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Highlights • This is the first mathematical model of RA which quantitatively describes the progression of the disease. • The model explains the roles of macrophages, inflammatory fibroblasts, and T cells in the degradation of cartilage in a joint. • Example is given how to achieve the same efficacy in stabilization of the cartilage while decreasing negative side-effects. Abstract Rheumatoid arthritis (RA) is a common autoimmune disease that mainly affects the joints. It is characterized by synovial inflammation, which may result in cartilage and bone destruction. The present paper develops a mathematical model of chronic RA. The model is represented by a system of partial differential equations (PDEs) in the synovial fluid, the synovial membrane, and the cartilage. The model characterizes the progression of the disease in terms of the degradation of the cartilage. More precisely, we assume a simplified geometry in which the synovial membrane and the cartilage are planar layers adjacent to each other. We then quantify the state of the disease by how much the cartilage layer has decreased, or, equivalently, how much the synovial layer has increased. The model is used to evaluate treatments of RA by currently used drugs, as well as by experimental drugs. [ABSTRACT FROM AUTHOR]

• A model is proposed based on the key role of Regulatory T cells in chemoimmunotherapy. • The model's fixed points are obtained and the bifurcation diagrams are discussed. • New features of personalizing a cancer mathematical model are presented. • Considering Regulatory T cells leads to identifying critical patient conditions more precisely. Studying the mathematical dynamics of cancer has gained the attention of bioengineers in the past three decades. Different kinds of modelling considering various aspects of treatment have been proposed. In this paper, the key role of Regulatory T cells is discussed and a model in ordinary differential equation (ODE) form is proposed by adding this state to the system dynamics considering chemoimmunotherapy treatment. Regulatory T cells are considered as one of the main tumor cells' tactics to deceive the body's immune system. The improved model is verified mathematically and biologically and fits all criteria in both fields. The results show that entering Regulatory T cells state on cancer mathematical modelling for simulating body cells for chemoimmunotherapy provides a way to identify critical cases more carefully, which a simplified model is unable to accomplish. This point emphasizes the fact that this state must be present in cancer modelling to anticipate immune response more accurately. The advanced system fixed points are obtained by the Newton method and bifurcation diagrams are derived and discussed. New features and remarks are proposed during the journey of developing more accurate models that have the best fit with laboratory data. The sensitivity chart of the model is illustrated and novel aspects of discussions are made with the aim of personalizing a model for a patient and identifying critical conditions based on the chart before any treatment begins. This point enables physicians to determine whether critical conditions have occurred for a patient in a specific treatment or not. [ABSTRACT FROM AUTHOR]

• An individual-based model describing interactions between tumour and CD8+ T cells. • The model takes into account tumour antigen expression and presentation. • The degree of tumour heterogeneity has an impact on immune response efficacy. • Wider ranges of antigens promote immune escape of tumours. • Lower levels of antigen presentation diminish the immune action. Intra-tumour heterogeneity (ITH) has a strong impact on the efficacy of the immune response against solid tumours. The number of sub-populations of cancer cells expressing different antigens and the percentage of immunogenic cells (i.e. tumour cells that are effectively targeted by immune cells) in a tumour are both expressions of ITH. Here, we present a spatially explicit stochastic individual-based model of the interaction dynamics between tumour cells and CD8+ T cells, which makes it possible to dissect out the specific impact of these two expressions of ITH on anti-tumour immune response. The set-up of numerical simulations of the model is defined so as to mimic scenarios considered in previous experimental studies. Moreover, the ability of the model to qualitatively reproduce experimental observations of successful and unsuccessful immune surveillance is demonstrated. First, the results of numerical simulations of this model indicate that the presence of a larger number of sub-populations of tumour cells that express different antigens is associated with a reduced ability of CD8+ T cells to mount an effective anti-tumour immune response. Secondly, the presence of a larger percentage of tumour cells that are not effectively targeted by CD8+ T cells may reduce the effectiveness of anti-tumour immunity. Ultimately, the mathematical model presented in this paper may provide a framework to help biologists and clinicians to better understand the mechanisms that are responsible for the emergence of different outcomes of immunotherapy. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we sought to identify the CD4+ T-cell dynamics in the course of HIV infection in response to continuous and intermittent intravenous courses of interleukin-2 (IL-2), the principal cytokine responsible for progression of CD4+ T-lymphocytes from the G1 to the S phase of the cell cycle. Based on multivariate regression models, previous literature has concluded that the increase in survival of CD4+ T-cell appears to be the critical mechanism leading to sustained CD4+ T-cell levels in HIV-infected patients receiving intermittent IL-2 therapy. Underscored by comprehensive mathematical modeling, a major finding of the present work is related to the fact that, rather than due to any increase in survival of CD4+ T-cells, the expressive, selective and sustained CD4+ T-cell expansions following IL-2 administration may be related to the role of IL-2 in modulating the dynamics of Fas-dependent apoptotic pathways, such as activation-induced cell death (AICD) or HIV-specific apoptotic routes triggered by viral proteins. [Copyright &y& Elsevier]

Abstract: Epstein-Barr virus (EBV) can infect both B cells and epithelial cells. Infection of B cells enables the virus to persist within a host while infection of epithelial cells is suggested to amplify viral output. Data from a recent study have shown that the virus shedding in EBV positive individuals is relatively stable over short periods of time but varies significantly over long periods. The mechanisms underlying the regulation of virus shedding within a host are not fully understood. In this paper, we construct a model of ordinary differential equations to study the dynamics of virus shedding into the saliva of infected hosts. Infection of epithelial cells is further separated into infection by virus released from B cells and virus released from epithelial cells. We use the model to investigate whether the long-term variation and short-term stability of virus shedding can be generated by three possible factors: stochastic variations in the number of epithelial cells susceptible to virus released from infected B cells, to virus released from infected epithelial cells, or random variation in the probability that CD8+ T cells encounter and successfully kill infected cells. The results support all three factors to explain the long-term variation but only the first and third factors to explain the short-term stability of virus shedding into saliva. Our analysis also shows that clearance of virus shedding is possible only when there is no virus reactivation from B cells. [Copyright &y& Elsevier]

Abstract: Mathematical models of T cell population dynamics after infection typically assume that T cells differentiate according to a linear process in which they first become effector cells, and then after some time, differentiate further into memory cells. In this paper, we offer a different mathematical model which can equally well capture T cell dynamics, using data from lymphocytic choriomeningitis (LCMV) infection. Our model assumes that memory cells are intermediates that further differentiate into effector cells only from additional or stronger antigenic stimulation. Our assumption naturally leads to a testable prediction about the generation of T cell memory—that the memory phenotype of T cells should be present in detectable numbers during the expansion phase of the response. We use our model to estimate a rate of differentiation from memory type cells to effectors. We argue that this differentiation assumption, where memory cells are intermediates, captures recent experimental work on T cell differentiation, and hence this new mathematical model could be helpful in doing further studies of T cell population dynamics. We also propose a method of distinguishing the models by examining the ratio of memory T cells detectable long after an infection to the peak numbers of T cells at the end of the expansion phase. [Copyright &y& Elsevier]