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S-curves (or sigmoid functions) are commonly used to model the evolution of social or biological systems over time [1]. These functions start with exponential growth, then increase linearly, and finally level off (therefore end up looking like a wonky s). Many things that we think of as exponential functions will actually follow an s-curve (otherwise the system would reach infinity). One famous example is the adoption of a new technology. The graph below shows the percentage of US adults who own a smartphone over time, with a best-fit s-curve imposed on the top. In this case the exponential growth occurs because of the way publicity and supply are rolled out. However, there are only a limited number of potential consumers (some of whom will never get a smartphone) and so the growth gradually slows to zero.

Another example, and the reason that these curves have been back in the news, is the propagation of disease. In this case the exponential growth occurs when the virus is new, such that most people encountering it will not have developed immunity. The level-off occurs because the virus is no longer encountering people without immunity (either due to ‘herd immunity’ or isolation of those infected). The graph below shows the number of deaths in China from the SARS outbreak in 2003, again with a best-fit s-curve.

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