Predicting the impact of border control on malaria transmission: a simulated focal screen and treat campaign

Background South Africa is one of many countries committed to malaria elimination with a target of 2018 and all malaria-endemic provinces, including Mpumalanga, are increasing efforts towards this ambitious goal. The reduction of imported infections is a vital element of an elimination strategy, particularly if a country is already experiencing high levels of imported infections. Border control of malaria is one tool that may be considered. Methods A metapopulation, non-linear stochastic ordinary differential equation model is used to simulate malaria transmission in Mpumalanga and Maputo province, Mozambique (the source of the majority of imported infections) to predict the impact of a focal screen and treat campaign at the Mpumalanga–Maputo border. This campaign is simulated by nesting an individual-based model for the focal screen and treat campaign within the metapopulation transmission model. Results The model predicts that such a campaign, simulated for different levels of resources, coverage and take-up rates with a variety of screening tools, will not eliminate malaria on its own, but will reduce transmission substantially. Making the campaign mandatory decreases transmission further though sub-patent infections are likely to remain undetected if the diagnostic tool is not adequately sensitive. Replacing screening and treating with mass drug administration results in substantially larger decreases as all (including sub-patent) infections are treated before movement into Mpumalanga. Conclusions The reduction of imported cases will be vital to any future malaria control or elimination strategy. This simulation predicts that FSAT at the Mpumalanga–Maputo border will be unable to eliminate local malaria on its own, but may still play a key role in detecting and treating imported infections before they enter the country. Thus FSAT may form part of an integrated elimination strategy where a variety of interventions are employed together to achieve malaria elimination. Electronic supplementary material The online version of this article (doi:10.1186/s12936-015-0776-2) contains supplementary material, which is available to authorized users.

Population migration between the different patches is characterised by three sets of movements: 1. Movement may occur between any two of the the five Mpumalanga patches i and j at a rate 1 if i or j = 6 (local movement only) where (x i , y i ) and (x j , y j ) are the centroid coordinates for patches i and j respectively. This movement is weighted inversely by distance so that movement between South African patches that are closer together occurs at a higher rate than those further apart.
2. Movement may occur when South African citizens cross the border into Maputo (from patch i = 1 − 5 in sub-patch 1 to patch 6 in sub-patch 3) and return (patch 6 in sub-patch 3 to patch i = 1 − 5 in sub-patch 2) at a rate of 1 ζi, 6 where where (x i , y i ) and (x 6 , y 6 ) are the centroid coordinates for patches i and 6 respectively. This movement is weighted inversely by distance so that movement between the South African patches and Maputo that are closer together occurs at a higher rate than those further apart.
3. Movement may also occur when Mozambican citizens cross the border into Mpumalanga (from patch 6 in sub-patch 1 to patch j = 1 − 5 in sub-patch 3) and return (patch j = 1 − 5 in sub-patch 3 to patch 6 in sub-patch 2) at a rate of 1 6,j where where (x 6 , y 6 ) and (x j , y j ) are the centroid coordinates for patches 6 and j respectively. This movement is weighted inversely by distance so that movement between the South African patches and Maputo that are closer together occurs at a higher rate than those further apart.
This leads to the following set of differential equations.
Sub-patch 1 (Local population): For each patch i with population movement (local or foreign) to − µS i,1 (1) Births in patch i (2) Local incidence arising from sub-patch 1 (3) Recovery of treated infectious stage infections at a rate dependent on the time to seek treatment and the time to recovery (4) Return to full susceptibility at a rate determined by the duration of clinical immunity (5) Assimilation of population in sub-patch 2 (locals having returned from foreign travel) back into sub-patch 1 from whence they originated.
New infections arising from sub-patch 2 due to local transmission and not infections contracted while travelling (20) Assimilation of population in sub-patch 2 (locals having returned from foreign travel) back into sub-patch 1 from whence they originated.      Table 1) The Poisson probability of observing x counts when the average number of counts per week is λ given by As the model is being fitted to time series data with N time bins, λ, the expected number of counts per bin is a function of time. Assuming the independence of data from different time bins, the likelihood reduces to and the log likelihood becomes The model is fitted to 16 sets of data for each weekly time bin: treated cases for three sub-patches in five Mpumalanga municipalities and treated cases for Maputo. Under the assumption of independence, the log likelihood to be maximised is The log-likelihood is negated and minimised using the hydroPSO function implementing a version of the Particle Swarm Optimisation algorithm in the R package hydroPSO v0.3-3 [18,19]. Particle Swarm Optimisation is a global stochastic optimisation technique initially inspired by social behaviour of birds and fish [20,21]. It shares similarities with evolutionary optimisation techniques like Genetic Algorithms (GA) but explores the multi-dimensional solution space on the basis of individual and global best-known "particle positions" without evolution operators. Problems are optimised by moving particles (the population of candidate solutions) around the search-space based on the particles' position and velocity.
Particle movements are a function of local best positions and other best particle positions in the search-space. Thus the particles "swarm" towards the best solutions in the search-space.
The parameters estimated through the model fitting process are presented in Table 1 The 95% uncertainty range for weekly case predictions is shown.

Test of Particle Swarm Optimisation routine
A synthetic test of the PSO routine is presented to assess if the method is capable of estimating the true model parameters and data given the high dimensionality of model. A set of parameters is prescribed for the model, and the model is run stochastically with this set of parameters to generate a dataset ( Table 3).
The optimisation routine is then performed on the model to assess if the search algorithm is able to locate the underlying parameter values. Figure 2 shows that the synthetic dataset closely resembles the deterministic model output generated with the true parameter values with a few noticeable differences: Malaria incidence in Nkomazi and Maputo are lower in the synthetic dataset, and rate of movement between residents of Maputo and Umjindi and Nkomazi are lower in the synthetic dataset. Figure 3 shows that the red line (model with estimated parameters from PSO routine) very closely resembles the synthetic dataset (black) and the estimated parameter values are in the region of the "true" parameter values (Table 3). As to be expected, the estimated β values for Nkomazi and Maputo are below the true values and the rate of movement from Maputo to the local municipalities is lower than the true value.

Migration rate sensitivity analysis
A test of the sensitivity of results to changes in the effect of varying coverage, detection thresholds, take-up proportions and adherence is presented for different levels of migration. As in the main paper, the decrease in local infections was measured for each combination of factors and a linear model regressing these four factors on the decrease in local infections was fitted to assess sensitivity. The absolute standardised regression coefficients in Figure 4 suggest that regardless of the level of migration and holding the other factors constant, detection threshold in an FSAT campaign has the largest absolute impact on decreasing local infections, followed by coverage achieved and take-up proportion.