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• Two numerical methods are developed for solving the fractional-order HIV-1 infection model of CD4 + T-cells. • Two method (Bessel matrix method and Bessel quasilinearization method) are based on the novel Bessel polynomials. • Error and convergence analysis are studied. • Numerical applications and comparisons show that methods are effective. • In the comparison of the two methods used, the Bessel-QLM method gives better results. In this paper, two numerical methods based on the novel Bessel polynomials are developed to solve the fractional-order HIV-1 infection model of CD 4 + T-cells considering the impact of antiviral drug treatment. In first of these methods, by using the Bessel polynomial and collocation points, we transform the HIV problem into a system of nonlinear algebraic equations. And this method, which is the method of direct solution is called as Bessel matrix method. The second method, which is called the Bessel-QLM method converts firstly HIV problem to a sequence of linear equations by using the technique of quasilinearization and then the reduced problem is solved by the direct Bessel matrix method. Error and convergence analysis are studied for the Bessel method. Finally, the applications are made on the numerical examples and also the numerical results are compared with the results of other available techniques. It is observed from applications that the presented results are better than the results of other existing methods and also the Bessel-QLM method is more efficient than the direct Bessel method. [ABSTRACT FROM AUTHOR]

Based on some important biological meanings, we propose a class of HIV infection models incorporating both classical cell-free virus diffusion and direct cell-to-cell transmission. According to recent studies, the direct cell-to-cell transfer of HIV is a significantly more efficient mode of retroviral dissemination. In the first part of our analysis, we show that our model possesses non-negative solutions. Then, we derive sufficient conditions for the asymptotic stability of equilibriums. The analytical solutions are verified by simulation results. At the end of the paper, some important conclusions are given. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behavior of a tumor–immune system (T–IS) interaction model with two discrete delays, namely the immune activation delay for effector cells (ECs) and activation delay for helper T cells (HTCs). By analyzing the characteristic equations, we establish the stability of two equilibria (tumor-free equilibrium and immune-control equilibrium) and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Our results exhibit that both delays do not affect the stability of tumor-free equilibrium. However, they are able to destabilize the immune-control equilibrium and cause periodic solutions. We numerically illustrate how these two delays can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumors. The numerical simulation results show that the immune activation delay for HTCs can induce heteroclinic cycles to connect the tumor-free equilibrium and immune-control equilibrium. Furthermore, we observe that the immune activation delay for HTCs can even stabilize the unstable immune-control equilibrium. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a delayed six-dimensional human immunodeficiency virus (HIV) model with Cytotoxic T Lymphocytes (CTLs) immune response. Our model describes the interaction of HIV with two target cells: CD4+ T cells and macrophages. We derive that the global asymptotic attractivity of the model is completely determined by the basic reproduction number and the immune reproduction number for the viral infection. By constructing Lyapunov functionals, we have shown that the infection-free equilibrium , the immune-free equilibrium and the chronic-infection equilibrium are globally asymptotically attractive when and , respectively. [Copyright &y& Elsevier]

Abstract: A viral infection model with immune circadian rhythms is investigated in this paper. By employing the persistence theory, we establish a threshold between the extinction and the uniform persistence of the disease. These results can be used to explain the oscillation behaviors of virus population, which were observed in chronic HBV or HCV carriers. Further, numerical simulations indicate that the dynamics of the lytic component of cytotoxicity T cells (CTLs) is crucial to the outcome of a viral infection. [Copyright &y& Elsevier]

Abstract: It is well-known that the mathematical models provide very important information for the research of human immunodeficiency virus-type 1 and hepatitis C virus (HCV). However, the infection rate of almost all mathematical models is linear. The linearity shows the simple interaction between the T cells and the viral particles. In this paper, we consider the classical mathematical model with non-linear infection rate. The global dynamics of this model is rigorously established. We prove that, if the basic reproduction number , the HIV infection is cleared from the T-cells population; if , the HIV infection persists. Further, the existence of a non-trivial periodic solution is also studied by means of numerical simulation. [Copyright &y& Elsevier]

Abstract: Human T-cell lymphotropic virus I (HTLV-I) infection is linked to the development of adult T-cell leukemia/lymphoma (ATL), among many illness. The healthy CD4+ T cells infect HTLV-I through cell-to-cell contact with infected T-cells. The infected T cells can remain latent and harbor virus for several years before virus production occurs. Actively infected T cells can infect other T cells and can convert to ATL cells, whose growth is assumed to follow a classical logistic growth function. In this paper, we consider the classical mathematical model with saturation response of the infection rate. By stability analysis we obtained the condition for the infected T cells die out and the condition for HTLV-I infection becomes chronic. At the same time, we also obtained the condition for a unique endemic equilibrium is globally stable in the interior of the feasible region. [Copyright &y& Elsevier]