$2019$ circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours?
1. **Assigning Numbers to Regions**: 
   For each region \(i\), assign a number \(a_i\) such that all points in region \(i\) are contained inside exactly \(a_i\) of the 2019 circles. This number \(a_i\) represents the count of circles that enclose the region \(i\).

2. **Parity Argument**:
   We claim that no region \(j\) adjacent to \(i\) can have \(a_i \equiv a_j \pmod{2}\). To prove this, assume for contradiction that there exist two adjacent regions \(i\) and \(j\) such that \(a_i\) and \(a_j\) have the same parity (both even or both odd).

3. **Contradiction**:
   Since regions \(i\) and \(j\) are adjacent, they are separated by an arc of one of the circles. Crossing this arc changes the count of circles enclosing the region by exactly one. Therefore, \(a_i\) and \(a_j\) must differ by exactly one, which implies that one is even and the other is odd. This contradicts our assumption that \(a_i\) and \(a_j\) have the same parity.

4. **Coloring the Regions**:
   Based on the parity of \(a_i\), color all regions \(i\) with even \(a_i\) red, and color all regions \(j\) with odd \(a_j\) blue. From the previous argument, this ensures that no two adjacent regions share the same color.

5. **Minimum Number of Colors**:
   Since we have successfully colored the map using only two colors (red and blue) and it is clear that one color is insufficient (as adjacent regions must have different colors), we conclude that the minimum number of colors needed is 2.

\(\blacksquare\)

The final answer is \( \boxed{ 2 } \)