Modeling the dynamics of malaria with infected immigrants using fractal–fractional techniques with deep neural networks

Abstract The current manuscript investigates the dynamics of a fractal–fractional malaria disease model caused by the Plasmodium parasite, in the framework of Caputo operator. The model under consideration has six compartments, namely, four categories of humans and two of vector populations. By utilizing the fixed point theory, we establish the existence results and uniqueness of solution for the aforementioned model. The Ulam–Hyers (UH) approach is carried out to evaluate the stability of the considered system. For the approximate solution of the model, the fractional Adams–Bashforth iterative techniques have been employed to obtain the numerical solution for the specified system. To simulate the model behavior, we use various values for the fractional order and for the fractal dimension , based on real parameter values. The application of fractal–fractional calculus is shown to be beneficial in understanding real‐world problems and in managing the global situation of the pandemic. Numerical simulations illustrate the significance of arbitrary order derivatives, suggesting that fractal–fractional order offer more insights into the dynamics of the proposed malaria model. Multiple graphical representations are provided to illustrate the model dynamics across different fractional orders. Finally, we apply a deep neural network (DNN) to analyze the considered system data and demonstrate high accuracy during training, testing, and validation as well as absolute errors.