Three different goals for this major : providing in the same time an initiation to mathematical tools, modelling and to different aspects of biology. You will get at the end of the academic year
- Strong knowledges allowing to provide an analysis of a wide range of mathematical problems applied to biology and medecine.
- Knowledge in mathematical modelling (discrete, continuouss, deterministic, stochastic, hybrid and multi-levels models).
- Knowledges in biology on specific topics.
Courses of this major are associated to
– refresher courses at the begining of the academic year
– a curriculum core during the first semester
– two additional course to be chosen among those proposed in the other majors (see the welcome page for a complet list).
- The Bernoulli model : bases of mathematical modelling in epidemiology
- Hamer model
- Kermack – Mc Kendrick model
- SEIR, SIS, SIRS models
- Mathematical analysis of deterministic epidemiologic models
- Basis reproduction rate R0 via the second generation matrix
- Stabilité of the SIR, SIER, SIET models (with treatment)
- Models with several origins
- Host-vector model
- Numerical simulations on the investigated models
- Spatial models derivation
- Application to the rabies model in a fox population ( Källén et al.)
- Tools: ODE, structured PDE
- Application: compartmental models and reaction-diffusion models
Theoretical evolving biology
Scientific goal : Display a joint biologist and mathematician point of view on some important questions in evolving biology. The course will be organised according to bilogical topics and mathematical technics (random, deterministic, multi-level analysis).
- Classical models in population genetics (Wright-Fisher, Moran), genetic drift and selection, diffusion limit.
- Ancestral processes, Kingman coalescent process, recombination and selection graph, multi-sepcies coalescent.
- Adaptative dynamics, canonical equation, and evolutive equilibrium.
- Integro-differential models for quantitative genetics, asymptotic regime with weak variance, Hamilton-Jacobi equations.
- Applications to some recent research problems:
- in evolutive epidemiology
- in evolutive genomics and in molecular evolution
- application to a changing environment population adaptation.
Cellular dynamics and complex systems
- Regulation models of the cellular cycle :
- Goldbeter mitotic oscillator
- Tyson and Novak model
- Norel and Agur model
- Proliferation and quiescency two phases model
- Smith-Martin model
- Normal and pathological haematopoiesis modelisation
- Pathogens intra-cellular replication model, cellular dynamics and immunitary response
Part II :
Fokker-Planck equation, Stochastic simulation algorithm. Link with deterministic systems. Some examples of cellular proliferation examples.
Chapter 2: Nonlinear Systems of ODEs
Existence and unicity of the solutions, Hartman-Grobman Theorem, Linearisation and linear stability, fixed points classification, co-dimension 1 and 2 pitchfork bifurcation, saddle point, transcritical, Hopf, bistable systems. Numerical study with stability analysis and bifurcation continuation softwares. Some examples of the cellular population dynamic (HIV dynamic, tumoral growth, cellular cycle).
Chapter 3: Discrete Systems
Existence/uniciy of the solutions, Linearisation and linear stability, comparison with ODE, Poincaré aplication, Bifurcations de doublement de Period doubling bifurcation, chaos. Applications : logistic equation, Leslie matrices.
Chapter 4: Large Soupled Systems and Collective Dynamics
Oscillators (phase oscillator, Goodwin model), networks, oscillators synchronisation. Kuramoto investigation, periodic systems learning. Examples and numerical models for circadian oscillators synchronisation, cellular cycle synchronisation via the circadian clock.
Chapter 5: Selected Topics – numerical methods…
Spatial ecology modelling
Keywords : stochastic and deterministic mathematical modelling, spatial ecology, microscopic-macroscopic scale changes, reaction-diffusion, propagation phenomenon, patterns
0) Presentation of the applications and/or external talks.
1) Stochastic modelling of spatial population (3h), introduction to micro/macro scale changes (2h).
2) Fisher-KPP (maximum principle, progressives waves, propagation of solutions associated to initial conditions with compact support) (3h), extension to bistable equations (1h), extension to competitive Lotka-Volterra (1h).
3) Turing instability (5h)
The goal of this course is to present a stochastic and deterministic family of models used in particular in spatial ecology but also in other domain of mathematical biology as Darwinian evolution or morphogenesis. We will begin with microscopic stochastic models describing in a sharp way individual behaviors and life cycles in a given population. After a scale change, we will investigate macroscopical reaction-diffusion equation systems describing, in a first time, nex territory invasion and, in a second time, the apparition of spatially heterogeneous patterns.