Whenever a disease emerges, awareness in susceptibles prompts them to take preventive measures, which influence individuals’ behaviors. Therefore, we present and analyze a time-delayed epidemic model in which class of susceptible individuals is divided into three subclasses: unaware susceptibles, fully aware susceptibles, and partially aware susceptibles to the disease, respectively, which emphasizes to consider three explicit incidences. The saturated type of incidence rates and treatment rate of infectives are deliberated herein. The mathematical analysis shows that the model has two equilibria: disease-free and endemic. We derive the basic reproduction number \(R_0\) of the model and study the stability behavior of the model at both disease-free and endemic equilibria. Through analysis, it is demonstrated that the disease-free equilibrium is locally asymptotically stable when \(R_0<1\), unstable when \(R_0>1\), and linearly neutrally stable when \(R_0=1\) for the time delay \(\varrho >0\). Further, an undelayed epidemic model is studied when \(R_0=1\), which reveals that the model exhibits forward and backward bifurcations under specific conditions, which also has important implications in the study of disease transmission dynamics. Moreover, we investigate the stability behavior of the endemic equilibrium and show that Hopf bifurcation occurs near endemic equilibrium when we choose time delay as a bifurcation parameter. Lastly, numerical simulations are performed in support of our analytical results.

每当一种疾病出现时，易感人群的意识就会促使他们采取预防措施，从而影响个人的行为。因此，我们提出并分析了一个时间延迟的流行病模型，其中易感个体分为三个子类：不知道易感者、完全意识到易感者和部分意识到疾病易感者，该模型强调考虑三个明确的发病率。本文讨论了感染性疾病的发病率和治疗率的饱和类型。数学分析表明，该模型具有两个平衡：无病和地方病。我们推导出基本再生数\(R_0\)并研究模型在无病和地方性平衡下的稳定性行为。通过分析证明无病平衡在\(R_0<1\) 时局部渐近稳定，在\(R_0>1\)时不稳定，在\(R_0=1\)时线性中性稳定延迟\(\varrho >0\)。进一步研究了当\(R_0=1\)时的无延迟流行病模型，这表明该模型在特定条件下表现出前向和后向分叉，这对疾病传播动力学的研究也具有重要意义。此外，我们研究了地方性平衡的稳定性行为，并表明当我们选择时间延迟作为分岔参数时，Hopf 分岔发生在地方性平衡附近。最后，进行数值模拟以支持我们的分析结果。