A note on generation times in epidemic models

The time between the infection of a primary case and one of its secondary cases is called a generation time. The distribution (and mean) of the generation times is derived for a rather general class of epidemic models. The relation to assumptions on distributions of latency times and infectious times or more generally on random time varying infectiousness, is investigated. Serial times, defined as the times between occurrence of observable events in the progress of an infectious disease (e.g., the onset of clinical symptoms), are also considered.     

THE DISTRIBUTION OF CELL GENERATION TIMES

Two different distributions of generation times, log-normal and reciprocal normal, were compared by fitting them to experimental generation time distributions for bacteria, yeast, protozoa, and mammalian cells. In every case, the reciprocal normal distribution gave better agreement, whether the original frequency distributions or their cumulatives were compared. the log-normal distribution failed to give significant agreement for three of the original distributions. It is concluded that the reciprocal normal distribution of generation times, or more precisely, the truncated normal distribution of generation rates, is consistent with the published data for steady state populations of all kinds of cells that divide by regular binary fission. A model is suggested for this distribution of generation times, associating it with control of transport of materials into the cell, and with linear cell growth.     

INTRAPOPULATION KINETICS OF THE MITOTIC CYCLE

Data obtained with time lapse cinemicrographic techniques showed that the distribution of generation times for exponentially proliferating human amnion cells in culture is skewed to the right and that reciprocals of generation times appear normally distributed. As shown for bacteria, the true age distribution is much broader than theoretical distributions which fail to take into account the dispersion of generation times. By means of the technique utilizing autoradiographic detection of tritiated thymidine in cells whose mitotic histories were recorded by time lapse cinemicrography, it was shown that the G1 distribution is similar to the generation time distribution but is more variable. In our experiments, the G2 + prophase distribution resembled the generation time and G1 distributions. The data suggested two possibilities for S: either it is relatively constant, or it is inversely related to the lengths of G1 and G2 + prophase. Since G1 is more variable than the total cycle, and G2 + prophase more variable than the computed sum of S + G2 + prophase + metaphase, it was concluded that the relationships between parts of the cycle are non-random and that compensating mechanisms apparently help regulate the lengths of successive parts of the mitotic cycle in individual cells.     

Structural heterogeneity in populations of the budding yeast Saccharomyces cerevisiae.

Bud scar analysis integrated with mathematical analysis of DNA and protein distributions obtained by flow microfluorometry have been used to analyze the cell cycle of the budding yeast Saccharomyces cerevisiae. In populations of this yeast growing exponentially in batch at 30 degrees C on different carbon and nitrogen sources with duplication times between 75 and 314 min, the budded period is always shorter (approximately 5 to 10 min) than the sum of the S + G2 + M + G1* phases (determined by the Fried analysis of DNA distributions), and parent cells always show a prereplicative unbudded period. The analysis of protein distributions obtained by flow microfluorometry indicates that the protein level per cell required for bud emergence increases at each new generation of parent cells, as observed previously for cell volume. A wide heterogeneity of cell populations derives from this pattern of budding, since older (and less frequent) parent cells have shorter generation times and produce larger (and with shorter cycle times) daughter cells. A possible molecular mechanism for the observed increase with genealogical age of the critical protein level required for bud emergence is discussed.     

DETERMINATION OF STEADY STATE GENERATION TIME DISTRIBUTIONS FROM LABELLED MITOSIS EXPERIMENTAL DATA

The proper application of detailed deterministic cell kinetic models depends on the way in which cells are assigned their generation times. A method is presented for the determination of population generation time distributions from labelled mitoses experiments. the model assumes that the generation time of each new cell is a function of both the steady-state generation time distribution function of the population, and also the generation time frequency-function of the previous generation of cells. This approach is applied to two different cell types to successfully simulate extended labelled mitoses curves using a population balance model with constant maturation rates.     

Distributions of time to fixation of neutral genes

Exact distributions of times to fixation are derived for neutral alleles in a discrete generation, constant census model that accommodates specified variations in gametic contributions per parent. This enables an evaluation of the performance of the effective number concept, and of diffusion equation approximations, used to characterize the time scale of drift events. Following simultaneous comparisons of modes, medians, means and standard deviations of exact and approximate distributions of times to fixation, it is concluded that diffusion equation methods, in conjunction with the appropriate effective number, do produce accurate results.     

Effects of asymmetric division on a stochastic model of the cell division cycle

The stochastic model of cell division formulated by Alt and Tyson is generalized to the case of imprecise binary fission. Closed-form expressions are derived for the generation-time distribution, the birth-size and division-size distributions, the beta curve, and the correlation coefficient of generation times of sister cells. The theoretical results are compared to observations of cell division statistics in a culture of fission yeast.     

The correlation between infectivity and incubation period of measles, estimated from households with two cases

The generation time of an infectious disease is the time between infection of a primary case and infection of a secondary case by the primary case. Its distribution plays a key role in understanding the dynamics of infectious diseases in populations, e.g. in estimating the basic reproduction number. Moreover, the generation time and incubation period distributions together characterize the effectiveness of control by isolation and quarantine. In modelling studies, a relation between the two is often not made specific, but a correlation is biologically plausible. However, it is difficult to establish such correlation, because of the unobservable nature of infection events. We have quantified a joint distribution of generation time and incubation period by a novel estimation method for household data with two susceptible individuals, consisting of time intervals between disease onsets of two measles cases. We used two such datasets, and a separate incubation period dataset. Results indicate that the mean incubation period and the generation time of measles are positively correlated, and that both lie in the range of 11-12 days, suggesting that infectiousness of measles cases increases significantly around the time of symptom onset. The correlation between times from infection to secondary transmission and to symptom onset could critically affect the predicted effectiveness of isolation and quarantine.     

The effective number of a population that varies cyclically in size. II. Overlapping generations

We consider haploid and dioecious age-structured populations that vary over time in cycles of length k. Results are obtained for both autosomal and sex-linked loci if the population is dioecious. It is assumed that k is small in comparison with numbers of haploid individuals (or of numbers of males and females) in any generation of a cycle. The inbreeding effective population size N(e) is then approximately given by the expression [T summation operator (k-1)(j=0)1/[N(e)(j)T(j)]](-1), where N(e)(j) and T(j) are, respectively, the effective population size and generation interval that would hold if the population was at all times generated in the same way as at time j. The constant T, which is the effective overall generation interval, is defined to be k times the harmonic mean of the quantities T(j). Our expressions for T and N(e), in terms of N(e)(j) and T(j), are general, but the N(e)(j)s are derived under the assumption that offspring are produced according to Poisson distributions.     

Intrinsic and realized generation intervals in infectious-disease transmission

The generation interval is the interval between the time when an individual is infected by an infector and the time when this infector was infected. Its distribution underpins estimates of the reproductive number and hence informs public health strategies. Empirical generation-interval distributions are often derived from contact-tracing data. But linking observed generation intervals to the underlying generation interval required for modelling purposes is surprisingly not straightforward, and misspecifications can lead to incorrect estimates of the reproductive number, with the potential to misguide interventions to stop or slow an epidemic. Here, we clarify the theoretical framework for three conceptually different generation-interval distributions: the ‘intrinsic’ one typically used in mathematical models and the ‘forward’ and ‘backward’ ones typically observed from contact-tracing data, looking, respectively, forward or backward in time. We explain how the relationship between these distributions changes as an epidemic progresses and discuss how empirical generation-interval data can be used to correctly inform mathematical models.     

Coordinate initiation of chromosome and minichromosome replication in Escherichia coli.

Escherichia coli minichromosomes harboring as little as 327 base pairs of DNA from the chromosomal origin of replication (oriC) were found to replicate in a discrete burst during the division cycle of cells growing with generation times between 25 and 60 min at 37 degrees C. The mean cell age at minichromosome replication coincided with the mean age at initiation of chromosome replication at all growth rates, and furthermore, the age distributions of the two events were indistinguishable. It is concluded that initiation of replication from oriC is controlled in the same manner on minichromosomes and chromosomes over the entire range of growth rates and that the timing mechanism acts within the minimal oriC nucleotide sequence required for replication.     

The three-loop model: a neural network for the generation of saccadic reaction times

This paper presents a computer simulation of the three-loop model for the temporal aspects of the generation of visually guided saccadic eye movements. The intention is to reproduce complex experimental reaction time distributions by a simple neural network. The operating elements are artificial but realistic neurones. Four modules are constructed, each consisting of 16 neural elements. Within each module, the elements are connected in an all-to-all manner. The modules are working parallel and serial according to the anatomically and physiologically identified visuomotor pathways including the superior colliculus, the frontal eye fields, and the parietal cortex. Two transient-sustained input lines drive the network: one represents the visual activity produced by the onset of the saccade target, the other represents a central activity controlling the preparation of saccades, e.g. the end of active fixation. The model works completely deterministically; its stochastic output is a consequence of the stochastic properties of the input only. Simulations show how multimodal distributions of saccadic reaction times are produced as a natural consequence of the model structure. The gap effect on saccadic reaction times is correctly produced by the model: depending only on the gap duration (all model parameters unchanged) express, fast-regular, and slow-regular saccades are obtained in different numbers. In agreement with the experiments, bi- or trimodal distributions are produced only for medium gap durations (around 200 ms), while for shorter or longer gaps the express mode disappears and the distributions turn bi- or even unimodal. The effect of varying the strength of the transient-sustained components and the ongoing activity driving the hierarchically highest module are considered to account for the interindividual variability of the latency distributions obtained from different subjects, effects of different instructions to the same subject, and the observation of express makers (subjects who produce exclusively express saccades). How the model can be extended to describe the spatial aspects of the saccade system will be discussed as well as the effects of training and/or rapid adaptation to experimental conditions.     

CLONE SIZE ANALYSIS: A PARAMETER IN THE STUDY OF CELL POPULATION KINETICS

The method of clone size analysis is described. Such an analysis provides a measure of the distribution of generation times in a cell population. Treatment by ionizing irradiation leads to perturbation of the generation times of a cell population and such perturbations are shown by changes in the clone size distribution. Distributions were compared after neutron irradiation and β-irradiation with those after X-rays; and also between X-irradiation under aerobic and anoxic conditions. At the same level of cell survival no differences were found between the patterns of cell survival with respect to generation times. Other uses of clone size analysis are discussed.     

PHYLOGENETIC CONSTRAINTS IN KEY FUNCTIONAL TRAITS BEHIND SPECIES’ CLIMATE NICHES: PATTERNS OF DESICCATION AND COLD RESISTANCE ACROSS 95 DROSOPHILA SPECIES

Species distributions are often constrained by climatic tolerances that are ultimately determined by evolutionary history and/or adaptive capacity, but these factors have rarely been partitioned. Here, we experimentally determined two key climatic niche traits (desiccation and cold resistance) for 92–95 Drosophila species and assessed their importance for geographic distributions, while controlling for acclimation, phylogeny, and spatial autocorrelation. Employing an array of phylogenetic analyses, we documented moderate‐to‐strong phylogenetic signal in both desiccation and cold resistance. Desiccation and cold resistance were clearly linked to species distributions because significant associations between traits and climatic variables persisted even after controlling for phylogeny. We used different methods to untangle whether phylogenetic signal reflected phylogenetically related species adapted to similar environments or alternatively phylogenetic inertia. For desiccation resistance, weak phylogenetic inertia was detected; ancestral trait reconstruction, however, revealed a deep divergence that could be traced back to the genus level. Despite drosophilids’ high evolutionary potential related to short generation times and high population sizes, cold resistance was found to have a moderate‐to‐high level of phylogenetic inertia, suggesting that evolutionary responses are likely to be slow. Together these findings suggest species distributions are governed by evolutionarily conservative climate responses, with limited scope for rapid adaptive responses to future climate change.     

The kinetic significance of cell size : II. Size distributions of resting and proliferating cells during interphase

This second part in a two part report describes the kinetic, cell size and nuclear size characteristics of S phase cells and cells with greatly protracted generation times (‘resting’ cells) in a cell line of human lymphoid cells. The median cell and nuclear sizes of S phase cells were greater than the corresponding median sizes observed in the whole population. Resting cells (operationally defined as unlabelled cells after 5 days of continuous labelling with [³H]TdR) have cell and nuclear size distributions overlapping with the cell and nuclear size distributions of the whole population. These resting cells are kinetically characterized by means of the observed labelling index vs time data during continuous labelling. The implication of these results are discussed.     

Single generation cycles and delayed feedback cycles are not separate phenomena

We study a simple model for generation cycles, which are oscillations with a period of one or a few generation times of the species. The model is formulated in terms of a single delay-differential equation for the population density of an adult stage, with recruitment to the adult stage depending on the intensity of competition during the juvenile phase. This model is a simplified version of a group of models proposed by Gurney and Nisbet, who were the first to distinguish between single-generation cycles and delayed-feedback cycles. According to these authors, the two oscillation types are caused by different mechanisms and have periods in different intervals, which are one to two generation times for single-generation cycles and two to four generation times for delayed-feedback cycles. By abolishing the strict coupling between the maturation time and the time delay between competition and its effect on the population dynamics, we find that single-generation cycles and delayed-feedback cycles occur in the same model version, with a gradual transition between the two as the model parameters are varied over a sufficiently large range. Furthermore, cycle periods are not bounded to lie within single octaves. This implies that a clear distinction between different types of generation cycles is not possible. Cycles of all periods and even chaos can be generated by varying the parameters that determine the time during which individuals from different cohorts compete with each other. This suggests that life-cycle features in the juvenile stage and during the transition to the adult stage are important determinants of the dynamics of density limited populations.     

In vitro determination of generation times for Entodinium exiguum, Ophryoscolex purkynjei and Eudiplodinium maggii

ABSTRACT. Most previously reported generation times for rumen ciliate protozoa are longer than would be required to prevent their being flushed out of the rumen. In an earlier study from this lab, using a sequential transfer procedure, generation times between 12 and 13 h were determined for both Epidinium caudatum and Entodinium caudatum . This would permit these species to be maintained in a rumen with a fluid volume turnover rate as rapid as twice a day. In this study, generation times were estimated for Entodinium exiguum (13.2 h), Eudiplodinium maggii (26.8 h), and Ophryoscolex purkynjei (29 h), by sequential transfer at both 12 and 24 h time periods. The generation time for E. exiguum is lower than reported for this and other Entodinium species as determined by logarithmic growth from a small inoculum, but similar to that obtained for Ent. caudatum using sequential transfer. Eudiplodinium maggii and O. purkynjei generation times are similar to previous estimates of 24- and 24–48 h, respectively. However, it was observed that after an adaptation period of 36 to 48 h (generally 3–4 transfers) cell concentrations decreased and generation times were markedly decreased, i.e. 12.2 h for Ent. exiguum , 15.0 h for E. maggii and 12.8 h for O. purkynjei . In a separate study, varying both the concentration of Epidinium and the quantity of substrate fed per cell had no effect on generation time.     

Genetics of the polycross

Summary Rates and patterns of male gamete incorporation for a polycross mating design were studied for two independent years of pollination in Norway spruce, Picea abies (L) Karst. Segregation distortion in a subset of maternal clones was documented for one locus. We have proposed a model, involving the existence of a linked lethal allele, which accounts for these observations. Significant temporal and maternal clonal differences were observed in the rates at which single locus and multilocus gametes were incorporated. Striking differences in apparent fertility existed among four clones which produced unique multilocus gametes. One clone, in particular, was shown to be contributing three times as many gametes to the next generation as predicted by the hypothesis of equal clonal male contribution. These deviations from expectation were also detected in the genotypic distributions of the resultant filial generation. Ramifications of these results on family structures in the filial generation, effective size of the male population, and possible bias in inferences of genetic differences and parameter estimation are discussed.     

Connection between stochastic and deterministic modelling of microbial growth

We present in this paper various links between individual and population cell growth. Deterministic models of the lag and subsequent growth of a bacterial population and their connection with stochastic models for the lag and subsequent generation times of individual cells are analysed. We derived the individual lag time distribution inherent in population growth models, which shows that the Baranyi model allows a wide range of shapes for individual lag time distribution. We demonstrate that individual cell lag time distributions cannot be retrieved from population growth data. We also present the results of our investigation on the effect of the mean and variance of the individual lag time and the initial cell number on the mean and variance of the population lag time. These relationships are analysed theoretically, and their consequence for predictive microbiology research is discussed.     

The selection for early and late pupariation in the flesh-fly, Sarcophaga argyrostoma, and its effect on the incidence of pupal diapause

ABSTRACT. Artificial selection from the first and last larvae of Sarcophaga argyrostoma (Robineau-Desvoidy) to form puparia gives rise to two strains, 'fast' (F) and 'slow' (S), 'fast' pupariating about 3 days earlier than 'slow' in continuous light, 25°C. The two strains differ only in the time from larval wandering to pupariation; other aspects of development are identical. In light-dark cycles (1°C) the distributions of pupariation times in 'fast' are unimodal and nearly normal, whereas those for 'slow' are multimodal and with a marked skew, especially in short daylengths. Pupariation times in 'fast'×'slow' hybrids, an F₂ generation, and a backcross [(F×S)×S] are intermediate between 'fast' and 'slow' but incline towards 'fast'. It is concluded that control of pupariation time is polygenic, and that 'slow' contain considerable residual variation. When compared with the unselected stock, 'fast' and 'slow' both produce a reduced incidence of pupal diapause in short daylengths, and 'fast' show a shorter critical daylength. These effects are interpreted in terms of a modified version of Gibbs' (1975) photoperiodic 'counter' hypothesis.