Measles remains a significant public health threat despite widespread vaccination, with recent resurgences driven by vaccine hesitancy and coverage gaps. Existing mathematical models often fail to capture the substantial temporal heterogeneity in incubation periods, vaccine-induced protection, and recovery processes that characterize measles transmission. We develop and analyze an SVEIR epidemic model incorporating four independent distributed time delays with exponential survival factors, capturing the realistic variability in these epidemiological processes. The model features compartment-specific mortality rates, disease-induced mortality, and imperfect vaccination with failure probability θ. Using next-generation matrix methods adapted for delay kernels, we derive the delay-dependent reproduction number R0d and prove, via systematic construction of Volterra-type Lyapunov functionals, that it constitutes a sharp threshold: the disease-free equilibrium is globally asymptotically stable when R0d≤1, while a unique endemic equilibrium emerges and is globally stable when R0d>1. Normalized forward sensitivity analysis reveals that the transmission rate β and recruitment rate Λ exhibit maximal positive elasticity, while the vaccination rate p, vaccine failure probability θ, and incubation delay τ3 possess the largest negative elasticities. Critically, τ3 exerts exponential influence via e−n3τ3, making interventions that delay infectiousness—such as post-exposure prophylaxis—unusually potent. We derive an explicit expression for the critical delay τ3cr at which R0d=1, demonstrating that prolonging the effective incubation period sufficiently can shift the system from endemic persistence to extinction. Numerical simulations using Dirac delta kernels confirm all theoretical predictions. These findings provide three actionable insights for public health: (1) maintaining high vaccination coverage among new birth cohorts remains paramount; (2) improving vaccine quality (reducing θ) yields substantial returns; and (3) the incubation delay represents a quantifiable, measurable target for evaluating the population-level impact of time-sensitive interventions. The framework is broadly applicable to infectious diseases characterized by significant temporal heterogeneity.
Alharbi et al. (Sun,) studied this question.