Application of Tikhonov’s regularization method to solve inverse problems for two population models

Tikhonov’s regularization method is applied to numerical solution of inverse problems for two population models. For the first model we solve the inverse problem that involves simultaneous determination of the mortality rate and the initial distribution of individuals given supplementary information on population density. For the second model we determine the growth rate of the individuals given additional information about their density. Examples of numerical solution are presented for both inverse problems. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 5–14, 2006.     

Mathematical models of bipolar disorder

We use limit cycle oscillators to model bipolar II disorder, which is characterized by alternating hypomanic and depressive episodes and afflicts about 1% of the United States adult population. We consider two non-linear oscillator models of a single bipolar patient. In both frameworks, we begin with an untreated individual and examine the mathematical effects and resulting biological consequences of treatment. We also briefly consider the dynamics of interacting bipolar II individuals using weakly-coupled, weakly-damped harmonic oscillators. We discuss how the proposed models can be used as a framework for refined models that incorporate additional biological data. We conclude with a discussion of possible generalizations of our work, as there are several biologically-motivated extensions that can be readily incorporated into the series of models presented here.     

CHAOS IN AN INDIVIDUAL-LEVEL PREDATOR-PREY MODEL

A deterministic predator-prey model describing populations as collections of autonomous individuals was used to investigate population dynamics as an emergent property of individual behaviors and actions in a simulated environment. The model's behavior was clearly chaotic for both the two species interaction and for the prey population alone. Furthermore, estimated parameters from a simple difference equation model fitted to the single-species simulation were nowhere near the chaotic region for the equation. Findings indicate that chaos may very well be characteristic of biological populations despite the findings of several empirical studies in the past, and that chaotic models can be constructed from direct observation of, and experimentation with, biological populations by focusing on individuals and their behaviors, rather than on population parameters.     

Tree-valued resampling dynamics Martingale problems and applications

The measure-valued Fleming–Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the “individuals” in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with the Gromov-weak topology serves as state space for tree-valued processes. We use well-posed martingale problems to construct the tree-valued resampling dynamics of the evolving genealogies for both the finite population Moran model and the infinite population Fleming–Viot diffusion. We show that sufficient information about any ultra-metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled “individuals”. We give explicit formulas for the evolution of the Laplace transform of the distribution of finite subtrees under the tree-valued Fleming–Viot dynamics.     

Making and breaking power laws in evolutionary algorithm population dynamics

Deepening our understanding of the characteristics and behaviors of population-based search algorithms remains an important ongoing challenge in Evolutionary Computation. To date however, most studies of Evolutionary Algorithms have only been able to take place within tightly restricted experimental conditions. For instance, many analytical methods can only be applied to canonical algorithmic forms or can only evaluate evolution over simple test functions. Analysis of EA behavior under more complex conditions is needed to broaden our understanding of this population-based search process. This paper presents an approach to analyzing EA behavior that can be applied to a diverse range of algorithm designs and environmental conditions. The approach is based on evaluating an individual’s impact on population dynamics using metrics derived from genealogical graphs. From experiments conducted over a broad range of conditions, some important conclusions are drawn in this study. First, it is determined that very few individuals in an EA population have a significant influence on future population dynamics with the impact size fitting a power law distribution. The power law distribution indicates there is a non-negligible probability that single individuals will dominate the entire population, irrespective of population size. Two EA design features are however found to cause strong changes to this aspect of EA behavior: (1) the population topology and (2) the introduction of completely new individuals. If the EA population topology has a long path length or if new (i.e. historically uncoupled) individuals are continually inserted into the population, then power law deviations are observed for large impact sizes. It is concluded that such EA designs can not be dominated by a small number of individuals and hence should theoretically be capable of exhibiting higher degrees of parallel search behavior.     

Size-structured populations: immigration, (bi)stability and the net growth rate

We consider a class of physiologically structured population models, a first order nonlinear partial differential equation equipped with a nonlocal boundary condition, with a constant external inflow of individuals. We prove that the linearised system is governed by a quasicontraction semigroup. We also establish that linear stability of equilibrium solutions is governed by a generalised net reproduction function. In a special case of the model ingredients we discuss the nonlinear dynamics of the system when the spectral bound of the linearised operator equals zero, i.e. when linearisation does not decide stability. This allows us to demonstrate, through a concrete example, how immigration might be beneficial to the population. In particular, we show that from a nonlinearly unstable positive equilibrium a linearly stable and unstable pair of equilibria bifurcates. In fact, the linearised system exhibits bistability, for a certain range of values of the external inflow, induced potentially by Allée-effect.     

Dynamics of Lotka-Volterra Competition Systems with Large Interaction

We show that under certain additional hypothesis, two population competing species models in bounded domains with diffusion and large interaction have simple dynamics. In particular, the solutions approach stationary states as t tends to infinity.     

A mathematical model of epidemics with screening and variable infectivity

Two models for the dynamics of an epidemic of S-I-R type are described in which the active population is randomly screened. Infectivity is not required to be constant in one of them. The positive screened individuals move into the class of “removed” together with the immune. Global existence and uniqueness results are established.     

Efficient Data Augmentation for Fitting Stochastic Epidemic Models to Prevalence Data

Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic epidemic model parameters, in which measurements are augmented with subject-level disease histories. In our MCMC algorithm, we propose each new subject-level path, conditional on the data, using a time-inhomogenous continuous-time Markov process with rates determined by the infection histories of other individuals. The method is general, and may be applied to a broad class of epidemic models with only minimal modifications to the model dynamics and/or emission distribution. We present our algorithm in the context of multiple stochastic epidemic models in which the data are binomially sampled prevalence counts, and apply our method to data from an outbreak of influenza in a British boarding school. Supplementary material for this article is available online.     

The role of constraints in a segregation model: The symmetric case

In this paper we study the effects of constraints on the dynamics of an adaptive segregation model introduced by Bischi and Merlone (2011) [3]. The model is described by a two dimensional piecewise smooth dynamical system in discrete time. It models the dynamics of entry and exit of two populations into a system, whose members have a limited tolerance about the presence of individuals of the other group. The constraints are given by the upper limits for the number of individuals of a population that are allowed to enter the system. They represent possible exogenous controls imposed by an authority in order to regulate the system. Using analytical, geometric and numerical methods, we investigate the border collision bifurcations generated by these constraints assuming that the two groups have similar characteristics and have the same level of tolerance toward the members of the other group. We also discuss the policy implications of the constraints to avoid segregation.     

Modeling influenza progression within a continuous-attribute heterogeneous population

We consider three attributes of an individual that are critical in determining the temporal dynamics of pandemic influenza: social activity, proneness to infection, and proneness to shed virus and spread infection. These attributes differ by individual, resulting in a heterogeneous population. We develop discrete-time models that depict the evolution of the disease in the presence of such population heterogeneity. For every individual, the value for each of the three describing attributes is viewed as an experimental value of a continuous random variable. The methodology is simple yet general, extending more traditional discrete compartmental models that depict population heterogeneity. Illustrative numerical examples show how individuals who have much larger-than-average values for one or more of the attributes drive the influenza wave, especially in the early generations of the pandemic. This heterogeneity-driven pandemic physics carries important policy implications. We conclude by using contact data in four European countries to demonstrate empirical uses of our model.     

USE OF AGE‐ AND STAGE‐STRUCTURED MATRIX MODELS TO PREDICT LIFE HISTORY SCHEDULES FOR SEMELPAROUS POPULATIONS

Many studies of semelparous salmon populations use Leslie matrices that classify individuals on the basis of age alone and do not explicitly impose death upon reproduction. Although these models may suffice for studying long‐term population dynamics (like asymptotic growth rate), they do not accurately represent the diversity of individual life history outcomes in semelparous populations. Cohorts breeding at different ages have different life history traits (e.g., age at first reproduction and remaining life expectancy) that are obscured in Leslie models and this distorts our understanding of life history diversity and its importance for semelparous population dynamics. We present a simple transformation that uses age‐specific breeding probabilities to reconfigure Leslie matrices as explicitly semelparous models. Explicitly semelparous models conserve asymptotic measures like population growth rate, vital rate elasticities, life expectancy at birth, and generation time but also better predict life history schedules and reproductive values. Strictly age‐classified Leslie models underestimate ages at first reproduction and mean ages at death for older breeders but overestimate mean ages at death for early breeders. Leslie models also slightly overestimate variance in lifetime reproductive success, and underestimate entropy exhibited by life history outcomes.     

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

In this article, we propose and study a generalized Ricker–Beverton–Holt competition model subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using the theory of monotone dynamics and the properties of critical curves for non-invertible maps, our analysis show that our model has relatively simple dynamics, i.e. almost every trajectory converges to a locally asymptotically stable equilibrium if the intensity of intra-specific competition intensity exceeds that of inter-specific competition. This equilibrium dynamics is also possible when the intensity of intra-specific competition intensity is less than that of inter-specific competition but under conditions that the maximum intrinsic growth rate of one species is not too large. The coexistence of two competing species occurs only if the system has four interior equilibria. We provide an approximation to the basins of the boundary attractors (i.e. the extinction of one or both species) where our results suggests that contest species are more prone to extinction than scramble ones are at low densities. In addition, in comparison to the dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the persistent attractor, whereas scramble competition models may have the extinction of both species as its only attractor under certain conditions, i.e. the essential extinction of two species due to strong Allee effects; (ii) Scramble competition models like Ricker type models can have much more complicated dynamical structure of interior attractors than contest ones like Beverton–Holt type models have; and (iii) Scramble competition models like Ricker type competition models may be more likely to promote the coexistence of two species at low and high densities under certain conditions: At low densities, weak Allee effects decrease the fitness of resident species so that the other species is able to invade at its low densities; While at high densities, scramble competition can bring the current high population density to a lower population density but is above the Allee threshold in the next season, which may rescue a species that has essential extinction caused by strong Allee effects. Our results may have potential to be useful for conservation biology: For example, if one endangered species is facing essential extinction due to strong Allee effects, then we may rescue this species by bringing another competing species subject to scramble competition and Allee effects under certain conditions.     

From Individuals to Populations: A Symbolic Process Algebra Approach to Epidemiology

Is it possible to symbolically express and analyse an individual-based model of disease spread, including realistic population dynamics? This problem is addressed through the use of process algebra and a novel method for transforming process algebra into Mean Field Equations. A number of stochastic models of population growth are presented, exploring different representations based on alternative views of individual behaviour. The overall population dynamics in terms of mean field equations are derived using a formal and rigorous rewriting based method. These equations are easily compared with the traditionally used deterministic Ordinary Differential Equation models and allow evaluation of those ODE models, challenging their assumptions about system dynamics. The utility of our approach for epidemiology is confirmed by constructing a model combining population growth with disease spread and fitting it to data on HIV in the UK population. This work was supported by EPSRC through a Doctoral Training Grant (CM, from 2004–2007), and through System Dynamics from Individual Interactions: A process algebra approach to epidemiology (EP/E006280/1, all authors, 2007–2010).     

离散的SI和SIS传染病模型的研究

为了描述个体的死亡、染病者的恢复以及疾病的传染,引入了相应的概率.基于总种群中个体数量为常数的假设,根据染病者能否恢复分别建立了具有生命动力学的离散SI和SIS传染病模型.所得到的结果显示:它们具有与相应连续模型相同的动力学性态,并确定了各自的阈值.在它们的阈值之下,传染病最终将灭绝;在它们的阈值之上,传染病将会发展成为地方病,染病者的数量将趋向于一确定的正常数.     

Mathematical study of a risk-structured two-group model for Chlamydia transmission dynamics

A new two-group deterministic model for Chlamydia trachomatis, which stratifies the entire population based on risk of acquiring or transmitting infection, is designed and analyzed to gain insight into its transmission dynamics. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium (DFE) co-exists with one or more stable endemic equilibria when the associated reproduction number is less than unity. Unlike in some of the earlier modeling studies on Chlamydia transmission dynamics in a population, this study shows that the backward bifurcation phenomenon persists even if individuals who recovered from Chlamydia infection do not get re-infected. However, it is shown that the phenomenon can be removed if all the susceptible individuals are equally likely to acquire infection (i.e., for the case where the susceptible male and female populations are not stratified according to risk of acquiring infection). In such a case, the DFE of the resulting (reduced) model is globally-asymptotically stable when the associated reproduction number is less than unity and no re-infection of recovered individuals occurs. Thus, this study shows that stratifying the two-sex Chlamydia transmission model, presented in [1], according to the risk of acquiring or transmitting infection induces the phenomenon of backward bifurcation regardless of whether or not the re-infection of recovered individuals occurs.     

Pattern formation and control of spatiotemporal chaos in a reaction diffusion prey–predator system supplying additional food

Spatiotemporal dynamics of a predator–prey system in presence of spatial diffusion is investigated in presence of additional food exists for predators. Conditions for stability of Hopf as well as Turing patterns in a spatial domain are determined by making use of the linear stability analysis. Impact of additional food is clear from these conditions. Numerical simulation results are presented in order to validate the analytical findings. Finally numerical simulations are carried out around the steady state under zero flux boundary conditions. With the help of numerical simulations, the different types of spatial patterns (including stationary spatial pattern, oscillatory pattern, and spatiotemporal chaos) are identified in this diffusive predator–prey system in presence of additional food, depending on the quantity, quality of the additional food and the spatial domain and other parameters of the model. The key observation is that spatiotemporal chaos can be controlled supplying suitable additional food to predator. These investigations may be useful to understand complex spatiotemporal dynamics of population dynamical models in presence of additional food.     

Colony and evolutionary dynamics of a two‐stage model with brood cannibalism and division of labor in social insects

Division of labor (DOL) is a major factor for the great success of social insects because it increases the efficiency of a social group where different individuals perform different tasks repeatedly and presumably with increased performance. Cannibalism plays an important role in regulating colony growth and development by regulating the number of individuals in a colony and increasing survival by providing access to essential nutrients and minimizing competition among colony mates. To understand the synergy effects of DOL and cannibalistic behavior on colony dynamic outcomes, we propose and study a compartmental two‐stage model using ecological and evolutionary game theory settings. Our analytical results of the ecological and evolutionary models suggest that: (1) A noncannibalistic colony can survive if the efficiency of energy investment reflecting the DOL is greater than the relative death rate of the older population. (2) A cannibalistic colony can die out if both the efficiency of energy investment and the relative cannibalism rate (where each is also reflecting the DOL) are too large; or if the relative cannibalism rate alone is too small. (3) From our numerical analysis, cannibalism can increase or reduce the colony's total population size, which greatly depends on the benefit of egg cannibalism increasing or decreasing of adult's lifespan. (4) A cannibalistic and noncannibalistic colony can experience bistability due to cooperative behavior. (5) In the evolutionary settings, DOL can prevent colony death and natural selection can preserve strong Allee effects by selecting the traits with the largest investment on brood care and the lowest cannibalism rate. (6) Evolutionary dynamics may increase the fitness of the colony, i.e., the successful production of workforce which results in the increase of total worker population size, colony survival, and reproduction. Our results suggest both cannibalism and DOLs are adaptive strategies that increase the size of the worker population, and therefore, persistence of the colony.     

On analytical and numerical approaches to division and label structured population models

Even among cells in the same population, the concentration of a protein or cellular constituent can vary considerably. This heterogeneity can arise from several sources, including differences in kinetic rates between cells and distribution of cellular constituents through cell division. Compartmental models have been used to describe the distribution of the number of divisions undergone by cells in a population. More recently, such models have been coupled with the dynamics of intracellular labels and analytical solutions to the division and label structured population equations have been found. However, such approaches have thus far focused on simple models of intracellular dynamics such as the decay of an intracellular label. In this work, we demonstrate that analytical solutions are possible for more general forms of intracellular dynamics offering the promise to lend mathematical insight into population dynamics in more realistic biological settings.     

On the application of Wishart process to the pricing of equity derivatives: the multi-asset case

Given the inherent complexity of financial markets, a wide area of research in the field of mathematical finance is devoted to develop accurate models for the pricing of contingent claims. Focusing on the stochastic volatility approach (i.e. we assume to describe asset volatility as an additional stochastic process), it appears desirable to introduce reliable dynamics in order to take into account the presence of several assets involved in the definition of multi-asset payoffs. In this article we deal with the multi asset Wishart Affine Stochastic Correlation model, that makes use of Wishart process to describe the stochastic variance covariance matrix of assets return. The resulting parametrization turns out to be a genuine multi-asset extension of the Heston model: each asset is exactly described by a single instance of the Heston dynamics while the joint behaviour is enriched by cross-assets and cross-variances stochastic correlation, all wrapped in an affine modeling. In this framework, we propose a fast and accurate calibration procedure, and two Monte Carlo simulation schemes.