Modeling and Analysis of SIRR Model (Ebola Transmission Dynamics Model) with Delay Differential Equation

Ebola virus disease (EVD) is a severe and often fatal illness with high transmission potential and recurring outbreaks. Traditional compartmental models often neglect biologically important delays, such as the latent period before an infected individual becomes infectious, limiting their ability to capture real-world epidemic patterns. Including such delays can provide a more accurate understanding of outbreak persistence and control strategies.

In this study, we develop and analyze a novel deterministic SIRR model that captures the complex transmission dynamics of Ebola by explicitly combining nonlinear incidence rates with a delay differential equation framework. Unlike traditional models, this approach integrates a biologically motivated delay to represent the latent period before infectiousness, providing a more realistic depiction of disease spread. The basic reproduction number (R ₀) is derived using the next-generation matrix, and local stability for disease-free and endemic equilibria is established. Using center manifold theory, we investigate transcritical bifurcation at R ₀ = 1, while Hopf bifurcation analysis determines when delays trigger oscillatory epidemics. Sensitivity analysis identifies parameters most influencing R ₀, and numerical simulations are performed using the fourth-order Runge-Kutta method.

The main novelty of this work lies in its detailed investigation of how delays influence outbreak persistence and can trigger oscillatory epidemics, patterns often observed in practice but rarely captured by classic models. For R ₀< 1, the disease-free equilibrium is locally asymptotically stable; for R ₀> 1, an endemic equilibrium emerges. Increasing delays destabilizes the system, amplifying peak infections, prolonging outbreaks, and producing sustained oscillations. Isolation of recovered individuals (c) significantly reduces R_0, while transmission rate (β), recruitment rate (Λ), and isolation transition rate (ρ) are identified as the most sensitive parameters.

Accounting for delayed recovery dynamics is crucial for accurately predicting outbreak patterns and designing effective interventions. This delay-based, nonlinear-incidence model offers a robust analytical and computational framework for guiding public health strategies, with direct implications for reducing transmission, shortening outbreak duration, and preventing epidemic resurgence.