Dynamics of an epidemic model with imperfect vaccinations on complex networks

Vaccination is commonly used for reducing the spread of infectious diseases; however, we know that not all vaccinations are completely effective. Thus, it is important to study the impacts of vaccine failure on the spreading dynamics of infectious diseases. In this study, we investigate the dynamics of an epidemic model with three types of imperfect vaccinations on complex networks. More specifically, the present model follows a susceptible-infected-susceptible process with a vaccinated compartment that permit leaky, all-or-nothing, and waning vaccines. A threshold value R-v is first presented to ensure the disease-free equilibrium is asymptotically stable. Next, we derive a necessary and sufficient criterion that assures the presence of a backward (or subcritical) bifurcation when R-v = 1. From this criterion, we can observe that the leaky vaccine plays an important role in leading to such a bifurcation, and interestingly, we also find that this condition is independent of the network structure. The disease is also demonstrated to persist whenever R-v > 1. Numerical simulations are conducted to validate the theoretical results. Our results show that imperfect vaccination can cause a backward bifurcation, which usually has serious consequences for disease control. Thus, appropriate infection control policies need to be further developed.