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Concepts


:The Basic Reproduction Number , ''R''0 = the number of other individuals each infected individual will infect in a population that has no Immunity to the disease;

: ''S'' = the proportion of the population (given as a decimal between 0 and 1) who are susceptible to the disease (that is, not immune).

A


L



Assumptions

  • We assume a Rectangular Age Distribution , such as that which is typically found in developed countries where there is a low infant mortality and much of the population lives to the life expectancy. In developed countries this assumption is often well justified.

  • We also assume homogeneous mixing of the population. That is, that the individuals of the population under scrutiny assort and Make Contact at random and do not mix mostly in a smaller subgroup. This assumption is rarely justified as, when dealing with a country such as the UK, most people in London, say, only make contact with other Londoners. If we deal only with London, then there will be smaller subgroups such as the Turkish community or teenagers (just to give two examples) who will mix with each other more than people outside their group. However, homogeneous mixing is a necessary assumption to make the maths simple.


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 maths 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:

:
\ {R_0} imes {S} = {1}


The basic reproduction number (''R''0) of the disease assuming everyone is susceptible, multiplied by the proportion of the population that actually is 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:

:
{S} = \frac {A} {L}.


But the mathematical definition of the endemic steady state can be rearranged to give:

:
{S} = \frac {1} {R_0}.


And therefore, since things equal to the same thing are equal to eachother:

:
\frac {1} {R_0} = \frac {A} {L}

:
{R_0} = \frac {L} {A}.


This provides us with a simple way to estimate the parameter ''R''0 using easily available data.


In a population with an exponential age distribution

For a population with an Exponential Age Distribution , it turns out that

:
{R_0} = {1} + \frac {L} {A}.


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:

:
\ {R_0} imes {S} = {1}.


''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:

: \ {R_0} imes ({1}-{q}) = {1},
: {1}-{q} = \frac {1} {R_0},
: {q} = {1} - \frac {1} {R_0}.

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.

: {q_c} = {1} - \frac {1} {R_0}


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 unforseen problems.

Suppose that a proportion of the population ''q'' (where ''q'' < ''qc'') is immunised at birth against an infection with ''R''0>1. The vaccination programme changes ''R''0 to ''R''''q'' where

:
\ {R_q} = {R_0}{({1}-{q})}.


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 ''A''q in those who have been left unvaccinated.

Recall the relation that linked R0, ''A'' and ''L''. Assuming that life expectancy has not changed, now:

: \ {R_q} = \frac {L} {A_q},
: \ {A_q} = \frac {L} {R_q},