Air carrier Captain reported food poisoning symptoms during flight along with other crew members. A dead heading pilot was asked to help out as the crew decided to divert.

Narrative

Approximately 2:30 minutes after departure the IRO (International Reserve Officer) informed me that he wasn't feeling well. He said he had symptoms of food poisoning. I told him to keep me informed and continue on his break as needed. He returned to the cockpit around the three hour mark. Feeling better; but still not well.Thirty minutes later I informed him that I wasn't feeling well either and had the same symptoms as he did.A Flight Attendant joined us in the cockpit as I then went back to alleviate myself of some of the food poisoning symptoms. The symptoms seem to come in waves around every twenty minutes. I then returned to the cockpit not feeling well. At that point we requested the presence of the DH (Dead Heading) Captain. Myself the PF (Pilot Flying) and the DH Captain discussed a possible diversion plan. Initially we adopted a wait-and-see attitude. At that time our choices for a divert were ZZZZ1 and ZZZZ As this was not a dire emergency I suggested we continue toward ZZZZ.At that point I need to leave the cockpit. Sometime after that [ATC was advised of our medical situation] and the new destination would be ZZZ. A Flight Attendant kept me informed in the bunk area of the new intended destination and arrival time. Due to weather and the PF being familiar with ZZZ that was chosen as the best possible solution. I joined the cockpit crew 15 minutes prior to landing. The safest option at this point to minimize human error was to have the DH Captain continue as the PM (Pilot Monitoring); the FO (First Officer) continue as PF and me in the [Observer] seat. Our landing weight was 1;000 lbs. over max so we chose not to dump fuel. All the necessary checklists were performed and the descent; arrival; and landing were performed without incident.The paramedics attended to myself the IRO and the Purser. We stayed at a local hotel until the following day.