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Mathematical modelling in epidemiology
It is possible to mathematically model the progress of most infectious diseases to discover the likely outcome of an epidemic or to help manage them by vaccination. This article uses some basic assumptions and some simple mathematics to find parameters for various infectious diseases and to use those parameters to make useful calculations about the effects of a mass vaccination programme.
Whenever we are modelling anything mathematically, whether in epidemiology or otherwise, we would be wise to remember that a mathematical model is only as good as the assumptions on which it is based. If a model makes predictions which are out of line with observed results and the mathematics is correct, we must go back and change our initial assumptions in order to make the model useful.
The endemic steady state
An infectious disease is said to be endemic when it can be sustained in a population without the need for external inputs. This means that, on average, each infected person is infecting exactly one other person (any more and the number of people infected will grow exponentially and there will be an epidemic, any less and the disease will die out). In mathematical terms, that is:
The basic reproduction number (R0) of the disease, assuming everyone is susceptible, multiplied by the proportion of the population that is actually susceptible (S) must be one (since those who are not susceptible do not feature in our calculations as they cannot contract the disease). Notice that this relation means that for a disease to be in the endemic steady state, the higher the basic reproduction number, the lower the proportion of the population susceptible must be, and vice versa; a mathematical basis for a result that might have been intuitively obvious.
The first assumption (above) lets us say that everyone in the population lives to age L and then dies. If the average age of infection is A, then on average individuals younger than A are susceptible and those older than A are immune (or infectious). Thus the proportion of the population that is susceptible is given by:
But the mathematical definition of the endemic steady state can be rearranged to give:
And therefore, since things equal to the same thing are equal to each other:
This provides us with a simple way to estimate the parameter R0 using easily available data.
In a population with an exponential age distribution
For a population with an exponential age distribution, it turns out that
The mathematics required to calculate this is a little more complicated than that above, and thus beyond the scope of this article. However, this does allow you to work out the basic reproduction number of a disease given A and L in either type of population distribution.
The mathematics of mass vaccination
If the proportion of the population that is immune exceeds the herd immunity level for the disease, then the disease can no longer persist in the population. Thus, if this level can be exceeded by vaccination, the disease can be eliminated. An example of this being successfully achieved worldwide is the global eradication of smallpox, with the last wild case in 1977. Currently, the WHO is carrying out a similar campaign of vaccination in an attempt to eradicate polio.
The herd immunity level will be denoted q. Recall that, for a stable state:
S will be (1 − q), since q is the proportion of the population that is immune and q + S must equal one (since in this simplified model, everyone is either susceptible or immune). Then:
Remember that this is the threshold level. If the proportion of immune individuals exceeds this level due to a mass vaccination programme, the disease will die out.
We have just calculated the critical immunisation threshold (denoted qc). It is the minimum proportion of the population that must be immunised at birth (or close to birth) in order for the infection to die out in the population.
When a mass vaccination programme cannot exceed the herd immunity
If the vaccine used is insufficiently effective or the required coverage cannot be reached (for example due to popular resistance) the programme may not be able to exceed qc. Such a programme can, however, disturb the balance of the infection without eliminating it, often causing unforeseen problems.
Suppose that a proportion of the population q (where q < qc) is immunised at birth against an infection with R0>1. The vaccination programme changes R0 to Rq where
This change occurs simply because there are now fewer susceptibles in the population who can be infected. Rq is simply R0 minus those that would normally be infected but that cannot be now since they are immune.
As a consequence of this lower basic reproduction number, the average age of infection A will also change to some new value Aq in those who have been left unvaccinated.
Recall the relation that linked R0, A and L. Assuming that life expectancy has not changed, now:
But R0 = L/A so:
Thus the vaccination programme will produce an increase in the average age of infection, another mathematical justification for a result that might have been intuitively obvious. Unvaccinated individuals now experience a reduced force of infection due to the presence of the vaccinated group.
However, it is important to consider this effect when vaccinating against diseases which increase in severity with age. A vaccination programme against such a disease that does not exceed qc may cause more deaths and complications than there were before the programme was brought into force as individuals will be catching the disease later in life. These unforeseen outcomes of a vaccination programme are called perverse effects.
When a mass vaccination programme exceeds the herd immunity
If a vaccination programme causes the proportion of immune individuals in a population to exceed the critical threshold for a significant length of time, transmission of the infectious disease in that population will gradually come to a halt. This is known as elimination of the infection and is different from eradication.
Other articles that treat infectious diseases mathematically
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Mathematical_modelling_in_epidemiology". A list of authors is available in Wikipedia.|