A Filippov model describing the effects of media coverage and quarantine on the spread of human influenza

Mass-media reports on an epidemic or pandemic have the potential to modify human behaviour and affect social attitudes. Here we construct a Filippov model to evaluate the effects" of media coverage and quarantine on the transmission dynamics of influenza. We first choose a piecewise smooth incidence rate to represent media reports being triggered once the number of infected individuals exceeds a certain critical level I-C1. Further, if the number of infected cases increases and exceeds another larger threshold value I-C2 (> I-C1), we consider that the incidence rate tends to a saturation level due to the protection measures taken by individuals; meanwhile, we begin to quarantine susceptible individuals when the number of susceptible individuals is larger than a threshold value S-c. Then, for each susceptible threshold value S-c, the global properties of the Filippov model with regard to the existence and stability of all possible equilibria and sliding-mode dynamics are examined, as we vary the infected threshold values I-C1 and I-C2. We show generically that the Filippov system stabilizes at either the endemic equilibrium of the subsystem or the pseudoequilibrium on the switching surface or the endemic equilibrium E-C = (S-C, I-C2), depending on the choice of the threshold values. The findings suggest that proper combinations of infected and susceptible threshold values can maintain the number of infected individuals either below a certain threshold level or at a previously given level.