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

VI Workshop on Computational Data Analysis and Numerical Methods: Book of Abstracts

The VI Workshop on Computational Data Analysis and Numerical Methods (WCDANM) is going to be held on June 27-29, 2019, in the Department of Mathematics of the University of Beira Interior (UBI), Covilhã, Portugal and it is a unique opportunity to disseminate scientific research related to the areas of Mathematics in general, with particular relevance to the areas of Computational Data Analysis and Numerical Methods in theoretical and/or practical field, using new techniques, giving especial emphasis to applications in Medicine, Biology, Biotechnology, Engineering, Industry, Environmental Sciences, Finance, Insurance, Management and Administration. The meeting will provide a forum for discussion and debate of ideas with interest to the scientific community in general. With this meeting new scientific collaborations among colleagues, namely new collaborations in Masters and PhD projects are expected. The event is open to the entire scientific community (with or without communication/poster)

Stochastic and deterministic models for age-structured populations with genetically variable traits

Understanding how stochastic and non-linear deterministic processes interact is a major challenge in population dynamics theory. After a short review, we introduce a stochastic individual-centered particle model to describe the evolution in continuous time of a population with (continuous) age and trait structures. The individuals reproduce asexually, age, interact and die. The 'trait' is an individual heritable property (d-dimensional vector) that may influence birth and death rates and interactions between individuals, and vary by mutation. In a large population limit, the random process converges to the solution of a Gurtin-McCamy type PDE. We show that the random model has a long time behavior that differs from its deterministic limit. However, the results on the limiting PDE and large deviation techniques \textit{\`a la} Freidlin-Wentzell provide estimates of the extinction time and a better understanding of the long time behavior of the stochastic process. This has applications to the theory of adaptive dynamics used in evolutionary biology. We present simulations for two biological problems involving life-history trait evolution when body size is plastic and individual growth is taken into account.Comment: This work is a proceeding of the CANUM 2008 conferenc

Agent-Based Modeling of Host-Pathogen Systems: The Successes and Challenges

Agent-based models have been employed to describe numerous processes in immunology. Simulations based on these types of models have been used to enhance our understanding of immunology and disease pathology. We review various agent-based models relevant to host-pathogen systems and discuss their contributions to our understanding of biological processes. We then point out some limitations and challenges of agent-based models and encourage efforts towards reproducibility and model validation.Comment: LaTeX, 12 pages, 1 EPS figure, uses document class REVTeX 4, and packages hyperref, xspace, graphics, amsmath, verbatim, and SIunit

HOW THE MUTATIONAL-SELECTION INTERPLAY ORGANIZES THE FITNESS LANDSCAPE

Fundamental questions posed in classical genetics since early 20th century are still fundamental in today post genomic age. What has changed is the availability of huge amount of molecular genetics information on a broad spectrum of species and a more powerful and rich methodological approach, particularly that one based on statistical mechanics and dynamical system theory which is providing unprecedented prediction power. Here we focus on the behavior of basic life forms such as bacteria and viruses which have small genomes and short generation times. We show that central issues of the evolutionary theory, i.e. how genotype, phenotype and fitness are related, the effect of positive and negative natural selection, the specie formation could be described by simple models which allow predictions and validation using experimental data

Statistical mechanics and thermodynamics of viral evolution

This paper analyzes a simplified model of viral infection and evolution using the 'grand canonical ensemble' and formalisms from statistical mechanics and thermodynamics to enumerate all possible viruses and to derive thermodynamic variables for the system. We model the infection process as a series of energy barriers determined by the genetic states of the virus and host as a function of immune response and system temperature. We find a phase transition between a positive temperature regime of normal replication and a negative temperature 'disordered' phase of the virus. These phases define different regimes in which different genetic strategies are favored. Perhaps most importantly, it demonstrates that the system has a real thermodynamic temperature. For normal replication, this temperature is linearly related to effective temperature. The strength of immune response rescales temperature but does not change the observed linear relationship. For all temperatures and immunities studied, we find a universal curve relating the order parameter to viral evolvability. Real viruses have finite length RNA segments that encode for proteins which determine their fitness; hence the methods put forth here could be refined to apply to real biological systems, perhaps providing insight into immune escape, the emergence of novel pathogens and other results of viral evolution.Comment: 39 pages (55 pages including supplement), 9 figures, 11 supplemental figure