The effect of filters on spam mail

We provide a model to study the effect of improving the quality of the filters that users use, and the increase of the cost of mailing spam messages, on the total number of messages that spammers send. We show that increasing the cost of mailing spam messages have the unequivocal effect of reducing the total number of spam messages sent by spammers. On the other hand, improving the quality of the filters have an ambiguous effect: users who uses the improved filter receive less spam messages, but the total number of messages sent by spammers might increase, and other users, who keep their old filters, might receive more spam messages.