General model
An autoregressive, additive, geostatistical, linear mixed model was applied to the malaria surveys and the surrounding conflict locations. Autoregressive components are often used in spatiotemporal analyses of malaria prevalence [
43,
44], and enable relative (to its initial value) measures of the
PfPR variation to be produced. Spatial and spatiotemporal autocorrelation has been found to be significant in other conflicts [
11,
42,
45] and malaria studies [
31,
44,
46].
A preliminary analysis was carried out to test if the differences in
PfPR (in the 2–10 age range) before (
PfPR
b) and after (
PfPR
a) conflicts,
ΔP, were more accurately explained by the model than using
PfPR
a as the dependent variable. The results showed that using
ΔP increased the explanatory power of the model by 12 %. In addition, fitting
ΔP enabled removal of some uncertainties due to the transformation of
PfPR
a to
ΔP in the post-modelling stage. Therefore, for each conflict location,
ΔP was calculated as:
$$ \varDelta P = Pf{\text{PR}}_{\text{b}} - Pf{\text{PR}}_{\text{a}} - S_{m} $$
(1a)
$$ S_{m} = A + \left( {B\sin \left( {\theta \cdot m} \right)} \right)\quad \quad m = 1, \ldots ,T $$
(1b)
where
PfPR
a and
PfPR
b are obtained from each malaria survey within 5° and 10 months from the conflict location. In other words only
PfPR collected in different times at the same location are considered (6205 malaria surveys). In fact, taking into account single surveys (
PfPR surveyed only once in a location) and averaging their values with those from other surveys within 5° and 10 months from a conflict event, reduced the explanatory power of the model (−26 %).
The values of 5° (parameter
\( {\varphi} \)) and 10 months (parameter
ρ) were obtained from a Monte Carlo simulation in which, at each iteration, 20 % of the
ΔP values were randomized in space and time. At each of the 10,000 iterations, the gamma variance for each combination of spatial and temporal lags (spatial lags,
\( {\varphi} \), spanning from 0.1° to 10° between conflict location and malaria surveys; and temporal lags,
ρ, spanning from 1 to 100 months between conflict starting date,
c, and the time of the malaria surveys) were calculated by fitting the γ-variances (known as experimental variogram):
$$ {{\gamma \left( {\varphi ,\rho } \right) = \frac{1}{2}\frac{1}{n\left( \varphi \right)}\frac{1}{{n\left( {c + \rho } \right)}}\frac{1}{{n\left( {c - \rho } \right)}}\sum\limits_{i = 1}^{n\left( \varphi \right)} {\sum\limits_{ii = 1}^{{n\left( {c + \rho } \right)}} {\sum\limits_{iii = 1}^{{n\left( {c - \rho } \right)}} {\left( {PfPR_{i,ii} - PfPR_{i,iii} } \right)}^{2} } } }} $$
(2)
with a non-separable, spatiotemporal, exponential function and applying the non-linear minimization method
nml [
47]:
$$ {{\hat{\gamma }\left( {\varphi ,\rho } \right) = \exp \left( { - \frac{d}{\varphi } + \frac{d}{\varphi }\frac{h}{\rho } - \frac{h}{\rho }} \right)}} $$
(3)
where
d and
h are the spatial Haversine distance and the temporal distance, respectively, between conflicts and malaria settings. Five degrees and 10 months are the average spatial and temporal lags obtained from the Monte Carlo simulation. Equation [
2] differs from the canonical equation of the variogram in the lag parameters, as they express the spatial and temporal distances between malaria survey locations and the conflict and not the distance between malaria surveys.
The parameter
S
m
in Eq. (
1b) is the seasonality in the
PfPR modelled as sinusoidal function of the scaling parameter
A, the parameter controlling the amplitude (
B) and the parameter controlling the phase (
θ); finally,
m is the moment along the time series of total length
T. To obtain
S
m
, the average monthly
PfPR from 1997 to 2010 was fitted with a sinusoidal curve (
1b). The parameters
A = 0.31,
B = 0.1 and
θ = 1.8 were obtained through applying a least square method to the time series. While the average correction is only 0.04 of the prevalence rate, the use of the de-seasonalized data in Eq. (
4a) improved the Akaike Information Criterion (AIC) from −17,356 to −19,516.
A method for accounting for the effects of seasonality was implemented in order to avoid the situation where the period before and after conflict may include a seasonal effect. For example, if
PfPR tends to be higher due to seasonality when conflict occurs and lower when measured ‘after conflict’ then the outcome (a measurable decline in
PfPR) may simply reflect seasonal effects. Other methods, i.e., the use of covariates in order to simulate seasonality (i.e., precipitation and temperature) may be useful [
48], however, this does not solve intrinsic periodicities in endemic-malaria countries, and requires user-defined qualitative (i.e., spatial scale of the seasonality for pre-defined regions) and quantitative parameterization of the model. The complexity of modelling malaria seasonality in Africa, in terms of the time of the year, amplitude and phase is discussed elsewhere [
49]. Global, rather than local, corrections for the periodicities in the data are typically applied [
43,
44], but more often the seasonal component is not modelled or removed in national or sub-national prevalence mapping studies [
50]. Finally, the use of an explicit seasonality component, rather than including the seasonality in the covariance function is due to the use of a correlation matrix dependent on the distance from conflict settlings to malaria surveys and not from malaria surveys to malaria surveys (see below term
\( \hat{\gamma } \) in Eq.
4c).
The
PfPR differences at each conflict location and malaria survey were fitted using an additive, geostatistical, linear mixed model containing an autoregressive component (
PfPR
b
); a matrix of covariates
X (fixed effects); a matrix of random effects
W; a spatiotemporal correlation effect,
Z; and an error component
ε:
$$ \Delta P_{q,t} = \beta_{0} PfPR_{b(q,t)} + \beta_{1} X + bW + Z + \varepsilon $$
(4a)
$$ b\sim N\left( {0,\varSigma_{b} } \right) $$
(4b)
$$ Z \sim N\left( {0,\sigma_{z}^{2} \hat{\gamma }\left( {\varphi ,\rho } \right)} \right) $$
(4c)
$$ \varepsilon \sim N\left( {0,\sigma_{e}^{2} {\rm I}} \right) $$
(4d)
$$ \Delta P_{q,t} \sim N\left( {\beta_{0} PfPR_{b(q,t)} + \beta_{1} X,W\varSigma_{b} W^{T} + \sigma_{z}^{2} \hat{\gamma }\left( {\varphi ,\rho } \right) + \sigma_{e}^{2} {\rm I}} \right) $$
(4e)
$$ \varSigma_{b} = \sigma_{b}^{2} {\rm I} $$
(4f)
where the subscripts
q and
t indicate the location and the time of the conflict event, respectively;
β
0
and
β
1
are the regression coefficients for
PfPR
b and (a vector of coefficients) for
X, respectively;
b is a one column vector of aspatial normally distributed random effects with mean zero and covariance matrix
Σ
b, given by the product of the variance
\( \sigma_{b}^{2} \) and the identity matrix
I [
51];
Z is a one column vector with spatiotemporal normally distributed random effects with mean zero and a covariance matrix given by the product of the spatial variance,
\( \sigma_{z}^{2} \), and the correlation matrix,
\( \hat{\gamma } \). As shown above,
\( \hat{\gamma } \) is expressed as a function of the spatial correlation parameter,
\( {\varphi} \), defining the spatial range of
ΔP from the conflict location; and the temporal correlation,
ρ, defining the temporal range of Δ
P from the conflict location (Eq.
3). Finally,
\( \varepsilon \) is the independent and identically normally distributed error, with error variance
\( \sigma_{e}^{2} \)
I. The covariates in
X are: number of conflicts experienced at the malaria survey location, its distance from conflicts and duration of the conflicts at the conflict location. In addition, the typology of conflicts is considered in
X in form of dummy variables (1, Battle-Government regains territory; 2, Battle-no change of territory; 3, Battle-non-state actor overtakes territory; 4, Headquarters or base established; 5, Non-violent activity by a conflict actor; 6, Non-violent transfer of territory; 7, Riots/protests; 8, Violence against civilians; see [
52] for conflict-type definition). Environmental and socio-economic variables are not taken into account because this analysis focuses only on the relationships between spatial and temporal dimensions in conflicts and malaria prevalence surveys (point to point analyses). Including non-conflict variables can likely explain additional variance in the differences in
PfPR but it is unlikely to alter the contribution of the conflict variables in prevalence changes. In order to account for country characteristics, the random effect is the country of the malaria survey location and the macro area. For the latter, a dichotomous variable with values ‘East Africa’ and ‘West Africa’ was employed due to the differences in prevalence sample sizes [
53].
Model (
4a) was the best model from various alternatives where different types of dependent variables (
PfPR, at malaria locations and conflict locations), fixed effects (e.g., longitude and latitude of conflict, longitude and latitude of malaria surveys), spatiotemporal effects (e.g., without
Z, only spatial, using parameters obtained from the canonical variogram of malaria or from the variogram of conflicts) and random effects (only country or macro-area, no random effects) were considered.