ReaDiNet 2021: An Online Conference on Recent
Topics in Reaction-Diffusion System, Biology, Medicine
and Chemistry

October 25th to 29th, 2021

Abstracts

Henri Berestycki, EHESS
Epidemiology modelling with social diffusion and the Covid-19 pandemic
This talk is a tribute to the memory of M. Mimura. In epidemiology, diffusion arises
under several guises and leads to reaction-diffusion systems. I will present here some recent work using such systems and aiming at explaining the appearance of plateaus andrebounds in epidemics dynamics. I will first describe observations obtained from wastewater measurements that exhibit these dynamical features. Then I will discuss the epidemiological model with heterogeneity and social diffusion that allows us to explain them. I
report here on joint work with Benoit Desjardins, Bruno Heintz and Jean-Marc Oury. Ref.
https://www.nature.com/articles/s41598-021-97077-x

Meriem Bouguezzi, Université Paris Saclay
Mathematical modeling and analysis of a pit corrosion phenomenon
Steel corrosion plays a central role in different technological fields. The prediction of longterm corrosion behavior of stainless steel is needed to ensure that its physical integrity will
not be affected during its expected life time. In this talk, we focus on pitting corrosion ofsteels in contact with a solution of sodium chloride (NaCl). We present both a complete
model which describes a pure iron dissolution in sodium chloride as well as a simple case ofa corrosion phenomenon. We prove that under rather general hypotheses on the initial data,
the solution of this iron dissolution model converges to a self-similar profile as t → +∞.
This result is proved for an equivalent one-dimensional formulation of the physical model
known as a one-phase Stefan problem. In order to prove the convergence result, we apply
a comparison principle together with suitable upper and lower solutions. This is joint workwith D. Hilhorst, F. Lequien, H. Matano, Y.Miyamoto, F.Rouillard, J-F. Scheid.

Chueh-Hsin Chang, Tunghai University
Computing Evans functions for traveling waves of reaction-diffusion systems via fundamental series solutions
In this talk, we consider a general approach of the asymptotic stability of traveling wave
solutions of two-components reaction diffusion system through the use of Evans function
which is defined as the Wronskian of the fundamental solutions of the linearized operators.
By suitable change of variables, the eigenvalue problem can be transformed into an ODE
with two finite regular singular points. The fundamental solutions of the ODE are derived as
infinite series. Then, the Evans function is represented as the Wronskian of the fundamental
series solutions. The number of the point spectrum lying in the right half of the complex
plane can be numerically identified by counting the winding numbers of the images of suitablecontours mapped by the Evans function. This is a joint work with Cheng-Hsiung Hsu and
Tzi-Sheng Yang.

Chiun-Chuan Chen, National Taiwan University
Lotka-Volterra competition systems of two and three species
In this talk, we consider the wave phenomenon of two and three species Lotka-Volterracompetition systems and present the results collaborated with and inspired by Professor
Masayasu Mimura and his collaborators during the past years. In particular, we investigate
the problems related to exact solutions, the sign of wave speeds, 2-species singular limit
problems with “latent heat” effect, and 3-species non-monotone waves.

2

Lorenzo Contento, Universitat Bonn
Integrative modelling of reported case numbers and seroprevalence reveals
time-dependent test efficiency and infectious contacts
Mathematical models have been widely used during the ongoing SARS-CoV-2 pandemic
for data interpretation, forecasting, and policy making. However, most models are basedon officially reported case numbers, which depend on test availability and test strategies.
The time dependence of these factors renders interpretation difficult and might even resultin estimation biases. Here, we present a computational modelling framework that allows
for the integration of reported case numbers with seroprevalence estimates obtained from
representative population cohorts. To account for the time dependence of infection andtesting rates, we embed flexible splines in an epidemiological model. The parameters ofthese splines are estimated, along with the other parameters, from the available data using a
Bayesian approach. The application of this approach to the official case numbers reported forMunich (Germany) and the seroprevalence reported by the prospective COVID-19 Cohort
Munich (KoCo19) provides first estimates for the time dependence of the under- reporting
factor. Furthermore, we estimate how the effectiveness of non- pharmaceutical interventions
and of the testing strategy evolves over time. Overall, our results show that the integration of
temporally highly resolved and representative data is beneficial for accurate epidemiologicalanalyses.

Shin-Ichiro Ei, Hokkaido University
Interaction of non-radial camphor tips
In this talk, we theoretically investigate the interaction between two non-radially symmetric camphor particles, and derive the equation describing the motion as an ordinary
differential system for the locations and the rotations. In particular, slightly modified
non-radially symmetric cases from radial symmetry is investigated, and we obtain explicit
motions under fairly general settings. As one example, we theoretically show that elliptically deformed camphor particles interact parallel with significant axes. Such theoretically predicted motions are also checked by actual experiments and numerical simulations.
This work is basically due to the work: S.-I. Ei, H. Kitahata, Y. Koyano and M. Nagayama, Interaction of non-radially symmetric camphor particles, Physica D 366 (2018),
10-26. https://doi.org/10.1016/j.physd.2017.11.004

3

Jong-Shenq Guo, Tamkang University
Liouville-type theorems for some reaction-diffusion systems
Reaction-diffusion systems arise in many ecological systems and epidemic models. In
this talk, we shall first discuss Liouville-type theorems for entire solutions of some reaction–diffusion systems. Then, based on these theorems, we shall present some applications
on the spreading dynamics of solutions to some interesting systems from ecology and epidemiology. This talk is based on recent joint works with Arnaud Ducrot, Thomas Gilettiand Masahiko Shimojo.

Amirali Hannani, Université Paris Dauphine - PSL
Hydrodynamic limit for a disordered quantum harmonic chain
Chains of quantum harmonic oscillators can be toy models for various physical ad biological phenomena, such as heat conduction in solids, photosynthesis processes, and energy
transfer in random media. In this talk, I present a hydrodynamic limit, in the hyperbolic
space-time scaling, for a one-dimensional unpinned chain of quantum harmonic oscillatorswith random masses. To the best of my knowledge, this is among the first examples where
one can rigorously prove the hydrodynamic limit for a quantum system. I show that the
distribution of the elongation, momentum, and energy averaged under the proper Gibbs
state converges to the solution of the Euler equation. There are two main phenomena inthis chain that let us deduce this result. First is the Anderson localization, which decouples
the mechanical and thermal energy, closing the equation for energy and indicating that thetemperature profile does not evolve in time. The second phenomenon is similar to some sortof decay of correlation phenomena that let us circumvent the difficulties arising from the factthat our Gibbs state is not a product state due to the system’s quantum nature.

Hyung-Ju Hwang, POSTECH Mathematics
Neural network approach in mathematical biology
We discuss neural network approach to two problems which may arise in mathematicalbiology. First, we look into the chemotaxis model and the neural net can be used to findthe approximate solutions of the PDE. Also, the Neural Net approximation can be easily
applied to the inverse problem. It was confirmed that even when the coefficients of the PDE
4equation were unknown, prediction with high accuracy was achieved. Next, we focus on how
to approximate traveling wave solutions for various kinds of partial differential equations via
artificial neural networks. We propose a novel method to approximate both the travelingwave solution and the unknown wave speed via a neural network and an additional freeparameter.

Satoru Iwasaki, Osaka University
Standing waves of reaction-diffusion equations on an infinite metric graphIn this talk, we deal with bistable reaction-diffusion equations in a domain of a metricgraph with two vertices, that is, a domain of multiple half-lines with two junctions connectedby a line segment. We reveal relations between the length of the line segment and existence ofstanding front waves. From numerical simulations, it is observed that those standing waves
play a role to block the front propagation.

Hyuntae Jo, POSTECH Basic Science Research Institute
COVID19 – Mathematical modeling and machine learningThis presentation includes the following two topics. First of all, we consider a spread
model of COVID-19 with time-dependent parameters via deep learning. We developed a
SIR model with time-dependent parameters via deep learning methods. Furthermore, wevalidated the model with the conventional model to confirm its convergent nature. Next,
We also developed a machine learning model that predicts the mortality of infected patientsby using basic patients information such as age, residence, comorbidity, and past medical
history. Furthermore, we aim to establish a medical system that allows patients to checktheir own severity, and informs them to visit the appropriate clinic center by referring to thepast treatment details of other patients with similar severity.

Jinsu Kim, POSTECH Mathematics
Stochastically modeled reaction networks: stability and mixing times
A reaction network is a graphical configuration that describes an interaction between
species (molecules). If the abundances of the network system are small, then the randomness
5inherent in the molecular interactions is important to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain.
One of the challenging issues facing researchers who study biological systems is the often extraordinarily complicated structure of their interaction networks. Thus, how to characterize
network structures that induce characteristic behaviors of the system dynamics is one of the
major open questions in this literature. In this talk, I will provide an analytic approach tofind a class of reaction networks whose associated Markov process has a stationary distribution. Moreover, I will also talk about the convergence rate for the process to its stationary
distribution with the mixing time.

Yong-Jung Kim, KAIST
Modeling cross-diffusion using “a link between microscopic and macroscopic
models of self-organized aggregation”
Masayasu Mimura published a paper titled “a link between microscopic and macroscopic
models of self-organized aggregation” with his coauthors T. Funaki, H. Izuhara, and C. Urabe
in 2012. He kindly introduced me to the paper and his idea of modeling cross-diffusion as afast reaction limit. In this memorial talk for Mayan, I will introduce the paper, his idea, andrecent related results with my coauthors on modeling cross-diffusion through the link.

Hélène Leman, ENS Lyon
Dynamics of a plant and pollinator community
In this work, we are interested in modeling the ecological dynamics of a plant and pollinator community, using a microscopic individual-based model. We then use different renormalizations of this process to study the dynamics of the community under different assumptions:
we will be interested in the assumption of large populations and in the one of a large numberof species. Our aim is to understand the resilience of the community after the deletions ofcertain species or populations, depending on the structure of their links.

6

Elisabeth Logak, CY Cergy Paris Université
Analysis of a coupled within- and between-host model with virus mutations
In a joint work with D. Hilhorst (Univ. Paris-Saclay), D. Pontier and D. Fouchet (Univ.
Claude Bernard Lyon 1), we consider a nested host-parasite model which links the withinhost virus dynamics with the epidemic transmission between hosts. The within-host model
is a nonlocal ODE system, where virus mutations are described by an integral kernel. The
between-host model is a structured SI-type renewal equation where the transmission ratedepends on the density of viruses obtained from the within-host model. I will present themodel and preliminary analytical results on global existence and large time asymptotics.

Hirokazu Ninomiya, Meiji University
Area-preserving curvature flow of a convex plane curve in an inhomogeneous
mediumArea-preserving curvature flows in a two-dimensional homogeneous medium have beenstudied for several decades. In 1986, Gage showed that an initially convex closed curve
remains convex and converges to a circle as time goes to infinity. However, in many applications, the medium is not homogeneous and the closed curve moves towards a more favorable
environment. The properties of the medium are described by a signal function, which is atwice continuously differentiable function defined in two-dimensional space. In this talk, we
consider the area preserving flow in an inhomogeneous medium. First, I will explain theproperties of stationary solutions. Then I will show the global existence of convex closed
curves under certain conditions for a signal function. Finally I will explain the dynamics of
curves.

Sungrim Seirin-Lee, Kyoto University
Dynamics of cells and polarity in early development
The human body is composed of 250 300 kinds of cells with different functions and sizes,
which are diversified from a single fertilized egg cell during early development. One of thewidespread mechanisms for generating cell diversity is asymmetric cell division. A mother cell
divides into two dissimilar daughter cells, which subsequently develop completely different
properties. The phenomenon of asymmetric cell division is the ultimate spatiotemporal
7

control of the cell and it delivers integrated understanding of how life self-organizes itself. In
recent decades, asymmetric cell division has been extensively explored using Caenorhabditis
elegans embryos as a biological model system. In this talk, I will broadly introduce thedynamics of polarity formation and cells dynamics of C. elegans embryo with recent results.

Yoshitaro Tanaka, Future University Hakodate
Effective nonlocal kernels on reaction–diffusion networks
A new method to derive an essential integral kernel from any given reaction–diffusion
network is proposed. Any network describing metabolites or signals with arbitrary manyfactors can be reduced to a single or a simpler system of integro-differential equations called“effective equation” including the reduced integral kernel (called “effective kernel”) in the
convolution type. As one typical example, the Mexican hat shaped kernel is theoreticallyderived from two component activator-inhibitor systems. It is also shown that a three component system with quite different appearance from activator-inhibitor systems is reduced
to an effective equation with the Mexican hat shaped kernel. It means that the two different systems have essentially the same effective equations and that they exhibit essentiallythe same spatial and temporal patterns. Moreover, we will introduce other applications forpattern formations with this method.

Je-Chiang Tsai, National Tsing Hua University
Traveling waves of reaction-diffusion models of farmers and hunter-gathersin the Neolithic transition
We will report traveling wave solutions of two reaction-diffusion models of farmers andhunter-gathers in the Neolithic transition: one is the Ammerman and Cavalli-Sforza model,
and the other is the Aoki-Shida-Shigesada model.

Keiichi Ueda, University of Toyama
Emergence of synchronized oscillations by self-tuning of parameters
We propose a parameter-tuning algorithm based on a selection principle. The algorithm allows the emergence of synchronization between populations of oscillators through autonomous
8changes of the intrinsic parameters. The coupled oscillator systems with the algorithm exhibit self-recovery of the synchronized state after the existing synchronized state is broken
suddenly. We also propose a model described by reaction-diffusion systems for the self-tuningof parameters.

Feng-Bin Wang, Chang Gung University
Applications of nonlinear eigenvalue problems to ecological models with internal storage
Competition for resources is a fundamental topic in theoretical ecology. Population dynamics are coupled to dynamics of one or more resources by assuming a constant quota ofnutrient per individual. In fact, quotas may vary, leading to variable-internal-stores models. When nutrient is taken up, it is stored internally, and population growth is a positive
function of stored nutrient. The competitors live in a flowing habitat with both advection
and diffusion, where the nutrient is supplied in the upstream flow, and all constituents flowout at the downstream end. The main difficulties in mathematical analysis for such systems
are caused by the singularity at the extinction state. In this talk, we first investigate thenonlinear eigenvalue problem in the special positive cones of functions motivated by the ratio
dependence. Then the threshold type result on the extinction/persistence of the species canbe determined by the principal eigenvalue of our nonlinear eigenvalue problem. When the
habitat is infinite, we shall attempt to study the travelling wave and spreading speeds.

Chang-Hong Wu, National Yang Ming Chiao Tung University
The formation of spreading front: the singular limit of three-component
reaction-diffusion models
In recent decades, mathematical models in population dynamics with free boundary are
considered (e.g., Mimura et al., Jpn J Appl Math 2:151–186, 1985; Du and Lin, SIAM J
Math Anal 42:377–405, 2010), where the free boundary is used to determine spreading front.
To understand the formation of fronts, we will consider the singular limit of three-component
reaction-diffusion models and give some interpretations for spreading front from the modelingviewpoint. Finally, as an application, we revisit the issue of the spread of the grey squirrel
in the UK and estimate the spreading speed of the grey squirrel based on our results. Thisis joint work with Hirofumi Izuhara and Harunori Monobe.

9Changwook Yoon, Chungnam National University
Remarks on a chemotaxis system with Fokker-Planck type diffusion
In this talk, we consider a chemotaxis system with Fokker-Planck type nonlinear diffusion,
which describes the directed movement of the living organisms. The idea of the model is thatthe organisms determine their motility depending on the circumstances. The advantage ofthe nonlinear diffusion lies in allowing the global existence of solutions in terms of L2 boundedness. We will apply this model to the cell aggregation and signal absorption phenomena
and explore some recent results on analytical properties of solutions.

10