Complex dynamics in tick-borne disease transmission: A Filippov-type control strategy model with multiple time delays

This paper presents a tick-borne disease transmission model with a Filippov-type control strategy that involves spraying insecticides to kill ticks once the number of infected hosts exceeds a certain threshold. The model also incorporates two delays in disease transmission: an internal delay ${\tau }_1,$ representing the maturation period of pathogens inside ticks, and an external delay ${\tau }_2,$ accounting for the time from a host being bitten by an infected tick to becoming infectious. Theoretical analysis deduces that the endemic equilibrium of the delayed Filippov system may undergo a Hopf bifurcation as the delays exceed critical levels. Furthermore, based on Filippov’s convex analysis, the sliding mode dynamics of the system are explored. The results indicate that depending on the threshold levels, the system’s solutions eventually converge to either the regular equilibrium of the two subsystems, a pseudo-equilibrium on the sliding mode, or a stable periodic solution. From a numerical perspective, the system undergoes different boundary focus bifurcation under different time delays and thresholds. Moreover, variations in the delay can lead to the emergence of a global sliding bifurcation on the sliding mode. Therefore, a Filippov system with multiple delays provides new insights and directions for controlling the spread of tick-borne diseases.