- Luiss Open - https://open.luiss.it/en -

A planning problem in controlling the Covid-19 pandemic

Epidemiological studies developed in the 1930s on the transmission of the virus serve as the starting point of our analysis. In particular, we adopt a modified version of the SIR epidemiology model which divides the agents between:
– those susceptible to be infected S(t), in other words, those at risk of contracting the virus;
– those infected I(t);
– those recovered R(t), healed or died.
It is a dynamic model. In particular, it is a model used by epidemiological statisticians that explains how a system that starts with a certain rate of infected behaves over time. For data parameters, the model makes certain notions very precise, such as “at what point herd immunity is acquired” (the situation in which the percentage of those infected tends to 0 even with a return to circulation by the population).

An economist must know and be able to use epidemiological models considering that, as in the case of COVID-19, it is not only a matter of public health – and the spread of a virus – but also a problem that impacts the economic dimension of a country. In fact, the first measure adopted, the lockdown – the forced quarantine of citizens – has enormous economic consequences.
The policy maker must know the dynamics mechanism of the virus when adopting a policy. From this perspective, the goal of minimizing virus-related deaths must be managed together with the goal of not bringing the economic system to collapse (i.e., not bringing people to death due to other evils, such as hunger, social arrest or any other situation that may arise from a limitation of freedom of movement of this type).

Based on these considerations, we have combined two elements: on the one hand, the dynamics of virus transmission, on the other, its interaction with economic behavior and, in particular, with production.

Some standard economic analysis tools have been applied to the model, such as those of “optimal control”. In particular, the policy maker adopts an optimal lockdown policy based on the dual objective of minimizing both the number of deaths related to the virus and the economic costs deriving from the lockdown.
The model also permits an evaluation of the benefits of serological tests or a future vaccine (tools that enable management of the course of the virus much more efficiently). Nonetheless, even epidemiologists highlight the uncertainty of the characteristic parameters of this infection (for example, the case fatality rate, or how fatal it is, the speed of propagation, etc.) because the measurements were not aimed at detecting the sizes that interest us (for example, the swabs were only used on the hospitalized and not through a “sample” mechanism) and were rather limited in their extent.
There is thus significant uncertainty about the basic parameters (regarding the fatality of the disease, how many are infected) which must be considered in the interpretation of the results of the model. In any case, the model remains more reliable than impressions or sensations because it allows us to derive conclusions based on a coherent analysis whose hypotheses are explicit.

Some parameters are related to the spread of the epidemic. They are parameters coming from the “medical field”, for example the so-called parameter R or R0. We use data from the World Health Organization, the samples made in Austria on the Princess Diamond cruise ship (it is important to have a more or less complete – albeit particular – sample of a ship), the Vo’ Euganeo sample, the Faroe Islands samples. Among these, the so-called case fatality rate (or “how many among those infected die”), is particularly significant. This rate is not constant because it depends on congestion effects on the health system (e.g. if hospitalized at the same time – as happened in Lombardy – and under duress in the intensive units, patients are unable to be given the treatments that would be guaranteed under normal conditions; therefore the “fatality rate” increases significantly). One of the component “blocks” of the model is composed of a function that defines the degree of fatality of the virus, which depends on the function on the number of infected. The function gives the policy maker a valid reason to disperse the infection rate over time. This is a dimension of the problem which is important in determining the optimal policy to be implemented.
It should also be noted that the lockdown policy has different effects. On the one hand, the spread of the virus slows down due to a reduction in physical contact between citizens: in this way, the epidemiological model controls R0. On the other hand, the policy maker influences the country’s production (e.g. in the basic hypothesis, if the individual is at home, he/she does not produce, or produces less).

The policy maker chooses how much to slow down the spread of the virus while also factoring in the economic cost that quarantine entails.
The value of life represents another relevant dimension of the analysis. Our study, in particular, takes into account statistical measurements of life (such as the value of a statistical life- VSL). We used a benchmark value of 30 times annual GDP per capita value, exploring cases where this value rises up to 130 times.
Another dimension of the study is to understand whether or not – among the tools available to the policy maker – there is the possibility of foreseeing the immunity test. Various types of tests exist, in particular the cd. “Community Card”, a serological test that permits a determination of the possibility for the citizen to return to work – after contracting the disease – given that he or she has become immune. Our analysis considers, in particular, two scenarios to quantify the value of the test:
– scenario 1: lack of test
– scenario 2: presence of test
Between the two scenarios, the second reduces the costs of the shock by approximately 2% of GDP (for Italy this equates to roughly € 30 billion), while quantifying and clarifying how important it would be to consider non-indiscriminate lockdown policies. The presence of a test enables those in charge to keep in lockdown only those who are susceptible – because they haven’t contracted the virus – or those who have recovered. Naturally, the optimal lockdown policy varies between the two scenarios. In the case of the presence of the test, the lockdown lasts longer, as it only applies to the fraction of people affected by the virus. In the case envisaged by scenario 1 (similar to the current Italian or American situation), following 40 days of closure, the lockdown policy is removed very quickly because continuing to maintain it would be extremely expensive (many people who have recovered are kept in lockdown and therefore the economic cost is very high).

Keeping in mind the limitations discussed above – related to the uncertainty about the value of certain basic parameters and to the linear structure of the model (the model does not, for example, provide a stratification of the behavior of infections by age group) – the model provides an order of magnitude on how “reasonable” is to keep the economy in lockdown. The analysis is based on a pessimistic parameterization of the model. In this way, we have considered fatality rates in the upper part of the estimates that have been taken into consideration (for example, in the sampling of the population of Vo’ Euganeo, we have assumed a case fatality rate of 1 %). Our analysis also significantly assesses the effects of congestion (within our model, with 40% of infected, the fatality rate increases from 1% to 3%).
The results obtained arise from a pessimistic but not “foolish” scenario, which considers the value of life to be approximately 20 times that of annual GDP per capita. This value is used by statisticians when fatalities are concentrated around the elderly.
The model defines two main predictions:
I) What type of lockdown does the policy maker implement in the absence of a test? The lockdown is applied to the whole population without distinction, as in the Italian case. In this case, the optimal policy lasts 60 days: two months of gradual intensity lockdown. The infection begins with a rate of 1% and after 20 days the policy maker acts through a very severe lockdown – which involves about 70% of the population – that lasts about 20 days; then the planner gradually releases the measure (after 50 days, 20% of the population is in lockdown) which ends after 60 days.
There are two consequences of this policy: a great output cost – or of lost income equivalent to losing 8% of one year’s GDP (a situation similar to the financial crisis of 2008) – and a great welfare cost- higher than the economic one – that includes the cost of “lost” lives. Therefore, in the absence of the test, the output cost is equal to 8% of GDP and the cost of lives exceeds 20% of GDP.
II) What type of lockdown does the policy maker implement if he has a test available to recognize the infected who have recovered and who can gradually return to work? The presence of the test “softens” the hardness of the lockdown, which lasts longer because the healed can return to produce. In this way, the lockdown rate is separated from the fraction of the population that is in lockdown: the lockdown applies only to the infected and potentially infected, while the healed no longer remain at home. The policy acts as follows: 20 days after the arrival of the virus, a strong lockdown of 70% of the population is applied and is maintained for about 25 days. After this period, the lockdown is gradually relaxed. Whereas in the first scenario, after 60 days, the whole population exits from the lockdown, in the second scenario, the policy is much more gradual: 100 days after the start of the closure, the lockdown rate – for infected and potentially infected – is still at 20 % and the percentage of the population in lockdown is almost 10%.

The risks of a major economic crisis are enormous. One aspect that the model does not take into consideration – but that exists in reality – is that forced closure destroys an intangible capital: know-how and the ability to work do not readily adapt to a new situation. If Italy has faced numerous problems since the 2008 financial crisis, the risks of the current situation are even greater.
In addition, the country – already financially weak – will have a much higher debt than the previous one (for this reason, in the model, there is no consideration of these persistent elements of the lockdown, nor of the public debt). Policy makers should keep in mind that the economy is not that flexible and could suffer a permanent trauma.