  The production of prompt $D^+$ and $D^+_{s}$ mesons is studied in proton-lead
collisions at a centre-of-mass energy of $\sqrt {s_{\mathrm{NN}}}=5.02\,$TeV.
The data sample corresponding to an integrated luminosity of
$(1.58\pm0.02)\mathrm{nb}^{-1}$ is collected by the LHCb experiment at the LHC.
The differential production cross-sections are measured using $D^+$ and
$D^+_{s}$ candidates with transverse momentum in the range of $0<p_{\mathrm{T}}
<14\,\mathrm{GeV}/c$ and rapidities in the ranges of $1.5<y^*<4.0$ and
$-5.0<y^*<-2.5$ in the nucleon-nucleon centre-of-mass system. For both
particles, the nuclear modification factor and the forward-backward production
ratio are determined. These results are compared with theoretical models that
include initial-state nuclear effects. In addition, measurements of the
cross-section ratios between $D^+$, $D^+_{s}$ and $D^0$ mesons are presented,
providing a baseline for studying the charm hadronization in lead-lead
collisions at LHC energies.
