Workshop (14/7/2023): Some Contemporary problems in Mathematical Biology
Invited Speakers:
- Cung The Anh (Hanoi National University of Education)
- Benoît Perthame (Sorbonne Université)
- Nguyen Huu Du (Vietnam National University) (To be confirmed)
- Fugo Takasu (Nara Women's University, Japan)
- Hatem Zaag (Université Sorbonne Paris Nord)
- Lauren M. Childs ( Virginia Tech, USA )
- Vu Thai Luan (Mississippi State University, USA) (online)
- Nguyen The Toan (Vietnam National University Hanoi)
1. Cung The Anh
Title: Global existence and decay rates of solutions to semilinear structurally damped σ-evolution equations
Abstract:
We first study the decay rates of solutions to the Cauchy problem for the linear damped σ-evolution equation that are expressed in terms of the decay characters of the initial data. Then, by using properties of decay characters and the fixed point method, we investigate the global existence and decay rates of small data solutions to the corresponding semilinear problem.
This is a joint work with Pham Trieu Duong and Tang Trung Loc.
2. Benoît Perthame
Title: Structured equations in biology; relative entropy, Monge-Kantorovich distance
Abstract:
Models arising in biology are often written in terms of Ordinary Differential Equations. The celebrated paper of Kermack-McKendrick (1927), founding mathematical epidemiology, showed the necessity to include parameters in order to describe the state of the individuals, as time elapsed after infection. During the 70s, many mathematical studies were developed when equations are structured by age, size, more generally a physiological trait. The renewal, growth-fragmentation are the more standard equations.
The talk will present structured equations, show that a universal generalized relative entropy property is available in the linear case, which imposes relaxation to a steady state under non-degeneracy conditions. In the nonlinear cases, it might be that periodic solutions occur, which can be interpreted in biological terms, e.g., as network activity in the neuroscience.
When the equations are conservation laws, a variant of the Monge-Kantorovich distance also gives a general non-expansion property of solutions.
3. Nguyen Huu Du (VNU University of Science, Hanoi, Vietnam)
Title: Permanence and Extinction of Stochastic Models in Eco-Systems and Epidemiology
Abstract:
In this talk, we deal with the long term behavior of a stochastic models described the evolution of species in an eco-system or of the spread of infectious disease for SIR epidemic models.
We present sufficient conditions that are very close to the necessary conditions for the permanence or extinction of the underlying system by constructing a threshold value. Also, we develop the ergodicity of the system by characterizing the support of a unique invariant probability measure and show that the transition probabilities converge in total variation norm to the invariant measure with polynomial rate.
Email address: nhdu@visam.edu.vn
4. Fugo Takasu
Title: Spatial population dynamics as point pattern dynamics - how can we understand point pattern dynamics mathematically?
Abstract:
Spatial population dynamics has been largely studied by reaction-diffusion models as PDE. In this talk, I introduce another approach based on a mapped point pattern or simply "point pattern". A point pattern is defined as a collection of points in a space. In this approach, an individual is represented as a point and each point gives birth, dies, and moves with a certain rule and point pattern dynamically changes. A point pattern can be characterized by the number of points (1st oder structure), the number of pairs displaced by a distance (2nd order), etc. A mathematical tool to study point pattern dynamics, the method of moments, has been proposed that focuses on the dynamics of the 1st and the 2nd order structure as a set of integro-differential equations. All ODE models can be easily extended to point pattern dynamics PPD and simulation is technically easy to carry out. Based on the method of moments, we can derive dynamical system of 1st and 2nd order (and further 3rd order). However, it remains not clear how to solve the derived integro-differential equations and how these equations are related to reaction-diffusion models as PDE. I give several examples to explore how PPD and PDE are linked together.
5. Hatem Zaag
Title: How can mathematics help with Inflammatory Bowel Diseases?
Abstract:
Inflammatory Bowel Diseases (IBD) have been constantly arising since the middle of the 20th century. They seem to be related to the increase in living standards. Among them, Ulcerative Colitis (UC) and Crohn’s disease share similar severe symptoms, altering the quality of daily life. The study of IBD involves many biochemical techniques, together with biomedical imaging. The measurement of IBD’s severity relies on the judgement of doctors, which may vary from one individual to another. In LAGA, the Math Department of université Sorbonne Paris Nord (USPN), relying on our partnership within the inflamex Laboratory of Excellence (Labex), in particular with doctors and biologists from Bichat and Beaujon hospitals, we address the question of proposing an automatic scoring system. In this work, we combine imaging and Partial Differential Equations techniques in order to propose such a scoring.
6. Lauren M. Childs ( Virginia Tech, USA )
Title: Modeling Infectious Disease Dynamics: A Case Study of Malaria Immunity
Abstract:
The importance of understanding, predicting, and controlling infectious disease has become increasingly evident during the current COVID-19 pandemic. In particular, the pandemic highlighted the need for interpretable, quantitative models that link mechanism with data while accounting for variability. Despite significant effort and advances, infectious disease dynamics remain incompletely understood, in part due to the lack of heterogeneity considered in immunological, ecological, and epidemiological aspects, which produce complicated, non-linear feedbacks. In this talk, we will focus on an age-structured PDE model of malaria, one of the deadliest infectious diseases globally. Our novel model of malaria specifically tracks acquisition and loss of immunity across a population. We study the role of vaccination and immunity feedback on severe disease and malaria incidence, through a combination of our analytical calculation of the basic reproduction number (R0) and numerical simulations. Using demographic and immunological data, we parameterize our model to simulate realistic scenarios in Kenya. Our work sheds new light on the role of natural- and vaccine-acquired immunity in malaria dynamics in the presence of demographic effects.
7. Vu Thai Luan
Title: Efficient exponential methods for genetic regulatory systems
Abstract:
Genetic regulatory systems are networks of molecular interactions that control gene expression, which is crucial for many biological processes, including protein synthesis and the functioning of all living organisms (such as metabolism, cell signalling, and immune response). These systems can be modelled by a set of coupled nonlinear differential equations. In this talk, we construct and analyze a new second-order partitioned exponential method for simulating genetic regulatory systems. For systems with high nonlinearities, a novel splitting technique is proposed to improve the stability and increase accuracy and efficiency of the method. Alternatively, one can perform dynamic linearization of the system. In particular, we additionally suggest a second- and fourth-order exponential methods for such systems depending on their structure and properties. The linear stability analysis of the proposed methods is also presented. Finally, we present numerical experiments on various models, including a 2-gene and 3-gene repressilator models and a budding yeast cell cycle model, to demonstrate the effectiveness of our proposed methods.
8. Nguyen The Toan (VNU University of Science, Hanoi, Vietnam)
Title: Machine learning application to biomedicine research at the VNU Key Laboratory for Multiscale simulation of Complex Systems
Abstract:
In this talk, several applications of machine learning methods (computer vision, graph neural networks and variable autoencoders) to various problems in computational biomedicine is presented. In the context of drug design, protein interactions, biophysical simulations. These methods offer a complementary and sometime advantageous analyses to existing bioinformatic and biophysical approaches. Specific examples are researches on covid-19, Gout disease, morphine-based analgesic compounds being done at the VNU Key Laboratory for Multiscale simulation of Complex Systems, VNU University of Science, Hanoi, Vietnam.