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Irena Cieślińska: What do mathematicians know about the epidemic?

PROF. TOMASZ LIPNIACKI: From a modeling perspective, the most important data that can be relied on are the increasing number of identified cases.

How quickly the disease spreads depends on the infectivity of the virus itself, as well as the nature, or rather the culture, of the given territory - how densely populated the country is, what habits its inhabitants have, whether they eat out, how they greet each other, how far apart they stand from each other when talking etc.

The disease spreads when one patient infects more than one person. It is simple to visualize: if the patient infects two people, and each of them infects the next two, the epidemic is spreading exponentially. The growth rate still depends on the incubation period of the virus, which determines the minimum time separating the infection of a"donor" and the infection of his or her "recipient".

Because we can't really change the nature of the virus itself, in order to stop the epidemic we need to alter the culture - reduce people-to-people contacts or isolate infected people faster.

How drastic should the restrictions be for Poland to succeed in fighting the epidemic?

- Our estimates show that the Chinese have reduced their number of contacts 30-fold. In the initial period in China, the number of identified cases increased by 40%. daily, i.e. it doubled every two days; but 10 days after the introduction of the quarantine on January 23 it stopped growing.

The Chinese government implemented extremely strict rules that covered over 50 million people.

- The Chinese sat at home for two months. They did not go to work, only one person from the family could go out once every two or three days, and that was only to do shopping. Everyone had to wear masks.

Please compare this with what was happening in Italy. There, the initial increase in diagnosed cases between February 21 and 28 reached 50% daily. As a result of the implementation of increasingly stringent restrictions, the rate began to decrease and is currently below 10 percent. (data from March 22-26), while the number of new daily cases stopped growing.

Prof. Tomasz LipniackiProf. Tomasz Lipniacki tl

Mathematically, this means that the Italians reduced the average number of contacts more than sixfold. It is important to remember that when we talk about diagnosed patients, we are actually describing infections from three to seven days ago, so in all likelihood the increase in total number of cases in Italy has already slowed down even further.

However, Italy is nevertheless some distance away from the severity of restrictions adopted by the Chinese. As an example, in northern Italy they started by closing bars and pubs, but ... only from 6pm onwards. Now the restrictions are much stricter, although still quite on the same level as in Wuhan during the more than two months of quarantine.

Our model allows you to estimate by a factor of how many you need to limit the number of contacts so as to, assuming a given increase in daily incidence, lead to a scenario in which the number of new infections begins to decrease.

Of course, this will not happen immediately. Since the virus incubation time is about five days, you have to wait about ten days for the quarantine effect. Firstly, because of the increase in newly diagnosed infected people who contracted the virus three to seven days before the restrictions were introduced. Secondly, even if someone is quarantined, he or she can still infect other household members, so the first round of transmission of the infection can take place at home.

What can the model tell us about the situation in Poland?

- First of all, we can see that in almost all European countries and in the United States the disease develops with more or less similar dynamics. In Poland, the beginning is milder, the rate of growth of patients is slightly lower, but the restrictions we have adopted so far will not cause the epidemic to stop. So we will probably follow a similar trajectory to the Italians.

And what if our milder, as you put it, early phase is due to the fact that we have been underestimating the number of new infections? Simply put - do we perform enough tests to detect all cases?

- It doesn't matter much for the model. What we need for our calculations is not so much the number of infections itself, but the rate at which this number is increasing. Our model shows that if one person infects an average of 10 people during the epidemic growth phase, the number of unidentified cases exceeds nine times the number of those who have symptoms or have gone through the disease.

In a word, what we detect is just the tip of the iceberg. But this peak grows at the same rate as the mountain itself.

In epidemiological models, the key figure is the basic reproduction number, called R0, which estimates how many people (on average) contract the virus from a single infected person. It depends on the virus, but it is also greater the more contacts a given community maintains. Conversely, the sooner infected persons are quarantined, the faster R0 decreases. And the ability to effectively select people to be placed under quarantine obviously depends on the number of tests done.

So when you start doing more tests, you will end up isolating a lot of people, but thanks to this isolation, the rate spread of coronavirus will slow down.

However, in the long run having a slow increase in the incidence rate and a large number of unidentified cases is irreconcilable. 

Based on your model, would you be capable of predicting how long the pandemic will last and how many people will be infected?

- Europe is looking at the history of the epidemic in China and comes to the conclusion that "with us it will end the same." But the restrictions introduced in China were incomparably stricter.

Restrictions in Europe are slowing down the growth rate of coronavirus, but they are not enough to fully arrest the growth of the epidemic, much less lead to a significant decrease in the number of new daily cases.

There is a long way to go from a situation in which the number of daily cases does not increase to one in which it drops by 10 percent per each day (as has been the case in China from February 3 for a month)  - according to our calculations it requires a further threefold reduction in the intensity of contacts.

It seems that the situation is out of control in Europe. It is too late. Whatever we do in Europe, the number of cases will exceed several million. And even if a drastic quarantine is introduced, it will have to be maintained for a very long time, pursuing a common European policy, until the epidemic withdraws enough to loosen the bans (in China some of them are still in force).

If it is not possible to stop the epidemic, maybe it is enough to moderate its progress?

- Such considerations have recently appeared in the public debate: that the rate of growth of the epidemic should be lowered so that the peak of incidence does not exceed the efficiency of health care - the so-called flattening of the curve. This is completely unrealistic in our opinion.

We counted it for a country of 50 million. If the number of new infections increased at a rate of 10 percent a day, at the peak of the epidemic one would expect a million new cases one day! And in a more optimistic version of flattening, when the increase is only 5 percent, during the summit there will be up to 300,000 a day new patients.

This is completely beyond the possibilities of any healthcare system. It should be remembered that slowing down the epidemic also postpones the moment when the peak comes. And if our goal is to focus on this slower growth, we would have to wait eight months for the peak of the epidemic. But to keep such a slow, 5% daily increase in the number of cases, quarantine would have to remain in place throughout this entire time.

Eight months of quarantine, right?

- Yes, eight months of restrictions comparable to those currently in place in Italy, where the growth rate has dropped below 10 percent. We can also try to predict how many people will be infected during the worst week and the worst month, in total. With a 10% daily increase in total cases, the latter amounts to half the population.

It is obvious that we do not have to choose whether our health service will cope with it or not. You can see that under any such scenario the epidemic will be impossible to manage.

But the models we are talking about at the moment do not take into account one factor important for human societies, namely the factor of fear.

Fear?

- We proposed two scenarios in which we take into account that at some point people, regardless of bans, also just start to be afraid. Because at the beginning, when infections are low, there are those who think to themselves: "Well, the chance that something will happen to me if I go out with my friends for a beer is infinitesimal." And they go for this beer, which causes an exponential increase in the epidemic. But after some time, when they see that some people from their broader social circle are dying, their attitude switches to: "Heck, it’s getting dangerous out there, I should stay at home. At the very least, I should be in touch with no more than two people, and preferably on Skype”.

Therefore, we assumed that once a country reaches a certain number of new daily occurrences, let’s call this number H, we voluntarily limit our contacts by a factor of two.

Does the fear model work in Italy?

- In a simulation we have run to demonstrate our fear model, we assumed that H = 10 thousand. Even now, there are fewer new cases per day, about 5,000. But maybe the Italians are already starting to get a little scared, the streets on webcams seem deserted.

I also don't know the Italian society all that well, even though I have two Italian colleagues who work in my laboratory. I have a friend Franco, he lives in a small town in the province of Emilia-Romagna. By the time the quarantine was already supposedly strict, he went out to buy an e-cigarette liquid. The police stopped his car.  He figured if he were to honestly confess his reason to be outside, they would send him back home. It was not a serious enough reason, and in any case the current epidemic of a pulmonary virus would be a good reason to stop smoking. So instead he told the policemen that he was on his way to get a good, high quality fish. He said that the stores nearby did not have anything decent on offer, which made a trip to the port necessary. So they let him go.

Of course, people get scared at a different point depending on their country’s culture. There are those that accept greater risk. The number of homicides in Brazil is at least 50 times higher per capita than in Poland. If we suddenly reached such a level in Poland, we would stop leaving the house. And the Brazilians are going out nonetheless.

I look at your simulations and see that the number of daily cases stabilizes and then slowly decreases over a few years. Somehow I can't imagine five years of fear and confinement at home.

- I would like to emphasize that such policy ideals are not something I propose or encourage. On our part, this is simply an attempt to understand and describe what is happening and what can happen. And please note that this is not a hopeless situation in the sense that, considering the fear factor, we are not dealing with a million new patients a day, but rather with tens of thousands. And after a while people go through the disease and become immune, hence this slow decline in the number of cases.

Nonetheless, the healthcare systems are unable to keep the situation under control.

- How else could it be? In Poland, we have 38 million people. If there were 50,000 new infections per day, it would take a little over a year for half the country to become infected. So even if we manage to spread out the epidemic throughout a whole year, it is still beyond the scope of any healthcare system. In turn, if we further flatten the peak of the epidemic to 10 thousand new cases per day, we will extend the duration of the epidemic even further, requiring quarantine and other restrictive measures for up to five years! However, there is a good chance that a vaccine will be invented and disseminated during this time.

I will definitely not be able to stay confined in my apartment for five years.

- Our models also take that into account! You can say this: people got scared and stopped going out. But they sit at home for a month, two, three and say: "It's hopeless, but it doesn't look like anything is going to change, and you have to live your life and earn money somehow!" In other words, the threat becomes more banal, we get used to it.

On the one hand, we have a fear factor, and on the other, people are beginning to accept an ever higher level of risk. This was the case with the Spanish flu, leading to subsequent waves of epidemics. It seemed to be fading, people would be more daring and start having more contacts, and the epidemic would come back. And so it lasted for two years, until a significant part of the population got sick and subsequently immune.

Therefore, the most likely scenario is that fear will stop the pace of the epidemic, but it will recur in waves after people and governments are getting used to the risks. Young people who have the most contacts will also be the most willing to take risks, which will additionally immunize the population, because those who can potentially pass the virus to the largest number of friends, after becoming ill, will become immune.

To tell the truth, when arranging this conversation, I was hoping to hear something more optimistic.

- Maybe we should look at it from a different perspective. It is currently possible to calculate from the Italian data that the mortality rate is 10%. (March 26 - 8,000 deaths in 80,000 cases). Most probably the number of infections is much higher,  which reduces actual mortality, but some people have no symptoms at all, or have mild symptoms and after a few attempts give up on trying to call the hospital.

After passing the disease, these people acquire immunity, which contributes to the slowdown of the epidemic. Therefore, while no one knows what proportion of the population will die, it is probably much less than 10 percent.

If in two years 2 percent of the population dies due to coronavirus, this will mean doubling the total mortality rate, because in most European countries just 1% die of old age. population per year (in Poland last year 405,000). Thus, during the epidemic the number of deaths will be twice as large as under normal circumstances.

This is obviously terrible, but it is not yet a demographic collapse. At the same time, if the epidemic reaches that magnitude the healthcare sector would be paralyzed and the only thing it could probably do is to issue death certificates.

Apparently, we are diametrically opposed in our definition of what optimism is! I got the impression that the forecasts of other researchers are not as dire as yours.

- The difference between our simulations and many others, which are collected on the page gabgoh.github.io/COVID/index.html , where there is a calculator that allows you to simulate the development of an epidemic with different parameters, is that we have a different, and in my opinion more realistic, way of modeling the distribution of the virus incubation time, i.e. the time after which an infected person can infect others.

Everyone agrees that it takes about five days.

- It is true. But what people are not quite aware of is that the dynamics of the epidemic depend heavily on the distribution of that time. It's not like everyone is starting to infect on the fifth day. Some begin to infect on the third day, others after a week. And it is important to consider the average value of incubation time as well as its distribution. We have adopted a distribution which stipulates there is almost no chance that someone will pass on the virus within the first two days after infection.

This seems to be consistent with epidemiological data. Other models, for simplicity reasons, allow for the possibility of almost immediate infection and this changes the dynamics of the epidemic. As a result, from the same epidemiological data, our model estimates twice as many people being infected by one patient as the other models. 

This is the basic reproduction number that you explained earlier. When it is greater than one - the number of patients increases. Less than one - the epidemic is ending.

- That's it. If you know how many people are infected by one patient, you can estimate how much you need to limit people-to-people contacts so that this indicator falls below unity. Our estimates show that for a freely developing epidemic, i.e. when the number of cases increases by 40 percent per day, i.e. doubles every two days, reproduction is close to 10.

In Poland, over the past week (March 19-26), the number of cases has increased from 355 to 1221, which means an average growth rate of 23 percent per day and gives a current reproduction rate of about four. “Current” here means up-to-date with the situation from three to seven days ago, when people who have symptoms today become infected.

This means that further drastic reduction of contacts is necessary to stop the growth of the epidemic and even greater effort to put it out. In this sense, in fact, our forecast is less optimistic.

But it may be more realistic.

- The value of a basic reproduction number is to provide key information for authorities by showing how drastic the measures have to be to achieve a given result. In Poland, scientists have difficulty reaching those in power, but Polish government is copying what other countries are doing.

In England or Germany, epidemiology specialists have a greater impact on what the politicians do. And if these specialists are convinced that it is enough to reduce the number of contacts five times, then they only propose measures that will get us to this number, and not below, since more drastic interventions would be even deadlier for the economy.

Therefore, I would not exclude that the restrictions imposed by the Italians, which were followed by the rest of Europe, were too mild precisely because the basic reproduction number was incorrectly calculated.

We cannot expect the pandemic to go down the Chinese path without taking equally drastic measures as China did. Expecting otherwise strikes me as magical thinking on our part. But still, I hold on to a glimmer of hope that I will be proven wrong.

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