Nonlinear dynamics of infectious diseases via information-induced vaccination and saturated treatment

The present study of a nonlinear compartmental SVIR model accounts for the dynamics of infectious diseases in population. In this model, the effect of information on vaccination coverage is quantified when medical resources are limited during the epidemic outbreak. Model analysis is performed and the global asymptotic stability of the disease free equilibrium is established when treatment is available for all infective individuals. However, the existence of multiple endemic equilibria is observed due to saturation in medical treatment and information-induced vaccination coverage. If the basic reproduction number is greater than unity, a unique endemic equilibrium for the model is obtained under some parametric conditions. Using the geometric approach, the global asymptotic stability of this endemic equilibrium is established under a parametric constraint. Further, when medical resources are limited, the existence Hopf bifurcation is shown analytically, which infers the oscillatory persistence of disease within the population. Occurrence of Hopf–Hopf bifurcation is also investigated numerically. At first Hopf bifurcation threshold, an endemic equilibrium loses its stability and produces oscillations, and further, it regains its stability at another Hopf bifurcation threshold when oscillations disappear. Hence, multiple stability switches are observed due to the saturated treatment and information-induced vaccination. The effect of the fading of memory also plays a crucial role and gives rise to complex dynamical behaviour and also affects the stability properties. Numerically, different types of backward bifurcation are also examined. Extensive numerical experimentations are also carried out to further explore the model system.