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Measurement of heavy-flavour production in small systems can be used to testthe Quantum ChromoDynamic (QCD) models. In this manuscript, the productioncross section of D mesons at midrapidity and open heavy-flavour decay muons(HFM) measured at forward rapidity in pp collisions at $\sqrt{s}$ = 5.02 TeV,and open heavy-flavour decay electrons (HFE) measured at midrapidity in ppcollisions at $\sqrt{s}$ = 13 TeV with ALICE detector will be presented. Theself-normalized yield of open heavy-flavour decay electrons and muons as afunction of multiplicity in pp and p$-$Pb collisions will be presented.Finally, the latest result on nuclear modification factor ($Q_{\rm pPb}$) of Dmesons and the $\nu_{2}$ of open heavy-flavour decay electrons and muons inp$-$Pb collisions will be discussed.

Study of open heavy-flavour hadron production in pp and p$-$Pb collisions with ALICE