Stochastic demography and the neutral substitution rate in class-structured populations.

The neutral rate of allelic substitution is analyzed for a class-structured population subject to a stationary stochastic demographic process. The substitution rate is shown to be generally equal to the effective mutation rate, and under overlapping generations it can be expressed as the effective mutation rate in newborns when measured in units of average generation time. With uniform mutation rate across classes the substitution rate reduces to the mutation rate.

Stochastic time-concentration activity models for cytotoxicity in 3D brain cell cultures.

BACKGROUND: In vitro aggregating brain cell cultures containing all types of brain cells have been shown to be useful for neurotoxicological investigations. The cultures are used for the detection of nervous system-specific effects of compounds by measuring multiple endpoints, including changes in enzyme activities. Concentration-dependent neurotoxicity is determined at several time points. METHODS: A Markov model was set up to describe the dynamics of brain cell populations exposed to potentially neurotoxic compounds. Brain cells were assumed to be either in a healthy or stressed state, with only stressed cells being susceptible to cell death. Cells may have switched between these states or died with concentration-dependent transition rates. Since cell numbers were not directly measurable, intracellular lactate dehydrogenase (LDH) activity was used as a surrogate. Assuming that changes in cell numbers are proportional to changes in intracellular LDH activity, stochastic enzyme activity models were derived. Maximum likelihood and least squares regression techniques were applied for estimation of the transition rates. Likelihood ratio tests were performed to test hypotheses about the transition rates. Simulation studies were used to investigate the performance of the transition rate estimators and to analyze the error rates of the likelihood ratio tests. The stochastic time-concentration activity model was applied to intracellular LDH activity measurements after 7 and 14 days of continuous exposure to propofol. The model describes transitions from healthy to stressed cells and from stressed cells to death. RESULTS: The model predicted that propofol would affect stressed cells more than healthy cells. Increasing propofol concentration from 10 to 100 μM reduced the mean waiting time for transition to the stressed state by 50%, from 14 to 7 days, whereas the mean duration to cellular death reduced more dramatically from 2.7 days to 6.5 hours. CONCLUSION: The proposed stochastic modeling approach can be used to discriminate between different biological hypotheses regarding the effect of a compound on the transition rates. The effects of different compounds on the transition rate estimates can be quantitatively compared. Data can be extrapolated at late measurement time points to investigate whether costs and time-consuming long-term experiments could possibly be eliminated.

Conditioning of stochastic 3-D fracture networks to hydrological and geophysical data

The geometry and connectivity of fractures exert a strong influence on the flow and transport properties of fracture networks. We present a novel approach to stochastically generate three-dimensional discrete networks of connected fractures that are conditioned to hydrological and geophysical data. A hierarchical rejection sampling algorithm is used to draw realizations from the posterior probability density function at different conditioning levels. The method is applied to a well-studied granitic formation using data acquired within two boreholes located 6 m apart. The prior models include 27 fractures with their geometry (position and orientation) bounded by information derived from single-hole ground-penetrating radar (GPR) data acquired during saline tracer tests and optical televiewer logs. Eleven cross-hole hydraulic connections between fractures in neighboring boreholes and the order in which the tracer arrives at different fractures are used for conditioning. Furthermore, the networks are conditioned to the observed relative hydraulic importance of the different hydraulic connections by numerically simulating the flow response. Among the conditioning data considered, constraints on the relative flow contributions were the most effective in determining the variability among the network realizations. Nevertheless, we find that the posterior model space is strongly determined by the imposed prior bounds. Strong prior bounds were derived from GPR measurements and helped to make the approach computationally feasible. We analyze a set of 230 posterior realizations that reproduce all data given their uncertainties assuming the same uniform transmissivity in all fractures. The posterior models provide valuable statistics on length scales and density of connected fractures, as well as their connectivity. In an additional analysis, effective transmissivity estimates of the posterior realizations indicate a strong influence of the DFN structure, in that it induces large variations of equivalent transmissivities between realizations. The transmissivity estimates agree well with previous estimates at the site based on pumping, flowmeter and temperature data.

Stochastic stability and the evolution of coordination in spatially structured populations.

Animals can often coordinate their actions to achieve mutually beneficial outcomes. However, this can result in a social dilemma when uncertainty about the behavior of partners creates multiple fitness peaks. Strategies that minimize risk ("risk dominant") instead of maximizing reward ("payoff dominant") are favored in economic models when individuals learn behaviors that increase their payoffs. Specifically, such strategies are shown to be "stochastically stable" (a refinement of evolutionary stability). Here, we extend the notion of stochastic stability to biological models of continuous phenotypes at a mutation-selection-drift balance. This allows us to make a unique prediction for long-term evolution in games with multiple equilibria. We show how genetic relatedness due to limited dispersal and scaled to account for local competition can crucially affect the stochastically-stable outcome of coordination games. We find that positive relatedness (weak local competition) increases the chance the payoff dominant strategy is stochastically stable, even when it is not risk dominant. Conversely, negative relatedness (strong local competition) increases the chance that strategies evolve that are neither payoff nor risk dominant. Extending our results to large multiplayer coordination games we find that negative relatedness can create competition so extreme that the game effectively changes to a hawk-dove game and a stochastically stable polymorphism between the alternative strategies evolves. These results demonstrate the usefulness of stochastic stability in characterizing long-term evolution of continuous phenotypes: the outcomes of multiplayer games can be reduced to the generic equilibria of two-player games and the effect of spatial structure can be analyzed readily.

Rates of cultural change and patterns of cultural accumulation in stochastic models of social transmission.

Cultural variation in a population is affected by the rate of occurrence of cultural innovations, whether such innovations are preferred or eschewed, how they are transmitted between individuals in the population, and the size of the population. An innovation, such as a modification in an attribute of a handaxe, may be lost or may become a property of all handaxes, which we call "fixation of the innovation." Alternatively, several innovations may attain appreciable frequencies, in which case properties of the frequency distribution-for example, of handaxe measurements-is important. Here we apply the Moran model from the stochastic theory of population genetics to study the evolution of cultural innovations. We obtain the probability that an initially rare innovation becomes fixed, and the expected time this takes. When variation in cultural traits is due to recurrent innovation, copy error, and sampling from generation to generation, we describe properties of this variation, such as the level of heterogeneity expected in the population. For all of these, we determine the effect of the mode of social transmission: conformist, where there is a tendency for each naïve newborn to copy the most popular variant; pro-novelty bias, where the newborn prefers a specific variant if it exists among those it samples; one-to-many transmission, where the variant one individual carries is copied by all newborns while that individual remains alive. We compare our findings with those predicted by prevailing theories for rates of cultural change and the distribution of cultural variation.

Stochastic Population Projection for Germany - based on the Quadratic Spline approach to modelling age specific fertility rates

This contribution builds upon a former paper by the authors (Lipps and Betz 2004), in which a stochastic population projection for East- and West Germany is performed. Aim was to forecast relevant population parameters and their distribution in a consistent way. We now present some modifications, which have been modelled since. First, population parameters for the entire German population are modelled. In order to overcome the modelling problem of the structural break in the East during reunification, we show that the adaptation process of the relevant figures by the East can be considered to be completed by now. As a consequence, German parameters can be modelled just by using the West German historic patterns, with the start-off population of entire Germany. Second, a new model to simulate age specific fertility rates is presented, based on a quadratic spline approach. This offers a higher flexibility to model various age specific fertility curves. The simulation results are compared with the scenario based official forecasts for Germany in 2050. Exemplary for some population parameters (e.g. dependency ratio), it can be shown that the range spanned by the medium and extreme variants correspond to the s-intervals in the stochastic framework. It seems therefore more appropriate to treat this range as a s-interval covering about two thirds of the true distribution.

On the problematic of reserving and stochastic loss models

Abstract Traditionally, the common reserving methods used by the non-life actuaries are based on the assumption that future claims are going to behave in the same way as they did in the past. There are two main sources of variability in the processus of development of the claims: the variability of the speed with which the claims are settled and the variability between the severity of the claims from different accident years. High changes in these processes will generate distortions in the estimation of the claims reserves. The main objective of this thesis is to provide an indicator which firstly identifies and quantifies these two influences and secondly to determine which model is adequate for a specific situation. Two stochastic models were analysed and the predictive distributions of the future claims were obtained. The main advantage of the stochastic models is that they provide measures of variability of the reserves estimates. The first model (PDM) combines one conjugate family Dirichlet - Multinomial with the Poisson distribution. The second model (NBDM) improves the first one by combining two conjugate families Poisson -Gamma (for distribution of the ultimate amounts) and Dirichlet Multinomial (for distribution of the incremental claims payments). It was found that the second model allows to find the speed variability in the reporting process and development of the claims severity as function of two above mentioned distributions' parameters. These are the shape parameter of the Gamma distribution and the Dirichlet parameter. Depending on the relation between them we can decide on the adequacy of the claims reserve estimation method. The parameters have been estimated by the Methods of Moments and Maximum Likelihood. The results were tested using chosen simulation data and then using real data originating from the three lines of business: Property/Casualty, General Liability, and Accident Insurance. These data include different developments and specificities. The outcome of the thesis shows that when the Dirichlet parameter is greater than the shape parameter of the Gamma, resulting in a model with positive correlation between the past and future claims payments, suggests the Chain-Ladder method as appropriate for the claims reserve estimation. In terms of claims reserves, if the cumulated payments are high the positive correlation will imply high expectations for the future payments resulting in high claims reserves estimates. The negative correlation appears when the Dirichlet parameter is lower than the shape parameter of the Gamma, meaning low expected future payments for the same high observed cumulated payments. This corresponds to the situation when claims are reported rapidly and fewer claims remain expected subsequently. The extreme case appears in the situation when all claims are reported at the same time leading to expectations for the future payments of zero or equal to the aggregated amount of the ultimate paid claims. For this latter case, the Chain-Ladder is not recommended.

Essays on the treatment of cash flows under stochastic interest rates

Abstract This paper shows how to calculate recursively the moments of the accumulated and discounted value of cash flows when the instantaneous rates of return follow a conditional ARMA process with normally distributed innovations. We investigate various moment based approaches to approximate the distribution of the accumulated value of cash flows and we assess their performance through stochastic Monte-Carlo simulations. We discuss the potential use in insurance and especially in the context of Asset-Liability Management of pension funds.

Stochastic models and numerical algorithms for a class of regulatory gene networks.

Regulatory gene networks contain generic modules, like those involving feedback loops, which are essential for the regulation of many biological functions (Guido et al. in Nature 439:856-860, 2006). We consider a class of self-regulated genes which are the building blocks of many regulatory gene networks, and study the steady-state distribution of the associated Gillespie algorithm by providing efficient numerical algorithms. We also study a regulatory gene network of interest in gene therapy, using mean-field models with time delays. Convergence of the related time-nonhomogeneous Markov chain is established for a class of linear catalytic networks with feedback loops.

An allele-specific, stochastic gene expression process controls the expression of multiple Ly49 family genes and generates a diverse, MHC-specific NK cell receptor repertoire.

Mouse NK cells express MHC class I-specific inhibitory Ly49 receptors. Since these receptors display distinct ligand specificities and are clonally distributed, their expression generates a diverse NK cell receptor repertoire specific for MHC class I molecules. We have previously found that the Dd (or Dk)-specific Ly49A receptor is usually expressed from a single allele. However, a small fraction of short-term NK cell clones expressed both Ly49A alleles, suggesting that the two Ly49A alleles are independently and randomly expressed. Here we show that the genes for two additional Ly49 receptors (Ly49C and Ly49G2) are also expressed in a (predominantly) mono-allelic fashion. Since single NK cells can co-express multiple Ly49 receptors, we also investigated whether mono-allelic expression from within the tightly linked Ly49 gene cluster is coordinate or independent. Our clonal analysis suggests that the expression of alleles of distinct Ly49 genes is not coordinate. Thus Ly49 alleles are apparently independently and randomly chosen for stable expression, a process that directly restricts the number of Ly49 receptors expressed per single NK cell. We propose that the Ly49 receptor repertoire specific for MHC class I is generated by an allele-specific, stochastic gene expression process that acts on the entire Ly49 gene cluster.