Mathematical modeling of epidemiology

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Most infectious diseases can be modeled mathematically to study or predict their epidemiological behavior . Using a few basic assumptions, parameters for various infectious diseases can be found with which, for example, calculations can be made about the effects of vaccination programs .

Concepts

The base reproduction number is the number of secondary cases that an infected person creates in a given population. It is assumed that there is no immunity in the population .

(from English susceptibles ) is the proportion of the population that is not immune to the disease. This is a decimal number between 0 and 1.

indicates the average age at which the disease affects a population.

(from English Life expectancy ) describes the average life expectancy in the population.

Assumptions

  • A rectangular age pyramid is assumed, as is typically found in developed countries with low child mortality rates and frequent reaching life expectancy.
  • A homogeneous mixture of the population is assumed. This means that the examined individuals make contacts by chance and are not limited predominantly to a smaller group. This requirement is seldom justified, but it is necessary to simplify the math.

The endemic status

An infectious disease is endemic when it exists continuously within a population without external influences. This means that, on average, every sick person infects exactly one other person . If this value were lower, the disease would die out; if it were larger, it would develop into an epidemic due to exponential growth . Mathematically speaking, this means:

For a disease with a high basic reproductive number (assuming no immunity) to remain endemic, the number of those actually susceptible must therefore inevitably be low.

With the assumption made above about the age pyramid, it can be assumed that every individual in the population will exactly reach life expectancy and then die. If the average age is the infection , on average younger individuals are susceptible, while older individuals have already been immunized (or are still infectious) from previous infection. Consequently, the proportion of those susceptible to the disease is:

In the endemic case, however, the following also applies:

This applies

,

which enables an estimation of the basic reproduction number through easily ascertainable data.

For a population with an exponential age pyramid it can be seen that

.

The mathematics used here is more complex and therefore outside the scope of this consideration.

The math of vaccinations

If the immunized proportion of the population (or “vaccination coverage” ) is above the level necessary for herd immunity , a disease cannot remain in an endemic state within that population. An example of global success in this way is the eradication of smallpox , the last case of which was documented in Somalia in 1977 . The WHO is currently running a similar vaccination strategy to eradicate polio .

The degree of collective immunity is referred to as . As for an endemic condition

must be fulfilled , because is the immune fraction of the population and (since in this simplified model each individual is either susceptible or immune). Then:

This is the threshold for collective immunity, which must be exceeded in order for the disease to die out. The value calculated here is the critical immunization threshold . It is the minimal proportion of the population that needs to be vaccinated at birth (or shortly afterwards) for the disease to die out in the given population.

Vaccination program below the critical immunization threshold

If the serums used are not sufficiently effective or cannot be applied on a sufficiently broad front, for example due to social resistance (see for example MMR vaccine ), the vaccination program is not able to outperform. However, such a program can upset the infection balance and cause unforeseen problems.

Assume that the portion of the population immunized at birth is (where ) and the disease has the base reproduction number . Then the vaccination program changed to , wherein

.

This change is simply due to the reduced number of potentially vulnerable people. is nothing else than without those individuals who would be infected under normal circumstances but will not become infected due to vaccination.

Because of this lower base reproduction number, the average age among those not vaccinated also changes to a value .

According to the above relation, which , and connected, applies (assuming constant life expectancy):

However , it follows that:

Thus, the vaccination program increases the mean age of infection. Unvaccinated individuals are now subject to a reduced infection rate due to the presence of the vaccinated group.

However, this effect is disadvantageous in diseases, the course of which becomes more severe with increasing age. If the probability of a fatal outcome is high, an unsuccessful vaccination program can, in extreme cases, claim more victims among the unvaccinated than would have happened without a vaccination program.

Immunization programs above the critical immunization threshold

If a vaccination program exceeds the critical immunization threshold of a population for a significant period, the disease within that population is stopped. If this elimination is done worldwide, it will ultimately lead to the eradication of the disease.

See also

  • SI model (contagion without recovery)
  • SIS model (spread of infectious diseases without building up immunity)
  • SIR model (spread of infectious diseases with immunity formation)
  • SEIR model (spread of contagious diseases with immunity development, in which infected people are not immediately infectious)
  • Base reproduction number
  • Dynamic system (mathematical generic term)

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