Course 1: Arnaud Ducrot
Lecture 1: Introduction to Linear Population Dynamics
This lecture will present several examples of linear mathematical models in population dynamics. This lecture corresponds to the first chapter 1 in .
Lecture 2: Positivity and the Perron–Frobenius Theorem
The second lecture will be devoted to the consequences of the Perron-Frobenius theorem. The lecture corresponds to the chapter 4 in  .
Lecture 3: Monotone Semiflows
The third lecture will be devoted to monotone semiflow. This lecture corresponds to the chapter 8 in .
Lecture 4: Semiflows, ω-limit Sets, α-limit Sets, Attraction, and Dissipation
The fourth lecture will be devoted the notion of omega-limit sets. This lecture corresponds to the chapter 1 in .
 Ducrot, A., Griette, Q., Liu, Z., & Magal, P. (2022). Differential Equations and Population Dynamics I: Introductory Approaches. Lecture Notes on Mathematical Modelling in the Life Sciences, Springer Cham.
 Ducrot, A., Griette, Q., Liu, Z., & Magal, P. Differential Equations and Population Dynamics II: Advanced Approaches, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer Cham (in preparation)
Abstract: Outbreaks of Hepatitis A within the population of men who have sex with men (MSM) have been observed since the 1980s in countries with low incidence of hepatitis A. In the general population, hepatitis A virus (HAV) transmissions occur mainly through non-sexual routes, while epidemiological studies have pointed out the importance of sexual transmission in outbreaks of HAV within MSM populations. In this talk, we present an edge-based compartmental model (EBCM) including two routes of transmission in order to understand the role of sexual and non-sexual transmissions of HAV within the MSM population and its spillover to the general population. In this model, non-sexual transmission is modeled through the law of mass action, and sexual transmission is modeled through a sexual contact network. An EBCM is a low-dimensional system of ODEs that represents the transmission dynamics in the large population limit. In addition, we illustrate the model through numerical simulations and by fitting it to the data of hepatitis A outbreaks started within MSM populations in Australia (1991-1992) and in the Netherlands (2017-2018).
Abstract: In the projection or explanation of the Covid-19 data based on the standard SEIR model, the first scientific conviction that appears is to consider that the transmission rate should not be constant. In this sense, we will do a bibliographical review of how a variable beta rate has been installed in both tactical and strategic order models, up to installing a dynamic law for this rate (under the hypothesis of behavioral changes in the affected populations, see ). It is shown that this introduction is a simple route to capture the geometry of the main epidemiological curves, at least in the period before the supply of vaccines and the appearance of viral variants.References
Abstract: A basic mathematical model in epidemiology is the SIR (Susceptible–Infected–Removed) model, which is commonly used to characterize and study the dynamics of the spread of some infectious diseases. In this work, we study the dynamical behavior of a modified SIR epidemiological model by introducing feedback effects. We will see how a negative and positive feedback effects in SIR models (, ) can promotes more changes to the propagation of the disease than other parameters. Finally, we will also show with numerical simulations how a delay () in the feedback effect causes very interesting qualitative changes of the system with epidemiological significance.
 LÓPEZ-CRUZ R. Global stability of an SAIRD epidemiological model with negative feedback. Advances in Continuous and Discrete Models. 2022 May 12;2022(1):41.
 LV Y, CHEN L, CHEN F, LI Z. Stability and bifurcation in an SI epidemic model with additive Allee effect and time delay. International Journal of Bifurcation and Chaos. 2021 Mar 30;31(04):2150060.
 KUMAR A. AND NILAM. Stability of a Time Delayed SIR Epidemic Model Along with Nonlinear Incidence Rate and Holling Type-II Treatment Rate, International Journal of Computational Methods, Vol. 15, No. 1 (2018)
Abstract: Among the molecules believed to play an important role in the origin of life on Earth, the first RNAs and the first peptides were the source of mutually positive interactions. RNAs likely served as a template for the formation of peptides, while peptides protected RNAs from denaturation. We propose a possible mechanism of such interactions and show its combinatorial properties. We describe evidence from the present genomes of a circular RNA proposed to have existed at the center of the early mechanism of the peptide biosynthesis. Its remnants still exist in present-day genomes of many species, and their occurrence frequency could serve as a quantitative marker of the evolutionary age of these genomes.