Summer School – Mathematical and Theoretical Biology

# Program

**Dynamical models of Cancer** See abstract

David Dingli (Mayo Clinic, USA).

Adaptive dynamics and the evolution of pathogens See abstract

Eva Kisdi (University of Helsinki, Finland).

Modelling Meso-evolution: adaptive dynamics and beyond See abstract

Hans Metz (Leiden University, The Netherlands) .Click here for supporting material.

Stochastic and Deterministic Processes in Spatial Population Dynamics See abstract

Sergei Petrovskii (University of Leicester, UK).

Mathematical Models in Hemodynamics See abstract

Adelia Sequeira (Universidade Técnica de Lisboa, Portugal).

Ecology and Eco-epidemiology See abstract

Ezio Venturino (Universitá di Torino, Italy).

## Abstracts

**Dynamical models of Cancer (David Dingli, Mayo Clinic, USA):** These lectures will be divided into 2 parts : i) normal hematopoiesis and ii) troubled hematopoiesis. I will set up a mathematical model of hematopoiesis which allows one to estimate the number of active stem cells and their rate of replication, as well as the number of different intermediate stages of differentiation of hematopoietic cells together with their respective rates of replication. Next I show how this description of normal hematopoiesis can be used to understand hematopoietic disorders, namely: Paroxysmal Nocturnal Hemoglobinuria (PNH), Chronic Myeloid Leukemia (CML) and Cyclic Neutropenia (CN).

**Adaptive dynamics and the evolution of pathogens (Eva Kisdi, University of Helsinki, Finland):** Adaptive dynamics links population dynamics to evolution driven by natural selection via possibly complex ecological interactions. The main emphasis is on dynamical phenomena such as the origin and divergence of new lineages by evolutionary branching. The general theory of adaptive dynamics has been applied to analyze evolution in many and diverse ecological settings. This course will introduce the basic mathematical theory of adaptive dynamics illustrated with examples from the evolution of pathogens.

**Modelling Meso-evolution: adaptive dynamics and beyond (Hans Metz, Leiden University, The Netherlands):** Picking up on the lectures by Eva Kisdi I will in my first lectures give, for a very general class of eco-evolutionary models, a derivation of the so-called canonical equation of adaptive dynamics, a differential equation for the evolutionary change of the dominant trait value in the population. In addition I will give one lecture on the question under what conditions evolutionary outcomes satisfy an optimization principle, and one lecture on how adaptive dynamics fits in the total of evolutionary arguments.

** Stochastic and Deterministic Processes in Spatial Population Dynamics (Sergei Petrovskii, University of Leicester, UK):** Dynamics of ecological systems arises as a result of the interplay between deterministic and stochastic factors, yet the relative importance of those factors as well as their implications are not always easy to understand. Mathematical modeling is a powerful and convenient approach that allows distinguishing between the impacts of different factors. However, the choice of an adequate mathematical framework is by no means straightforward. In this brief course, I will start with revisiting some mathematical tools that are in common use to describe inherently stochastic population dynamics. The focus will be on spatial phenomena and hence I use diffusion as an instructive (albeit not the only possible) example. The main message from this part of the course is that, while the dynamics may be stochastic, the corresponding mathematical model can still be perfectly deterministic; immediate examples are given by the diffusion equation and the Fokker-Planck equation. I will then proceed to the spatiotemporal population dynamics on a multi-generation time scale. The corresponding generic mathematical model is given by stochastic partial differential equations. I will examine its properties with a special attention to the possibility of chaotic dynamics and to reveal the situations when the effects of stochastisity or ‘noise’ can enhance or hamper the onset of chaos. I will conclude with showing how this approach can improve our understanding of ecosystems properties.

**Mathematical Models in Hemodynamics (Adelia Sequeira, Universidade Técnica de Lisboa, Portugal):** Mathematical modeling and numerical simulation can provide an invaluable tool for the interpretation and analysis of the circulatory system functionality, in both physiological and pathological situations. However, although many substantial achievements have been made, most of the difficulties are still on the ground and represent major challenges for the coming years. In this course we introduce some recent mathematical models of the cardiovascular system and comment on their significance to yield realistic and accurate numerical results. They include fluid-structure interaction (FSI) models to account for blood flow in compliant vessels and the geometrical multiscale approach to simulate the reciprocal interactions between local and systemic hemodynamics. Recent results on a sensitivity hemodynamic analysis to vascular geometry and blood flow modeling in patient specific cerebral aneurysms will also be presented.

**Ecology and Eco-epidemiology (Ezio Venturino, Universitá di Torino, Italy):** The course will concentrate at first on some issues in ecology. We consider invasion models and the role the Allee effet has on the development of patches in predator-prey models. Also a new model for viral invasion of bacteria is presented. We provide mathematical means for managing public resources and biologically controlling fruit orchards. Models for the latter study both wanderer and web builder spiders. In predator-prey systems, the effect of prey herd behavior is examined. In the second part of the course, ecoepidemic models will be presented. In this context, we will analyze mainly models of interacting populations among which a contagious disease spreads. Among the possible populations interactions, we will consider the classical predator-prey and competing models as well as symbiotic systems. In the former case, different situations arise if the disease is assumed to spread among the predators or the prey. The herd behavior in population systems can be considered also in the framework of ecoepidemic models. The interactions between populations and the disease incidence are both nonlinear terms, and could be modelled in several ways, from mildly to highly nonlinear functions. The predators’ switching feeding, as well as selective hunting among sound and infected prey, will also be considered. Several other situations of epidemics spread will consider the following issues: the disease progress could be modelled via stages; the disease could also overcome the species barrier; the disease could be carried by a vector; there could be a disease incubation period. Finally, also the spatial version of some of these models will be taken into account. .