Stability and bifurcation analysis of an infectious disease model with different optimal control strategies

This paper deals with the non-linear Susceptible–Infected–Hospitalized–Recovered model with Holling type II incidence rate, treatment with saturated type functional response for the prevention and control of disease with limited healthcare facilities. The well-posedness of the model is ensured with the help of the non-negativity and boundedness of the solution of the system. The feasibility of the model with DFE (Disease-free equilibrium) and EE (endemic equilibrium) is analysed. The local and global stability are discussed with the help of the computed basic reproduction number R0. At R0=1, we use the Centre manifold theory to analyse the transcritical bifurcation exhibited by the system. It is found that the disease is not eradicated even if R0<1 due to the occurrence of backward bifurcation. The occurrence condition of Hopf bifurcation is obtained. The optimal control theory is used to analyse the effects of the minimum possible medical facilities, hospital beds, and awareness creation on the population dynamics. The Hamiltonian function is constructed with the extended optimal control model and solved by Pontryagin’s maximum principle to get the minimum possible expenditure. Different types of control strategies are shown by numerical simulation. The sensitivity analysis is discussed with the help of a crucial parameter that depends on the reproduction number. Further, the model is simulated numerically to support the theoretical studies. This paper emphasizes the significance of treatment intensity, the total number of hospital bed available and their occupancy rate as vital parameters for prevention of disease prevalence.