Dynamic behaviors of a modified SIR model with nonlinear incidence and recovery rates

Alshammari, Fehaid Salem; Khan, Muhammad Altaf

A complex SIR epidemic dynamical model using nonlinear incidence rate and nonlinear recovery rate is established to consider the impact of available hospital beds and interventions reduction on the spread of infectious disease. Rigorous mathematical results have been established for the model from the point of view of stability and bifurcation. The model has two equilibrium points when the basic reproduction number R 0 > 1 ; a disease-free equilibrium E 0 and a disease endemic equilibrium E 1 . We use LaSalle’s invariance principle and Lyapunov’s direct method to prove that E 0 is globally asymptotically stable if the basic reproduction number R 0 < 1 , and E 1 is globally asymptotically stable if R 0 > 1 , under some conditions on the model parameters. The existence and nonexistence of limit cycles are investigated under certain conditions on model parameters. The model exhibits Hopf bifurcation near the disease endemic equilibrium. We further show the occurring of backward bifurcation for the model when there is limited number of hospital beds. Finally, some numerical results are represented to validate the analytical results.