### Entomological study of ATSB

A Phase II entomological study (previously reported in this journal [16]) was undertaken in 14 villages in central Mali. The climate in this region is highly seasonal, with high rainfall in the rainy season (peaking in September) and very dry conditions in the dry season (December–March). Full details of the study and outcomes are reported elsewhere and are summarized here for completeness.

Fourteen villages were selected to participate in the study. In the first year of the study (April 2016 to May 2017), baseline entomological data were collected in all 14 villages. They were randomly sorted into two groups of seven, with one group designated as the intervention (ATSBs + standard of care) group and one as the control (standard of care) group. ATSBs were then deployed in the intervention villages in June 2017 with two bait stations containing the insecticide dinotefuran being placed on the outer walls of each building, and entomological data collected through to December 2017. To estimate the feeding rate on the ATSBs, 1-day tests using stained bait were carried out at monthly intervals in the control villages. As a means of estimating the bait-feeding rate, simple tests were carried out in which attractive sugar baits without toxic additives (ASBs) were temporarily introduced to villages where ATSBs were not used. These baits contained a harmless dye which allowed captured mosquitoes in the relevant villages to be separated into those which had fed on the ASBs and those which had not.

Mosquitoes were collected monthly in each village using Centre for Disease Control (CDC) UV light traps, Malaise traps and pyrethroid spray catch (PSC) inside houses. Here the data from CDC traps is used as a measure of mosquito density. In addition, human landing catch (HLC) measurements were carried out indoors and outdoors by four volunteers (two indoors in separate homes and two outdoors at least 5 m apart from the indoor volunteers). A random sample of the captured mosquitoes were examined to determine the proportion containing viable sporozoites and, therefore, onwardly infectious; this sporozoite rate is multiplied by the number caught per volunteer per unit time in HLC experiments to estimate the entomological inoculation rate (EIR).

### Estimating the impact of ATSBs on mosquito density and EIR

To estimate the impact of the ATSBs on the two entomological endpoints—mosquito count and EIR—a non-linear model was formulated to capture the seasonal variation in outcomes in addition to the effect of the intervention whilst accounting for the village cluster-level variability. The non-linear model with mosquito count outcomes from the CDC light traps had convergence difficulties. *M*_{i} denotes the mosquito population count in village *i* where *i* = *1, 2…14* (7 treatment and 7 control villages) at time *t* [indexing the month of the year in which count of mosquito outcome was obtained with July being the first month (t = 1) and November the fifth month (t = 5)].

Equation 1a shows the trigonometric function, which captures the seasonal variation in the mosquito population density for an individual village where *a*_{M}*, b*_{M}*, c*_{M}*, d*_{M} are parameters to be estimated. The term *R*_{M} denotes the treatment effect coefficient for ATSB (the fractional decrease in population density) while *δ*_{T} is the predictor that identifies treatment assignment at village level (coded as 1 and 0 for treatment and control villages respectively). The variation in count between villages was captured by a Poisson distribution with its mean based on Eq. 1a (Eq. 1b).

$$M\left( {\text{t}} \right) = \left( {1 - {R_M}{\delta_T}} \right)({{\text{a}}_M}\sin \left( {{{\text{b}}_M}{\text{t}} - {{\text{c}}_M}} \right) + {{\text{d}}_M})$$

(1a)

$${{\text{M}}_{\text{i}}}\left( {\text{t}} \right)\ \sim{\text{Poi}}({\uplambda } = {{\text{e}}^{\left( {{\text{r}}{_{{\text{M}},{\text{i}}}} + \left( {1 - {{\upbeta }_M}{{\updelta }_{\text{T}}}} \right)} \right)}} \cdot \left( {{{\text{a}}_M}\sin \left( {{{\text{b}}_M}{\text{t}} - {{\text{c}}_M}} \right) + {{\text{d}}_M})} \right)$$

(1b)

The terms *r*_{M,i} denotes the random intercept that captures the correlation of mosquito population count at village level assumed to be be normally distributed as *r*_{M,i} ~ *N(0,σ*_{M,r}^{2}*)* where *σ*_{M,r}^{2} is a level 2 variance component to be estimated. The reduction *R*_{M} in mosquito count can be approximated by the formula \({R}_{M}\sim \left(\mathrm{exp}\left(1\right)-\mathrm{exp}(1-{\beta }_{M})\right)/\mathrm{exp}(1)\) where *β*_{M} is the effect size parameter estimated from fitting the statistical model from Eq. 1b.

The EIR is modelled by a normal distribution with its parameters captured by the model shown in Eq. 2. The mean of the normal distribution was defined by the term \(\left(1-{R}_{EIR}{\delta }_{T}\right)({a}_{EIR}\mathrm{sin}\left({b}_{EIR}t-{c}_{EIR}\right)+{d}_{EIR})+ {r}_{EIR,i}\) and the residual error variance by *ε*_{i}.

$$EI{R_i}\left( t \right) = \left( {1 - {R_{EIR}}{\delta_T}} \right)({a_{EIR}}\sin \left( {{b_{EIR}}t - {c_{EIR}}} \right) + {d_{EIR}}) + \;{r_{EIR,i}} + {\varepsilon_i}$$

(2)

Due to the nature of the data, the time *t* was measured in months in which the EIR value was obtained. The months were coded as t = 1 to 7 representing the months from June to December. As with Eq. (1), the fixed effect part of Eq. 2 containing the term *a*_{EIR}* sin(b*_{EIR}*t – c*_{EIR}*)* + *d*_{EIR} is a trigonometric function that captures the seasonal variation in the EIR where *a*_{EIR}*, b*_{EIR}*, c*_{EIR}*, d*_{EIR} are parameters to be estimated. The term *R*_{EIR} denotes the treatment effect coefficient for ATSB (the fractional decrease in EIR) while *δ*_{T} is the predictor that identifies treatment assignment at village level (coded as 1 and 0 for treatment and control villages, respectively). The terms *r*_{EIR,i} and *ε*_{i} denote the random intercept that captures the correlation of EIR at village level and the residual variability, respectively. These are assumed to be normally distributed as *r*_{EIR,i} ~ *N(0,σ*_{EIR,r}^{2}*)* and *ε*_{ij} ~ *N(0,σ *_{EIR,e}^{2}*)* where *σ *_{EIR,r}^{2} and *σ *_{EIR,e}^{2} are variance components to be estimated.

In both cases, the model was fitted to the mosquito catch and EIR data using PROC NLMIXED with the adaptive Gauss-Hermite quadrature method [20] in Statistical Analysis System (SAS) software version 9.4 [21] to obtain the point estimate for the parameter *R* together with its corresponding 95% confidence interval based on a two-sided p-value for the null hypothesis \({H}_{0} :R=0\) *versus* the alternative\({H}_{A} :R\ne 0\). The parameter values extrapolated from population and EIR data are shown in Additional file 1.

### Estimating the excess mortality

Equation 3a expresses the rate of change of the mosquito catch in ATSB villages *M*_{EXP} following the introduction of ATSBs. This expression, based on the approach taken by Marshall et al*.* [8], is a simplified version of the more detailed mosquito population model, used here as a means of relating the function fitted to the observed data (Eq. 1) to mosquito mortality parameters.

$$\frac{{d{M_{EXP}}\left( t \right)}}{dt} = {\mu_{BASE}}{M_{EQ}}\left( t \right) - \left( {{\mu_{BASE}} + {\mu_{ATSB}}} \right){M_{EXP}}\left( t \right)$$

(3a)

$$\frac{{d{M_{CON}}\left( t \right)}}{dt} = {\mu_{BASE}}{M_{EQ}}\left( t \right) - {\mu_{BASE}}{M_{CON}}\left( t \right)$$

(3b)

Here *M*_{EQ} is the seasonally varying equilibrium mosquito catch, *µ*_{BASE} is the baseline adult mosquito appearance and death rate in the absence of ATSBs, and *µ*_{ATSB} is the excess mortality due to ATSBs. *µ*_{BASE} is given by the natural mosquito death rate *µ*_{NAT} added to any additional mortality due to vector control interventions present in both control and ATSB villages. In control villages, the mosquito catch *M*_{CON} is given by the same equation with *µ*_{ATSB} set to 0 (Eq. 3b).

From Eq. 1a, the average mosquito catch rate in the control and ATSB arms can be written as shown in Eqs. 4a, b.

$${M_{CON}}\left( t \right) = {a_M}\;{\text{sin}}\left( {{b_M}t - {c_M}} \right) + {d_M}$$

(4a)

$${M_{EXP}}\left( t \right) = \left( {1 - {R_M}} \right){M_{CON}}\left( t \right) = \left( {1 - R} \right)\left[ {{a_M}\;{\text{sin}}\left( {{b_M}t - {c_M}} \right) + {d_M}} \right]$$

(4b)

The relationship between *M*_{EXP} and *M*_{CON} (Eq. 4b) can be substituted into Eq. 3a (Eq. 5a). Equation 5a and Eq. 3b can then re-arranged to give two different expressions for *M*_{EQ} (Eqs. 5b–c). These can then be equated in order to express the relationship between *M*_{CON}, *R*_{M}, *µ*_{BASE} and *µ*_{ATSB} (Eq. 5d).

$$\left( {1 - {R_M}} \right)\frac{{d{M_{CON}}}}{dt} = {\mu_{BASE}}{M_{EQ}} - \left( {1 - {R_M}} \right)\left( {{\mu_{BASE}} + {\mu_{ATSB}}} \right){M_{CON}}$$

(5a)

$${M_{EQ}} = \frac{{1 - {R_M}}}{{{\mu_{BASE}}}}\left( {\frac{{d{M_{CON}}}}{dt} + \left( {{\mu_{BASE}} + {\mu_{ATSB}}} \right){M_{CON}}} \right)$$

(5b)

$${M_{EQ}} = \frac{1}{{{\mu_{BASE}}}}\frac{{d{M_{CON}}}}{dt} + {M_{CON}}$$

(5c)

$${\mu_{ATSB}} = \frac{{R_M}}{{1 - {R_M}}}\left( {\frac{1}{{{M_{CON}}}}\frac{{d{M_{CON}}}}{dt} + {\mu_{BASE}}} \right)$$

(5d)

Equation 5d can be rewritten as follows with Eq. 4a used to substitute for *M*_{CON}, to give an estimate of *µ*_{ATSB} in terms of the estimated parameters *a*_{M}, *b*_{M}, *c*_{M}, *d*_{M}, *R*_{M} and the base death rate:

$${\mu_{ATSB}} = \frac{{R_M}}{{1 - {R_M}}}\left( {\frac{{{a_M}{b_M}\;{\text{cos}}\left( {{b_M}t - {c_M}} \right)}}{{{a_M}\;{\text{sin}}\left( {{b_M}t - {c_M}} \right) + {d_M}}} + {\mu_{BASE}}} \right)$$

(6)

### Estimating the impact of ATSBs on malaria prevalence and incidence

An existing detailed model [17,18,19, 22] of malaria was used for simulations of the effects of ATSBs on malaria infection levels in human populations. In the model, individuals begin life susceptible to *P. falciparum* infection and are exposed to infectious bites at a rate that depends on local mosquito density and infectivity. Newborn infants passively acquire maternal immunity, which decays in the first 6 months of life. After exposure, individuals are susceptible to clinical disease and may progress through a range of infection categories (clinical infection, asymptomatic infection, subpatent infection, treated and prophylaxis). As they age, the risk of developing disease declines through natural acquisition of immunity, at a rate that depends on the rate of continued exposure. At older ages, parasitaemia levels fall so that a high proportion of asymptomatic infections become sub-microscopic. Full mosquito-population dynamics were included in the model to capture the effects of vector control in preventing transmission, killing adult female mosquitoes, and the resulting reduction in egg-laying. The model has previously been fitted to existing data on the relationship between rainfall (the seasonal parameter found to give best fit to data [22]), mosquito abundance, entomological inoculation rate (the rate at which people receive infectious bites), parasite prevalence and clinical disease incidence in order to establish parameter values. Full mathematical details of the model and a complete parameter list are included in Additional file 1.

The effect of ATSBs was included in the model by modifying the death rate of mosquitoes from *µ*_{BASE} to *µ*_{BASE} + *µ*_{ATSB} as shown in the previous section. Note that this differs from the modelling of other common vector control interventions such as LLINs and IRS, where direct reduction in biting rate must also be incorporated and additional mortality is affected by biting rate [23].

The initial conditions for a study were created by generating characteristics (proportions of humans in different infection categories, immunity levels) at steady state under particular levels of adult mosquito density, then after an extended period of time with particular seasonal variation in adult mosquito density. ATSBs were then introduced to modify the mosquito death rate, resulting in reduced mosquito populations due to direct death and reduced larval birth rate. As noted above, the population of infectious mosquitoes decreases more significantly than the overall population, due to increased death rates causing fewer infected mosquitoes to survive for the duration of the parasite incubation period. This in turn caused reductions in EIR which in turn reduced the number of new infections. Benchmark data values including malaria prevalence and clinical incidence were recorded at regular intervals and the results compared with the same data values under control conditions (where the mosquito death rate is simply equal to the natural value *µ*_{NAT}) to measure the effectiveness of ATSBs.