A Mixed Fractional‑Order Stochastic Malaria Model with Immune Memory and Extinction Thresholds: Implications for Elimination Strategies in Nigeria

Despite vector control efforts, Nigeria remains highly endemic for malaria. Classical epidemiological models ignore both environmental variability and immunological memory, and these two factors critical to malaria transmission dynamics. This study introduces a novel mixed fractional-order stochastic malaria model incorporating human immune memory (Caputo derivatives, α = 0.9) and environmental stochasticity in mosquito dynamics (integer-order SDEs with multiplicative noise). We prove well-posedness, positivity, boundedness, and global Mittag–Leffler stability of the disease-free equilibrium when R₀ < 1. Using a stochastic next-generation operator, we derive the stochastic reproduction threshold  showing that Rₛ < 1 ensures almost-sure extinction. Calibrated to Nigerian malaria surveillance data, the fractional model delays epidemic peaks by 12–18 days and reduces peak prevalence by 15–20% relative to integer-order models. Although the deterministic model predicts persistence at R₀ = 35, environmental fluctuations induce extinction probabilities of 0%, 18%, and 73.2% at σ = 0.05, 0.15, and 0.30, respectively. Sensitivity analysis identifies environmental variability and transmission rates as the dominant determinants of elimination feasibility. Our findings demonstrate that reliance solely on deterministic R₀ underestimates elimination opportunities. We propose policy recommendations for Nigeria, including seasonal intervention windows and monitoring prevalence variability as indicators of elimination potential.