  In this paper, by introducing two temporal-derivative-dependent auxiliary
variables, a linearized and decoupled fourth-order compact finite difference
method is developed and analyzed for the nonlinear coupled bacterial systems.
The temporal-spatial error splitting technique and discrete energy method are
employed to prove the unconditional stability and convergence of the method in
discrete maximum norm. Furthermore, to improve the computational efficiency, an
alternating direction implicit (ADI) compact difference algorithm is proposed,
and the unconditional stability and optimal-order maximum-norm error estimate
for the ADI scheme are also strictly established. Finally, several numerical
experiments are conducted to validate the theoretical convergence and to
simulate the phenomena of bacterial extinction as well as the formation of
endemic diseases.
