Mathematical models for the effects of hypertension and stress on kidney and their uncertainty

Abstract

Determining the future state of individual patients, with regard to the susceptibility to life-threatening condition is a crucial problem in the public health. Chronic Kidney Disease (CKD) is a perilous disease, which is characterized by gradual loss of kidney function. This paper presents a set of mathematical models for CKD by applying both deterministic and stochastic approaches. Specifically, hypertension and stress are considered as the main causes of damaging and removing of nephrons thereby reducing the Glomerular Filtration Rate (GFR), which leads to irreparable damages to the kidney function and may result in renal failure, or even kidney loss. This paper analyzes the equilibria for both deterministic and stochastic models. In this regard, mathematical theorems related to stability and stochastic stability of the models are provided. The other important issue, which is discussed in the presented work, is the uncertainty problem for stochastic models. We have used simulations in MATLAB to calculate the uncertainty of the stochastic models. In addition, we have provided a framework for the future prediction of proposed models in specific cases. Finally, the models and the theoretical results are verified in application by applying them to the real clinical data.

Keywords: Chronic kidney disease (CKD); Entropy; Stochastic modeling; Stochastic stability.