Classic Science: A Contribution to the Mathematical Theory of Epidemics

Describing a model of infectious disease, as done in the field of mathematical epidemiology, helps scientists and doctors maximize the effectiveness while lowering cost of treating a population of individuals.  To do this, however, one must first be able to describe how an infection moves through a population or, more accurately, how the population changes over time due to infectious agents.

This has been done using dynamic differential equation models for a long time now (and they can get quite complex).  The most common of these models is called the SIR model (which stands for Susceptible -> Infected -> Recovered) representing the dynamic conditions of individuals within a general population as they “flow through” these different disease states at various rates.

The first ‘discovers’ of the simple SIR model was W. O. Kermack and A. G. McKendrick in 1927, in a paper published in the Proceedings of the Royal society of London. Their model is, ex post, appropriately titled The Kermack-McKendrick model.

The paper itself goes through some complex derivations but the results are suprisingly simple and intuitive. In a closed population, N, where there is no population growth, they assume instantaneous disease incubation, homogeneous population with no age, spatial, or social structure and fixed transmission and recovery rates.  The result is this:

The susceptible population, which is the total population at t=0, decreases as an infection spreads and individuals become infected.  This occurs at the constant rate of β as a function of the number of infected individuals (the source of new infections) times the number of susceptible individuals (where the infectious agents are spreading to).  This means that to sustain an epidemic a disease requires a healthy reservoir of susceptibles or risk “burning” itself out.  This equation looks like this:

The dynamics of the infected population depends on the growth of new infections minus the recovery rate (or even death rate) or infected individuals, in much the same way that was described for the change in susceptible population:

Finally, the infected population recovers at a constant rate, γ, at some fraction of the population.

They get a pretty simple model, which they solve mathematically, but, with all these assumptions, can they get any useful data?

Yes!

If you can’t read the chart’s description, this is an epidemic of the plauge on the “Island of Bombay” which fits the The Kermack-McKendrick model for epidemics quite precisely.  When the infection gets introduced, although there are plenty of susceptible individuals around, the infection grows slowly (because the number of infected individuals starts out small).  However, the exponential trend takes off as the disease creates many new infections (which in turns create more new infections) .  But then the disease level crashes as it runs out of new individuals, as individuals recover or, in this case, they die.  Whereupon the epidemic is over.

Can we describe or predict the time evolution of an epidemic in a more general way?

Yes (according to the Kermack-McKendrick model)!

The so-called time evolution of the model is described by the so-called epidemiological threshold:

Ro can be thought of as the ‘potential’ for new infections, or the number of secondary infections caused by a primary infection.  When Ro < 1, the number of secondary infections is falling, and so the infection is dying out.  When Ro > 1 each infected individual will infect more than 1 other person, thereby causing the infection to spread positively within a population.

This quantity Ro (the details of which differ depending on the specific model being used) has been described as one of the most important variables in epidemeology, with consideration for disease transmission potential and therefore disease evolution (in more complex considerations).

Despite the fact that the model’s assumptions may not accurately reflect most real human diseases, Kermack and McKendrick come up with genuinely interesting and relevant conclusions.   They find that a population level requires some threshhold value in order for an epidemic to occur, a population size under this threshhold (which is disease specific) will not experience and epidemic.  An infection in a population that exceeds this threshhold will reduce the populationas far below the threshhold as it was above it.  Small increases in the infectivity rate can lead to dramatic changes in the dynamics of an epidemic, making them stronger.  Epidemics generally end well before the entire population of susceptibles has been exhausted.  Similar results are seen for diseases transmitted through an intermediate host (like vector bourne diseases).

Although this differential equation model lacks the accuracy of more complex models which relax assumptions and new types of stochastic models, the field of epidemeology has a lot to thank Kermack and McKendrick, for basically creating a new branch, bridging biology, mathematics and medicine.

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