Quorum　sensing,　a　widespread　phenomenon　in　bacteria　that　is　used　to　coordinate　gene　expression　among　local　populations,　intervenes　in　the　competition　between　bacteria　and　the　immune　system.　The　domain　of　attraction　of　the　bacteria-free　equilibrium　results　from　a　linear　matrix　inequality　optimization　with　a　multivariate　polynomial　objective　under　constraints.　The　Bogdanov-Takens　singularity　and　bifurcation,　including　a　saddle-node　bifurcation,　a　Hopf　bifurcation,　and　a　homoclinic　bifurcation,　are　obtained　from　normal　form　theory.　The　normal　form　of　a　bifurcation　is　a　simple　dynamical　system　which　is　equivalent　to　all　systems　exhibiting　this　bifurcation.