## 20 February 2013

### Life expectancy, GDP and HIV prevalence

What explains the variation in life expectancy between different countries? The GDP is of course very important: the LE increases roughly linearly with its logarithm. To remove this dependence I fitted the data with a sigmoidal function:

$$LE (GDP) = LE_{\text{min}} + ( LE_{\text{max}} - LE_{\text{min}}) \frac{1}{2} [1+\text{erf} (\alpha \log (\beta \, GDP))] \qquad \qquad (1)$$

shown as solid line in Figure 1.

Figure 1: Life expectancy at birth (UN World Population Prospects 2010) versus GDP by country. Overall value (male and female). Color corresponds to the geographical region.

The fit is slightly better than with a simple logarithm and the function (1) makes more sense: the LE cannot increase (or decrease) indefinitely. To give an idea of the dispersion I also plotted (as dashed lines) the same curve but with parameter $$\Delta _{LE} = LE_{\text{max}} - LE_{\text{min}}$$ equal to 0.8 and 1.2 of its optimal value.

In Figure 2 I normalize the LE by the model (after subtracting the baseline) and plot the result as a function of the HIV prevalence. $$y$$-values of 0.8, 1.0 and 1.2 correspond to the lines in Figure 1.

Figure 2: Life expectancy versus HIV prevalence by country. Color corresponds to the geographical region.

The normalized LE is strongly influenced by the HIV prevalence: All countries below 0.8 in the former are above 1% in the latter, except for Afghanistan (not shown in Figure 2), where the ongoing war might explain the reduced LE.

How can one understand the few LE values above 1.2?