Models for the spread of universally fatal diseases

In the formulation of models of S-I-R type for the spread of communicable diseases it is necessary to distinguish between diseases with recovery with full immunity and diseases with permanent removal by death. We consider models which include nonlinear population dynamics with permanent removal. The principal result is that the stability of endemic equilibrium may depend on the population dynamics and on the distribution of infective periods; sustained oscillations are possible in some cases.     

Synchronized oscillations in growing cell populations are explained by demographic noise

Understanding synchrony in growing populations is important for applications as diverse as epidemiology and cancer treatment. Recent experiments employing fluorescent reporters in melanoma cell lines have uncovered growing subpopulations exhibiting sustained oscillations, with nearby cells appearing to synchronize their cycles. In this study, we demonstrate that the behavior observed is consistent with long-lasting transient phenomenon initiated and amplified by the finite-sample effects and demographic noise. We present a novel mathematical analysis of a multistage model of cell growth, which accurately reproduces the synchronized oscillations. As part of the analysis, we elucidate the transient and asymptotic phases of the dynamics and derive an analytical formula to quantify the effect of demographic noise in the appearance of the oscillations. The implications of these findings are broad, such as providing insight into experimental protocols that are used to study the growth of asynchronous populations and, in particular, those investigations relating to anticancer drug discovery.     

The spread and persistence of infectious diseases in structured populations

A basic assumption of many epidemic models is that populations are composed of a homogeneous group of randomly mixing individuals. This is not a realistic assumption. Most actual populations are divided into a number of subpopulations, within which there may be relatively random mixing, but among which there is nonrandom mixing. As a consequence of the structuring of the population, there are several sources of heterogeneity within populations that can affect the course of an infection through the population. Two of these sources of heterogeneity are differences in contact number between subpopulations, and differences in the patterns of contact among subpopulations. A model for the spread of a disease in such a population is described. The model considers two levels of interaction: interactions between individuals within a subpopulation because of geographic proximity, and interactions between individuals of the same or different subpopulations because of attendance at common social functions. Because of this structure, it is possible to analyze with the model both heterogeneity in contact number and variation in the patterns of contact. A stability analysis of the model is presented which shows that there is a unique threshold for disease maintenance. Below the threshold the disease goes extinct, and the equilibrium is globally asymptotically stable. Above the threshold, the extinction equilibrium is unstable, and there is a unique endemic equilibrium. The analysis presents a sufficient condition for disease maintenance, which determines critical subpopulation sizes above which the disease cannot go extinct. The condition is a simple inequality relating the removal rate of infectives to the infection rate of susceptibles. In addition, bounds on the actual threshold and the effect of symmetry in the interaction matrix on the threshold are presented.     

An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations

The thresholds for mathematical epidemiology models specify the critical conditions for an epidemic to grow or die out. The reproductive number can provide significant insight into the transmission dynamics of a disease and can guide strategies to control its spread. We define the mean number of contacts, the mean duration of infection, and the mean transmission probability appropriately for certain epidemiological models, and construct a simplified formulation of the reproductive number as the product of these quantities. When the spread of the epidemic depends strongly upon the heterogeneity of the populations, the epidemiological models must account for this heterogeneity, and the expressions for the reproductive number become correspondingly more complex. We formulate several models with different heterogeneous structures and demonstrate how to define the mean quantities for an explicit expression for the reproductive number. In complex heterogeneous models, it seems necessary to define the reproductive number for each structured subgroup or cohort and then use the average of these reproductive numbers weighted by their heterogeneity to estimate the reproductive number for the total population.     

Survival of small populations under demographic stochasticity.

We estimate the mean time to extinction of small populations in an environment with constant carrying capacity but under stochastic demography. In particular, we investigate the interaction of stochastic variation in fecundity and sex ratio under several different schemes of density dependent population growth regimes. The methods used include Markov chain theory, Monte Carlo simulations, and numerical simulations based on Markov chain theory. We find a strongly enhanced extinction risk if stochasticity in sex ratio and fluctuating population size act simultaneously as compared to the case where each mechanism acts alone. The distribution of extinction times deviates slightly from a geometric one, in particular for short extinction times. We also find that whether maximization of intrinsic growth rate decreases the risk of extinction or not depends strongly on the population regulation mechanism. If the population growth regime reduces populations above the carrying capacity to a size below the carrying capacity for large r (overshooting) then the extinction risk increases if the growth rate deviates from an optimal r-value.     

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

Density-dependent parameters and demographic equilibrium in open populations

Populations near their equilibrium are expected to show density-dependence through a negative feedback on at least one demographic parameter, e.g. survival and/or productivity. Nevertheless, it is not always clear which vital rate is affected the most, and even less whether this dependence holds in open populations in which immigration and emigration are also important. We assessed the relative importance of population density in the variation of local survival, recruitment, proportion of transients (emigrants) and productivity through the analysis of detailed life-histories of 4286  seabirds from a colony that reached an apparent demographic equilibrium after a period of exponential increase. We provide evidence that the role of population density and resource availability changes according to the demographic parameter considered. Estimates indicated that transients increased from 5% to 20% over the study period, suggesting an average turnover of about 1400 individuals per year. The parameters most influenced by population density alone were local survival and probability of transience. Recruitment was negatively associated with population density during the increasing phase but unexpected high values were also recorded at high population levels. These high values were explained by a combination of population size and food availability. Mean productivity varied with food availability, independently from population variations. The population density alone explained up to a third of the yearly variation of the vital rates considered, suggesting that open populations are equally influenced by stochastic and density-independent events (such as environmental perturbations) than by intrinsic (i.e. density-dependent) factors.     

Signatures of demographic bottlenecks in European wolf populations

Monitoring the loss of genetic diversity in wild populations after a bottleneck event is a priority in conservation and management plans. Here, we used diverse molecular markers to search for signatures of demographic bottlenecks in two wolf populations; an isolated population from the Iberian Peninsula and a non-isolated population from European Russia. Autosomal, mtDNA and Y-chromosomal diversity and the effective population size (N_e) were significantly lower in the Iberian population. Neutrality tests using mtDNA sequences, such as R_2, Fu and Li’s F*, Tajima’s D and Fu’s F_s, were positively significant in the Iberian population, suggesting a population decline, but were not significant for the Russian population, likely due to its larger effective population size. However, three tests using autosomal data confirmed the occurrence of the genetic bottleneck in both populations. The M-ratio test was the only one providing significant results for both populations. Given the lack of consistency among the different tests, we recommend using multiple approaches to investigate possible past bottlenecks. The small effective population size (about 50) in the Iberian Peninsula compared to the presumed extant population size could indicate that the bottleneck was more powerful than initially suspected or an overestimation of the current population. The risks associated with small effective population sizes suggest that the genetic change in this population should be closely monitored in the future. On the other hand, the relatively small effective population size for Russian wolves (a few hundred individuals) could indicate some fragmentation, contrary to what is commonly assumed.     

Sexual selection and alternative mating behaviours generate demographic stochasticity in small populations.

Recent theory predicts that environmental variation and small population size facilitate the coexistence of alternative phenotypes despite unequal mean fitness. However, traditional studies of reproductive strategies often assume that the stability of alternative mating behaviours relies on equal male fitness. We present results from field observations and experimental manipulations of thermal resources on territories demonstrating the coexistence of alternative reproductive behaviours with unequal fitness. The side-blotched lizard Uta stansburiana exhibits two alternative strategies for territoriality: "usurp" and "defend". Paternity analysis revealed significantly greater mean fitness for "usurpers" than "defenders" in our study of natural variation. Moreover, variance in fitness was significantly higher for usurpers on both experimental and natural plots, implying that "usurp" is a risky strategy with potentially large pay-offs or none at all. We show theoretically that significantly higher variance in usurper fitness can allow for coexistence with defenders despite higher mean fitness of usurpers. This coexistence is facilitated by small population size. Our results have general implications for the evolution of alternative strategies and the maintenance of genetic diversity in small populations.     

Animal movements and the spread of infectious diseases

Domestic and wild animal population movements are important in the spread of disease. There are many recent examples of disease spread that have occurred as a result of intentional movements of livestock or wildlife. Understanding the volume of these movements and the risks associated with them is fundamental in elucidating the epidemiology of these diseases, some of which might entail zoonotic risks. The importance of the worldwide animal trade is reviewed and the role of the unregulated trade in animals is highlighted. A range of key examples are discussed in which animal movements have resulted in the introduction of pathogens to previously disease-free areas. Measures based on heightened surveillance are proposed that mitigate the risks of new pathogen introductions.     

Persistence and spread of stage-structured populations in heterogeneous landscapes

Journal of Mathematical Biology - Conditions for population persistence in heterogeneous landscapes and formulas for population spread rates are important tools for conservation ecology and...     

Urbanization and the spread of diseases of affluence in China

We quantify, track and explain the distribution of overweight and of hypertension across Chinese provinces differentiated by their degree of urbanicity over the period 1991–2004. We construct an index of urbanicity from longitudinal data on community characteristics from the China Health and Nutrition Survey and compute, for the first time, a rank-based measure of inequality in disease risk factors by degree of urbanicity. Prevalence rates of overweight and hypertension almost doubled between 1991 and 2004 and these disease risk factors became less concentrated in more urbanized areas. Decomposition analysis reveals that one-half of the urbanicity-related inequality in overweight is directly attributable to community level characteristics, while for hypertension the contribution of such characteristics increased from 20% in 1991 to 62% in 2004. At the individual level, lower engagement in physical activity and farming explain more than half of the urban concentration of overweight and a rising share (28%) of the greater prevalence of hypertension in more urbanized areas. Higher incomes explain around one-tenth of the urban concentration of both overweight and hypertension, while the education advantage of urban populations has a similar sized offsetting effect.     

Dispersals as demographic processes: testing and describing the spread of the Neolithic in the Balkans

Although population history and dispersal are back at the forefront of the archaeological agenda, they are often studied in relative isolation. This contribution aims at combining both dimensions, as population dispersal is, by definition, a demographic process. Using a case study drawn from the Early Neolithic of South-Eastern Europe, we use radiocarbon dates to jointly investigate changes in speed and population size linked to the new food production economy and demonstrate that the spread of farming in this region corresponds to a density-dependent dispersal process. The implications of this characterization are evaluated in the discussion.This article is part of the theme issue ‘Cross-disciplinary approaches to prehistoric demography’.     

Amino acid uptake in growing and arrested human diploid cell populations.

Human diploid fibroblasts cultured in vitro weremonitored for amino acid uptake in preconfluent and confluent cultures under conditions amenable to growth and in confluent cultures arrested in an essentially nonmitotic state. Preconfluent and confluent cultures in growth medium showed similar uptake patterns for leucine and alpha-aminoisobutyrate; arrested cultures exhibited a reduced uptake of both amino acids. Kinetic measurements revealed a 4-fold reduction in apparent Vmax for alpha-aminoisobutyrate influx in arrested cultures. These results suggest that the culture conditions used in this study to produce restrictions in mitotic activity likewise influence amino acid accumulation.     

The diffusive spread of alleles in heterogeneous populations

The spread of genes and individuals through space in populations is relevant in many biological contexts. I study, via systems of reaction-diffusion equations, the spatial spread of advantageous alleles through structured populations. The results show that the temporally asymptotic rate of spread of an advantageous allele, a kind of invasion speed, can be approximated for a class of linear partial differential equations via a relatively simple formula, c = 2 square root of (rD), that is reminiscent of a classic formula attributed to R. A. Fisher. The parameters r and D represent an asymptotic growth rate and an average diffusion rate, respectively, and can be interpreted in terms of eigenvalues and eigenvectors that depend on the population's demographic structure. The results can be applied, under certain conditions, to a wide class of nonlinear partial differential equations that are relevant to a variety of ecological and evolutionary scenarios in population biology. I illustrate the approach for computing invasion speed with three examples that allow for heterogeneous dispersal rates among different classes of individuals within model populations.