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Dr. Adquate Mhlanga

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The Program for Experimental and Theoretical Modeling, Division of Hepatology, Department of Medicine, Stritch School of Medicine, Loyola University Chicago, Maywood, IL 84101, USA
Interests: viral dynamics; nonlinear mixed-effects models; mathematical epidemiology; population dynamics; ecology and evolution; data science; model development

Special Issue Information

Dear Colleagues,

This aim of this issue is to explore in detail the application of differential equations (partial differential equations, delay differential equations, stochastic differential equations, and ordinary differential equations) in understanding and controlling infectious diseases. The focus will be on how these mathematical models describe and predict the dynamics of epidemics and pandemics, specifically on the spread and curtailing of various infectious diseases. Key themes include advancements in classic epidemiological models, such as SIR and SEIR, and the introduction of new methods to handle greater complexity, such as spatial dispersal, heterogeneous mixing patterns, agent-based models, and stochastic effects. This issue also welcomes models that address real-world challenges, such as limited healthcare resources and vaccination optimization. Additionally, we seek to highlight models that incorporate environmental and socioeconomic factors influencing disease transmission. Special consideration will be given to models that aid in understanding disease resurgence after initial control efforts. This issue intends to bridge theory and practice by demonstrating how these models can inform public health policy and intervention strategies. Contributions that advance theoretical understanding of stability and bifurcation in disease-free and endemic equilibria are also welcomed. The role of delay differential equations in capturing time-lagged effects in disease transmission dynamics will be examined, alongside the application of optimal control theory. Expected submissions include original research articles, review articles, case studies, simulation studies, as well as perspectives and commentaries.

Dr. Adquate Mhlanga
Guest Editor

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Research

33 pages, 947 KB

Open AccessArticle

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

by Mohammed H. Alharbi and Ali Rashash Alzahrani

Mathematics 2026, 14(7), 1219; https://doi.org/10.3390/math14071219 - 5 Apr 2026

Viewed by 324

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

R_{0}^{d}

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

R_{0}^{d} \leq 1

, while a unique endemic equilibrium emerges and is globally stable when

R_{0}^{d} > 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

τ_{3}

possess the largest negative elasticities. Critically,

τ_{3}

exerts exponential influence via

e^{- n_{3} τ_{3}}

, making interventions that delay infectiousness—such as post-exposure prophylaxis—unusually potent. We derive an explicit expression for the critical delay

τ_{3}^{cr}

at which

R_{0}^{d} = 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. Full article

(This article belongs to the Special Issue Advances in Epidemiological and Biological Systems Modeling)

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43 pages, 1173 KB

Open AccessArticle

A New Hybrid Stochastic SIS Co-Infection Model with Two Primary Strains Under Markov Regime Switching and Lévy Jumps

by Yassine Sabbar and Saud Fahad Aldosary

Mathematics 2026, 14(3), 445; https://doi.org/10.3390/math14030445 - 27 Jan 2026

Viewed by 296

Abstract

We study a hybrid stochastic SIS co-infection model for two primary strains and a co-infected class with Crowley–Martin incidence, Markovian regime switching, and Lévy jumps. The model is a four-dimensional regime-switching Lévy-driven SDE system with state-dependent diffusion and jump coefficients. Under natural integrability conditions on the jumps and a mild structural assumption on removal rates, we prove uniform high-order moment bounds for the total population, establish pathwise sublinear growth, and derive strong laws of large numbers for all Brownian and Lévy martingales, reducing the long-time analysis to deterministic time averages. Using logarithmic Lyapunov functionals for the infective classes, we introduce four noise-corrected effective growth parameters

λ_{1}, \dots, λ_{4}

and two interaction matrices

A, B

that encode the combined impact of Crowley–Martin saturation, regime switching, and jump noise. In terms of explicit inequalities involving

λ_{k}

and the entries of

A, B

, we obtain sharp almost-sure criteria for extinction of all infectives, persistence with competitive exclusion, and coexistence in mean of both primary strains, together with the induced long-term behaviour of the co-infected class. Numerical simulations with regime switching and compensated Poisson jumps illustrate and support these thresholds. This provides, to our knowledge, the first rigorous extinction-exclusion-coexistence theory for a multi-strain SIS co-infection model under the joint influence of Crowley–Martin incidence, Markov switching, and Lévy perturbations. Full article

(This article belongs to the Special Issue Advances in Epidemiological and Biological Systems Modeling)

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