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Mathematics and Biology: a Roundtrip in the Light of Suns and Stars
The basic reproduction number in periodic or random environments
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.
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
Within-host dynamics of viral infections
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.
An extension of the classification of evolutionarily singular strategies in Adaptive Dynamics
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 , we consider an extended framework in which the selection gradient exists (so the definition of evolutionary singularities extends verbatim), but where the invasion fitness may lack the smoothness necessary for the classification `a la Geritz et al. In this talk, we present the classification of singular strategies with
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.
 Boldin, B., Diekmann, O.: Superinfections can induce evolutionarily stable coexistence of pathogens. Journal of Mathematical Biology 56(5), pp. 635–672 (2008)
 Geritz, S.A.H., Kisdi, E., Meszena, G., Metz, J.A.J.: Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12, pp. 35–57 (1998)
 Metz, J.A.J., Geritz, S.A.H., Meszena, G., Jacobs, F.J.A., van Heerwaarden, J.S.: Adaptive dynamics: A geometrical study of the consequences of nearly faithful reproduction. Stochastic and Spatial structures of Dynamical Systems (S.J. van Strien and S.M. Verduyn Lunel eds.) pp. 183–231 (1996)
Understanding the transmission dynamics of OXA-48 in K. pneumoniae and E.coli during an outbreak in a Dutch hospital.
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.
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
). 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.
Mathematics of structured populations: past, present and future
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.
Sebastiaan Janssens, joint
work with S.A. van Gils and Yu.A. Kuznetsov
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.
Mathematical Modelling in Biology
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.
Yuri Kuznetsov, joint work with Stephan van Gils, Sebastiaan Jansens, and Sid Visser
Local bifurcations in Neural Field Equations
Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star
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.
The interplay of infectivity that decreases with virulence and limited cross-immunity: (toy) models for respiratory disease evolution.
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.
Self-Organization in Bacterial Colonies
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 Bacillus subtilis and E. coli. and discuss how colonial patterns occur in a self-organized way, by using macroscopic PDE models,.
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 Daphnia–phytoplankton interactions suggest that the concept of ontogenetic asymmetry may help elucidate the population level consequences of contaminant impacts mediated by different physiological modes of action.
A converter from time-periodicity to spatial-periodicity
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.
Andrea Pugliese, joint work with Alberto Gandolfi and Carmela Sinisgalli
Epidemic models structured by within-host immune response
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 ,
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 .
Epidemic models and threshold quantities
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,
André de Roos
When size does matter: Ontogenetic symmetry and asymmetry in energetics
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.
Elements of bifurcation theory for bursting patterns in multifunctional Central Pattern Generator models.
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.
Angela Stevens, joint work with Jan Fuhrmann
Mathematical Modeling of Polymerization and Depolymerization of Actin Filaments within the Cellular Cytoskeleton
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.
Horst R. Thieme
Pieter Trapman, joint work with Frank Ball, Tom Britton, Jean-Stephane Dhersin and Viet Chi Tran
Ignoring population structure, does it matter?
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.
Sjoerd Verduyn Lunel
Wasserstein distances in the analysis of time series and dynamical systems
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.
Lumping networks of spiking and bursting neurons
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.
Complicated motion caused by state-dependent delay
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.