Stability analysis and numerical simulation of a Caputo-type fractional SIRS model with three time delays

Abstract

A susceptible–infectious–recovered–susceptible (SIRS) epidemic model with waning immunity is formulated in the Caputo fractional-derivative framework and augmented with three delays: latency (\(\tau\)), recovery (\(\delta\)), and immunity loss (\(\xi\)). We establish well-posedness (non-negativity, boundedness, positive invariance of a feasible region) and derive the basic reproduction number \(\mathcal{R}_0\) by the next-generation approach, showing the endemic equilibrium exists iff \(\mathcal{R}_0>1\). Linearisation yields a transcendental characteristic equation whose delay dependence collapses to two effective delays, \(\tau\) and the loop delay \(\rho=\tau+\delta+\xi\). Using a Rouché-type argument on the boundary of the fractional stability sector, we prove the endemic equilibrium is delay-independently locally asymptotically stable whenever an explicit condition on the characteristic coefficients holds; this is established analytically at the sector vertex, yielding a closed-form margin \(\beta I^{*}\mu(\mu+\sigma+\gamma)>0\), and verified numerically elsewhere over an extensive parameter window. Consequently, unlike the oscillatory behaviour often reported for delayed epidemic models, the present model exhibits no delay-induced Hopf bifurcation for the parameter ranges considered: delays lengthen the transient but do not destabilise the endemic state, and a smaller fractional order is mildly stabilising. The integer-order limit recovers a classical delayed SIRS system, extending known delay-independent stability results for related SIR models to fractional order and multiple delays. All findings are corroborated by a delay-adapted fractional Adams–Bashforth–Moulton predictor–corrector scheme, whose second-order accuracy is confirmed by an empirical self-convergence study and comparison with a classical solver.