fractional Model of HIV-1 Infection in Framework of Different Fractional Derivatives

In this paper, we have extended the model of HIV-1 infection to the fractional mathematical model using Caputo-Fabrizio and Atangana-Baleanu fractional derivative operators. A detailed proof for the existence and the uniqueness of the solution of fractional mathematical model of HIV-1 infection in Atangana-Baleanu sense is presented. Numerical approach is used to find and study the behavior of the solution of the stated model using different derivative operators, and the graphical comparison between the solutions obtained for the Caputo-Fabrizio and the Atangana-Baleanu operator is presented to see which fractional derivative operator is more efficient.

Fractional numerical simulation of mathematical model of HIV-1 infection with stem cell therapy

This paper introduces fractional-order into a mathematical model of HIV infection of CD4$ ^{+} $ T-cells combining with the rate of multiply uninfected CD4$ ^{+} $ T-cells through mitosis and stem cell therapy. The paper shows the theoretical studies including positivity and stability of the solution. In addition, the numerical solutions are obtained and illustrated. The results show that the stem cell's therapy increases the quality of a HIV patient's life only for short time. This results are consistent with medical case studies.

Oscillatory and complex behaviour of Caputo-Fabrizio fractional order HIV-1 infection model

<abstract><p>HIV-1 infection is a dangerous diseases like Cancer, AIDS, etc. Many mathematical models have been introduced in the literature, which are investigated with different approaches. In this article, we generalize the HIV-1 model through nonsingular fractional operator. The non-integer mathematical model of HIV-1 infection under the Caputo-Fabrizio derivative is presented in this paper. The concept of Picard-Lindelof and fixed-point theory are used to address the existence of a unique solution to the HIV-1 model under the suggested operator. Also, the stability of the suggested model is proved through the Picard iteration and fixed point theory approach. The model's approximate solution is constructed through three steps Adams-Bashforth numerical method. Numerical simulations are provided for different values of fractional-order to study the complex dynamics of the model. Lastly, we provide the oscillatory and chaotic behavior of the proposed model for various fractional orders.</p></abstract>

Investigating the effect of pyroptosis on the slow CD4+ T cell depletion in HIV-1 infection, by dynamical analysis of its discontinuous mathematical model

HIV infection is one of the most serious causes of death throughout the world. CD4+ T cells which play an important role in immune protection, are the primary targets for HIV infection. The hallmark of HIV infection is the progressive loss in population of CD4+ T cells. However, the pathway causing this slow T cell decline is poorly understood [16]. This paper studies a discontinuous mathematical model for HIV-1 infection, to investigate the effect of pyroptosis on the disease. For this purpose, we use the theory of discontinuous dynamical systems. In this way, we can better analyze the dynamical behavior of the HIV-1 system. Especially, considering the dynamics of the system on its discontinuity boundary enables us to obtain more comprehensive results rather than the previous researches. A stability region for the system, corresponding to its equilibria on the discontinuity boundary, will be determined. In such a parametric region, the trajectories of the system will be trapped on the discontinuity manifold forever. It is also shown that in the obtained stability region, the disease can lead to a steady state in which the population of uninfected T cells and viruses will preserve at a constant level of cytokines. This means that the pyroptosis will be restricted and the disease cannot progress for a long time. Some numerical simulations based on clinical and experimental data are given which are in good agreement with our theoretical results.

Analysis of mathematical model of HIV-1 infection of CD4+ T cells with CTL response and antiretroviral treatment

A mathematical model is developed to capture the spread of HIV-1 infection within host cells caused by the contact of cell to cell and CTL response. In this paper, we propose a mathematical model of HIV-1 infection in CD4+T cells taking into account viral transmission from cell to cell and CTL response. The HIV transmission from cell to cell is one of the main factor in the spread of HIV infection and CTL response determines viral set point. We analyse the model to investigate the existence and stability of the equilibria. We analyse the local stability of disease free equilibrium by linearization, while the global stability of endemic equilibrium of the system by constructing Lyapunov function. Numerical simulations are presented to find the effectiveness of antiretroviral treatment in different scenarios and to the implication of CTL response in controlling the progression of HIV-1 infection.

A model incorporating combined RTIs and PIs therapy during early HIV-1 infection

We develop a within host mathematical model of HIV-1 infection describing the effects of combined RTIs and PIs treatments on early HIV-1 infection when treatment is captured using periodic functions of pharmacokinetics type. We use an alternative of the basic reproduction number to analyze endemicity level of HIV-1 infection. Various treatment scenarios incorporating perfect and imperfect drug adherence in drug administration are explored. Our results show that pharmacokinetics treatment is a more realistic way of administering the treatment. Apart from confirming that PIs drugs are more effective than RTIs drugs and that combined RTIs and PIs therapy is more effective than monotherapy of RTIs or PIs, our results show that imperfect drug adherence leads to the increase of viral in the absence of mutation even though the drug is good.

A mathematical model of HIV-1 infection within host cell to cell viral transmissions with RTI and discrete delays

A within host viral infection model with both virus to cell and cell to cell transmissions and three discrete delays using reverse transcriptase inhibitors is investigated. The effect of time delay on stability of the equilibria of the infection model has been studied. By choosing the immune delay as the bifurcation parameter, we establish the sufficient condition for local stability of chronic infection equilibrium, Hopf bifurcation occurs when the immune delay passes a critical value. The global stability analysis of the model is carried out in terms of the basic reproduction number $$R_0$$ . If $$R_0 < 1$$ , the infection-free (semi-trivial) equilibrium is globally stable for both ODE and DDE model; if $$R_0>1$$ , the chronic infection (positive) equilibrium exists and is globally stable for both ODE and DDE model. Numerical simulations are obtained to validate our analytical findings by varying the system parameter.

Stability analysis of HIV-1 model with multiple delays

A mathematical model for HIV-1 infection with multiple delays is proposed. These delays account for (i) the delay in contact process between the uninfected cells virus, (ii) a latent period between the time target cells which are contacted by the virus particles and the time the virions enter the cells, and (iii) a virus production period for new virions to be produced within and released from the infected cells. For this model, the basic reproductive number is identified and its threshold property is discussed. The uninfected and infected steady states are shown to be locally as well as globally asymptotically stable. The value of the basic reproductive number shows that increasing any one of these delays will decrease this number. This may suggest a new direction for new drugs that can prolong the infection process and spreading of virus. The proved results have potential applications in HIV-1 therapy.

Stability and Hopf Bifurcation in a HIV-1 System with Multitime Delays

In this paper, we propose a mathematical model for HIV-1 infection with three time delays. The model examines a viral-therapy for controlling infections by using an engineered virus to selectively eliminate infected cells. In our model, three time delays represent the latent period of pathogen virus, pathogen virus production period and recombinant (genetically modified) virus production period, respectively. Detailed theoretical analysis have demonstrated that the values of three delays can affect the stability of equilibrium solutions, can also lead to Hopf bifurcation and oscillated solutions of the system. Moreover, we give the conditions for the existence of stable positive equilibrium solution and Hopf bifurcation. Further, the properties of Hopf bifurcation are discussed. These theoretical results indicate that the delays play an important role in determining the dynamic behavior quantitatively. Therefore, it is a fact that delays are very important, which should not be missed in controlling HIV-1 infections.

Bifurcation analysis of HIV infection model with antibody and cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response

In this paper, a mathematical model for HIV‐1 infection with antibody, cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response is investigated. The stability of the infection‐free and infected steady states is investigated. The basic reproduction number R0 is identified for the proposed system. If R0 &lt; 1, then there is an infection‐free steady state, which is locally asymptotically stable. Further, the infected steady state is locally asymptotically stable for R0 &gt; 1 in the absence of immune response delay. We use Nyquist criterion to estimate the length of the delay for which stability continues to hold. Also the existence of the Hopf bifurcation is investigated by using immune response delay as a bifurcation parameter. Numerical simulations are presented to justify the analytical results. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Impact of delay on HIV-1 dynamics of fighting a virus with another virus.

In this paper, we propose a mathematical model for HIV-1 infection with intracellular delay. The model examines a viral-therapy for controlling infections through recombining HIV-1 virus with a genetically modified virus. For this model, the basic reproduction number R0 are identified and its threshold properties are discussed. When R0<1, the infection-free equilibrium E0 is globally asymptotically stable. When R0>1, E0 becomes unstable and there occurs the single-infection equilibrium Es, and E0 and Es exchange their stability at the transcritical point R0=1. If 1<R0<R1, where R1 is a positive constant explicitly depending on the model parameters, Es is globally asymptotically stable, while when R0>R1, Es loses its stability to the double-infection equilibrium Ed. There exist a constant R2 such that Ed is asymptotically stable if R1<R0<R2, and Es and Ed exchange their stability at the transcritical point R0=R1. We use one numerical example to determine the largest range of R0 for the local stability of Ed and existence of Hopf bifurcation. Some simulations are performed to support the theoretical results. These results show that the delay plays an important role in determining the dynamic behaviour of the system. In the normal range of values, the delay may change the dynamic behaviour quantitatively, such as greatly reducing the amplitudes of oscillations, or even qualitatively changes the dynamical behaviour such as revoking oscillating solutions to equilibrium solutions. This suggests that the delay is a very important fact which should not be missed in HIV-1 modelling.

Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model

Infection with HIV-1, degrading the human immune system and recent advances of drug therapy to arrest HIV-1 infection, has generated considerable research interest in the area. Bonhoeffer et al. (1997) [1], introduced a population model representing long term dynamics of HIV infection in response to available drug therapies. We consider a similar type of approximate model incorporating time delay in the process of infection on the healthy T cells which, in turn, implies inclusion of a similar delay in the process of viral replication. The model is studied both analytically and numerically. We also include a similar delay in the killing rate of infected CD4+ T cells by Cytotoxic T-Lymphocyte (CTL) and in the stimulation of CTL and analyse two resulting models numerically.The models with no time delay present have two equilibria: one where there is no infection and a non-trivial equilibrium where the infection can persist. If there is no time delay then the non-trivial equilibrium is locally asymptotically stable. Both our analytical results (for the first model) and our numerical results (for all three models) indicate that introduction of a time delay can destabilize the non-trivial equilibrium. The numerical results indicate that such destabilization occurs at realistic time delays and that there is a threshold time delay beneath which the equilibrium with infection present is locally asymptotically stable and above which this equilibrium is unstable and exhibits oscillatory solutions of increasing amplitude.

A mathematical model of HIV-1 infection including the saturation effect of healthy cell proliferation

A mathematical model of HIV-1 infection including the saturation effect of healthy cell proliferationIn this paper we derive a model describing the dynamics of HIV-1 infection in tissue culture where the infection spreads directly from infected cells to healthy cells trough cell-to-cell contact. We assume that the infection rate between healthy and infected cells is a saturating function of cell concentration. Our analysis shows that if the basic reproduction number does not exceed unity then infected cells are cleared and the disease dies out. Otherwise, the infection is persistent with the existence of an infected equilibrium. Numerical simulations indicate that, depending on the fraction of cells surviving the incubation period, the solutions approach either an infected steady state or a periodic orbit.

A delayed HIV-1 infection model with Beddington–DeAngelis functional response

In this paper, a mathematical model for HIV-1 infection with intracellular delay and Beddington–DeAngelis functional response is investigated. We obtain a necessary and sufficient condition for the global stability of the infection-free equilibrium and give some sufficient conditions for the local stability of the infected equilibrium.

Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay

In this paper, we consider a mathematical model for HIV-1infection with intracellular delay and cell-mediated immuneresponse. A novel feature is that both cytotoxic T lymphocytes(CTLs) and the intracellular delay are incorporated into themodel. We obtain a necessary and sufficient condition for theglobal stability of the infection-free equilibrium and givesufficient conditions for the local stability of the two infectionequilibria: one without CTLs being activated and the other with.We also perform some numerical simulations which support theobtained theoretical results. These results show that largerintracellular delay may help eradicate the virus, while theactivation of CTLs can only help reduce the virus load andincrease the healthy CD$_4^+$ cells population in the long termsense.

Impact of delays in cell infection and virus production on HIV-1 dynamics

Analysed is a mathematical model for HIV-1 infection with two delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells. For this model, the basic reproduction number is identified and its threshold property is discussed: the uninfected steady state is proved to be globally asymptotically stable if and unstable if . In the latter case, an infected steady state occurs and is proved to be locally asymptotically stable. The formula for shows that increasing either of the two delays will decrease . This may suggest a new direction for new drugs-drugs that can prolong the latent peri od and/or slow down the virus production process.