On a mathematical model of immune competition

AbstractThis work deals with the qualitative analysis of a nonlinear integro-differential model of immune competition with special attention to the dynamics of tumor cells contrasted by the immune system. The analysis gives evidence of how initial conditions and parameters influence the asymptotic behavior of the solutions

On a Mathematical Model of Immune Competition

This work deals with the qualitative analysis of a nonlinear integro-differential model of immune competition with special attention to the dynamics of tumor cells contrasted by the immune system. The analysis gives evidence of how initial conditions and parameters influence the asymptotic behavior of the solutions

The Suppression of Immune System Disorders by Passive Attrition

Exposure to infectious diseases has an unexpected benefit of inhibiting autoimmune diseases and allergies. This is one of many fundamental fitness tradeoffs associated with immune system architecture. The immune system attacks pathogens, but also may (inappropriately) attack the host. Exposure to pathogens can suppress the deleterious response, at the price of illness and the decay of immunity to previous diseases. This “hygiene hypothesis” has been associated with several possible underlying biological mechanisms. This study focuses on physiological constraints that lead to competition for survival between immune system cell types. Competition maintains a relatively constant total number of cells within each niche. The constraint implies that adding cells conferring new immunity requires loss (passive attrition) of some cells conferring previous immunities. We consider passive attrition as a mechanism to prevent the initial proliferation of autoreactive cells, thus preventing autoimmune disease. We see that this protection is a general property of homeostatic regulation and we look specifically at both the IL-15 and IL-7 regulated niches to make quantitative predictions using a mathematical model. This mathematical model yields insight into the dynamics of the “Hygiene Hypothesis,” and makes quantitative predictions for experiments testing the ability of passive attrition to suppress immune system disorders. The model also makes a prediction of an anti-correlation between prevalence of immune system disorders and passive attrition rates

Chronic Inflammation in the Epidermis: A Mathematical Model

The epidermal tissue is the outmost component of the skin that plays an important role as a first barrier system in preventing the invasion of various environmental agents, such as bacteria. Recent studies have identified the importance of microbial competition between harmful and beneficial bacteria and the diversity of the skin surface on our health. We develop mathematical models (M1 and M2 models) for the inflammation process using ordinary differential equations and delay differential equations. In this paper, we study microbial community dynamics via transcription factors, protease and extracellular cytokines. We investigate possible mechanisms to induce community composition shift and analyze the vigorous competition dynamics between harmful and beneficial bacteria through immune activities. We found that the activation of proteases from the transcription factor within a cell plays a significant role in the regulation of bacterial persistence in the M1 model. The competition model (M2) predicts that different cytokine clearance levels may lead to a harmful bacteria persisting system, a bad bacteria-free state and the co-existence of harmful and good bacterial populations in Type I dynamics, while a bi-stable system without co-existence is illustrated in the Type II dynamics. This illustrates a possible phenotypic switch among harmful and good bacterial populations in a microenvironment. We also found that large time delays in the activation of immune responses on the dynamics of those bacterial populations lead to the onset of oscillations in harmful bacteria and immune activities. The mathematical model suggests possible annihilation of time-delay-driven oscillations by therapeutic drugs.ope

A Mathematical Model of Immune Response to Tumor Invasion

Recent experimental studies by Diefenbach et al. [1] have brought to light new information about how the immune system of the mouse responds to the presence of a tumor. In the Diefenbach studies, tumor cells are modified to express higher levels of immune stimulating NKG2D ligands. Experimental results show that sufficiently high levels of ligand expression create a significant barrier to tumor establishment in the mouse. Additionally, ligand transduced tumor cells stimulate protective immunity to tumor rechallenge. Based on the results of the Diefenbach experiments, we have developed a mathematical model of tumor growth to address some of the questions that arise regarding the mechanisms involved in the immune response to a tumor challenge. The model focuses on the interaction of the NK and CD8+ T cells with various tumor cell lines using a system of differential equations. We propose new forms for the tumor-immune competition terms, and validate these forms through comparison with the experimental data of [1]

Outcome prediction in mathematical models of immune response to infection

Clinicians need to predict patient outcomes with high accuracy as early as possible after disease inception. In this manuscript, we show that patient-to-patient variability sets a fundamental limit on outcome prediction accuracy for a general class of mathematical models for the immune response to infection. However, accuracy can be increased at the expense of delayed prognosis. We investigate several systems of ordinary differential equations (ODEs) that model the host immune response to a pathogen load. Advantages of systems of ODEs for investigating the immune response to infection include the ability to collect data on large numbers of `virtual patients', each with a given set of model parameters, and obtain many time points during the course of the infection. We implement patient-to-patient variability $v$ in the ODE models by randomly selecting the model parameters from Gaussian distributions with variance $v$ that are centered on physiological values. We use logistic regression with one-versus-all classification to predict the discrete steady-state outcomes of the system. We find that the prediction algorithm achieves near $100\%$ accuracy for $v=0$, and the accuracy decreases with increasing $v$ for all ODE models studied. The fact that multiple steady-state outcomes can be obtained for a given initial condition, i.e. the basins of attraction overlap in the space of initial conditions, limits the prediction accuracy for $v>0$. Increasing the elapsed time of the variables used to train and test the classifier, increases the prediction accuracy, while adding explicit external noise to the ODE models decreases the prediction accuracy. Our results quantify the competition between early prognosis and high prediction accuracy that is frequently encountered by clinicians.Comment: 14 pages, 7 figure

Qualitative analysis of kinetic-based models for tumor-immune system interaction

A mathematical model, based on a mesoscopic approach, describing the competition between tumor cells and immune system in terms of kinetic integro-differential equations is presented. Four interacting populations are considered, representing, respectively, tumors cells, cells of the host environment, cells of the immune system, and interleukins, which are capable to modify the tumor-immune system interaction and to contribute to destroy tumor cells. The internal state variable (activity) measures the capability of a cell of prevailing in a binary interaction. Under suitable assumptions, a closed set of autonomous ordinary differential equations is then derived by a moment procedure and two three-dimensional reduced systems are obtained in some partial quasi-steady state approximations. Their qualitative analysis is finally performed, with particular attention to equilibria and their stability, bifurcations, and their meaning. Results are obtained on asymptotically autonomous dynamical systems, and also on the occurrence of a particular backward bifurcation

Cancer development: a population theoretical perspective

In the history of life, immune system and cancer have been engaged in an evolutionary arms race driven by the twin forces of mutation and selection. Ideally therapies should be a resolutive weapon, but, despite great progresses during the last 50 years or so, the race still goes on. The aim of this paper is to present a mathematical model, which can be used as in silico laboratory, to provide some indication on the effectiveness of therapies. Here we focus on two cancer populations competing for resources and subjected to the action of two types of immune system cells: thus the model results in a system of 4 differential equation that is analytically and computationally studied to elucidate its properties and emerging behaviors. At the beginning, some speci.c subsystems are analyzed and the effects of different therapies simulated; in particular .rst the system comprising a single cancer and immune cells type is considered and next the case of two cancer clones in absence of the immune cells. The complete model is then presented, which yields a rich variety of behaviors; in particular it is shown that for strong intertumoral competition, and high recognition levels by the immune system, stable stationary states are replaced by sustained oscillations. Finally some conclusion about therapy effectiveness are drawn, based on the results of simulations

Immune-mediated competition in rodent malaria is most likely caused by induced changes in innate immune clearance of merozoites

Malarial infections are often genetically diverse, leading to competitive interactions between parasites. A quantitative understanding of the competition between strains is essential to understand a wide range of issues, including the evolution of virulence and drug resistance. In this study, we use dynamical-model based Bayesian inference to investigate the cause of competitive suppression of an avirulent clone of Plasmodium chabaudi (AS) by a virulent clone (AJ) in immuno-deficient and competent mice. We test whether competitive suppression is caused by clone-specific differences in one or more of the following processes: adaptive immune clearance of merozoites and parasitised red blood cells (RBCs), background loss of merozoites and parasitised RBCs, RBC age preference, RBC infection rate, burst size, and within-RBC interference. These processes were parameterised in dynamical mathematical models and fitted to experimental data. We found that just one parameter μ, the ratio of background loss rate of merozoites to invasion rate of mature RBCs, needed to be clone-specific to predict the data. Interestingly, μ was found to be the same for both clones in single-clone infections, but different between the clones in mixed infections. The size of this difference was largest in immuno-competent mice and smallest in immuno-deficient mice. This explains why competitive suppression was alleviated in immuno-deficient mice. We found that competitive suppression acts early in infection, even before the day of peak parasitaemia. These results lead us to argue that the innate immune response clearing merozoites is the most likely, but not necessarily the only, mediator of competitive interactions between virulent and avirulent clones. Moreover, in mixed infections we predict there to be an interaction between the clones and the innate immune response which induces changes in the strength of its clearance of merozoites. What this interaction is unknown, but future refinement of the model, challenged with other datasets, may lead to its discovery