The term 'Arms Race' is used to describe any competition where there is only relative goal of staying ahead of other contenders in rank.
It is also possible that a huge deal of time, energy and money is spend on such a race and yet ending up in the same situation as the race had never been started.
In our context i.e. in context of weapons, this can be proved very lethal for the competitors. One assumes that if one country increases its weapons, another country will do the same. Sequentially, the first country responds by storing more weapons. Subsequently spending more than what they can afford. This leads to shortage of money, wastage of time, wastage of space to store more and bigger weapon and subsequently more chances of war in future.
The purpose of developing a mathematical model is to decide where the arms race would lead both the countries to i.e if there will be disarmament or a stabilized arms race or a run away arms race.