27 May, 2020

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Covid Transmission: Simple Mathematics Explain The Danger Of Crowded Situations

By Kirthi Tennakone

Prof. Kirthi Tennakone

Reducing density of crowds and movement in such situations and isolation of infected or those suspicious of exposure, effectively limit the progression of coronavirus infection. Imposition of such measures and education of the general public is an absolutely a must to contain the infection. 

Simple mathematics fathomable to most, explains how and when the infection expands into epidemic proportions. Attempting to grasp this idea would be an opportunity for students as well as laymen confined to their homes to refresh math they had learnt in the school and clear the myth that mathematics is difficult.

The tutorial below provides the mathematics prerequisite needed to understand the problem.

Mathematics Tutorial 

In mathematics numbers are denoted by letters of the alphabet. For example, the letter N could represent the population density (number persons per square kilometer) of Colombo or the price of one-kilogram rice in rupees. For simplicity, the product of two numbers, say N and M (N x M) is denoted by NM or equivalently MN and the ratio of N is to M by N/M.  The reader may have heard the statement that some quantity N is proportional to another quantity M. This means, the ratio N/M is always the same irrespective of the values N and M that are allowed and when expressed as an equation, proportionality is expressed as N/M = k (a constant) or equivalently N = kM.

Figure 1. Exponential growth of a population. Plot of population density (N) vs

Many quantities we are familiar, change with time and the rate of change depends on the time at which you measure it. How do we measure the rate of change of some quantity, say the population N of a city? If the population at certain time t is N and after lapse of very small interval of time denoted symbolically as dt, it changes to N + dN, where dN denotes the increase in population. The quantity dN/dt measures the rate of increase of the population at time t. Frequently the growth of a population at time t is proportional to the population at that instant of time itself, represented by the equation, dN/dt = kN, where k is a positive constant. In such situations referred to as exponential growth, N varies with time as shown in the figure below.

Equipped with above background knowledge you will be able to follow the mathematical argument of the following section, which explain why the movement in a dense crowd poses a severe danger in the present situation of the risk of COVID.

Mathematical explanation

The primary mode of transmission of COVID is believed to be the release of virus carrying droplets by coughing or sneezing of infected individuals. These droplets disperse around a range of the order of one meter (L = 1/000 of a kilometer). Suppose in one day you have moved a total distance of D kilometers in a crowed of N persons per kilometer square meter and among them a number M per square meter are carriers of the COVID virus. Figure.1 below pictures the crowd, grey dots representing healthy people and black dots the carriers of the virus.  It is clear from the figure, your critical range of potential exposure, extending a distance L to right or left has swept an area 2LD, shown by the contour bounded by broken lines and you are exposed to infected persons covering this area which is 2LDM.

Your chance of catching the disease in one day of roaming in the crowd is proportional to 2LDM and therefore expressible as 2LDMP, where P is a constant of proportionality. The number P takes into account the fact that, every one exposed to the virus would not get infected.  As there are N persons per unit area in the crowed largely in excess of infected individuals, each moving similarly, the rate of growth of the infection in the population per unit area is N times 2LDMP, which is 2PLDNM. 

The infected persons are also removed at a rate proportional to their density in the population owing to immunity, isolation or death, giving the removal rate as kM, where k is a constant. Therefore, the net rate (infected rate – removal rate) dM/dt of the increase of infected persons in the population per unit area can be written as, dM/dt =   2PLDNM  – kM  =  (2PLDN – k )M           

The above equation tells you, if 2PLDN is greater than k, the disease grows to epidemic proportions. Evidently, the more you move in a dense population, you provoke conditions favorable for induction of an epidemic. The epidemic can be contained by limiting crowding of people (reducing N) and movements in such environments (reducing D) and effective isolation of the infected (increasing k).

The average distance D a person walks in his social environment is around 4.75 km day and initially, the rate of removal of infected individuals is negligible compared to rate of infection (first term of equation written in the previous section). The important unknown P in this simple theory, deduced from the curve similar to Figure.1 for the initial epidemic in Wuhan China is of the order of 1/100.

How many people does one single infected individual pass the infection?

If an infected individual move around in a population of density N per km square to attend routine matters as usual and if D is the total distance moved during a day. His critical range 2L of possible passing of the infection will trace an area 2DL. As the number of people in this area is 2DLN, the number likely to catch the disease would be 2DLNP per one day. An infected individual continues to be contagious for about 14 days. Therefore the number of people he is likely to pass the infection is 28DLNP. When we set D = 4.75 km, L = (2 x 1/1000) km, N =1500, P =1/100, the above number turns out be nearly 3.  An infected person associated with a population of density 1500 per square kilometer will transmit the disease to about 2 persons on average. When population density doubles this number is also doubled.

The above characteristic number “ How many people does one single infected individual pass the infection”, measuring the intensity of an outbreak is referred to as reproduction number or R0 by epidemiologists. George Macdonald – British physician cum mathematician noted the importance of keeping R0 below 1 to control an epidemic and suggested a successful strategy to eradicate malaria in Sri Lanka

The model described above is not applicable to a circumstance where number of healthy persons and few infected confines to closed environment. Here, unless adequate precautions are taken, the numbers infected will be much larger. 

Mathematics indicates that the current recommendations for containing the pandemic, needs to be strictly adhered.  

*Prof. K. Tennakone, National Institute of Fundamental Studies, Hantana Road, Kandy.

Acknowledgement 

The author wish to thank Prof. Saman Seneweera, Director National Institute of Fundamental Studies (NIFS) for encouragement and Dr. Sanath Wijesinghe, Massachusetts, United States for suggestions and carefully reading the articles. Discussions with Prof. Lakshman Dissanayake (NIFS) and Prof.Ajith De Silva, University of West Georgia, United States were invaluable in drafting this note. He is indebted to Kasun Jayanath Wimalasena for checking the calculation and attending to the drawings. 

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Latest comments

  • 2
    0

    Mathematicians are also an asset,apart from Medical folks to understand and tackle Corona Virus.
    Way to go Prof; A very clear analysis.

    • 4
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      The key to minimize spreading the virus is testing and testing and nothing but testing.

      Germany 50,000 tests per day and targeting 100,000 tests per day
      UK 10,000 tests per day
      Sri Lanka 100 tests per day.

      But Police is apprehending 200-300 curfew violators daily!!

      Please increase the testing.

      • 4
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        Nimal Perera,

        Yes. Testing, testing, and testing. To solve , must identify the problem and the extent of the problem.

        Both South Korea and USA got the first COVID-19 case on January 21, 2020. South Koreans started large-scale testing starting on Jan 28. USA did not do large-scale testing until 45 days later, and President Gon Donald Trumph was fiddling around and claiming no problem.

        Every body knows, the rest of the story.

        • 2
          5

          Kirthi Tenn: your WHO math models are useless, when the GMOA mafia is massaging the Corona 19 figures and an incompetent and heartless government has put the hammer down on society without any planning or concern for those with few resources. It is one thing for this doctor and the middle class to stay at home, work using the Internet, and muddle through teaching their children from home; it is another for the billions of migrant labourers and day labourers, people who live hand to mouth, and people who have no homes. Lockdowns, quarantines, social distancing – these words mean nothing for the billions of people who work hard each day to socially reproduce the world and to produce the millions of commodities; they have not benefited from their work, but they have certainly enriched the few who are now hiding with their wealth behind their curtains, afraid of the reality that made them rich.
          More people die of dengue and malnutirition related illness than Covid which is a fake narrative in Tropical countries

          • 0
            0

            Don Stanley
            I am in agreement you in general, although, I may disagree with you somewhat that mathematical modelling is useless, which in my humble opinion, helps to forecast worst case scenarios to make appropriate preparations, but then again, in my limited understanding of mathematics, the validity depends on the accuracy of data & the domain within which the data is represented, therefore, serves only as a guide & cannot be applied universally. Statistics are useless if inaccurate & not may relate to the actual situation at ground level. Statistics can be made to distort or enhance the situation by those compiling it.
            However, I agree with you totally that more people die from diseases, such as, diabetes & kidney failures & the health professionals should do more, at least, to educate masses to manage such health issues. I believe the curfew is an unnecessary burden imposed on average people, although, many of my SL friends disagree. A recent article in CT by a retired ex GMOA President, Dr Janapriya, exposed the unethical & even false representation of facts by the current GMOA president & in the TV discussion attached, clearly indicates that even the police spokesman admits the purpose of the curfew is lost when people congregate in search of necessities. The proposals to provide door step delivery of essentials is unsustainable & will not cover the entire population. The only other country, as far as I am aware, to impose curfew is India but it caused chaos with people defying the curfew in a stampede to find food & in bus & train stations, trying to find their way back to their villages. If the curfew continues indefinitely, there will be riots, which are likely to be clamped down brutally by the military & the subsequent deaths, as well as, from hunger & malnutrition as a result of the curfew will be more than from the virus,

        • 3
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          Prof. Kirthi Tennakon,

          Dr. Anthony Fauci: Coronavirus Outbreak ‘Is Going to Get Worse Before It Gets Better’ | TODAY

          https://youtu.be/jHdblyM_3hs

          Well, two months ago President Gon Donald Trumph said no virus problem. President Gon Gotabaya Rajapaksa said no problem, on March 17, 2020.

          Sri Lanka has the 4th death. Still many infected imbeciles out there loose, infecting others

          It must be hard to model the behavior of imbeciles.

    • 3
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      Prof. Kirthi Tennekoon,

      Thanks for the article succinctly explaining the mathematics of the COVID-19 spread, the importance of the population density, as it impacts the probability of coming into contact with an infected, symptomatic or asymptomatic person, and the likelihood of getting infected.

      Now think about the situation, inside a bus, train, in a queue, plane or a gathering of faithful asking for divine blessings. Their population density is very high and the time is quite long as well, equivalent to roaming long distances. The Ro in such situations will be quite high.

      The best protection, is to stay away from other people. Then, have personal protection equipment, masks, that can stop the airborne droplets, and wash the hands, and wear gloves, if possible Do not get contaminated when you remove them. You can go to the countryside, where the population density is low. The farmers can be in the field doing their work, with a low probability of contamination.

      Let’s hope a good fraction of the populace, mean measured IQ 79, understands the mathematics succinctly presented here.

      Do you think, Wimal Buruwansa, the “mathematician “ of the parliament of 2/2=0, fame, who is a minister of the present government understands your COVID1-19 spreading mathematics? Do the saffron clad monks to whom the imbeciles prostrate, the priests and the Ulama, understand?

      • 4
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        Lemon Puff Veeraya had declared that he will carry out a Covid-19 genocide the same way he did it to Tamils.

        You might have heard that sharp mind is the sharpest sword. That is the path of science. But the 8th grade murderer didn’t learn any science. So his model does not include any differentiation or Integration.

  • 5
    1

    “https://www.youtube.com/watch?time_continue=45&v=X1oVeeCkocs&feature=emb_title”

    Are the Rajapaksa’s getting ready for another Tsunami donation distribution?

    • 7
      1

      Always. Rajafucksas pray to various aliens to bring disasters to SL so they can make money.

    • 2
      6

      Idiot Mr. Furt…..
      /
      Article is a mathematical representation of community transmission of Covid.
      Not some political bullshit.
      If you don’t understand it keep your butthole mouth shut.

      • 4
        1

        S.C. Shitqual – Please keep your rear hole shut and get off these columns. You have raised your ugly head again.

  • 0
    0

    Mathematical explanation to lift curfew division wise istead of all town to avoid could represent the population density Supermarkets and ( bank ATM Machine) want you to know: there’s absolutely no need to panic. Instead of catching the disease in one day of roaming in the crowd.

  • 1
    2

    Do our UNP Opposition Leader and his Yahapalana TEAM Mk 11 understand what the good Professor is on about?..

    BTW I didn’t see any members of this TEAM ,at the Meeting with the Prez to beat Corona Virus..
    Wonder why….

  • 3
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    I have reread this excellent reasoning of Prof: Tennakone [Possibly Maths Dept: Peradeniya in the good old days].

    These are curfew days. When it is lifted people rush out shopping and avoid the other normal chores. Therefore is not the value of D= 4.75 km [Approximately 3 miles ] on the high side?. If there is no curfew, then the distance traversed per day D could be ASSUMED to be as given in the equation.

    • 0
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      Many in SL believe the curfew is necessary to control people who are ignorant & selfish. In my humble opinion, we all have basic intelligence & the capacity to understand, therefore, can be educated of the severity & the need to obey instructions for greater good. In UK too, people took advantage of a sunny day to go on picnics when the govt. first declared ‘a holiday’ to stay at home but with constant education through the media & other means, has conveyed the message that people should not congregate as it is the only way the virus can be contained. Therefore, not more than 2 people can be together in public unless from the same family & are asked to keep a distance of 2′ in public, as well as, not going on non essential journeys. The police fine those who disobey the law but we are free to shop for essentials locally which are open as usual & there are no crowds as only 1 person is allowed in. There maybe short queues outside sometimes, keeping the required distance, as only a limited number is allowed inside at a time but no panic buying. Yesterday afternoon I went to my local supermarket, there were no queues outside & I was able to do my shopping quickly with no shortage of fresh food & on the way back, I bought some bird seed from a pet shop & a cycle tube for my daughter’s cycle from a DIY store. People are allowed to walk their dogs, jog or cycle but not in groups of more than 2 persons, as long as, the required distance is kept with others in public spaces..
      Lockdown does not mean a curfew but controlling the problem sensibly. Only India is the other country that has imposed a curfew to my knowledge & it is not working.

  • 2
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    Prof. Kirthi Tennanoon,

    “The model described above is not applicable to a circumstance where number of healthy persons and few infected confines to closed environment. Here, unless adequate precautions are taken, the numbers infected will be much larger.“

    Thanks. How do you get this message to a populace, where the mean measured IQ is 79, and have many unproven beliefs.

    The Manayakas, the head monks, want all temples to, chant pirith, for 7 days for protection against the virus. Can you model the futility of this exercise.

    Can you use mathematics to bust their unproven beliefs, such as prostrating to monks, to statues, pilgrimaged etc.?

  • 2
    1

    One of the best- fit mathematical model for the spread of any epidemic is the simple SIR (Susceptible- Infected-Recovered) model.
    The SIR model is defined by a set of inter-linked differential equations. This is a simple system of differential equations that qualitatively behalves in a reasonable way. It defines 3 interlinked functions S(t), I(t) and R(t) where:

    S(t): No susceptible @ time t
    I(t): No infected @ time, t and
    R(t): No recovered @ time t

    The SIR model interrelates the first derivatives of these functions and the functions themselves.

    Studying this near the beginning gives a value called Ro which governs the early growth rate which is exponential growth and is a standard metric in epidemiology. Indeed, real world data shows the current COVID-19 endemic is well modelled by this kind of exponential growth.

    Further variables may be introduced (as appropriate) to the system if there are more pivot variables such as:
    1. Exposed (E (t)) to address any latency between contacting the virus and getting infected.
    2. Adjust for new births and deaths
    3. Effect of vaccines
    4. New mutations of the virus effecting the no susceptible
    5. Other localised variables etc.

    However, note that all these are “toy” models, and developing a precise mathematical model is far more complicated mainly, to the many pivot variable acting on the different functions.

    In order to define accurate model that is able to give a real predictive handle of any particular viral epidemic, extensive research needs to be conducted to identify the specific features, and the localized effect of such features, and the final model be simulated to test against real-world historical data.

    I don’t intend to explain the theory of the SIR model since any interested person may google the topic and find loads of info.

    • 4
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      Thanks Darshana. The purpose of this note is not teach elementary epidemiology in terms of a SIR model , but to put things in a simplistic way for students and laymen to ponder. What is needed is people to realize the gravity of present problem and also people to think out-of-box and mathematically. Laymen and students cannot fathom R0 in terms of a SIR model . Similarly most epidemiologists cannot understand the deeper meaning of RO, , which can be rigorously defined only in terms of theory of stochastic distributions . We are well aware that simple models will not lead to full understanding of the situation

      • 2
        0

        Prof. Tennakone,

        “Laymen and students cannot fathom R0 in terms of SIR model,”

        The Mahanaykas, a stone’s throw from the Institute of Fundamental studies, Hantana, overseeing the Temple of the (Canine) Tooth, have asked the Temples to chant pirith for 7-days, to tame COVID-19 virus. Will it have an impact on R0?

        I am very curious, as to how you are going to model the spread of the virus in the simple model or the SIR model, for the chanting of purith.

        M(t) :Cumulative Number of pirith chants, @time t.

        Since, in practice, this will have no effect, unless they gather the imbeciles to one place, what value would you use for M(t)?
        =0, for no- effect

        > 1, because of clumping of the imbeciles.

        The COVID-19 has really exposed the bluff of the monks, Priests and the Ulama

        Present your simple math to the above.

  • 1
    0

    Professor, explaining how coronavirus spread through the communities to even decision makers in SL with any mathematical modeling is out of stretch. Isn’t it?

    Otherwise, how could they decide for holing a national election later this month?

    NSF could have tried to use a very familiar concept of fart theory to explain the science starting with decision makers then politicians before reaching out to laymen.

    Fart theory says that:

    If the fart is voluminous blare then we say what a nasty pig and take shelter in isolated places.

    If the fart is a hiss type then we just look around with exploring our options to get out of that place.

    If we don’t isolate then we have no option but to inhale the sulfur particles through our nose.

    Then the final pitch is that coronavirus won’t make a toot or hiss and we won’t even smell it. This approach might resonate well with our patriotic leaders and their sidekicks.

    If you want to coat the message with a little bit of political flavor then throw some hiss type politicians’ names to people from the other side of the political views!

  • 4
    0

    Mathematical modelling of physical phenomena is only as precise as the data.
    When several parameters enter the picture and where the significance of each is not proportionally represented, any model is bound to crash.
    I have come across this in computational modelling of flows in industrial appliances. But there is always the ‘fudge factor’ that helps to match prediction with observation. (It is thus more of ‘postdiction’ than prediction.)
    Can any mathematical genius explain the wide variation in data for victims in Europe?
    It is a challenge to model any situation, and that should be supported. But to assume that there are ‘mathematical answers’ for everything is dangerous. A critical mind filters fact from fantasy.

    • 2
      0

      AI and data mining can help to predict and overcome challenges during the outbreaks. Some AI based solution providers are already helping governments.
      You might find this article interesting…
      “From chaos to coherence: Managing pandemics with data
      Can data analytics prevent future pandemics?”

      https://expectexceptional.economist.com/managing-pandemics-with-data.html

      • 3
        0

        Thanks ,AI based solutions will be very useful

      • 1
        0

        AI
        When a problem seems complex, one should try some commonsense.
        Even the best AI uses human inputs that can be flawed,

        • 0
          0

          It’s depending on how AI related resources are being used to resolve a complex problem in short time with minimal cost.
          It doesn’t mean AI will provide a solution to every complex problem.
          One example is how number of teams is now using AI to create a short list of molecular configurations (<100) out of millions of possibilities so that they can focus on those targeted configurations to speed up the drug development process. That is the good side of AI.
          https://spectrum.ieee.org/the-human-os/artificial-intelligence/medical-ai/companies-ai-coronavirus
          Attempts by big brothers to use AI resources to monitoring billions of smart phones or any networks in real time to create a short list of targets for further investigation to any specific query is not good news to friends of human rights that are associated with other acronym for AI.
          We like it or not, this is what we all will get into to manage big data coming from faster communication networks or molecular structures!

    • 4
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      The model was not an attempt to gain a complete understanding of the pandemic. Neither mathematics nor any other thinking will provide full answers. The model quantitatively demonstrate the factors that allow infection to reach epidemic proportions. Mathematical models have helped to eliminate epidemics. Fantasy and imaginations pave way to reality and enabling finding solutions problem. All major innovations , to begin with were dreams. Fantasy is higher level of intellectualism than blindly entering data into a computer (with no capability in mathematics) and deriving conclusions – Far more dangerous!

      • 3
        0

        Prof. Tennakone,

        Can AI and mathematical modeling predict the behavior of imbeciles to external stimuli? Corona virus infection is just one example. Their behavior to the curfews, and incitement by racists, religious fundamentalists and politicians?

        It looks like all the politicians have to go is say LTTE, and scare the imbeciles to vote for them. Then they go and prostrate to Monks.

  • 1
    0

    The model was not an attempt to gain a complete understanding of the pandemic. Neither mathematics nor any other thinking will provide full answers. The model quantitatively demonstrate the factors that allow infection to reach epidemic proportions. Mathematical models have helped to eliminate epidemics. Fantasy and imaginations pave way to reality and enabling finding solutions problem. All major innovations , to begin with were dreams. Fantasy is higher level of intellectualism than blindly entering data into a computer (with no capability in mathematics) and deriving conclusions – Far more dangerous!

  • 1
    0

    I present the COVID doom in India – made to order by Modi
    The Disaster of the Chaiwala – again !
    The penta-gonia has been completed.The disaster of the Chaiwala from Demonetisation,GST,Sir-Jee-Kal Strike,Balakot and then came , the COVID (The Fifth Pincer)
    The words of Beruni are prophetic,in this regard.
    – The Hindus believe that there is no country but theirs, no nation like theirs, no kings like theirs, no religion like theirs, no science like theirs.
    – They are haughty, foolishly vain, self-conceited, and stolid.
    – They are by nature niggardly in communicating that which they know, and they take the greatest possible care to withhold it from men of another caste among their own people,still much more, of course, from any foreigner
    Chaiwala though he was special,along with his countrymen -aka, from the Lost Tribe of Israel.Those fantas-ical land – statistical contortions by the Indians, that the numbers were doubling at the lowest rate in Hindoo-land – lapped up by the lackeys in the Hindoo Media – made them feel mighty special – just like the sons of Rama. This is what happens Hindoos study Statistics and Maths.dindooohindoo
    The Chaiwala had a simple formula.He did not do any testing at all.He waited for people to come to test,and then die.At least 30000 people die in India every day and Indians,like tardigrades, are infested with so many diseases – that no one takes a cold/cough seriously – until it is time to die – a euphemism for visiting the Indian doctor. The Chaiwala’s dream punch line would have been – that his dubious curfew ,which was unplanned, and w/o any application of mind – worked.

  • 1
    0

    The Chaiwala Tea Seller PM of India has found the solution to COVID.The man was born to doom Hindoosthan and the Hindoos and the world.dindooohindoo

    His solution is to switch off the lights for 9 minutes at 9 pm and light candles,torches and cell phone lights – the sign of impending and inevitable doom.

    https://www.business-standard.com/article/current-affairs/coronavirus-on-pm-modi-s-exhortation-india-to-light-lamps-at-9-pm-today-120040500319_1.html

    His aim is to plunge the Indians into the darkness of the shit hole – in which they are already resident.The Darkness of death in a morgue
    or a grave.

    The Indian PM’s mother was and is a dishwasher.

    I week earlier he asked Indians to beat utensils in public on the roads

    https://theprint.in/india/after-bartan-bajao-pm-modi-asks-indians-to-light-lamps-candles-as-symbols-of-hope/394181/

    Another Indian Genius is using Cow Piss on a large scale

    https://www.deccanherald.com/national/national-politics/cow-urine-for-covid-19-becomes-bone-of-contention-in-west-bengal-bjp-815839.html

    When Babar was at the doors of Hindoosthan – Hindoos were praying to Shiva and sacrifcing goats,bulls and temple prostitutes.Rest is History

    I really worry about the US and EU health systems and the Indians working in it

    1300 years ago, a man called Beruni made a prophetcic statement on the worth of the Indian race

    The words of Beruni are prophetic,in this regard.

    – The Hindus believe that there is no country but theirs, no nation like theirs, no kings like theirs, no religion like theirs, no science like theirs.
    – They are haughty, foolishly vain, self-conceited, and stolid.
    – They are by nature niggardly in communicating that which they know, and they take the greatest possible care to withhold it from men of another caste among their own people,still much more, of course, from any foreigner

    • 0
      0

      Samir sardana,

      The Indians , mean measured IQ 81, is good fodder for the Prime minister, who follows Hinduism, just like the Para-Sinhala Para-Buddhist Rajapaksas and imbeciles, mean IQ 79, who,believe in unproven reincarnation and rebirth, in Buddhism.

      The Chinese COVID-19 simply does not care.

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