Servaas Houben, Kees Van Heugten and Rob Smit are three Dutch actuaries who have been doing their own modelling of the impact of Covid-19. They have now produced a paper on the likely impact of the vaccination which is fascinating and provocative. Though the translation is at times a little brutal, I am publishing it as it has been sent me by Kees. I am sure it will be of more than actuarial interest! Consult their website for more detail of their work.
Effective vaccination strategies for Sars-Cov-2
In November 2020 it became clear that society can expect a vaccine for Sars-Cov-2 with an expected effectiveness of more than 90%, meaning that the vaccine will work for 90% of the vaccinated people. There is still a lot unknown regarding the vaccine, like for example the period it will work and if there are no long-term effects using the vaccine. But due to the good news the question arises ‘what is an effective vaccination strategy’?
In this article we will go into more detail regarding this question. We assume 90% efficacy for every individual in a population.
Effective vaccination strategy
Before one can start analyzing we need to define what we mean by ‘effective’ with regards to vaccinating. An effective vaccination strategy can be defined as:
- A strategy that runs out the virus. Due to the vaccinations there will be a form of herd immunity that results in an effective reproduction factor of less than 1, that will finally make the virus vanish
- A strategy that protects the vulnerable. Since the vulnerable are protected by the vaccine, the other members in society can be unvaccinated. In this case the virus will not vanish, but it cannot harm the vulnerable. The level of vulnerability is higher for the higher ages.
In the following chapters we analyze these strategies.
The Markov Model
Www.crownactuaries.eu use a Markov model to describe the behavior of the virus on population levels. This model consists of the several statuses a group can be in. Within the population we have a group of:
- Healthy individuals
- Infected individuals
- Sick individuals
- Immune individuals
- Deceased individuals
There are three levels of sickness (light, medium or severe) and three periods of immunity, depending on the level of sickness. The chance of dying depends on the level of sickness. The number of newly infected is determined based on the reproduction factor of the virus and the number of people that can be infected (the healthy population).
For the chances of becoming sick we use the following probabilities:
And the chances of dying when being light, medium or severely sick (within a 4-day period}:
Combining the two tables tells us for example that when someone is 80-years-old there is a 10% chance of becoming severely sick , getting hospitalized and going to the ICU. When one is hospitalized or in the ICU you will have an 8% chance of dying in 4 days.
Starting with 109.000 infected (the estimated number in the Netherlands end November) and if we let the virus go, so not taking any measures, and assuming that the infected are divided in line with the population of The Netherlands, the graph below shows the development of the number of severe sick people. In the long run the number stabilizes around 47.000 seriously sick people that will be in the hospital or ICU (permanently). We excluded mortality to see the effect of the development of the virus better.
Vaccination strategies (The Markov model)
4.1 Vaccinating the full population
With a vaccine that is 90% effective, only 10% of the population can be infected. With a base reproduction factor of 2.5 this will decrease to 0.25, since most of the time an infected person will meet someone that cannot be infected. This gives the following picture.
The number of severe sick people increases to a max of 2,594. After that the number of severely ill decreases. Due to the effective reproduction number of 0.25, the number is too low to let the virus grow again, so it vanishes.
4.2 Vaccination of the older people
In this strategy we vaccinate only people from 70 years and older, being 21% of the population. If we assume random contacts in the population the effective reproduction factor will be 2.03, considering that the vaccine is 90% effective. The chances to become severe sick for 70, 80 and 90 will decrease as well, since this age groups are vaccinated. The number of severe sick people will be less than in the situation where there is no control. The steady state number of severe sick will be around 18.000. For the hospital capacity in the Netherlands this number will be too high.
If we decrease the age limit to 60 years, we still have a steady state number of severe sick people of 8,455. For age 50 we have 4,438 severe sick people.
If we include the 40-year-old, we get an effective reproduction factor of less than 1.08. The number of severe sick increase but stabilizes up till 974, see the picture below:
With the population from 40 being vaccinated, 63% of the population is vaccinated. Due to the efficacy of 90% of the vaccine this results in an effective reproduction rate of 1.08%. The steady state number of severe sick people should not lead to a capacity problem in the hospitals. Using this strategy, we choose for the virus not to vanish, but to protect the vulnerable and manage the hospital capacity.
The stochastic model
In an earlier article we described how a Markov model could be replaced by a stochastic model. In that way you can consider that the virus doesn´t spread symmetrically, but more in super spread events. For that we used a discrete gamma distribution, and we added a ´cross infection´ table. Once infected we do not assume a random pick in the population, but we have defined a limitation where the main part of the infections takes randomly place within the own age group. In this chapter we investigate if a vaccination strategy gives different results using the stochastic model
5.1 Vaccinating 70+
The first result we present is the strategy described in Chapter 4.2. In this strategy we vaccinate only people from 70 years and older, being 21% of the population. With an efficacy of 90% this leads to the underlying pictures. We show two pictures where the first one is the outcome of the simulation using 10 thousand model points and the second one a transformation towards the total Dutch population.
The conclusion is that the outcome is similar in comparison with the Markov model. The effective reproduction factor will eventually come towards 2 but that will take a while.
As mentioned in chapter 4, this strategy will lead to too many Covid-19 patients in the hospitals.
5.2 Vaccinating 60+
Currently the Government has decided to start the vaccination program above 60 years. Assuming we let the virus go, this gives the following result.
In this simulation the results are in line with the use of the Markov model. There is a need for more than 8,000 hospitalizations, which is still too much for the current capacity in the hospitals.
5.3 Vaccinating 40+
How many do we need to vaccinate to go back to the normal life assuming that the efficacy is 90%?
If we decided to vaccinate everyone older than 40 years old, it would be possible to quickly enter a steady state of 1,000 hospitalized. Although this amount of required beds is still high this level is acceptable. We see the same result as with the Markov model.
The difference between the pictures shown here and on 4.2 lies in the fact that in the stochastic model the persons who is vaccinated still runs through the whole infection chain and will therefore influence result in hospital beds. In other words, once being vaccinated you can still become sick and eventually end up in the hospital, where in the Markov model this is ´simulated´ by reducing the R and the chances of being hospitalized.
In this simulation you first will reach a level of 2.500 beds. After this peak it will move towards its steady state. The reason for this increase at first is due the fact that this simulation has a different starting point as can be seen in the picture below, which results in the increasing slope of the hospital capacity during the first month.
The overall conclusion is that only with the assumption that the vaccine has a 90% efficacy, vaccinating everyone above 40 years old makes it possible to not take additional measures.
Vaccination strategies continued (using the stochastic model)
What if the vaccine works so well that it not only protects the person becoming infected, but also stops an infected person from spreading the virus? This scenario will lead to a different outcome of the vaccination strategy.
Currently there is no vaccine that stops the spreading of the virus, so this simulation describes a possible solution once a vaccine with this property is available in the near future. If we assume that we have a vaccine like this, what would then be the optimal strategy?
Vaccinating everyone over 40 gives us this result :
The result shows similar behavior to that shown in paragraph 4.2.
This strategy unfortunately takes a lot of vaccinations. Can we find a better way?
Could it be possible to lower the amount of vaccinations and at the same time lower the R and therefore stop the virus from spreading? If we only vaccinate the ones that are infected, been in the hospital, tested positive and the asymptomatic infections, this is possible.
Knowing this we can start applying the impact on the hospital beds. This results in the following pictures:
These pictures show the impact on hospital beds and total number of infected persons using different reproduction factors.
Reducing measures while starting with vaccinations is not an optimal strategy. The R would rise to 2.5 and the amount of beds needed will rise to an intolerable amount. This scenario shows however that if capacity would not be an issue the peak will be very high after 1.5 months. Every time you put more restrictions on society the R will drop but it will take longer to end the cycle.
One solution seems to be working well. Keeping pressure on the hospitals and vaccinating every person that becomes ill will result in relative more vaccinations in the beginning but eventually stops the cycle within 100 days. (R=1 as is scenario).
The pictures above give more insight in how the infections and the R will decrease towards 0.7 in period 11 slowly grows back towards 0.9. Eventually bringing the amount of infections almost to zero. To do this only 4% of the population needs to be vaccinated. This is divided over the different ages.
The underling pictures also show the impact on vaccination in the other cases. In the case of R =2.5 10% of the population would be required, if R=1.5 than 7% and for the current situation (R=1.04) this would be 5%.
We have been investigating the impact of the vaccine. We assumed an overall efficacy of 90%. With that number we started our calculations. It is clear, unless we vaccinate everyone above 40 years old, we cannot let go of the current restrictions. That needs a huge amount of vaccinations and it also needs time ; besides that vaccines are scarce.
A limitation of the model is that we can only differentiate by age. It is known that people with underlying conditions are at most risk. Assuming that the vaccine is effective for these people vaccinating them will be effective as well. This strategy needs less vaccinations. We will further investigate this possibility.
Unfortunately, the vaccine does not also stop people infecting others. If that happens, we have a different situation. It still shows that it does not work to just select a part of the society. But if you choose wisely in every part of society the R will be lowered, so the virus will eventually disappear. Our calculations show that if we only vaccinate people that have tested positive, sick ones and the ones that are asymptomatic, only 4% of the total population needs to be vaccinated.
These results are theoretical. We need to be able to find all the infected persons, so many tests are needed and also it has to be proven that the vaccines are not only protecting the vaccinated but those surrounding them (by stopping the spread).
More research needs to be done to find out how effective the vaccines turn out to be.
But it is a scenario to keep in mind.