In the present article, the dynamics of a novel combination of ratio-dependent incidence rate and saturated treatment rate in susceptible-infected-recovered disease compartmental model has been presented. The ratio-dependent incidence rate has been incorporated into the model to monitor the situation when ratio of the number of infectives to that of the susceptibles is getting higher. The saturated treatment rate of the infected population has been considered as Holling type II functional, which explains the limitation in treatment availability. From the mathematical analysis of the model, two types of equilibria of the model have been obtained, which are named as disease-free equilibrium (DFE) and endemic equilibrium (EE). The local stability behavior of equilibria has been investigated by the basic reproduction number \( \left( {R_{0} } \right) \), center manifold theory and Routh–Hurwitz criterion. It has been investigated that the DFE is locally asymptotically stable when \( R_{0} < 1 \), and when \( R_{0} = 1 \), the DFE exhibits either a forward bifurcation or a backward bifurcation under some conditions. The local stability behavior of the EE has also been analyzed, and some conditions are obtained for the same. Finally, some numerical computations have been performed in support of our theoretical results.

中文翻译：

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具有比率依赖发生率和饱和处理的流行病模型的稳定性行为研究

在本文中，已经提出了在感病感染恢复的疾病区室模型中比率依赖的发生率和饱和治疗率的新颖组合的动力学。比率相关的发生率已被纳入模型中，以监视传染性数与易感性数之比越来越高的情况。感染人群的饱和治疗率被认为是Holling II型功能性疾病，这解释了治疗可用性的局限性。通过对该模型的数学分析，获得了该模型的两种平衡，分别称为无病平衡（DFE）和地方性平衡（EE）。通过基本复制数研究了平衡的局部稳定性行为\（\ left（{R_ {0}} \ right）\），中心流形理论和Routh–Hurwitz准则。研究表明，当\（R_ {0} <1 \）时，DFE是局部渐近稳定的；当\（R_ {0} = 1 \）时，DFE在某些条件下表现出前向分叉或后向分叉。还对EE的局部稳定性行为进行了分析，并获得了一些条件。最后，已经进行了一些数值计算以支持我们的理论结果。