Consider a supercritical Crump‒Jagers process in which all births are at integer times (the lattice case). Let μ̂(z) be the generating function of the intensity of the offspring process, and consider the complex roots of μ̂(z)=1. The root of smallest absolute value is e-α=1∕m, where α>0 is the Malthusian parameter; let γ* be the root of second smallest absolute value. Subject to some technical conditions, the second-order fluctuations of the age distribution exhibit one of three types of behaviour: (i) when γ*>e-α∕2=m-1∕2, they are asymptotically normal; (ii) when γ*=e-α∕2, they are still asymptotically normal, but with a larger variance; and (iii) when γ*<e-α∕2, the fluctuations are in general oscillatory and (degenerate cases excluded) do not converge in distribution. This trichotomy is similar to what has been observed in related situations, such as some other branching processes and for Pólya urns. The results lead to a symbolic calculus describing the limits. The asymptotic results also apply to the total of other (random) characteristics of the population.

We consider the distribution of the age of an individual picked uniformly at random at some fixed time in a linear birth-and-death process. By exploiting a bijection between the birth-and-death tree and a contour process, we derive the cumulative distribution function for this distribution. In the critical and supercritical cases, we also give rates for the convergence in terms of the total variation and other metrics towards the appropriate exponential distribution.

Age distributions of sulfur cinquefoil populations were determined on sites that were historically grazed, cultivated, and mechanically disturbed. From 12 sites, a total of 279 reproductively active plants were collected and aged by using herbchronology (counting rings in the secondary root xylem of the root crown) to (1) estimate the age structure of the populations, (2) relate plant size and flower production to plant age, and (3) examine the relation of population age structure to environmental variables and disturbance history. Results indicated that the mean age for all sampled plants was 3.5 (± 1.74 SD) yr and ranged from 1 to 10 yr. Age was not related to number of flowers, plant size (number of stems per plant or plant height), or site disturbance type but was positively correlated with site elevation (P < 0.001). The pooled age distribution from all 12 sites was right-skewed with fewer old plants than young plants. We conclude that sulfur cinquefoil plants sampled in northeast Oregon are able to colonize, establish, and reproduce at disturbed sites rapidly. We suggest that herbchronology may be a useful technique to improve understanding of invasion biology and ecology for invasive plant species that form annual rings.

Privet has escaped from cultivation and is invading natural areas throughout eastern North America. Understanding the pattern of invasion over time could help us develop more efficient management strategies. We studied the invasion history and spatial distribution pattern of privet by mapping age and spatial data for established patches in a 132-ha (326 ac) forested natural area in northeast Ohio. We determined the age of 331 geo-referenced patches by counting annual rings, and mapped them with corresponding land habitat. Age distribution and cumulative number of privet patches over about 40 yr showed three phases of invasion. The initial 19-yr lag phase was characterized as a dispersed spatial pattern (based on nearest neighbor analysis), with patches located mostly at edges of different habitats and open places. In a second phase of about 15 yr, an average of 19 patches were initiated yearly, in a pattern that trended towards clustered. The final phase began around 2007, as the rate of new patch establishment declined, possibly because of saturation of the suitable habitat. Establishment of new patches was not associated with specific habitats. Aggregation of patches with similar ages increased after 1998 and became significantly clustered. Mapping of clusters of old and young patches identified invasion hot spots and barriers. Results affirmed that the best time for invasive control is during the lag phase. By monitoring edge habitats associated with early establishment, managers might detect and control early invaders and delay the onset of the expansion phase.

We investigated the seasonality of age-specific tuberculosis (TB) in Japan. To allow the development of TB control strategies for different age groups we used a time-series analysis, including a spectral analysis and least squares method, to analyse the monthly age-specific numbers of newly registered cases of all forms of active TB in Japan from January 1998 to December 2013. The time-series data are reported in 10-year age groups: 0–9, 10–19, …, 70–79, and ⩾80 years. We defined the contribution ratio of the 1-year cycle, Q1, as the contribution of the amplitude of a 1-year cycle to the whole amplitude of the time-series data. The Q1 values in the age groups corresponding to adolescence and middle life (10–39 years) and old age (⩾70 years) were high. The peaks in the active TB epidemics for the ⩾70 years age group occurred in August and September, 1–2 months behind the peaks for the 10–39 years age group (June and July). An active TB epidemic might be attributable to travel by public transport and irregular employment in the 10–39 years age group and immune system suppression by low winter temperatures in the ⩾70 years age group.

Population aging has become a subject of widespread popular and academic interest. In providing an assessment of the extent of aging that is in prospect, this paper makes available a number of projections of the age-sex distribution of the Canadian population for the 45-year period 1996 to 2041 and comparisons with the previous 45-year period. The projections combine assumptions relating to fertility, mortality, and migration so as to produce future populations characterized as “medium,” “old,” “young,” “high immigration” and “low immigration”. Supplementary calculations include growth rates, for both the total population and selected age groups, and various types of dependency ratios, including ones with a range of age-differentiated weights. It is concluded that substantial aging of the Canadian population appears virtually certain but, based on the demographic evidence, the “dependency burden” is likely to remain below the peak levels attained during the baby boom.

Sixty-five has long been used to define the beginning of “old age”. Yet it is clear that the definition is arbitrary, and with continuing reductions in mortality and morbidity rates it will become increasingly inappropriate as time passes. We consider how the definition might be modified to reflect changes in life table probabilities, and how the future numbers and proportions in “old age” would be affected. In a similar manner, we consider also the redefinition of the “oldest old” from a current definition of 85 and over.

This study examines the effects of host age and sex on the frequency distribution of Brugia malayi infections in the human host. Microfilarial (mf) counts for a large data base on the epidemiology of brugian filariasis in Shertallai, Kerala, South India are analysed. Frequency distributions of microfilarial counts partitioned by age are successfully described by zero-truncated negative binomial distributions, fitted by maximum likelihood. This analysis provides estimates of the proportion of mf-positive individuals who are identified as negative due to sampling errors, allowing the construction of corrected mf age–prevalence curves, which indicate that the observed prevalence may under-estimate the true figures by between 18 and 47%. There is no evidence from these results for a decrease in the degree of over-dispersion of parasite frequency distributions with host age, such as might be produced by the acquired immunity to infection. This departure from the pattern in bancroftian filariasis (where there is evidence of such decreases in over-dispersion; Das et al. 1990) is discussed in terms of the long history of filariasis control (and consequently low infection prevalence) in Shertallai.

This paper examines the effects of host age and sex on the frequency distribution of Wuchereria bancrofti infections in the human host. Microfilarial counts from a large data base on the epidemiology of bancroftian filariasis in Pondicherry, South India are analysed. Frequency distributions of microfilarial counts divided by age are successfully described by zero-truncated negative binomial distributions, fitted by maximum likelihood. Parameter estimates from the fits indicate a significant trend of decreasing overdispersion with age in the distributions above age 10; this pattern provides indirect evidence for the operation of density-dependent constraints on microfilarial intensity. The analysis also provides estimates of the proportion of mf-positive individuals who are identified as negative due to sampling errors (around 5% of the total negatives). This allows the construction of corrected mf age–prevalence curves, which indicate that the observed prevalence may underestimate the true figures by between 25% and 100%. The age distribution of mf-negative individuals in the population is discussed in terms of current hypotheses about the interaction between disease and infection.

During the last 26 years the annual number of diagnosed vestibular schwannomas (VS) has been increasing. The aim of this study is to describe and analyse this increase. Since 1976, 1446 new cases of VS have been diagnosed at the authors’ centre. Special focus was on the age at diagnosis, the localization and the size of the tumour. The size of the tumour was registered as either intrameatal or with the largest extrameatal diameter. The annual number of diagnosed VS has increased from 26 in 1976 to 101 in 2001. The size of the diagnosed tumours has decreased from a median of 35 mm in 1979 to 10 mm in 2001. In the first years large and giant tumours dominated, in contrast to recent years in which intrameatal and small tumours dominated. The median age at the time of diagnosis has been almost unchanged through the period (median 55 years). If the decreasing size of the tumour and the increasing incidence of VS can be explained only by earlier diagnosis and easier access to magnetic resonance (MR) scanning it should be expected that the median age at the time of diagnosis would decrease simultaneously. In this study, the median age at the time of diagnosis has been almost unchanged throughout the 26-year period. This paradox can be explained by the fact that, with easier access to MR scanning, the examination has been offered also to elderly patients, in whom the small and intrameatal tumours dominate.

Between May 1984 and December 1985, the population of immature stages of Anopheles arabiensis was sampled by the dip method at sites close to two villages in Mwea Irrigation Scheme, Kenya. Larval presence corresponded to rainy seasons and the flooding phase of the rice cycle. No larvae were found during the postharvest dry season. The instar composition, age distribution and survivorship curves for the different stages were determined. Pupal productivity for Mwea rice fields was estimated to be 1% of the total immature population.

Let be a supercritical Bellman-Harris process with finite offspring mean. Cohn [17] has shown that there always exist constants Ct such that limt→∞Zt/Ct = W almost surely for some non-degenerate random variable W. In this paper we give an alternative proof, based on the study of (Zt) as a point process. Our methods are to some extent analytical and parallel Seneta's [18] and Heyde's [11] approaches in the case of the Galton–Watson process. We further identify Ct as 1/(–log Ft(–1)(γ)), where Ft(γ) = E(γzt), i.e. the norming constants found by Seneta [18] for the Galton–Watson process, apply also to the Bellman-Harris process. Finally we derive a weak law of large numbers for W, prove that W is continuous on (0,∞) and show that W has [0,∞) as its support.

As a consequence of the convergence of the age distribution, several other limit results follow for supercritical Bellman-Harris processes. These concern the distribution of split times, the growth ‘rate' of the process, and the distribution of ancestral lifelengths.

The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.

The paper deals with a stochastic organization model described by a generalized Poisson input process and given probabilities of entering individuals obtaining certain attributes in this organization. The distribution of attributes at any given instant is derived. Examples illustrate the application of obtained results.