The contact rate is defined as the average number of contacts adequate for disease transmission by an individual per unit time and it is usually assumed to be constant in time. However, in reality, the contact rate is not always constant throughout the year due to different factors such as population behavior, environmental factors and many others. In the case of serious diseases with a high level of infection, the population tends to reduce their contacts in the hope of reducing the risk of infection. Therefore, it is more realistic to consider it to be a function of time. In particular, the study of models with contact rates decreasing in time is well worth exploring. In this paper, an SIR model with a time-varying contact rate is considered. A new form of a contact rate that decreases in time from its initial value till it reaches a certain level and then remains constant is proposed. The proposed form includes two important parameters, which represent how far and how fast the contact rate is reduced. These two parameters are found to play important roles in disease dynamics. The existence and local stability of the equilibria of the model are analyzed. Results on the global stability of disease-free equilibrium and transcritical bifurcation are proved. Numerical simulations are presented to illustrate the theoretical results and to demonstrate the effect of the model parameters related to the behavior of the contact rate on the model dynamics. Finally, comparisons between the constant, variable contact rate and variable contact rate with delay in response cases are presented.

中文翻译：

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具有时变接触率的 SIR 模型

接触率被定义为一个人每单位时间内足以传播疾病的平均接触次数，通常假定它在时间上是恒定的。然而，实际上，由于人口行为、环境因素和许多其他因素等不同因素，全年接触率并不总是恒定的。在感染率高的严重疾病的情况下，人群倾向于减少接触，以期降低感染风险。因此，将其视为时间的函数更为现实。特别是接触率随时间下降的模型的研究非常值得探索。在本文中，考虑了具有时变接触率的 SIR 模型。提出了一种新的接触率形式，该接触率从初始值随时间递减，直到达到一定水平，然后保持不变。提议的表格包括两个重要参数，它们表示接触率降低的幅度和速度。这两个参数被发现在疾病动态中发挥重要作用。分析了模型平衡点的存在性和局部稳定性。证明了无病平衡和跨临界分岔的全局稳定性结果。数值模拟用于说明理论结果并证明与接触率行为相关的模型参数对模型动力学的影响。最后，比较了恒定、可变接触率和可变接触率在响应案例中的延迟。