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