Confidence intervals for estimated mean FOI values in simulated risk scenarios with homogeneous exposure, before and during IRS interventions with three different coverage levels.

The times of the local transmission events follow a Gamma distribution, and transmission is seasonal with the spatial setting corresponding to a closed system. We provide FOI estimates based respectively on the true MOI values, on the MOI estimates with the Bayesian method and a bootstrap imputation approach correcting for all aforementioned errors, and on the MOI estimates with one source of error at a time corrected by either the Bayesian method or the bootstrap imputation approach, namely under the missing data error, the under-sampling of var genes, and the antimalarial treatment issue. To obtain the true mean FOI per host per year, we divide the total number of infections acquired by the population by the total number of individual hosts in the population. Minimum, 5% quantile, median, 95% quantile, and maximum values are shown in the boxplot.

Confidence intervals for the estimated mean FOI values in simulated risk scenarios with heterogeneous exposure, before and during IRS interventions with three different coverage levels.

The times of the local transmission events follow a Gamma distribution, and the transmission is seasonal with the spatial setting corresponding to a semi-open system. We provide FOI estimates based respectively on the true MOI values, on the MOI estimates with the Bayesian method and a bootstrap imputation approach correcting for all aforementioned errors, and on the MOI estimates with one source of error at a time corrected by either the Bayesian method or the bootstrap imputation approach, namely under the missing data error, the under-sampling of var genes only, and the antimalarial treatment issue. To obtain the true mean FOI per host per year, we divide the total number of infections acquired by the population by the total number of individual hosts in the population. Minimum, 5% quantile, median, 95% quantile, and maximum values are shown in the boxplot.

Confidence intervals for the estimated mean FOI values in Ghana surveys across a transient three-round IRS intervention based on the combination of surveys from the wet/high-transmission and dry/low-transmission seasons

(A) The estimated FOI values when considering that antimalarial treatment does not affect infection status and MOI values in treated individuals (assumption 1). (B) The estimated FOI values when considering instead that antimalarial treatment affects infection status and MOI values in treated individuals (assumption 2). High, mid, and low detactability correspond to 0%, 5%, and 10% of PCR-negative individuals carrying infections respectively. Minimum, 5% quantile, median, 95% quantile, and maximum are shown in the boxplot.

Saturation in FOI with EIR and the non-linear relationship between these two quantities from previous field studies.

(A) and (B) includes our empirical estimates under the assumption that antimalarial treatment does not affect infection status and MOI values in treated individuals. (C) and (D) includes our estimates under the assumption that the treatment affects such quantities. The points correspond to paired EIR-FOI values from the literature, summarized in (Smith et al., 2010), and the crosses indicate the range of those values when several estimates of the EIR or the FOI were reported or estimated for the same location. Namely, a line or a cross was plotted with its center at the arithmetic mean and with legs that connect the center to each one of the estimates, and similarly for the scenario when ranges were reported for both variables. The red function curve is the best-fit to these paired EIR-FOI values from (Smith et al., 2010). The black hollow diamond and circle correspond to the Ghana data, for our respective estimates of FOI with the two methods and the EIR measure in the field by the entomological team (Tiedje et al., 2022).

(A) Each simulation follows three stages: a “pre-IRS” period during which the transmission in the local population reaches a stationary state, followed by an “IRS” intervention period of three years which reduces transmission rate (in what we call transient IRS), and a “post-IRS” period when transmission rates go back to their original levels. We let the system run for some years to reach a (semi-) stationary state before applying IRS interventions. After initial seeding, closed systems do not receive migrant genomes from the regional pool. Semi-open systems are explicit simulations of two individual local populations coupled via migration events. Regionally-open systems receive continuous migrant genomes from the regional pool throughout the simulation course. (B) Transmission intensity or effective contact rate (Appendix 1-Simulation data) varies as a function of time, from pre-, to during, to post-intervention. We simulate three different coverages of perturbations, ranging from low-coverage one which reduces the system’s transmission by around 20% to high-coverage one which reduces the system’s transmission by around 70-75%. (C) We incorporate heterogeneity in transmission across individual hosts by varying two things. First, we consider different statistical distributions for times of local transmission events: exponential and Gamma. Second, we consider homogeneous and heterogeneous exposure risks. For the latter, we set of the total population as being at high-risk and receiving more than 90% of all bites whereas the rest population receives only less than 10% of all bites. (D) The measurement error which describes potential sequencing errors of var genes: histogram of the number of non-upsA (i.e., upsB and upsC) DBLα var gene types per repertoire. The molecular sequences were previously sequenced from infections that are expected to be monoclonal (as they had less than or equal to 45 non-upsA DBLα types), sampled during six cross-sectional surveys made from 2012 to 2016 in Bongo District. (E) Study design showing the timing of the four age-stratified cross-sectional surveys conducted in Bongo District, Ghana at the end of the wet/high-transmission seasons (blue circles) and at the end of the dry/low-transmission seasons (gold circles). The study can be broken down into two phases: (1) Pre-IRS: Survey 1 (S1) October 2012 and Survey 2 (S2) May/June 2013; (2) Right post-IRS: Survey 3 (S3) October 2015 and Survey 4 (S4) May/June 2016. IRS were implemented against a background of widespread LLIN usage which were distributed across Bongo District between 2010 – 2012 and again in 2016 (Tiedje et al., 2023; Gogue et al., 2020).

Comparison of the estimated MOI distribution under only one type of error among those commonly encountered in the collection of empirical data (missing data, under-sampling of var genes, and usage of antimalarial drugs) versus under all the types of the errors together and the distribution of true MOI values (indicated by different shades of the red). The scenarios shown here are simulated as semi-open systems under seasonal transmission. Individuals are under heterogeneous exposure risk (Appendix 1-Figure 5C) and the arrival of infection follows a Gamma distribution. The Bayesian varcoding method and the bootstrap imputation approach are able to address all types of the errors. Results of the comparison across other simulated scenarios can be found in supplementary file 1-MOImethodsPerformance.xlsx

Comparison of the estimated MOI distribution based on the original varcoding method, the Bayesian approach, and the true values. The scenarios shown here correspond to a semi-open system under seasonal transmission. Arrival of infection follows a Gamma distribution. Both the Bayesian varcoding approach and the original method perform well, with the former showing a clear improvement in capturing the tail of high MOI values. Results of the comparison across the remaining different simulated scenarios can be found in supplementary file 2-BayesianImprovement.xlsx.

The distribution of MOI estimates based on the Bayesian varcoding method and the bootstrap imputation approach for longitudinal surveys in Bongo District, from pre-IRS to right post-IRS. See Appendix 1-Figure 5E for the study design of the IRS intervention. The wet/high season survey for pre-IRS phase was collected in 2012 and denoted as S1 in Figure 5E. The dry/low season survey for pre-IRS phase was collected in 2013 and denoted as S2. The wet/high season survey for right post-IRS phase was collected in 2015, corresponding to S3. The dry/low season survey for right post-IRS phase was collected in 2016 and denoted as S4. As explained in the methods, the empirical surveys suffer from missing data, i.e., only microscopy-positive individuals have their var genes typed and sequenced. We quantify the number of individuals with missing MOI by utilizing the empirically measured detectability/sensitivity of microscopy relative to PCR, and assuming a range for the PCR detectability. We consider a high detectability in which PCR picks up all infected individuals, to a mid value in which 5% PCR-negative individuals actually carry infections, to a low value in which 10% PCR-negative individuals carry infections. The empirical surveys also suffer from missing data due to other factors than the detection limit of the selected method, for example, low DNA quality. We impute MOI values for all those individuals with missing MOI by sampling from MOI estimates of microscopy-positive individuals with their var sequenced and typed successfully who are not treated. In addition, some individuals may seek and get antimalarial treatment in a previous window of time. We consider here the extreme scenario that antimalarial treatment has no impact on treated individuals’ infection status and MOI values, and use their infection status and MOI values directly for FOI inference.

The distribution of MOI estimates based on the Bayesian varcoding method and the bootstrap imputation approach for longitudinal surveys in Bongo District, from pre-IRS, to right post-IRS, to post-IRS/SMC. We assume here the other extreme scenario that antimalarial treatment shifts and changes treated individuals’ infection status and MOI values completely. We thus impute MOI values for those treated individuals by sampling from MOI estimates of microscopy-positive individuals with their var sequenced and typed successfully who are not treated.

The true and estimated FOI by the two-moment approach and Little’s law for additional simulated homogeneous exposure risk scenarios, under non-seasonal transmission, for a closed system. The times of the local transmission events are Gamma-distributed. The true mean FOI per host per year is calculated by dividing the total number of infections acquired by the population by the total number of individual hosts in the population. Minimum, 5% quantile, median, 95% quantile, and maximum values are shown in the boxplot.

The true and estimated FOI by the two-moment approach and Little’s law for simulated heterogeneous exposure risk scenarios under non-seasonal transmission in a semi-open system. The times of the local transmission events are Gamma-distributed. The true mean FOI per host per year is calculated by dividing the total number of infections acquired by the population by the total number of individual hosts in the population. Minimum, 5% quantile, median, 95% quantile, and maximum values are shown in the boxplot.

The true and estimated FOI by the two-moment approach and Little’s law for simulated homogeneous exposure risk scenarios under seasonal transmission in a regionally-open system. The times of the local transmission events are Gamma-distributed. The true mean FOI per host per year is calculated by dividing the total number of infections acquired by the population by the total number of individual hosts in the population. Minimum, 5% quantile, median, 95% quantile, and maximum values are shown in the boxplot.

The true and estimated FOI by the two-moment approach and Little’s law for simulated homogeneous exposure risk scenarios under non-seasonal transmission in a regionally-open system. The times of the local transmission events are Gamma-distributed. The true mean FOI per host per year is calculated by dividing the total number of infections acquired by the population by the total number of individual hosts in the population. Minimum, 5% quantile, median, 95% quantile, and maximum values are shown in the boxplot.

The true and estimated FOI by the two-moment approach and Little’s law for simulated homogeneous exposure risk scenarios and seasonal transmission. The spatial setting corresponds to the closed system. The times of the local transmission events follow a exponential distribution. The true mean FOI per host per year is calculated by dividing the total number of infections acquired by the population by the total number of individual hosts in the population. Minimum, 5% quantile, median, 95% quantile, and maximum values are shown in the boxplot.

Magnitudes of the estimated variance by the two-moment approximation method for different simulation scenarios and the field data in Bongo District, Ghana.