According to the quick info on this search link:
H1N1 (swine) flu started with pig, bird, and human flu viruses mixing DNA into a 2009 pandemic strain. Since then, H1N1 has been a fixture of annual flu reporting, and according to the link above, it's now considered a seasonal flu right next to H3N2 influenza A.
Additional references: spread and mutation patterns
According to the study in the link above, H3N2 flu strains are constantly being transmitted and mutated around in east and southeast Asia, quickly spreading to other nations (usually through plane travel) in their respective flu seasons. Each strain of H3N2 typically causes a wave of infections outside the east/southeast Asia region and then dies out (source: the Vox link above).
H1N1 behaves more like an influenza B strain in mutation and infection patterns (source: the Business Insider news link above). It doesn't mutate as quickly and tends to remain local, although it can persist for years in the same geographic neighborhood.
The metrics: Define the number of flu infections for a year in a country as the sum of the total number of times a person caught the flu in that country. For example, if you got the flu (different strains) 3 times in your home country over a season, you add 3 to this metric for your country. This number of points gets counted over every person in each country.
Overseas infection rule: If you are visiting another country and are infected with a flu virus, this counts against the country you are visiting for purposes of the above metric.
International travel rule: Flu infections received on a plane, international mass commercial transit, or a port used for such transit count against the country where the source originated. For example, if a visitor from Country 1 with an active flu infection travels via plane to Country 2 and infects a number of people there at an airport terminal, all of the people infected at the terminal are counted against Country 1.
Define the total number of global infections in a year as the above metric summed up across all countries for the given year.
The question: Does the existence of H1N1 as a seasonal strain significantly increase the total number of global flu infections each year according to the above metric? In particular, does H1N1 throw in additional misery (in terms of total number of global flu infections) each year, or would the slack have been made up through extra spread and dominance of traditional flu strains if H1N1 never existed?