Front Propagation in Random Media

This PhD thesis deals with the problem of the propagation of fronts under random circumstances. A statistical model to represent the motion of fronts when are evolving in a media characterized by microscopical randomness is discussed and expanded, in order to cope with three distinct applications: wild-land fire simulation, turbulent premixed combustion, biofilm modeling. In the studied formalism, the position of the average front is computed by making use of a sharp-front evolution method, such as the level set method. The microscopical spread of particles which takes place around the average front is given by the probability density function linked to the underlying diffusive process, that is supposedly known in advance. The adopted statistical front propagation framework allowed a deeper understanding of any studied field of application. The application of this model introduced eventually parameters whose impact on the physical observables of the front spread have been studied with Uncertainty Quantification and Sensitivity Analysis tools. In particular, metamodels for the front propagation system have been constructed in a non intrusive way, by making use of generalized Polynomial Chaos expansions and Gaussian Processes.The Thesis received funding from Basque Government through the BERC 2014-2017 program. It was also funded by the Spanish Ministry of Economy and Competitiveness MINECO via the BCAM Severo Ochoa SEV-2013-0323 accreditation. The PhD is fundend by La Caixa Foundation through the PhD grant “La Caixa 2014”. Funding from “Programma Operativo Nazionale Ricerca e Innovazione” (PONRI 2014-2020) , “Innotavive PhDs with Industrial Characterization” is kindly acknowledged for a research visit at the department of Mathematics and Applications “Renato Caccioppoli” of University “Federico II” of Naples

Front propagation in random media.

244 p.This PhD thesis deals with the problem of the propagation of fronts under random circumstances. Astatistical model to represent the motion of fronts when are evolving in a media characterized bymicroscopical randomness is discussed and expanded, in order to cope with three distinctapplications: wild-land fire simulation, turbulent premixed combustion, biofilm modeling. In thestudied formalism, the position of the average front is computed by making use of a sharp-frontevolution method, such as the level set method. The microscopical spread of particles which takesplace around the average front is given by the probability density function linked to the underlyingdiffusive process, that is supposedly known in advance. The adopted statistical front propagationframework allowed a deeper understanding of any studied field of application. The application ofthis model introduced eventually parameters whose impact on the physical observables of the frontspread have been studied with Uncertainty Quantification and Sensitivity Analysis tools. Inparticular, metamodels for the front propagation system have been constructed in a non intrusiveway, by making use of generalized Polynomial Chaos expansions and Gaussian Processes.bcam:basque center for applied mathematic

Mathematical models for chemotaxis and their applications in self-organisation phenomena

Chemotaxis is a fundamental guidance mechanism of cells and organisms, responsible for attracting microbes to food, embryonic cells into developing tissues, immune cells to infection sites, animals towards potential mates, and mathematicians into biology. The Patlak-Keller-Segel (PKS) system forms part of the bedrock of mathematical biology, a go-to-choice for modellers and analysts alike. For the former it is simple yet recapitulates numerous phenomena; the latter are attracted to these rich dynamics. Here I review the adoption of PKS systems when explaining self-organisation processes. I consider their foundation, returning to the initial efforts of Patlak and Keller and Segel, and briefly describe their patterning properties. Applications of PKS systems are considered in their diverse areas, including microbiology, development, immunology, cancer, ecology and crime. In each case a historical perspective is provided on the evidence for chemotactic behaviour, followed by a review of modelling efforts; a compendium of the models is included as an Appendix. Finally, a half-serious/half-tongue-in-cheek model is developed to explain how cliques form in academia. Assumptions in which scholars alter their research line according to available problems leads to clustering of academics and the formation of "hot" research topics.Comment: 35 pages, 8 figures, Submitted to Journal of Theoretical Biolog

Modelling population and disease dynamics in complex ecological systems

Mathematical models are a theoretical tool used to understand ecological processes. In this thesis we create mathematical frameworks to describe and evaluate four ecological systems. In the first case study we extend a host-pathogen framework to include a maternal effect which increases the disease resistance of offspring when the maternal environment is poor. Maternal effects impacting life-history traits have been shown to increase the propensity for population cycles. Our contrasting results show maternal effects acting on disease resistance stabilise host-pathogen systems. The second case study examines the impact infection may have on population estimates using Capture-Mark-Recapture (CMR) studies. We show that the estimates using the statistical Program Capture are accurate when capture rates are infection dependent. The final two case studies use spatial, individual-based, stochastic models to simulate disease spread and the colonisation of the Eurasian red squirrel (Sciurus vul- garis) on real-life landscapes. Using novel techniques we highlight the role habitat connectivity has on the dispersal routes which influence the spread of disease and re-population dynamics. Moreover the inclusion of seasonality shows that squirrel population dynamics are driven by the multi-year signal of resources.Heriot-Watt UniversityScottish Rural University Colleg

Multi-stage stochastic optimization and reinforcement learning for forestry epidemic and covid-19 control planning

This dissertation focuses on developing new modeling and solution approaches based on multi-stage stochastic programming and reinforcement learning for tackling biological invasions in forests and human populations. Emerald Ash Borer (EAB) is the nemesis of ash trees. This research introduces a multi-stage stochastic mixed-integer programming model to assist forest agencies in managing emerald ash borer insects throughout the U.S. and maximize the public benets of preserving healthy ash trees. This work is then extended to present the first risk-averse multi-stage stochastic mixed-integer program in the invasive species management literature to account for extreme events. Significant computational achievements are obtained using a scenario dominance decomposition and cutting plane algorithm.The results of this work provide crucial insights and decision strategies for optimal resource allocation among surveillance, treatment, and removal of ash trees, leading to a better and healthier environment for future generations. This dissertation also addresses the computational difficulty of solving one of the most difficult classes of combinatorial optimization problems, the Multi-Dimensional Knapsack Problem (MKP). A novel 2-Dimensional (2D) deep reinforcement learning (DRL) framework is developed to represent and solve combinatorial optimization problems focusing on MKP. The DRL framework trains different agents for making sequential decisions and finding the optimal solution while still satisfying the resource constraints of the problem. To our knowledge, this is the first DRL model of its kind where a 2D environment is formulated, and an element of the DRL solution matrix represents an item of the MKP. Our DRL framework shows that it can solve medium-sized and large-sized instances at least 45 and 10 times faster in CPU solution time, respectively, with a maximum solution gap of 0.28% compared to the solution performance of CPLEX. Applying this methodology, yet another recent epidemic problem is tackled, that of COVID-19. This research investigates a reinforcement learning approach tailored with an agent-based simulation model to simulate the disease growth and optimize decision-making during an epidemic. This framework is validated using the COVID-19 data from the Center for Disease Control and Prevention (CDC). Research results provide important insights into government response to COVID-19 and vaccination strategies

Dynamical Models of Biology and Medicine

Mathematical and computational modeling approaches in biological and medical research are experiencing rapid growth globally. This Special Issue Book intends to scratch the surface of this exciting phenomenon. The subject areas covered involve general mathematical methods and their applications in biology and medicine, with an emphasis on work related to mathematical and computational modeling of the complex dynamics observed in biological and medical research. Fourteen rigorously reviewed papers were included in this Special Issue. These papers cover several timely topics relating to classical population biology, fundamental biology, and modern medicine. While the authors of these papers dealt with very different modeling questions, they were all motivated by specific applications in biology and medicine and employed innovative mathematical and computational methods to study the complex dynamics of their models. We hope that these papers detail case studies that will inspire many additional mathematical modeling efforts in biology and medicin

Consumer-Resource Dynamics: Quantity, Quality, and Allocation

CITATION: Getz, W. M. & Owen-Smith, N. 2011. Consumer-resource dynamics : quantity, quality, and allocation. PLoS ONE, 6(1): e14539, doi:10.1371/journal.pone.0014539.The original publication is available at http://journals.plos.org/plosoneBackground: The dominant paradigm for modeling the complexities of interacting populations and food webs is a system of coupled ordinary differential equations in which the state of each species, population, or functional trophic group is represented by an aggregated numbers-density or biomass-density variable. Here, using the metaphysiological approach to model consumer-resource interactions, we formulate a two-state paradigm that represents each population or group in a food web in terms of both its quantity and quality. Methodology and Principal Findings: The formulation includes an allocation function controlling the relative proportion of extracted resources to increasing quantity versus elevating quality. Since lower quality individuals senesce more rapidly than higher quality individuals, an optimal allocation proportion exists and we derive an expression for how this proportion depends on population parameters that determine the senescence rate, the per-capita mortality rate, and the effects of these rates on the dynamics of the quality variable. We demonstrate that oscillations do not arise in our model from quantity-quality interactions alone, but require consumer-resource interactions across trophic levels that can be stabilized through judicious resource allocation strategies. Analysis and simulations provide compelling arguments for the necessity of populations to evolve quality-related dynamics in the form of maternal effects, storage or other appropriate structures. They also indicate that resource allocation switching between investments in abundance versus quality provide a powerful mechanism for promoting the stability of consumer-resource interactions in seasonally forcing environments. Conclusions/Significance: Our simulations show that physiological inefficiencies associated with this switching can be favored by selection due to the diminished exposure of inefficient consumers to strong oscillations associated with the wellknown paradox of enrichment. Also our results demonstrate how allocation switching can explain observed growth patterns in experimental microbial cultures and discuss how our formulation can address questions that cannot be answered using the quantity-only paradigms that currently predominate. © 2011 Getz, Owen-Smith.http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0014539Publisher's versio

Multiannual infestation patterns of grapevine plant inhabiting Scaphoideus titanus (Hemiptera: Cicadellidae) leafhoppers

The Nearctic leafhopper Scaphoideus titanus Ball (Hemiptera: Cicadellidae) was accidentally introduced in Europe, where it became the vector of the ‘Candidatus Phytoplasma vitis' phytoplasma causing the ‘Flavescence dorée' disease of grapevine plants. A time-varying distributed delay model, simulating the successive occurrences of egg hatching, nymph presence, and adult emergence, is extended here to represent multi-generation infestation patterns of grapevine plants inhabited by eggs, nymphs, and adults. The model extension includes intrinsic mortality, mortality caused by plant dormancy, and low temperatures, development of diapausing and post-diapausing eggs, fecundity rates, and adult longevity. Field observations and published data were used to estimate parameters. The model was validated with five years canopy infestation data from five vineyards not subjected to insecticide treatments and found to have satisfactory explicative and predictive qualities. The model output is most sensitive to a 10% variation in the upper threshold and in the shape parameters of the survivorship function and least sensitive to a 10% variation in the shape parameters of the development function and the survivorship level. Recommendations are made to take into account other factors than temperature and plant phenology and include a wider geographical area in further model developmen