  The induced diffusion of tracers in a bacterial suspension is studied
theoretically and experimentally at low bacterial concentrations. Considering
the swimmer-tracer hydrodynamic interactions at low-Reynolds number and using a
kinetic theory approach, it is shown that the induced diffusion coefficient is
proportional to the swimmer concentration, their mean velocity and a
coefficient $\beta$, as observed experimentally. The coefficient $\beta$ scales
as the tracer-swimmer cross section times the mean square displacement produced
by single scatterings. The displacements depend on the swimmer propulsion
forces. Considering simple swimmer models (acting on the fluid as two monopoles
or as a force dipole) it is shown that $\beta$ increases for decreasing
swimming efficiencies. Close to solid surfaces the swimming efficiency degrades
and, consequently, the induced diffusion increase. Experiments on W wild-type
{\em Escherichia coli} in a Hele-Shaw cell under buoyant conditions are
performed to measure the induced diffusion on tracers near surfaces. The
modification of the suspension pH vary the swimmers' velocity in a wide range
allowing to extract the $\beta$ coefficient with precision. It is found that
the solid surfaces modify the induced diffusion: decreasing the confinement
height of the cell, $\beta$ increases by a factor 4. The theoretical model
reproduces this increase although there are quantitative differences, probably
attributed to the simplicity of the swimmer models.
