Course A: Adaptive evolution and concentrations in nonlocal PDEs
Lecturer: Professor Benoît Perthame

Living systems are subject to constant evolution through the two processes, stated by C. Darwin,  selection of the fittest and mutations. The goal of this series of lectures is to formalize them in a self-contained mathematical formalism. Exemples include:   resistance to treatment, monomorphism vs dimorphism, polymorphism and continuous distribution of traits, evolution of dispersion. We use a large population formalism based on integro-differential equations. Mathematical methods are based on singular perturbations of reaction-diffusion equations, concentration effects, constrained Hamilton-Jacobi equations, effective Hamiltonians, Turing type instabilities.

Ch 1. Principles of adaptation/evolution modeling. The constrained  Hamilton-Jacobi equations.

Ch 2. Dynamics of the fittest trait and ESS

Ch 3. Example: resistance to drugs.

Ch 4. Evolution without proliferating advantage.

Ch 5.  Generalist or specialist?

Course B: Singularity formation in PDE models related to biology
Lecturer: Professor Hatem Zaag

Partial Differential Equations are commonly used to model real-world phenomena, including biology. This is the case of the Keller-Segel system, which was set in the 1970’ as a model for chemotaxis: the movement of cells, bacteria, or amoeba, under the influence of some chemical. Under some conditions, we may see finite-time aggregation. Mathematically, the density goes to infinity in that case. This phenomenon is referred to as finite-time blow-up. In this lecture series, we will first derive the Keller-Segel system, from biophysical considerations. Then, we will review some basic properties of blow-up for that system. Other mathematical models exhibiting blow-ups will be considered. This will give the opportunity to review important methods in singularity formation study.

Course C: Extending dynamical system models to individual-based models - simulation and its analytical treatment
Lecturer: Professor Takasu Fugo

Many mathematical models of population dynamics are dynamical system described as ODE, an "equation-based" approach where population size is real-valued. ODE models are deterministic and target un-structured populations where only population size does matter. On the other hand, "individual-based" approach or IBM mechanistically describes how each individual gives birth and dies according to a certain rule. In IBM simulation, it is easy to assign various properties to each individual (age, height, susceptible, infectious, etc.) to study dynamics of structured populations. In this lecture, I introduce how an ODE model can be extended to an IBM where population size is integer-valued. Population dynamics in the IBM is inherently stochastic and simulation is easily carried out by generating pseudo-random numbers. Starting from the classical birth and death process (each individual gives birth and dies with constant birth and death rate), I explain how to simulate and analytically deal with stochastic process using master-equation. Other examples include logistic growth model, epidemic SIS, SIR models. I finally extend an ODE model to spatial population dynamics as point pattern dynamics (each point has location in space and interact with points nearby). Knowledge and skill you learn in my lecture will open up a potential research where IBMs are linked with corresponding well-studied ODE models.

I recommend participants to bring laptop PC with a programming environment installed such as python3.

Asistants : Dr. Võ Hoàng Hưng, Dương Giao Kỵ