If an owner of cattle refuses to slaughter exposed, nonreacting bovine animals pursuant to the provisions of Article 7 (commencing with Section 10421) when requested by the director to do so, the cattle owner shall then be responsible for paying to the director the full costs of the department in eradicating brucellosis from his herd. Such responsibility shall begin with the refusal to slaughter exposed, nonreacting animals and shall continue until brucellosis is eradicated from the herd.