Mathematical analysis of a disease transmission model with quarantine, isolation and an imperfect vaccine

A new mathematical model for the transmission dynamics of a disease subject to the quarantine (of latent cases) and isolation (of symptomatic cases) and an imperfect vaccine is designed and analyzed. The model undergoes a backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. It is shown that the backward bifurcation phenomenon can be removed if the vaccine is perfect or if mass action incidence, instead of standard incidence, is used in the model formulation. Further, the model has a unique endemic equilibrium when the threshold quantity exceeds unity. A nonlinear Lyapunov function, of the GohVolterra type, is used to show that the endemic equilibrium is globally-asymptotically stable for a special case. Numerical simulations of the model show that the singular use of a quarantine/isolation strategy may lead to the effective disease control (or elimination) if its effectiveness level is at least moderately high enough. The combined use of the quarantine/isolation strategy with a vaccination strategy will eliminate the disease, even for the low efficacy level of the universal strategy considered in this study. It is further shown that the imperfect vaccine could induce a positive or negative population-level impact depending on the size (or sign) of a certain associated epidemiological threshold.