Caravan Undifferentiated
Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Commercial
Flight Crew Last 90 Days 70; Flight Crew Total 7150; Flight Crew Type 3500
Communication Breakdown; Physiological - Other; Fatigue; Situational Awareness; Time Pressure
Conflict NMAC; Deviation / Discrepancy - Procedural Published Material / Policy; Flight Deck / Cabin / Aircraft Event Illness / Injury
Air Traffic Control Issued Advisory / Alert; Flight Crew Took Evasive Action
Human Factors; Aircraft; Airspace Structure; Environment - Non Weather Related
Departing a Class D airport behind another IFR aircraft; which resulted in a rare runway heading departure.  Given the somewhat remote location I'm usually given an 'on course' take off clearance which normally results in an immediate turn to the south when departing the northbound runway.  Instead I'm climbing out runway heading.While climbing around Vx airspeed to avoid flying too far out of the way I was handed off to Center.  After radar contact I was given a climb to 10;000 ft. MSL and my choice of turn for on course.  The Controller then began speaking with other flights.I began accelerating to cruise climb while heading back south over the airport and then was notified by Center about traffic over the airport.  My first thought was it's the King Air that departed after me flying in the pattern.  Instead he said it's a Cessna orbiting the airport at 5500 ft. MSL.  At this point I was climbing out of 5000 ft. MSL.That got my attention; but due to factors I'll explain later I was slow to take action...I began pushing the nose over but did not completely stop my climb.  Instead I replied 'looking for traffic'.The other aircraft was also on Center; and he was given the same traffic advisory and replied with the same vague answer.  At this point all three of us in this conversation have been vague about direction of flight; location and intentions.  A few seconds later the Supervisor took over and said we are heading right at each other; less than half a mile and both of you turn right.Before his sentence was even finished I had begun a diving right turn as my altitude by this point was approximately 5500 ft. MSL. I had stopped monitoring the altimeter with my eyes outside.  It was only at this point that both of us gained sight of each other with wings up in the turn.  I believe I was descending out of 5200-5300 ft. MSL after a quick glance inside.  The other aircraft responded 'traffic in sight'.Contributing factors - I was flying tired and somewhat sick.  I had been up most of the night with on and off stomach cramps; and by [early morning] I was on the toilet.  While I had dozed throughout the night I never remember actually waking up from sleep.  My spouse was also tossing and turning; and attempting to sleep on the couch downstairs proved futile due to a noisy pet.  My show time this morning was [one hour later than normal].I began trying to talk myself into flying - it's a later show time than normal; I've (unfortunately) flown after lousy sleep before; the weather has been nice all week.  After arriving at the airport I decided to sit in the plane and doze while waiting for freight.  Then I began to feel somewhat nauseous and clammy.  Now I'm considering getting back in my car and heading home; which with staffing issues and short term notice will definitely result in a canceled flight.After eating a banana the nausea went away and I was left with contending with fatigue.  The weather was clear and a million; and attempting to feel like a responsible pilot I said if the weather was bad I would definitely cancel; but given the nice day I'll 'tough it out'.And yet; most mid-air collisions happen in day VFR.  Should that have actually been a warning sign?  Yes the weather is good; which means more aircraft will be out flying.  A nice 'easy' day with an 800 ft. ceiling would probably be a safer situation.Given this same situation while healthy I probably would have taken charge and leveled off at 5000 ft. MSL.  But my reactions were slow.  How would an in flight emergency have played out on this day?  Flying for a living there's the unspoken pressure to get the job done.  Most of us have pushed it with personal health and weather - if we canceled every time there was forecast moderate icing we would spend half the winter on the ground.We all have our personal limits; and today I was very borderline.  Yes; the flight would have been canceled.  Yes; the office would be stressed out.  But the company would survive; and so would I.
C-208 Captain reported an NMAC after takeoff while in the airport traffic area.