A week of cold with an intermittent dry cough and no fever is almost always post-viral airway irritation. The virus clears in 4-5 days but the cough lingers 2-3 weeks. No antibiotic needed for this.
Next Steps
Steam twice a day, warm fluids, saline nasal spray, and levocetirizine at night for a few days. If you start running fever, get breathless, cough up blood or it crosses three weeks, you need a chest X-ray and a proper exam. Happy to do a quick teleconsult on Practo if it isn't settling.