The raw number of new infections in a given year is
New infections[n]=β[n]∗N[n]I[n]∗S[n]where:
- S[n]=number of susceptible in given year
- I[n]=number of infected in given year
- N[n]=total population in given year
But the data I have (from UNAIDS) gives HIVIncidence in "per 1000" so:
HIVIncidence[n]=New infections[n]∗N[n]1000⟺HIVIncidence[n]=β[n]∗N[n]I[n]∗N[n]S[n]∗1000
- N[n]I[n]=HIVPrevalence is the prevalence that I have from UNAIDS as a proportion (from 0 to 1)
- N[n]S[n]=1−N[n]I[n]=1−HIVPrevalence (same from 0 to 1 depending on above)
I let β absorb the factor of 1000 and I end up with:
β=HIVPrevalence∗(1−HIVPrevalence)HIVIncidenceExample:
2 countries with same β=10−3 and HIVPrevalence=20%=0.2 but different total populations:
- Country A has population NA=109
- Country B has population NB=105
New infections:
- Country A then has 10−3∗0.2∗0.8∗109=1.6∗105 new infections
- Country B then has 10−3∗0.2∗0.8∗105=16 new infections
Incidence in "per 1000":
- Country A has 1.6∗105∗109103=0.16
- Country B has 1.6∗101∗105103=0.16