Stability and Bifurcation of a Dual-Strain Epidemic Model with Multiple Time Delays

In this paper, we analyze a dual-strain epidemic model with six compartments and two discrete delays, [Formula: see text] for the original strain and [Formula: see text] for the mutant strain. Strain-specific reproduction numbers [Formula: see text] and [Formula: see text] yield the overall threshold [Formula: see text]. Lyapunov functional is used to show that the disease-free state is globally stable when [Formula: see text]. If one reproduction number exceeds unity while the other does not, the corresponding single-strain endemic state is globally stable. When both exceed unity coexisting equilibria appear; their stability depends on a quasipolynomial that includes the delays. Explicit Hopf bifurcation conditions identify critical delays [Formula: see text] and [Formula: see text] where periodic oscillations arise. Simulations confirm these thresholds and indicate that longer latencies reduce overall infection. Time delays therefore act as key control parameters for persistence, coexistence, and oscillations in multistrain epidemics, guiding effective intervention design that accounts for mutation and vaccination.