A novel model fitted to multiple life stages of malaria for assessing efficacy of transmission-blocking interventions

Background Transmission-blocking interventions (TBIs) aim to eliminate malaria by reducing transmission of the parasite between the host and the invertebrate vector. TBIs include transmission-blocking drugs and vaccines that, when given to humans, are taken up by mosquitoes and inhibit parasitic development within the vector. Accurate methodologies are key to assess TBI efficacy to ensure that only the most potent candidates progress to expensive and time-consuming clinical trials. Measuring intervention efficacy can be problematic because there is substantial variation in the number of parasites in both the host and vector populations, which can impact transmission even in laboratory settings. Methods A statistically robust empirical method is introduced for estimating intervention efficacy from standardised population assay experiments. This method will be more reliable than simple summary statistics as it captures changes in parasite density in different life-stages. It also allows efficacy estimates at a finer resolution than previous methods enabling the impact of the intervention over successive generations to be tracked. A major advantage of the new methodology is that it makes no assumptions on the population dynamics of infection. This enables both host-to-vector and vector-to-host transmission to be density-dependent (or other) processes and generates easy-to-understand estimates of intervention efficacy. Results This method increases the precision of intervention efficacy estimates and demonstrates that relying on changes in infection prevalence (the proportion of infected hosts) alone may be insufficient to capture the impact of TBIs, which also suppress parasite density in secondarily infected hosts. Conclusions The method indicates that potentially useful, partially effective TBIs may require multiple infection cycles before substantial reductions in prevalence are observed, despite more rapidly suppressing parasite density. Accurate models to quantify efficacy will have important implications for understanding how TBI candidates might perform in field situations and how they should be evaluated in clinical trials. Electronic supplementary material The online version of this article (doi:10.1186/s12936-017-1782-3) contains supplementary material, which is available to authorized users.

1 and [1]. A statistical model was fitted to the observed data and experimental structure using a Bayesian posterior distribution in Stan [2]. The model predictions (posterior draws) for the parasite densities of respective life stages were then used to calculate the efficacy of ATV.
For each treatment regime (ATV-32% and control), mouse-to-mouse transmission operated as described previously [1,3], the experiment is graphically demonstrated in Fig. 1 and described in the figure legend. All care and handling of animals strictly followed the Guidelines for Animal Care and Use prepared by Imperial College London, and was performed under the UK Home Office Licences 70/7185 and 70/8788. Initial parasite density was measured by counting the number of infected red blood cells in the mice (N infected erythrocytes out of a total subsample of 1200 cells).
Mosquitoes were dissected to assess the number of oocysts in the mosquito population. The sporozoite measurement was additionally binned into specified ranges (scores of 0-4 representing 0, 1-10, 11-100, 101-1000, 1000+ sporozoites, respectively). The structure of the data (from [1] and listed in S2.1) resulted in 4 complete scenarios whereby malaria was transmitted mouse-to-mouse via mosquitoes.

Statistical methods
The data consist of measurements of the life stages in mosquito midguts, salivary glands and mice. In this experiment, it was not possible to measure the number of sporozoites reaching the salivary glands from each oocyst or the number of bloodstage infections in mice resulting from injected sporozoites. These relationships are uncertain and are incorporated as nuisance parameters in the model. Each of these stages are modelled sequentially, similarly to a hidden Markov model. The number of parasites in each stage is represented by a zero-inflated negative binomial distribution to account for both infected and uninfected individuals.
A bi-modal structure is fitted using a zero-inflation parameter, π. This parameter determines the proportion of mice or mosquitoes that are uninfected and therefore cannot transmit. The shape of the relationships between different parasite life stages are unknown. This is accounted for by including a random effects component that allows the mean number of parasites in each group to vary according to the observed data.

Seeding mouse population
Let each treatment arm (t = 0 for controls, 1 for ATV treatment) inform the mean µ and dispersion φ parameters for the distribution of parasite densities (N infected erythrocytes out of 1200 cells). At the start of the experiment (transmission cycle i = 0), µ and φ describe the parasite densities in mice N0 that have been injected with high numbers of parasites. A negative binomial distribution is fitted to the initial mice population for each treatment arm and these are convoluted into a global distribution as there is no biological reason for treatment arms to have different numbers of injected parasites at this stage, therefore, Throughout, the negative binomial is parameterised by a mean parameter and a parameter that controls for overdispersion of the variance relative to the square of the mean (described as neg_binomial_2 in the RStan manual, [2]). The model parameters are fit on the log scale, the data are kept as linear counts. After the initial mice stage, the transmission is looped through the respective mosquito to mouse cycles for as many cycles i as there are data, for each control or treatment arm. At each subsequent life stage a new estimate of µ and φ describe the parasite density.

Oocyst counts in mosquitoes
In the experiment, the mosquito population becomes infected by feeding on all 5 mice simultaneously; any mosquito could feed on any mouse. Therefore, there is no effect of biting rate for the cohort of mosquitoes that all feed on all mice. Following the experimental design ( Fig. 1), the variation that is observed between biting rates Where δ is logical and depends on whether the predicted parasite density in mice N is zero; (and see neg_binomial_2 in the RStan manual, [2]). Here, Oimt are the measured data on whether a mosquito has oocysts (1) or not (0) and oimt and τimt are parameters of the latent distribution.

Sporozoite counts in mosquitoes
A sub-sample of the mosquito population is randomly selected to infect the next generation of mice. The number of sporozoites remaining in the salivary glands The probability of sporozoites in a given bin, b, will be equal to the cumulative distribution frequency (cdf) up to that bin (b) minus the cdf up to the preceding bin (b -1), for example: The sporozoite count in mosquitoes is then estimated as: allow the relationship between sporozoite score and asexual parasitemia to be determined by these data.
The probability of infections in mice is a mixture of the uninfected and infected distributions and informed by the incremental log probability. So that: Where, and Nimt indicates where mice do (1) or do not (0) have parasitic infection.

Model fitting
All parameters were fitted jointly using a Bayesian posterior distribution in RStan (version 2.13.1, [2]). To ensure robust fits, a non-centred parameterisation method was employed [5,6]. The model parameter fitting was achieved using a Hamiltonian Monte Carlo method [2], burn-in was 500 and the subsequent 500 samples from each chain (n = 4) were used for the posterior predictive checks. The model code that can be used in R is provided in supporting information S2, with the accompanying data (S2.1). The parameter estimates are supplied in S3.

Model output
Here The 95% credible intervals were presented as 1.96 x the standard error for the efficacy estimates. These were calculated directly from the posterior predictive outputs (Table 1).
To justify model assumptions, it was important to investigate the difference between the overall distribution of sporozoite scores in control and treatment groups, parasitemia (%) and gametocytemia (%). Data exploration was conducted in R, version 3.2.2 [7].