Fig 1.
The lumped age-class model of mosquito life history.
Mosquitoes are divided into four life stages: egg, larva, pupa and adult. The durations of the sub-adult stages are TE, TL and TP for eggs, larvae and pupae, respectively. Sex is modeled at the adult stage, with half of pupae developing into females and half developing into males. Daily mortality rates vary by life stage—μE, μL, μP and μA for eggs, larvae, pupae and adults, respectively. Density-dependent mortality occurs at the larval stage and is a function of the total number of larvae, NL. Females mate once upon emergence, and retain the genetic material from that mating event for the remainder of their lives. Males mate at a rate equal to the female emergence rate. Females lay eggs at a rate, β.
Table 1.
Demographic and life history parameters for Aedes aegypti mosquitoes.
Table 2.
Kinship categories, sampled life stages, sampling times, and probability symbols used in close-kin mark-recapture analysis.
Fig 2.
Schematic representation of parent-offspring kinship probabilities.
Parameters and state variables are as defined in Table 1 and §2.1. Subscript 1 refers to the parent (blue), and subscript 2 refers to the offspring (purple, the perspective from which probabilities are calculated). Circles represent living individuals and squares represent sampled individuals. Parents are sampled on day t1, eggs are laid on day y2, and offspring are sampled on day t2. Offspring kinship probabilities are the ratio of the expected number of surviving offspring from a given adult on day t2, and the expected number of surviving offspring from all adult females on this day. The expected number of surviving offspring from all adult females requires considering days of egg-laying consistent with larval ages at sampling in the range [0, TL) (for larval offspring) (A), or adult ages at sampling in the range [0, TA) (for adult offspring) (C). Calculating the expected number of surviving larval offspring on day t2 from an adult female requires considering days of egg-laying, y2, consistent with maternal ages at egg-laying in the range [0, TA), and with larval offspring ages at sampling in the range [0, TL) (B). For father-adult offspring pairs, this requires considering days of mating, ti, and egg-laying, y2, consistent with maternal ages at egg-laying, and paternal and adult offspring ages at sampling in the range [0, TA) (D).
Fig 3.
Schematic representation of sibling kinship probabilities.
Parameters and state variables are as defined in Table 1 and §2.1. Subscript 1 refers to the reference sibling (blue), and subscript 2 refers to the sibling from whose perspective the probabilities are calculated (purple). Circles represent living individuals and squares represent sampled individuals. The reference sibling is sampled on day t1 and laid on day y1. Sibling 2 is sampled on day t2 and laid on day y2. Sibling kinship probabilities are the ratio of the expected number of surviving siblings of a given individual on day t2, and the expected number of surviving offspring from all adult females on this day. Calculating the expected number of surviving larval full-siblings of a larva requires considering days of their mother emerging as an adult, ti, and of egg-laying, y1 and y2, that are consistent with maternal ages at egg-laying in the range [0, TA), and with larval ages at sampling in the range [0, TL) (A). Calculating the expected number of surviving larval half-siblings of a larva requires considering days of their father emerging as an adult, tj, their mothers emerging as adults, ti and tk, and of egg-laying, y1 and y2, that are consistent with paternal ages at mating in the range [0, TA), maternal ages at egg-laying in the range [0, TA), and larval ages at sampling in the range [0, TA) (B).
Fig 4.
Kinship probabilities versus time between samples.
Selected kinship probabilities are depicted as a function of time between samples, t2 − t1. Parent-offspring kinship probabilities are the probability that an individual sampled on day t2 is an offspring of a given adult sampled on day t1. Sibling kinship probabilities are the probability that an individual sampled on day t2 is a sibling of a given individual sampled on day t1. Each probability is calculated as the reproductive output having that relationship divided by the total reproductive output of all adult females in the population, as described in Table 2. The modeled population consists of 3,000 adult Ae. aegypti with bionomic parameters listed in Table 1.
Fig 5.
Sampling schemes to estimate NA and μA for Ae. aegypti.
Violin plots depict estimates of NA and μA for sampling scenarios described in §3.1. The simulated population consists of 3,000 adult Ae. aegypti with bionomic parameters listed in Table 1. Boxes depict median and interquartile ranges of 100 simulation-and-analysis replicates for each scenario, thin lines represent 5% and 95% quantiles, points represent outliers, and kernel density plots are superimposed. The default sampling scheme consists of 1,000 individuals sampled as ca. 11–12 individuals per day over three months. In panels (A-B), sampled larval, adult female and adult male life stage proportions are varied in 25% increments and limited to scenarios where the number of sampled adult females exceeds the number of sampled adult males (e.g., “75F25M” represents a sample consisting of 75% adult females and 25% adult males, and “50F25M25L” represents a sample consisting of 50% adult females, 25% adult males, and 25% larvae). The case of 100% sampled pupae is also included. In panels (C-D), all sampled individuals are adult females, and four sampling frequencies are considered—daily, biweekly, weekly and fortnightly. In panels (E-F), biweekly sampling is adopted, and sampling durations of 1–4 months are explored. In panels (G-H), a sampling duration of three months is adopted, and total sample sizes of 500, 1,000, 1,500 and 2,000 adult females are explored.
Fig 6.
Pseudo-likelihood components to estimate NA and μA for Ae. aegypti.
Violin plots depict estimates of NA (A) and μA (B) for the optimal sampling scheme determined in Fig 4 (1,000 adult females collected biweekly over a three month period, i.e., ca. 40 adult females per collection) and various included pseudo-likelihood components. The simulated population consists of 3,000 adult Ae. aegypti with bionomic parameters listed in Table 1. Boxes depict median and interquartile ranges of 100 simulation-and-analysis replicates for each scenario, thin lines represent 5% and 95% quantiles, points represent outliers, and kernel density plots are superimposed. Adult parameter estimates inferred from combined parent-offspring and full-sibling pseudo-likelihood components are more accurate than those inferred from either pseudo-likelihood component in isolation and more accurate than those inferred by inclusion of half-sibling pairs. In panels (C-D), parent-offspring and full-sibling pseudo-likelihood components are used and cases of biweekly sampling and sampling every two days are compared, both where the sampling day is known and where the sampling day is only known within the interval between samples.
Fig 7.
Sampling schemes to estimate μL and TL for Ae. aegypti.
Violin plots depict estimates of μL and TL for sampling scenarios described in §3.2. The simulated population consists of 3,000 adult Ae. aegypti with bionomic parameters listed in Table 1. Boxes depict median and interquartile ranges of 100 simulation-and-analysis replicates for each scenario, thin lines represent 5% and 95% quantiles, points represent outliers, and kernel density plots are superimposed. The default sampling scheme consists of 1,000 adult females and supplemental larvae sampled daily over a three month period. In panels (A-B), total larval sample sizes of 500, 1,000, 2,000 and 4,000 are explored. In panels (C-D), a larval sample size of 4,000 is adopted, and four sampling frequencies are considered—daily, biweekly, weekly and fortnightly. In panels (E-F), biweekly sampling is adopted, and sampling durations of 1–4 months are explored. The optimal sampling scheme consists of 4,000 larvae and 1,000 adult females collected biweekly over a three month period. In panels (G-H), the optimal sampling scheme is adopted and cases of biweekly sampling and sampling every two days are compared, both where the sampling day is known and where the sampling day is only known within the interval between samples.
Fig 8.
Application of CKMR methods to infer intervention impact.
CKMR methods are applied to a simulated population of 3,000 adult Ae. aegypti with bionomic parameters listed in Table 1 and the optimal sampling scheme for estimating adult parameters (a total of 1,000 adult females sampled biweekly over three months). Fogging is simulated as an intervention that elevates adult mortality. In panels (A-B), violin plots depict estimates of NA and μA for fogging-induced adult mortality rates of 0.01–0.10 per day (increased in 0.01 per day increments). Boxes depict median and interquartile ranges of 100 simulation-and-analysis replicates for each intervention-induced mortality rate, thin lines represent 5% and 95% quantiles, points represent outliers, and kernel density plots are superimposed. In panel (B), actual intervention-modified adult mortality rates are denoted by red lines. Results depict a pattern of increasing estimated μA and decreasing estimated NA in response to increasing fogging-induced mortality rates. In panel (C), the statistical power to detect an increase in μA or a decrease in NA is depicted, assuming a type I error rate of 5%.