An appropriate pest management SI model with biological and chemical control concern

In this work, we consider an SI model for pest management, and concern about releasing infective pests and spraying pesticides at different fixed moments. We prove that all solutions of (*)dS(t)dt=rS(t)1-S(t)+θI(t)K-βS(t)I(t),t≠(n+l-1)τ,t≠nτ,dI(t)dt=βS(t)I(t)-wI(t),t≠(n+l-1)τ,t≠nτ,▵S(t)=-μ1S(t),t=(n+l-1)τ,▵I(t)=-μ2I(t),t=(n+l-1)τ,▵S(t)=0,t=nτ,▵I(t)=μ,t=nτ,0