Measles - Vaccinations for measles was implemented in 1968. Death from measles had already declined by 99% before the vaccination was implemented according to the graph.
The graph shows that a lot of measles deaths had already declined before vaccination. That means that vaccination is not the only method for combating disease. Which of course, is known.
If we look at the data from 1950-1979 we see the following:
1950 221
1951 317
1952 141
1953 245
1954 50
1955 176
1956 30
1957 95
1958 49
1959 98
1960 80
1961 152
1962 39
1963 166
1964 73
1965 115
1966 80
1967 99
1968 51 vaccination started
1969 36
1970 42
1971 28
1972 29
1973 33
1974 20
1975 16
1976 14
1977 23
1978 20
1979 6
The data is also missing the last twenty years - when the MMR vaccine made it to the UK (your data is just for England, but we can ignore that because the data for England and the UK is essentially the same.) I don't have 1988-1992 but here is 1993 onwards:
1993 0
1994 0
1995 0
1996 0
1997 0
1998 0
1999 0
2000 0
2001 0
2002 0 (numbers using vaccine started to fall)
2003 0
2004 0
2005 0
2006 1 Victim had pre-existing lung condition.
It also misses out the number of cases of measles. Which before 1988 was between 50,000-100,000. In recent times this is closer to 10,000. A drop of 90% I don't see any reference to the location of outbreaks and measles take up. A lot of cases are from traveling families (not families that go on holiday but 'gypsies') - such as with the single death we've had this year or from areas which have had low take up in vaccines.
It also is suspicious in picking a small wealthy island nation with a notorious history of eradicating disease (eg rabies, which exists in France is basically extinct in the UK).
If we look at global statistics we get a different picture entirely. In Africa, for example (where 50% of measle deaths happen), measles is the fifth biggest killer.
Deaths in Africa (and globally) have been reducing significantly, in line with improved vaccination programs.
I have seen similar stats for whooping cough. Cases increase as immunization decreases and vice versa.
The unsourced stuff here does have sources, but my browser is having a fit. If you don't accept something, I'm sure I'll be able to find the source again. This was just what I could find in thirty minutes that leads me to be suspicious of the data being presented by the anti-vaccine crowd.