Analysis of a vector-borne disease model with impulsive perturbation and reinfection

Abstract

We formulate a mathematical model for vector-borne disease with impulsive perturbation based on indoor residual spraying. The dynamical properties are studied theoretically and numerically. It is shown that with consideration of impulsive spraying at fixed times, there exists a disease-free periodic solution that is locally stable when the threshold $${\mathcal {R}}_0$$ is less than unity, otherwise the disease is uniformly persistent if $${\mathcal {R}}_0>1$$. Furthermore, the bifurcation analysis is performed, revealing the possible existence of nontrivial periodic solution bifurcated from the disease-free periodic solution at $${\mathcal {R}}_0=1$$. In addition to simulations of parameter sensitivity, when implementing impulsive spraying once the number of infected humans exceeds a threshold level, the effectiveness of such state-dependent control is also conducted numerically.