The predominant hydrogen-containing species in Ti${\mathrm{O}}_{2}$ (rutile) are O${\mathrm{H}}^{\ensuremath{-}}$ (hydroxyl) ions, in which the oxygen occupies a regular oxygen ion site, and the O-H bond is perpendicular to the $c$ axis. It is proposed that diffusion of hydrogen parallel to the $c$ axis proceeds by a proton jump from one ${\mathrm{O}}^{2\ensuremath{-}}$ ion to another along the channel as represented by O${\mathrm{H}}^{\ensuremath{-}}$\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}${\mathrm{O}}^{2\ensuremath{-}}$ \ensuremath{\rightarrow} ${\mathrm{O}}^{2\ensuremath{-}}$\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}${\mathrm{H}}^{+}$\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}${\mathrm{O}}^{2\ensuremath{-}}$ \ensuremath{\rightarrow} ${\mathrm{O}}^{2\ensuremath{-}}$\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}H${\mathrm{O}}^{\ensuremath{-}}$. It is also proposed that diffusion perpendicular to the $c$ axis proceeds by a rotation of the O${\mathrm{H}}^{\ensuremath{-}}$ bond to move the proton from one channel to an adjacent channel, followed by a proton jump to another ${\mathrm{O}}^{2\ensuremath{-}}$ ion in the same channel. From a potential-energy model, which includes a Morse function to represent the O${\mathrm{H}}^{\ensuremath{-}}$ bond, as well as electrostatic and repulsive terms, the activation energies for hydrogen and tritium diffusion parallel to the $c$ axis were calculated to be (including a zero-point energy correction) 0.60 and 0.69 eV, respectively, in good agreement with the respective experimental values of 0.59 and 0.75 eV. The calculated activation energy for diffusion perpendicular to the $c$ axis was 1.23 eV (no zero-point energy correction), as compared to the experimental values of 1.28 and 1.11 eV, respectively, for hydrogen and tritium. The calculated equilibrium orientation of the O${\mathrm{H}}^{\ensuremath{-}}$ ion in Ti${\mathrm{O}}_{2}$ and the calculated stretching frequency of this species were also in good agreement with the respective experimental results.