Qualitative analysis and numerical investigations of time-fractional Zika virus model arising in population dynamics

Epidemic models play a crucial role in population dynamics, offering valuable insights into disease transmission while aiding in epidemic prediction and control. In this paper, we analyze the mathematical model of the time-fractional Zika virus transmission for human and mosquito populations. The fractional derivative is considered in the Caputo sense of order $α\in(0,1).$ We begin by conducting a qualitative analysis using the stability theory of differential equations. The existence and uniqueness of the solution are established, and the model's stability is examined through Hyers-Ulam stability analysis. Furthermore, an efficient difference scheme utilizing the standard L1 technique is developed to simulate the model and analyze the solution's behavior under key parameters. The resulting nonlinear algebraic system is solved using the Newton-Raphson method. Finally, illustrative examples are presented to validate the theoretical findings. Graphical results indicate that the fractional model provides deeper insights and a better understanding of disease dynamics. These findings aid in controlling the virus through contact precautions and recommended therapies while also helping to predict its future spread.