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Scoping

A scoping exercise was carried out to assess the feasibility of a synthesis of MR studies with respect to survival. A query of the Guerra et al. [9] database indicated the number of mark–recapture studies with or without known-age releases, which is presented in Table 1. Though known-age studies are a minority, 51 were available with potential information. The set of ages-at-release in any study was usually small: single-age release, more occasionally two or three. A prime requirement was that MR data were available that could be put into the form of a capture history matrix or m-array (these data structures are explained further in the section hereafter). Scoping found that publications never reported the capture history matrix and rarely the full m-array but often reported m-array information directly in a tabular form or indirectly in graphical form. Single-release experiments were much more common and would usually yield information for a single-row m-array. Recapture was usually carried out with similar apparatus and effort from occasion to occasion. Mosquitoes were usually killed on recapture; experiments with re-release of recaptured mosquitoes were very unusual. The exercise concluded that there were no strong obstacles to a pooled analysis of studies making use of an age-dependent CJS model, and the assumption of time-independent capture probability was defensible.

Data

The ‘capture history’ of the cohort(s) can be put in matrix form. This mark–recapture matrix forms a summary of the set of individual capture histories in the experiment, of which there are ({2}^{T}-1) possibilities where T is the number of recapture occasions. Under the assumptions of mark–recapture, an even more concise summary is provided by the ‘reduced m-array’ (henceforth ‘m-array’), which counts the numbers of mosquitoes released at i and next caught at j, without regard to the capture history prior to i or subsequent to j. The m-array is the usual form of reporting recapture data in publications.

When only a single release is carried out, the data reported can be structured as a single-row m-array, and this is the most common format. In some experiments, the release and/or recapture occasions are irregularly spaced.

An example m-array is shown here from Takagi et al. [14], in which three cohorts of 4-day old marked mosquitoes were released on three successive days (days 22, 23 and 24):

The first to penultimate columns show a set of counts ({m}_{ij}) of animals released at occasion i and subsequently caught at occasion j. The final column shows the numbers released but not re-caught again over the course of the experiment. In this example, the final column was calculated from the supplied numbers released and numbers re-caught. No recaptures were attempted on days 32 and 35, so the columns are empty.

A much fuller expression of the study information contains m-arrays by age (the ‘full’ or ‘generalised’ m-array, McRea and Morgan [13]). This structure is a set of counts with ({m}_{ij}left(aright)) denoting the number of individuals which, when released on day i at age a days, are next captured at j. Accompanying it is an array ({R}_{i}left(aright)), which is the number of individuals released on day i of age a days. The generalised m-array is almost never reported directly but could in principle be surmised from the study report. A reduced m-array for an experiment with cohorts of known age can be expressed as a generalised m-array.

In the example above, the cohorts released were all of the same age on each occasion (multiple release, single age). An alternative is to release multiple ages simultaneously (single release, multiple ages), for which a full m-array data structure is required. Harrington et al. [15] simultaneously released ‘young’ and ‘old’ cohorts (3 and 13 days old, respectively) in Puerto Rico. The data can be formed to give the following R-array.

and generalised m-array:

Finally, different ages may be released at each occasion (single or multiple release and multiple age). For example, Eldridge and Reeves [16] released cohorts of ages 5, 1 and 2 on days 1, 4 and 7, respectively.

The MR and release data were extracted from the original studies and then entered into R as arrays.

A study may report more than one MR experiment, with releases separated in time and space. For example, Reisen et al. [17] reported separate experiments carried out in different months. When release cohorts overlap, for our study, a multiple-release m-array was formed where possible. Recaptures from separate releases were sometimes accumulated in the source publication in such a way that they could not be disaggregated and had to be treated as a single-release experiment, e.g. in Ref. [18].

Other aggregation or conditioning factors include experimental site, sex, mosquito species and age. For example, Reisen et al. [17] released and recaptured male and female cohorts of Cu. tarsalis and Cu. quinquefasciatus. However, the recapture data were only reported fully for Cu. tarsalis females. Table 2 presents a brief summary of study and dataset-level characteristics.

Searching

The references identified by Guerra et al. [9] were supplemented by a much smaller set of references collected ad hoc by the author (n = 26), and combined with a Web of Science search from 2014 to date [title terms: mosquito AND (surviv* OR longevity OR mortality)] to create the complete reference set, n = 188.

A flow diagram of the search and selection process is provided in Additional File 1: Supplementary Fig. S1 (Appendix S2). Studies in which a survival-related experimental negative intervention (e.g. genetic modification) was apparent from the title were excluded. Studies in which age of release was unknown were excluded. For example, Takken et al. [19] captured mosquitoes from houses with aspirators prior to marking, so the ages of these adults at release were not known. Studies without useable MR information were excluded. For example, the authors of Marini et al. [20] reported recapture numbers in aggregated form, e.g. in the first experiment as 2–5, 6–9 and 10–21 day totals; these data could not be put into the usual m-array form by recapture day, and the study was therefore excluded.

None of the selected datasets were exclusively concerned with males, and the majority contained female-specific MR information, so the analysis followed Ref. [9] in confining results to females. Similarly, the vast majority of datasets were in the three genera Anopheles, Aedes and Culex; other genera were excluded.

Studies were then filtered according to conditions established by simulation (details below). For example, Takagi et al. [14] released three laboratory-reared cohorts 4–5 days after emergence. Criteria established on the basis of simulation results excluded this single-age study because of the older age of the mosquito cohorts and the inadequate size (< 500) of the release cohorts (296, 161 and 249).

After exclusions, there were 73 MR datasets with ages known at release and, from these, 30 datasets of female mosquitoes with suitable MR information and experimental characteristics. The references supplying the final datasets are listed in Additional File 2.

Analysis

After the selection of studies described above, analysis is carried out in two stages. In the first stage, the parameters of a mark recapture model are estimated for each selected study, which include survival and capture parameters. The capture parameters have a modelling function, but the survival parameters are of primary interest. Study-specific capture probabilities are ascribed to each study, allowing study characteristics (experimental design, local conditions, etc.) to influence the data. For example, a study in which recapture uses baited recapture is allowed a different (probably higher) recapture probability to one that does not. In this way, important heterogeneity is modelled. In the second stage, the EL and its variance are estimated from the survival parameters, and this outcome is analysed by conventional meta-analysis.

In the first stage, each study is analysed using the CJS model. In our analysis, the Weibull survival curve determines the values of the discrete survival parameters in the CJS model, so the parameters in the likelihood are reduced from a potentially large set of discrete survival parameters to the small set of Weibull parameters that they map to. A summary of symbols used is presented in Table 3.

Analysis uses the age-specific CJS likelihood equation [13, p. 74] written here as:

where ({R}_{i}left(aright)) and ({m}_{ij}left(aright)) are data arrays with examples given in the Data section, and for a mosquito of age a when released at occasion i, ({nu }_{ij}left(aright)) is the probability of next recapture at j, and ({chi }_{i}left(aright)) is the probability, of not being caught afterwards, so (chi_{i} left( a right) = 1 – mathop sum limits_{j} nu_{ij} left( a right).)

This is a multinomial likelihood and, leaving age aside for the purposes of explanation, includes:

• the probability of no recapture (({chi }_{i})) raised to the power of the numbers not recaptured (({R}_{i}-sum {m}_{ij})); hence, the first term, and

• the probabilities of recapture (({nu }_{ij})) raised to the power of the number of recaptures (({m}_{ij})); hence, the second term.

The parameters in the current analysis are relatively simple compared with the general form: p is the probability of capture on any recapture occasion, which is assumed time-independent, and (underset{_}{phi }) is a vector of probabilities, with element (phi left[kright]) the probability of surviving from age k to k + 1.

Then,

Conventionally, discrete survival probabilities (({phi }_{k})) are used in MR analyses (see Ref. [13]). The analysis for age-dependence when interest lies in discrete age classes is set out by Pollock et al. [21], as is common in some fields (e.g. birds: immature and mature). Analysis with many age classes requires many parameters and associated limits on precision. Parametric age-dependence on a continuum has been additionally utilised in this paper because it provides a more compact parameterisation and potentially increased precision. The parameter vector for an individual study under the discrete survival formulation (with a time-independent capture model) is (left(p,{phi }_{1},…{phi }_{k}…right)), whereas under the compact parameterisation it is (for the Weibull survival model) (underset{_}{theta }=left(p,alpha ,eta right)).

A Weibull survival model is assumed with shape and scale parameters (alpha) and (eta). There are several textbook parameterisations of the Weibull, and the one adopted here corresponds to that coded in R. Note that the symbol for the Weibull scale parameter ((sigma)) in the R parameterisation is replaced in this paper with (eta) because (sigma) is also commonly used for measures of dispersion. The Weibull distribution is fairly flexible though monotonic, and it includes the exponential as a special case when (alpha =1).

For the Weibull, the continuous survival function is:

The conditional survival over a time step is (Sleft(k+1right)/Sleft(kright)) [22, p. 31], so the equation:

connects the continuous survival model with the discrete apparent survival of the CJS, in which (phi left(kright)) represents the probability of an animal alive at age (k) surviving to (k+1).

Weibull parameters are restricted to (alpha >0) and (eta >0). These constraints were implemented by numerical fitting with the Nelder–Mead method on the transformed variables (text{log}left(alpha right)) and (text{log}left(eta right)).

The EL of a mosquito is given by (int Sleft(tright)text{d}t) and has an analytic solution for the Weibull model for known parameter values. To incorporate the parameter uncertainty in estimates of (alpha) and (eta), further analysis is required. The calculation in this paper of the variance of the conditional mean of the EL under a Weibull model is described in Additional File 1: Appendix S3.

Meta-analysis was carried out using the metafor package in R. The meta-analysis on expected lifetimes used a log transformation for this positive-value outcome, with inverse-variance weighting. The variance of the log-transformed EL was approximated using the ‘delta method’, that is:

The pooled estimate from the meta-analysis used a random-effects model to account for heterogeneity, which incorporates extra ‘between-study’ variation in the estimates.

Each study receives its own (constant) capture probability, which means there are as many capture parameters as studies; however, it is the survival parameters that are of primary interest and the capture parameters serve a modelling function only. In the analysis with genus as a moderator, there is an average for each group (e.g. for genus Anopheles) shared by those studies.

Three sensitivity analyses were carried out with alternative constraints:

1. 1.

   For the overall model, with 0 < p < 0.05 and (alpha ge) 1. The analysis asserts increasing or constant mortality with age, and rules out capture probabilities > 0.05, which may be implausible.

2. 2.

   For the genus-specific model, 0 < p < 0.05 and (alpha) >0.1. The boundary on low values of (alpha) is a practical step to help avoid numerical difficulties, as discussed elsewhere.

3. 3.

   For the genus-specific model, a meta-analysis excluding any studies with (alpha <1), where simulations showed estimation, produced a high root mean square error (RSME) (Additional File 1: Appendix S4).

Simulations

Simulations of known-age MR experiments were carried out using known parameter values and known age, with time-independent capture probability and age-dependent survival. Four Weibull-derived survival models were used to generate simulated data, one exponential ((alpha =1)), two with larger shoulders and increasing mortality with age ((alpha >1)) and one where mortality fell with age ((alpha <1)). These survival curves are shown in Additional File 1: Supplementary Fig. S2 along with their parameter values. The capture probability was set to 0.01 throughout, and there were 1000 runs in each scenario. When summarising scenarios, simulated data were trimmed where (widehat{alpha }) > 30 or (widehat{eta })> 30. The proportion of simulations with these outliers was 0.13.

In broad terms, the simulations showed that bias and variance reduces with more recapture occasions and larger cohort sizes, with younger release ages, and with more releases. The following inclusion criteria were adopted, when mosquitoes are released at known age, to give broadly accurate estimates (details below): releases at young age (le 3), a sufficient number of recapture occasions (ge 8) and

1. 1.

   With single release, cohort size (Rge 1000)

2. 2.

   With multiple releases, (Rge 500)

The heuristic reasoning for allowing smaller cohort sizes for multiple release experiments (500 versus 1000 for single release) is that the resulting loss of efficiency from the smaller cohorts is somewhat balanced by further releases made within the same experiment. Studies that did not meet the inclusion criteria were excluded from the meta-analysis (see Appendix S2).

The statistical performance of the main outcome of interest in the present study (text{EL}), along with results for (alpha) and (eta), is summarised in Appendix S4. Under the inclusion criteria, it can be seen that the bias of (widehat{text{EL}}) is low (magnitude (lesssim) 0.5). Furthermore the RMSE of EL is rather smaller than the RMSE of (alpha) when (alpha gtrsim 2) (scenarios a and b). However, when (alpha <1) (scenario d), the RMSE of (widehat{text{EL}}) is large. The main conclusion of the simulations is that the bias of (widehat{text{EL}}) may be reduced to low-moderate levels by the inclusion criteria but that the RMSE of (widehat{text{EL}}) is especially high when (alpha <1). This is the region where Weibull variables are inherently most variable, and any estimates can be very imprecise.