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In this paper, a fractional order HIV/HTLV co-infection model with HIV-specific antibody immune response is established. Two cases are considered: constant control and optimal control. For the constant control system, the existence and uniqueness of the positive solutions are proved, and then the sufficient conditions for the existence and stability of five equilibriums are obtained. For the second case, the Pontryagin's Maximum Principle is used to analyze the optimal control, and the formula of the optimal solution are derived. After that, some numerical simulations are performed to validate the theoretical prediction. Numerical simulations indicate that in the case of HIV/HTLV co-infection, the concentration of CD4+T cells is no longer suitable as an effective reference data for understanding the development process of the disease. On the contrary, the number of HIV virus particles should be used as an important indicator for reference. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a multiple delayed HIV-1 infection model which describes the interactions of humoral, cytotoxic T lymphocyte (CTL) immune responses, and two modes of transmission that are the classical virus-to-cell infection and the direct cell-to-cell transmission are investigated. The model incorporates three delays, including the delays of cell infection, virus production and activation of immune response. We first prove the well-posedness of the model, and calculate the biological existence of equilibria and the reproduction numbers, which contain virus infection, humoral immune response, CTL immune response, CTL immune competition, and humoral immune competition. Further, the threshold conditions for the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response, and interior are established by utilizing linearization method and the Lyapunov functionals. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics. [ABSTRACT FROM AUTHOR]

Latently infected CD 4 + T cells represent one of the major obstacles to HIV eradication even after receiving prolonged highly active anti-retroviral therapy (HAART). Long-term use of HAART causes the emergence of drug-resistant virus which is then involved in HIV transmission. In this paper, we develop mathematical HIV models with staged disease progression by incorporating entry inhibitor and latently infected cells. We find that entry inhibitor has the same effect as protease inhibitor on the model dynamics and therefore would benefit HIV patients who developed resistance to many of current anti-HIV medications. Numerical simulations illustrate the theoretical results and show that the virus and latently infected cells reach an infected steady state in the absence of treatment and are eliminated under treatment whereas the model including homeostatic proliferation of latently infected cells maintains the virus at low level during suppressive treatment. Therefore, complete cure of HIV needs complete eradication of latent reservoirs. [ABSTRACT FROM AUTHOR]

In this paper, a model of HIV infection taking into account cell-free and cell-to-cell transmissions is investigated under latency. The viral production rate of infectious CD4 T cells and the activation rate of latently infected CD4 T cells play a major role in the evolution of the disease. A system of nonlinear differential equations is derived to capture the dynamics and systematically analyzed thanks to the spectral theory of dynamical systems and a Routh-Hurwitz criterion. In particular, numerical simulations will be performed in order to extract the critical epidemiological features of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we introduce ordinary and delay differential equations to describe the interactions between a malignant tumour and the immune systemin vivoin the presence of human immunodeficiency virus (HIV) infection of CD4+T-cells. In the delay model, we take into account the time lags required by the healthy effector cell components to recognize the pathogens and tumour cells. The models consist of four populations: tumour cells, healthy effector cells (CD4+T-cells), effector cells infected by HIVs and free viral particles. The presence of delay term in the model leads to a notable increase in the complexity of the observed behaviour. We investigate the qualitative behaviour of the models and find the conditions that guarantee the asymptotic stability of the steady states. Numerical simulations are provided to illustrate and extend the theoretical results. The obtained results are consistent with the real phenomena and give a better understanding of cancer immunity and viral oncogenesis. [ABSTRACT FROM PUBLISHER]

In this paper, neglecting the effects of free virus, we consider a simple model of cell-to-cell spread of HIV-1. We discuss the impact of the CD8+ cell non-cytotoxic anti-viral response (CNAR) and cytotoxic T lymphocyte (CTL) activity on infection by HIV-1. Two types of models are considered: the ordinary differential equation (ODE) model and the discrete time delay differential equation (DDE) system. The steady states of the ODE model are globally asymptotically stable respectively under two threshold criteria. In the DDE model, the global stability of the infected steady state of the ODE model becomes only ultimately stable. Moreover, at a certain interval of the time delay, the DDE model will produce Hopf bifurcation or periodic solutions. The introduction of CTL and CNAR will change the values of these steady states and induce fluctuations in the cell concentration. It also affects the critical value of the time delay and is of utility in the interpretation of typical HIV-dynamics data resulting from clinical trials. The DDE model produces sustained infective oscillations in some real parameter values, which is different from the result of the similar cell-free viral spread model with delay. [ABSTRACT FROM AUTHOR]

Recent studies demonstrate that resource competition is an essential component of T-cell proliferation in HIV progression, which can contribute instructively to the disease development. In this paper, we formulate an age-structured within-host HIV model, in the form of a hyperbolic partial differential equation (PDE) for infected target cells coupled with two ordinary differential equations for uninfected T-cells and the virions, to explore the effects of both the T-cell competition and viral shedding variations on the viral dynamics. The basic reproduction number is derived for a general viral production rate which determines the local stability of the infection-free equilibrium. Two special forms of viral production rates, which are extensively investigated in previous literature, the delayed exponential distribution and a step function rate, are further investigated, where the original system can be reduced into systems of delay differential equations. It is confirmed that there exists a unique positive equilibrium for two special viral production rates when the basic reproduction number is greater than one. However, the model exhibits the phenomenon of backward bifurcation, where two positive steady states coexist with the infection-free equilibrium when the basic reproduction number is less than one. [ABSTRACT FROM AUTHOR]

We consider a Human Immunodeficiency Virus (HIV) model with a logistic growth term and continue the analysis of the previous article. We now take the viral diffusion in a two-dimensional environment. The model consists of two ODEs for the concentrations of the target T cells, the infected cells, and a parabolic PDE for the virus particles. We study the stability of the uninfected and infected equilibria, the occurrence of Hopf bifurcation and the stability of the periodic solutions. [ABSTRACT FROM AUTHOR]