Computational Methods and Results for Structured Multiscale Models of Tumor Invasion

We present multiscale models of cancer tumor invasion with components at the molecular, cellular, and tissue levels. We provide biological justifications for the model components, present computational results from the model, and discuss the scientific-computing methodology used to solve the model equations. The models and methodology presented in this paper form the basis for developing and treating increasingly complex, mechanistic models of tumor invasion that will be more predictive and less phenomenological. Because many of the features of the cancer models, such as taxis, aging and growth, are seen in other biological systems, the models and methods discussed here also provide a template for handling a broader range of biological problems

Mixed approximation of a population diffusion equation

AbstractA numerical method is proposed to approximate the solution of a nonlinear and nonlocal system of integro-differential equations describing age-dependent population dynamics with spatial diffusion. A finite difference method along the characteristic age-time direction combined with mixed finite elements in the spatial variable is used for the approximation. Optimal order error estimates are derived for the relevant variables. Using nonnegativity of the discrete solution, a stability of the method is also proved

Numerical simulation of a susceptible-exposed-infectious space-continuous model for the spread of rabies in raccoons across a realistic landscape

We introduce a numerical model for the spread of a lethal infectious disease in wildlife. The reference model is a Susceptible-Exposed-Infectious system where the spatial component of the dynamics is modelled by a diffusion process. The goal is to develop a model to be used for real geographical scenarios, so we do not rely upon simplifying assumptions on the shape of the region of interest. For this reason, space discretization is carried out with the finite element method on an unstructured triangulation. A diffusion term is designed to take into account landscape heterogeneities such as mountains and waterways. Numerical simulations are carried out for rabies epidemics among raccoons in New York state. A qualitative comparison of numerical results to available data from real-world epidemics is discussed

Numerical approximation of density dependent diffusion in age-structured population dynamics

summary:We study a numerical method for the diffusion of an age-structured population in a spatial environment. We extend the method proposed in [2] for linear diffusion problem, to the nonlinear case, where the diffusion coefficients depend on the total population. We integrate separately the age and time variables by finite differences and we discretize the space variable by finite elements. We provide stability and convergence results and we illustrate our approach with some numerical result

The cultural psychology of obesity: diffusion of pathological norms from Western to East Asian societies

We examine the accelerating worldwide obesity epidemic using a mathematical model relating a cognitive hypothalamic-pituitary-adrenal axis tuned by embedding cultural context to a signal of chronic, structured, psychosocial threat. The obesity epidemic emerges as a distorted physiological image of ratcheting social pathology involving massive, policy-driven, economic and social 'structural adjustment' causing increasing individual, family, and community insecurity. The resulting, broadly developmental, disorder, while stratified by expected divisions of class, ethnicity, and culture, is nonetheless relentlessly engulfing even affluent majority populations across the globe. The progression of analogous epidemics in affluent Western and East Asian socieities is particularly noteworthy since these enjoy markedly different cultural structures known to influence even such fundamental psychophysical phenomena as change blindness. Indeed, until recently population patterns of obesity were quite different for these cultures. We attribute the entrainment of East Asian societies into the obesity epidemic to the diffusion of Western socioeconomic practices whose imposed resource uncertainties and exacerbation of social and economic divisions constitute powerful threat signals. We find that individual-oriented 'therapeutic' interventions will be largely ineffective since the therapeutic process itself (e.g. relinace on drug treatments) embodies the very threats causing the epidemic

Epidemic processes in complex networks

In recent years the research community has accumulated overwhelming evidence for the emergence of complex and heterogeneous connectivity patterns in a wide range of biological and sociotechnical systems. The complex properties of real-world networks have a profound impact on the behavior of equilibrium and nonequilibrium phenomena occurring in various systems, and the study of epidemic spreading is central to our understanding of the unfolding of dynamical processes in complex networks. The theoretical analysis of epidemic spreading in heterogeneous networks requires the development of novel analytical frameworks, and it has produced results of conceptual and practical relevance. A coherent and comprehensive review of the vast research activity concerning epidemic processes is presented, detailing the successful theoretical approaches as well as making their limits and assumptions clear. Physicists, mathematicians, epidemiologists, computer, and social scientists share a common interest in studying epidemic spreading and rely on similar models for the description of the diffusion of pathogens, knowledge, and innovation. For this reason, while focusing on the main results and the paradigmatic models in infectious disease modeling, the major results concerning generalized social contagion processes are also presented. Finally, the research activity at the forefront in the study of epidemic spreading in coevolving, coupled, and time-varying networks is reported.Comment: 62 pages, 15 figures, final versio

A numerical method for the stability analysis of linear age-structured models with nonlocal diffusion

We numerically investigate the stability of linear age-structured population models with nonlocal diffusion, which arise naturally in describing dynamics of infectious diseases. Compared to Laplace diffusion, the analysis of models with nonlocal diffusion is more challenging since the associated semigroups have no regularizing properties in the spatial variable. Nevertheless, the asymptotic stability of the null equilibrium is determined by the spectrum of the infinitesimal generator associated to the semigroup. We propose to approximate the leading part of this spectrum by first reformulating the problem via integration of the age-state and then by discretizing the generator combining a spectral projection in space with a pseudospectral collocation in age. A rigorous convergence analysis is provided in the case of separable model coefficients. Results are confirmed experimentally and numerical tests are presented also for the more general instance.Comment: 23 pages, 11 figure

Structured Psychosocial Stress and Therapeutic Intervention: Toward a Realistic Biological Medicine

Using generalized 'language of thought' arguments appropriate to interacting cognitive modules, we explore how disease states can interact with medical treatment, including, but not limited to, drug therapy. The feedback between treatment and response creates a kind of idiotypic 'hall of mirrors' generating a pattern of 'efficacy', 'treatment failure', and 'adverse reactions' which will, from a Rate Distortion perspective, embody a distorted image of externally-imposed structured psychosocial stress. This analysis, unlike current pharmacogenetics, does not either reify 'race' or blame the victim by using genetic structure to place the locus-of-control within a group or individual. Rather, it suggests that a comparatively simple series of questions to identify longitudinal and cross-sectional stressors may provide more effective guidance for specification of individual therapy than complicated genotyping strategies of dubious meaning. These latter are likely to be both very expensive and utterly blind to the impact of structured psychosocial stress -- a euphemism for various forms of racism and ethnic cleansing -- which, we contend, is often a principal determinant of treatment outcome at both individual and community levels of organization. We propose, to effectively address 'health disparities' between populations, and in contrast to current biomedical ideology based on a simplistic genetic determinism, a richer program of biological medicine reflecting Lewontin's 'triple helix' of genes, environment, and development, a program more in concert with the realities of a basic human biology marked by hypersociality unusual in vertibrates. Aggressive social, economic, and other policies of affirmative action to redress the persisting burdens of history would be an integral component of any such project