where \(\mu_{A_{i}^{l}}\left(x_{i}\right)\) is the membership function of the fuzzy set \(A_{i}^{l}\), \(\mu_{B^{l}}\left(y\right)\) is the membership function of the fuzzy set \(B^{l}\), and\(\mu_{B^{{}^{\prime}}}\left(y\right)\) is the membership function of the output fuzzy set.
In this paper, the designed Mamdani fuzzy system has one input-one output. The age of the patient is considered as a language variable and a Mamdani fuzzy system is designed for controlling chemo-drug dose. The input of the fuzzy system is the age of the patient and its output is the upper limit of the chemo-drug dose. The age of the patient is divided into four groups: Baby, young, middle-aged, and old. The chemo-drug dose is considered in three groups: Low, medium and high. The membership functions and the rule-base are shown in Figure 5 and Table 2, respectively.
Mixed treatment simulation
By considering the parameters’ value in Table 1, the treatment must be imposed in the whole life of the patient because of relapsing cancer after stopping the treatment (see Figure 1). Due to the instability of the tumor-free equilibrium point, the dynamics modification is essential for finite-duration treatment, which is done by immunotherapy. The vaccine therapy modifies the dynamics of the system by changing the two parameters \(\sigma\) and \(\delta\). In other words, the phase plane of the system changes from Figure 1 to Figure 3. After immunotherapy, the optimal chemotherapy pushes the trajectory of the system toward the equilibrium point E1.
The proposed therapeutic approach is simulated by considering the patient’s age. The simulation results are presented for three cases: young, middle-age and old patients with no specific disease. In each case the immunotherapy is first implemented. By immunotherapy the free system dynamics changes from Figure 1 to Figure 4 and then to Figure 3. Chemotherapy is then implemented by using the pseudo-spectral method.
Case 1: A young patient
In this case, a 20-year-old patient is considered.  The fuzzy system presented a maximum chemo-drug dose as 8.2. As shown in Figure 6, the vaccine has modified the system’s dynamics, and other equilibrium points are removed from the system’s phase plane.  The task of chemotherapy is to push the system’s trajectory toward the tumor free equilibrium point E1. As shown in this figure, the trajectory of the system is pushed to the point E1 and the imposed chmoe-drug approaches to zero.
Case 2: A middle-age patient
In this case, a 45-year-old patient is considered. The combined treatment of vaccine therapy and chemotherapy is applied. The fuzzy system presented a maximum chemo-drug dose as 4.1. As shown in Figure 7, the vaccine has modified the system’s dynamics and the system’s trajectory is guided to the equilibrium point E1. When the trajectory of the system placed in this area the chemotherapy is ceased. In this situation, the new modified free system is able to guide the trajectory of the system toward the equilibrium point E1 without any external treatment.
Case 3: An old patient
In this section, a 60-year-old patient is considered. The fuzzy system implemented the maximum dose of the chemotherapy drug as 1.6. The behavior of the system is shown in Figure 8. After the modification of the dynamics of the system by immunotherapy, the chemotherapy pushes the trajectory of the system to the the equilibrium point E1. The chemo-drug dose doses not converge to zero. However, the treatment can be stopped due to the stability of the tumor free equilibrium point.
Figure 1 shows that the system’s dynamics correction is necessary in order to have a finite-duration treatment. Hence, some of the therapeutic inputs should modify the system’s dynamic. In other words, some therapeutic inputs should have a permanent effect on the system’s parameters. Hence, a chemo-immonutherpy is regarded, which despite other researches, the immunotherapy modifyies the dynamics of the system.
By comparing Figures 6, 7 and 8, the chemo-drug dose is smaller in the old patient. However, the duration of chemotherapy is longer in the old patient. Chemotherapy is ceased in a shorter duration in the younger patient.
Compared with other works such as [9, 12, 14, 28, 29], the dynamics modification during the treatment is considered as a necessary phase for finite duration treatment. Also, the special condition of the patient is regarded by designing a fuzzy system. The simulation results show the effectiveness of the pseudo-spectral controller and its ease of implimrntation.
Conclusion
A mathematical model of cancer is ectended by adding chemotherapy and immunotherapy terms in a new approach. The immunotherapy affects on the parameters of the system and reinforcing the immune system’s ability. Chemotherapy pushes the trajectory of the system toward the tumor-free equilibrium point. The chemotherapy protocol is derived using a pseudo-spectral method. The continuous –time nonlinear model of the cancer is converted into an NLP problem and is solved. To restrict the maximum chemo-drug dose a Mamdani fuzzy system is designed. In this fuzzy system the maximum chemo-drug dose based on the age of the patient is calculated. Three cases with different age are regarded to simulation. In all cases the dynamics of the system is modified, and the trajectoty of the system is guided toward the desired equilibrium point. The maximum chemo-drug dose in the young patient is higher than other patients due to his/her immune system’s ability to recovery. Moreover, the treatment duration in the young patient is shorter. The simulation results show the effectiveness of the proposed treatment strategy and ease of implementation of the presented pseudo-spectral method to design a stabilizing controller.
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