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The time fractional-order nonlinear model of HIV-1 infection of T-cells with and without the rate of virus-producing cells’ death is very important in mathematical modeling. This paper presents proofs of the existence and uniqueness theorems using the Banach fixed point theorem, along with new approximate analytical solutions to two nonlinear fractional Equationuations characterizing HIV-1 infection in $CD{4^ + }$CD4+ T-cells. We use a combination of two methods, namely the natural transform method to deal with the linear terms and the Adomian decomposition method to deal with the nonlinear terms. There are two nonlinear numerical examples presented in this work, namely, we find approximate solutions to the time fractional-order nonlinear model of HIV-1 infection of T cells with and without the rate of virus-producing cells’ death using an effective method called the Adomian natural decomposition method (ANDM). The present approach, which has numerous applications in the science and engineering fields, is a great alternative to the many existing methods for solving systems of differential Equationuations. It also holds great promise for additional real-world applications. The Mathematica 12 package is used to carry out some of the nonlinear term calculations and the graphs presented in this work. [ABSTRACT FROM AUTHOR]

In this paper, we incorporate immune systems containing Cytotoxic T lymphocyte and humoral immunity into a general human immunodeficiency viruses infection model, which also considers logistic growth for target cells and both modes of spread, cell-to-cell and cell-free represents, by linear functions. We derive five threshold parameters which are used to study the existence of equilibria. By considering the characteristic equations, the local stability of disease-free and immune-free equilibria is investigated. Lyapunov functions and LaSalle's invariance are constructed to prove the global stability of all steady states. Global dynamics of the human immunodeficiency viruses model can be accurately expressed by threshold parameters; also the existence of Hopf bifurcation is discussed and the results are expressed in the form of a proposition. Furthermore, numerical simulations confirm the corresponding theoretical results. [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]

It is well known that dynamical systems are very useful tools to study viral diseases such as HIV, HBV, HCV, Ebola and Influenza. This paper focuses on a mathematical model of the cell-to-cell and the cell-free spread of HIV with both linear and nonlinear functional responses and logistic target cell growth. The reproduction number of each mode of transmission has been calculated and their sum has been considered as the basic reproduction number. Based on the values of the reproduction number, the local and global stabilities of the rest points are investigated. Choosing a suitable bifurcation parameter, some conditions for the occurrence of Hopf bifurcation are also obtained. Moreover, numerical simulations are presented to support the analytical results. Finally, to study the effect of the drug on the disease process, some control conditions are determined. Since two modes of transmission and both linear and nonlinear functional responses have been included in this manuscript, our obtained results are a generalization of those in the literature. Moreover, the results are obtained with weaker assumptions in comparison with the previous ones. [ABSTRACT FROM AUTHOR]

In recent decades, AIDS has been one of the main challenges facing the medical community around the world. Due to the large human deaths of this disease, researchers have tried to study the dynamic behaviors of the infectious factor of this disease in the form of mathematical models in addition to clinical trials. In this paper, we study a new mathematical model in which the dynamics of CD 4 + T-cells under the effect of HIV-1 infection are investigated in the context of a generalized fractal-fractional structure for the first time. The kernel of these new fractal-fractional operators is of the generalized Mittag-Leffler type. From an analytical point of view, we first derive some results on the existence theory and then the uniqueness criterion. After that, the stability of the given fractal-fractional system is reviewed under four different cases. Next, from a numerical point of view, we obtain two numerical algorithms for approximating the solutions of the system via the Adams-Bashforth method and Newton polynomials method. We simulate our results via these two algorithms and compare both of them. The numerical results reveal some stability and a situation of lacking a visible order in the early days of the disease dynamics when one uses the Newton polynomial. [ABSTRACT FROM AUTHOR]

The paper is focused on stochastic modelling of the process of transition of infected cells and virions of HIV-1 infection between two lymph nodes. The model is based on the following assumptions: (1) the duration of transition of infected cells and virions between two lymph nodes is set using a time-dependent function, (2) infected cells produce virions in the process of transition between two lymph nodes, (3) infected cells and virions may die when moving between two lymph nodes. The methods of the theory of branching random processes are used to study analytically the model variables. An algorithm for statistical modelling of the number of infected cells and virions in the second lymph node is presented. The results of computational experiments studying the distribution law of the number of virions produced by one infected cell depending on the duration of movement between two lymph nodes are presented. [ABSTRACT FROM AUTHOR]