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Possible distribution scenarios for COVID-19



With my head in my hand, I sadly watched the endless news feed of panic, watching the disturbing news from Italy, and caught no less unpleasant rumors about the possibility of stopping transport with us. Someone in the comments actively inflamed: “Are all these measures necessary? But how will they help fight the virus? ”, Others called for preparing for the worst, while others even denied the threat. I wanted to more clearly and reasonably understand what was happening.

Having devoted the evening to the problem, I decided to answer with the help of mathematics the following questions:

  • Are the measures effective? How much?
  • Will it be like in Italy? Is the worst thing yet to come?
  • How likely is it now to meet an infected person in a vehicle?
  • Will the new virus be completely overcome?

As a result, I created a small program for modeling ( binary , source ) and got results with it - encouraging, but ambiguous. For details - please, under cat.

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Disclaimer


I am not a medic or biologist - I am an engineer. My specialization is the tasks of situational modeling and queuing. Real scientists create mathematically accurate models based on dozens of reliable indicators and hundreds of factors studied. My work is semi-empirical models. Their goal is to give a forecast in conditions of incomplete information - in which direction can the situation begin to develop if factor X increases and factor Y decreases. Such models, being rather crude, are derived "from the first principles" and allow us to understand the trend of an event.


Allows you to understand the trend - but not the exact figure. Why, I strongly DO NOT RECOMMEND using my simplified model for writing news by the method of "drove the latest indicators for my hometown, the predicted mortality rate has grown by 2% - now we all die!".

As a basis, I used the classic epidemic spread model, introducing several factors and assumptions specific to COVID-19 into it. Below I will talk in more detail about the model, then I will comment on the results (the word “forecasts” here will be too bold), and after that there will be conclusions. If you are not particularly interested in technical details, feel free to scroll to section 2.


1. Principles and Assumptions



Some of the numbers and indicators I have used are extremely arbitrary, and the model itself is simplified. You can take the source of my program and make any changes to independently verify your own hypotheses.

The model considers one city where, at the time of the beginning, a completely healthy population lives. The source of infection is residents returning to the city with a certain frequency from abroad, carrying the coronavirus during the incubation period.

In my model, the disease has three stages:

  1. Initial - asymptomatic incubation period ("passively" sick),
  2. Active - with a clear manifestation of symptoms ("actively" sick),
  3. Terminal - the symptoms are manifested actively, the patient has a risk of dying if they do not get help.

The transition from the initial stage to the active, and from the active to the terminal one occurs with some predetermined probability - and there is also the probability of a (small) independent recovery at any stage. In my model, the “active” patient is considered to be one who has pronounced symptoms, but continues to go about his business - while the “passive” patient either does not observe them, or quietly and peacefully sits at home “with a cold”.


Infection occurs at random, and its mechanism is pretty cleverly designed. At each of the cycles of the simulation, any of the patients has two checks for the infection of another person (conditionally - in transport and at work / study). In active and terminal patients, the probability of infecting someone is the same and quite high, in “passive” patients it is low.
The surrounding urban environment also contributes to the likelihood of infecting a random person - high population density, high social activity, as well as incipient panic increase it. The inhabitants of the city have a social life whose activity is randomly uneven; many, I think, noticed that on some days the bus that approached in the morning was jam-packed, on others there are empty seats. Quarantine measures, respectively, reduce social activity.

Oh yes, panic. Excitement, lines, clogged shops. Everything is simple here - by a given day it increases linearly to the indicated value, and then linearly decreases with the same speed. However, her contribution is not too significant.


Source: Reuters


City services also do not sleep. Every step, all "terminally" patients, of course, go to hospitals (if there is room for them) - it is believed that in this state they no longer need to be looked for.

A certain percentage of the remaining patients (primarily “active”) are randomly searched for - and they are also sent to hospitals. If there is no longer any place in hospitals, then such a patient is transferred to a “passive” state - he sits at home with a strict order to minimize communication, which makes it less infectious to others.

In the hospital, each patient has (one per clock cycle of simulation) a chance to recover and die. Diagnostic and treatment measures, accordingly, change these numbers.

Cured in a hospital or cured, he again falls into the general pool of healthy residents of the city - since the coronavirus is characterized (according to the latest news) by the lack of immunity being developed, then re-infection is also possible.

Simulation takes place at 12 o'clock - for each day there are two cycles of simulation, conditionally corresponding to the morning and evening conditions of the city.

In the process of modeling each measure, the following occurs:
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The most interesting part is the measures taken by the authorities in order to eliminate the epidemic. There are many possible measures, and you can try many different combinations, or move the application a few days earlier or later to see how the picture of infection changes.

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2. Options for the development of the situation


So we come to the most interesting. Once again, what is shown below is qualitative in nature, not quantitative. I do not solve the problem of “how much?” - just showing how the situation might look.

Our simple model is ready - let's simulate different options. To do this, I provided a set of presets in the "Counteraction" tab:



A) “Worst-case scenario"



There are no measures. This is a scenario of denying or ignoring the threat to the last - which, fortunately, has not come true almost anywhere (perhaps countries like Iran come to mind). At least Russia was ready and escaped.

Hypothesis: The scenario is this: the new virus was not taken seriously. They didn’t tell anyone about him. The authorities did not take any measures, people calmly become infected, and there is silence in the news. To do this, lower the "Quality of detection of infected" somewhere to 5%, and on the tab "Counteraction" disable all items. If you want more accuracy, then on the tab "About the program" check the box "Exact mode" (yes, I unsuccessfully hid it) . The graphs in the program are scrolled with the mouse and approximated with a wheel.
The results look devastating:


All schedules are clickable.

During the first few days, the infection grows like an avalanche, and most of the infected ones probably will not even detect this. And then here it is - that terrible exponential growth of infection, between the 30th and 60th days. Moreover, even increasing the number of hospitals here will not save - most of the population is “passive” carriers. The number of patients in the "active" stage is almost unchanged due to the established dynamics - hospitals treat as much as possible, people recover themselves, someone goes to the terminal stage. The situation begins to look great like a seasonal flu epidemic, when almost everyone seems to cough and sore . Yes, that's just coronavirus is not flu at all, the cured do not develop immunity and get sick again.

A serious increase in mortality begins on the 60th day, and mortality breaks through the border of 1% of the population of our entire small town per day ! Truly an apocalyptic scenario.

Fortunately, this scenario is unlikely to have a chance of developing up to a duration of 100 days, if only because who doesn’t catch it when the death count goes to hundreds? Such a thing in the long run, perhaps only somewhere in the cultural Middle Ages, where the disease is considered a punishment of heaven, and neither the authorities nor the people themselves begin to take ABSOLUTELY NO safety precautions during such a time, joyfully continuing to arrange concerts and ride in a crowded metro.

CONCLUSION:Interestingly, even here there is no picture drawn by the imagination of some citizens - cities that are instantly dying out under zero. The exhibitor rests against a certain “ceiling”, there is no one to infect, and there are quite a few victims - but the situation is stable.


INTERESTING: What's next, what's next, the number of patients in the incubation period is growing much faster than the number of actively ill people infected. It reaches dozens of undetected for one identified patient. In addition, the number of "active" patients is constantly decreasing due to diagnosis and hospitalization - while the number of "passive" continues and continues to increase, if the "active" are not treated in a timely manner.

Hypothesis: Is it possible to pacify an epidemic that has already broken out of control? Suppose that from the 65th day the country is fully quarantined, the borders are closed, 4 times as many hospitals are working, patients are being intensively treated, and from the 85th the tests will be delivered. CONCLUSION: Yes, the results, so to speak, are positive, but with a catch. Mortality after the introduction of all these measures is falling extremely slowly (hospitals are all crammed, all the same cannot be cured), and stubbornly keeps up. In this case, only vaccination can save the situation. So, if you do not intercept in time, then later, even with great efforts, it is difficult to correct the situation.







B) The scenario "caught on late"


Most likely, something like this is now unfolding in Italy. We looked at the beginning of the spread of the virus, and then, as we realized, we applied everything at once, but the infected were already decent.

Hypothesis: The measures are as follows:

  • Quarantine for visitors was introduced 20 days after the start, a dozen more - the borders were completely closed.
  • Two weeks later, schools closed, and soon more hospitals were equipped to cope with the flood of those infected.
  • After (on the 55th day) they announced full quarantine.
  • On the 80th, viral tests were brought up.

We set the simulation time to 120 days to see the “tail” of the epidemic: Please note that the population on the “Dynamics of Disease” chart even seems to have “grown up” at the end — residents in hospitals returned to the general pool. CONCLUSION: The good news is that there is no exponential growth here - they just manage to intercept it at the very beginning, until the infection process has not yet gotten out of control:









As you can see, the chart has two interesting points - around the 55th day and around the 80th. Before the introduction of quarantine, the virus could not be effectively stopped, and in the worst days, up to 0.1% of the population died. After quarantine was introduced, the infected gradually passed into the “active” stage, were detected and cured - so 0.5% of the town’s population died during the entire epidemic period, and the epidemic was completely crushed. If quarantine is introduced five days later, mortality will double already.


Hypothesis: And if antivirus tests are still not delivered? We look: The chart naturally lost its second peak, mortality increased by 25%, and the epidemic ... stretched. Now, in 120 days, the end has not yet come. CONCLUSION: This supports the very necessity of the tests in this situation!







B) Optimal measures



Taking timely measures, but without unnecessary severity and heavy blows to the economy. Probably the most optimal in the current situation for most countries - including Russia, too.

HYPOTHESIS: The authorities are not going to put our city “under the hood”, they are rather reluctant to close everything, and immediately resort to the necessary measures. And this scenario presented the most surprises: the number of infected is very small, the number of deaths can be counted on the fingers. A good scenario, which does not hit too hard on the economy - many establishments are closed, but transport runs and people work. It is interesting that the closure of borders, where new patients continue to come from, does not play a special role. I was amazed when I learned that even so, that way, but in the worst days



around 0.5% of all residents go infected.

CONCLUSION: That is, every two hundred in the city will be infected - and nothing, with the right measures, it does not cause terrible consequences. There is no epidemic, the city is not dying, hospitals are empty. With the right organization of affairs, the coronavirus should quietly go into the background, supported (within the framework of my model) by the influx of infected from other countries.


HYPOTHESIS: But what if the tests and the vaccine do not appear at all and the borders are not closed? CONCLUSION: The same background, but worse. The number of cases remains noticeable, but while the struggle is on, everything is under control. There is no destructive flash.





D) The toughest measures


For reinsurance - maybe “close everything”? Let's introduce all reasonable measures almost as soon as we hear about the virus, and all strict ones a little later. A little more, and buses will stop walking, and soon we will put everyone home.

HYPOTHESIS: Will it be possible to save more human lives by full quarantine introduced before the virus begins to seriously walk around the population? CONCLUSION: Indeed, we have achieved a slightly lower mortality rate - by several dozen people. By completely stopping the economy. At the same time, they still didn’t get rid of the last infections in 100 days - here and there, a person with a virus jumps in. And this is with tests and the vaccine!






findings


Based on all my calculations, we can conclude the following:

A) The measures taken in our country are effective . What is being done is not panic vanity with attempts to do at least somehow and at least something. If I calculated correctly, then the epidemic, within the framework of this set of measures, will be contained within fairly tight borders for about a month. Perhaps the measures will need to be supported a little longer - for example, until the summer, where the vaccine will arrive in time. Fine.

B) We are unlikely to get rid of COVID-19 as we, for example, got rid of Ebola. What famous microbiologists are now saying is fully confirmed in my model. The virus will go “into the people” and become what unpleasant diseases like polio have become for us - a dangerous but rare disease.Plus, they say that the virus can mutate - but I can’t say anything about it, my job is mathematics. C

) Around us there may already be a certain number of infected people who do not know about it. This number will increase over time, and will be maintained no lower than a certain level - at least for some time. The figure "every two hundred" seems to me somewhat overstated. The ratio of “one detected case to ten not found” also seems to me to be overestimated.

But keep in mind that almost anyone can become infected (including yourself, carrying the virus on your feet) - and don’t even know about it. Taking into account point B, hiding and waiting is useless - just take care of those at risk, especially diligently!


However, keep the paranoia in check.

D) Most likely, the "Italian version" we have already slipped. Honor and praise for the operational response of various services. And, if something doesn’t go very wrong, we won’t need to enter full quarantine and stop the transport.


In general, to summarize, I can say that the calculation is rather encouraging. We cannot completely get rid of the virus even with the toughest measures - but we can quite successfully reduce the number of cases to units per month. There is no reason to panic, but now we need to maintain attention and concentration. The victory over the virus is achieved by a series of careful and consistent measures.

Do not be ill!

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