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.

📊 Statistics & Risk Management Seminar Series

NOVA MATH | NOVA School of Science and Technology

🎓 Talk: Nonparametric Estimation of the Extreme Quantiles of Compound Frequency Distributions

👤 Speaker: Prof. Helgard Raubenheimer, North-West University, South Africa

📅 Date: Wednesday, 11 November 2026

🕑 Time: 14:00–15:00 (Lisbon time, WET / UTC+0 / GMT)

💻 Format: Online (Google Meet)

🔗 Join us: Click here to join the seminar

📆 Add to Calendar: Google Calendar | Outlook / Apple Calendar (see attached .ics file)

📝 About the Talk

Accurately estimating extreme losses is a major challenge in quantitative risk management, particularly when assessing the reserves needed to cover rare but potentially severe events. In this seminar, Prof. Raubenheimer will present a nonparametric approach to estimating extreme quantiles of compound loss distributions. By combining extreme value theory with single-loss and perturbative approximations, the proposed methodology aims to improve estimation accuracy without relying on potentially inadequate parametric assumptions about the tails of loss distributions. Simulation results highlight the potential of second-order multiplier estimators for practical applications in operational risk management.

📄 Abstract

Estimation of operational risk reserves is still widely done using the loss distribution approach. The accuracy of the estimation depends heavily on the accuracy with which the extreme quantiles of the aggregate loss distributions are estimated. Various approaches have been proposed to estimate the extreme quantiles of this compound distribution, including estimators based on the single-loss and perturbative approximations that rely on estimating an even more extreme quantile of the underlying severity distribution. However, estimation of these extreme quantiles may be inaccurate due to fitting a parametric severity distribution that fits the bulk of the data well but struggles to capture the tail behavioural characteristics of the distribution that generated the loss data. To circumvent this problem, we propose estimating nonparametrically a less extreme or lower quantile of the severity distribution, hopefully with better accuracy, and then multiplying this lower quantile by a certain factor to obtain an estimate of the required extreme quantile of the compound distribution. The factor or multiplier is derived by using extreme value theory and the single-loss and perturbative approximations and estimated nonparametrically. This approach is evaluated by means of a simulation study, which suggests that the second-order multiplier estimator based on the second-order perturbative approximation is a good choice for practical applications.

👨‍🏫 About the Speaker

Prof. Helgard Raubenheimer is a Professor at North-West University (NWU), South Africa, and Director of the Centre for Business Mathematics and Informatics (CBMI). He joined CBMI in 2003, became Head of the SAS Laboratory in 2016, and was appointed Director of CBMI in 2022. He holds a PhD in Risk Analysis from NWU. His expertise lies in quantitative risk management, with extensive experience in academic research, postgraduate education, industry collaboration, and consulting. He has led numerous industry-directed research projects and delivered specialised professional training to financial services organisations, both locally and internationally. His work focuses on bridging the gap between academic research and practical applications in the financial services industry.

Academic Profiles:

• Google Scholar: https://scholar.google.co.za/citations?hl=en&user=7YJp9uwAAAAJ

• LinkedIn: https://za.linkedin.com/in/helgard-raubenheimer-8b59111a

• ORCID: https://orcid.org/0000-0002-4126-1739

✨ Free entry | All are welcome!

We warmly invite researchers, students, and professionals interested in statistics, extreme value theory, and quantitative risk management to join us for this seminar.

We look forward to seeing you online for an engaging scientific discussion!

Organizers:  Mina Norouzirad and Isabel Natário