  This paper investigates a novel nonlinear singular fractional SI model with
the $\Phi_p$ operator and the Mittag-Leffler kernel. The initial investigation
includes the existence, uniqueness, boundedness, and non-negativity of the
solution. We then establish Hyers-Ulam stability for the proposed model in
Banach space. Optimal control analysis is performed to minimize the spread of
infection and maximize the population of susceptible individuals. Finally, the
theoretical results are supported by numerical simulations.
