For developing preordain capability in health care, it is important to understand the healthcare system. To achieve this demeanour, it is important to model the system mathematically and define the automata of the system. It is serendipitous that quantitative analysis of the major healthcare system can be done using the mathematical model of the healthcare system. This review paper discusses a few of the mathematical models. The paper is categorized into five sections. Section 1 discusses the pioneer work on the mathematical modelling of the heart which laid the foundation stone for the remaining works including model- and automata-based analysis of the heart and pacemaker. The second section discusses the parameters associated with the modelling of the heart and further extends onto the mathematical modelling which is followed by the aberrant heart conditions and use of pacemaker to control the conditions. The section further discusses one of the major concerns of health, i.e., increasing rate of cancer globally. Referring to the report of Global Cancer Observatory (GCO), an International Cancer Research Agency (WHO), there are 36 types of cancer reported from 185 countries worldwide, in 2020 which has also been published in American Cancer Society Journal. The proliferating rate of cancer needs to be correlated with the population. The third section discusses the model of change of population with the augmentation of the cancer cases. It also discusses the prediction of cancer cases in future for prescient vision and analysis and for initiating adaptive measures to control cancer.