09/07/2020

On the Science & Mathematics Blog, strategies to curb Covid-19.

Foto: Agência Brasil

Reproduction from the IMPA Science & Mathematics blog, from O Globo, coordinated by Claudio Landim.

Ricardo Takahashi, Full Professor at the Institute of Exact Sciences, UFMG

During this period in which society has been forced to seek ways to deal with the COVID-19 pandemic, several public managers have been surprised by what appears to be a very strange behavior of the disease: at one moment, hospitals have a comfortable occupancy rate, with 50% of beds available, and less than a week later they are full, unable to receive any more patients. At this point, new people with severe cases of the disease simply do not receive treatment.

My goal today is to show that this behavior of COVID-19 is not mysterious at all; on the contrary, it arises because this disease has a mathematical characteristic that other everyday things we know also have: a dynamic with a time delay . A familiar example for many of us is the shower with a drum heater – that shower where two taps, one for cold water (coming from the water tank) and the other for hot water (coming from the drum heater), serve to control the temperature of the shower. Those who are not used to using it usually get a shock the first time they use it: when opening the hot tap, the water that comes out of the shower is cold. The person then opens the hot tap further, and the water continues to come out cold. They open the hot tap to the maximum and close the cold tap completely, and what continues to come out is cold water, until the hot water finally arrives, and the jet of water that appears is boiling. At that moment, the unfortunate bather tries to close the hot tap and open the cold one. The flow of boiling water continues for quite some time, while our bather closes the hot water tap more and more, and opens the cold water tap. Of course, after a while the effect appears: the jet of water suddenly stops being boiling water and becomes ice-cold water again… This time delay between the command (opening or closing a tap) and its effect (the temperature of the bath water) deceives the bather, leading him to cause a large fluctuation in the water temperature, which will keep changing from boiling to ice-cold and vice versa until he realizes that he has to take this delay into account to calculate his actions.

COVID-19 has characteristics similar to that shower. Between the moment a person is exposed to the virus, contracting the infection, and the moment symptoms begin to appear, typically three days pass. Then, between the moment symptoms appear and the moment they worsen enough to require hospitalization, an average of seven more days pass. Therefore, there is a delay of about ten days between the moment a person contracts the infection and the moment they will occupy a hospital bed. The example of the showerman struggling with the shower illustrates what will happen with the COVID-19 epidemic if managers do not take this delay into account when acting.

The tragic events in the Lombardy region of Italy illustrate well what can happen. When the epidemic began to circulate there, the authorities maintained that everything was under control until, around March 10th, the hospitals began to become overwhelmed. The government then decreed a belated lockdown . Let's trace a timeline to understand what happened:

February 21: 19 people had been confirmed to have the disease in Italy.
March Mondays: With the number of infected people growing exponentially, on this date there were just over 2000 diagnosed infected individuals. At this time, these people did not yet require hospitalization.
March 12th: Of the approximately 2,000 people infected ten days earlier, about 20% (400 people) developed severe symptoms and sought medical attention. With this sudden increase in patients, hospital occupancy became a concern. The government then decided it was time to start controlling the epidemic and decreed a lockdown . However, as the epidemic had continued to grow until that date without any action being taken, by this point there were already more than 15,000 diagnosed cases of infection.
March 2Mondays: The number of new daily diagnosed cases of the disease stopped increasing the day after the start of the lockdown . However, people who had already been infected continued to experience worsening symptoms, and the number of patients seeking hospital care continued to grow each day. Of the 15,000 people who were infected ten days earlier, about 3,000 developed severe symptoms and sought care in the already overwhelmed healthcare system. This was just the beginning of the collapse of the hospital system in Italy.

Several of the "easing protocols" that have been released by municipal and state governments in our country have, as their main indicator of action, the monitoring of hospital bed occupancy rates. But, as we saw above, the naive idea that it would be possible to count the existing vacant beds in hospitals and adopt containment measures when they are close to being exhausted can be quite dangerous.

But what can be done then? Is there a safe way to avoid all of us being in a quarantine that will last a year or more? The answer is: yes, there is a way. There are some reasonable alternatives, which depend on the manager being willing to use knowledge to guide their actions. I will discuss three possibilities here that are easy to explain:

It is possible to implement the easing of restrictions very cautiously, opening only a few activities at a time, and waiting two weeks before taking the next step, which will only be taken if hospitals have stable occupancy. This is exactly what a bather does after learning how the shower works: they turn on the tap slowly and wait to see the water temperature change before touching it again.
The major problem with the previous alternative is the delay in taking action. However, as we have seen, this delay stems from measuring the effect of easing restrictions by observing bed occupancy. This means it is necessary to wait for the interval between the moment there is an increase in the number of infections and the moment when infected people are hospitalized. There is a smart way to reduce this delay: instead of counting the number of occupied beds, we start counting the number of newly infected people. To do this, simply test for COVID-19 as soon as people start showing symptoms, instead of waiting until they are in serious condition. The delay in being able to start measuring the effect of such a easing measure decreases from about ten days to about three days. There is the added advantage that it becomes possible to care for people, in many cases preventing the worsening of their condition.

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