Bifurcations and dynamics of a Filippov epidemic model with nonlinear threshold control policy and medical-resource constraints

The infected population triggered threshold policy has been widely used in the prevention and control of infectious diseases. This study proposes a Filippov model with nonlinear threshold function and saturated treatment function to describe the newly infected population triggered control strategy and medical resource constraints. The existence conditions for sliding segments, sliding dynamics and different type of equilibria are derived and the local and global discontinuity-induced bifurcations including the boundary equilibrium bifurcation, tangency bifurcation and limit cycle bifurcations are theoretically and numerically analyzed. Particularly, both persistence and non-smooth fold can be observed in the boundary equilibrium bifurcations. The results show that the proposed model may have multiple sliding segments, which is caused by the nonlinear threshold function. Numerical analysis shows that when there is an unstable focus for the control system, a homoclinic loop with a boundary saddle may occur along with the boundary saddle bifurcation, and then it becomes a sliding limit cycle. This is a novel phenomenon. The numerical global behavior analysis reveals that the threshold control policy has no advantage if the newly infected population threshold is above the equilibrium level of the free system. However, if the threshold level is preset too low, a worse case may occur, in which the (newly) infected population fluctuate in a large range and the peak exceeds the equilibrium level of the free system. Thus our results suggest that suitable threshold level should be carefully chosen so that the ideal attractor with a low number or a small range of fluctuations of (newly) infected individuals is approximated.