Mathematical Model and Analysis of Population Dynamics for Heart Failure and Heart Transplant

Abstract

Heart failure is a non-communicable disease that causes significant morbidity and mortality worldwide. Heart transplant is a vital treatment option for prolonging the lives of patients with heart failure. This study aims to develop a mathematical model that describes the population dynamics of heart failure and examines its stability. This model consists of three nonlinear ordinary differential equations that describe the interaction between individuals at risk of heart failure, heart failure patients, and heart transplant patients. The positivity of the solution is proven. The Routh-Hurwitz criteria are used to analyze the local stability of the equilibrium point. The global stability of the equilibrium point is established using a suitable Lyapunov function. The analysis reveals that the model has only one non-negative equilibrium point, which is found to be locally and globally asymptotically stable. Numerical simulations of the model using MATLAB are provided to verify and illustrate the analytical results. The findings of the study show that the parameters that should be controlled are the rate of acquired risk factors later in life, the probability of reversing modifiable risk factors, the progression rate from at-risk individuals to heart failure patients, the availability of heart transplant resources, the success rate of transplants, the rate of failed transplants, and the saturation factor. Control measures include implementing educational and vaccination programs, promoting lifestyle changes, conducting regular screenings, and expanding heart transplant resources. Controlling these parameters can reduce the at-risk population, decrease progression to heart failure, and improve transplant outcomes, thereby contributing to a lower overall prevalence of heart failure.