Forecasting the Coronavirus Outbreak
Imagine that you have a finite population of people who are at risk of getting infected. You introduce one infected person into that population. What will happen?
The growth rate of infections in that population are determined by two key factors, namely the number of people already infected, and the number of uninfected.
Consequently, the higher number of infected and uninfected, the higher the growth rate. However, as the number of infected increases and the number of uninfected decreases (through death or recovery), the growth rate inevitably slows down, and finally reaches an upper limit.
For a transmission to take place, an infected has to encounter an uninfected. If the population size is N, and the number of infected is n, the probability of randomly picking one infected and one uninfected is p = n / N * (N - n) / N. If the number of infections at t equals I(t), the infection rate is proportional to the number of infected as
(1)
where k is some proportionality constant, accounting for the chance of transmission during an “encounter” between two people, the frequency of encounters between people etc.
An encounter here can be direct or indirect, e.g. through sneezing, touching a contaminated surface etc. The purpose of quarantine is hence to limit the size of the “population”, and to make it as small as possible.
However, due to long-distance travel etc., the pool of people at risk also gradually increases over time with some growth factor g. As the global community also gradually puts preventive measures in place, it tends to level off over time and is therefore regressive, hypothetically proportional to the square root of time, i.e. Hence, the current population size at time t is therefore
(2)
Consequently, the final number of infections as a function of time therefore equals to
(3)
Curve fitting with actual Wuhan coronavirus data, we estimate that N = 7e4, k = 2.1e4, g = 1.015, and I(1) = 23. This yields the following results:
