Fractional-order epidemic modeling with a deep neural network framework.

Infectious diseases still represent a significant world threat because of their contagiousness, changeability and immense influence on the health care systems of the population. The effective approach to public health planning requires accurate modeling of infectious diseases, especially to capture complex dynamics and memory-dependent effects that are not always taken into account by traditional models. This work suggested a new fractional-order [Formula: see text] (Susceptible-Infectious-Treated-Recovered) epidemiological model that has a saturated incidence function and uses the Caputo derivative, which enables the system to retain the effects of long-term memory. A qualitative test is carried out to determine whether the solutions are positive, bounded and biologically feasible, such that the results of the model remain of epidemiological use. The basic reproduction number is calculated and evaluated to measure the capacity of the disease to be transmitted and to ascertain the conditions needed to control an outbreak. The equilibrium points are also determined and their stability behaviour is also analyzed to understand what the system does in the long run, as well as to describe the conditions under which the disease either persists or dies off. The model is numerically solved by the Adams-Bashforth-Moulton predictor-corrector method, which is the most appropriate numerical scheme to resolve a model with nonlocal dynamics caused by fractional derivatives, where it is not expected that a closed-form solution can be found. In order to improve computational efficiency and offer an independent validation of the numerical output, the Bayesian Regularized Deep Neural Network (BR-DNN) is used as a data-driven surrogate solver. In order to prove the impact of the fractional order on the dynamics of transmission and treatment of diseases, numerical experiments are designed to show the importance of the effects of memory in the development of epidemics. The validation of predictions is done using absolute error plots, the regression analysis, the function fitting graphs, the histogram and the evaluation of mean squared error. Numerical and BR-DNN solutions are in high agreement, which proves the validity of the suggested framework.