Treatment of breast cancer (positive for HER2, i.e., ERBB2) is described by a mathematical model involving non-linear ordinary differential equations with a hidden hierarchy. To reveal the hierarchy of dynamical variables of the system being considered, we applied the singular perturbed vector field (SPVF) method, where a system of equations can be decomposed to fast and slow sub-systems with explicit small parameters. This new form of the model, which is called a singular perturbed system, enables us to apply a semi-analytical method called the method of directly defining inverse mapping (MDDiM), which is based on the homotopy analysis asymptotic method. We introduced the treatment protocol in explicit form, through an analytical function that describes the exact dose and intervals between treatments in a cyclical manner. In addition, a new algorithm for the optimal dosage that causes tumour shrinkage is presented in this study. Furthermore, we took the concept of protocol optimisation a step further and derived a differential equation that represents vaccination depending on tumour size and yields an optimal protocol of different doses at every time point. We introduced the treatment protocol in explicit form, through an analytical function that describes the exact dose and intervals between treatments in a cyclical manner. In addition, a new algorithm for finding the optimal dosage that causes tumour shrinkage is presented in this study. Additionally, we took the concept of protocol optimisation a step further and derived a differential equation that represents vaccination depending on tumour size and yields an optimal protocol of different doses at every time point.

乳腺癌（HER2 阳性，即 ERBB2）的治疗由涉及具有隐藏层次的非线性常微分方程的数学模型描述。为了揭示所考虑系统的动态变量的层次结构，我们应用了奇异扰动矢量场 (SPVF) 方法，其中方程组可以分解为具有显式小参数的快速和慢速子系统。这种新形式的模型称为奇异摄动系统，使我们能够应用一种称为直接定义逆映射方法 (MDDiM) 的半解析方法，该方法基于同伦分析渐近方法。我们通过一个分析函数以明确的形式引入了治疗方案，该函数以循环方式描述了治疗之间的确切剂量和间隔。此外，本研究提出了一种新算法，用于确定导致肿瘤缩小的最佳剂量。此外，我们进一步采用了方案优化的概念，并推导出了一个微分方程，该方程代表根据肿瘤大小的疫苗接种，并在每个时间点产生不同剂量的最佳方案。我们通过一个分析函数以明确的形式引入了治疗方案，该函数以循环方式描述了治疗之间的确切剂量和间隔。此外，本研究还提出了一种用于寻找导致肿瘤缩小的最佳剂量的新算法。此外，我们进一步采用了方案优化的概念，并推导出了一个微分方程，该微分方程代表取决于肿瘤大小的疫苗接种，并在每个时间点产生不同剂量的最佳方案。