Effect of Behavioral Awareness Delay on the Dynamics of a Discrete-Time SIR Model for Mpox in Kenya: Stability and Neimark–Sacker Bifurcation Analysis
Vivian Nyakio Gachie *
Department of Mathematics, Multimedia University of Kenya, Nairobi, Kenya.
Iyaya C. Wanjala
Department of Mathematics, Multimedia University of Kenya, Nairobi, Kenya.
Joy Mutegi
Department of Mathematics, Multimedia University of Kenya, Nairobi, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Aims/ Objectives: To investigate how delayed behavioral awareness affects the dynamics of a discrete-time SIR model for Mpox transmission, with an emphasis on local stability and the occurrence of a Neimark–Sacker bifurcation.
Study Design: Theoretical mathematical modeling study with analytical stability analysis and numerical simulation. Place and Duration of Study: Multimedia University of Kenya, Faculty of Science and Technology; research conducted between August 2025 and August 2026.
Methodology: A normalized discrete-time SIR model incorporating a population-level behavioral awareness index Bn representing a combination of being informed about Mpox infection risk and adopting protective behavior. A discrete delay t was introduced in the behavioral-awareness response to the infection. The disease-free and endemic equilibria were derived and their local stability was analyzed using linearization and the Jury criterion. For t > 0, the system was linearized about the endemic equilibrium and the spectral radius p examined numerically in MATLAB as t varied. Time-series simulations and phase portraits were used to characterize the resulting dynamics. Time-series simulations and phase portraits were generated to illustrate the dynamics.
Results: For the selected baseline parameter configuration (R0 = 2.07), the endemic equilibrium was locally asymptotically stable for shorter delays. As the delay t increased, a complex-conjugate pair of characteristic roots crossed the unit circle at a critical delay of approximately 9–10 weeks. Numerical simulations showed sustained oscillations, consistent with a Neimark–Sacker bifurcation.
Conclusion: Delayed behavioral awareness can destabilize the endemic equilibrium and induce oscillatory Mpox dynamics through a Neimark–Sacker bifurcation. The findings support the importance of timely behavioral response and sustained risk communication. The estimated 9–10 week threshold is a model-dependent result under the selected parameter configuration and should not be interpreted as an empirical national estimate for Kenya.
Keywords: Mpox, discrete-time SIR model, behavioral awareness delay, Kenya, Neimark-Sacker bifurcation, local stability analysis, spectral radius