SVEIR epidemic model with a delay in diagnosis is studied in a constant and variable environment. The mathematical analysis shows that the dynamics of the model in the constant environment are completely determined by the magnitude of the delay-induced reproduction number Rα. We established that if Rα<1, the disease-free equilibrium is globally asymptotically stable, and when Rα>1 the endemic equilibrium is globally asymptotically stable. In the variable environment, the model undergoes a transcritical bifurcation for Rα=1 leading to changes in the stability of the equilibrium points. The analytical effect of the delays in epidemic diagnosis is investigated. A minimum diagnosisrateαminhas been determined to face or control the disease effectively. Finally, numerical illustrations were presented tosupport the theoretical results

sensitivity, delays in diagnosis, changing environment, Lyapunov function