Stability Analysis and Optimal Control of a Population Model of Hypertension

Abstract

Hypertension is one of the most common chronic diseases that lead to serious health complications if not treated properly. The purpose of this study is to investigate the population dynamics of hypertension disease with ordinary differential equations and optimal control. A hypertensive population model is developed where the population is categorized into three compartments: non-hypertensive individuals, hypertensive individuals without complications, and hypertensive individuals with complications. The non-negativity and boundedness of the solutions are analyzed. The model has a unique equilibrium point, which is proven to be locally asymptotically stable. After that, an advanced version of the model is proposed where a control variable representing the hypertension intervention strategies is introduced. In the control-based model, we seek to minimize the number of hypertensive individuals and relative costs of implementing the intervention strategies. Numerical simulations are carried out to show the impact of key parameters such as the rate of developing hypertension, recovery rate of complications, and the relative-cost weighting parameter. The results show that implementing intervention strategies can significantly reduce the number of hypertension cases (with and without complications). This study demonstrates how mathematical modeling can help in understanding hypertension progression in the population and planning effective healthcare strategies.