Mathematical models for immunology:current state of the art and future research directions

The advances in genetics and biochemistry that have taken place over the last 10 years led to significant advances in experimental and clinical immunology. In turn, this has led to the development of new mathematical models to investigate qualitatively and quantitatively various open questions in immunology. In this study we present a review of some research areas in mathematical immunology that evolved over the last 10 years. To this end, we take a step-by-step approach in discussing a range of models derived to study the dynamics of both the innate and immune responses at the molecular, cellular and tissue scales. To emphasise the use of mathematics in modelling in this area, we also review some of the mathematical tools used to investigate these models. Finally, we discuss some future trends in both experimental immunology and mathematical immunology for the upcoming years

Behaviour of Solutions to Marchuk's Model Depending on a Time Delay

Marchuk's model of an immune reaction is a system of differential equations with a time delay. The aim of this paper is to study the behaviour of solutions to Marchuk's model depending upon the delay of immune reaction and the history of an illness. We study Marchuk's model without delays, with aconstant delay and with an infinite delay. A continuous dependence on thedelay is considered. Bifurcation points are found using computer simulations

Logistic Equations in Tumour Growth Modelling

The aim of this paper is to present some approaches to tumour growth modelling using the logistic equation. As the first approach the well-known ordinary differential equation is used to model the EAT in mice. For the same kind of tumour, a logistic equation with time delay is also used. As the second approach, a logistic equation with diffusion is proposed. In this case a delay argument in the reaction term is also considered. Some mathematical properties of the presented models are studied in the paper. The results are illustrated using computer simulations

Impact of Time Delay in Perceptual Decision-Making: Neuronal Population Modeling Approach

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Delay can stabilize: Love affairs dynamics

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A modified van der Pol equation with delay in a description of the heart action

In this paper, a modified van der Pol equation is considered as a description of the heart action. This model has a number of interesting properties allowing reconstruction of phenomena observed in physiological experiments as well as in Holter electrocardiographic recordings. Our aim is to study periodic solutions of the modified van der Pol equation and take into consideration the influence of feedback and delay which occur in the normal heart action mode as well as in pathological modes. Usage of certain values for feedback and delay parameters allows simulating the heart action when an accessory conducting pathway is present (Wolff–Parkinson–White syndrome)

Optimal Protocols for the Anti-VEGF Tumor Treatment

Cancer treatment using the antiangiogenic agents targets the evolution of the tumor vasculature. The aim is to significantly reduce supplies of oxygen and nutrients, and thus starve the tumor and induce its regression. In the paper we consider well established family of tumor angiogenesis models together with their recently proposed modification, that increases accuracy in the case of treatment using VEGF antibodies. We consider the optimal control problem of minimizing the tumor volume when the maximal admissible drug dose (the total amount of used drug) and the final level of vascularization are also taken into account. We investigate the solution of that problem for a fixed therapy duration. We show that the optimal strategy consists of the drug-free, full-dose and singular (with intermediate values of the control variable) intervals. Moreover, no bang-bang switch of the control is possible, that is the change from the no-dose to full-dose protocol (or in opposite direction) occurs on the interval with the singular control. For two particular models, proposed by Hahnfeldt et al. and Ergun et al., we provide additional theorems about the optimal control structure. We investigate the optimal controls numerically using the customized software written in MATLAB®, which we make freely available for download. Utilized numerical scheme is based on the composition of the well known gradient and shooting methods