Room:  1.19 do VII 
 
Abstract: The study and characterization of the subvarieties of W (variety of semigroups whose square is completely regular) when W is a E-semigroup (the set of its idempotents, E(S), is a subsemigroup[1]) has been proposed by Borralho and Kinyon[3] following a question from Araújo et al.[2].
Although in its formulation this problem had been limited to semigroups where the idempotents commute, called E-commutative semigroups[1], ∀e, f ∈ E(S), e.f = f.e, in this work, I will also present results for the E-semigroups in W. These can be defined by the additional condition ∀e, f ∈ E(S), f.e ∈ W(e.f), where W(a) represents the set of weak inverses of a, W(a) = {x : x.a.x = x}.
When idempotents form a subsemigroup, we call it a band. If they commute, E(S) is
a semilattice of idempotents; otherwise, it is a semilattice of rectangular bands. Although the former is a particular case of the latter, this presentation treats both situations distinctly, as they are different structures with different properties.

The point of view we adopt to study the structure of a relatively complex semigroup is to first decompose it into a collection of subsemigroups, each of which has a somewhat
simpler structure, and then compose a complicated and “bigger” semigroup from simpler
ones.
In this work, this is accomplished in 3 steps:
1. Dividing the semigroup S in disjunct closed classes Sα of known structure.
2. Defining a “gross” structure by studying the relation between these classes induced
by the semigroup operation, Sα.Sβ → Sγ.
3. Studying a “fine” structure that relates the positions of two elements from different
classes to the position of their product. For any S ∈ W and any a ∈ S, a2 lies in a group; thus S is an epigroup, also called group-bound in Howie[4]. Then S splits into unipotency classes, Ke, defined by elements whose square lies in Ge, the largest subgroup whose idempotent is e[5]. In W, these unipotency classes are disjoint subsemigroups of S and can be indexed by the elements of the semilattice E(S). Starting from S 2 and taking into account that E(S2) = E(S), we refer to the conditions for being able to define homomorphisms in S, φα,β : Sα → Sβ for all α ≥ β of the semilattice E(S), and so define a multiplication of the elements of the
different unipotency classes. This way, we can study the “fine” structure of S ∈ W when
their idempotents form a subsemigroup.
Except as stated otherwise, the basic concepts used here can be found in Howie[4] and
Shevrin[5].
References
[1] J. Almeida, J.-E. Pin, and P. Weil, Semigroups whose idempotents form a subsemigroup, Math.
Proc. Camb. Phi. Soc. 111 (1992), 241-253.
[2] J. Ara ́ujo, M Kinyon, J. Konieczny, and A. Malheiro, Four notions of conjugacy for abstract
semigroups, in: Proceedings of the Royal Society of Edinburgh Section A: Mathematics 147(6) (2017),
1169-1214.
[3] M. Borralho and M. Kinyon, Variants of epigroups and primary conjugacy, Commun. Algebra
48(12) (2020), 5465-5473.
[4] J.M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs, New Series 12, (Oxford University Press, 1995).
[5] L.N. Shevrin, Epigroups, in Structural theory of automata, semigroups and universal algebra (Springer, 2005) 331-380.

The talk will concern some recent results dealing with spaces of functions with 'generalized' bounded variation and applications to continuum mechanics and imaging problems.

Abstract:

Probability and mathematical modelling offer complementary ways of understanding biological and health-related phenomena. While statistical distributions help describe individual variability and observed data, dynamical models provide insight into how diseases evolve and how interventions may influence their spread.

In this talk, I will present selected results from our recent research, beginning with flexible lifetime distributions, including the XLindley, New XLindley, Zeghdoudi, and related families. I will discuss their mathematical properties, mixture representations, statistical inference, and applications to lifetime and reliability data. I will then show how continuous distributions can be used to construct count models capable of accounting for heterogeneity and overdispersion.

The second part will focus on discrete epidemic models, particularly SIRS systems. We will discuss equilibrium points, stability conditions, and bifurcation phenomena, illustrating how changes in model parameters can lead to oscillations and more complex epidemic behaviour. I will also present recent work on optimal control strategies that take these dynamical features into account, including applications to COVID-19 data.

Finally, I will briefly introduce ongoing research on an SEIR model for tuberculosis, incorporating nonlinear transmission mechanisms and treatment limitations.

Abstract:

We introduce a stochastic Fisher–KPP equations on Riemannian manifolds, combining reaction–diffusion dynamics with random environmental fluctuations. The model is formulated using the Laplace–Beltrami operator and multiplicative stochastic noise. Particular discussion will be devoted to the positivity and invariant regions, and the influence of stochastic perturbations on propagation phenomena.

Keywords: Stochastic Fisher–KPP equation, Riemannian manifolds, stochastic partial differential equations, Laplace–Beltrami operator, invariant regions,

Abstract: This course provides an introduction to deterministic and stochastic SIR epidemic models. The deterministic framework is developed through ordinary differential equations, focusing on epidemic thresholds, the basic reproduction number, epidemic peaks, and final epidemic size. The stochastic extension introduces Brownian motion, Itô stochastic differential equations, and the influence of random fluctuations on disease dynamics. Fundamental mathematical properties, including positivity, population conservation, and extinction conditions, are discussed. Numerical simulations using the Euler and Euler–Maruyama methods illustrate the theoretical results and highlight the differences between deterministic and stochastic epidemic dynamics.

Keywords: SIR model, mathematical epidemiology, stochastic differential equations, Itô calculus, epidemic thresholds, numerical simulations.

6 hours mini-course, 2 sessions of 3 hours each
 
Lecture 1: The Continuous Deterministic SIR Model 
Lecture 2: Threshold, Epidemic Peak, and Final Size 
Lecture 3: Formulation of a Continuous Stochastic SIR Model 
Lecture 4: Existence, Uniqueness, Conservation, and Positivity
Lecture 5: Extinction, Persistence and Noise-Adjusted Thresholds
Lecture 6: Numerical Simulation and Model Comparison

Abstract: This mini-course introduces statistical and mathematical tools for studying lifetime data, event counts, and epidemic dynamics. The emphasis will be on understanding the mathematical ideas behind the models and how they can be applied to biological and epidemiological problems.

We will first explore flexible lifetime distributions, focusing on Lindley-type models, Gamma-mixture representations, survival and hazard functions, and reliability properties. We will then consider count-data models, particularly mixed-Poisson distributions, and discuss how they account for overdispersion and population heterogeneity.

The second part will focus on continuous and discrete SIR/SIRS epidemic models. We will examine the basic reproduction number, disease-free and endemic equilibria, local stability, and bifurcation phenomena. We will also discuss how mathematical analysis can help understand epidemic persistence, oscillations, and control strategies.

The course will combine mathematical explanations, worked examples, and exercises. It is intended for graduate students, PhD students, and researchers interested in probability, statistics, differential equations, and mathematical biology.

4h lecture in 2 sessions of 2h each

 Session 1
Flexible Lifetime and Count-Data Models

- Lindley-type distributions and Gamma-mixture representations.
- Survival and hazard functions and reliability properties.
- Statistical inference and model comparison.
- Mixed-Poisson distributions, overdispersion, and biomedical count data.

Session 2 (1h30mm)
Mathematical Modelling of Epidemic Dynamics

- Continuous and discrete SIR/SIRS models.
- Basic reproduction number and epidemic thresholds.
- Disease-free and endemic equilibria.
- Stability analysis and bifurcation phenomena.
- Epidemic control and illustrative examples.