Global Dynamics for a Distributed Delay SVEIR Model for Measles Transmission with Imperfect Vaccination: A Threshold Analysis

Mohammed H. Alharbi; Ali Rashash Alzahrani

doi:10.3390/math14071219

journal article Open Access Apr 05, 2026

Global Dynamics for a Distributed Delay SVEIR Model for Measles Transmission with Imperfect Vaccination: A Threshold Analysis

Mohammed H. Alharbi

Ali Rashash Alzahrani

Mathematics Vol. 14 No. 7 pp. 1219 · MDPI AG

View at Publisher Save 10.3390/math14071219

Abstract

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.

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Details

Published: Apr 05, 2026
Vol/Issue: 14(7)
Pages: 1219
License: View

Authors

M

Mohammed H. Alharbi

Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia

A

Ali Rashash Alzahrani

Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah 24382, Saudi Arabia

Funding

Umm Al-Qura University Award: 26UQU4320088GSSR01

Cite This Article

Mohammed H. Alharbi, Ali Rashash Alzahrani (2026). Global Dynamics for a Distributed Delay SVEIR Model for Measles Transmission with Imperfect Vaccination: A Threshold Analysis. Mathematics, 14(7), 1219. https://doi.org/10.3390/math14071219

Global Dynamics for a Distributed Delay SVEIR Model for Measles Transmission with Imperfect Vaccination: A Threshold Analysis

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