[Nathan]

People don't have the immunity to the new type of influenza,

While immunity is a noun, it is not often used with 'the'.

"People don't have immunity to the new type of influenza"
"People lack immunity to the new type of influenza "

Another word that is often used with diseases is 'strain', which means 'type of'.

"People don't have immunity to the new(est) strain of influenza"

so it's easy to infect.

With the sentence like this, you are saying we give influenza something. So you may wish to have it like this:
"so we are easy to infect."
"so we are infected easily."
"so we can be infected easily."
"so we catch it easily."

or

"so it infects us easily."
"so it can infect us easily."

No one is immune to the new type of influenza.

This part is fine as is.