By Chong Qi and Ramon A Wyss
The case of Sweden dealing with Covid19 has caused quite some attention and discussion in Europe. We have used a standard epidemiology model to fit to existing data in Sweden. Our fit from April 2 reproduces the observed data well. It appears that Sweden has reached the peak during Easter. The results show that Sweden is hit less hard by the pandemic than countries in Southern Europe, while relying on its citizens to maintain social distancing.
We are all familiar with the story of the inventor of the game of chess and the king who was so fond of the play that he wanted to reward the inventor. The inventor had – as the king thought – a very modest request: Give me one grain of rice for the first square of the play, then double it in the second square, and double it again in the third and so on until you reach the last square, knowing that the play of chess has 64 squares. The king realized eventually that after two rows, i.e. 16 squares, the number of rice grains exceeded 70 000 grains. After another row in the play, we need to switch to kilos. Obviously, there would not be enough rice in the world to full fill the inventors request.
The story exemplifies exponential growth – from a very modest increase, the increase becomes so large and rapid with the number of squares that it is almost unconceivable for the human mind.
Unfortunately, when we transmit infectious diseases, the numerics are the same. In average, one person – patient 0 we may call him Lars – will infect two others, Emma and Anders in a given period of time. Once Emma and Anders are infected, each will transmit the disease to two other persons, Lena, Olle, Sven and Maja. Now, they in turn will transmit the disease to another 8 persons. Already after having passed through 23 generations of transmitters, we will have surpassed Sweden’s entire population. No health care system can accompany this kind of growth, see the blue curve on the figure below.
While nature provides a mechanism to transmit an infection to ever more people, there is a similar mechanism that eventually slows this process down. The pool of people that can be infected is limited – hence once a certain number of people have been infected, Emma will not be able to transmit the disease to Lena and Olle anymore, since Olle has been already infected. At this stage, the exponential growth is slowing down. Once roughly half of the available pool of people has been infected, one infected person will on average only transmit the disease to another single person in the same span of time. This is the point where the yellow curve in the figure below is changing direction from increase to slow down. The number of infected people will still increase further, but the rate is slowing down for each subsequent day. Eventually, we reach a rate of infections where the growth has fully leveled out and one speaks about herd immunity, see the yellow curve on the same figure. One can reach herd immunity much earlier if the society is healthy and only a small portion of the total population is susceptible to the disease e.g. one million (black curve). Similarly, by imposing quarantine you will reduce the growth through the same mechanism: the pool of people is reduced and there are not enough people around to anymore sustain the exponential growth. Likewise, vaccination will stop the exponential growth. These different causes result in the same mechanism, as shown in the figure.
One may ask why Lars is transmitting the disease only to two persons at a given time, while not 10 or 20. The number 2 for transmission is arbitrary. When we deal with large numbers, like large sets of populations, while each individual will transmit the infection to a different number of persons, what is determining the spread is how a larger group of people during a given time till transmit the infection to a group of non-infected people.
Hence, to make a realistic calculation for Sweden e.g., one needs to know the number of people that has been infected and how much this number is increasing e.g. during one day. A common model that is used in epidemiology identifies two groups of people: the group of infected people (I) and the group that is susceptible to infection (S). The model is used frequently and called the SI-model of epidemiology, see e.g. See link See Footnote 1.)
One can also include in the fit the number of deceased people (D) and the model is also called SID.
To fit to realistic data and to be able to make prediction of the pandemic spread in one country (or region or the world) three parameters have to be fitted from the data. One crucial parameter is a time constant that tells how long it takes for one group of people in average to infect the next group of people. This time constant also decides between different diseases infectiousness. Polio e.g. is much more infectious than Covid-19 and hence the curve will rise much more rapid than the one we shown for Sweden. Similarly, introducing social distancing will slow down the spread and increase the time constant.
To make a reliable fit, it is necessary to have many data points, i.e. first after a certain time of the spreading of the disease will the model be able to make predictions. Our fit to the data until April2 became stable and has reproduced the data for Sweden.
Our fit indicates that, with the measures that Sweden has taken, the total number of fatalities will stay below 2500. The curve named ‘derivative‘, reveals when the exponential growth slows down – the derivative’s peak indicate the day where we have reached the largest number of newly infected people – after that day, the number will decrease. Our predictions say that the growth of infections has reached it’s peak around Easter.
1 (One may add a third group, the group of people that has recovered (R) and one speaks of the SIR model then. In the case of Covid 19 following the development over a short period of time it is not necessary)
From the number of infected people, one can derive the number of deaths due to the infectious disease. It will be proportional to the number of infections and will be shifted in time – the maximum number of deaths will be some days later than the maximum of newly infected people – the peak of the ‘derivative‘ will be shifted when it comes to the number of deaths by a couple of days, reflecting how in average the disease will result in death for a fraction of the infected people. The real data will fluctuate up and down, reflecting changes due to many effects. The purpose of the model is to mimic the data as good as possible and to predict trends not yet seen in the data.
The SI model and using the data until April 02, we were able to make the predictions shown in the figure. So far, the model is working well. The crudeness of the model implies that it is rather robust – but may miss details. E.g., new initiatives to slow down the infection will decrease the pool of susceptible people.
The simple model that we have been using was initially fitted to data from China by researchers from Spain, Prof Amaro, see this link
It has subsequently been fitted to data of Italy and Spain and France with good results. Our results show that Sweden by no means is sticking out in terms of number of reported deaths per capita, compared to Italy, Spain, France etc. Sweden certainly has more cases than other Nordic countries. Still, in those countries, the disease may come back more easily than in Sweden in a second wave as it has happened in e.g. Singapore and Hongkong. Until there will be a vaccine, new outbreak of Covid 19 are bound to occur. It is therefore premature compare the different Nordic countries and the effectiveness of measures that have been made.
The data of Covid-19 shows that the SI model captures the main trend. It works well to calculate the spread and in particular predict the down turn. The model enables to
see the mechanism behind the spread of the infection, the impact on society and estimates when we can reach a more normal situation. It is an open model, implying that policy makers and the public can understand trends and engage in informed decision making. Too many predictions have been made without giving details what is behind those. Transparent data and open models are key for a democratic society and will counterbalance theories of conspiracy.
The different institutions that present their daily prognosis should be open with the parameters of their model. This will enable the scientific and public community to engage in the discussion and understanding the premises of different models. It is time to be transparent, using open models and software, so that we can develop a deeper knowledge and understanding of what is going on – let our citizens become informed.
Latest comparison between countries from ‘our world in data’:
Details of our model can be found at https://arxiv.org/abs/2004.01575 with daily updates at: http://www.nuclear.kth.se/covid19/
Photo: Wikipedia Commons – McGeddon