Global Hopf bifurcation of a two-delay epidemic model with media coverage and asymptomatic infection

Abstract

The influence of media coverage on disease transmission has attracted more and more attention, but the impact mechanism of media coverage with asymptomatic infection needs to be studied urgently. Therefore, a SIsIaS−M model with two delays and asymptomatic infection is considered. The two kinds of delays are respectively caused by the lag time of media reports about the number of patients and the lag time of audience feedback after media coverage. By using the geometric approach based on the third compound matrix, the existence of a threshold for the average duration of media coverage ensures that the endemic equilibrium in delay-free differential system is globally asymptotically stable. In the delay system the uniqueness, positivity and boundedness of solutions are investigated, and the system is persistent when R0>1. Through stability analysis, we find that the disease can be eliminated when the proportion of symptomatic patients in total patients is above a threshold. By choosing the sum of two delays as a bifurcation parameter, we analyze the stability switches of the endemic equilibrium, and discuss the onset and termination of Hopf bifurcations of periodic solutions. The properties of local Hopf bifurcation are also obtained. Furthermore, the global Hopf bifurcations are unbounded and a sufficient condition for the system to have multiple periodic solutions is obtained. Three parameters related to media coverage are found in numerical simulations to have thresholds that guarantee the stability of the endemic equilibrium. In numerical simulations, if media coverage is highly responsive to the number of patients or it has a strong influence on individual behavioural changes, this can lead to periodic oscillations. Conversely, although this makes the disease stable, the number of patients who reach steady state is higher. This is the same result as for the dissipation rate of media coverage, and again there is a threshold that makes the endemic equilibrium stable.