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I can understand, Doxycycline is a very commonly used medicine in dermatology. Be careful if you want to continue this medicine.
It is not necessary that a person will have stomach pain always, or it is with the first dose. It can occur anytime to anyone, just like in your case. Finasteride and Escitalopram are relatively safe. They do not hurt the stomach as such.
Got your point, so if you want to continue it, take it just after meals with a full glass of water, do not lie down immediately after taking medicine, and have at least five to ten minutes walk. Meanwhile, continue Omeprazole 20 mg twice daily along with regular Gaviscon. If you feel well, continue the above plan. If you feel persistent pain still, there are two options. Either increase the dose of Omeprazole to 40 mg twice daily or stop Doxycycline at that time.
Wishing you good health.