Mathematical model for prediction of Tuberculosis in Nigeria using hybrid fractional differential equations and artificial neural network methods

In this paper, we developed a hybrid Fractional Order Differential Equation (FODE) and Artificial Neural Network (ANN) Model to study the transmission dynamics of Tuberculosis (TB) in Nigeria. The data used for the analysis were obtained from the TB report by the World Health Organisation (WHO) TB Data Base, Nigeria Dash Board, from the years 2010–-2020. For the comparative analysis in this work, two approaches were presented: a system of four FODEs for the TB model formulated in the Caputo sense, and the Hybrid FODE-ANN framework. These FODEs were discretized, and the parameter values were numerically estimated using the Grünwald-Letnikov method while the Hybrid FODE-ANN framework features a NN architecture with one input layer, 15 hidden layers of 100 neurons each, and a hyperbolic tangent (tanh) activation function. Training of the NN involves minimizing a loss function combining data fit and system constraints, optimized using the Adam and L-BFGS algorithms, achieving a high degree of accuracy with an MSE of 6.005×10⁻⁶. The result of FODEs shows an R-square estimation accuracy of 0.9968 but was not sufficiently reliable for predictions. Key findings using the Hybrid FODE-ANN framework reveal a steady decline in the susceptible population, reflecting continuous exposure to TB, and an increasing transmission rate, as estimated by the model. Predictions for the exposed, infected, and recovered compartments fit the observed data, with notable exponential growth in infections and recoveries post-2020. Fractional-order parameters, dynamically estimated during training, demonstrate the dynamical behaviour of TB progression under the hybrid framework. These results highlight the urgent need for enhanced TB control measures in Nigeria, including scaled-up vaccination programs, early diagnosis, isolation protocols, public health awareness, and targeted interventions for high-risk groups.