Analyzing steady-state equilibria and bifurcations in a time-delayed SIR epidemic model featuring Crowley-Martin...

This article presents a time-delayed SIR epidemiological model that has been quantitatively examined. The model incorporates a logistic growth function for the susceptible population, a Crowley-Martin type incidence, and Holling type II treatment rates. We investigated two separate time delays. The first delay refers to the rate at which new infections occur, allowing us to evaluate the impact of the latent period. The second delay relates to the rate of treatment for those who have contracted the infection, which allows us to examine the consequences of postponed access to therapy. The investigation of the steady-state behavior of the model emphasizes two equilibria, namely, the infection-free equilibrium and the endemic equilibrium. The determination of critical values involves the use of the fundamental reproduction number, denoted $R_0$ , which serves as a predictive measure to determine the potential elimination of a disease within a specific population. Using the fundamental reproduction number, it can be shown that infection-free equilibrium exhibits local asymptotic stability when the value of $R_0$ is less than 1. In contrast, when $R_0$ exceeds 1, the infection-free equilibrium becomes unstable in the context of the time-delayed system. Furthermore, an analysis of the steady-state dynamics of the endemic equilibrium indicates the appearance of oscillations and periodic solutions with the Hopf bifurcation for all feasible combinations of two-time delays as the bifurcated parameter. In sensitivity analysis, a sensitivity index is utilized to evaluate the relative modification in the fundamental reproduction number caused by each parameter. In summary, numerical simulations are employed to offer empirical evidence for the theoretical findings.