So far we have learned a number of techniques for solving a differential equation that is given to us. But very often, in the real world, we need to infer the underlying differential equation by looking at the data. This can be a much more difficult task.

We illustrate this by using Covid-19 data from New Zealand, as shown in Fig. 25.17.

a. Plot the selected data points in a computer program of your choice.

b. On the same graph plot the graph of rN(1 − N/K) (where N is the number of new cases per day) and choose r and K so that the function looks as much like the data as possible.
For now, just do this by hand, fiddling around with r and K until you get the best possible agreement. By the way, you won’t be able to get very good agreement at all between the function and the data.

c. Since the agreement is poor, what can you conclude about how well the initial spread of Covid-19 is described by the logistic equation dN dt = rN 
1 −
N K 
?

d. So let’s try a better model. The shape of the data in Fig. 25.17 suggests that it could be described well by a function of the form f (N) = rNa e

−bNc

. Again, plot the data and the function f (N) on the same graph and fiddle around the constants (r,

a, b and

c) until you get good agreement with the data.

Hint: r = 0.006, a = 4.5, b = 0.04 and c = 1.5 is a pretty good place to start.

e. What differential equation would you use to model the initial spread of Covid-19 in New Zealand?