1. Introduction

2. Material and Methods

2.7. Data Modeling

In this study, statistical, machine learning, and deep learning models such as Negative Binomial Regression (NBR), Seasonal AutoRegressive Integrated Moving Average with Exogenous Regressors (SARIMAX), Extreme Gradient Boosting (XGBoost) Regression, and Long Short-Term Memory (LSTM) were deployed to predict dengue cases in BRVT. The overall framework of the study and step-by-step modeling workflow is illustrated in Figure 1. Dengue cases were treated as the dependent or response variable, while the climatic variables and their respective lags were used as independent variables.

Figure 1. Overall process of DF case prediction study.

Figure 1. Overall process of DF case prediction study.

2.7.1. Negative Binomial Regression (NBR)

Negative Binomial Regression (NBR) was selected to predict dengue case counts due to the overdispersed nature of the data, where the variance substantially exceeded the mean. NBR is preferable to Poisson regression in such cases, as it introduces a dispersion parameter to account for overdispersion [13].

The NBR model relates the expected number of dengue cases (${\mu }_i$) to the predictor variables ($X_i$) through a log link function:

$$\log \left({\mu }_i\right)={\beta }_0+\sum _{k=1}^p{\beta }_k{X}_{ik}$$

where: ${\mu }_i$ is the expected count of dengue cases, $X_{ik}$ are the predictor variables (e.g., temperature, humidity, lagged climatic variables), ${\beta }_k$ are the regression coefficients, ${\beta }_0$ is the intercept.

NBR introduces a dispersion parameter α to account for overdispersion, which modifies the variance function:

$$Var\left(Y_i\right)={\mu }_i+\alpha {\mu }_i^2$$

This variance function allows the model to adjust for the extra variability observed in dengue case counts [13].

Model fitting was performed using maximum likelihood estimation (MLE). The dispersion parameter α was tuned via cross-validation. A grid search was conducted over a range of α values (0.001 to 2.0) to optimize model performance based on the Mean Absolute Error (MAE) on the test set.

The steps for model tuning included: (i) Train-Test Split—Data was used for training (80%), and data was used for testing (20%) to evaluate the model's predictive performance; (ii) Cross-Validation—A rolling window time-series cross-validation approach was employed to prevent data leakage and ensure temporal consistency [14]; (iii) Hyperparameter Optimization— GridSearchCV was used to identify the best-fitting model by minimizing MAE [15].

2.7.2. Seasonal AutoRegressive Integrated Moving Average with eXogenous Regressors (SARIMAX)

The SARIMAX model was selected for dengue case prediction due to its ability to model both temporal dynamics and the influence of exogenous variables, such as climate factors. SARIMAX extends the classical ARIMA (Autoregressive Integrated Moving Average) model by incorporating seasonal components and exogenous variables, which makes it particularly well-suited for modeling seasonal data with external influences.

The SARIMAX model is defined by the following components:

AR (Autoregressive) component: This component captures the relationship between an observation and its previous values.

I (Integrated) component: This accounts for differencing the series to make it stationary, reducing the impact of trends.

MA (Moving Average) component: This models the error terms as a linear combination of past error terms.

Seasonality: SARIMAX accounts for seasonality through seasonal AR, I, and MA terms.

The general SARIMAX model is written as:

$$Y_t=\sum _{i=1}^p{\varnothing }_i{Y}_{t-i}+\sum _{j=1}^q{\theta }_j{ϵ}_{t-j}+X_t\beta +{ϵ}_t$$

where: $Y_t$ is the dengue case count at time t, ${\varnothing }_i$ are the autoregressive coefficients, ${\theta }_j$ are the moving average coefficients, ${ϵ}_t$ is the error term (white noise), $X_t$ represents the exogenous variables (e.g., climate variables such as temperature, humidity, precipitation), β is the vector of coefficients associated with the exogenous variables.

For the seasonal component, the model is extended by introducing seasonal autoregressive (SAR), seasonal moving average (SMA), and seasonal differencing terms, denoted as P, D, and Q, respectively. The seasonal component is added with periodicity s, representing the number of time steps in a season.

The full SARIMAX model is typically written as:

$${SARIMA(p,d,q)(P,D,Q)}_s$$

where: p, d, q represent the non-seasonal parameters (autoregressive, differencing, and moving average), P, D, Q are the seasonal counterparts, s is the length of the seasonal cycle.

The exogenous variables ($X_t$) included in the SARIMAX model were climate factors such as temperature, humidity, and precipitation, along with their lagged versions. These variables were selected based on their known impact on dengue transmission and mosquito population dynamics [16]. Lagged versions of these variables (1 to 4 weeks) were included to capture delayed climatic effects on dengue incidence.

The SARIMAX model was tuned using a grid search approach to identify the optimal values for the parameters p, d, q, P, D, Q, and s. The following steps were undertaken: (i) Train-Test Split—The dataset was split into training and testing sets. The training set was used to fit the model, while the test set was used for out-of-sample evaluation; (ii) Grid Search—A grid search was conducted over a range of possible values for the hyperparameters p, d, q, P, D, and Q, along with seasonal periodicity s = 52, to find the optimal combination that minimized the Akaike Information Criterion (AIC) [17]; (iii) Cross-Validation—Time series cross-validation using a rolling window approach was implemented to ensure that temporal dependencies were preserved and to prevent data leakage during model evaluation [14].

2.7.3. Extreme Gradient Boosting (XGBoost) Regression

XGBoost (Extreme Gradient Boosting) was selected for this study due to its efficiency and performance in handling large, complex datasets with non-linear relationships. XGBoost is a supervised machine learning algorithm that builds an ensemble of decision trees using a boosting technique to improve prediction accuracy. The algorithm works by iteratively adding weak learners (trees) to minimize the residual errors from previous iterations.

XGBoost operates by minimizing a loss function, typically the Mean Squared Error (MSE) for regression tasks, along with a regularization term to control model complexity and prevent overfitting. The objective function is given by:

$$Obj\left(\theta \right)=\sum _{i=1}^{n}L(y_i,{\widehat{y}}_i)+\sum _{k=1}^K\mathsf{\Omega }\left(f_k\right)$$

where: $L(y_i,{\widehat{y}}_i)$ represents the loss function (e.g., squared error) between the actual values $y_i$ and predicted values ${\widehat{y}}_i$, $\mathsf{\Omega }\left(f_k\right)$ is the regularization term applied to each decision tree $f_k$, which penalizes the complexity of the model to avoid overfitting [18].

The regularization term is defined as:

$$\mathsf{\Omega }\left(f\right)=\gamma T+{\displaystyle \frac{1}{2}}\lambda \sum _{j=1}^T{w}_j^2$$

where T is the number of leaves in the tree, $w_j$ are the leaf weights, γ and λ are regularization hyperparameters that control the trade-off between model complexity and training performance.

The XGBoost model includes several key hyperparameters that were tuned using grid search to optimize the model's performance: (i) Learning Rate (η)—Controls the step size at each iteration of boosting. A lower learning rate requires more trees to achieve convergence but typically leads to better generalization; (ii) Max Depth—Controls the maximum depth of each decision tree. A deeper tree captures more complex patterns but increases the risk of overfitting; (iii) Subsample—Fraction of the training data used to build each tree. Subsampling helps prevent overfitting by introducing randomness into the training process; (iv) Gamma (γ)—Minimum loss reduction required to make a further partition on a leaf node of the tree; (v) Min Child Weight—Minimum sum of instance weights needed in a child node.

The hyperparameters were optimized using GridSearchCV with 5-fold cross-validation to ensure robustness. The combination of hyperparameters that minimized the Root Mean Square Error (RMSE) was selected for the final model [18].

2.7.4. Long Short-Term Memory (LSTM)

Long short-term memory (LSTM) is the most effective approach for time series forecasting [19], where classical linear methods are difficult to use multivariate forecasting issues. Among various machine learning methods, the recurrent neural network (RNN) uses feedback loops; subsequently, the network can learn the sequence of information. However, standard RNN fails to remember the long-term information; hence, the present study proposes LSTM network for dengue prediction.

LSTM is the recurrent neural network (RNN) architecture and it was designed by the Hochreiter and Schmidhuber to address the vanishing and exploding gradient problems of traditional RNNs [20]. These vanishing and gradient issues hinder the model accuracy. The LSTM consists of different blocks called memory cells. Memory block contains memory cells and gates. Memory cells are able to remember and manipulating the temporal state of the memory or network by self-connections and the process of memory or information controlled through three gates. The first gate called as “Forget” gate (fj), which is responsible for removing information. The second gate is “Input” gate (ij), which is responsible for addition of information to the cell state. The final gate is “Output” gate (oj), which selects the useful information from current cell and shows it as an output [21]. These forget gate, input gate, and output gate of LSTM are described by [20] using the following notations mentioned below:

$$f_j=\sigma (w_f·\left[h_{j-1},x_j\right]+b_f)$$

$$i_j=\sigma (w_i·\left[h_{j-1},x_j\right]+b_i)$$

$$o_j=\sigma \left(w_o·\left[h_{j-1},x_j\right]+b_o\right)$$

$$\widehat{C}=\tan h\left(w_c·\left[h_{j-1},x_j\right]+b_c\right)$$

$$C_j=f_j*C_{j-1}+i_j*\widehat{C}$$

$$h_j=o_j\ast \tan h\left(C_j\right)$$

where $f,i,o$ are forget gate, input gate and output gate respectively. The input gate is used to decide the information to be stored in the cell, forget gate is used to decide what kind of information is to be dropped from cell and output gate is used to decide output from the cell. $\sigma$ represents the logistic sigmoid function. $w,b$, and $h$ are weights, biases, and value of hidden layers respectively. $\widehat{C}$ and $C$ are vector of new candidate values and cell state respectively.

3. Results

4. Discussion

5. Conclusion

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