Botswana is semi-arid to arid with few permanent water bodies. The country is flat, mostly between 900 and 1200 m altitude. The rainy season is from November to March. Vegetation ranges from desert scrub-land in the South-West, where annual rainfall is <300 mm, through grassland, to wooded savannah in the North, which receives >500 mm rain annually. Mean annual temperatures are between 18 and 23°C. Botswana today has a total population of about 1.6 million; population density over two thirds of the country being <1 per square km 12. The population according to the 1971 census was 630379 with an approximate 3.1% annual increase 13 which if extrapolated back in time translates to around 470000 in 1961/62. In 1975 80% of the population lived in the eastern part of the country,

Malaria risk is highest in the tropical North (figure 1). Indoor residual spraying was introduced in 1946 on a limited scale. Coverage was gradually improved culminating in a comprehensive vector control program in the 1980's 14, but even by 1953 indoor residual spraying for mosquito control was a "regular feature" in risk areas, apparently mainly in towns, along rivers and apparently excluding rural areas "remote from regular medical supervision", but with good results 15. Larval control was also implemented when mosquito breeding was detected. Malaria prevalence decreased markedly after 1944, again between 1961/62 and 1974, and further thereafter 14. By 1960 no prevalence above 70% was measured, suggesting meso-to hypo-endemic conditions. Further South, transmission is hypo-endemic and epidemic, and over large areas entirely absent. Incidence, like the climate, is strongly seasonal, peaking around March/April 16. The gradient in malaria broadly follows the environmental gradients described before.

Archived malaria prevalence data were collated within the MARA project, as described by Omumbo et al 17. In Botswana geographical coordinates could be obtained for 613 out of a total of 1063 age-specific prevalence surveys. Of these, 20 did not report sample sizes and were excluded. Here we used only the 1961/62 national survey (figure 1) to develop a historical malaria risk map. For the 1–14 year age group, 122 prevalence results were available, for 118 unique locations across the country, progressively from August 1961 to May 1962. Surveys in different regions were carried out during different months (figure 2). The total number examined was 17149; the mean sample size was 141 per survey (range 2–831). The design effect was calculated in Stata 18.

Forty-nine variables representing different summaries and transformations of the eight environmental data themes (see table 1), were included in the study: elevation 19, surface water 20, land cover 21, long-term monthly mean rainfall, temperature 22, vapour pressure 23, and normalized difference vegetation index (NDVI) at 8 km 24 and 1 km 25 resolution.

Themes with monthly values (rainfall, temperature, NDVI and vapor pressure) were plotted against logit-transformed malaria prevalence, logit(p). Based on observed temporal patterns in the scatter plots, months were aggregated for "summer" (December to March) and "winter" (April to October). Different annual summary indices were also calculated for each theme. Calculations of some of the variables are shown in the appendix.

Distance from water bodies was calculated by projecting maps of perennial and non-perennial water bodies onto a 200 × 200 m grid and calculating for each grid cell the euclidian distance to the nearest water body. Values were transformed by adding a value of 100 m to each pixel and deriving the natural logarithm.

For land cover, the thirteen United States Geological Survey land cover classes occurring in Botswana were re-grouped into two categories, broadly corresponding to drier and moister land cover types. Most data points were found in "grassland" and "savannah" with only isolated surveys in the other land cover types. Prevalence was generally higher in "savannah" than in "grassland" areas. Other obviously drier and lower risk land cover types ("barren or sparsely vegetated", "shrub land", "urban or built-up") were therefore included with "grassland" in a "low risk" category, while other clearly moister classes ("herbaceous wetland", "water bodies", "evergreen broadleaf forest") were included with the higher-risk "savannah" category. Other minor land cover types were included in the category alongside which they mostly commonly occurred ("grassland/crop land mosaic" was mainly found scattered among "grassland"; "dryland crop land and pasture" and "mixed" among "savannah").

Values were extracted from the data grids for each geographical location where a malaria survey result was available.

We carried out a staged approach during model formulation. A flow chart of the variable selection procedure is shown in figure 3.

SD = ∑ m = 1 12 ( y ^ − y m ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGtbWucqqGebarcqGH9aqpdaGcaaqaamaaqahabaGaeiikaGIafmyEaKNbaKaacqGHsislcqWG5bqEdaWgaaWcbaGaemyBa0gabeaakiabcMcaPmaaCaaaleqabaGaeGOmaidaaaqaaiabd2gaTjabg2da9iabigdaXaqaaiabigdaXiabikdaYaqdcqGHris5aaWcbeaaaaa@3FD8@

where y_m = monthly value and y^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieaacuWF5bqEgaqcaaaa@2E3C@ = mean of all y_m.

Proportional SD = ∑ m = 1 12 ( 0.0833 − p y m ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGqbaucqqGYbGCcqqGVbWBcqqGWbaCcqqGVbWBcqqGYbGCcqqG0baDcqqGPbqAcqqGVbWBcqqGUbGBcqqGHbqycqqGSbaBcqqGGaaicqqGtbWucqqGebarcqGH9aqpdaGcaaqaamaaqahabaGaeiikaGIaeGimaaJaeiOla4IaeGimaaJaeGioaGJaeG4mamJaeG4mamJaeyOeI0IaemiCaaNaemyEaK3aaSbaaSqaaiabd2gaTbqabaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaeaacqWGTbqBcqGH9aqpcqaIXaqmaeaacqaIXaqmcqaIYaGma0GaeyyeIuoaaSqabaaaaa@5689@

where py_m = y_m/y_tot; y_tot = ∑y_m, and 0.0833 is the mean of all py_m (= 1/12)

Effective temperature = [8 * annual mean + 14 * annual range]/[8 + annual range]

Monthly rainfall is expressed as a vector (r_m, θ_m), rainfall being the magnitude (r) of the vector and the month its angle (θ) expressed in units of arc:

θ_m = m2π/12

where m is the month, so that January = 1 and December = 12.

The twelve monthly vectors are added to calculate the total vector (r_t, θ_t):

r t = ( ∑ m = 1 12 r m cos ⁡ θ m ) 2 + ( ∑ m = 1 12 r m sin ⁡ θ m ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCdaWgaaWcbaGaemiDaqhabeaakiabg2da9maakaaabaWaaeWaaeaadaaeWbqaaiabdkhaYnaaBaaaleaacqWGTbqBaeqaaOGagi4yamMaei4Ba8Maei4CamhcciGae8hUde3aaSbaaSqaaiabd2gaTbqabaaabaGaemyBa0Maeyypa0JaeGymaedabaGaeGymaeJaeGOmaidaniabggHiLdaakiaawIcacaGLPaaadaahaaWcbeqaaiabikdaYaaakiabgUcaRmaabmaabaWaaabCaeaacqWGYbGCdaWgaaWcbaGaemyBa0gabeaakiGbcohaZjabcMgaPjabc6gaUjab=H7aXnaaBaaaleaacqWGTbqBaeqaaaqaaiabd2gaTjabg2da9iabigdaXaqaaiabigdaXiabikdaYaqdcqGHris5aaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaaabeaaaaa@5AF4@

θ t = tan ⁡ − 1 ( ∑ m = 1 12 r m sin ⁡ θ m ∑ m = 1 12 r m cos ⁡ θ m ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5CB9@

The concentration index C is calculated as:

C = 100r_t/annual total

Concentration is 100% if all the rain falls in one month and 0% if all months have equal amount of rain.

θ_t is the mean peak month around which rainfall is concentrated.

Bayesian geostatistical model formulation has been described by a number of authors 29303132. Following these authors, the model is specified as follows:

Y_ji represents the binary response corresponding to the infection status of child j at site i (the survey site) taking value 1 if the child tested positive and 0 otherwise. The Y_ji are conditionally independent Bernoulli variables with infection probability p_i at location i.

The p_i are defined via a generalised linear mixed model, to take account of spatial dependence:

logit(p_i) = X_iβ+S(ℓ_i)

where β represents the regression coefficients for a set of known covariates X at all locations ℓ_i of the study area;

S = (S(ℓ₁),...., S(ℓ_n))^T denotes the values of the (unobserved) Gaussian spatial process S(·) at sample locations ℓ_i;

σ² = Var{S(ℓ)}, and Φ is a parameter of the correlation function ρ(d_ij, Φ), in our case exp(-d_ij/Φ), where d_ij is the distance between locations ℓ_i and ℓ_j.

For β flat priors were specified respectively (defaults in geoRglm) and for σ² a Scaled-Inverse chisquare distribution(χ²_ScI) with five degrees of freedom and a mean of 0.5. For Φ a discrete exponential prior with mean of 0.04 and 1000 discretisation points in the interval 0.0001 to 2 was specified.

Convergence was assessed by inspecting plots of traces of simulations for individual parameters. The first 50,000 iterations were discarded; thereafter simulations were run for 250,000 iterations. Every 50th sample was retained. For each model parameter the median and 2.5 and 97.5 percentiles were calculated from the 5,000 MCMC simulations.

Models were compared by calculating the deviance information criterion (DIC) for each model 42. Spatial prediction using Bayesian kriging was carried out for a grid of 2300 locations which correspond to the entire surface of Botswana. For each prediction location a posterior sample of MCMC simulations was generated taking account of the estimates of regression coefficients and the spatial effects at each location, and of the uncertainty of each parameter. This process is described in detail elsewhere 293132, and was carried out using geoR 30.