modeling in life sciences - szte bolyai intézetknipl/mpe2013/modelinginlife... · 2013. 11....
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Workshop on
Modeling in Life Sciences
November 30, 2013
Informatorium, Szent-Györgyi Albert Agora
organized by
Bolyai Institute, University of Szeged
in the framework of
Mathematics of Planet Earth 2013
FuturICT.hu project
FuturICT.hu www.futurict.szte.hu
The project „Infocommunication technologies and the society of future
(FuturICT.hu)” was established by the cooperation of four Hungarian research and
higher education institutes, and received a grant of almost 1.6 billion forints (100%
intensity rate) from the TÁMOP-4.2.2.C project of the New Széchenyi Plan.
The leader of the consortium is the University of Szeged. Further participants are the
Eötvös Loránd University, BME VIKING Plc. and DEAK Plc. The two-year-long project
supports more than hundred researchers working in ten subprojects.
Mathematics of Planet Earth mpe2013.org
MPE2013 is born from the will of the world mathematical community to learn more
about the challenges faced by our planet and the underlying mathematical
problems, and to increase the research effort on these issues. Indeed, the recent
tendencies have increased the pressure to comprehend the planet and its
environment: growing population competing for the same global resources,
increased frequency and intensity of dramatic meteorological events, and evidence
pointing to longer term patterns of general climate change. Mathematicians have an
expertise in modeling and solving problems. MPE2013 creates exceptional
opportunities for long-term partnerships, both inside the mathematical sciences and
with other related scientific disciplines. It will allow training a new generation of
researchers working on scientific problems related to climate change and
sustainability.
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Program
9:50 Gergely Röst: Opening
10:00 Keynote:
John G. Milton: When balance control fails: mathematical insights into falling
10:45 Keynote:
Viktor Müller: HIV competition dynamics over sexual networks
11:30 Maria Vittoria Barbarossa: Waning immunity and state-dependent delays
11:50 Katalin Ozogány: Hierarchical structures motivated by living communities
12:10 Lunch break
13:40 Keynote:
Ulf Dieckmann: Adaptive dynamics theory – Understanding life-history evolution, niche construction, and speciation
14:25 Ákos Bede-Fazekas: Modeling the climate envelope of some European vector species
14:45 Attila Trájer: Modeling the seasonality of Lyme borreliosis in Hungary
15:05 Live from Montréal:
Jacques Bélair: Mathematics of Planet Earth 2013
15:25 Poster session & Coffee break
16:15 Keynote:
Antal Berényi: Decoding of oscillatory dynamics in neuronal networks
17:00 Keynote:
Christina Kuttler: Who is there? – Mathematical modeling of bacterial communication
17:45 Balázs Papp: A network-level view of 'underground' metabolism
19:00 Fish soup dinner
Posters
Maria Vittoria Barbarossa: Delay equations explain Quorum sensing of P. putida
Attila Dénes: Risk of infectious disease outbreaks by imported cases with application to the European Football Championship 2012
Diána H. Knipl: Epidemic spread and variation of peak times in connected regions due to travel related infections
Pamela Moschini: A semi-discrete model for vector-borne diseases
Kyeongah Nah: Modeling of P. vivax malaria with bimodal incubation time
Yukihiko Nakata: Global dynamics of two-compartment models for cell production systems with regulatory mechanisms
Gergely Röst: Endemic bubbles generated by delayed behavioral response in epidemic models
László Székely: Can Brazilian waxing kill the pubic louse?
Mónika Szűcs: A mathematical model of oncolytic virus therapy
Zsolt Vizi: Backward bifurcation for pulse vaccination
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Talks
Waning immunity and state-dependent delays
Maria Vittoria Barbarossa and Gergely Röst
University of Szeged, Szeged, Hungary
When the body gets infected by a virus (e.g., measles, rubella, ...), the immune system develops a
certain resistance against it. As a matter of fact disease-induced immunity tends to wane and, long
time after recovery, an individual might become again susceptible to the virus. Exposure to the
pathogen boosts the immune system, thus prolonging the time in which the individual is immune.
In this work we focus on the feedback mechanism which makes possible for certain individuals to
have lifelong immunity, being regularly exposed to the infection. The mathematical model is based
on a system of differential equations with state-dependent delay. We shall consider the effects of
waning immunity and immune system boosting on epidemics outbreaks.
Modeling the climate envelope of some European vector species
Ákos Bede-Fazekas and Attila Trájer
Corvinus University of Budapest, Budapest, Hungary
Climate Envelope Modeling (CEM) – also known as correlative modeling – is a Species Distribution
Modeling (SDM) method assuming that the climatic envelope of the studied species, calculated in
the reference period from the species distribution, is static. This assumption enables the
extrapolation in temporal terms. By statistical methods or Artificial Intelligence (AI) algorithms, the
model can find the correspondence between the climatic environment and the species
distribution and draw the potential distribution in the reference period or in the future. A simple
Climate Envelope Model was run to examine whether the climate of Europe in the 21st century will
be suitable for some important Diptera vectors. Several Phlebotomus species (sand flies) are able
to transmit the Leishmania infantum protozoa, causative agent of the zoonosis called
leishmaniasis. Aedes albopictus (tiger mosquito) is one vector of the Dirofilaria immitis nematode
(roundworm), agent of the zoonosis called dirofilariasis. The distribution map of these species
were obtained from the European Centre for Disease Prevention and Control (ECDC) and REMO
regional climate model (RCM) was used for acquiring the climatic data of the reference and
prediction periods. The model results showed remarkable future potential expansion. The future
(and in some case the present) climate of Southern Hungary seems to be suitable for Phlebotomus
ariasi, Ph. neglectus, Ph. perfiliewi, Ph. perniciosus, Ph. tobbi, and Aedes albopictus. The model
proved the first lethal canine dirofilariasis case in Southern Hungary (Pécs, April 2013).
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Live from Montréal
Mathematics of Planet Earth 2013
Jacques Bélair
Université de Montréal, Montréal, Canada
In this video talk we will give a short overview on the Mathematics of Planet Earth 2013 project.
Keynote lecture
Decoding of oscillatory dynamics in neuronal networks
Antal Berényi
University of Szeged, Szeged, Hungary
Knowledge of how neural activity is conveyed along neuronal pathways in the behaving animal is a
requisite for the understanding of neuronal computation. In principle, this can be achieved by
monitoring the activity of multiple neurons which are monosynaptically connected to each other.
The orderly cytoarchitecture and the known unidirectional trisynaptic pathway between the
hippocampal subfields provide an excellent model system for investigating the transmission of
activity within a neural circuit. Understanding the input-output transformation as a complex
relationship between the synaptic inputs to neurons and their spiking outputs is fundamental for
deciphering a circuit’s computations. To this end, we recorded local field potentials (LFP) and
spikes at 512 locations in two 1.5 mm X 2.1 mm grid, covering parts of dentate gyrus, CA3-CA1,
and subicular region, using two high density silicon probes (8 shanks, with 32 sites at 50 µm
intervals each probe) in 5 rats, performing various spatial navigation and behavioral tasks. The
equally spaced and dense recording sites allowed a smooth and detailed reconstruction of the
hippocampal neuroanatomy based purely on electrophysiological data. As it was expected many
principal cells had well-defined place fields so that the firing rates and the phase of the spikes
relative to the theta cycle well defined the position of the rat on the track (O’Keefe and Nadel,
1978). While spike outputs of neurons can be detected with relative ease from extracellular
recordings, synaptic and subthreshold activity remains obscured and undifferentiated within the
oscillations (e.g. theta) of the LFP. We show that even within a small volume of the hippocampus,
the spatiotemporal structure of the local field potential evolves in a complex yet reproducible
sequence as a rat runs through its environment, and contains precise information about the
animal’s position. We identify position-locked sparse features underlying the wave structure using
unsupervised learning algorithms. The theta oscillation-coordinated activity encodes behavior as
robustly as output spikes of neuron populations. Our work provides a framework for defining the
coding properties of ongoing population activity within neural circuits and suggests that the LFP is
a promising signal for brain-computer interfaces.
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Keynote lecture
Adaptive dynamics theory – Understanding life-history evolution,
niche construction, and speciation
Ulf Dieckmann
International Institute for Applied Systems Analysis, Laxenburg, Austria
Providing a modern extension of classical evolutionary game theory, the theory of adaptive
dynamics allows deriving the fitness landscapes governing adaptive evolution from the underlying
ecological processes. This facilitates the analysis of adaptation in quantitative traits under natural
conditions, accounting for arbitrary forms of population structure and density regulation. Adaptive
dynamics theory highlights the importance of non-optimizing evolution and contributes to
understanding surprising evolutionary phenomena such as evolutionary branching, evolutionary
slowing down, evolutionary suicide, and evolutionary cycling. This, in turn, enables innovative
insights into life-history evolution, niche construction, and speciation, underscoring the need for
integrative treatments of ecological and evolutionary dynamics.
Keynote lecture
Who is there? – Mathematical modeling of bacterial communication
Christina Kuttler
Technische Universität München, Munich, Germany
Many bacteria developed a possibility to recognize aspects of their environment or to
communicate with each other by chemical signals. One important case is the so-called Quorum
sensing, a regulatory mechanism for gene expression. Bacteria can measure their own cell density
and the surrounding space by means of this signaling pathway. This system can be considered on
different scales: The intracellular regulation system - dependent on the bacterial species – often
contains several interconnected pathways, which allow for different qualitative (and quantitative)
behavior. Furthermore, some processes underlie a delay. This leads to ODE or DDE systems and
we can analyze the qualitative and quantitative behavior and compare it to experiments in batch
culture and continuous culture.
Another aspect concerns the observability of these processes and what we can learn from the
experimental findings. For the description of the intercellular communication, we can use, e.g., a
reaction-diffusion system, which allows us to consider single cells in space.
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Keynote lecture
When balance control fails: mathematical insights into falling
John G. Milton
Claremont College, Claremont CA, USA
Falls are a leading cause of morbidity and mortality in the elderly. Over 50% of falls occur during
transition from standing to walking suggesting that there may be a dynamic basis. Current
mathematical approaches to balance control are motivated by the observation that time-delayed
feedback can stabilize the upright position of a pendulum. Two experimental measurements,
namely the time delay and the minimum length of stick that can be stabilized, provide further
insight into the nature of the control mechanism. However, since these approaches rely on a local
stability analysis the question “How can the pendulum fall?” is not addressed. A key concept for
falling is the relationship between balancing dynamics and the edge of the basin of attraction for
the upright position. Experimental observations suggest that for stick balancing at the fingertip the
size of the basin of attraction is of the same order as the variance of the noisy perturbations.
Similarly the transition from standing to walking requires that the center of mass of the individual
be displaced outside the basin of attraction for the standing. Here I discuss the transient
behaviors that arise from the interplay between delay and noise in dynamical systems that are
tuned near, or at, the edge of stability. A surprising conclusion is that it may be possible to predict
a fall before it occurs.
Keynote lecture
HIV competition dynamics over sexual networks
Bence Ferdinandy1, Enys Mones1, Tamás Vicsek1 and Viktor Müller1,2
1Eötvös Loránd University, Budapest, Hungary 2Hungarian Academy of Sciences, Budapest, Hungary
Background: The global phylogeography of HIV is characterized by compartmentalized local
epidemics that are typically dominated by a single subtype, which indicates strong founder effects.
We hypothesized that the competition of viral strains at the epidemic level may be characterized
by an advantage of the “resident” strain that was the first to colonize a population. Such an effect
would slow down the invasion of new strains and thus also the diversification of the epidemic.
Methods: We developed a stochastic modeling framework to simulate HIV epidemics over
dynamic contact networks. We simulated epidemics in which the second strain was introduced
into a population where the first strain had established a steady-state epidemic, and assessed
whether and on which time scale the second strain was able to spread in the population.
Simulations were parameterized based on empirical data; we tested scenarios with varying levels
of overall prevalence, and varying differences in the transmission efficiency of both strains.
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Results: The spread of the second strain occurred on a much slower time scale compared to the
initial expansion of the first strain. With strains of equal transmission efficiency, the second strain
was unable to invade on a time scale relevant for the history of the HIV pandemic. To become
dominant over a time scale of decades, the second strain needed considerable (>10%) advantage
in transmission efficiency over the resident strain. We also tested how possible mechanisms of
interference contributed to the inhibition effect.
Conclusions: Our simulation confirmed asymmetrical competition dynamics of HIV at the
population level, with an advantage of the first successful strain in the population. This effect may
explain the global phylogeography of the virus and influence the future evolution of the pandemic.
Hierarchical structures motivated by living communities
Katalin Ozogány, Tamás Nepusz and Tamás Vicsek
Eötvös Loránd University, Budapest, Hungary
Social networks of living beings motivated our model for simulating the emergence of hierarchy.
The model is knowledge based where non cooperative individuals living in a changing environment
try to find a good response by making estimates. Since individuals are able to follow the more
capable ones, leader-follower relationships spontaneously emerge between them thus forming a
leader hierarchy. Introducing simple rules inspired by social animals and human communities can
lead to a modular structure, where smaller sub-units emerge based on similarities or some
cohesive forces. Multiple levels of hierarchical organization can be observed, since these sub-units
(associated with denser subgraphs) also constitute a hierarchical network resulting an effective
flow of information.
A network-level view of 'underground' metabolism
Balázs Papp
Biological Research Centre, Szeged, Hungary
Enzymes frequently display low-level catalytic side activities with no clear physiological role. Such
‘underground’ reactions can provide raw material for the evolution of novel catalytic functions.
However, it remains unknown how these reactions could generate evolutionary novelties in the
context of the entire metabolic network and if so, what factors may influence the realization of
these novelties during evolution. Here we computationally reconstructed a metabolic network of
Escherichia coli that includes known catalytic side activities. Due to chemical constraints, these
underground reactions are non-randomly distributed in the network and half of them can be
wired into the native metabolic network. They generally contribute to novel metabolic pathways
producing key biomass precursors, underscoring their potential biological relevance at the
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network level. However, we identified two factors that might limit the evolution of these novel
pathways in nature. First, underground reactions tend to introduce toxic metabolites into the
network. Second, despite their seamless integration into the network, most underground activities
do not provide advantage in any of a wide range of nutrient conditions.
We confirm using high-throughput experimental gene overexpression screens that amplification of
underground activities rarely confers growth under novel carbon sources. However, despite their
low frequency, such metabolic novelties are computationally predictable based on our knowledge
of underground metabolism. Taken together, our study indicates that a large fraction of the
biochemically feasible raw material have remained unexploited by adaptive evolution.
Modeling the seasonality of Lyme borreliosis in Hungary
Attila Trájer and Ákos Bede-Fazekas
Pannon University, Veszprém, Hungary
We found that the annual Lyme borreliosis (hence: LB) incidence doubled during a 13 year period
in Hungary. Our aim was to understand the most important factors which determine the LB season
in Hungary and explain the apparent contradiction between the annual unimodal LB incidence and
the bimodal Ixodes ricinus tick activity run in Hungary by distinguishing the temperature
dependent seasonal human and tick activity, the temperature-independent factors, and the
multiplicative effect of human outdoor activity in summer holiday, using data from Hungary in the
period of 1998–2012.
This distinction was verified by modeling the Lyme incidence based on the separated factors, and
comparing the run of the observed and modeled incidence. The human outdoor activity showed a
similar exponential correlation with ambient temperature to that the relative incidence did. It was
proved that summer holiday has great influence on Lyme incidence.
To better understand the role of LB as indicator disease we modeled the temperature-related,
mainly questing hard tick influenced spring and summer part of the Lyme season in Hungary in
1998–2010. Our model was based on the “wait and see” strategy of ticks and the probability
distribution of the latency of early manifestation forms as ECM and neuroborreliosis. We found
that the onset of the early symptoms show a log normal probability distribution which is in
accordance with the literary data of the 3 days to 2 months latency with peak in the 2nd and 3th
weeks. The model can explain the apparent contradiction between the observed April–May peak
of tick season, the serological LB peak in the Hungarian population and of the distribution of the
onset of the LB cases.
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Posters
Delay equations explain Quorum sensing of P. putida
Maria Vittoria Barbarossa1 and Christina Kuttler2
1University of Szeged, Szeged, Hungary 2Technische Universität München, Munich, Germany
The bacterial strain Pseudomonas putida IsoF, isolated from a tomato rhizosphere, possesses a
Quorum sensing regulation system, which allows the bacteria to recognize aspects of their
environment or to communicate with each other by the so-called autoinducer molecules.
In an experimental study the time series of the autoinducer (AHL) production did not show the
expected behavior, as it was observed for other bacterial species by indirect measurements.
Our approach supports the hypotheses of the existence of a further enzyme, which degrades the
AHLs into an inactive form. As numerical simulations show, the delay model can explain the AHL
dynamics observed in the experiments, thus supporting the biological hypotheses.
Further we could prove that the system shows a typical bistable behavior, choosing, e.g., bacterial
population density or abiotic degradation rate as exemplary bifurcation parameters. With a
particular choice of the parameter values in the delay model an oscillatory behavior was found.
Risk of infectious disease outbreaks by imported cases with application
to the European Football Championship 2012
Attila Dénes1, Péter Kevei1, Hiroshi Nishiura2 and Gergely Röst1
1University of Szeged, Szeged, Hungary 2University of Tokyo, Tokyo, Japan
The European Centre for Disease Prevention and Control called the attention in March 2012 to the
risk of measles in Ukraine among visitors to the 2012 UEFA European Football Championship.
Large populations of supporters travelled to various locations in Poland and Ukraine, depending
on the schedule of Euro 2012 and the outcome of the games, possibly carrying the disease from
one location to another. We propose a novel two-phase multitype branching process model with
immigration to describe the risk of a major epidemic in connection with large-scale sports-related
mass gathering events. By analytic means, we calculate the expected number and the variance of
imported cases and the probability of a major epidemic caused by the imported cases in their
home country. Applying our model to the case study of Euro 2012, we demonstrate that the
results of the football games can be highly influential to the risk of measles outbreaks in the home
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countries of supporters. To prevent imported epidemics, it should be emphasized that vaccinating
travelers would most efficiently reduce the risk of epidemic, while requiring the minimum doses of
vaccines as compared to other vaccination strategies. Our theoretical framework can be applied to
other future sport tournaments too.
Epidemic spread and variation of peak times in connected regions due
to travel related infections
Diána H. Knipl1, Gergely Röst1 and Jianhong Wu2
1University of Szeged, Szeged, Hungary 2York University, Toronto, Canada
National boundaries never hindered infectious diseases to reach distant territories, however the
speed at which an infectious agent can spread around the world via the global airline
transportation network has significantly increased during the last decades. We introduce an SEAIR-
based, antigravity-type model to investigate the spread of an infectious disease in two regions,
which are connected by transportation. As a submodel, an age-structured system is constructed to
incorporate the possibility of disease transmission during travel, where age is the time elapsed
since the start of the travel. The model is equivalent to a large system of differential equations
with dynamically defined delayed feedback. After describing fundamental, but biologically
relevant properties of the system, we detail the calculation of the basic reproduction number and
obtain disease transmission dynamics results in terms of R0. We parameterize our model for
influenza and use real demographic and air travel data for the numerical simulations. To
understand the role of the different characteristics of the regions played in the propagation of the
disease, three distinct origin-destination pairs are considered. The model is also fitted to the first
wave of the A(H1N1) 2009 pandemic influenza in Mexico and Canada. Our results highlight the
importance of including travel time and disease dynamics during travel in the model: the invasion
of disease free regions is highly expedited by the elevated transmission potential during
transportation.
A semi-discrete model for vector-borne diseases
Pamela Moschini
University of Trento, Trento, Italy
In this work a semi-discrete model for the transmission dynamics of vector-borne diseases is
presented. The transmission in a host population with a vector population is modeled in an SIS/SIR
epidemiological framework, where contact between host and vector is assumed to occur only
during the summer of each year. Two types of threshold for the spread of the disease are
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obtained: one (R0) measures the number of new infected generated by one infected individual; the
other one (S0) gives the average number of infected vectors produced at the start of the year by a
vector that was infected at the start of the previous year. From the local and global behavior
analysis, we show that if S0 < 1 the disease free equilibrium is locally stable and globally attractive,
if S0 > 1 the disease free equilibrium is locally unstable and there exists an endemic equilibrium
which is globally attractive in the SIS case. S0 is compared to the definition of the basic
reproduction number R0 for periodic continuous-time models, proposed by Bacaër and co-workers
showing that they yield the same threshold. Finally, simulations are performed showing typical
behavior for the SIS and SIR cases.
Modeling of P. vivax malaria with bimodal incubation time
Kyeongah Nah and Gergely Röst
University of Szeged, Szeged, Hungary
Malaria parasites are transmitted between mosquitoes and humans. If an infectious mosquito
bites a host, symptoms occur after a certain incubation period. The incubation period can vary
depending on the species of parasite or the regions. In particular, incubation period of
Plasmodium vivax – the malaria inducing parasite species most prevalent in temperate zones in
Korea – shows bimodal distribution, with short term and long-term incubation periods. In this
poster presentation, we compare transmission models for P. vivax malaria having different
expression for the incubation period.
Global dynamics of two-compartment models for cell production
systems with regulatory mechanisms
Philipp Getto1, Anna Marciniak-Czochra2, Yukihiko Nakata3 and María dM Vivanco4
1Technische Universität Dresden, Dresden, Germany 2University of Heidelberg, Heidelberg, Germany
3University of Szeged, Szeged, Hungary 4CIC bioGUNE, Derio, Spain
We present a global stability analysis of two-compartment models of a hierarchical cell production
system with a nonlinear regulatory feedback loop. The models describe cell differentiation
processes with the stem cell division rate or the self-renewal fraction regulated by the number of
mature cells. Using global stability analysis, we compare different regulatory mechanisms. For
both models, we show that there exists a unique positive equilibrium that is globally
asymptotically stable if and only if the respective reproduction numbers exceed one.
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Endemic bubbles generated by delayed behavioral response in epidemic
models
Maoxing Liu1,2, Eduardo Liz3, Gergely Röst1 and Gabriella Vas1
1University of Szeged, Szeged, Hungary 2North University of China, Taiyuan, China
3University of Vigo, Vigo, Spain
Many attempts have been made in the past to capture the phenomenon that people modify their
behavior during an epidemic outbreak. This reaction can be triggered by directly experiencing the
rising number of infections, media coverage, or intervention policies. In this talk we show that a
delayed activation of such a response can lead to some surprising dynamics. For an SIS type
process, if the delayed response occurs with a jump in the contact rate when the density of
infection reaches some threshold, we show that for some interval of reproduction numbers, the
system is oscillatory. The oscillation frequency is a discrete Lyapunov functional and there exists a
unique slowly oscillatory periodic solution with strong attractivity properties. We also construct
rapidly oscillatory periodic solutions of any frequency. In the case of continuously decreasing
transmission rate, if the response is not too strong, the system preserves global stability. However,
for sharp delayed response, we can observe stability switches as the basic reproduction number is
increasing. First, the stability is passed from the disease free equilibrium to an endemic
equilibrium via transcritical bifurcation as usual, but a further increase of the reproduction number
causes oscillations, which later disappear for higher values of the reproduction number, forming
an interesting structure in the bifurcation diagram what we call endemic bubble.
Can Brazilian waxing kill the pubic louse?
Attila Dénes1, Gergely Röst1 and László Székely2
1University of Szeged, Szeged, Hungary 2Szent István University, Gödöllő, Hungary
The pubic louse is an obligate ectoparasite of humans which lives mostly in pubic hair. Lice infect a
new host only by close contact between individuals, usually through sexual activity. According to
recent media reports, the pubic louse (Pthirus pubis) has almost been driven to extinction in
several countries. This was attributed to the gaining popularity of pubic hair removal among
females, which also provides protection againt lice infestation. To capture this interesting
phenomenon, we introduce a mathematical model, that is SIS type for males and SIRS type for
females, where the R compartment contains females having no (or only a little) pubic hair. We
identify the basic reproduction number R0 which turns out to be a threshold parameter and give
an analysis of the global dynamics via Lyapunov functions. Finally, by parameter estimation and
data analysis we try to answer the question whether the cultural phenomenon of pubic hair
removal can indeed eradicate Pthirus pubis.
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A mathematical model of oncolytic virus therapy
Mónika Szűcs
University of Szeged, Szeged, Hungary
Malignant neoplasms, commonly known as cancer, are a range of widespread diseases in which
abnormal cells divide uncontrollably and can invade healthy body tissue. Treatments for malignant
neoplasm have been importantly improved in the last decades. Nowadays, cancer patients usually
undergo chemotherapy, radiation therapy and surgery. However, these methods affect both
healthy and cancer cells, whereas an oncolytic virus preferentially infects and kills cancer cells.
In this work we deal with impulsive differential equations modeling tumor growth controlled by
oncolytic virus. Analysis of the mathematical model allows us to identify critical parameters and
suggests how to determine an effective treatment.
Backward bifurcation for pulse vaccination
Gergely Röst and Zsolt Vizi
University of Szeged, Szeged, Hungary
We investigate the types of appearing bifurcations in SIVS model with pulse vaccination strategy.
First we compute the disease-free periodic solution and prove its global asymptotic stability in the
disease-free subspace. We identify the corresponding control reproduction number Rc and prove
that the disease-free periodic solution is locally asymptotically stable if Rc < 1. Furthermore, under
some additional conditions it is globally asymptotically stable, too. For Rc > 1 we prove the uniform
persistence of the disease.
Our main result is that nontrivial endemic periodic solutions are bifurcating from the disease free
periodic solution as Rc passes the threshold value one. A complete bifurcation analysis is provided
for the associated nonlinear fixed point equation. We show that backward bifurcation of periodic
orbits is possible, and give explicit conditions to determine whether the bifurcation is backward or
forward. The main mathematical tools are comparison principles and Lyapunov–Schmidt
reduction.
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List of participants
Maria Vittoria
Barbarossa
Bolyai Institute University of Szeged Szeged, Hungary
Ákos
Bede-Fazekas
Department of Garden and Open Space Design Faculty of Landscape Architecture Corvinus University of Budapest Budapest, Hungary
Antal
Berényi
Department of Physiology University of Szeged Szeged, Hungary
Jacques
Bélair
Department of Mathematics and Statistics Université de Montréal Montréal, Canada
Attila
Dénes
Bolyai Institute University of Szeged Szeged, Hungary
Ulf
Dieckmann
Evolution and Ecology Program International Institute for Applied Systems Analysis Laxenburg, Austria
Ábel
Garab
Bolyai Institute University of Szeged Szeged, Hungary
János
Karsai
Bolyai Institute University of Szeged Szeged, Hungary
Diana
H. Knipl
MTA-SZTE Analysis and Stochastics Research Group Bolyai Institute Szeged, Hungary
Christina
Kuttler
Department of Mathematical Modelling Technische Universität München Munich, Germany
John G.
Milton
Keck Science Department Claremont University Claremont, California, USA
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Pamela
Moschini
Department of Mathematics University of Trento Trento, Italy
Viktor
Müller
Research Group of Theoretical Biology and Evolutionary Ecology Eötvös Loránd University and Hungarian Academy of Sciences Budapest, Hungary
Kyeongah
Nah
Bolyai Institute University of Szeged Szeged, Hungary
Yukihiko
Nakata
Bolyai Institute University of Szeged Szeged, Hungary
Katalin
Ozogány
Department of Biological Physics Institute of Physics, Eötvös Loránd University Budapest, Hungary
Gergely
Röst
Bolyai Institute University of Szeged Szeged, Hungary
Balázs
Papp
Biological Research Centre Hungarian Academy of Sciences Szeged, Hungary
László
Székely
Department of Mathematics Szent István University Gödöllő, Hungary
Mónika
Szűcs
Department of Medical Physics and Informatics University of Szeged Szeged, Hungary
Attila
Trájer
Department of Limnology University of Pannonia Veszprém, Hungary
Zsolt
Vizi
Bolyai Institute University of Szeged Szeged, Hungary
Bolyai Institute www.math.u-szeged.hu
The Bolyai Institute – the mathematical institute of the University of Szeged – was founded in 1921
by the two world-famed professors of mathematical analysis, Frigyes Riesz and Alfréd Haar. Since
then, the institute has become one of the most important centers for mathematics in Hungary,
where several internationally renowned researchers have been working. More than 50
mathematicians – including five members of the Hungarian Academy of Sciences – work in the six
departments: Algebra and Number Theory, Applied and Numerical Mathematics, Analysis,
Geometry, Set Theory and Mathematical Logic, and Stochastics. The institute has a mathematical
library with about 50000 volumes. The distinguished international journals Acta Scientiarum
Mathematicarum (founded by Riesz and Haar) and the Electronic Journal of Qualitative Theory of
Differential Equations as well as several mathematical textbooks are published by the institute.
EPIDELAY Research Group
www.epidelay.u-szeged.hu The aim of the EPIDELAY project is to develop and analyse infinite dimensional dynamical models
for the transmission dynamics and propagation of infectious diseases. We use an integrated
approach which spans from the abstract theory of functional differential equations to the practical
problems of epidemiology, with serious implications to public health policy, prevention, control
and mitigation strategies in cases such as the previous H1N1 pandemic.
Delay differential equations are one of the most powerful mathematical modelling tools and they
arise naturally in various applications from life sciences to engineering and physics, whenever
temporal delays are important. In abstract terms, functional differential equations describe
dynamical systems, when their evolution depends on the solution at prior times. The central
theme of this project is to forge strong links between the abstract theory of delay differential
equations and practical aspects of epidemiology. Our research will combine competencies in
different fields of mathematics and embrace theoretical issues as well as real life applications.
In particular, the theory of equations with state dependent delays is extremely challenging.
Developing new theories in this area and connecting them to relevant applications may have a
significant impact on infectious disease modeling.
Infocommunication technologies and the society of future
TÁMOP-4.2.2.C-11/1/KONV-2012-0013
Workshop on Modeling in Life Sciences November 30, 2013
Informatorium, Szent-Györgyi Albert Agora, 23 Kálvária Avenue, Szeged, Hungary
Organized by the Bolyai Institute, University of Szeged in the framework of
Mathematics of Planet Earth 2013 and the FuturICT.hu project.
Organizers:
Gergely Röst (chair)
Maria Vittoria Barbarossa
Attila Dénes
Ábel Garab
János Karsai
Diána H. Knipl
Contact:
EPIDELAY Research Group
Bolyai Institute
University of Szeged
Aradi vértanúk tere 1, Szeged
H-6720 Hungary
www.math.u-szeged.hu
www.epidelay.u-szeged.hu
www.model.u-szeged.hu