# Winter School on Mathematical Modelling in Epidemiology and Medicine 2023

## Courses

### Course 1: Arnaud Ducrot

#### Threshold phenomenon for bistable equations with diffusion

Abstract: In these lectures we shall discuss some threshold properties of bistable equations with diffusion, both with Laplace operator and nonlocal convolution operator. For homogeneous solutions, bistable equation roughly means that small population densities yield extinction while larger initial densities lead to the persistence of the population. When a spatial variable is considered, namely for non-homogeneous initial data, the threshold phenomenon is much more complex and the aim of these lectures is to shed some light of such phenomenon, both for local and nonlocal diffusion. First, under suitable assumptions on the stability of the homogenous steady states, using travelling wave solutions, we will explain that the solution spreads when the initial data is a step function with a sufficiently large length and height. Next we present an asymptotic analysis of the length when the height of the step initial data is slightly larger than the bistable threshold.

### Course 2: Pierre Magal

#### Differential Equations and Population Dynamics

Abstract: This series of lectures is based on the book [1] and [2].

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 [1].

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 [1] .

Lecture 3: Monotone Semiflows
The third lecture will be devoted to monotone semiflow.  This lecture corresponds to the chapter 8 in [1].

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 [2].

References:
[1] 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.
[2] 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)

### Course 3: Alain Miranville

#### Mathematical models for brain tumors

Abstract: Our aim in this course is to discuss several mathematical models for brain tumors and cancers. A special attention will be devoted to the energetic mechanisms in the brain, in particular lactate, which play a major role in the cancer development. We will also discuss the introduction of therapies.

### Course 4: Sergei Trofimchuk

#### Traveling waves in the KPP-Fisher delayed equation: basic theory and applications

Abstract: This course presents a smooth introduction, based on the personal interests of this lecturer, to the the theory of scalar delayed population models and traveling waves in their spatial versions (assuming Fickian diffusion in homogeneous medium). Four lectures are developed around the particular case of delayed logistic equation with diffusion (also known as the KPP-Fisher delayed equation or Hutchinson's diffusive equation):
1. Scalar (single-species) delayed models of population dynamics: Wright's equation, the food-limited equation, Nicholson's blowflies equation. Global stability and periodic solutions. Stability, chaos and happiness.
2. Traveling waves in the classical (non-delayed) KPP-Fisher equation: existence, uniqueness, stability, spreading. Monostable and bistable waves. Pushed and pulled waves.
3. Traveling waves in the delayed KPP-Fisher equation: existence, uniqueness, oscillations vs monotonicity, stability. Wavefronts and semi-wavefronts.
4. Further extensions of presented considerations: i) for a family of scalar non-local Kolmogorov ecological equations; ii) for the diffusive Nicholson's equation (existence of non-oscillating non-monotone wavefronts); iii) (Bonus) for the Belousov-Zhabotinsky reaction or for the scalar Gurtin-MacCamy population model.

## Talks

### Talk 1: Fabio Lima

#### An epidemic model with sexual and non-sexual transmissions for hepatitis A.

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).

### Talk 2: Melba Vertel (online)

#### Desarrollo de un modelo estadístico - matemático de predicción del riesgo de leptospirosis spp para Colombia basado en variables epidemiológicas (in Spanish)

Keyword: zoonosis, patógeno, enfermedades transmitidas por alimentos, minería de datos.

### Talk 3: Fernando Córdova-Lepe

#### On the ways to consider a variable transmission rate in a pandemic strategic SEIR model

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 [1]). 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
[1] Córdova-Lepe F, Vogt-Geisse K (2022) Adding a reaction-restoration type transmission rate dynamic-law to the basic SEIR COVID-19 model. PLoS ONE 17(6): e0269843. https://doi.org/10.1371/journal.pone.0269843

### Talk 4: Roxana López Cruz (online)

#### Feedback effects on stability in epidemic models

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 ([1], [2]) can promotes more changes to the propagation of the disease than other parameters. Finally, we will also show with numerical simulations how a delay ([3]) in the feedback effect causes very interesting qualitative changes of the system with epidemiological significance.

References

[1] 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.
[2] 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.
[3] 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)

### Talk 5: Jacques Demongeot (online)

#### Proposal of a mechanism of the origin of life

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.

References
1. J. Demongeot, A. Moreira, A circular RNA at the origin of life. J. Theor. Biol. 249, 314-324 (2007).
2. J. Demongeot, V. Norris, Emergence of a “Cyclosome” in a primitive network capable of building “infinite” proteins. Life 9, 51 (2019).
3. J. Demongeot, A. Moreira, H. Seligmann, Negative CG dinucleotide bias: An explanation based on feedback loops between Arginine codon assignments and theoretical minimal RNA rings. Bioessays 43, 2000071 (2021).
4. J. Demongeot, H. Seligmann, Evolution of tRNA subelement accretion from small and large ribosomal RNAs. Biosystems 193, 104796 (2022).
5. J. Demongeot, M. Thellier, Primitive oligomeric RNAs at the origins of life on Earth. IJMS 24, 2274 (2023).

### Talk 6: Abraham Solar

#### An application of Halanay inequality to stability of traveling waves in R-D equations

Abstract: In this presentation we show an integral version of Halanay inequality and we apply this result to obtain stability of traveling waves in several reaction-diffusion equations with delay. We also discuss the scope of this method.