After the lockdown period due to the COVID-19 virus, we will try to link the theory to the mathematical modelling of disease spreading during the outbreak. We start with the famous #R_0#-value that you have heard so much about in the news.

First a few notations: let us assume that the number of patients with COVID-19 at time #t# is denoted with #N(t)#. At the beginning of an epidemic, the number of patients often increases exponentially, because almost everyone is susceptible (you will learn more about why this is in weeks 4-8). So the number of patients is given by

\[ N(t) = N_0 e^{K\cdot t} = N_0 g^t, \]

where #N_0# is the number of patients at time #t=0#. The growth factor #g# is related to #R_0#, the basic reproduction number, however, it is not the same (to be precise: #R_0# is the number of others a person infects over the total course of his or her contagious period, whereas #g# is the number of others a person infects per unit of time). However, for simplicity, we will now assume the two are the same (so basically we are assuming an infected person always stays contagious). The growth rate #K# of the disease denotes how fast the number of infected persons grows over time.

The number #R_0# can be modelled as follows:

\[ R_0 = p D \alpha S,\]

where #D# is the amount of time a person is contagious, #\alpha# is the number of people per unit of time a person meets, #p# is the probability that during this meeting, the other person is infected and #S# is the fraction of the population that is susceptible to the virus. Of course there are many assumptions here (that everyone is the same: has the same number and type of contacts, everyone is equally susceptible, simple exponential growth dynamics, etc).