The answer is that it depends on the time available for the compound interest to work. In this particular example, situation B becomes the preferable after some 38 years of accumulation. The top graph is on a linear scale, the bottom one shows why for a constant percentage decline, a logarithmic scale is often chosen to display it graphically: a constant percentage change gives a straight line on a logarithmic scale.

The annual risk of infection can be likened to compound interest. The initial capital corresponds to the intercept parameter and corresponds to the current level of risk of infection. The interest corresponds to the slope and corresponds to the decline in the risk of infection. Of course, in the banking analogy, the slope should be positive. For the risk of infection, the slope should be negative. On the long term, the slope is much more important than the intercept (example Afghanistan compared to other countries in the Eastern Mediterranean region of WHO).

Investing in tuberculosis control is a long-term task and the more rapid the decline in the risk of infection, the more rapid the situation improves.

In western Europe, the average annual decline has been more than 10% per year. Even if the annual risk can be reduced by only 4% per year, the problem is halved in 17 years (100 * 0.96¹⁷= 50%). With a 3% average annual decline, it would take 23 years.

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