From the final print statement in the original code, it seems the intention is to mark a random sample of the population as infected, the size of this sample being input by the user as $answerinfect. If this interpretation is correct, the OP’s code does not work correctly, even setting aside the infinite loop, because each iteration of the outer for loop infects 2 distinct individuals, so the outcome is anywhere from $answerinfect to (2 * $answerinfect) infections (depending on the amount of overlap between loop iterations).

The following code incorporates the suggestions of aitap and frozenwithjoy, and insures that exactly $answerinfect individuals are infected. I have assumed (1) that the status field records infection, 0 for uninfected and 1 for infected; and (2) that the initial population contains no infected individuals.

#! perl
use strict;
use warnings;
use Data::Dumper;
use constant { WELL => 0,        SICK   => 1,    };

my  @pop = ( { name => 'Fred',   status => WELL, },
             { name => 'Wilma',  status => WELL, },
             { name => 'Barney', status => WELL, },
             { name => 'Betty',  status => WELL, }, );

(my $pop = scalar @pop) > 0                    or die "Empty populatio
+n\n";
print "How many random infections? (Enter a number between 1 and $pop)
+:\n";
chomp(my $answerinfect = <>);
($answerinfect > 0) && ($answerinfect <= $pop) or die "Invalid selecti
+on\n";

for (1 .. $answerinfect)
{
    for (my $new_infection = 0; !$new_infection;)
    {
        my $contact = int(rand($pop));

        if ($pop[$contact]{status} == WELL)
        {
            $pop[$contact]{status} = SICK;
            $new_infection         = 1;
        }
    }        
}

print Dumper(\@pop);
printf "You have infected %d %s: ", $answerinfect,
                                   ($answerinfect == 1) ? 'person' : '
+people';
print join(', ', sort map { $_->{name} } grep { $_->{status} == SICK }
+ @pop);
[download]

Typical output:

How many random infections? (Enter a number between 1 and 4):
2
$VAR1 = [
          {
            'status' => 0,
            'name' => 'Fred'
          },
          {
            'status' => 0,
            'name' => 'Wilma'
          },
          {
            'status' => 1,
            'name' => 'Barney'
          },
          {
            'status' => 1,
            'name' => 'Betty'
          }
        ];
You have infected 2 people: Barney, Betty
[download]

Some notes:

• The introduction of some elementary error checking helps prevent the generation of an infinite loop.
• Named constants improve readability and code clarity.
• The code to update a person from WELL to SICK has been moved from within the randompick_infect subroutine into the inner loop in the main code. As a consequence, randompick_infect’s remaining code is now short enough to be inlined. In any case, with the revised logic this subroutine would now be called only once anyway.
• The do {} while construct has been replaced with a for loop. Given Perl’s rules for lexical scoping (i.e., my variables), a for or while loop is often a simpler alternative to the do {} while construct. (And see perlfunc on the limitations of do loops in Perl.)

HTH,