Mathematical model of epidemic infection spread and its application for political decisions in the case of the COVID-19 crisis.

by Leonid Sakharov

Initial parameters:

Population:     People infected at day zerro:
Contentious after been exposed (days):   from till
Incubation period before hospitalized after been exposed (days):   from till
Part of exposed to be hospitalized:
Part of exposed with mortal outcome:
Mortal outcome after be exposed (days):   from till
Reproductive number (R0): after day 0:
after day R0=
after day R0=
after day R0=
after day R0=
Projection for future (days):



The mathematical modeling of how many people will be infected by a contagious pathogenic agent like a virus or germ as time passes is quite simple. You might have heard from your TV that sophisticated models from respectable Universities project this or that and should believe their words. Nevertheless the idea behind most of mathematical models can be trivial and you will easily understand the logic behind them. Of course there are no limits for improvements and detalization of any mathematical model but when you see on television bell like smooth curve be sure that this result comes from a very basic model.

These models could be easy to neglect as an intellectual game until the realization that they provide rationale for political decisions that could drastically change the lives of millions by virtually imprisoning everyone in quarantine ordered by scared politicians. If you have a brain at least you can educate yourself by reading this article about what is the real foundation for the order to wear masks, why quarantining could not immediately stop this nightmare or what is the second wave and where it is coming from.

The model presented here is based on the very most common assumptions. The validity of these assumptions you can judge for yourself. If assumptions are reasonably close to reality the model would produce numbers close to reality and warn about possibility of an unacceptable end game.

The model.

At any given day there are number of infections individuals who can infect certain number of healthy ones. After been exposed healthy person becomes contagious for a certain number of days until he or she completely overcome sickness. This time we will call an actively infectious period; Tb - is number of days after exposure the person became actively infectious and Te - is the number of days till complete recovery. Obviously Te > Tb - specific values of variables are coming from common knowledge about fly like deceases and safely can be presumed Tb= 2 and Te=21. If you think that is significantly different from reality - just change any number in the interactive model as you please and have different chart. The same is about any other specific numbers - play with them as you want.

Number of the newly infected people at any current day is calculated by the core recurring formula of the model:

Ni = ((P - I)/P) * (R0/(Te-Tb)) * Na

where - P - is the whole population where we model the epidemic; I - already infected population, R0 - coefficient of reproduction that gives number of people that actively infected person will infect until became healthy again, Na - number of actively infected peoples.

The formula above is in general self-explanatory. There are several notes to make about it. The important assumption in the formula is that I - number of already infected people will not be infected and sick again (develop immunity) it is true for most of known diseased but not yet established with certainty for COVID-19. If it will be proven wrong for COVID-19 the situation is even more serious than we are taught. Na - is the number of actively infections persons, the term (R0/(Te-Tb)) - gives coefficient of reproduction for actively infected person for one day. It means that if for whole time when one sick person can transfer infection (19 days) to 2 persons if R0=2 for one day it will infect for average 0.105.

As soon we will have a time line for the number of newly infected cases the calculation of hospitalized and dead is pretty straightforward and intuitively self-explanatory by applying to number of infected explicit coefficients to become very sick and coefficient of mortality. In case of COVID-19 there is some information about mortality coefficient but it is not too reliable because of lack of experimental information about how many infected people overcame infection without any symptoms. This is the key unknown.

The most influential parameter of the model is the coefficient of reproduction R0. Absent vaccine and medicine cure from the atypical pneumonia of COVID-19 diminishing a coefficient of reproduction is the only tool of society to fight epidemic. This is the exact reason to order social distancing - it will help for sure. The problem is that on a way civilization can commit total suicide in attempt to salvage one or two percent of most vulnerable. Cure can be worse than disease. It is as guillotine is the best cure from headache and dirty thoughts. It is the question what price society is ready to pay for saving lives from coronovirus. Are you ready to pay one thousand dollars to save your neighbor? Most possible. What about 0.03% of country population, complete strangers to you, for whole your 401 pension saving plus losing your home? It actually can be the choice for COVID-19. In case of influence the society decided to accept human loses in favor economic stability. In case COVID-19 political situation can force us to destroy everything and save nobody at all.

There are several basic possible scenarios of COVID-19 epidemic depend on political response that is coming from the model.

Do nothing. Let it go.

In this scenario virus is spreading naturally among society without any attempts to slow it down. No social distancing. No closed business. Minimal economic impact. In several months there whole nightmare is over. But about one million old people will be no more. Whole health care system is collapsed for half of a year. Relatives of dead blame politicians for cruelty. CNN has record audience. Click to see this scenario on the chart Do nothing scenario. In three months after first infected the country with size of USA will have start of mass death that will end with about 1.5 million additional deaths after first ten thousand buried in about three months. That will be it. If to compare to average mortality that is 0.2 millions per month the mortality rate these three months would be three times larger.

Total shutdown.

Here after first mass death from the decease the government would close everything what it can. Full scale quarantine. And what most important here the population is in full cooperation. In this scenario after 100 days after first infected the coefficient of reproduction is forced to value significantly less than 1. Click to see this scenario with push down coefficient of reproduction to R0=0.5. As it seen on the chart there would be only about 15 thousand deaths due to the virus if these draconian measures kept for about half year. Then, the virus would be wiped out from this society but without any guarantee not too return outside because only 3 millions would be infected and have immunity. It is just buying time to develop a vaccine. The economic cost can be staggering and possibly for nothing.

Mild social distancing.

This scenario is differing from total shutdown by severity of quarantine. Let say that coefficient of reproduction is down to the value higher than 1. Click to see the scenario. It would take about year and a half to end epidemic but with a 100 thousand deaths.

Second wave scenario.

Let say that in after strong quarantine the government would loosen restrictions on social distancing a couple times in hope that infection is no longer presents the real danger. If it is done too soon there could be situation when second wave of death will be much higher than first one. Click to see one of the imaginable scenarios of catastrophic second wave COVID-19 appearance.

In conclusion. In spite of trivial mathematical background the predictions such model as shown here could be done reliable only if parameters of the model are well experimentally obtained. It is not yet done for COVID-19. There are no exact measurement of coefficient of reproduction and how social distancing affect it, mortality rate is just a guess because a lots of folks display no symptoms and do not diagnosed as infected by COVID-19. There is no guarantee that a person after overcoming infection will be immune long enough to help society to wipe out infection at all. There are too many unknowns to make a scientifically optimized decision.

What is almost for sure that panic will not help. And erratically changing strategy could only prolong disaster.

Apr. 20, 2020; 22:03 EST

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