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.

在本研究中，提出了一个具有易感者逻辑增长、Crowley-Martin 型发病率和 Holling III 型治疗率的时滞 SIR 流行模型，并对其进行了数学分析。我们考虑了两个明确的时间延迟：一个是新感染的发生率以衡量潜伏期的影响，另一个是感染者的治疗率以分析后期治疗可用性的影响。针对两个平衡点分析模型的稳定性行为：无病平衡 (DFE) 和地方病平衡 (EE)。我们推导出阈值量，基本再生数\(R_0\)，这决定了宿主人群中传染病的根除或持续存在。使用基本再生数，我们表明当\(R_0< 1\)时 DFE 局部渐近稳定，当\(R_0= 1\)时线性中性稳定，当\(R_0> 1\)时不稳定-延迟系统。我们在没有潜伏期的情况下分析系统，揭示了\(R_0= 1\)处的前向分岔，这意味着保持\(R_0\)低于统一可以减少疾病。此外，研究了 EE 的稳定性行为，证明了通过 Hopf 分岔出现振荡和周期解，将两个时间延迟的每个可能分组作为分岔参数。总之，进行了支持理论发现的数值模拟。