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## Mathematics and Biology: a Roundtrip in the Light of Suns and Stars |

We
discuss how the concept of basic reproduction number may be adapted to periodic
or Markovian environments. The models may or may not
include demographic stochasticity. In particular we
shall present recent work on continuous-time models with both demographic and
environmental stochasticity. Bibliography: N. Bacaër, E. Ait
Dads: On the probability of extinction in a periodic environment. J Math Biol, doi:10.1007/s00285-012-0623-9
N. Bacaër, M. Khaladi: On
the basic reproduction number in a random environment. J Math Biol,
doi:10.1007/s00285-012-0611-0
Viruses
make many roundtrips between many generations of selection within one host and
a single generation of selection at the population level. At the within-host
level, a virus evolves mutations to escape from the unique immune responses
mounted by an individual host. Because of the massive heterogeneity of MHC
molecules in natural populations, these adaptive mutations are expected to be
detrimental in future hosts. The
within-host viral fitness determines virulence and infectiousness, and hence
the basic reproduction number (R0) at the population level. The
aim of our research is to investigate the implications of the above described
intertwined selection mechanisms.
The existing classification of evolutionarily singular
strategies in Adaptive Dynamics [2,3] assumes an
invasion exponent that is differentiable twice as a function of both the
resident and the invading trait. Motivated by nested models for studying the
evolution of infectious diseases [1], we consider an extended framework in
which the selection gradient exists (so the definition of evolutionary
singularities extends respect to convergence stability and invadability
and determine the condition for the existence of nearby dimorphisms. The
extended setting allows for a new type of evolutionary singularity: a so called
one-sided ESS that is invadable by mutant strategies
on one side of the singularity but uninvadable by
mutants on the other side. A more detailed analysis of the regions of mutual invadability in the vicinity of a one-sided ESS reveals
that two isoclines (one of each type) spring in a tangent manner from the
singular point at the diagonal of the Mutual Invadability
Plot. The way in which the isoclines unfold determines whether these one-sided
ESSs act as ESSs or as branching points. Contrary to the standard setting of
Adaptive Dynamics, the fate of dimorphisms nearby a singular strategy can, in
general, not be deduced by considering just the monomorphic invasion exponent.
To conclude the classification of evolutionary singularities in an extended
setting, we present a computable condition that enables us to determine the
relative position of the isoclines and thus allows us to predict the fate of
dimorphisms nearby a one-sided ESS. The talk is based on joint work with Odo
Diekmann. References: [1] Boldin,
B., Diekmann, O.: [2] Geritz,
S.A.H., Kisdi, E., Meszena,
G., Metz, J.A.J.: [3] Metz, J.A.J., Geritz, S.A.H., Meszena, G., Jacobs, F.J.A., van Heerwaarden,
J.S.:
Nosocomial
outbreaks of carbapenemase-producing Enterobacteriaceae such as OXA-48 are increasing. OXA-48
can reveal different phenotypes ranging from highly resistant (mostly in K. pneumoniae) to susceptible (e.g., in E. coli) to carbapenems. Using longitudinal PCR-based OXA-48 screening
data from an outbreak of OXA-48 producing Enterobacteriaceae
in a Dutch hospital involving 118 patients, we assessed the within-host
horizontal gene transfer rate and duration of colonization of OXA-48.
Furthermore, we investigated the likelihood of development of an OXA-48
reservoir in E. coli and the need for PCR-based screening.
The aim is to analyze a cell population model, in which the
maturation of stem cells is regulated by the mature cell population [1]. Hence
the maturation delay of a cell is dependent on the state of the cell
population. The resulting model is a differential equation with a state
dependent delay. For such equations the set of initial conditions should be
reduced to a submanifold of co-dimension one of C1
[4]. Existence of maximal and global solutions is analyzed by a combination of
methods from [2,3,4].References: [1] T. Alarcon, Ph. Getto, A. Marciniak-Czochra and Maria dM. Vivanco. A model for stem cell population dynamics with regulated maturation delay. Discr. Cont. Dyn. Sys. b. Supplement 2011, pp.32-43. [2] O. Diekmann, S. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations, Functional-, Complex-, and Nonlinear Analysis. Springer Verlag, New York, 1995. [3] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations. Springer Verlag, New York, 1991. [4] F. Hartung, T. Krisztin, H.-O.- Walther and Jianhong Wu, Functional Differential Equations with state dependent delays: Theory and Applications, Chapter V in Handbook of Differential Equations: Ordinary Differential Equations, Volume 4, Elsevier.
Equilibria of structured cell population models with internal cell cycleJ.J.
Tyson and B. Novak have developed several widely studied models for the cell
cycle of budding yeast, fission yeast and other organisms (see for example
[1]). One of the striking features in the behaviour
of such models is the funnel effect, by which the internal structure of the cell
in the later phases of its cycle, shortly
before division, is
nearly independent from its structure in the earlier phases of
the cycle. This mathematical property of
the models makes biological sense. Bibliography
In
this talk I give an overview of the history of structured population dynamics,
present the state-of-the- art and formulate some important future directions of
research.
Quiescent phases in ecological models tend to stabilize equilibria against the onset of oscillations - when all
interacting species go quiescent and become active with the same rates. If the
species differ with respect to these rates, then equilibria
may be destabilized. These bifurcation phenomena - closely related to Turing
instability - are studied in general and in ecological and epidemic models, delay
equations and reaction-diffusion equations.
Stabilization by quiescent phases can be set in a wider scenario
that shows that in some sense spatial heterogeneity stabilizes. Numerical simulations show that periodic orbits shrink when
quiescent phases are introduced. Rigorous results can be shown for convex orbits.
In
the 1980's work by Clément, Diekmann, Gyllenberg, Heijmans
and Thieme initiated the treatment of classical
functional differential equations using methods from dual perturbation theory,
also known as sun-star calculus. Recently it has become apparent that these
methods are equally well suited for the treatment of other classes of delay
equations (DE), see for example Tuesday's talk by Yu.A. Kuznetsov on "Local bifurcations in Neural Field Equations". After
a brief review of the basic setting we discuss recent developments concerning
various classes of equations such as abstract DE and DE with unbounded
operators in the right-hand side. We will also comment on the validity of the
center manifold theorem in the non-sun-reflexive case which is important for
local bifurcation analysis.
In the book by Diekmann,
Heesterbeek and Britton there is mention of the
methods we and others have developed to estimate transmission parameters from
observed transmission chains (“experiments”) with animals. Basically, we have
two types of data: interval data where we assume that the number of infected
and infectious individuals during a time interval is constant and we observe
the number of new cases, and final size data where we assume that the infection
chain has ended either because of depletion of susceptible individuals or
because all infectiousness has ended and we observe the number of individuals
that became infected. From interval data we estimate the transmission rate
parameter and from the final size data the reproduction ratio. With these
methods we can also compare different type of animals for example vaccinated
groups with not vaccinated groups. However, there are also situations where
we have to estimate several transmission parameters from heterogeneous groups.
Such heterogeneity can occur because
animals of the same type (for example vaccinated in the same way)
respond differently or because the large number of types does not allow us to
study each type separately (for example if we know the genotype and want to
find host traits for susceptibility and infectivity). Such heterogeneity can be dealt with in the
statistical models that we use to estimate transmission parameters. I will
discuss the possible ways of estimating several transmission parameters from
observed transmission chains, which assumptions are made, and how we can test
that these assumptions are valid.
We
discuss different approaches to model biological systems in different detail,
based on a distinction between bottom-up and top-down approaches. Basically
classical mathematical biology was based on top-down approaches, where
functional forms entering the equations were based on heuristic reasoning with
a sub- sequent consistency analysis, for example based on conservation laws. In
contrast, bottom-up approaches are based on the tradition of reaction systems
and statistical mechanics. The idea is to close the system description by
assuming fundamental basic entities (like 'molecules') and rules describing
their state transition. The macroscopic description of the system is then based
on a statistical analysis of microscopic ensembles.
In
pair formation models the duration of contacts between pairs of individuals is
explicitly taken into account. It is assumed that contacts, which possibly lead
to transmission of an infection, take place repeatedly between the same pair of
individuals. From the point of view of a pathogen those contacts occurring
after a transmission event are lost for further disease transmission. Also,
within monogamous pairs of susceptible individuals transmission cannot take
place at all. These models are useful for studying transmission dynamics of
sexually transmitted infections, where partnership duration and the duration of
infectiousness are long, possibly in the order of the lifetime of individuals.
Pair formation models have been used to describe transmission of HIV. An
important question in HIV epidemiology is what proportion of cases are produced during the highly infectious acute infection.
This of course depends on partner change rates and thus interacts with
partnership duration. For curable STIs long term partnerships may be a source
of reinfection for an individual who has previously cleared the infection or
has been treated. We discuss how the possibility of reinfection within
partnerships may impact on disease dynamics. Finally, we present first steps in
extending the pair formation approach to a situation where concurrent
partnerships are possible.
Neural field models with transmission delay may be cast as
abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus)
provides a
natural framework for the analysis of a broad class of delay equations, among which
DDE. In particular, it may be used advantageously for the
investigation of stability and bifurcation of steady states. After introducing
the neural field model in its basic functional analytic setting and discussing
its spectral properties, an example will be elaborated where the spectrum and
the resolvent can be found explicitly. Under certain
conditions the associated equilibrium may exhibit a Hopf
bifurcation. Furthermore, two Hopf curves may
intersect in a double Hopf point in a two-dimensional
parameter space. We provide general formulas for the corresponding critical
normal form coefficients, evaluate these numerically, and verify the results by
simulations.
Models
for the evolution of virulence traditionally assume a trade-off between inverse
disease-induced mortality rate and infectivity, resulting in intermediate
virulence. The underlying intuition is that faster growing agent populations do
both more damage and produce more infective particles. This
intuition implicitly assumes a well-mixed host body. In reality both damage and
infectivity depend mainly on the location in the body where the agents lodge.
This is related i.a. to the surface proteins that
allow agents to dock on and penetrate into different cell types. The typical
example is respiratory diseases where more deeply seated ones are both less
infective and more harmful. With the other standard assumption, full
cross-immunity between disease strains, this would lead to evolution towards
the tip of the nose. In reality cross-immunity depends on surface antigens and
hence is at least in part connected to depth. In this talk I discuss a simple adaptive
dynamics style model taking on board the aforementioned considerations. In doing so I will also shortly review salient aspects of the
adaptive dynamics toolbox. Some, probably robust, biological conclusions
are (1) higher host population densities are conducive to a higher disease
diversity, (2) disease diversity should be higher in the upper air passages
than lower in the lungs, (3) emerging respiratory diseases will usually combine
a high virulence with a low infectivity.
Turning to biological
systems, collaborative research of experimental and theoretical works has
gradually discovered the mechanism how self-organized patterns are generated in
far from equilibrium systems. It has been reported that genetics does not
always reveal the occurrence of such patterns and that surprisingly, even
simple systems may generate regular as well as irregular patterns in a
self-organized way In this lecture, I focus on
bacterial colonies of References:
The use of engineered nanomaterials (ENMs) (particles smaller
than 100nm in at least one dimension) is growting rapidly. Release of ENMs into soil and water is
inevitable, and the ecological consequences are uncertain. There is a near
limitless combination of ENMs, organisms,and environments of potential
importance, but there are limited resources for ecological studies which are
commonly expensive and time consuming. Progress in understanding ecological
implications of ENMs in the environment thus requires theory that relate
readily obtainable laboratory data on suborganismal processes to population and
ecosystem dynamics. I shall describe an
approach based on Dynamic Energy Budget (DEB) theory. At its core is a dynamic model of the
physiological performance of an individual organism, with a low-dimensional
system of ordinary differential equations describing the rates at which an
organism assimilates and utilizes energy and elemental matter for maintenance,
growth, reproduction, development, and reducing the risk of mortality, as well
as the rates of excetion of metabolic products to the environment. Ecotoxicological applications of DEB theory
may require additional submodels describing contaminant exchange with the
environment and chemical transformations within an organism. Toxic effect
submodels specify how the basic DEB model parameters change. The connection to population dynamics is made
through “structured” or “individual-based” modeling techniques. A brief outline of DEB theory will preceed a
brief overview of three recent applications to nanotoxicology: (i) a study of the response of bacteria to
Cd-Se quantum dots illustrates the importance for model testing of data on
suborganismmal dynamics; (ii) a model of
the response of freshwater phytoplankton populations to silver nanoparticles
(AgNPs) demonstrates the importance for population dynamics of a feedback
mechansim involving metabolic “waste” products;
(iii) structured and individual based models being used to design
experiments on the effects of AgNPs on
Heterogeneity is one of the most important and ubiquitous types
of external perturbations. We study a spontaneous pulse generating mechanism
caused by a heterogeneity of the jump type. Such a
pulse generator (PG) has attracted considerable interest in relation to
potential computational abilities of pulse waves in physiological signal
processing. We investigate rstly the conditions for
the onset of PGs, and secondly we show the bifurcational
origin of their complex ordered sequence of generating manners. To explore the
global bifurcation structure of heterogeneity-induced ordered patterns (HIOPs)
including PGs, we devise numerical frameworks to trace the long-term behaviors
of PGs as periodic solutions and we detect the associated terminal homoclinic orbits that are homoclinic
to a special type of HIOPs with a hyperbolic saddle. Such numerical approaches
assist in identifying a candidate for the organizing center producing a variety
of PGs as a codimension two gluing bifurcation, in
which two homoclinic trajectries
associated with pulse emission and breathing motions form a butterfly
configuration.
Several recent papers have introduced explicit
modelling of hosts' immune response in epidemic
dynamics, giving rise to "nested epidemic models" especially in order
to discuss the evolution of hosts and pathogens [1,2]. If, as assumed in [1],
pathogen load at infection is fixed, the model can reduce to an age-of-infection
structure, and its qualitative behaviour follows the
usual properties of epidemic models, although host heterogeneity in within-host
parameters can give rise to relevant evolutionary consequences [3]. References: J. theor.
Biol.
218: 289--308, 2002.[2] Mideo N., Alizon S., Day T., Linking within- and between-host dynamics in the evolutionary epidemiology of infectious diseases. TREE 23:511--517, 2008.[3] Pugliese A., The role of host population heterogeneity in the evolution of virulence, J. Biol. Dynamics 5: 104-119, 2011.
The
dynamics of the Kermack-McKendrick model, and its
special case the SIR model, are characterised by the
basic reproduction number, R0. In particular, the value of R0 determines the
initial rate of increase in infection incidence and the final size of the
epidemic. It also provides a measure of the control effort required to prevent
an epidemic, or to eliminate an existing infection from a population. For
structured models R0 is defined as the spectral radius of the Next Generation
Matrix (NGM). However, it is not always sensible to average over different host
types or states at infection, so an alternative threshold quantity the Type Reproduction
Number T has been defined. The value of T provides a measure of the effort
required when control is targeted. Another complication arises on food webs,
where there is an interaction between Ecological Stability and Epidemiological Stability.
The construction of the appropriate NGM leads to threshold quantities for these
situations. Finally, as R0 is one of the first quantities to be estimated at
the start of an epidemic, there is an error associated with its estimate. This
potential variation around predicted quantities may be accounted for in a
number of ways, including by constructing deterministic stochastic solutions. The
ideas presented in this talk were developed by, or in collaboration with, or inspired
by, Odo Diekmann.
Body size (≡ biomass) is the dominant
determinant of population dynamical processes such as giving birth or dying in
almost all species, with often drastically different behaviour
occurring in different parts of the growth trajectory, while the latter is
largely determined by food availability at the different life stages. This
leads to the question under what conditions unstructured population models,
formulated in terms of total population biomass, still do a fair job. To
contribute to answering this question we first analyze the conditions under
which a size-structured model collapses to a dynamically equivalent
unstructured one in terms of total biomass. The only
biologically meaningful case where this occurs is when body size does not
affect any of the population dynamic processes,
this is the case if and only if the mass-specific ingestion rate, the
mass-specific biomass production and the mortality rate of the individuals are
independent of size, a condition to which we refer as “ontogenetic
symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium
biomass-body size spectrum is proportional to 1/size, a form that has been
conjectured for marine size spectra and subsequently has been used as
prior assumption in theoretical papers dealing with the latter. As a next step
we consider an archetypical class of models in which reproduction takes over
from growth upon reaching an adult body size, in order to determine how
quickly discrepancies from ontogenetic symmetry lead to relevant novel population
dynamical phenomena. The phenomena considered are biomass overcompensation,
when additional imposed mortality leads, rather unexpectedly, to an
increase in the equilibrium biomass of either the juveniles or the adults (a
phenomenon with potentially big consequences for predators of the species), and
the occurrence of two types of size-structure driven oscillations,
juvenile-driven cycles with separated extended cohorts, and adult-driven cycles
in which periodically a front of relatively steeply decreasing frequencies
moves up the size distribution. A small discrepancy from symmetry can
already lead to biomass overcompensation; size-structure driven cycles only
occur for somewhat larger discrepancies.
We identify and describe the principal bifurcations of bursting
rhythms in multi-functional central pattern generators (CPG) composed of three
neurons connected by fast inhibitory or excitatory synapses. We develop a set of computational tools that reduce high-order
dynamics in biologically relevant CPG models to low-dimensional return mappings
that measure the phase lags between cells. We examine bifurcations of fixed
points and invariant curves in such mappings as coupling properties of the
synapses are varied. These bifurcations correspond to changes in the
availability of the network's phase locked rhythmic activities such as periodic
and aperiodic bursting patterns. As such, our findings provide a systematic
basis for understanding plausible biophysical mechanisms for the regulation of,
and switching between, motor patterns generated by various animals.
A
large variety of eucaryotic cells move actively along
different substrates. The actin cytoskeleton of the cells plays an important
role in this process. In the talk a minimal hyperbolic-parabolic free boundary
model for the reorganization of the cytoskeleton is introduced and analyzed.
Short time well posedness is proved, and the
emergence of Dirac measures, which can be interpreted as polymerization fronts
for active cell motion.
In the early nineties, Odo Diekmann and coworkers (Klaus Dietz, Hans Heesterbeek, Hans Metz) promoted the spectral radius of the next generation operator as the basic reproduction number of a structured population (or basic replacement ratio for infections therein). In structured populations with two sexes, the next generation operator (as approximation at the extinction equilibrium) may no longer be a linear bounded positive operator but rather a (positively) homogeneous operator (of degree one) that is bounded and order preserving. Various concepts of spectral radii are developed and compared for operators of this kind, and extensions of the Krein-Rutman theorem concerning existence of a positive eigenvector and eigenfunctional are addressed.
In
epidemiology we are often interested in R0 for emerging infectious diseases. We
will discuss how this quantity depends on the population structure and
relatively easy to obtain parameters, such as the shape of the infectivity
profile and the Malthusian parameter.
A
new approach based on Wasserstein distances, which are numerical costs of an
optimal transportation problem, allows to analyze
nonlinear phenomena in a robust manner. The long-term behavior is reconstructed
from time series, resulting in a probability distribution over phase space.
Each pair of probability distributions is then assigned a numerical distance
that quantifies the differences in their dynamical properties. From the
totality of all these distances a low-dimensional representation in a Euclidean
space is derived. This representation shows the functional relationships
between the dynamical systems under study. It allows to
assess synchronization properties and also offers a new way of numerical
bifurcation analysis.
Recent
development of new mathematical tools for studying lumped models with a spatial
component, i.e. neural fields, allows for the incorporation of features that
were previously excluded for they made the mathematical analysis intractable;
e.g. distance dependent transmission delays. Now that traditional neural
fields, requiring extensive simplifications of both the single cell dynamics
and the network architecture, are about to be fully characterized, the time
seems right to consider more involved formulations of neural fields. Here we
propose an extension to the prevalent framework of neural fields that
facilitates the inclusion of relevant spiking behaviors observed in single
neurons, e.g. tonically bursting and rebound spikes/bursts;
features surmised to be incompatible for lumping. This new setting is dependent
on the firing rate reduction of a single neuron, i.e. a model that reproduces
the rate at which spikes are generated rather than the generation of individual
spikes. Results are consistent with traditional reductions based on
integrate-and-fire neurons both with and without spike frequency adaptation,
but the main outcome is the formulation of a neural field based on Izhikevich neurons. Although we show a clear correspondence
between the original network and the reduction, the reduction still needs
refinements; both on the level of modeling as well as the mathematical
analysis.
We begin with the simplest linear differential equation for
negative feedback with a constant time lag and make this time lag
state-dependent outside a neighbourhood of zero in
such a way that the new equation has a homoclinic
solution. In the appropriate state space, which is a manifold of differentiable
functions, the intersection of stable and unstable manifolds along the homoclinic curve is minimal. Close to the homoclinic loop chaotic motion is expected, as in Shilnikov's earlier example of a vectorfield
in dimension 4, with pairs of complex conjugate eigenvalues of its
linearization at equilibrium in each halfplane. [Back] |