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Generally they are correct in a case like bed bugs. If you can prove there is an infestation coming from other apartments or that they were there when you moved in then you could likely hold them responsible but it is unlikely other than that.
You state "I pay for pest control every month". Do you pay that to the apartment complex, to a pest control company, etc.?
You would have to look at 1) the lease for the apartment and 2) the pest control agreement.
As far as the actual law, if you didn't have it done then anyone else who gets bedbugs could sue you for the damages, the apartment complex could sue you for the cost of them getting the place exterminated when you leave or if they get a complaint from any of the adjoining apartments they could sue you for the cost they will incur in treating those and possibly additional apartments to prevent the spread.
It is also conceivable that the issue could be considered a "threat to public health" and they could use it as grounds for an attempted eviction. Most leases have language in there that covers this in general terms and even if the language isn't present many judges will allow evictions if they find that there is a public health issue. I've never seen it with bedbugs but I have seen it with general cleanliness issues where there are roaches spreading, smells, etc.
It is vague. It hints that you are just supposed to cooperate but the first and third sentences could be read to place the burden on you unless you can somehow prove that you weren't the cause of the bedbugs coming into the apartment in the first place.
What the landlord's lawyer will argue is that you violated the first sentence by bringing bedbugs into the apartment. Then, when the landlord told you to spray for them that was their "treatment plan" and if you don't do it then you are in default.
You might be able to beat them if you went to court but if you lost you would have to pay for their lawyer plus the costs of the treatment (and their lawyer would likely be two to three times the cost of the treatment).