Abstract
ABSTRACTWe propose a new method to solve a system of complex ordinary differential equations (ODEs) with hidden hierarchy. Given a complex system of the ODE, the hierarchy of the system is generally hidden. Once we reveal the hierarchy of the system, the system can be reduced into subsystems called slow and fast subsystems. This division of slow and fast subsystems reduces the analysis and hence reduces the computation time, which can be expensive. In our new method, we first apply the singularly perturbed vector field method that is the global quasi-linearization method. This method exposes the hierarchy of a given complex system. Subsequently, we apply a version of the homotopy analysis method called the method of directly defining the inverse mapping. We applied our new method to the immunotherapy of advanced prostate cancer.
Highlights
Prostate cancer is typical among men over 50, and diagnosis among young people is rare
We propose a new method to solve a system of complex ordinary differential equations (ODEs) with hidden hierarchy
We presented the mathematical model of evolution and treatment of prostate cancer
Summary
Prostate cancer is typical among men over 50, and diagnosis among young people is rare. Prostate cancer grows extremely slowly; among the elderly especially, it often does not cause any symptoms. Prostate cancer can grow rapidly and even spread to other organs in the body, especially to the bones [2]. Since prostate cancer grows slowly, the onset of symptoms can occur years after the tumour has appeared. Typical symptoms are difficulty in urinating, increased urinary frequency, especially at night, pain during urination, blood in urine or sexual contact (rare). One should seek new coordinates and a representation of the governing equations (of the original model) in the form of an SPS Once these coordinates are found, the original system can be decomposed into slow and fast subsystems that allow for asymptotic and analytical methods to be applied. Shown below is the formalized general framework theory of the SPVF
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