jmc

We graph the points $(a,b)$ that satisfy each inequality. The graph of $a^2+b^2<16$ is the set of points inside the circle centered at the origin with radius 4.

From $a^2+b^2<8a,$ $$(a-4{)}^2+b^2<16.$$This represents the inside of the circle centered at $(4,0)$ with radius 4.

From $a^2+b^2<8b,$ $$a^2+(b-4{)}^2<16.$$This represents the inside of the circle centered at $(0,4)$ with radius 4.



We see that there are $\boxed{6}$ lattice points that lie inside all three circles.