Mathematical analysis of a disease-resistant model with imperfect vaccine, quarantine and treatment

In this paper, we develop a new disease-resistant mathematical model with a fraction of the susceptible class under imperfect vaccine and treatment of both the symptomatic and quarantine classes. With standard incidence when the associated reproduction threshold is less than unity, the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium. It is then proved that this phenomenon vanishes either when the vaccine is assumed to be 100% potent and perfect or the Standard Incidence is replaced with a Mass Action Incidence in the model development. Furthermore, the model has a unique endemic and disease-free equilibria. Using a suitable Lyapunov function, the endemic equilibrium and disease free equilibrium are proved to be globally-asymptotically stable depending on whether the control reproduction number is less or greater than unity. Some numerical simulations are presented to validate the analytic results.