Stability switches, periodic oscillations and global stability in an infectious disease model with multiple time delays

Abstract

A delay differential equation model of an infectious disease is considered and analyzed. In this model, the impact of information due to the presence of infection is considered explicitly. As information propagation is dependent on the prevalence of the disease, the delay in reporting the prevalence is an important factor. Further, the time lag in waning immunity related to protective measures (such as vaccination, self-protection, responsive behaviour etc.) is also accounted. Qualitative analysis of the equilibrium points of the model is executed and it is observed that when the basic reproduction number is less unity, the local stability of the disease free equilibrium (DFE) depends on the rate of immunity loss as well as on the time delay for the waning of immunity. If the delay in immunity loss is less than a threshold quantity, the DFE is stable, whereas, it loses its stability when the delay parameter crosses the threshold value. When, the basic reproduction number is greater than unity, the unique endemic equilibrium point is found locally stable irrespective of the delay effect under certain parametric conditions. Further, we have analyzed the model system for different scenarios of both delays (i.e., no delay, only one delay, and both delay present). Due to these delays, oscillatory nature of the population is obtained with the help of Hopf bifurcation analysis in each scenario. Moreover, at two different time delays (delay in information's propagation), the emergence of multiple stability switches is investigated for the model system which is termed as Hopf-Hopf (double) bifurcation. Also, the global stability of the endemic equilibrium point is established under some parametric conditions by constructing a suitable Lyapunov function irrespective of time lags. In order to support and explore qualitative results, exhaustive numerical experimentations are carried out which lead to important biological insights and also, these results are compared with existing results.

Keywords

• information
• behavioural response
• two discrete delays
• hopf-hopf bifurcation
• stability switch