Data was obtained by a field study in the locality of Bancoumana, located in the Sudanese savannah zone of the Upper Niger valley (district of Kati) about 60 km south-west of Bamako, the capital of Mali (Figure 1). This locality covers an area of 2.5 km² and has a population of 8,000 inhabitants. A dynamic cohort was constituted in June 1996 and followed up until June 2001. The study included 173 of the 340 households, selected at random from each of the four geographic blocks of the village, using a stratified sampling. In each household, all children aged 0 to 12 years were followed up, constituting the dynamic cohort (for more information, see 5). The surveys 22 were carried out at the rate of about one survey every two months during the rainy season and one every three months during the dry season. The intervals between surveys were defined on the basis of the previous knowledge of the seasonal transmission 4142. For each survey, a blood sample was taken and parasitaemia assessed. A trained team of biologists carried out microscopy to search for P. falciparum and its gametocytes in Giemsa-stained thick blood films. Biological diagnostic was subjected to quality control. Infection was defined as the presence of the parasite in the thick blood film. The time series of incidences of P. falciparum parasitaemia and gametocytaemia were analysed in the present work in order to provide a dynamical model of malaria transmission.

Community permission and individual Informed Consent were obtained according to the stepwise process described by Diallo et al 43.

For remotely sensed data the 15-day composite NDVI provided by the GIMMS group (Global Inventory Monitoring and Modelling Studies) at NASA/GSFC (National Aeronautics and Space Administration/Goddard Space Flight Center) was used (Figure 1). NDVI was derived from channels 1 and 2 of the NOAA AVHRR (Advanced Very High Resolution Radiometer) satellite series 7, 9, 11, 14, 16 and 17. NDVI data were acquired over 25 years from July 1981 to December 2006. Data obtained from the focus on Bancoumana were used into the model transmission.

NDVI was calculated as the normalized difference of corrected reflectance of the NIR (near infrared ranged from 0.725–1.10 μm) and visible (ranged from 0.58–0.68 μm) channels using AVHRR GAC (Global Area Coverage, 4 km resolution) data. The 15-day composites were generated by selecting the maximum value of NDVI, in order to minimize contamination by clouds. Spatial resolution was re-sampled to 8 km × 8 km pixels. The NDVI GIMMS data set was improved using the navigation procedure provided by El Saleous et al 44, the calibration of visible and NIR channels 45. The solar zenith angle values from AVHRR sensor were also corrected 46. Effects of stratospheric aerosols due to volcanic eruptions of El Chichon (1981) and Mount Pinatubo (1991), during April 82-December 84 and June 91-December 93, have been corrected using the method developed by Vermote et al 47. No correction has been applied to correct for atmospheric effects due to water vapour, Rayleigh scattering or stratospheric ozone.

An additional Quality Control was applied to the NDVI data set to filter unrealistic values (i.e. values larger than 1 or smaller than -1). NDVI values retrieved from spline interpolation or average seasonal profile have been considered as missing data. For each fortnight, data were calculated using the maximum NDVI value of 15-day composites, in order to provide time series of vegetation characteristics, in the locality of Bancoumana.

The statistical relationship between NDVI and incidence of P. falciparum infection was assessed by classical ARIMA time series analysis 4849 after logarithmic transformation of the incidences. These established statistical models have been used to model time series, by breakdown into tendency, cyclic and accidental components, and also to identify significant predictor 50. Observed NDVI was introduced in the ARIMA analysis as a covariate and tested, and temporal delays were also analysed.

ROC (Receiver Operating Characteristic) analysis was used to determine an NDVI threshold predicting an increase in the parasitaemia incidence. The quality of this threshold was assessed by AUC test (Area Under the ROC Curve) and by the DOR (Diagnostic Odd Ratio) [see for example 5152]. Statistical analysis was performed using SPSS 15.0^® (SPSS Inc., Chicago, Ill., USA). A significance level of α = 0.05 was used for hypothesis tests.

Malaria transmission was modelled using a deterministic approach. A SIRS-type model 1923 was adapted to Bancoumana's data. The model was built on the MacDonald equations, specifying states for infected-not-contagious and contagious children (such as Bailey's model 20) and adding a resistant state (such as Dutertre's model 23). The first state S was defined as the proportion of susceptible children. The second state I represented the proportion of infected but not contagious children, i.e. children without gametocytaemia. The third state G represented the production of contagious children, i.e. children with gametocytaemia. Indeed, the transmission needs two parasitic cycles, an asexual cycle in human and a sexual cycle in Anopheles, this latter is made possible by gametocyte production in human. The last state R represented the proportion of children "resistant" to infection, i.e. children were considered as resistant during the effectiveness of curative treatment (Figure 2). The transition from state S to state I depended on vectorial and climatic factors (i(t)), and children were considered without effective immunity. Demographic factors as human natality and mortality have been neglected. Infected but not contagious children (state I) could become contagious with a parameter η₁ (production of gametocytes) or resistant with a parameter γ (curative treatment). Contagious children (state G) could loose their contagiousness with a parameter η₂, or could become resistant (with the parameter γ). The parameter δ represented the inverse of the duration of the treatment effectiveness.

The vectorial part of the cycle was modelled with a two-state model: the state of susceptible Anopheles (A_s) and the state of contagious Anopheles (A_i). The transition took place when susceptible Anopheles had a blood meal on contagious children (G), with a parameter i_m(t) depending on vectorial and climatic factors. Vectorial parameters were density (μ), length of the gonotrophic cycle (ν), contagiousness (β), aggressiveness (α), and mortality (ξ). Human contagiousness (ζ) has been added to the model. Model equations have been written as follows:

where VI(t) was the vegetation index (NDVI) and represented environmental factor modelling. Environmental factors influenced vector mortality ξ, length ν of the gonotrophic cycle, vectorial aggressiveness α, with a time lag θ.

Parameter estimations were issued from a review of published works 53. The parameter values have been bounded within the range of published estimations and have to minimize quality indexes (RMSE and MAPE, see later). Furthermore, these values were validated by senior entomologists and parasitologists. Initial conditions were estimated from observed data (June 1996) (Table 1). Anopheles mortality and NDVI values were related by a functional form modelling a slow decrease of mortality when NDVI increased. This relationship provided also a high mortality constant rate for the lowest values of NDVI (during dry season), below a constant threshold (τ). The addition of 1 to the denominator permitted to avoid null values. χ represented the indicator function:

The transmission rates (i(t) and i_m(t)) were also related to NDVI values by a functional form modelling the increase of transmission when NDVI increased and low transmission constant rate during dry season.

The basic reproductive number z₀ has been calculated from these equations:

Note that the basic reproductive number was null for low values of NDVI. No transmission could occur if climatic factors do not favour the normal behaviour of Anopheles.

Environmental factors were considered as an extrinsic variable of the Bancoumana's model. Thus, these factors have been independently modelled. Among the environmental factors related to malaria, observations from 1981 to 2001 were analysed. The extrinsic NDVI variable VI(t) was simulated using a hidden Markov chain model. Observations from 2002 to 2006 served as external validation.

Hidden Markov models (HMM) were introduced by Baum and Petrie at the end of the 60's 5455. This family of stochastic models has been then developed both theoretically (for example 565758) and in terms of applications particularly in hydrology and climatology sciences 596061. These methods make the assumption that the observed data are generated by an underlying finite mixture of distributions, itself organized in a Markov chain (Figure 3). Used for sequence analysis, they provide a classification model of sequence parts. Indeed, the hidden variable can be interpreted as a class of the observed variables.

The hidden Markov model {(S_k, O_k)} is constituted by a set of finite states S_k, k ∈ {1, K}, associated to a probability distribution. Discrete time transitions between these hidden states are provided by transition probabilities, and the resulting time sequence of states (S_t, t > 0) is a homogeneous Markov chain of recursivity order 1:

At time t, for a given state S_t = k, an observation O_t = o is issued following the probability distribution associated to this state, the emission probability p(O_t = o/S_t = k). Then, the sequence of observations (O_t, t > 0) is a sequence of random variables conditionally independent, given the sequence of hidden states.

Such a model is defined by:

• p(S_{t = 1} = k)_{k ∈ {1, ..., K}}, initial probabilities (at time t = 1)

• p(S_t+1 = j/S_t = i)_{(i, j) ∈ {1, ..., K}}², ∀ t, elements of the matrix of transition probabilities

• p(O_t = o/S_t = k)_{k ∈ {1, ..., K}}, emission probabilities

Following this approach, a HMM of NDVI was designed, where the hidden states represented the monthly evolution of climate and environment. An emission probability represented the probability that an NDVI value occurred at a time t, given the environment of a determinate month. A transition probability represented the probability of an environmental change. The EM algorithm was used for the estimation of emission and transition probabilities and then simulating NDVI. The choice of the hidden states (1 state representing each month, 2 months or one season) was conducted by the quality indexes (RMSE and MAPE see below).

Quality of the predictions was performed using the root mean squared error (RMSE) and the mean absolute percentage error (MAPE) defined as follows:

where h was the time-lag of prediction, the prediction at time t, and X_t the observed value at time t.

The complete model was implemented using Matlab 7.0.4 ^®, (Mathworks, Inc., Natick Massachusetts, USA)

The seasonal pattern of P. falciparum incidence was significantly explained by NDVI (Figure 4), with a delay of 15 days (p = 0.001). The value of the adjusted R2 (R_adj2 = 89%) was relatively high, and the quality indexes were relatively low (RMSE = 0.04, MAPE = 5.61). Thus, the statistical model, using NDVI as covariate, showed a satisfactory goodness-of-fit. The known decrease in infection from year to year was significant (p = 0.001), but remained weak (-0.109 after logarithmic transformation, standard deviation SD = 0.031).

The NDVI values observed around Bancoumana were less than 0.34 during the dry season and the highest values (>0.52) have been observed during the rainy season. The ROC analysis has provided an NDVI threshold of 0.361. Beyond this threshold, the odd ratio of an increase in the parasitaemia incidence was significant, estimated at DOR = 2.64 (CI95% [1.26;5.52]).

The area under the ROC curve was 0.65 (CI95% [0.54;0.74]), significantly different from 0.5 (p = 0.007) (Figure 5).

The probabilities of changes in environmental characteristics (the transition probabilities from one month to another) were null if these 2 months were not contiguous. Indeed, environmental characteristics cannot change suddenly. Persistence of environment was also possible. Indeed, environmental characteristics may persist from one month to the next, with a probability of 50%. An environmental change from one month to the next was also possible, with a probability of 50%. These transmission probability were constant, whatever the months were. These estimations reflected the seasonal nature of the phenomenon: persistence of environmental characteristics between 2 contiguous months or progressive changes.

Probabilities of observing specific values of NDVI, the emission probabilities estimated for each month (Figure 6), were not high and reflected also the seasonal changes in NDVI regimes. Indeed, the probabilities that high NDVI values occur in January, February or March were nil and small NDVI values could occur with non-zero probabilities; for example there was 35.0% of chance observing an NDVI between 0.2 and 0.25 during March (cumulative probability). A contrario, the probabilities that high NDVI values occur were important during August and September; for example there was about 47.6% of chance observing an NDVI between 0.6 and 0.65 during September (cumulative probability).

The choice of hidden classes reflecting the monthly scale of seasonal changes was conducted by MAPE and RMSE (Table 2) between predictions and validation set values (2002–2006 NDVI).

The external validation showed the smallest values of MAPE (0.178) and RMSE of (59.63) for a monthly scale of seasonal changes. The model predicted adequately seasonal variations (Figure 7) and then could be used as environmental factor for malaria modelling.

The deterministic transmission model, with stochastic environmental factor, predicted an endemo-epidemic pattern of malaria infection. Indeed, incidences of parasitaemia fluctuated around 70 per 100 inhabitants per 15-days. The model provided a seasonality pattern of incidences, with low values for the dry seasons (about 65%) and high values for the rainy seasons (75%). These oscillations of predicted incidences were similar to observed values (Figure 8 and 9). Quality indexes have shown the smallest values (MAPE = 0.07, RMSE = 0.01 for parasitaemia) for a monthly model of environmental changes (Table 3). Incidences of gametocytaemia fluctuated around 5 per 100 inhabitants per 15-days. Oscillations of predicted gametocytaemia incidences were less pronounced as observed incidences, but indexes have also shown low values (MAPE = 0.01, RMSE = 0.001 for gametocytaemia).

Incidences of parasitaemia and gametocytaemia were adequately modelled, using the observed NDVI as well as the HMM model of NDVI (Figure 8 and 9). Indeed, seasonal variations and mean value of incidences were similar using both NDVI data.