Mathematical Investigation of delay-driven dynamics in Q fever

Abstract

The introduction of discrete time delays in epidemic models may cause Hopf bifurcation, leading to recurrent outbreaks. Understanding how these delays influence the onset and suppression of such oscillatory behavior is crucial for predicting epidemic persistence and developing effective control strategies. We proposed a delayed deterministic model for Q fever transmission, with incubation, treatment, and immunisation lags. We establish the model's epidemiological validity and derive the basic reproduction number ($\mathcal{R}_0$) via the Diekmann--van Gils next-generation operator to assess disease persistence. Analytical and numerical bifurcation analyses identify the critical delay thresholds marking the onset of Hopf bifurcation. The theoretical predictions are validated by numerical simulations, which also show how changes in significant delay parameters affect system stability. According to the results, vaccinated livestock must wait a short time before developing complete immunity, and symmetric increases in incubation delays significantly raise the threshold before system stability is eventually achieved. The results offer vital information for creating prompt disease control strategies and agree with the literature that incubation delays and treatment have a stabilizing effect in averting epidemic growth.