Quantitative Immunology for Physicists

The adaptive immune system is a dynamical, self-organized multiscale system that protects vertebrates from both pathogens and internal irregularities, such as tumours. For these reason it fascinates physicists, yet the multitude of different cells, molecules and sub-systems is often also petrifying. Despite this complexity, as experiments on different scales of the adaptive immune system become more quantitative, many physicists have made both theoretical and experimental contributions that help predict the behaviour of ensembles of cells and molecules that participate in an immune response. Here we review some recent contributions with an emphasis on quantitative questions and methodologies. We also provide a more general methods section that presents some of the wide array of theoretical tools used in the field.Comment: 78 page revie

Mathematical and statistical methods for single cell data

The availability of single-cell data has increased rapidly in recent years and presents interesting new challenges in the analysis of such data and the modelling of the processes that generate it. In this thesis, we attempt to deal with some of those challenges by developing and exploring mathematical and statistical models for the evolution of population distributions over time, and methods for using aggregated single-cell data from individual patients in predictive diagnostic models of disease. In the first part of the thesis, we explore structured population models – a class of partial differential equations for describing the evolution of individual-level cell properties in a population over time. We begin by analysing an age-structured model of cell growth in which rates of proliferation and cell death are controlled by an external resource. We follow this with a method for extracting properties of a more general class of structured population models directly from single-cell data. In the final part of the thesis, we develop a flexible Bayesian statistical framework for building predictive models from possibly high-dimensional data collected from patients using single-cell technologies and find that the performance is promising compared to a number of existing methods

Computational and mathematical modelling of plant species interactions in a harsh climate

This thesis will consider the following assumptions which are based on a few insights about the artic climate: (1)the artic climate can be characterised by a growing season called summer and a dormat season called winter (2)in the summer season growing conditions are reasonably favourable and species are more likely to compete for plentiful resources (3)in the winter season there would be no further growth and the plant populations would instead by subjected to fierce weather events such as storms which is more likely to lead to the destruction of some or all of the biomass. Under these assumptions, is it possible to find those change in the environment that might cause mutualism (see section 1.9.2) from competition (see section 1.9.1) to change? The primary aim of this thesis to to provide a prototype simulation of growth of two plant species in the artic that: (1)take account of different models for summer and winter seasons (2)permits the effects of changing climate to be seen on each type of plant species interaction

Exploiting the synergy between carboplatin and ABT-737 in the treatment of ovarian carcinomas

Platinum drug-resistance in ovarian cancers is a major factor contributing to chemotherapeutic resistance of recurrent disease. Members of the Bcl-2 family such as the anti-apoptotic protein Bcl-XL have been shown to play a role in this resistance. Consequently, concurrent inhibition of Bcl-XL in combination with standard chemotherapy may improve treatment outcomes for ovarian cancer patients. Here, we develop a mathematical model to investigate the potential of combination therapy with ABT-737, a small molecule inhibitor of Bcl-XL, and carboplatin, a platinum-based drug, on a simulated tumor xenograft. The model is calibrated against in vivo\ud experimental data, wherein tumor xenografts were established in mice and treated with ABT-737 and carboplatin on a fixed periodic schedule, alone or in combination, and tumor sizes recorded regularly. We show that the validated model can be used to predict the minimum drug load that will achieve a predetermined level of tumor growth inhibition, thereby maximizing the synergy between the two drugs. Our simulations suggest that the time of infusion of each carboplatin dose is a critical parameter, with an 8-hour infusion of carboplatin administered each week combined with a daily bolus dose of ABT-737 predicted to minimize residual disease. We also investigate the potential of ABT-737 co-therapy with carboplatin to prevent or delay the onset of carboplatin-resistance under two scenarios. When resistance is acquired as a result of aberrant DNA-damage repair in cells treated with carboplatin, the model is used to identify drug delivery schedules that induce tumor remission with even low doses of combination therapy. When resistance is intrinsic, due to a pre-existing cohort of resistant cells, tumor remission is no longer feasible, but our model can be used to identify dosing strategies that extend disease-free survival periods. These results underscore the potential of our model to accelerate the development of novel therapeutics such as ABT-737, by predicting optimal treatment strategies when these drugs are given in combination with currently approved cancer medications

New developments in Functional and Fractional Differential Equations and in Lie Symmetry

Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

Models of Delay Differential Equations

This book gathers a number of selected contributions aimed at providing a balanced picture of the main research lines in the realm of delay differential equations and their applications to mathematical modelling. The contributions have been carefully selected so that they cover interesting theoretical and practical analysis performed in the deterministic and the stochastic settings. The reader will find a complete overview of recent advances in ordinary and partial delay differential equations with applications in other multidisciplinary areas such as Finance, Epidemiology or Engineerin