Abstract
In this article, a nonlinear fractional bi-susceptible [Formula: see text] model is developed to mathematically study the deadly Coronavirus disease (Covid-19), employing the Atangana-Baleanu derivative in Caputo sense (ABC). A more profound comprehension of the system's intricate dynamics using fractional-order derivative is explored as the primary focus of constructing this model. The fundamental properties such as positivity and boundedness, of an epidemic model have been proven, ensuring that the model accurately reflects the realistic behavior of disease spread within a population. The asymptotic stabilities of the dynamical system at its two main equilibrium states are determined by the essential conditions imposed on the threshold parameter. The analytical results acquired are validated and the significance of the ABC fractional derivative is highlighted by employing a recently proposed Toufik-Atangana numerical technique. A quantitative analysis of the model is conducted by adjusting vaccination and hospitalization rates using constant control techniques. It is suggested by numerical experiments that the Covid-19 pandemic elimination can be expedited by adopting both control measures with appropriate awareness. The model parameters with the highest sensitivity are identified by performing a sensitivity analysis. An optimal control problem is formulated, accompanied by the corresponding Pontryagin-type optimality conditions, aiming to ascertain the most efficient time-dependent controls for susceptible and infected individuals. The effectiveness and efficiency of optimally designed control strategies are showcased through numerical simulations conducted before and after the optimization process. These simulations illustrate the effectiveness of these control strategies in mitigating both financial expenses and infection rates. The novelty of the current study is attributed to the application of the structure-preserving Toufik-Atangana numerical scheme, utilized in a backward-in-time manner, to comprehensively analyze the optimally designed model. Overall, the study's merit is found in its comprehensive approach to modeling, analysis, and control of the Covid-19 pandemic, incorporating advanced mathematical techniques and practical implications for disease management.