Dynamics of a Tick-Borne Disease Model with Birth Pulse and Pesticide Pulse at Different Moments

Abstract

Tick-borne diseases pose a potential risk to public health, which is influenced by the stage structure and seasonal reproduction of tick populations. In this paper, a model that explains the transmission dynamics of pathogens among ticks and hosts is formulated and analyzed, considering birth pulse and pesticide pulse on tick population at different moments. Using the stroboscopic mapping for the disease-free system, we prove a globally asymptotically stable positive periodic solution exists when the pulsed pesticide spraying intensity is less than a critical threshold. Applying the comparison theorem for the impulsive differential system, the conditions for global attraction of the disease-free periodic solution to the investigated system are given. Moreover, we demonstrate the persistence of the studied system and give numerical simulations to verify it. Ultimately, we discuss the case with multiple pesticide sprays and conclude that fewer sprays are more favorable for disease extinction.