A Caputo Fractional Stochastic Delay Model of Poliomyelitis Calibrated to Surveillance Data from Pakistan

Pakistan is one of the last two countries where wild poliovirus type 1 (WPV1) remains endemic. We correct structural and parametric problems in the vaccination–sanitation poliomyelitis model of Bornaa, Seidu and Makinde and extend it to a Caputo fractional stochastic model with a delay representing the environmental persistence of shed virus. For the deterministic fractional delay model we prove well-posedness, positivity and boundedness, derive the basic reproduction number R0, show that the disease-free equilibrium is locally asymptotically stable if R0<1 for every fractional order and delay, and establish existence of endemic equilibria. In the Itô case (α=1) we prove global positivity and almost-sure extinction under a threshold condition that is uniform in the noise intensities. A drift-implicit product-integration midpoint scheme attains an observed order close to 1+α and remains stable where explicit fractional methods diverge. Calibrated by maximum likelihood to annual WPV1 cases in Pakistan for 2010–2025, the fractional model (α^=0.95) places the turning points of the series better than the integer-order model, but the evidence for memory is weak (borderline likelihood-ratio p≈0.017, a wide profile interval for α, and an AICc favouring α=1), so α remains partly a modelling assumption. Conditional on the fractional fit, restoring 2015–2018 campaign intensity and reducing transmission by 20% through sanitation would cut the projected 2027–2035 burden by about 74%.