Vincenzo Verardi (UNAMUR-CRED, FNRS, ULB).
In this note, we will look at a very simple model to sketch how the COVID 19 epidemic could evolve over time (focusing on Belgium). We will use a model called \(SIR\) (and more precisely its Kermack-McKendrick version). This model could be complexified substantially to incorporate more specificities of the virus and of the transmission mechanism. To simplify things as much as possible, we however prefer to stick for the moment to its simplest version. In a future note we will propose a more elaborate model considering both symptomatic and asymptomatic cases and considering various age classes.
The \(SIR\) model divides the population into three groups of individuals: \(S\), \(I\) and \(R\). Group \(S\) is the group of susceptible individuals (i.e. those individuals that are at risk of being contaminated). For the case of COVID-19, at the beginning of the epidemic \(S\) is the entire population given that nobody has anti-bodies (it is indeed a new virus for which no vaccine is available). Group \(I\) is the group of individuals that have been contaminated recently and that are infectious. Finally, the result or removed group \(R\) is the group of individuals that were contaminated but that had an outcome (either a recovery of death). They are not infectious anymore.
The sizes of these groups evolve over time as the virus spreads. The size of S decreases when people get contaminated and move into the infectious group \(I\). When individuals recover or die, they then move from the infectious group \(I\) to the removed group \(R\). The evolution of the sizes of these groups can be modelled by a system of 3 differential equations:
These variations are very simple to interpret:
The first equation states that the size of \(S\) decreases by the number of newly contaminated individuals which is simply the infection rate (\(β\)) multiplied by the number of susceptible individuals (\(S\)) that encountered infectious individuals (\(I\)).
The second equation states that the number of infectious individuals (\(I\)) will be increased by the newly contaminated individuals (\(βSI\)) minus the previously infectious individuals that had an outcome and moved to group \(R\) (i.e. the removal rate γ multiplied by the infectious individuals \(I\)).
Finally, the last equation states that the removed group increases by the number of individuals that were infectious that had an outcome (\(γI\)). In the case of COVID 19, before the beginning of the epidemic the size of \(S\) is the entire population (as nobody is immune to the new virus). Then, once a first individual is contaminated, \(S\) decreases by one unit and \(I\) increases by one unit. This is the beginning of the dynamic of the epidemic. After some time, this infectious individual contaminates new individuals before recovering (or dying). In the meantime, the newly contaminated individual start spreading the virus and the epidemic starts.
A crucial parameter in an epidemic is the average number of people who will catch a disease directly from one contagious person (this is the so-called reproduction number, generally noted as \(R_0\), which is closely linked to \(β\) and \(γ\)). In the case of COVID-19, it has been estimated that the reproduction number is approximately 2.7, \(γ\) is approximately 0.2 and \(β\) is approximately 0.54.
In the subsections here below, we look at the evolution of \(S\), \(I\) and \(R\) over time (days) considering different scenarios (after normalizing the entire population to 1 to work with percentages). Only lines related to group \(I\) are activated. The other can be activated by clicking on the legend on their reference.
In the cases of non-intervention, the virus spreads very quickly. The number of infected individuals increases and at day 50, about 28% of the population is infectious. If 10% of these need to be hospitalized, even for a small country like Belgium results are scary: about 300 000 individuals have to be hospitalized. Knowing that approximately 10% of hospitalized patients will need intensive care, the medical sector is completely overwhelmed (slightly more than 2000 beds are available in intensive cares in Belgium). At the end of the epidemic, 90% of the population will have been contaminated as can be seen in the (plain green) “result” line if activated. The collective immunity will have been reached but the cost in terms of lives would be enormous (remember that is is estimated that the mortality rate of the disease is about 4.5%).
Click on the legend to add or remove lines