Abstract Memory effects of epidemics play a vital role in mathematical models of infectious diseases. In this research study, an epidemiological SEIR (Susceptible, Exposed, Infectious, Removed) type model for the rubella epidemic has been proposed via classical and fractional order Caputo differential operators while assuming the periodic transmission rate β ( t ) . The Caputo model has been investigated for the existence and uniqueness of its solutions via fixed point theory while the unique non-negative solution remains bounded within the biologically feasible region. Later, the non-fixed biological parameters of the classical and the Caputo model are obtained via nonlinear least squares fitting technique taking the real monthly cases for the rubella epidemic in Pakistan for the period 2010–2012. The performance rate of the Caputo model is 35% higher than that of the model with integer order derivative. The numerical simulations are obtained under different cases and it is highly suggested that the infectious rate σ must be controlled as much as possible to eradicate the rubella epidemic.

中文翻译：

---

基于巴基斯坦真实数据的标准和分数卡普托算子下风疹流行的周期性动态

摘要 流行病的记忆效应在传染病的数学模型中起着至关重要的作用。在这项研究中，通过经典和分数阶 Caputo 微分算子在假设周期性传播率 β ( t ) 的情况下，提出了风疹流行的流行病学 SEIR（易感、暴露、传染、移除）类型模型。Caputo 模型已经通过不动点理论研究了其解的存在性和唯一性，而唯一的非负解仍然在生物学可行区域内。随后，以巴基斯坦2010-2012年风疹流行每月真实病例为样本，通过非线性最小二乘拟合技术得到经典模型和Caputo模型的非固定生物学参数。Caputo 模型的性能比具有整数阶导数的模型高 35%。数值模拟是在不同情况下获得的，强烈建议必须尽可能控制传染率σ以根除风疹流行。