You should review Florida Statute 83.51 to see what obligations a landlord has to maintain the premises. If your landlord has been neglecting these duties, (which includes pest control) you might have an argument that such negligence or breach of statutory duty has caused your damages. If, however, your landlord has been routinely providing pest control, I doubt a single swarming would constitute any negligence or breach of duty by the landlord. The duty to provide pest control is also for the premises not your personal property. Unfortunately, Florida is one of the many states that has termite swarming periodically throughout the year. Nothing I am aware of can prevent such swarming.
You may want to contact a pest control company if you believe your furniture has been affected by the swarming. It is possible for them to take your furniture out of your apartment and treat it in small applications.
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Landlords are obligated to take reasonable steps to control pests. What is reasonable is somewhat subjective. However, if they are swarming as you describe, it seems there efforts are inadequate. If you have property that is damaged due to the landlord's violation of the law, they are liable, no matter if they have a disclaimer saying otherwise.
If you wish to terminate the lease, you should proceed with a 7 day notice. The clerk of court has forms that you can fill out. Detail the termite issues in the apartment on the notice, and send it certified mail to your landlord, keeping a copy for yourself. Not counting the day of receipt, the landlord has seven days to fix the issues in the notice.
If the issues are not addressed in seven days, you may reduce the rent in proportion to the reduction of the value of the property. In other words, if you would pay $500/month for the apartment in good condition, but the termites reduce your use and enjoyment by 20%, you would start paying $400 a month. You may also choose to terminate the lease if the issues render the apartment wholly uninhabitable.
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