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

Measurement of heavy-flavour production in small systems can be used to test the Quantum ChromoDynamic (QCD) models. In this manuscript, the production cross 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 pp collisions at $\sqrt{s}$ = 13 TeV with ALICE detector will be presented. The self-normalized yield of open heavy-flavour decay electrons and muons as a function of multiplicity in pp and p$-$Pb collisions will be presented. Finally, the latest result on nuclear modification factor ($Q_{\rm pPb}$) of D mesons and the $\nu_{2}$ of open heavy-flavour decay electrons and muons in p$-$Pb collisions will be discussed.