Stability analysis and optimal control of a generalized SIR epidemic model with harmonic mean type of incidence and nonlinear recovery rates

A challenging SIR epidemiological dynamic model with harmonic rate of incidence and nonlinear rate of recovery is developed to examine the effects of available beds in hospitals and intervention decrease upon the propagation of viral disease. The incorporation of the harmonic mean as an incidence rate is the novelty of the present manuscript. The assumed incidence rate is less sensitive to the larger values of the variables and proves more advantageous for highly skewed data compared to the bi-linear and monod types of incidence rates. For model's stability, precise mathematical conclusions have been produced. The model has two states of equilibrium: an infection-free equilibrium (E₀), whenever the threshold number assumes values less than one, and a disease-present state (E₁), whenever R₀>1. We employ the principle of LaSalle invariance and Lyapunov's direct technique to demonstrate that the basic threshold quantity R₀<1 indicates the global asymptotic stability of E₀. Also whenever 1<R₀ then the equilibrium point E₁ is stable, under specific parametric conditions. The optimal control strategies for the described model are chosen with the aid of maximum principle of Pontryagin. To support the analytical conclusions, some numerical results are presented. The study's findings show that the model can accurately reflect the complex dynamics of numerous epidemic diseases, resulting in a considerable decrease in infection transmission.