The stage-structured epidemic: linking disease and demography with a multi-state matrix approach model

Stage-structured epidemic models provide a way to connect the interacting processes of infection and demography. Reproduction and development can replenish the pool of susceptible hosts, and demographic structure leads to heterogeneous transmission and disease risk. Epidemics, in turn, can increase mortality or reduce fertility of the host population. Here we present a framework that integrates both demography and epidemiology in models for stage-structured epidemics. We use the vec-permutation matrix approach to classify individuals jointly by their demographic stage and infection status. We describe demographic and epidemic processes as alternating in time with a periodic matrix model. The application of matrix calculus to this framework allows for the calculation of R₀{\mathcal{R}_0} and sensitivity analysis.     

The Predictive Power of R 0 in an Epidemic Probabilistic System

An important issue in theoretical epidemiology is the epidemic thresholdphenomenon, which specify the conditions for an epidemic to grow or die out.In standard (mean-field-like) compartmental models the concept of the basic reproductive number, R ₀, has been systematically employed as apredictor for epidemic spread and as an analytical tool to study thethreshold conditions. Despite the importance of this quantity, there are nogeneral formulation of R ₀ when one considers the spread of a disease ina generic finite population, involving, for instance, arbitrary topology ofinter-individual interactions and heterogeneous mixing of susceptible andimmune individuals. The goal of this work is to study this concept in ageneralized stochastic system described in terms of global and localvariables. In particular, the dependence of R ₀ on the space ofparameters that define the model is investigated; it is found that near ofthe `classical' epidemic threshold transition the uncertainty about thestrength of the epidemic process still is significantly large. Theforecasting attributes of R ₀ for a discrete finite system is discussedand generalized; in particular, it is shown that, for a discrete finitesystem, the pretentious predictive power of R ₀ is significantlyreduced.     

Tests of Independence and Zero Correlations Among P Random Variables

Suppose it is desired to determine whether there is an association between any pair of p random variables. A common approach is to test H₀ : R = I, where R is the usual population correlation matrix. A closely related approach is to test H₀ : R_pb = I, where R_pb is the matrix of percentage bend correlations. In so far as type I errors are a concern, at a minimum any test of H₀ should have a type I error probability close to the nominal level when all pairs of random variables are independent. Currently, the Gupta-Rathie method is relatively successful at controlling the probability of a type I error when testing H₀: R = I, but as illustrated in this paper, it can fail when sampling from nonnormal distributions. The main goal in this paper is to describe a new test of H₀: R_pb = I that continues to give reasonable control over the probability of a type I error in the situations where the Gupta-Rathie method fails. Even under normality, the new method has advantages when the sample size is small relative to p. Moreover, when there is dependence, but all correlations are equal to zero, the new method continues to give good control over the probability of a type I error while the Gupta-Rathie method does not. The paper also reports simulation results on a bootstrap confidence interval for the percentage bend correlation.     

Population Dynamics of Evolutionary Change: Demographic Parameters as Indicators of Fitness

I evaluated demographic parameters as indicators of fitness by calculating the net reproductive rate (R₀), exponential rate of change (r), lifetime reproductive success (LRS), and Malthusian parameter (m) for nine genotypes and four phenotypes (two alleles at each of two independent loci) of an age-structured population. The given starting conditions included age-specific survival rates of males and females and age-specific fecundity of females for each genotype (to simplify the problem I presumed no differences in survivorship or fecundity of genotypes with the same phenotype) and the same age structure for each genotype. The prevailing genotype had the greatestm, but it did not have the greatestr,R₀, or LRS, or even the greatest survivorship of either juveniles or adults, or the greatest fecundity. This result indicates thatmis the only correct measure of fitness (i.e., as a predictor of which genotype should prevail from among a group of genotypes) and that comparisons ofr,R₀, LRS, juvenile or adult survival rates, or fecundity may be misleading indicators of which genotype should prevail (i.e., be most “fit”) over time (i.e., be selected for).     

Genealogy with seasonality,the basic reproduction number,and the influenza pandemic

The basic reproduction number R ₀ has been used in population biology, especially in epidemiology, for several decades. But a suitable definition in the case of models with periodic coefficients was given only in recent years. The definition involves the spectral radius of an integral operator. As in the study of structured epidemic models in a constant environment, there is a need to emphasize the biological meaning of this spectral radius. In this paper we show that R ₀ for periodic models is still an asymptotic per generation growth rate. We also emphasize the difference between this theoretical R ₀ for periodic models and the “reproduction number” obtained by fitting an exponential to the beginning of an epidemic curve. This difference has been overlooked in recent studies of the H1N1 influenza pandemic.     

Respiratory carbon use and carbon storage in mid‐rotation loblolly pine (Pinus taeda L.) plantations: the effect of site resources on the stand carbon balance

We used estimates of autotrophic respiration (R_A), net primary productivity (NPP) and soil CO₂ evolution (S_ff), to develop component carbon budgets for 12‐year‐old loblolly pine plantations during the fifth year of a fertilization and irrigation experiment. Annual carbon use in R_A was 7.5, 9.0, 15.0, and 15.1 Mg C ha^?1 in control (C), irrigated (I), fertilized (F) and irrigated and fertilized (IF) treatments, respectively. Foliage, fine root and perennial woody tissue (stem, branch, coarse and taproot) respiration accounted for, respectively, 37%, 24%, and 39% of R_A in C and I treatments and 38%, 12% and 50% of R_A in F and IF treatments. Annual gross primary production (GPP=NPP+R_A) ranged from 13.1 to 26.6 Mg C ha^?1. The I, F, and IF treatments resulted in a 21, 94, and 103% increase in GPP, respectively, compared to the C treatment. Despite large treatment differences in NPP, R_A, and carbon allocation, carbon use efficiency (CUE=NPP/GPP) averaged 0.42 and was unaffected by manipulating site resources. Ecosystem respiration (R_E), the sum of S_ff, and above ground R_A, ranged from 12.8 to 20.2 Mg C ha^?1 yr^?1. S_ff contributed the largest proportion of R_E, but the relative importance of S_ff decreased from 0.63 in C treatments to 0.47 in IF treatments because of increased aboveground R_A. Aboveground woody tissue R_A was 15% of R_E in C and I treatments compared to 25% of R_E in F and IF treatments. Net ecosystem productivity (NEP=GPP‐R_E) was roughly 0 in the C and I treatments and 6.4 Mg C ha^?1 yr^?1 in F and IF treatments, indicating that non‐fertilized treatments were neither a source nor a sink for atmospheric carbon while fertilized treatments were carbon sinks. In these young stands, NEP is tightly linked to NPP; increased ecosystem carbon storage results mainly from an increase in foliage and perennial woody biomass.     

Identifying non-invasible habitats for marine copepods using temperature-dependent <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript>

If a non-indigenous species is to thrive and become invasive it must first persist under its new set of environmental conditions. Net reproductive rate (R ₀) represents the average number of female offspring produced by a female over its lifetime, and has been used as a metric of population persistence. We modeled R ₀ as a function of ambient water temperature (T) for the invasive marine calanoid copepod Pseudodiaptomus marinus, which is introduced to west coast of North America from East Asia by ship ballast water. The model was based on temperature-dependent stage-structured population dynamics given by a system of ordinary differential equations. We proposed a methodology to identify habitats that are non-invasible for P. marinus using the threshold of R ₀(T) < 1 in order to identify potentially invasible habitats. We parameterized the model using published data on P. marinus and applied R ₀(T) to identify the range of non-invasible habitats in a global scale based on sea surface temperature data. The model predictions matched the field evidence of species occurrences well.     

Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations

In nonlinear matrix models, strong Allee effects typically arise when the fundamental bifurcation of positive equilibria from the extinction equilibrium at r=1 (or R₀=1) is backward. This occurs when positive feedback (component Allee) effects are dominant at low densities and negative feedback effects are dominant at high densities. This scenario allows population survival when r (or equivalently R₀) is less than 1, provided population densities are sufficiently high. For r>1 (or equivalently R₀>1) the extinction equilibrium is unstable and a strong Allee effect cannot occur. We give criteria sufficient for a strong Allee effect to occur in a general nonlinear matrix model. A juvenile–adult example model illustrates the criteria as well as some other possible phenomena concerning strong Allee effects (such as positive cycles instead of equilibria).     

Demography of Oenocarpus bataua and implications for sustainable harvest of its fruit in western Amazon

Oenocarpus bataua is the seventh most abundant tree in the Amazon and one of the most used palms in the region. The main resource obtained from the species is the fruits that are harvested from wild populations for human consumption. Across its distribution area adults are most frequently felled to obtain the racemes, which may affect the palm's populations. In this paper we studied the demography of two populations of Oenocarpus bataua to assess the harvest potential of its fruits and the density variation in different habitats in the western Amazon to estimate fruit yields in different forest types. Non-inundated lands held the greatest densities with an average of 11 adults ha⁻¹ (0–132 adults ha⁻¹). The population finite growth rate (λ) in Amacayacu, Colombia, was 0.9103 because of slow growth and low survival of stemless individuals and low recruitment. On the contrary, in Yasuní, Ecuador, we found a growing population with λ = 1.0368. According to our simulations, adult felling reduced transient population growth (λ_t) in both populations, especially when harvest was frequent even at low intensities. In Amacayacu a simulated harvest of 60 % year⁻¹ of the fruits by climbing did not modify λ_t substantially, while in Yasuní, a regime of 80 % of annual harvest did not diminish λ_t below one and the initial number of adults. The results help to understand the demography of useful palms and to address sustainable management. For instance high yields can be obtained by shifting to non-destructive harvest techniques that can meet the increasing demand and maintain the populations.     

Isolation and characterisation of 11 tetranucleotide microsatellite loci in the common genet (Genetta genetta)

We report the isolation and characterization of 11 polymorphic tetranucleotide microsatellite loci in the common genet (Genetta genetta) from genomic libraries enriched for (AAAG) _n and (AGAT) _n repeat sequences. We chose to develop tetranucleotide repeats because they can be scored less ambiguously. In a sample of 25 individuals, we observed between four and thirteen alleles per locus and their observed and expected heterozygosities ranged from 0.60 to 0.84 and from 0.68 to 0.92, respectively. All genotypic frequencies conformed to Hardy–Weinberg equilibrium expectations and there were no instances of linkage disequilibrium detected between pairs of loci. These loci will be of use in studies of population genetics, historical demography, and molecular ecology of the common genet.     

The basic reproductive number of tick–borne encephalitis virus

Tick-borne encephalitis virus (TBEV) is reciprocally transmitted between Ixodes ricinus ticks and small mammals. Recently, transmission between co-feeding ticks has been postulated as an epidemiological by important mechanism of perpetuating the agent. To empirically examine the question whether the “traditional” mode of transmission is sufficient to maintain enzootic TBEV transmission, the basic reproductive number R₀ of TBEV could be estimated under this model for sites in which TBEV is enzootic. I propose an empirical estimator of R₀ for TBEV which is based on longitudinal stage–specific local tick infestation densities assessed by live trapping of small mammals. A Gibbs sampler–based 95%–credibility interval is presented. When applied to published field data from TBEV enzootic sites sub–critical R₀ estimates are obtained for both sites. I discuss potential shortcomings of this method and possible implications of these findings on the discussion of supplemental mechanisms of transmission. I thank Marc Lipsitch (Harvard School of Public Health, Boston, MA) who helped with an early version of this paper. I also thank Dr. Andrea Pugliese and two anonymous referees for their insightful comments and helpful suggestions.     

Variation in the content of hypericins in four generations of seed progeny of <Emphasis Type="Italic">Hypericum perforatum</Emphasis> somaclones

The content of hypericins in in vitro regenerated Hypericum perforatum plants (R ₀) and four generations of their seed progeny (R ₁–R ₄) was compared. The mean content of hypericins in field-grown plants over the period 1992–2002 gradually increased under selection, and in the R ₄ generation it was almost seven-times higher than that in the R ₀ somaclones. Significant difference between hypericin content in diploids and tetraploids was detected in R ₀, R ₁ and R ₃ generations. Hypericin content in four diploid and tetraploid lineages originated from a single somaclone was genotype dependent. To eliminate the influence of environmental conditions during different growing seasons, we used seeds of selected R ₀–R ₃ plants to derive R′₁ to R′₄ generations cultivated during the same years. In this case no statistically significant difference in hypericin content was found between the R′₁–R′₄ generations. Apomictically and sexually derived plants were distinguished by PCR using variable numbers of tandem repeats (VNTR) primers. The content of hypericins in apomictically derived progenies was compared.     

Suitability of four different prey species for Dichochrysa tacta (Neuroptera: Chrysopidae)

The prey suitability for generalist insect predators is an important factor in efficient mass rearing. This work constitutes the first biological study of the predatory lacewing Dichochrysa tacta. We tested the effects of four different prey species on the biological parameters of the predatory lacewing D. tacta under laboratory conditions. These prey species consisted of Ephestia kuehniella (Zeller) and the aphids: Aphis fabae, Aphis punicae and Macrosiphum rosae. The results showed that prey species elicited a significant effect on preimaginal development times, survival, adult longevity and fecundity. In contrast, the progeny sex ratio, egg hatch and egg duration were not significantly affected by prey species. Eggs of E. kuehniella were the most suitable prey among those tested, and led to high survival rates, short development times for the preimaginal stages, long adult longevity, high fecundity, highest intrinsic rate of increase (r_m) and highest net reproductive rate (R₀) of D. tacta. Nymphs of both A. punicae and M. rosae were also relatively favourable prey, whereas nymphs of A. fabae were less suitable and resulted in reduced development, survival, adult longevity, fecundity, r_m and R₀. These results could be useful for optimising the mass rearing of D. tacta as an indigenous predator and for understanding its population dynamics in relation to different prey species in the field.     

Seasonal and spatial variation of woody tissue respiration in a <Emphasis Type="Italic">Pinus cembra</Emphasis> tree at the alpine timberline in the central Austrian Alps

Total stem, branch, twig, and coarse root respiration (R_t) of an adult Pinus cembra tree at the alpine timberline was measured continuously at ten positions from 7 October 2001 to 21 January 2003 with an automated multiplexing gas exchange system. There was a significant spatial variability in woody tissue respiration when expressed per unit surface area or per unit sapwood volume. Surface area related maintenance (R_m) respiration at 0°C ranged between 0.109 and 0.643 mol m^–2 s^–1 and there was no clear trend with respect to tissue type and diameter. Sapwood volume based R_m at 0°C by contrast, varied between 2.5 mol m^–3 s^–1 in the stem and 193.2 mol m^–3 s^–1 in thin twigs in the upper crown. Estimated Q₁₀ values ranged from 1.7 to 3.1. These Q₁₀ values were used along with R_m at 0°C and annual woody tissue temperature records to predict annual total R_m. Annual total R_m accounted for 73±6% of annual R_t in 2002.     

Krill (<Emphasis Type="Italic">Euphausia superba</Emphasis>) recruitment indices from the western Antarctic Peninsula: are they representative of larger regions?

This study presents data on krill demography west of the Antarctic Peninsula covering many years since 1982. The recruitment indices are compared with those for other mesoscale study regions collected over the last 25 years. We use these data to investigate whether results from such mesoscale surveys are representative of large-scale stocks or regional populations. Generally, the proportional recruitment indices for 1- (R₁) and 2-year-old (R₂) krill differ substantially between years for a given area. However, indices were in conformity with the results from other regional scientific surveys. Recruitment indices showed a significant correlation for age-class 1 krill between scientific surveys from the northern Bellingshausen Sea, the Elephant Island area and South Georgia. The correlation becomes weaker for R₂ recruitment indices. There was no correlation between krill recruitment indices from Atlantic and Indian Ocean survey sites. Problems of single-year outliers from Elephant Island are discussed, as well as the problem of "undersized" length classes of the age-1 group that occur in the samples of some years.     

A net reproductive number for periodic matrix models

We give a definition of a net reproductive number R ₀ for periodic matrix models of the type used to describe the dynamics of a structured population with periodic parameters. The definition is based on the familiar method of studying a periodic map by means of its (period-length) composite. This composite has an additive decomposition that permits a generalization of the Cushing–Zhou definition of R ₀ in the autonomous case. The value of R ₀ determines whether the population goes extinct (R ₀<1) or persists (R ₀>1). We discuss the biological interpretation of this definition and derive formulas for R ₀ for two cases: scalar periodic maps of arbitrary period and periodic Leslie models of period 2. We illustrate the use of the definition by means of several examples and by applications to case studies found in the literature. We also make some comparisons of this definition of R ₀ with another definition given recently by Bacaër.     

Demographic traits of Hippodamia variegata (Goeze) (Coleoptera: Coccinellidae) reared on Aphis fabae Scopoli (Hemiptera: Aphididae)

Lady beetles are the most important predatory species among arthropods; thus, studying their population parameters gives a clear picture about their life span. The demography of the lady beetle, Hippodamia variegata (Goeze) (Coleoptera: Coccinellidae), was studied under laboratory condition, on Aphis fabae at 27?°C based on the age-stage, two-sex life table. The means and standard errors were estimated by the bootstrap techniques. The results showed that the intrinsic rate of increase (r) and the finite rate of increase (λ) were 0.18 and 1.20 d^?1, respectively. The net reproduction rate (R ₀) and the gross reproduction rate were 232.49 and 442.48 offspring/individual, respectively. The mean generation time (T) was 29.03?days. The first peak of reproductive value in females was on the 18th day, which coincides with the total pre-oviposition period counted from birth. The life expectancy curve (e_xj ) of a new hatched female adult was 60?days. The results demonstrated that growth, survival and development of the predator can be described accurately by use of the age-stage, two-sex life table.     

Demographic consequences of age-structure in extreme environments: population models for arctic and alpine ptarmigan

Organisms living in arctic and alpine environments are increasingly impacted by human activities. To evaluate the potential impacts of global change, a better understanding of the demography of organisms in extreme environments is needed. In this study, we compare the age-specific demography of willow ptarmigan (Lagopus lagopus) breeding at arctic and subalpine sites, and white-tailed ptarmigan (L. leucurus) breeding at an alpine site. Rates of egg production improved with age at the alpine and subalpine sites, but the stochastic effects of nest and brood predation led to similar rates of annual fecundity among 1-, 2-, and 3+-year-old females. All populations had short generation times (T<2.7 years) and low net reproductive rates (R ₀<1.2). Stable age distributions were weighted towards 1-year-old females in willow ptarmigan (>59%), and to 3+-year-old females in white-tailed ptarmigan (>47%). High damping ratios (ρ>3.2) indicated that asymptotic estimates were likely to match natural age distributions. Sensitivity and elasticity values indicated that changes in juvenile survival would have the greatest impact on the finite rate of population change (λ) in willow ptarmigan, whereas changes to the survival of 3+-year-old females would have a greater effect in white-tailed ptarmigan. High survivorship buffers white-tailed ptarmigan in alpine environments against the potential effects of climate change on annual fecundity, but may make the species more sensitive to the effects of pollutants or harvesting on adult survival. Conversely, processes that reduce annual fecundity would have a greater impact on the population viability of willow ptarmigan in arctic and subalpine environments. If these same demographic patterns prove to be widespread among organisms in extreme environments, it may be possible to develop general recommendations for conservation of the biological resources of arctic and alpine ecosystems.