Abstracts:  NVTB - Meeting  2001

Pathogens that depend on direct contact between hosts to be transmitted will spread through the contact-network created from the social and spatial organization of their host population. This network environment may have effects on epidemics and pathogen evolution that differ radically from classical mean-field expectations. Simple, local epidemiological interactions are found to self-organize into a variety of spatio-temporal behaviors. Consequently, these different spatial organizations appear to have different evolutionary consequences.

Semi-arid vegetation can occur in remarkable patterns. Observed patterns in the field include typical regular-banded vegetation (``tiger bush''), irregular bands, arcs and spotted vegetation (``leopard bush''). We show with a simple but realistic mathematical model that all these patterns can originate without any underlying heterogeneity. On hillsides, where surface water flows downhill, regular-banded vegetation moves slowly up-slope. On flat ground, where surface water flow is not directed, irregular bands, arcs and spots are generated. Here, the net displacement of surface water to vegetated areas is due to the fact that water infiltration in vegetated ground is much faster than in bare soil. We show that spatial patterns can occur in parameter regions where also a stable homogeneous equilibrium exists. In addition, the patterns can occur when the homogeneous equilibrium predicts extinction of all vegetation. In a one-dimensional analysis we demonstrate how the various patterns are interlinked, and how they originate from the homogeneous equilibrium.

There are numerous examples of size-structured populations where individuals sequentially exploit several niches in the course of their life history. Efficient exploitation of such ontogenetic niches generally requires specific morphological adaptations. Since individuals have limited scope for changing their morphology as they grow, increased efficiency in one niche generally implies decreased efficiency in another. In this talk, potential evolutionary consequences of such life-history trade-offs are explored. In particular, life-history adaptation is studied in a simple size-structure population model in which individuals can exploit a primary niche while they are small, whereas a second niche becomes gradually accessible only beyond a certain body size. First, we study the ecological effect of the size at which the ontogenetic niche shift occurs. Second, we study evolution of the switch size incorporating the ecological feedback. We focus on the following questions:

• Does evolution lead to invasion of an unexploited niche?
• In a system with two niches, can life-history evolution lead to evolutionary branching and subsequent speciation?
• After branching, does evolution of the first-niche specialist drive the second-niche occupant to extinction?
• Does size-structure facilitate branching?

We consider a discrete time model of semelparous species dynamics, especially the case of two year classes (adults and juveniles). The main feature of this model is including competition between year classes via a variable which describes environmental conditions. One of the more important solution of the problem is a so-called Single Year Class Solution, in other words, competitive exclusion between the year classes, it means that in a year n the only year class is present, either adults or juveniles, while in the next year n+1 so is another year class. That happens due to the difference in sensitivity to and impact on environmental conditions for the different year classes, in particular, if competition between year classes is more severe than within a year class. The model allows also coexistence of both year classes, moreover there is exclusion between this solution and the Single Year Class solution.

Two ways to describe the dynamics of populations with age structure will be presented. The first approach is to set up a balance equation (PDE) for the time and age dependent population density with given per capita birth and death rates and then interpret the PDE as an abstract Cauchy problem which generates a semigroup of bounded linear operators on L₁(R₊). Alternatively one can start with two functions modeling survival and reproduction and then describe the dynamics as a semigroup of linear operators on a space of measures. The second method has the clear advantage, that it generalizes much easier to models incorporating more or different structure than age and, in particular, a variable individual growth rate.

Insects can be infected by bacteria, which cause death of all infected male progeny. These bacteria are transmitted vertically via the females of their host.

In previous models it has been shown that a perfectly transmitted malekiller is either eliminated from the system, or takes over and then destroys the host population. Malekilling can only be maintained when infected hosts have a fitness advantage above uninfected hosts and when the bacteria are not transmitted to all progeny. A population with a perfect malekiller can never exist. But experimental data shows that perfect malekilling does occur in the field.

We here use a spatial model (a cellular automaton) to study the biological system described above. A perfect malekiller can survive in this spatial setting, this is also true without assuming any fitness differences between infected and uninfected hosts. The viability of the system is due to the patterns that are formed. These different patterns cause different implicit advantages for infected or uninfected hosts. Perfect malekilling especially thrives when the hosts live relatively long..

I am interested in obtaining a better understanding of how disease persistence, the ability of a disease to remain endemic, depends on the way contacts between individuals are distributed in a host population consisting of subpopulations that are spatially separated (but not isolated from each other). For the case of a homogeneously mixing host population, disease persistence (as measured by the expected time to disease extinction from the quasi-stationary endemic state) has been studied recently by Van Herwaarden and Grasman (J. Math. Biol. 33 (1995) 581) and by Nasell (J. R. Statist. Soc. B 61 (1999) 309) using the stochastic SIR (susceptible-infected-removed) model. Grasman (Math. Biosci. 152 (1998) 13) has also considered an SIR model with spatial structure and migration between subpopulations. In this talk I will aim to get insight into 'meta-population disease persistence' by applying the diffusion approximation to the SIR model for a disease in a host population consisting of two subpopulations. I will also present an analytical approximation for the case of loosely linked subpopulations that relates the expected time to disease extinction in a two-patch system to that in a homogeneously mixing population.

Recently, vaccines against classical swine fever (CSF) have been developed for a better control of the transmission of classical swine fever virus (CSFV) in future epidemics. Experiments have revealed that these vaccines are well able to reduce transmission of CSFV between individual pigs. Next question is whether the vaccines will be successful in an outbreak situation. We present a mathematical model with which control measures can be compared in their ability to reduce transmission of CSFV between pig herds. Virus transmission within herds is modelled as deterministic exponential growth of the infectiousness of a herd after infection. Virus transmission between herds is modelled as a stochastic branching process. Two types of herds are distinguished (multiplier herds, where piglets are produced, and finishing herds, where these piglets are raised until they are slaughtered) to account for some heterogeneity in the between-herd contact process. Because measures may have a tr! ansient effect (e.g. single vaccination in finishing herds where the vaccinated pigs are in time replaced by unvaccinated animals), the basic reproduction ratio between herds Rh is not a useful parameter to judge control measures. Therefore we calculate the probability that the outbreak will spontaneously go extinct (minor outbreak). This presentation will cover the model and its assumptions and, with the example of a single vaccination at finishing herds, the method to calculate the probability of a minor outbreak, which is based on the method described by Diekmann and Heesterbeek (pp. 5-9) [1].

[1] - Diekmann & Heesterbeek, series 2000.
Mathematical Epidemiology of Infectious Diseases; Model Building, Analysis and Interpretation. Wiley, New York.

BSE (mad cow disease) is very difficult to diagnose, especially in early stages of the disease. Therefore active surveillance is necessary to determine the prevalence of this infection, which is feared to induce Creutzfeldt-Jakob disease in humans. A surveillance system could become more efficient by concentrating on groups with higher prevalence of the infection. Age structured modelling is used to analyse the age dependent prevalence of BSE. Essential input parameters are the distribution of the incubation period of the disease, the reproduction ratio of the infection in the past and demographic parameters of the cattle population. Using this information it can be calculated in which age groups BSE is most likely to be found, in the past at present and in the near future.

Generally a predator-prey system is modelled by two ordinary differential equations which describe the rate of changes of the biomasses. Consequently no chaotic behaviour occurs. In the popular Rosenzweig-MacArthur model, which replaced the Lotka-Volterra model, a stable equilibrium or a stable limit cycle exist. In this paper the prey consumes a non-viable nutrient whose dynamics is modelled explicitly and this gives an extra ordinary differential equation. For a predator==prey system under chemostat conditions where all parameter values are biologically meaningful, coexistence of multiple, also chaotic, attractors is possible in a narrow region of the two-parameter bifurcation diagram, where the chemostat controp parameters are the bifurcation parameters. Crisis-limited chaotic behaviour and a bifurcation point where two coexisting chaotic attractors merge will be discussed. The interior and boundary crises of this continuous-time predator-prey system look similar to those found for the discrete-time Hénon map. The link is via a Poincaré map for a suitable chosen Poincaré plane where the predator attains an extremum. Global homoclinic bifurcations occur with boundary and interior crises. To show this we will analyse the two-dimensional Hénon and an one-dimensional cubic map, both extensions of the classical quadratic map.

Classical food chain theory predicts that omnivory destabilizes food webs. Omnivory should thus be rare in theory, but appears to be widespread in reality. To understand this contradiction, a number of ecologists have reinvestigated the role of omnivory in ecology. They applied more mechanistically based models and considered both equilibrium and non-equilibrium dynamics. As a result, evidence of a stabilizing role of omnivory is accumulating. Contrary to omnivory, nutrient enrichment is still thought to have a destabilizing effect on ecosystems. If omnivory truly stabilizes food webs, the effects of nutrient enrichment may well be counteracted by omnivorous interactions. To date, no rigorous modelling study of the combined effects of enrichment and omnivory is available. We analyse a mechanistical model with variable degrees of omnivory and assess the effects of nutrient enrichment on this model. We find that the role of an omnivore in a simple food web is ambiguous; it depends strongly on both the specific feeding behavior of the omnivore and the nutrient supply in the environment.

In recent years, ``graph theory'' has established itself as an important mathematical tool in a wide variety of subjects, ranging from geography and operational research to genetics and electrical engineering. We investigate how ``graph theory'' can be used to characterize the structure of large fungal networks in soil. Therefore, a plane graph is constructed of which several properties are derived, like the adjacency matrix, incidence matrix, spanning trees, cycle rank, etc. We explore the importance of individual fungal strands in a network with respect to the distribution of nutrients from sources to sinks.

Organisms differ widely in the range of habitats they use and habitat specialists and generalists often coexist. However, models about habitat use generally predict that specialists will exclude generalists, and that coexistence of generalists and specialists requires stringent conditions. Why, then, are there generalists? I investigate coexistence of specialist and generalist species that have a metapopulation structure in a landscape consisting of patches of different habitat types. Generalists can use more types of habitat, whereas specialists can use only one type of habitat but have a local competitive advantage against generalists. It appeared that specialists and generalists can coexist when there are intermediate amounts of habitat. The generalists then use patches left empty by the specialists. No flexible habitat choice or temporally varying habitat qualities are needed. When there is too little habitat, only generalists exist and when there is too much habitat, prop! agule pressure of the specialists excludes the generalists. Generalists need more than one type of habitat to survive. Coexistence is enhanced by increased niche-width or colonization capacity of the generalists. When generalists have non-zero local competitive ability, their density is increased at the cost of the specialists. The proportion of generalists in the landscape is also increased by habitat destruction, increased disturbance or increased dispersal resistance, and by increased niche-width of the generalists. When they are able to use a broad range of habitats, generalists generate so much propagule pressure that only little local competitive power is needed to globally exclude the specialists. Hence, the question arises why there are specialists.

Short Introduction

The proteasome is a multisubunit cytoplamic protease that is involved both in the ubiquitin(Ub)-independent and Ub-dependent pathways of protein degradation. The protein degradation by the proteasome has a large influence on the host immune response. Cytotoxic T-cells recognize fragments of proteins, 8-10 amino acids long, presented by MHCI on the surface of an antigen presenting cell.

The eucaryotic proteasome complex consists of 28 protein subunits. These are organized in four rings. The outer two rings consist of alpha subunits, the inner two of beta's. The alphas seem to bind to caps that unfold the protein, whereas three of the betas do the cleaving. In eucaryots with an adaptive immune system the three active subunits are replaced by immuno-subunits induced by interferon-gamma. Due to it's central role in protein degradation, the proteasome structure seems to be strongly conserved.

Abstract

I have investigated the evolutionary tree of these subunits of several organisms by making an alignment of the protein sequences, and form that computing an evolutionary tree. I have also measured the dn/ds ratio of the different subunits. It appears that the alpha subunits are much more conserved than the beta's and that the Immuno-subunits are the least conserved. From this we can can conclude that the active immuno-subunits, have co-evolved with viruses.

To investigate this co-evolution, I examined neural network predicted peptides of 8-10 amino acids long from three common viruses and human proteins. The predictions were made for the normal and the immuno-proteasome. with this I tested three hypotheses

1. Peptides from immunoproteasome bind better to MHCI.
2. Immuno-proteasome makes peptides that fit better to MHCI.
3. Immuno-proteasome discriminates better between viral and self proteins, t.i. Immuno viral peptides look less like Immuno self peptides, than normal viral peptides look like normal self peptides.

A simple Levins-type patch model with predator and prey species leads to a very rich spectrum of possible equilibria as a function of the fraction of available habitat and the extinction and colonization parameters. In certain cases, the presence of predators leads to prey persistence at very low values of available habitat, where they would be extinct in the absence of predators. In addition, the spatially explicit version of the model has interesting properties at low values of available habitat: its reduced connectivity compared to the mean-field model can actually aid the persistence of the metapopulation in small isolated clusters for very long times.

Species with different body sizes exploit the environment at different scales. A large species (with a large window size) is restricted to those large patches in which the average resource density exceeds its requirements. In these patches the large species outcompetes all smaller ones due to its size advantage. The habitat leftover, however, may contain small patches with resource densities sufficiently high for a smaller species. This repeats itself at smaller scales and the result is an exclusive niche for each of a series of species. The relative abundance of small and large species reflects the interplay between geometric and biological factors. The geometric factors are (1) the fractal dimension of the resource distribution and (2) the variability of the resource density at a fixed window size w. The crucial biological property is (3) the relation between the minimum required resource density R and the species window size w (reflecting the allometric relationship between body size and metabolic rate). These three factors determine the shape and width of the species abundance curve.

Since introduction of vaccination in 1976, the Netherlands has faced recurrent measles outbreaks of varying size. There is no evidence of persistent transmission in between outbreaks, or persistence within socio-geographical clusters of people refusing vaccination on religious grounds. Recently, a very large outbreak of measles was observed - the largest since introduction of vaccination. Was this large outbreak in line with what we should have expected, given the fraction of children that are vaccinated? We answer this question by developing approximate relations between number of susceptibles at start of an epidemic and the expected probability of outbreak, and expected outbreak size, and we infer the expected interepidemic period for recurrent outbreaks without persistence. Results are compared with simulations of a stochastic SEIR model and with data.