# COVID-19/Mathematical Modelling/Logistical Growth

## Comparison Exponential Growth - logistical growth

In school you might be exposed to exponential growth. In terms of epidemiology the exponential growth model assumes that there are not limits of the growth. In fact for epidemiology the maximum of infected people is given by the total population. If we consider the other biological processes of growth (e.g. cell division) we have also limitations of growth (e.g. limits of resources, limits of space, ...). The logistical growth incorporates a capacity in the modelling. A logistic function or logistic curve is a common "S" shape (sigmoid curve), with equation:[1]

${\displaystyle f(x)={\frac {M}{1+e^{-k(x-x_{0})}}}}$

where

• ${\displaystyle e}$  = the base of exponential function, which is also known as Euler's number,
• ${\displaystyle x_{0}}$  = the ${\displaystyle x}$ -value of the sigmoid's midpoint, which is in epidemiology the point in time with maximum growth rate (maximum value of the derivation).
• ${\displaystyle M}$  = the curve's capacity of the growth, and
• ${\displaystyle k}$  = the logistic growth rate or steepness of the curve.[2]

## References

1. Wikipedia contributors (February 5, 2020). Logistic function.
2. Verhulst, Pierre-François (1838). "Notice sur la loi que la population poursuit dans son accroissement" (PDF). Correspondance Mathématique et Physique 10: 113–121. Retrieved 3 December 2014.