You didn't list a jurisdiction, and these things vary a good deal from locale to locale. For example, in the U.S., here's a state-by-state reference.
Note, in some states (like Florida), the answer depends on who introduced the pests. A landlord would be responsible unless the tenant brought the bugs in (say, in a mattress).
The TAA's addendum would provides structure for people on TAA leases. However, that doesn't apply to everyone (and doesn't sound like your situation). That said, Texas Property Code §92.052 discusses like conditions:
(a) A landlord shall make a diligent effort to repair or remedy a condition if:
(1) the tenant specifies the condition in a notice to the person to whom or to the place where rent is normally paid;
(2) the tenant is not delinquent in the payment of rent at the time notice is given; and
(3) the condition:
(A) materially affects the physical health or safety of an ordinary tenant; or
(B) arises from the landlord's failure to provide and maintain in good operating condition a device to supply hot water of a minimum temperature of 120 degrees Fahrenheit.
(b) Unless the condition was caused by normal wear and tear, the landlord does not have a duty during the lease term or a renewal or extension to repair or remedy a condition caused by:
(1) the tenant;
(2) a lawful occupant in the tenant's dwelling;
(3) a member of the tenant's family; or
(4) a guest or invitee of the tenant.
(c) This subchapter does not require the landlord:
(1) to furnish utilities from a utility company if as a practical matter the utility lines of the company are not reasonably available; or
(2) to furnish security guards.
(d) The tenant's notice under Subsection (a) must be in writing only if the tenant's lease is in writing and requires written notice.
See also TX Property Code §92.056.