Growth rate and basic reproduction number for population models with a simple periodic factor

For continuous-time population models with a periodic factor which is sinusoidal, both the growth rate and the basic reproduction number are shown to be the largest roots of simple equations involving continued fractions. As an example, we reconsider an SEIS model with a fixed latent period, an exponentially distributed infectious period and a sinusoidal contact rate studied in Williams and Dye [B.G. Williams, C. Dye, Infectious disease persistence when transmission varies seasonally, Math. Biosci. 145 (1997) 77]. We show that apart from a few exceptional parameter values, the epidemic threshold depends not only on the mean contact rate, but also on the amplitude of fluctuations.     

On vaccine efficacy and reproduction numbers

We consider the impact of a vaccination programme on the transmission potential of the infection in large populations. We define a measure of vaccine efficacy against transmission which combines the possibly random effect of the vaccine on individual susceptibility and infectiousness. This definition extends some previous work in this area to arbitrarily heterogeneous populations with one level of mixing, but leads us to question the usefulness of the concept of vaccine efficacy against infectiousness. We derive relationships between vaccine efficacy against transmission, vaccine coverage and reproduction numbers, which generalize existing results. In particular we show that the projected reproduction number RV does not depend on the details of the vaccine model, only on its overall effect on transmission. Explicit expressions for RV and the basic reproduction number R0 are obtained in a variety of settings. We define a measure of projected effectiveness of a vaccination programme PE=1-(RV/R0) and investigate its relationship with efficacy against transmission and vaccine coverage. We also study the effective reproduction number Re(t) at time t. Monitoring Re(t) over time is an important aspect of programme surveillance. Programme effectiveness PE is less sensitive than RV or the critical vaccination threshold to model assumptions. On the other hand Re(t) depends on the details of the vaccine model.     

Persistence and extinction in single-species reaction-diffusion models

Single-species reaction-diffusion models are analyzed to determine the effect of various diffusion mechanisms on species persistence or extinction.     

Directionality of stripes formed by anisotropic reaction-diffusion models

Turing mechanism explains the formation of striped patterns in a uniform field in which two substances interact locally and diffuse randomly. In a twin paper, to explain the directionality of stripes on fish skin in closely related species, we studied the effect of anisotropic diffusion of the two substances on the direction of stripes, in the cases in which both substances have high diffusivity in the same direction. In this paper, we study the direction of stripes in more general situations in which the diffusive direction may differ between the two substances. We derive a formula for the direction of stripes, based on a heuristic argument of unstable modes of deviation from the uniform steady state. We confirm the accuracy of the formula by computer simulations. When the diffusive direction is different between two substances, the directions of stripes in the spatial pattern change smoothly with the magnitude of anisotropy of two substances. When the diffusive direction of the two substances is the same, the stripes are formed either parallel or perpendicular to the common diffusive direction, depending on the relative magnitude of the anisotropy. The transition between these two phases occurs sharply.     

Chromosome numbers and reproduction inRubus subgen.Malachobatus

A review of current knowledge of chromosome numbers and modes of reproduction in the genusRubus L. is presented. Chromosome numbers from some species of subg.Malachobatus Focke together with results of crossing experiments are reported for the first time.     

Pattern formation in reaction-diffusion models with nonuniform domain growth

Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction-diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain. In this paper we generalize our recent results for reaction-diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary (apical) growth are unified in a consistent manner. The results have implications for all reaction-diffusion type models subject to underlying domain growth.     

Estimation of effective reproduction numbers for infectious diseases using serological survey data

The effective reproduction number of an infection, denoted Re, may be used to monitor the impact of a vaccination programme. If Re is maintained below 1, then sustained endemic transmission of the infection cannot occur. In this paper we discuss methods for estimating Re from serological survey data, allowing for age and individual heterogeneity. We describe semi-parametric and parametric models, and obtain an upper bound on Re when vaccine coverage and efficacy are not known. The methods are illustrated using data on mumps and rubella in England and Wales.     

Applications of computerized stochastic models of human reproduction and population growth in family planning evaluation

This paper reports on the results of a computer experiment demonstrating some of the capabilities of computerized versions of a stochastic model of population growth [4] and a stochastic model of human reproduction [5] in family planning evaluation. The paper is divided into five sections, with Sec. 1 being devoted to a discussion of some of the motivations underlying the paper and the limitations of the results. Section 2 contains the numerical specifications of the component distributions of the model, and in Sec. 3 an experimental design is defined whereby the interactions of two alternative family building schemes, late versus early marriage, and low versus high desired family size may be studied with respect to population growth. The experimental design consists of eight reproductive regimes, and in Sec. 4 graphs of the calculated net maternity functions of these regimes together with the corresponding Malthusian parameter, the crude birth rate, and the mean number of female offspring in a completed family are presented. Section 5 is devoted to two population projections designed to measure the impact on population growth of two experimental schemes of transition from a high reproductive regime to a lower one. The indicators of population change and distributions used to measure the impact of these two experimental transitions in reproductive regimes as a function of time in these projections were crude birth rate, annual rate of population growth, age-specific birth rates, age distribution, and mean total population size.     

Pattern sensitivity to boundary and initial conditions in reaction-diffusion models

We consider Turing-type reaction-diffusion equations and study (via computer simulations) how the relationship between initial conditions and the asymptotic steady state solutions varies as a function of the boundary conditions. The results indicate that boundary conditions which are non-homogeneous with respect to the kinetic steady state give rise to spatial patterns which are much less sensitive to variations in the initial conditions than those obtained with homogeneous boundary conditions, such as zero flux conditions. We also compare linear pattern predictions with the numerical solutions of the full nonlinear problem.This work supported in part by U.S. Army Grant DAJA 37-81-C-0220 and the Science and Engineering Research Council of Great Britain Grant GR/c/63595     

Density-structured models for plant population dynamics

Density-structured models are structured population models in which the state variable is the proportion of populations or sites in a small number of discrete density states. Although such models have rarely been used, they have the advantage that they are straightforward to parameterize, make few assumptions about population dynamics, and permit rapid data collection using coarse density assessment. In this article, we highlight their use in relating population dynamics to environmental variation and their robustness to measurement error. We show that density-structured models are able to accurately represent population dynamics under a wide range of conditions. We look at the effects of including a persistent seedbank and describe numerical approximations for the mean and variance of population size. For simulated data, we determine the extent to which the underlying continuous process may be inferred from density-structured data. Finally, we discuss issues of parameter estimation and applications for which these types of models may be useful.     

A general theory for target reproduction numbers with applications to ecology and epidemiology

Journal of Mathematical Biology - A general framework for threshold parameters in population dynamics is developed using the concept of target reproduction numbers. This framework identifies...     

Separable models for age-structured population genetics

This paper is concerned with the applications of nonlinear age-dependent dynamics to population genetics. Age-structured models are formulated for a single autosomal locus with an arbitrary number of alleles. The following cases are considered: a) haploid populations with selection and mutation; b) monoecious diploid populations with or without mutation reproducing by self-fertilization or by two types of random mating. The diploid models do not deal with selection. For these cases the genic and genotypic frequencies evolve towards time-persistent forms, whether the total population size tends towards exponential growth or not.     

Iteroparous reproduction strategies and population dynamics

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.     

Interpolation coding: A representation for numbers in neural models

A central task of perception can be defined as one of computing hierarchies of invariants. One way of representing such invariants in intermediate levels of abstraction in this hierarchy is to use discrete units. These have been termed value units. A problem with such an encoding is that there has not been a good way to represent accurate numerical quantities using these units. This paper remedies the deficiency by describing a scheme that interpolates values between units representing fixed numerical quantities. The scheme has nice properties: it extends across functional mappings and it allows different sources of evidence to be combined.This work was supported in part by the National Science Foundation under Grant DCR-8405720 and the National Institutes of Health under Public Health Service Grant 1R01NS22407-01     

Estimating individual and household reproduction numbers in an emerging epidemic

Reproduction numbers, defined as averages of the number of people infected by a typical case, play a central role in tracking infectious disease outbreaks. The aim of this paper is to develop methods for estimating reproduction numbers which are simple enough that they could be applied with limited data or in real time during an outbreak. I present a new estimator for the individual reproduction number, which describes the state of the epidemic at a point in time rather than tracking individuals over time, and discuss some potential benefits. Then, to capture more of the detail that micro-simulations have shown is important in outbreak dynamics, I analyse a model of transmission within and between households, and develop a method to estimate the household reproduction number, defined as the number of households infected by each infected household. This method is validated by numerical simulations of the spread of influenza and measles using historical data, and estimates are obtained for would-be emerging epidemics of these viruses. I argue that the household reproduction number is useful in assessing the impact of measures that target the household for isolation, quarantine, vaccination or prophylactic treatment, and measures such as social distancing and school or workplace closures which limit between-household transmission, all of which play a key role in current thinking on future infectious disease mitigation.     

Accounting for diffusion in agent based models of reaction-diffusion systems with application to cytoskeletal diffusion

Diffusion plays a key role in many biochemical reaction systems seen in nature. Scenarios where diffusion behavior is critical can be seen in the cell and subcellular compartments where molecular crowding limits the interaction between particles. We investigate the application of a computational method for modeling the diffusion of molecules and macromolecules in three-dimensional solutions using agent based modeling. This method allows for realistic modeling of a system of particles with different properties such as size, diffusion coefficients, and affinity as well as the environment properties such as viscosity and geometry. Simulations using these movement probabilities yield behavior that mimics natural diffusion. Using this modeling framework, we simulate the effects of molecular crowding on effective diffusion and have validated the results of our model using Langevin dynamics simulations and note that they are in good agreement with previous experimental data. Furthermore, we investigate an extension of this framework where single discrete cells can contain multiple particles of varying size in an effort to highlight errors that can arise from discretization that lead to the unnatural behavior of particles undergoing diffusion. Subsequently, we explore various algorithms that differ in how they handle the movement of multiple particles per cell and suggest an algorithm that properly accommodates multiple particles of various sizes per cell that can replicate the natural behavior of these particles diffusing. Finally, we use the present modeling framework to investigate the effect of structural geometry on the directionality of diffusion in the cell cytoskeleton with the observation that parallel orientation in the structural geometry of actin filaments of filopodia and the branched structure of lamellipodia can give directionality to diffusion at the filopodia-lamellipodia interface.     

Local and global stability for population models

In general, local stability does not imply global stability. We show that this is true even if one only considers population models.We show that a population model is globally stable if and only if it has no cycle of period 2. We also derive easy to test sufficient conditions for global stability. We demonstrate that these sufficient conditions are useful by showing that for a number of population models from the literature, local and global stability coincide.We suggest that the models from the literature are in some sense simple, and that this simplicity causes local and global stability to coincide.