It is possible to model mathematically 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.

Concepts

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R₀, the basic reproduction number

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

The average age at which the disease is contracted in a given population;

L

The average life expectancy in a given population.

Assumptions

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• 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 mathematics 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 mathematics is correct, we must go back and change our initial assumptions in order to make the model useful.

The endemic steady state

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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} \times {S} = {1}

The basic reproduction number (R₀) 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 each other:

$$

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

$$

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

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

$$

{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

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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} \times {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:

$\backslash \; \{R\_0\}\; \backslash times\; (\{1\}-\{q\})\; =\; \{1\},$

$\{1\}-\{q\}\; =\; \backslash frac\; \{1\}\; \{R\_0\},$

$\{q\}\; =\; \{1\}\; -\; \backslash frac\; \{1\}\; \{R\_0\}.$

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 q_c). 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.

$\{q\_c\}\; =\; \{1\}\; -\; \backslash frac\; \{1\}\; \{R\_0\}$

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 q_c. 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 < q_c) is immunised at birth against an infection with R₀>1. The vaccination programme changes R₀ 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. R_q 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:

$\backslash \; \{R\_q\}\; =\; \backslash frac\; \{L\}\; \{A\_q\},$

$\backslash \; \{A\_q\}\; =\; \backslash frac\; \{L\}\; \{R\_q\},$

$\backslash \; \{A\_q\}\; =\; \backslash frac\; \{L\}\; \{\{R\_0\}(\{1\}-\{q\})\}.$

But R₀ = L/A so:

$\backslash \; \{A\_q\}\; =\; \backslash frac\; \{L\}\; \{(\{L\}/\{A\})(\{1\}-\{q\})\},$

$\backslash \; \{A\_q\}\; =\; \backslash frac\; \{AL\}\; \{\{L\}(\{1\}-\{q\})\},$

$\backslash \; \{A\_q\}\; =\; \backslash frac\; \{A\}\; .$

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 q_c 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.

Elimination

Interruption of endemic transmission of an infectious disease, which occurs if each infected individual infects less than one other and is achieved by maintaining vaccination coverage to keep the proportion of immune individuals above the critical immunisation threshold.

Eradication

Reduction of infective organisms in the wild worldwide to zero. So far, this has only been achieved for smallpox. To get to eradication, elimination in all world regions must first be progressed through.

Other articles that treat infectious diseases mathematically

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• Compartmental models in epidemiology
• Endemic
• Epidemic
• Force of infection
• Perverse effects of vaccination
• Risk factor
• Sexual network
• Susceptible
• Transmission risks and rates

Literature

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• "Infectious Diseases of Humans" Roy M. Anderson and Robert M. May (ISBN:0-19-854040-X)
• "Smallpox and its eradication" Jenner

External links

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• Scientific American (March 2005) If Smallpox Strikes Portland ... (an article about "Episims")

Epidemiology | Mathematical biology

Mathematische Modellierung der Epidemiologie