Bifurcation analysis and optimal control of an epidemic model with limited number of hospital beds

Abstract

This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic’s future course when the public healthcare facilities, specifically the number of hospital beds, are limited. The feasibility and stability of the obtained equilibria are analyzed, and the basic reproduction number ([Formula: see text]) is obtained. We show that the system exhibits transcritical bifurcation. To show the existence of Bogdanov–Takens bifurcation, we have derived the normal form. We have also discussed a generalized Hopf (or Bautin) bifurcation at which the first Lyapunov coefficient evanescences. To show the existence of saddle-node bifurcation, we used Sotomayor’s theorem. Furthermore, we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure. An optimal control setting is studied analytically using optimal control theory, and numerical simulations of the optimal regimen are presented as well.