On the Rate of Growth of the Population of the United States since 1790 and Its Mathematical Representation

Stochastic methods in population forecasting

"This paper presents a stochastic version of the demographic cohort-component method of forecasting future population. In this model the sizes of future age-sex groups are non-linear functions of random future vital rates. An approximation to their joint distribution can be obtained using linear approximations or simulation. A stochastic formulation points to the need for new empirical work on both the autocorrelations and the cross-correlations of the vital rates. Problems of forecasting declining mortality and fluctuating fertility are contrasted. A volatility measure for fertility is presented. The model can be used to calculate approximate prediction intervals for births using data from deterministic cohort-component forecasts. The paper compares the use of expert opinion in mortality forecasting with simple extrapolation techniques to see how useful each approach has been in the past. Data from the United States suggest that expert opinion may have caused systematic bias in the forecasts."

Composite scoring--methods and predictive validity: insights from the Framingham Study

After three decades of epidemiologic research at Framingham and elsewhere, the risk factor concept is now firmly established. Atherosclerotic cardiovascular disease can now be predicted and highly vulnerable candidates identified from profiles derived from ordinary office procedures and simple laboratory tests [1]. Risk can be estimated over a 20--30-fold range, and close to half of the cardiovascular events are found to occur in a tenth of the population at highest multivariate risk. Categorical risk assessments focusing on the number of "risk factors" present also identify high-risk subjects but tend to overlook high-risk individuals with multiple marginal abnormalities. Multivariate cardiovascular risk profiles made up of the major cardiovascular risk factors can predict all of the major cardiovascular events, even in advanced age, with reasonable efficiency. Such multivariate risk assessments can be made convenient by reproduction of handbooks and use of small programmed calculators, software for personal computers, and slide rules to facilitate office and public health assessments. The sensitivity and specificity of these risk profiles can probably be improved by more detailed lipid information, including HDL-cholesterol [12], vital capacity determination, and other ECG abnormalities. General cardiovascular risk profiles can be devised to predict efficiently all of the major cardiovascular events.

Population dynamics of the badger (Meles meles) and the epidemiology of bovine tuberculosis (Mycobacterium bovis)

A survey and analysis is presented of the population biology of the badger ( Meles meles ) and the epidemiology of bovine tuberculosis ( Mycobacterium bovis ) within badger populations. Simple mathematical models are used to further our understanding of Special attention is given to the identification of areas in which current knowledge is inadequate, and to future research needs. The badger is shown to have a low intrinsic population growth rate, a not insignificant maturation delay to first breeding, to produce small litters of cubs which experience high rates of mortality in their first year of life but low rates thereafter, and to exhibit limited powers of dispersal. Population abundance is largely determined by habitat type and long term stability appears to arise primarily as a consequence of density-dependent constraints on fecundity. Such constraints are thought to only operate at densities close to the carrying capacity of the habitat. Cyclic fluctuations in abundance, with a period of between six and eight years, may occur in areas of moderate to poor habitat. Such fluctuations will be most apparent with respect to cub abundance as opposed to adult density. Bovine tuberculosis is endemic within many badger populations throughout regions of Britain but is particularly prevalent in areas of good badger habitat in the southwest of England. Current evidence suggests that badgers play a significant role in disease transmission to cattle. It is argued that the infection is able to persist in high, moderate and low density badger populations. The observed stability of the disease appears to be a consequence of ‘pseudo-vertical’ transmission (from parent to new born offspring), the long duration of infectiousness of infected animals (low disease-induced mortality rate), the presence of carriers and inactive cases and the social organization and behaviour of the host species. Disease prevalence is likely to be related to badger density although in a nonlinear manner. Control measures based on the removal of infected social groups of badgers in the southwest of England appear to have reduced the force of infection within badger populations by approximately 50%. The disease, however, remains endemic but at low levels of prevalence. The reduction in the force of infection has reduced the frequency of disease transmission to cattle herds. Eradication of the infection within badger populations may not be necessary for the short-term control of the infection in cattle. The persistence of low levels of infection in low density badger populations (suppressed by control measures), in areas of intensive cattle farming activity, however, presents a continual threat to cattle health in the long term . The ability of badger populations to recover from substantive reductions in density is poor, with a return time (to the pre-control state) of approximately five years. Small reductions in abundance, however, are likely to enhance net population growth rates as a consequence of the relaxation of density-dependent constraints on fecundity (the natural population regulatory mechanism). As such, rapid population growth to precontrol levels is predicted, following small reductions in density. As a consequence of the requirement for continual and substantive suppression of badger abundance (a renewable resource) in areas of intensive cattle farming it is suggested that alternative methods of disease control should be actively sought with a view to the design of more effective long-term control policies.

Energy flow and the persistence of a human population: a simulation analysis

The authors describe "a human ecosystem model, NUNOA, [which] simulates the yearly energy balance of individuals, families, and extended families in a hypothetical farming and herding community of Quechua Indians in the high Andes. The yearly energy demand of each family, based on the caloric requirements of its members, is computed by simulation of agricultural and herding activities in response to stochastic environmental conditions. The family energy balance is used in determining births, deaths, marriages, and resource sharing." The model can be used "to investigate the effect of changes in marriage patterns, resource sharing patterns, or subsistence activities on the ability of the human population to survive in the harsh Andean environment. Results from the model suggest that the substructuring of a population into extended families provides a mechanism for sheltering the population from control by exogenous influences. A population without substructures for resource sharing is shown to be unstable in such an unpredictable environment."

Generalizing Fisher's "reproductive value": linear differential and difference equations of "dilute" biological systems

R. A. Fisher's 1930 "reproductive value" is defined as the contribution made by a population's initial age elements to its asymptotically dominating exponential growth mode. For the Leslie discrete-time model, it is the characteristic row vector of the Leslie matrix, and for the integral-equation model of Lotka the similar eigenfunction. It generalizes neatly to a 2-sex model of linear differential equations, and to general n-variable linear systems. However, when resource limitations end the "dilute" stage of linearity, reproductive value loses positive definability. The present linear analysis prepares the way for generalizing reproductive value to nonlinear systems involving first-degree-homogeneous relationships.

Discrete demographic models with density-dependent vital rates

The standard Leslie model of population growth in an age structured population is modified so as to incorporate density-dependent feedback control on each parameter of the standard projection matrix. Under fairly general conditions, the population converges to a stable age-distribution and a constant population size. This steady-state solution is uniquely determined by the parameters of the model. In general, fertility damping results in a flatter age-distribution than yielded by the undamped Leslie model. General survival damping results in the Leslie age-distribution. Post-infant survival damping results in a very steep age-distribution. For populations with high intrinsic growth rates, these differences in stable age-distributions are pronounced. For populations of low intrinsic growth rate, the patterns are the same, but the differences in stable age-distribution are more subtle. The age-distribution usually converges rapidly to the steady-state array, although population size generally takes longer to approach a stable value. Convergence properties are described for a series of cases which show periodicity. Such cases arise from "periodic" behavior of certain fertility-damping strategies, and ultimately approach a stable steady-state, although convergence may be very slow. Although the model is very general, it can be considerably simplified in practice. Special cases, which can be constructed, are the Malthusian (Leslie) model and the Logistic model. As a generality, the model is approximately Logistic, once the age-distribution approaches the steady-state array. One may use this fact for purposes of population projection.

Modeling and Forecasting Populations by Time Series: The Swedish Case

Time series analysis techniques are used to model and to forecast populations. An autoregressive (AR) and a moving average (MA) model, which seem to fit the population of Sweden very well, are found. Forecasts are calculated using both models and are compared with the forecasts obtained by other methods. This comparison is very favorable for the time series models. Although our study is confined to the mid-year population of Sweden, there are good reasons to expect that the technique can be successfully applied to other population parameters.

Development and accuracy of projections of population and households in the united states

The history of the official U. S. projections of population and house-holds in recent decades is briefly reviewed, with particular attention to methodology and the relation of the methodology to the accuracy of the projections. The introduction of the cohort-component procedure in the 1930’s opened the way for separate analysis of the trend of the components of births, deaths, and net immigration in connection with making population projections. As a further development, the period-fertility method of projecting births gave way in the 1960’s to the cohort-fertility method. Consideration is now being given to various methods involving parity-progression procedures. Some alternative methods and problems of measuring the accuracy of population projections are then considered. The percent “error” in the projected population growth, by components and age, and the range from high to low expressed as a percent should also be examined in addition to the percent “error” in the total population. However accuracy is measured, the projections made in the 1930’s and 1940’s were often wide of the mark, and those made in the 1950’s and 1960’s failed to anticipate the sharp changes which occurred, even though the actual figures usually fell within the range projected. Elaboration of projection methodology has not resulted in any great increase in the precision of the projections, largely because birth rates have fluctuated widely, and the fluctuations have proven difficult, if not impossible, to predict. The projections of households have had a roughly similar history, and the methods and problems of evaluation are somewhat similar. Their development has been characterized by the introduction of alternative and changing “headship” rates and increasing disaggregation of the data and procedures. The paper concludes with some generalizations based on U. S. experience. Although refinement of methods may contribute little to accuracy, accuracy is only one aspect of the usefulness of projections. The need for conditional projections and their analytical usefulness are such that there is no question that we should confidently continue to make them.