Age-specific mortality analysis of the dry forest kissingbug, Rhodnius neglectus Jorge E. Rabinovich1*, Eliana L. Nieves1 & Luis F. Chaves21Centro de Estudios Parasitolo´gicos y de Vectores, Universidad Nacional de La Plata, La Plata, Prov. de Buenos Aires, Argen-tina, and 2Department of Environmental Studies, Emory University, Atlanta, GA, USA Key words: senescence, cohort studies, Chagas disease, mortality rate, Weibull model, Gompertzmodel, exponential model, logistic model, DeMoivre model, Heteroptera, Reduviidae Age-specific mortality patterns can be very different across insects with different life histories. Someholometabolous insects (like mosquitoes, fruit flies) show a pattern where mortality rate deceleratesat older ages, whereas other holometabolous insects (bruchid beetles) and hemimetabolous insects(cotton stainers, milkweed bugs, and kissing bugs) show an age-specific mortality pattern thatincreases through all ages. Kissing bugs are strictly hematophagous and are vectors of Trypanosomacruzi Chagas, the etiologic agent of Chagas disease. Here, we tested whether cohort data from the dryforest kissing bug, Rhodnius neglectus Lent (Hemiptera: Reduviidae), supports an increase of mortal-ity rate that decelerates with age. We analyzed the age-specific mortality pattern of a cohort of 250individuals of R. neglectus. We used a suite of seven models with different degrees of complexity, tomodel age-dependent forms of change in mortality rate increase in R. neglectus in the laboratory. Weused the Akaike model selection criterion to choose between models that consider absence or pres-ence of mortality deceleration. Five of the seven models (logistic, Gavrilovs, Gompertz, DeMoivre,and exponential) showed a statistically significant fit to the mortality rate. Weak late-age mortalitydeceleration in R. neglectus was supported by the best fit (logistic model), and this result is consistentwith predictions of the disposable soma theory of senescence.
with very different life histories. Holometabolous insects, such as mosquitoes (Styer et al., 2007), fruit flies (Curt- The understanding of senescence, that is, the increase in singer et al., 1992; Carey et al., 1992, 2005; Fukui et al., mortality rate with age because a decline in physiological 1993), but not a bruchid beetle (Tatar et al., 1993), have functioning, is a major goal of the research agenda in shown a pattern where mortality decelerates at older ages, evolutionary ecology (Williams, 1957; Abrams, 1993; eventually reaching a plateau. Studies in hemimetabolous Gavrilov & Gavrilova, 2001; Carey, 2003). The need to insects, including the cotton stainer (Dingle, 1966), the fully understand its underpinnings requires the use of the milkweed bug (Dingle, 1966), and two species of kissing comparative approach (Carey, 2001) which builds upon bugs (Chaves et al., 2004a,b), have shown a pattern where observations of a wide variety of organisms. Most of the studies have been focused on a small subset of species, with Kissing bugs have strict hematophagy across all of their the most comprehensive studies coming from insects, foraging ontogenetic stages, which is associated with a especially fruit flies (e.g., Carey, 2003). It has been observed mechanism that avoids the accumulation of oxidative sub- that mortality patterns can be very different across insects stances associated with aging (Grac¸a-Souza et al., 2006).
They are also vectors of Trypanosoma cruzi Chagas, theetiologic agent of Chagas disease (WHO, 1991). Thus, the *Correspondence: Jorge E. Rabinovich, Centro de Estudios Parasit- study of senescence in this group may contribute to the olo´gicos y de Vectores, Universidad Nacional de La Plata, Calle 2 No.
study of senescence as an evolutionary process and prove 584, 1900 La Plata, Prov. de Buenos Aires, Argentina. E-mail: jorge.ra- useful for pest population management. Here, we present a cohort study of the dry forest kissing bug, Rhodnius Ó 2010 The Authors Entomologia Experimentalis et Applicata 1–11, 2010 Journal compilation Ó 2010 The Netherlands Entomological Society neglectus Lent (Hemiptera: Reduviidae) focusing on the serving both as resting place and for climbing to the top at analysis of age-specific mortality and survivorship. As pre- vious studies on kissing bugs (using Rhodnius prolixus Sta˚l Each cohort was fed weekly using hens placed on a woo- and Rhodnius robustus Larrousse) were focused on the den box with holes at the bottom, through which the tops Gompertz and the positive-shaped Weibull models of the cohort jars could be tightly inserted. The insects (Chaves et al., 2004a,b) we compare these two models, climbed to the top and fed through the nylon mesh. Food which cannot account for mortality rate deceleration, to was offered for 1 h, and during the following hour the jars an additional suite of five models with various degrees of were horizontally exposed to a fan, to avoid the accumula- complexity, that can model mortality rate deceleration and tion of excessive moisture in the glass containers, which is different forms of mortality rate increase with age (Appen- common after the insects become engorged. The hens were dix A). Our aim is to test whether the instantaneous mor- sedated using an intramuscular dose of midazolam tality rate in R. neglectus decelerates with age. We compare (0.25 mg kg)1), a sedative commonly used to relax small model fitting using Akaike Information Criterion model animals before surgery. We allowed the hens to rest for selection to test the absence or presence of instantaneous 5–10 min after the sedative application, and then carefully rate of mortality deceleration in this species.
plucked them on one side on a surface not larger than adiameter of 7 cm (the jars’ mouth size). A towel was usedto restrain the hens to minimize stress from being restrained. This procedure was carried out by two people who would gently wrap the towel around the hen, letting Rhodnius neglectus is a species that has been observed in the head and legs stick out for better freedom of move- Northeast Brazil (Lent & Wygodzinsky, 1979; Carcavallo ments. This procedure was approved by the Animal Use et al., 1999; Galvao et al., 2003) and in Venezuela in the Institutional Review Board of IVIC (Venezuelan Institute State of Amazonas, until at least 1965 (Gamboa Cuadrado, for Scientific Research) and is in conformity with Venezu- 1973). This species is found in a region characterized by mean annual temperatures ranging between 19 and 30 °C After being exposed to the fan, each jar was opened to (average 23.3 °C), and a mean (±SD) precipitation of check the number of dead individuals (identified by 1 246 ± 429 mm year)1. Rhodnius neglectus has been instar); identification by sex was performed only for identified by Curto de Casas et al. (1999) to occupy the adults, and only the female population was recorded.
Holdridge Dry Forest and Very Dry Forest or Savannah Thus, data on mortality during the nymphal stages reflect a mix of the two sexes, whereas the adult stage reflects onlyfemales. In two instances, the strict weekly feeding and counting schedule could not be maintained (although The population of R. neglectus used was made available by they were never out of phase for more than 3 days), and Dr Rodolfo Carcavallo and came from the insectary of the as the biodemographic methods require a constant time Instituto Oswaldo Cruz (Rio de Janeiro, Brazil), but its unit for analysis, the recorded information was subjected exact geographical origin is not known. The study was car- to linear interpolation to keep the week as the time unit ried out in Caracas, Venezuela, in a climate room with for calculation of the age-specific mortality rates. Addi- constant conditions of temperature (26 ± 1 °C) and tionally, this procedure had to be applied at the beginning humidity (60 ± 10% r.h.). Regulation of temperature is of the cohort study (when the number of individuals was essential because of the strong effect it has on the develop- still high). This interpolation is a sound procedure, mental cycle and survival (Garcia da Silva & da Silva, because (1) when the cohorts are still in their first ages, all 1988). The ambient photoperiod was used: this did not models predict an exponential mortality, which for small need to be controlled, because at the latitude of Caracas its x values can be considered linear (due to the Taylor seasonal variation is very small (day length in December is expansion; Demidovich, 1973), and (2) the various mod- 11:29 hours, and in June 12:42 hours).
els do not discriminate mortality patterns at early ages of The design involved the follow up of five independent the cohort, that is, most of them coincide in their predic- cohorts initiated simultaneously. Each cohort was started tions for early ages, yet they can predict very different pat- with 100 recently laid (0- to 48-h old) eggs, kept in 150-ml glass containers until all viable eggs hatched. The firstinstars were transferred to 3.8-l jars, covered with nylon mesh, and with vertically placed strips of paper inside, The weekly recording of the numbers of bugs alive anddead was used to calculate, following Carey (2001), the basic parameters of a decrement life table: (1) survival as a well as a description of the models and parameters used, function of age (lx = Nx ⁄ N0, or fraction alive at age x), (2) age-specific (period) survival (px = lx+1 ⁄ lx, or fractionalive at age x surviving to x + 1), (3) age-specific (period) mortality (qx = 1 – px, or fraction alive at x dying prior to Carey et al. (1992) claim that survival curves are poorly suited for summarizing age-specific mortality patterns as The age-specific mortality, qx, is a discrete quantity, rep- compared with mortality rate curves, so we fitted models resenting mortality as the probability of dying over the to data using lx. For that purpose, the data used to fit the 1-week interval used. The continuous analog of this mea- models were converted from their initially lx format, into sure is referred to as the force of mortality or instanta- px = lx+1 ⁄ lx, and then to mortality rates by lx = -ln(px).
neous mortality rate and denoted as l(x); it is defined as All models were fitted using various procedures in the R the mortality rate representing the limiting value of the language (R Development Core Team, 2007). Because of age-specific mortality rate when the age interval to which the small number of mortality rate observations between the rate refers becomes infinitesimally short (Carey, 2001).
weeks 83 and 89, when the last bugs died, we only analyzed The age-specific mortality rate is preferred over age-spe- the first 83 weeks, because estimates of the mortality rate cific mortality, qx, because it is not bounded by unity, it is (lx) become unreliable given the small number of individ- independent of the size of the age intervals, and it is used uals that can face death (Carey et al., 1992). Despite the in numerous mortality models (Carey, 2001). The analyti- number of individuals decreasing with age, we did not cal relationship between the discrete form of mortality, qx, weight the data because the models used are non-linear and its continuous expression is l(x) = )ln px and regression models that constitute an alternative to a weigh- ing function in linear regression analyses. The resulting We first analyzed the five cohorts independently, and time series was analyzed both raw and smoothed; the rea- carried out a Friedman ANOVA and a Kendall coefficient son for smoothing before model fitting is that the data of concordance test to verify if the differences in mortality become easier to handle by the models by removing irreg- rates l(x) among cohorts were statistically significant.
ularities and inconsistencies, thus capturing better the Although mortality rates in older ages were not synchro- behavior of the mortality rate process being modeled nized, results showed that l(x) differences among (Carey, 2001). For smoothing we used the Kolmogorov– the five cohorts were not statistically significant Zurbenko adaptive filter (KZA) (Chaves et al., 2008), with (ANOVA: v2 = 5.902, P = 0.21; Coefficient of concor- parameter q = 2 (half the size of the smoothing window).
dance = 0.0168, Average rank r = 0.00547). The further The exponential model was fitted using least squares (Far- analysis of the age-specific mortality pattern was carried away, 2005) and confidence intervals obtained from the out after pooling the five cohorts, treating the data as a standard errors. For this model no differences were single cohort. The initial number of eggs in the pooled observed when compared with a weighted regression, with data was taken 250, because the average sex ratio was 0.44 weights proportional to the number of surviving individu- [$ ⁄ ($+#)] (which is not significantly different from 0.5; als. The Gavrilovs model was fitted using the Nelder–Mead two-sided t-test: P = 0.18; StatSoft, 2009). Similarly, optimization algorithm (Nelder & Mead, 1965) imple- unbiased 0.5 sex ratios have been found in other triato- mented with the R command ‘optim’, and 95% confidence mine species (Ronderos, 1972; Paz Rodrı´guez, 1996).
intervals were found using a non-parametric bootstrap Despite the five cohorts being pooled for their fitting to (Faraway, 2005) on 1 000 replications. All other models mortality rate models, the results of the individual were fitted by non-linear least squares (Bates & Watts, cohorts are also presented to provide visualization of the 1988), using the R command ‘nls’. For these models, SEs dispersion of the data of the cohorts around their pooled were computed with a non-parametric bootstrap similar to the one described for the Gavrilovs model.
In the analysis of the age-specific mortality pattern of the pool of the five cohorts, the following six mortality models were used: the exponential, deMoivre, Gompertz, To compare the performance of different models with var- Gompertz-Makeham, Weibull, and logistic models, using ious degrees of freedom (i.e., different number of fitted the algebraic presentation by Carey (2001), and a seventh parameters), we used the Akaike Criterion (AIC) (Akaike, three-parameter model based on reliability theory as pro- 1974), a method preferred over classical goodness-of-fit posed by Gavrilov & Gavrilova (2001); for simplicity the tests (such as the chi-squared or the G-tests) because it latter will be called hereafter ‘Gavrilovs’. The reasons for ‘penalizes’ models with a high number of parameters. The using a mortality analysis and not a survival analysis, as AIC evaluation of model performance (one of the most used goodness-of-fit indicators) represents the ‘informa-tion content’ of a given set of parameter estimates (Burn-ham & Anderson, 2002), and it is independent of themagnitude of the data points and of the number of obser-vations. It is calculated as: where Yobsi and Ycalci are the observed and predicted mor-tality rates for the ith observation, ‘p’ is the number ofparameters, and ‘n’ the number of observations. The smal-ler the value of the AIC, the more appropriate the model is.
Figure 1 Age-specific survival (lx) for each cohort of Rhodnius After converting the cohort age-specific survival (l neglectus and the pool of the five cohorts. Vertical dashed lines indicate the average periods of immature development. E, eggs; x and l(x), and despite the peaks of mortality in later ages 1–5, instars 1–5; A, female adults.
not being synchronized, the differences between theage-specific mortality rate l(x) among the five cohorts ofR. neglectus are not statistically significant (Table 1),indicating that the five cohorts may be pooled.
The age-specific female survival (lx) values for each cohort as well as the pool of the five cohorts are presentedin Figure 1. There is a sharp decline in survival in the firstdevelopmental stages (egg, and instars 1 and 2), then thedecline stabilizes until approximately the age of 45 weeks(about 30 weeks of adult female’s age) when another sharpdecline is observed; after age 45 there is a progressivedecline in survival. Figure 2 shows the weekly evolution ofthe age-specific mortality rate l(x) for the individualcohorts and for the pool among cohorts.
Some of the results of the fitting of the smoothed and non-smoothed data to the models are shown in Figure 3and Table 2. The Weibull model is not shown graphicallybecause it had a statistically non-significant fit to one of its Figure 2 Age-specific mortality rate [l(x)] patterns of each parameters. Figure 3 shows the predictions of the remain- cohort of Rhodnius neglectus (black dots) and the pool of the fivecohorts (white dots and solid line). The individual cohorts were ing five models (Gompertz, exponential, DeMoivre, Gavri- not identified separately to facilitate visibility.
lovs, and logistic) with observed pooled laboratory data of Table 1 Friedman ANOVA and Kendall coefficient of concor- the age-specific mortality rate l(x) of R. neglectus. The dance to test the differences in the weekly mortality rates [l(x)] parameter values of those five age-specific mortality rate sequence among the five cohorts of Rhodnius neglectus models tested are shown in Table 2 for the raw data andthe KZA smoothing. Figure 4 shows the maximum effort of egg laying. The quadratic fit for fecundity, given by the equation y = -0.0085x2 + 0.7386x - 5.4479 (r = 0.908), is shown only to ease visualization of the peak of the repro- ductive effort of R. neglectus females. The method for calculating fecundity (eggs ⁄ female ⁄ week) is given in Rabi- novich (1972). It is striking how the peak of reproductive ANOVA v2 = 5.902, d.f. = 4, P = 0.21; n = 88.
effort of R. neglectus coincides with the second early peak Coefficient of concordance = 0.0168, average rank r = 0.00547.
In recent years, some striking lack of fit to the Gompertzmodel has been found in large cohorts of experimentalanimals, primarily insects (Partridge & Mangel, 1999). Theresults of our study show weak evidence for late-age mor-tality deceleration in R. neglectus, in contrast to patternsreported for several Diptera, including the fruitfly Dro-sophila melanogaster Meigen (Curtsinger et al., 1992;Pletcher & Curtsinger, 1998), the medfly Ceratitis capitata(Wiedemann) (Carey et al., 1992, 1998), the mexfly Ana-strepha ludens (Lo¨w) (Carey et al., 2005), and the mosquitoAedes aegypti (L.) (Styer et al., 2007). Rhodnius neglectusage-specific mortality pattern seems to be more similar tothat reported for other kissing bugs such as: R. prolixus Figure 3 Age-specific mortality rate l(x) of observed laboratorydata (as pool of five cohorts) of Rhodnius neglectus and the pre- (Chaves et al., 2004a), R. robustus (Chaves et al., 2004b), dictions of the DeMoivre, exponential, Gompertz, logistic, and the milkweed bug, Oncopeltus fasciatus (Dallas) (Dingle, Gavrilovs models, for the smoothed data applied to l(x).
1966), the cotton stainer, Dysdercus fasciatus Signoret(Dingle, 1966), and the bruchid beetle Callosobruchusmaculatus (Fabricius) (Tatar et al., 1993). This result is To determine the model that represents best the age- consistent with the equilibrium predictions of the antago- specific mortality rate data of R. neglectus, the results of the nistic pleiotropy and mutation accumulation models of application of the Akaike model selection criterion to the senescence (decline in physiological functioning with age; seven mortality rate models are shown in Table 3 for the Abrams & Ludwig, 1995). These models are known as the KZA smoothed l(x) data. From the models that con- ‘disposable soma’ theory for the evolution of senescence formed significantly to the data (the Weibull and GM (Kirkwood & Rose, 1991). This theory states that senes- models did not have a significant fit), the logistic and the cence arises from an optimal balancing of resources Gompertz conformed best, and DeMoivre and exponential between reproduction and somatic repair. For instance, the worst. Similar results (not shown) were obtained with model simulation showed that only when reproductive output gradually approaches, but never reaches, a maxi-mum value as the amount of diverted resources increases,senescence is consistent with the Gompertz model(Abrams & Ludwig, 1995).
Table 2 Parameter values of the five age-specific mortality rate [l(x)] models (the Weibull and Gompertz-Makeham models were notincluded because their fit to the laboratory data was not statistically significant) ‘p’ is the number of parameters of each model. In brackets are the 95% confidence intervals; intervals that do not contain zero are statistic-ally significant with P<0.05, and have been presented graphically in Figure 3. KZA refers to the Kolmogorov–Zurbenko adaptive filtersmoothing method. The models are presented in increasing order of their number of parameters.
found in the beetle C. maculatus (Tatar et al., 1993) andthe kissing bug R. prolixus (Sulbaran & Chaves, 2006).
Williams (1957) claimed that greater rates of ‘extrinsic’(age- and condition-independent) mortality favored morerapid senescence. However, Abrams (1991) showed thatthe effects of the ‘extrinsic’ mortality differentially affectthe rate of senescence according to density-dependence.
Furthermore, Abrams (1991) showed that mortality pat-terns, contrary to Williams (1957) predictions, are possiblewhen density-dependence is present, and acts primarily onthe survival or fertility of later ages, or when most of thevariation in mortality rates is due to variation in non-extrinsic mortality. No experimental evaluation exists ondensity-dependent processes in R. neglectus. However,Rodrı´guez & Rabinovich (1980) showed in R. prolixus that Figure 4 Age-specific survival of the pool of five cohorts of density had a significant effect on the developmental rate Rhodnius neglectus from the egg stage and the maternity function(eggs ⁄ female ⁄ week) as an indicator of egg laying effort. The of second, third, and fourth instars, but not on first instars, maternity function was fitted to a quadratic equation (y = adult survivorship, instantaneous population parameters, )0.0085x2 + 0.7386x - 5.4479, r = 0.908) only to ease visualiza- or age-specific parameters. If similar patterns were present tion of the peak of the reproductive effort of R. neglectus females.
in R. neglectus, then this would confirm that in this speciesWilliams’ (1957) rapid senescence may be a response to Our results fit these predictions: we found a sharp greater rates of ‘extrinsic’ mortality.
decline in survival in the early developmental stages (egg In R. neglectus there is no apparent reduction of the and instars 1 and 2), then the decline stabilizes at the age of length of reproductive life as a consequence of high early 45 weeks (about 30 weeks of adult female’s age) when fecundity (J.E. Rabinovich, unpubl.), unlike R. prolixus another sharp decline is observed, possibly related to the (Sulbaran & Chaves, 2006) and several strains of D. mela- maximum effort of egg laying; after 45 weeks there is a nogaster (Giesel & Zettler, 1980; Partridge, 2001). On the progressive decline in survival. These two declines in l contrary, R. neglectus shows a clear exponential increase in correspond of course to the two strong earlier peaks of death rate in older ages, suggesting a Gompertz behavior, mortality rate l(x) as shown in Figure 3. Figure 4 further and possibly reflecting a minor effect of the trade-off supports the relationship between maximum effort of egg between reproduction and somatic repair, confirming a laying and the second early peak in the mortality rate. This possible larger effect of ‘extrinsic’ mortality.
apparent relationship between egg-laying effort and effects In contrast with the Gompertz model, in the Weibull on the age-specific mortality rate is similar to the one model the causes of death for young adults and old indi-viduals are different, independent, and additive, andinclude death causes due to catastrophic intrinsic sourceswith a probability that increases with age, and not as a Table 3 Relative ranking (1 = best, 7 = worst) of the fit of seven physiological function that declines with age as in the mortality rate models to the KZA smoothed laboratory data of Gompertz model (Gavrilov & Gavrilova, 2001; Ricklefs & Rhodnius neglectus using Akaike model selection criterion (AIC) Scheuerlein, 2002). Partly for these reasons, the Gompertzmodel is considered by most demographers as an empiri- cal model rather than a ‘law’ (Carey, 2003). Despite the Weibull function being usefully used in conjunction with failure-time models (in which failure depends on the occurrence of one or more rare events, such as genetic mutations or cell deaths; Ricklefs & Scheuerlein, 2002), the fit of this model to the R. neglectus mortality rate data was The only other instance of application of mortality Asterisks identify those models that had a statistically significant models to kissing bug species are the studies of Chaves et fit by the criterion of the 95% confidence intervals with the boot- al. (2004a,b). Only the Gompertz model was tested on the age-specific mortality rate of R. robustus, and the Gom- pertz and Weibull models were tested on the age-specific Triatoma infestans Klug, Triatoma brasiliensis Neiva, and mortality rate of R. prolixus. The results from these two Triatoma pseudomaculata Correia & Espinola, and Pan- studies are different to those of R. neglectus in terms of the strongylus megistus Burmeister was 37.5 and 45.9%, respec- parameter values of both models. This may be partially tively. The adaptation to a domestic environment causes a due to demographic evolution of these taxa: R. prolixus has similar effect on triatomines (Forattini, 1980) because a population intrinsic rate of natural increase (rm) of about conditions inside rural houses provide smaller predation 0.032 per week, almost double that of R. neglectus of 0.017 risks, smaller fluctuations of temperature and relative per week (J.E. Rabinovich, unpubl.) reflecting their adap- humidity that exclude deleterious extreme values, and bet- tation as results of genetic selection to different environ- ter chances of feeding. The values of the age-specific mor- ments. This result also confirms the soundness of the tality rate parameters estimated under domiciliary assumptions of Charlesworth (2000), supporting the use conditions will prove of importance in the epidemiology of rm to explain the evolution of senescence and life-his- of Chagas disease, and even decisive in the design of pest The weak late-age mortality deceleration in R. neglectus In a recent review of mammal and bird comparative was observed only through the fit of the logistic model of analyses of survival senescence by using life tables, Jones et mortality, the model selected as ‘best’ by the Akaike model al. (2008) have identified generalizations including the selection criterion. However, this late-age mortality decel- observation that mammals senesce faster than similar- eration process is extremely sensitive to the sample size of sized birds. Furthermore, McCoy & Gillooly (2008) devel- the cohort study, and our initial cohort size of 250 individ- oped a model of natural mortality (relating body size and uals was probably too small to permit a strong assertion temperature to biological rate processes) and tested it with about late-age mortality deceleration in R. neglectus. As extensive field data from plants, invertebrates, fish, birds, claimed by Gaines & Denny (1993), when mortality rate is and mammals; their results indicate that much of the het- close to zero it behaves as a threshold character; and to esti- erogeneity in natural mortality rates can be predicted, mate the points ‘below threshold mortality’ becomes diffi- explicitly and quantitatively, despite the many extrinsic cult and their measures are usually relatively inaccurate sources of mortality in natural systems, suggesting that (Promislow et al., 1999). Carey (2003) considers that an common rules govern mortality rates. We believe that estimate of the threshold mortality can be taken as 1 ⁄ N, comparative review is due for different orders of insects.
where N is the initial cohort size; in the case of R. neglectus Such a review not only has academic value, but it would the initial cohort size of 250 individuals would indicate a also impinge on important pest population management threshold mortality of 0.004, and mortality rate values and epidemiological areas. New age-dependent models of below this threshold exist in our laboratory data, both raw pathogen transmission show the importance of age and and smoothed. An increase in the initial cohort size is diffi- population age structure to transmission dynamics (Styer cult for kissing bugs, with a complex feeding behavior, the et al., 2007). These authors claim that, in the case of mos- need of large containers to avoid problems of overcrowd- quitoes, a departure from the age-independent mortality ing, and requirements of important numbers of adequate paradigm is essential for an accurate understanding of and live avian or mammal hosts for feeding.
mosquito biology and pathogen transmission.
However, we strongly suggest future studies with kissing Environmental heterogeneity, and particularly high bugs to consider this threshold mortality factor, which will environmental uncertainty (stochastic effects) affects envi- allow confirmation of whether hemimetabolous insects do ronmental circumstances that are very important in not decelerate their mortality at old ages, as our results insects, such as encounter rates with suitable oviposition suggest. As stated by Chaves et al. (2004a) in relation to sites, food availability, physiological state (e.g., reserves for R. prolixus, the mortality rate of this triatomine vector has producing oocytes, egg maturation rate, and somatic epidemiological importance through the demography of maintenance costs), and expected reproductive success its populations. In the case of R. neglectus, despite being that can lead to different patterns of behavior and rates of essentially sylvatic, it is considered to be in the process of mortality and reproduction (Partridge & Mangel, 1999), becoming a truly domestic species (Lent & Wygodzinsky and they should be incorporated in future studies for 1979). It has been shown in triatomines that important understanding senescence in insects.
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table are directly related: survival as a function of age (lx = Nx ⁄ N0), and age-specific mortality [qx = 1 – Pletcher SD & Curtsinger JW (1998) Mortality plateaus and the (lx+1 ⁄ lx)]. Thus we could have modeled survival (lx) or evolution of senescence: why are old-age mortality rates so mortality [qx; actually the instantaneous mortality rate, denoted as l(x); that is, the rate representing the limiting Promislow DEL, Tatar M, Pletcher S & Carey JR (1999) Below- value of the age-specific mortality rate when the age inter- threshold mortality: implications for studies in evolution, val becomes infinitesimally short]. There are several rea- ecology and demography. Journal of Evolutionary Biology 12: sons why we preferred to use a mortality analysis and not R Development Core Team (2007) R: A Language and Environ- the method of survival analysis: (1) survival analysis is ment for Statistical Computing. R Foundation for Statistical based upon models like Kaplan–Meier or Cox, which esti- Computing, Vienna, Austria. http: ⁄ ⁄ www.R-project.org ⁄ mate the death hazards but do not reflect any mechanismor process underlying the fitted curves as the models we use do (survival models can be viewed as ordinary regres- sion models in which the response variable is time), (2) survival rates are mainly important for projections or where lD(x) denotes the DeMoivre hazard function. The prognosis, and to estimate the importance of cause-spe- hazard rate increases towards infinity as x approaches the cific survival (e.g., to rank the effects of different causes of maximum attainable age x. This maximum age implies mortality), which is not the goal of our analysis, (3) when that the age-specific mortality rate increases faster than in no truncation or censoring occurs, the Kaplan–Meier the Gompertz model (see below), and in particular as x curve is equivalent to the empirical distribution, and (4) approaches x, when the mortality rate tends to approach we were directly interested in the mortality and not the In relation to the latter, despite the fact that mortality and survival are intimately related, there is an important Gompertz (1825), cited in Olshansky & Carnes, 1997) set difference: death is an event whereas survival is a ‘non- forth what is now recognized as the law of mortality. He event’, that is, the absence of the mortality event (Carey, noted that the differences between the common logarithm 2001). The advantage of the survival analysis method to of the number of individuals living in successive equal age deal with censored observations would not have improved intervals were almost identical during a significant portion our analysis, for in our study there were no censored of their life span, in general at least after the age of sexual observations (defining censored observations, as it is com- maturity or another predetermined age, such as the maxi- monly used in survival analysis, as those individuals who mum reproductive effort (Carey, 2001, 2003). The mortal- are lost to follow up being still alive or that remained alive ity rate (lx) function in Gompertz’s model has the when the study ended). Although we deleted observations between weeks 83 and 89 because the mortality rate (lx)becomes unreliable given the small number of individuals that can face death at those ages (see section ‘Fitting mod- where ‘a’ is a parameter representing the initial mortality els to data’ under ‘Materials and methods’), those observa- rate, and ‘b’ is the Gompertz parameter that represents the tions cannot be considered censored under our definition, rate of increase of mortality with age (also called the Gom- Makeham (1860) refined Gompertz’s law of mortality, noting that the logarithms of the probabilities of living This is the simplest mortality model that assumes that the from Gompertz’s formula, instead of proceeding in uni- mortality rate grows linearly with age x. The formula for form geometrical progression, increased at a faster pace at higher ages than at younger ages. He solved this problem by adding a ‘constant’ term, and redefining the Gompertz law as ‘the probabilities of living, increased or diminishedin a certain constant ratio, from a series whose logarithmsare in geometrical progression’. Later Makeham (1867) modified Gompertz’s formula by developing ‘a theory of The mortality rate in the DeMoivre model (1725, cited in partial forces of mortality’. The Gompertz-Makeham Kohler & Kohler, 2000) attempts to describe mathemati- cally the mortality patterns assuming that ‘the number oflives existing at any age is proportional to the number of years intercepted between the age given and the extremity where ‘c’ is the additional parameter introduced by Make- ham to represent the additional constant rate of mortality.
where x is the maximum attainable age in the population.
As with the Gompertz model, the Weibull model incorpo- Then the hazard rate, or the age-specific mortality rate at rates a minimum mortality rate suffered by young adults prior to the onset of their physiological decline. This isusually referred to as the initial mortality rate (l0). After this initial mortality the aging-related mortality rate For this model mortality rates level off at older ages. This increases exponentially as a multiple of the initial mortality model is similar to a frailty model sensu Vaupel et al.
l0. In the Weibull model, the aging-related component of mortality is a power function of age that is added to theinitial mortality rate: Gavrilov & Gavrilova (2001) proposed a general model for aging based on reliability principles. In this model it is where ‘a’ and ‘n’ are the parameters of the Weibull model.
assumed that mortality rate [l(x)] comes from the deteri- The value of ‘a’ determines the scale, and ‘n’ determines oration of redundant subunits (n) organized in blocks (m) the shape of the curve. The Weibull hazard function that compose individuals, at a constant failure rate (k), increases if n>0, decreases if n<0, and is constant if n = 0.
where the probability of an element being initially func-tional can vary from fully functional (q = 1) to highly likely to be unfunctional (q $ 0). In this model the initial The logistic model has a Gompertz-type mortality rate as a number of functional elements has a binomial distribu- baseline hazard, with hazard function (Carey, 2001): Only the three-parameter version of this model was tested (where m remains constant, i.e., m = 1).

Source: http://www.ecopaedia.com.ar/cv/pdf/Age_specific_mortality.pdf

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