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In this article, we propose and analyze a time-delayed susceptible–infected–recovered (SIR) mathematical model with nonlinear incidence rate and nonlinear treatment rate for the control of infectious diseases and epidemics. The incidence rate of infection is considered as Crowley–Martin functional type and the treatment rate is considered as Holling functional type II. The stability of the model is investigated for the disease-free equilibrium (DFE) and endemic equilibrium (EE) points. From the mathematical analysis of the model, we prove that the model is locally asymptotically stable for DFE when the basic reproduction numberR  0{R_0}    is less than unity(    R  0    &lt;  1  )   ({R_0} \lt 1)    and unstable whenR  0{R_0}    is greater than unity(    R  0    &gt;  1  )   ({R_0} \gt 1)    for time lagτ  ≥  0    \tau \ge 0   . The stability behavior of the model for DFE atR  0    =  1   {R_0} = 1    is investigated using Castillo-Chavez and Song theorem, which shows that the model exhibits forward bifurcation atR  0    =  1    {R_0} = 1   . We investigate the stability of the EE for time lagτ  ≥  0    \tau \ge 0   . We also discussed the Hopf bifurcation of EE numerically. Global stability of the model equilibria is also discussed. Furthermore, the model has been simulated numerically to exemplify analytical studies.

Dynamic Behavior of an SIR Epidemic Model along with Time Delay; Crowley–Martin Type Incidence Rate and Holling Type II Treatment Rate