Mathematical Modelling In Epidemiology Concepts Assumptions

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.

3 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 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:

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

This provides us with a simple way to estimate the parameter R₀ using easily available data.

3.1 In a population with an exponential age distribution

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.

4 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 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.

4.1 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 resistanceThe MMR vaccine is a combined vaccine for immunization against measles, mumps and rubella. It is generally administered to children around the age of 1 year, with a booster dose before starting school (i. It is widely used around the world; since its intr) 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 unforseen 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

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:

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 infectionIn epidemiology, force of infection (denoted λ) is the rate at which susceptible individuals become infected by an infectious disease. Because it takes account of susceptibility it can be used to compare the rate of transmission between different g 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 effectsPerverse effects of vaccination programmes manifest themselves when insufficient numbers of susceptibles are vaccinated to reach the critical threshold value (denoted q at which enough people are immune to the disease that its spread through the populatio.

4.2 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.

5 Other articles that treat infectious diseases mathematically

Compartmental models in epidemiologyA population comprises a large number of individuals, all of whom are different in various fields. In order to model the progress of an epidemic in such a population this diversity must be reduced to a few key characteristics which are relevant to the inf
Endemic
Epidemic
Force of infectionIn epidemiology, force of infection (denoted λ) is the rate at which susceptible individuals become infected by an infectious disease. Because it takes account of susceptibility it can be used to compare the rate of transmission between different g
Perverse effects of vaccination
Risk factor
Sexual network
Susceptible
Transmission risks and rates

Epidemiology

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1 Concepts

2 Assumptions