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The spreading of HIV or HTLV-I among the cells has received the great attention in recent modelling study to explore the virus infection dynamics. The co-infection of HIV and HTLV-I with the effect of Cytotoxic T-lymphocytes (CTLs) immune response is also important from epidemiological point of view. To identify the co-infection scenario of HIV and HTLV-I with the CTLs effect we proposed in this paper a six compartmental ODE-model with uninfected, HIV-infected, HTLV-I infected CD4+T cells and free HIV virus particles with HIV specific CTLs and HTLV-I specific CTLs. The rates of infection of the cases are considered here saturated type and proliferation rate of uninfected and HIV infected CD4+ T-cells are of logistic terms. To establish the well-posedness of the model we have shown that the solution of the proposed model is non-negative and bounded. We obtain the basic reproduction number which is the maximum of the HIV-related reproduction and the HTLV-I related reproduction number. Along with the disease free equilibrium point the system contains other seven endemic equilibrium points containing infection by single disease or both. Analytically, we establish the local and global stability conditions of the equilibrium points and also we establish that the system experiences transcritical bifurcation by the generation of only HIV or HTLV-I infected endemic equilibrium point. Using numerical simulations, we validate the theoretical results and found two infection paths, one initiating with HIV and other with HTLV-I, both cases ultimately become co-infected. Finally, using the optimal control analysis we found the optimal policy for treatment using AVR, RTI & PI for HIV or AZT for HTLV-I control and lastly concluded by some recommendations. • A co-infected HIV & HTLV-I infection model in vivo. • Proliferation of CD4+ T-cells are taken in logistic form. • Incidence rate is taken as saturated type. • Verified competitive exclusive principle. • Spreading path has been studied using Flow-diagram. • Three types of control strategies are investigated. [ABSTRACT FROM AUTHOR]

Highlights • We formulate a viral infection model driven by Lévy noise. • This model admits a unique global positive solution. • Sufficient conditions for extinction and persistence in mean are obtained. • Numerical simulations are introduced to illustrate the results. Abstract In this paper, we are interested in the study of a stochastic viral infection model with immune impairment driven by Lévy noise. First we prove the existence of a unique global solution to the model. By means of the Lyapunov method we study the stability of the equilibria. We present sufficient conditions for the extinction and persistence in mean. Furthermore, we present some numerical results to support the theoretical work. [ABSTRACT FROM AUTHOR]

This paper deals with the study of a virus dynamics model in order to get better insights into HIV infection within the body. The model incorporates therapeutic modalities such as reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs). RTIs prevent viral replication/entry within the infected CD4 + T cells while PIs block the virus assembly and thus further propagation and production of new virions. The proliferation of uninfected CD4 + T cells has been assumed to be as full logistic growth term to capture the dynamics of HIV virus. The model also considers two important components of the acquired immune response, namely the cytotoxic T lymphocyte (CTL) immune response (self stimulation due to infection and stimulation due to infected cells have been considered) and antibody immune response. Critical threshold conditions for the existence of equilibrium points have been determined. We studied the analytical behavior of these equilibrium points locally as well as globally using Lasalle’s invariance principle and Lyapunov’s direct method. We explored the sensitivity of the therapeutic drugs on the model system. Further, the behavior of the proposed model system has been studied numerically through simulation tools. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a Lotka–Volterra competition like model which consists of system of differential equations with piecewise constant arguments of delay to study of interaction between tumor cells and Cytotoxic T lymphocytes (CTLs). In order to get local and global behaviors of the system, we use Schur–Cohn criterion and constructed a Lyapunov function. Some algebraic conditions which satisfy local and global stability of the system are obtained. In addition, we investigate the possible bifurcation types for the system and observe that the system may undergo Neimark–Sacker bifurcation. Moreover, it is predicted a threshold value above which there is uncontrollable tumor growth, and below periodic solutions that leading to tumor dormant state occur. [ABSTRACT FROM AUTHOR]

• A reaction-diffusion fractional-order model for M1 oncolytic virotherapy with CTL immune response is presented. • Threshold parameters and equilibria are determined. • Stability analysis are rigorously established. • Numerical simulations are given to illustrate the analytical results. Despite many years of research, cancer continues to be a major worldwide concern. Several researchers proposed their mathematical models to understand and describe the dynamics of tumor cells under some cancer treatments. In this paper, we present a fractional partial differential equation system to describe the dynamics in space and time of the concentration of normal cells, tumor cells, nutrient, M1 virus, and cytotoxic T lymphocytes (CTL) cells, as well as the interaction between them. We study and analyze the equilibrium points of the presented system. The global stability of these equilibrium points is proved by constructing adequate Lyapunov functional for each equilibrium point. Furthermore, we give some numerical simulations to illustrate our results. [ABSTRACT FROM AUTHOR]

Type 1 diabetes (T1D), previously known as juvenile diabetes or insulin-dependent diabetes, is an autoimmune disease characterized by the insufficient (or lack of) production of insulin by the pancreas. Insulin is crucial to maintain blood sugar at healthy levels. High blood sugar damages the body and causes a variety of symptoms, ranging from severe thirst, fatigue, to urinary infections. The cells responsible for the production of insulin are the β -cells. In T1D, these are killed by an abnormal response of the immune system. Specific clones of cytotoxic T-cells invade the pancreatic islets of Langerhans, and eliminate them. T1D diabetes may develop in human immunodeficiency virus (HIV)-infected patients, though in rare situations. In this paper, we propose a cell model for the development of T1D in these patients, after immune restoration, during highly active antiretroviral therapy (HAART). The study includes the derivation of the qualitative properties of the model, and its comprehensive investigation via path-following methods, using the continuation platform COCO. In this way, the main theoretical predictions are verified in detail. Furthermore, this numerical part establishes accurate parameter thresholds to ensure an effective disease treatment in HIV-infected persons to prevent the development of T1D. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes an adaptive immunity HIV infection model. The model describes the interaction between healthy CD 4 + T cells, silent (latent) infected cells, active infected cells, free HIV particles, Cytotoxic T lymphocytes (CTLs) and antibodies. The healthy CD 4 + T cells can be infected when they are contacted by one of the following: (i) free HIV particles, and this is known as virus-to-cell (VTC) transmission (ii) silent infected cells, and we call this mode of infection as silent HIV-infected cell-to-cell (CTC) transmission, and (iii) active infected cells, and we call this mechanism active HIV-infected CTC transmission. The incidence rates of the healthy CD 4 + T cells with free HIV particles, silent infected cells, and active infected cells are given by general functions. Moreover, the production/proliferation and removal/death rates of all compartments are represented by general functions. The model is an improvement of the existing HIV infection models which have neglected the incidence between the silent infected cells and healthy CD 4 + T cells. We first show that the model is well-posed. Then, we show that the model has five equilibria and their existence are governed by five threshold parameters. Under a set of conditions on the general functions and the threshold parameters, we have proven the global asymptotic stability of all equilibria by using Lyapunov's method. We have illustrated the theoretical results via numerical simulations. We have studied the effect of CTC transmission on the dynamical behavior of the system. We have shown that inclusion of CTC transmission decreases the concentration of the healthy CD4 + T cells and increases the concentrations of the infected cells and free HIV particles. [ABSTRACT FROM AUTHOR]

• A mathematical model of CAR-T cell treatment for T-cell leukaemias is developed. • CAR-T cell treatment can control, but not eradicate, T-cell leukaemias. • The model describes the dynamics observed in vitro. Immunotherapy has the potential to change the way all cancer types are treated and cured. Cancer immunotherapies use elements of the patient immune system to attack tumor cells. One of the most successful types of immunotherapy is CAR-T cells. This treatment works by extracting patient's T-cells and adding to them an antigen receptor allowing tumor cells to be recognized and targeted. These new cells are called CAR-T cells and are re-infused back into the patient after expansion in-vitro. This approach has been successfully used to treat B-cell malignancies (B-cell leukemias and lymphomas). However, its application to the treatment of T-cell leukemias faces several problems. One of these is fratricide, since the CAR-T cells target both tumor and other CAR-T cells. This leads to nonlinear dynamical phenomena amenable to mathematical modeling. In this paper we construct a mathematical model describing the competition of CAR-T, tumor and normal T-cells and studied some basic properties of the model and its practical implications. Specifically, we found that the model reproduced the observed difficulties for in-vitro expansion of the therapeutic cells found in the laboratory. The mathematical model predicted that CAR-T cell expansion in the patient would be possible due to the initial presence of a large number of targets. We also show that, in the context of our mathematical approach, CAR-T cells could control tumor growth but not eradicate the disease. [ABSTRACT FROM AUTHOR]

• We propose a new model of fractional-order for the hepatitis C virus (HCV) infection. • The results obtained in this paper are a kind of the power-law and the exponential-law kernel. • The numerical solutions based upon the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation. • We investigate the effect of the variation of the fractional-order. • We obtain chaotic behaviors for (HCV) infection. Our main aim in this article is to introduce and investigate a new model of fractional-order model for hepatitis C virus (HCV) infection with immune response, in which the effect of dendritic cells (DC) and cytotoxic T lymphocytes (CTL) on HCV infection is considered. In the first stage, the HCV model was extended to Liouville-Caputo (LC) fractional derivative and Caputo-Fabrizio fractional derivative in the Caputo sense (CFC). In particular, we evaluate the approximate solutions of this model by means of the power law and the exponential law kernel. The approximate solutions are based upon the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation. We study the effect of the variation the fractional-order on the behavior of the solutions due to the presence of definitions of new fractional-calculus operators. We observe that the chaotic behavior and illustrate the chaotic of a chronic HCV infection model with immune response by plotting the solutions in the plane. Finally, some numerical simulations using with the aid of Mathematica were done for different values of fractional order. [ABSTRACT FROM AUTHOR]

Riesz wavelets in L 2 (R) have been proven as a useful tool in the context of both pure and numerical analysis in many applications, due to their well prevailing and recognized theory and its natural properties such as sparsity and stability which lead to a well-conditioned scheme. In this paper, an effective and accurate technique based on Riesz wavelets is presented for solving weakly singular type of fractional order integro-differential equations with applications to solve system of fractional order model that describe the dynamics of uninfected, infected and free virus carried out by cytotoxic T lymphocytes (CTL). The Riesz wavelet in this work is constructed via the smoothed pseudo-splines refinable functions. The advantage of using such wavelets, in the context of fractional and integro-differential equations, lies on the simple structure of the reduced systems and in the powerfulness of obtaining approximated solutions for such equations that have weakly singular kernels. The proposed method shows a good performance and high accuracy orders. [ABSTRACT FROM AUTHOR]

• Mathematical modelling of the interactions of cancer stem cells (CSCs), non-CSCs, cytotoxic T-cells (CTCs), and dendritic cells (DCs) to replicate the following experimental results: Grafting mice with CSCs produces more, and larger tumors than transfusing mice with nCSCs. Giving mice dendritic cell vaccines or CTC treatment prepared with CSCs is more effective than giving mice CTC treatment prepared with nCSCs. • Mathematical modelling of how chemotherapy and immunotherapy could work together to minimize tumor size and future tumor growth: Although chemotherapy is more effective at reducing current tumor burden, CSC-specific therapy reduces tumorogenicity, and decreases future tumor burden. The cancer stem cell hypothesis states that tumors are heterogeneous and comprised of several different cell types that have a range of reproductive potentials. Cancer stem cells (CSCs), represent one class of cells that has both reproductive potential and the ability to differentiate. These cells are thought to drive the progression of aggressive and recurring cancers since they give rise to all other constituent cells within a tumor. With the development of immunotherapy in the last decade, the specific targeting of CSCs has become feasible and presents a novel therapeutic approach. In this paper, we construct a mathematical model to study how specific components of the immune system, namely dendritic cells and cytotoxic T-cells interact with different cancer cell types (CSCs and non-CSCs). Using a system of ordinary differential equations, we model the effects of immunotherapy, specifically dendritic cell vaccines and T-cell adoptive therapy, on tumor growth, with and without chemotherapy. The model reproduces several results observed in the literature, including temporal measurements of tumor size from in vivo experiments, and it is used to predict the optimal treatment schedule when combining different treatment modalities. Importantly, the model also demonstrates that chemotherapy increases tumorigenicity whereas CSC-targeted immunotherapy decreases it. [ABSTRACT FROM AUTHOR]