Stability analysis of a logistic growth epidemic model with two explicit time-delays, the nonlinear incidence and treatment rates

In the present study, a time-delayed SIR epidemic model with a logistic growth of susceptibles, Crowley-Martin type incidence, and Holling type III treatment rates is proposed and analyzed mathematically. We consider two explicit time-delays: one in the incidence rate of new infection to measuring the impact of the latent period, and another in the treatment rate of infectives to analyzing the effect of late treatment availability. The stability behavior of the model is analyzed for two equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). We derive the threshold quantity, the basic reproduction number R-0, which determines the eradication or persistence of infectious diseases in the host population. Using the basic reproduction number, we show that the DFE is locally asymptotically stable when R-0 < 1, linearly neutrally stable when R-0 = 1, and unstable when R-0 > 1 for the time-delayed system. We analyze the system without a latent period, revealing the forward bifurcation at R-0 = 1, which implies that keeping R-0 below unity can diminish the disease. Further, the stability behavior for the EE is investigated, demonstrating the occurrence of oscillatory and periodic solutions through Hopf bifurcation concerning every possible grouping of two time-delays as the bifurcation parameter. To conclude, the numerical simulations in support of the theoretical findings are carried out.

Automated summary

This study proposes and analyzes a time-delayed SIR epidemic model with logistic growth of susceptibles, specific incidence rates, and treatment rates, investigating the stability behavior of the model and the impact of the basic reproduction number on disease spread.