Mumps is the most common cause of acquired unilateral deafness in children, in which hearing loss occurs at all auditory frequencies. We use a box model to model hearing loss in children caused by the mumps virus, and since the fractional-order derivative retains the effect of system memory, we use the Caputo–Fabrizio fractional derivative in this modeling. In the beginning, we compute the basic reproduction number ${\mathcal{R}}_0$ and equilibrium points of the system and investigate the stability of the system at the equilibrium point. By utilizing the Picard–Lindelof technique, we prove the existence an unique solution for given fractional $\mathcal{CF}$-system of hearing loss model and investigate the stability of iterative method by fixed point theory. The optimal control of the system is determined by considering the treatment as a control strategy to reduce the number of infected people. Using the Euler method for the fractional-order Caputo–Fabrizio derivative, the approximate solution of the system is calculated. We present a numerical simulation for the transmission of disease with respect to the transmission rate and the basic reproduction number in two cases $R_0<1$ and $R_0>1$. To investigate the effect of fractional order derivative on the behavior and value of each of the variables in Model 2, we calculate the results for several fractional order derivatives and compare the results. Also, considering the importance of reproduction number in the continuation of disease transmission, we analyze the sensitivity of $R_0$ respect to each of the model parameters and determine the impact of each parameter.

腮腺炎是儿童后天性单侧耳聋的最常见原因，其中所有听觉频率都发生听力损失。我们使用盒模型对腮腺炎病毒引起的儿童听力损失进行建模，并且由于分数阶导数保留了系统记忆的作用，因此在此模型中使用Caputo–Fabrizio分数导数。首先，我们计算基本繁殖数${\mathcal{[R}}_0$和系统的平衡点，并研究系统在平衡点的稳定性。通过使用Picard–Lindelof技术，我们证明了给定分数的唯一解$\mathcal{碳纤维}$系统的听力损失模型，并通过定点理论研究迭代方法的稳定性。通过将治疗作为减少感染人数的控制策略来确定系统的最佳控制。使用Euler方法求解Caputo–Fabrizio分数阶分数，可以计算出系统的近似解。我们针对两种情况下疾病的传播率和基本繁殖数量进行了数值模拟${\mathrm{[R}}_0<\mathrm{1个}$ 和 ${\mathrm{[R}}_0>\mathrm{1个}$。为了研究分数阶导数对模型2中每个变量的行为和值的影响，我们计算了几个分数阶导数的结果并进行了比较。此外，考虑到繁殖数量在疾病传播中的重要性，我们分析了${\mathrm{[R}}_0$ 尊重每个模型参数并确定每个参数的影响。