The raw number of new infections in a given year is

where:

• $S[n] = \text{number of susceptible in given year}$
• $I[n] = \text{number of infected in given year}$
• $N[n] = \text{total population in given year}$

But the data I have (from UNAIDS) gives $HIV_{Incidence}$ in "per 1000" so:

• $\frac{I[n]}{N[n]} = HIV_{Prevalence}$ is the prevalence that I have from UNAIDS as a proportion (from 0 to 1)
• $\frac{S[n]}{N[n]} = 1-\frac{I[n]}{N[n]} = 1-HIV_{Prevalence}$ (same from 0 to 1 depending on above)

I let $\beta$ absorb the factor of 1000 and I end up with:

Example:

2 countries with same $\beta = 10^{-3}$ and $HIV_{Prevalence} = 20\% = 0.2$ but different total populations:

• Country A has population $N_A = 10^9$
• Country B has population $N_B = 10^5$

New infections:

• Country A then has $10^{-3} * 0.2 * 0.8*10^9 = 1.6*10^5$ new infections
• Country B then has $10^{-3} * 0.2 * 0.8*10^5 = 16$ new infections

Incidence in "per 1000":

• Country A has $1.6*10^5 * \frac{10^3}{10^9} = 0.16$
• Country B has $1.6*10^1 * \frac{10^3}{10^5} = 0.16$