In this paper, a deterministic model for transmission of an epidemic has been proposed by dividing the total population into three subclasses, namely susceptible, infectious and recovered. The incidence rate of infection is taken as a nonlinear functional along with time delay, and treatment rate of infected is considered as Holling type III functional. We have structured a deterministic transmission model of the epidemic taking into account the factors that affect the epidemic transmission such as social and natural factors, inhibitory effects and numerous control measures. The delayed model has been analyzed mathematically for two equilibria, namely disease-free equilibrium (DFE) and endemic equilibrium. It is found that DFE is locally and globally asymptotically stable when the basic reproduction number \( (R_{0} ) \) is less than unity. It has also been shown that the delayed system for DFE at \( R_{0} = 1 \) is linearly neutrally stable. The existence of an endemic equilibrium has been shown and found that under some conditions, endemic equilibrium is locally asymptotically stable, and is globally asymptotically stable when \( R_{0} > 1 \). Further, the endemic equilibrium exhibits Hopf bifurcation under some conditions. Finally, an undelayed system has been analyzed, and it is shown that at \( R_{0} = 1 \), DFE exhibits a forward bifurcation.

中文翻译：

---

确定性时滞SIR流行模型：数学建模和分析。

在本文中，通过将总人口分为易感性，传染性和恢复性三个亚类，提出了一种流行病的确定性模型。感染的发生率被视为具有时间延迟的非线性功能，感染的治疗率被视为Holling III型功能。考虑到影响流行病传播的因素，例如社会和自然因素，抑制作用和许多控制措施，我们构建了流行病的确定性传播模型。对延迟模型进行了数学上的两个平衡分析，即无病平衡（DFE）和地方性平衡。发现当基本再现数\（（R_ {0}）\）时，DFE在局部和全局渐近稳定。不到团结。还显示了DFE在\（R_ {0} = 1 \）时的延迟系统是线性中性稳定的。已经证明了地方均衡的存在，并且发现在某些条件下，地方均衡在\（R_ {0}> 1 \）时是局部渐近稳定的，并且在全局渐近稳定。此外，地方性平衡在某些条件下表现出霍普夫分歧。最后，分析了一个无延迟的系统，结果表明，在\（R_ {0} = 1 \）处，DFE表现出正向分支。