Time series analysis of survival and oviposition cycle duration of Anopheles funestus (Giles) in Mozambique

Study site

The village of Furvela has been described in previous publications of the project (Charlwood, 2011, 2017; Charlwood & Braganca, 2012; Charlwood et al., 2021; Kampango et al., 2013). This is a subsistence farming community, with the main crops being maize, manioc, peanuts, and beans. The single rainy season extends from October to March, with approximately 1,200 mm of rain. Daily mean temperatures, recorded at the nearby site of Vilanculos vary between 18 °C (July) and 30 °C (December) (Charlwood, 2017).

Plasmodium falciparum malaria was highly endemic with infection rates about 80% in infants at the health post (Charlwood et al., 2021). Long-lasting insecticide treated mosquito nets (LLINs) were provided to 150 houses closest to the river valley in July 2007. Anopheles funestus was the primary vector in the village although three members of the An. gambiae complex were also present, especially at the start of the study (Charlwood, 2017).

Mosquito sampling

Mosquito collections began in June 2001 and continued until January 2009. Mosquitoes were caught in exit-traps, CDC light-traps, resting collections. Details of the trapping methods were described previously (Charlwood, 2017). The first 2 years of light-traps were based on a random set of houses. The later samples were a convenience sample and over-sampled the central parts of the village. Resting (and tent-trap) collections were undertaken on an ad hoc basis.

Field and laboratory techniques

Collected mosquitoes were separated into species or species group according to the keys of Gillies & DeMeillon (1968) and Gillies & Coetzee (1987). Females caught in exit-traps and resting collections were classified according to their abdominal appearance as being ‘unfed’, ‘part-fed’, ‘fully-fed/semi-gravid’ or ‘gravid’, according to the condition of the ovaries.

Dissections were undertaken on unfed mosquitoes collected from the light-traps. Parous status and the presence of ovariolar sacs were determined as described by Charlwood et al. (2003b) and Charlwood et al. (2018).

Estimates of survival and cycle duration based on aggregated data

If a whole annual seasonal cycle is representatively sampled, then the proportions of mosquitoes in different categories of parity, gravidity, and the presence of ovariolar sacs provide consistent and unbiased estimates of survival and cycle duration. Estimates made using this approach as follows:

${\mathrm{P}}_{\mathrm{o}}$, the survival of adult female mosquitoes per oviposition cycle was estimated by $P_o^{\left(1\right)}$, the proportion of host-seeking female mosquitoes in light-traps that are parous (Davidson, 1954).

${\theta }_r,$ the duration of the resting period was estimated from resting catch data using an extension of the approach used by Kulkarni et al. (2006) and Tchuinkam et al. (2010). The proportion of resting mosquitoes recorded as fed (i.e., having fed the previous night and still digesting the blood meal) among those that are either fed, semi-gravid, or gravid was estimated by:

(1) $$\begin{array}{c}{f}^{\left(1\right)}=\sum _t{\mathit{F}}_{\mathit{t}}^{\left(\mathit{R}\right)}/\sum _t\left({\mathit{F}}_{\mathit{t}}^{\left(\mathit{R}\right)}+{\mathit{G}}_{\mathit{t}}^{\left(\mathit{R}\right)}\right)\end{array}$$where $t$ represents the trap night and ${\mathit{F}}_{\mathit{t}}^{\left(\mathit{R}\right)}$ and ${\mathit{G}}_{\mathit{t}}^{\left(\mathit{R}\right)}$ represent the number of fed and semi-gravid/gravid mosquitoes caught in resting catches, respectively (Table 1). Neglecting mortality while resting, the duration of the resting period is then estimated by ${\theta }_r^{\left(1\right)}={\displaystyle \frac{1}{f^{\left(1\right)}}}$. Kulkarni et al. (2006) based on the assumption of negligible resting stage mortality.

$a_0$, the proportion of host-seeking mosquitoes that return on the same night as oviposition.

This was estimated by the sac rate, as described by Charlwood et al. (1985) and applied also by Charlwood et al. (1995, 2003a, 2003b, 1997), Molez, Desenfant & Jacques (1998).

${\theta }_o$, the duration of the full oviposition cycle in days.

This was calculated as a function of the resting period, ${\theta }_r$ and of $a_0$ (Charlwood et al., 2016):

(2) $$\begin{array}{c}{\theta }_o={\theta }_r+\left(1-a_0\right)/a_0\end{array}$$

Substitution of ${\theta }_r={\theta }_r^{\left(1\right)}$, gave an estimate ${\theta }_o^{\left(1\right)}$ for ${\theta }_o$.

$\mathrm{p}$, the daily survival of female mosquitoes

Assuming an exponential survival distribution, daily survival, p, is related to ${\mathrm{P}}_{\mathrm{o}}$ by:

(3) $$\begin{array}{c}\ln p={\displaystyle \frac{\ln \left({\mathrm{P}}_{\mathrm{o}}\right)}{{\theta }_o}}\end{array}$$

Substitution of ${\theta }_o={\theta }_o^{\left(1\right)}$, and $P_o=P_o^{\left(1\right)}$ gave an estimate $p^{\left(1\right)}$ for p.

Estimates of survival and cycle duration based on time-series analysis

The notation used for time-dependent quantities including the counts of numbers of mosquitoes in different categories is given in Table 1. Upper case Latin letters are used for total numbers of mosquitoes in any category caught in all traps on night t. Lower-case letters are used for average numbers per trap-night.

Data description

The overall sampling period spanned 8 years from January 2001 until January 2009, with the trapping effort and the numbers of An. funestus caught by each method given in Table 2. All of the An. funestus group mosquitoes examined morphologically had a single pale spot on the upper branch of the 5th vein and did not have a pale spot at the tip of the 6th vein and hence corresponded to An. funestus. All 71 females of the An. funestus group identified by PCR using the protocols of Koekemoer et al. (2002) were An. funestus (Charlwood, 2017). Since this is the most endophilic member of the species group, and that it was this behavior that was examined, we assume that this was the only member of the species group present in our collections.

Of the 71 unfed females dissected from resting collections 68 were virgins (Charlwood, Thompson & Madsen, 2003).

Densities of host-seeking mosquitoes were measured using light-trap collections throughout the period, and were highest in 2001, and then again in 2007 and 2008 (Fig. 1). Dissections were carried out on mosquitoes caught in light-traps between May 2007 and October 2008 on 87 distinct nights. Of 3,192 parous mosquitoes dissected, 1,927 (60.4%) had evident ovariolar sacs.

Resting catches were carried out intermittently throughout the research period, while exit-trap collections began in March 2003. Resting collections comprised mainly fed, gravid, or male mosquitoes, while many more unfed than fed mosquitoes were caught in exit-traps (Table 2). The majority of mosquitoes caught were in exit-traps and most of these were males. All female mosquitoes caught in exit-traps were analysed for blood meal status. The numbers of gravid females captured in exit-traps were comparable to those unfed or part-fed (Fig. 1).

The autocorrelations show the expected pattern of a decrease in correlation the longer the lag period (Fig. 2). The number of male mosquitoes in exit-traps was highly correlated with the number of mosquitoes that were still host-seeking (unfed or part-fed) on the same night. Lagged correlations between numbers of males and unfed/part-fed mosquitoes decreased with the time-interval (Fig. 2A). In contrast, there was only a small correlation between numbers of gravid mosquitoes in exit-traps and the numbers of unfed mosquitoes on the same night, and the lagged correlations were close to zero (Fig. 2A). All three categories of mosquitoes showed a strong temporal autocorrelation in mean numbers in exit-traps (Fig. 2B). This was highest for males (consistent with the same age-cohorts of male mosquitoes being represented on successive nights), lowest for gravid mosquitoes (each age cohort is gravid only once per cycle), and intermediate for unfed/part-fed mosquitoes, for which the autocorrelation is likely to reflect temporal correlations in environmental factors.

Additional, complementary, descriptions of the temporal patterns of mosquito densities and numbers of mosquitoes in different categories over time are provided by Charlwood (2017).

Evaluation of estimators using simulations

Simulated datasets were generated with the same numbers of traps sampled on each day as in the corresponding field data, and with autocorrelation structure based on that found in the data. These datasets were used to evaluate the performance of the different estimators.

The modified version of the Birley model (Eqs. (2)–(5)) fitted to the light-trap data (Fig. 3) performed well in recovering the parameter values used to generate the simulated datasets, accurately recovering both the survival per cycle (Fig. 3A) and the duration of the oviposition cycle (Fig. 3B), over the entire range of values simulated. The estimates of both variables appear to be unbiased, but the estimates of cycle duration displayed more of a tendency to deviate from the target (input) values.

The performance of the novel model fitted to exit-trap data for female mosquitoes (Eqs. (11)–(16)) depended on which of the parameters were assumed known from other data (e.g., from concurrent light-traps) with five different implementations evaluated (a–e in Table 3). These five implementations differed in which parameters were estimated, and which were assumed known from other data (Table 3).

By assuming externally determined values of female mosquito survival, ( $P_r$) and of the sac rate, $a_0$, implementation a corresponds to the external data available in Furvela. Implementation b makes use only of an external value for the sac rate, $a_0$. Since it is reasonable to treat $a_0$ as invariant, this provides a way of estimating $P_r$ for subsets of the full dataset, for instance in order to evaluate annual variations. Implementation c, in which both $a_0$ and the trapping efficiency, ${\mathrm{\Upsilon }}^{\left(1\right)}$, are assumed known, removes one important source of uncertainty from the other implementations, while implementations d and e evaluate whether misspecification of the values of $P$ and $a_0$ respectively, have important effects on the estimates of the other parameters.

None of the implementations gave estimates of ${\mathrm{\Upsilon }}^{\left(1\right)}$ that reliably agreed with the simulated values (CCC values were all less than 0.5 corresponding to considerable scatter in Fig. 4A). Implementations a, b, and e gave estimates with a positive bias while implementation d gave estimates with negative bias (Table 3). Precise and unbiased estimates of female mosquito survival were only obtained by fixing ${\mathrm{\Upsilon }}^{\left(1\right)}$ at its true value (implementation c) (Fig. 4B). When only $a_0$ was fixed at the true value (implementation b) $P_{\mathrm{r}}^{\left(2\right)}$ was often a poor estimate, with the bias (mean 1.9) having wide confidence intervals, although these overlapped with zero. The overall level of agreement between estimates and true values was nevertheless acceptable (CCC = 0.84) (Table 3). All five of the implementations deliver comparable performance in estimating cycle length ( ${\theta }_o^{\left(4\right)}$). These estimates are close to unbiased (with narrow confidence intervals overlapping 0 for the bias), and with an acceptable CCC of around 0.8, though for many of the individual simulations the estimates were quite poor (Fig. 4).

In addition to accuracy and a lack of bias, a further desirable property of an estimator is that it converges with the true value as the sample size is increased (consistency). This was not achieved by the estimates of trapping efficiency, for which there was no decrease in error as the size of the simulated dataset increased (Fig. 5A). The error did show the expected decrease with increased sample size in the estimates of survival per cycle. The error is already very small with 60 or more days with complete data, when the trapping efficiency is known. When trapping efficiency is estimated a very large number of nights of collection appear to be needed to ensure accuracy of the estimates (Fig. 5B). Although the methods do not require all the collection nights to be consecutive, a need for 2 months or more of data is likely to be a severe limitation. Irrespective of which of the other parameters were treated as known, the error in estimates of the duration of the resting period or of the full cycle decreased with an increase in sample size up to a value of about 100 days with complete data. Above this sample size there was no further improvement (Fig. 5C).

The performance of the novel model fitted to exit-trap data for male mosquitoes (Eqs. (17) and (18)) was evaluated for two implementations, differing in whether the trapping efficiency for males, ${\mathrm{\Upsilon }}^{\left(2\right)}$, was assumed to be known. When ${\mathrm{\Upsilon }}^{\left(2\right)}$ was estimated (implementation a) these estimates were very inaccurate and hardly correlated with the true value (Table 4 and Fig. 6).

In contrast, the estimates of male survival were reasonably accurate even when ${\mathrm{\Upsilon }}^{\left(2\right)}$ was unknown. They were very highly accurate when ${\mathrm{\Upsilon }}^{\left(2\right)}$ was known.

The plots of the squared error of the estimates against the sample size (Fig. 7) indicated the expected improvement with increased sample size for the implementation with fixed ${\mathrm{\Upsilon }}^{\left(2\right)}$. The relationship was less clear when ${\mathrm{\Upsilon }}^{\left(2\right)}$ was estimated.

Parameter estimation from aggregated data only

The estimates of life cycle parameters from aggregated data (Table 5) had narrow confidence intervals, reflecting the large number of observations. The estimated average duration of the resting period was close to 2 days, with slightly higher values obtained when resting period mortality was allowed for. Less than half of the female mosquitoes were estimated to survive the full cycle.

Parameter estimation from time-series data

Fitting of the model of Eqs. (6)–(9) to the field data from light-traps gave an estimate of $P_{\mathrm{o}}^{\left(2\right)}$ = 0.528, which is rather higher than the unadjusted parous rate of $P_{\mathrm{o}}^{\left(1\right)}$ = 0.481 (Table 6).

$P_{\mathrm{o}}^2$ = 0.528 is to be preferred to the crude parous rate as an estimate of survival per cycle, because it does not depend on the approximation that an integral number of seasonal cycles were representatively sampled. The small difference between the crude and model-based survival estimates is consistent with a modest level of unrepresentativeness in the sampling. The same model also gave an estimate of only 2.13 days for the oviposition cycle duration (Table 6) which is shorter than the estimate based on the resting collection data only. The corresponding estimate of daily survival of p = 0.74, corresponding to life expectancy of the adult female mosquito assuming an exponential survival model of ln(2)/ln(p) = 2.3 days (95% credible intervals [1.6–3.4]). The numbers of parous mosquitoes show only a limited degree of overdispersion and the estimate of the standard deviation of the duration of the oviposition cycle based on light-trap data, ${\sigma }_o^{(1)}$, is also low (Table 6).

The numbers of female mosquitoes captured in exit-traps were analysed with three different implementations of the model of Eqs. (10)–(15) (Table 7). Implementation a, which used the estimate of survival from the analysis of light-trap data, gave an estimate of ${\mathrm{\Upsilon }}^{\left(1\right)}=1.8$, implying that a gravid mosquito is almost twice as likely to be caught in exit-traps as an emergent female. In contrast, when the model was used to estimate $P$, very high survival estimates were obtained but the trapping efficiency was estimated to be about 0.4. In view of the evidence from simulations that fixing $P$ leads to overestimation of ${\mathrm{\Upsilon }}^{\left(1\right)}$, and of the absence of other evidence for a difference in trapping efficiency between emergent and gravid females, a third set of estimates were made, fixing ${\mathrm{\Upsilon }}^{\left(1\right)}$ at a value of 1. These gave intermediate levels of survival and similar estimates of the duration of the oviposition cycle to those obtained from the light-trap data.

The numbers of male mosquitoes captured in exit-traps were analysed with two different implementations of the model of Eqs. (16) and (17) (Table 8), with implementation b constraining the trapping efficiency for male mosquitoes to be the same as that for emergent females.

Analyses of the exit-trap data for female mosquitoes were made separately for each annual period (Fig. 8) using the same model implementations for the global analyses reported in Tables 7 and 8. When the trapping efficiency was not constrained, the estimates of ${\mathrm{\Upsilon }}^{\left(1\right)}$ for the different years varied considerably, with a maximum in 2005 suggesting that the introduction of LLINs was associated with a reduction in ${\mathrm{\Upsilon }}^{\left(1\right)}$ (Fig. 8A). Irrespective of assumptions about trapping efficiency or survival, the estimated of cycle duration decreased from 2003–2006 but then increased again (Fig. 8B) although these estimates all have wide credible intervals. Correspondingly, the estimated survival per cycle (Fig. 8C) was lower in the later years. When the trapping efficiency was fixed (implementation c) there was more of an indication of the expected decrease between the first and second halves of the study period in estimates both of survival per cycle and daily survival (Fig. 8D).

The corresponding analyses of data for male mosquitoes estimated uniformly very high daily survival ( $\pi$) when ${\mathrm{\Upsilon }}^{\left(2\right)}=1$ (Fig. 9B). When ${\mathrm{\Upsilon }}^{\left(2\right)}$ was allowed to vary, the estimates of this quantity were mostly substantially higher than unity, while the estimates of $\pi$ were about 0.7 (a little higher than the global estimate of 0.65 (Table 8)), with no indication of any trend over time.

The final analysis of the exit-trap data estimated separate values for cycle duration and survival by temperature (Fig. 10). The estimates of cycle duration were highest for weeks with mean temperatures of 23–25 °C, although all these estimates were very imprecise (Fig. 10A). The estimated female survival per cycle tended to increase with temperature but the confidence intervals were again broad (Fig. 10B).

Analysis of the linear trend on a logistic scale found that the odds of surviving the cycle were multiplied by 1.021 for every degree of mean weekly temperature increase (95% credible interval [0.991–1.051]). This modest trend was even less obvious when the results were expressed as survival per 24-h period (Fig. 10C). Male survival was almost independent of temperature (Fig. 10D), with the odds of surviving multiplied by only 1.002 per degree of mean weekly temperature increase (95% credible interval [0.967–1.040]).