Delayed information induces oscillations in a dynamical model for infectious disease

Abstract

During disease outbreak, it has been observed that information about the disease prevalence induces the individual’s behavioral changes. This information is usually assumed to be generated by the density of infective individuals and active mass media. The delay in reporting of these infective individuals may have its impact on generated information. Hence, to study the impact of delay on information generation, and therefore on the disease dynamics, a delay differential equation model is proposed and analyzed. The dynamics of information with delay effect is also modeled by a separate rate equation. Model analysis is performed and a unique infected equilibrium is obtained when the basic reproduction number ([Formula: see text]) is greater than one, whereas the disease free equilibrium always exists. When [Formula: see text], the disease free equilibrium is found to be locally stable independent of delay effect. The unique infected equilibrium is found to be locally stable till delay reaches a threshold value. The global stability of the unique infected equilibrium is also established under some parametric conditions by constructing a suitable Lyapunov function. The occurrence of Hopf bifurcation is observed when the delay in information crosses the threshold value. Analytically, the direction and stability of bifurcating periodic solutions is established. Further, we observed the occurrence of Hopf-Hopf bifurcation at two different delays. At first delay threshold, the endemic equilibrium loses its stability and produces periodic oscillations via Hopf bifurcation. It further regains its stability at second delay threshold via another Hopf bifurcation. Hence, the delay effect on information shows possibility of stability switches. Numerical experiments are carried out to support the obtained analytical results. Our study infers that the disease will show persistent oscillations if there is a significant time lag in reporting of infective after the disease outbreak. Thus, the delay in dissemination of information shows rich and complex dynamics in the model and provides important insights. We also observe numerically that the saturation in information plays a significant role on stability of infected equilibrium in presence of delay.