One proposed mechanism of the reaction of $\mathrm{HBr}$ with ${\mathrm{O}}_2$ is given here

$\begin{array}{l}\mathrm{HBr}+{\mathrm{O}}_2⟶\mathrm{HOOBr}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ (slow) }\\ \mathrm{HOOBr}+\mathrm{HBr}⟶2\mathrm{HOBr}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ (fast) }\\ \mathrm{HOBr}+\mathrm{HBr}⟶{\mathrm{H}}_2\mathrm{O}+{\mathrm{Br}}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ (fast) }\end{array}$

What is the equation for the overall reaction?

Mechanism given as,

$\begin{array}{l}\mathrm{HBr}+{\mathrm{O}}_2⟶\mathrm{HOOBr}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{(slow) }\\ \mathrm{HOOBr}+\mathrm{HBr}⟶2\mathrm{HOBr}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{(fast) }\\ \mathrm{HOBr}+\mathrm{HBr}⟶{\mathrm{H}}_2\mathrm{O}+{\mathrm{Br}}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{(fast) }\end{array}$

Multiply 3rd equation with 2 and add to let two equations. By this the overall reaction obtained will be,

$4\mathrm{HBr}+{\mathrm{O}}_2⟶2{\mathrm{H}}_2\mathrm{O}+2{\mathrm{Br}}_2$