2003 ford focus zetec se rhd starts then stalls

I am having a serious problem with my 2003 ford focus it has a zetec se engine, at first the car was losing power for a few seconds while driving the rpm would go all the way down and if I mash the gas pedal at times it would rev back up. It would also cut out when the car comes to a stop in traffic. So I started with replacing the fuel filter but the problem came back after 20 minutes of driving, I was really hoping it would have been a cheap fix but next I took out the gas tank check the strainers and replaced the fuel pump and installed new spark plugs NGK LTR5GP and the car drove with no problems for two days I thought all was well I was so relieved but now a new problem surfaced. I would turn the key and hear the fuel pump kick in for two seconds, start the car it will start normal then it would start to idle rough and die slowly. Someone had suggested that the initial start of the pump would pressurize the lines but after that the pump wont continue working and the car would just use the gas that it can and die out. So what would cause the pump not to work after initial start up? What can I do to solve this problem?