The Italian government has recently locked down the country after a surge in coronavirus cases. I analysed the evolutionary trend of the virus to assess whether it was reasonable to adopt the drastic measures which have been put in place.
The first available data report as low as 20 cases in the whole country, whereas 19 days later (12/03/2020), coronavirus infections reached 10,000 units.
The evolution of coronavirus cases in Italy seems to be following an exponential growth, which I have highlighted by plotting the number of cases over time on a logarithmic scale. When plotting on a logarithmic scale, each step on the y-axis is 10 times the step before. Thus, an exponential growth translates into a linear evolution. Figure 2 shows that the rate of growth was actually very high in the first days of February, it then declined and eventually flat-lined. The initial high rate of growth is likely due to the fact that coronavirus was already circulating in Italy before the 21st of February and many cases were initially discovered due to the accumulation of missed cases in the previous days. Given that the evolution of new coronavirus cases flat-lined in the last eight days, I run a linear regression on the last 8 data-points:
The implications of the estimated trend are worrying: should the rate of infections not decrease, it would only take less than 12 days to get from 10,000 to 100,000 cases and less than 24 days to reach 1,000,000 cases. Given the trend, all the restrictionary measures undertaken by the Italian government appear to be eminently essential.
A Simple Model to Study the Evolution of Coronavirus
In order to compute the evolution of the rate of infection from coronavirus, I theorised a simple model. Specifically, I assumed a country with a total population of 60,461,827 and closed borders, so that the only way for a citizen to take the virus is to be in contact with an infected citizen who has not been tested positive yet. This is because all those who test positive stop infecting other people (they are either hospitalised or quarantined). Similarly, those who either recover from the virus or die because of it cannot be a source of new infections. Finally, deaths from coronavirus represent the only way through which the total population reduces.
Therefore, in this model, the number of new cases depends on the rate at which not-infected individuals contract the virus, the rate at which the infect recover and the mortality rate of coronavirus. Mathematically:
The previous equation shows that the number of infected by coronavirus at time t+1 equals the number of infected at time t, minus the infected who either recovered or died (which happens at a rate r and d respectively), plus the not infected at time t who contract the virus (at a rate i). Daily data on the evolution of coronavirus in Italy are available from the 24th of February 2020 and are shown in the following table:
Following the equation above for the 26th of February, the data in Table 1 can be read as:
It is now possible to compute the daily recovery, infection and mortality rates of coronavirus. Results are shown in Table 2 below. The computation of mortality and recovery rates with aggregate data is much more common and do not suffer from the daily variance in the number of recovered and deaths as daily rates do. Therefore, I decided to add them as a reference. Notice that all the rates suffer from the fact that the virus spreads fast, but recovery and deaths take time to occur. For the sake of the example, let us assume that the actual rate of recovery is 100%, but it takes 20 days from the moment of being tested positive to recover. It follows that during the 20 days necessary for the infected to recover, the number of people who caught the virus will have increased considerably. Then, the estimated recovery rate (both at the aggregate and daily level) would result much smaller than it actually is. Thus, the most accurate way to compute the recovery rate would be to track every single patient, in order to see how many of them have recovered in 1 month time, 2 months time et cetera. Unfortunately, such data are not yet available in Italy.
Figure 3 shows graphically the results of Table 2. Recall that the rate of recovery and mortality show high volatility due to the variance in the number of new cases that are diagnosed with coronavirus each day.
Since the rate of new infections is important to uncover the speed at which new cases will appear in the future, the infection rate is displayed alone. Figure 4 shows the infection rate alone.
As Figure 4 would suggest an exponential trend in the rate of infections, I repeated the process previously adopted for overall cases. In other words, I have plotted the infection rate on a logarithmic scale (see Figure 5) and I run a linear regression through the data, thus obtaining:
If the draconian measures implemented by the Italian government will be effective in reducing contagions, the infection rate is expected to reduce and, in turn, the overall growth in the number of infected should decline as well. This model, because of its simplicity, can be therefore used to track the evolution of coronavirus in Italy.