Fractional-order two-infection SIR-SIR epidemic model with waning immunity
In this paper, we develop and analyze a six-dimensional Caputo fractional SIR-SIR epidemic model that incorporates waning immunity and secondary infection. The model captures the interaction between primary and secondary transmission, along with demographic effects and memory-dependent dynamics governed by the fractional-order parameter. The basic reproduction number is derived using the next-generation matrix approach and is shown to act as a threshold for disease persistence. The disease-free equilibrium is locally asymptotically stable when the reproduction number is less than one, while a unique endemic equilibrium exists when it exceeds one. The stability results are established using linearization and the Routh–Hurwitz criteria. The analysis further shows that the fractional-order parameter does not alter the threshold condition or the equilibrium structure, but it significantly influences transient dynamics. In particular, memory effects slow down convergence and enhance persistence as the fractional order decreases. Numerical simulations are performed using a predictor-corrector scheme and a semi-analytical method, demonstrating strong agreement and computational efficiency. Sensitivity analysis indicates that the transmission rate plays a dominant role in determining disease spread, while waning immunity parameters influence the intensity of secondary infection. Overall, the proposed fractional framework extends classical integer-order epidemic models by incorporating realistic memory effects and provides additional insight into the temporal evolution and control of infectious diseases.
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