What a simple mathematical model can tell us about Covid-19

There are certain characteristics of disease outbreaks which are common to many epidemics, and mathematicians and epidemiologists have captured some of these aspects in simple analytical models.  In this post I will introduce one such model and discuss what can be learned about the Covid-19 from it. 

One of the oldest and simplest mathematical models for an epidemic is the so-called S-I-R model introduced by Kermack and McKendrick in 1927.  And I think it provides a good first approximation to the dynamics of the current Coronavirus epidemic, and can provide some important insights. 

In this compartmental model there are three ‘compartments’ - ‘Susceptibles’, S;  ‘Infectives’, I; and ‘Removed’, R.  This last class includes those who have had the infection and are no longer susceptible either through acquired immunity or death.  The dynamics of the disease are described by the three equation of motion:  

dS/dt = −β S I/N

dI /dt = β S I/N - νI 

dR/dt = ν I,

where N is the total population size (= S+I+R) and β and ν are parameters. 

The left-hand sides are the time rates of change, e.g number of new Susceptibles  (or Infectives or Removed) per day.    The rate of infections (rate at which individuals move from Susceptible to Infective class) is β S I/N.  Note that it is assumed to depend on both the current number of Infectives, I,  in the population and the current number of Susceptibles, S.   For a fixed value of S, the number of infectives is growing exponentially at the rate (β S /N - ν) because then dI/dt = (β S /N - ν) I.    So the epidemic is growing fastest when S is largest i.e. when it is at its initial value S0 =  N when the disease is first introduced.  

The term ν I  represents the rate at which Infectives are removed either by recovery or death. 

Although there are two parameters in this system,  β and ν, it is easily shown by re-scaling that its behaviour depends only on the ratio 

R0 =  β/ν.  

This quantity is very important and is known as the basic reproduction number.  It represents the maximum expected number of new Infectives for any single infected individual.   As one might expect if R0 is less than one, no epidemic arises.  But if R0 is greater than one an epidemic breaks out.  

If one writes the right hand side of the second equation as (β S /N - ν) I, one can see that the number of infectives is growing (dI/dt > 0) if βS/N−ν > 0; and is decreasing if βS/N−ν < 0.  In other words the epidemic is growing all the time that S >N ν/β=N/R0; and is declining if S < N/R0.  The peak of the outbreak thus occurs when the number of susceptibles S is reduced to N/R0.  

This also tells us how many people need to become immune, in order to achieve so-called herd immunity.  Herd immunity occurs in a population when the number of susceptibles is not large enough to maintain an epidemic.  It can often be brought about by vaccination, but unfortunately not yet for Covid. 

From the calculations above an epidemic will not occur if  S < N/R0, or if the number with immunity is greater than N- N/R0 = N(R0-1)/R0 . 

Thus the percentage of the population required for herd immunity is 100(R0-1)/R0 %.  Estimates of R0 for Covid-19 vary, but a study from Wuhan published in Lancet put it at about 2.3.  A study of cruise ship infections had a similar estimate. This suggests that it would require immunity in about 57% of the population.  However there was uncertainty in the estimates.  The Lancet study gave a 95% confidence interval of 1.15 to 4.77, which suggests that to obtain herd immunity somewhere between 13% and 79% of the population would need to become immune. So there is still a great deal of uncertainty with regard to R0 and herd immunity. 

Also by writing the right hand side of the equation for the dynamics of I, one can see why an epidemic occurs only if the basic reproduction number R0 is greater than one.  For if initially S0 = N then, initially dI/dt =  (β S0/N - ν) I =  (β - ν) I  = ν(R0-1) I, which will be positive if R0 > 1 and negative otherwise.   

Explicit solution of the differential equations is is not easy, in terms of simple functions.  But qualitative behaviour and numerical solutions are easy to obtain.  The graph below shows typical trajectories (in the case R0 > 1) of the numbers of Infectives (green)  Susceptibles (blue) and Removed (red) are as shown in the graph. 

What is important is that the height of the peak, and the number uninfected when the epidemic is over both depend on R0.  The larger R0, the higher the peak and the smaller the number escaping the disease.  

So from the public health point of view reducing R0 is of major importance. This can be done by making the numerator, β, smaller or the denominator, ν, larger.  The first can be achieved by reducing contact between Infectives and Susceptibles (social distancing, self-quarantine) and the second by isolating Infectives.

Another conclusion from this model is that the epidemic ends through there being insufficient Infectives to keep it going, so that at the end there is a positive number of Susceptibles who never get the disease. 

Kermack and McKendrick successfully used the model to describe the behaviour of a plague outbreak in Bombay in 1906, and other historical epidemics for which data existed.  A graph from their paper is shown.

There are more sophisticated models of Covid-19 which include such things as age structure in the population, stochastic effects and more.  A video lecture describing some of these models is availability here 
https://www.youtube.com/watch?v=dLVLEZMzIOk

Who will be held responsible if Julian Assange dies?

Is the UK Government hoping that Julian Assange will die of Covid-19 and thus solve its extradition problem?  He is reported to be in frail health, and his request for bail to be released from confinement, because of the Covid risk, has been turned down. 

117 medical doctors, including several world prominent experts in the field, published a letter in the Lancet warning that Assange’s treatment amounts to torture and that he could die in jail.   Should Assange die in a UK prison, as the UN Special Rapporteur on Torture has warned, he will effectively have been tortured to death. Much of that torture will have taken place in a prison medical ward, on doctors’ watch. The medical profession cannot afford to stand silently by, on the wrong side of torture and the wrong side of history, while such a travesty unfolds.

From everything I have read about his extradition hearing, being held in Belmarsh Prison, it is a travesty of British justice and more like something one might expect in Russia (either in Soviet days or now).  

For example:  The International Bar Association’s Human Rights Institute (IBAHRI) condemns the reported mistreatment of Julian Assange during his United States extradition trial in February 2020, and urges the government of the United Kingdom to take action to protect him. According to his lawyers, Mr Assange was handcuffed 11 times; stripped naked twice and searched; his case files confiscated after the first day of the hearing; and had his request to sit with his lawyers during the trial, rather than in a dock surrounded by bulletproof glass, denied.

It is worth remembering that Julian Assange is a remand prisoner who has served his unprecedentedly long sentence for bail-jumping. His status is supposedly at present that of an innocent man facing charges.

After the pandemic recedes

As we struggle with the crisis of the spread of Coronavirus and the measures necessary to contain both it and the damage it has induced in the economy and elsewhere, it maybe worth thinking about how the world will look once it has receded.  Of one thing I am quite sure - that a very different world will emerge at the end of it.

Given that the Coronavirus originated in China and caused great damage there, it may sound sound counter-intuitive, but I believe that China will come out of the pandemic with its power and position in the world enhanced.

When the epidemic arose in Wuhan and quickly spread through Hubei province and threatened other parts of the country things looked very bad for China.  Lunar New Year celebrations were put on hold and much of the country was put on lockdown.  Factories and businesses closed and supply chains were interrupted, resulting no doubt in a big drop in first quarter output.  

Things didn’t look too good for the Communist Party leadership either, when it was revealed how Party officials in Wuhan had tried to cover up the outbreak and had punished the doctor who had first spoken out about it.  There were even reports of public unhappiness with the leadership Chairman Xi.   Coupled with the concurrent unrest in Hong Kong, the virus epidemic looked as if it might cause the ground to shift a bit under the feet of the autocratic leadership.

But now as we near the end of March, things look quite different.  China seems to have gotten on top of the virus with no new cases of community spread reported in Wuhan for several days.  Furthermore it is sending medical equipment and advice to various other hard-hit countries including Iran and Italy.  And it is Western countries that are struggling to control the virus and whose economies seem to be going into a tailspin.

Apart from the propaganda success, there is another way in which China is gaining.  The US and other Western countries have announced massive spending programs, at first in an attempt to prevent the collapse of the stock market and ensuing financial mayhem, and subsequently to help protect workers who cannot earn a paycheck through businesses going under or through enforced self isolation.  The deficits will be enormous - certainly involving many trillions of dollars.  And the governments incurring these massive deficits are, for the most part, already up to their eyes in debt, much of it incurred following the 2008 financial crisis, when banks and financial institutions were bailed out.  And the worst borrower by far has been the US Government, which has continued to fight wars around the world and dramatically increase its military budget, while at the same time, cutting taxes at home.  

I am not entirely confident that I understand how government deficit financing operates, but I think it is mostly through the selling of Treasury bills.  And who has the reserves to buy such bonds (i.e. to lend such gargantuan sums of money)?  I think the answer is China.  So when it all shakes down, the US (and other Western nations) will be massively in debt to China.

Perhaps the US Treasury can simply create money with a few keystrokes on computer? But history teaches that artificially creating money out of thin air, as was tried by the Ancien Regime before the French Revolution and by Weimar Germany, can lead to hyper-inflation and social and political dislocation.

So however these massive debts are financed, it looks to me as if the long-term winner will be China. The way China is bringing the epidemic under control with fewer deaths even than Italy, and a medical disaster emerging in the US, will all combine to make the authoritarian Chinese system seem attractive to many non-Western countries.  Coupled with a debt-ridden capitalist West, with its liberal democracy tainted by extreme inequality, and an electoral system that can result in the election of a narcissistic reality TV huckster, who didn’t even obtain a plurality of votes, the moral supremacy of the West looks definitely shaky.  

It could turn out that the Coronavirus will be seen as the catalyst which led to China  replacing the USA as the world’s number one power, just as World War II saw Britain being replaced by the US.  I can’t say I want it to happen, but in many ways the US, with the rest of the West tagging along, will have brought this unattractive outcome on itself.