Comprehensive Analysis of a Pertussis Model: Stability, Sensitivity, Bifurcation, and Disease Control Approaches

Abstract

Despite widespread vaccination programs, pertussis continues to pose a major public health challenge, primarily due to waning immunity, persistent transmission, and treatment delays. This study introduces an SVEITR model to explore the transmission dynamics and control strategies for pertussis. Unlike many existing pertussis models, this framework explicitly incorporates a treatment compartment alongside vaccination and recovery, allowing for a more thorough evaluation of disease control measures. The fundamental mathematical properties of the model are established, equilibrium points are derived, and the basic reproduction number R0 is calculated. Theoretical conditions for local and global stability of the equilibrium points are rigorously analyzed using eigenvalue theory, the Routh–Hurwitz criterion, the Castillo-Chavez methodology, and Lyapunov theory. Bifurcation analysis, conducted via center manifold theory, reveals a forward transcritical bifurcation at R0=1, confirming that disease eradication is possible when R0<1. Local and global sensitivity analyses identify the most influential epidemiological parameters affecting disease dynamics (i.e., transmission, vaccination, and treatment rates). Numerical simulations illustrate the effects of vaccination, transmission, and treatment rates on disease progression and delineate stable and unstable endemic regions. The findings indicate that increased vaccination coverage and prompt treatment significantly reduce disease prevalence, whereas higher transmission rates intensify outbreaks. The proposed model offers new insights into the combined effects of vaccination and treatment in controlling pertussis and provides a quantitative framework for designing effective public health intervention strategies.