Caputo fractional-order mumps dynamics with two-dose vaccination and multiple interventions

Mumps remains a public-health concern where vaccination is incomplete and close-contact transmission persists. We develop a dimensionally consistent Caputo fractional-order model with two-dose vaccination, public-health education, asymptomatic carriage, symptomatic infection, isolation, and recovery. The model is shown to be well posed, and a control reproduction number, R c , is derived. A unique mumps-present equilibrium exists for R c > 1 , while R c < 1 provides a sufficient local stability condition for the mumps-free equilibrium. Hyers–Ulam stability is also established on a finite interval under a Lipschitz condition. Numerical solutions are obtained with a fractional Adams–Bashforth–Moulton PECE scheme. Five-year scenario analyses show that the fractional order mainly changes the temporal response of the system rather than acting as an intervention. Increasing first-dose vaccination markedly redistributes individuals from the susceptible class into the vaccination pathway, whereas increasing second-dose vaccination primarily accelerates progression from first- to second-dose protection. In threshold-relevant transmission scenarios, stronger public-health education reduces early exposure and carrier burden, while faster isolation has its clearest direct effect on reducing symptomatic community infection and increasing transfer into isolation. The results are qualitative and scenario-based because the model is not calibrated to a specific outbreak. The framework provides a mathematical basis for studying how vaccination, behavioural prevention, isolation, and fractional memory jointly shape mumps dynamics.