The dynamics of T-cell fratricide: application of a robust approach to mathematical modelling in immunology

Modeling and analysis of a within-host HIV/HTLV-I co-infection

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that attack the CD4 + T cells and impair their functions. Both HIV and HTLV-I can be transmitted between individuals through direct contact with certain body fluids from infected individuals. Therefore, a person can be co-infected with both viruses. HIV causes acquired immunodeficiency syndrome (AIDS), while HTLV-I is the causative agent for adult T-cell leukemia (ATL) and HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP). Several mathematical models have been developed in the literature to describe the within-host dynamics of HIV and HTLV-I mono-infections. However, modeling a within-host dynamics of HIV/HTLV-I co-infection has not been involved. The present paper is concerned with the formulation and investigation of a new HIV/HTLV-I co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD4 + T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell transmission. On the other side, HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact through the virological synapse, and (ii) vertical transmission through the mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We define a set of threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and Lyapunov-LaSalle asymptotic stability theorem. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

Modeling and stability analysis of HIV/HTLV-I co-infection

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that infect the susceptible CD[Formula: see text]T cells. It is known that HIV and HTLV-I have in common a way of transmission through direct contact with certain body fluids related to infected individuals. Therefore, it is not surprising that a mono-infected person with one of these viruses can be co-infected with the other virus. In the literature, a great number of mathematical models has been presented to describe the within-host dynamics of HIV or HTLV-I mono-infection. However, the within-host dynamics of HIV/HTLV-I co-infection has not been modeled. In this paper, we develop a new within-host HIV/HTLV-I co-infection model. The model includes the impact of Cytotoxic T lymphocytes (CTLs) immune response, which is important to control the progression of viral co-infection. The model describes the interaction between susceptible CD[Formula: see text]T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. We first show the nonnegativity and boundedness of the model’s solutions and then we calculate all possible equilibria. We derive the threshold parameters which govern the existence and stability of all equilibria of the model. We prove the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle’s invariance principle. We have presented numerical simulations to illustrate the effectiveness of our main results. In addition, we discuss the effect of HTLV-I infection on the HIV-infected patients and vice versa.

HTLV/HIV Dual Infection: Modeling and Analysis

Human T-lymphotropic virus type I (HTLV-I) and human immunodeficiency virus (HIV) are two famous retroviruses that share similarities in their genomic organization, and differ in their life cycle as well. It is known that HTLV-I and HIV have in common a way of transmission via direct contact with certain body fluids related to infected patients. Thus, it is not surprising that a single-infected person with one of these viruses can be dually infected with the other virus. In the literature, many researchers have devoted significant efforts for modeling and analysis of HTLV or HIV single infection. However, the dynamics of HTLV/HIV dual infection has not been formulated. In the present paper, we formulate an HTLV/HIV dual infection model. The model includes the impact of the Cytotoxic T lymphocyte (CTLs) immune response, which is important to control the dual infection. The model describes the interaction between uninfected CD4+T cells, HIV-infected cells, HTLV-infected cells, free HIV particles, HIV-specific CTLs, and HTLV-specific CTLs. We establish that the solutions of the model are non-negative and bounded. We calculate all steady states of the model and deduce the threshold parameters which determine the existence and stability of the steady states. We prove the global asymptotic stability of all steady states by utilizing the Lyapunov function and Lyapunov–LaSalle asymptotic stability theorem. We solve the system numerically to illustrate the our main results. In addition, we compared between the dynamics of single and dual infections.

Shifted Boubaker Lagrangian approach for solving biological systems

Mathematical models and computer simulations are useful experimental tools for building and testing theories. Many mathematical models in biology can be formulated by a nonlinear system of ordinary differential equations. This work deals with the numerical solution of the hantavirus infection model, the human immunodeficiency virus (HIV) infection model of CD4[Formula: see text]T cells and the susceptible–infected–removed (SIR) epidemic model using a new reliable algorithm based on shifted Boubaker Lagrangian (SBL) method. This method reduces the solution of such system to a system of linear or nonlinear algebraic equations which are solved using the Newton iteration method. The obtained results of the proposed method show highly accurate and valid for an arbitrary finite interval. Also, those are compared with fourth-order Runge–Kutta (RK4) method and with the solutions obtained by some other methods in the literature.

An exponential collocation method for the solutions of the HIV infection model of CD4+T cells

In this paper, an exponential method is presented for the approximate solutions of the HIV infection model of CD4[Formula: see text]T. The method is based on exponential polynomials and collocation points. This model problem corresponds to a system of nonlinear ordinary differential equations. Matrix relations are constructed for the exponential functions. By aid of these matrix relations and the collocation points, the proposed technique transforms the model problem into a system of nonlinear algebraic equations. By solving the system of the algebraic equations, the unknown coefficients are computed and thus the approximate solutions are obtained. The applications of the method for the considered problem are given and the comparisons are made with the other methods.

Computational analysis of the model describing HIV infection of CD4+T Cells

An analysis of the model underpinning the description of the spread of HIV infection of CD4⁺T cells is examined in detail in this work. Investigations of the disease free and endemic equilibrium are done using the method of Jacobian matrix. An iteration technique, namely, the homotopy decomposition method (HDM), is implemented to give an approximate solution of nonlinear ordinary differential equation systems. The technique is described and illustrated with numerical examples. The approximated solution obtained via HDM is compared with those obtained via other methods to prove the trustworthiness of HDM. Moreover, the lessening and simplicity in calculations furnish HDM with a broader applicability.

Model-based optimal immunization for antibody production in birds

A dynamic model of the immune response in poultry was developed in order to enhance antibody production. Efficient production of antibodies is very valuable for researchers and physicians since they are used for other molecules detection. Large amounts of poultry-based antibodies are found in birds' eggs. However, inoculation timetables are based on empirical data. A seven differential equation system represents cellular and molecular populations of the humoral immune response in poultry. Model parameters are presented and simulation results reflect the typical immune responses. Finally, a genetic algorithm was designed in order to optimize antibody production.

Cytotoxic T lymphocytes kill multiple targets simultaneously via spatiotemporal uncoupling of lytic and stimulatory synapses

A longstanding paradox in the activation of cytotoxic T lymphocytes (CTL) arises from the observation that CTL recognize and rapidly destroy target cells with exquisite sensitivity despite the fact that cytokine production requires sustained signaling at the immunological synapse. Here we solve this paradox by showing that CTL establish sustained synapses with targets offering strong antigenic stimuli and that these synapses persist after target cell death. Simultaneously, CTL polarize lytic granules toward different cells without discrimination regarding antigenic potential. Our results show that spatiotemporal uncoupling of immunological synapse and lytic granule secretion allows multiple killing and sustained signaling by individual CTL. This unique mechanism of responding to multiple contacts provides remarkable efficiency to CTL function.

Dynamics of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells

Stilianakis and Seydel (Bull. Math. Biol., 1999) proposed an ODE model that describes the T-cell dynamics of human T-cell lymphotropic virus I (HTLV-I) infection and the development of adult T-cell leukemia (ATL). Their model consists of four components: uninfected healthy CD4+ T-cells, latently infected CD4+ T-cells, actively infected CD4+ T-cells, and ATL cells. Mathematical analysis that completely determines the global dynamics of this model has been done by Wang et al. (Math. Biosci., 2002). In this note, we first modify the parameters of the model to distinguish between contact and infectivity rates. Then we introduce a discrete time delay to the model to describe the time between emission of contagious particles by active CD4+ T-cells and infection of pure cells. Using the results in Culshaw and Ruan (Math. Biosci., 2000) in the analysis of time delay with respect to cell-free viral spread of HIV, we study the effect of time delay on the stability of the endemically infected equilibrium. Numerical simulations are presented to illustrate the results.

Human T cell lymphotropic virus type I (HTLV-I)-specific CD4+ T cells: immunodominance hierarchy and preferential infection with HTLV-I

CD4(+) T cells predominate in early lesions in the CNS in the inflammatory disease human lymphotropic T cell virus type I (HTLV-I)-associated myelopathy/tropical spastic paraparesis (HAM/TSP), but the pathogenesis of the disease remains unclear and the HTLV-I-specific CD4(+) T cell response has been little studied. We quantified the IFN-gamma-producing HTLV-I-specific CD4(+) T cells, in patients with HAM/TSP and in asymptomatic carriers with high proviral load, to test two hypotheses: that HAM/TSP patients and asymptomatic HTLV-I carriers with a similar proviral load differ in the immunodominance hierarchy or the total frequency of specific CD4(+) T cells, and that HTLV-I-specific CD4(+) T cells are preferentially infected with HTLV-I. The strongest CD4(+) T cell response in both HAM/TSP patients and asymptomatic carriers was specific to Env. This contrasts with the immunodominance of Tax in the HTLV-I-specific CD8(+) T cell response. The median frequency of HTLV-I-specific IFN-gamma(+) CD4(+) T cells was 25-fold greater in patients with HAM/TSP (p = 0.0023, Mann-Whitney) than in asymptomatic HTLV-I carriers with a similar proviral load. Furthermore, the frequency of CD4(+) T cells infected with HTLV-I (expressing Tax protein) was significantly greater (p = 0.0152, Mann-Whitney) among HTLV-I-specific cells than CMV-specific cells. These data were confirmed by quantitative PCR for HTLV-I DNA. We conclude that the high frequency of specific CD4(+) T cells was associated with the disease HAM/TSP, and did not simply reflect the higher proviral load that is usually found in HAM/TSP patients. Finally, we conclude that HTLV-I-specific CD4(+) T cells are preferentially infected with HTLV-I.

Human T-lymphotropic virus type 1 (HTLV-1): persistence and immune control

The human retrovirus human T-lymphotropic virus type 1 (HTLV-1) is associated with two distinct types of disease: the malignancy known as adult T-cell leukemia and a range of chronic inflammatory conditions including the central nervous system disease HTLV-1-associated myelopathy/tropical spastic paraparesis (HAM/TSP). Until recently, it was believed that HTLV-1 was largely latent in vivo. However, evidence from a number of types of experiments shows that HTLV-1 persistently expresses its genes, and that the "set point" of an individual's proviral load of HTLV-1 is mainly determined by the efficiency of that individual's cellular immune response to the virus. These conclusions have two main consequences. First, HTLV-1 may be vulnerable to antiretroviral drug therapy or immunotherapy. Second, HTLV-1 infection has become a useful system to analyze the determinants of the efficiency of the antiviral immune response.

An introduction to lymphocyte and viral dynamics: the power and limitations of mathematical analysis

Mathematics is a useful tool in the analysis and understanding of population dynamic aspects of the immune response. However, the power of mathematical modelling in immunology is frequently limited by the shortage of experimental data. Here, we review the contribution of mathematics to two areas of immunology. We highlight the problem caused by lack of knowledge of the system, which can greatly restrict the use of mathematics and lead to errors caused by model-specific results.