Mathematical modelling of infectious disease

Most infectious diseases can be mathematically modeled to investigate their epidemiological behavior or to predict. This article introduces some possible ways of thinking in mathematical epidemiology and uses some basic assumptions and mathematics to find parameters for various infectious diseases. These parameters can be, for example, calculations of the impact of vaccination programs set up. The mathematics used here is deliberately kept simple. At the end of the article there are references to some models, some of which have a slightly more sophisticated mathematical elaboration of the subject.

Concepts

The base rate of reproduction is the number of secondary events that produces an infected in a given population. Here, it is assumed that there is no immunity in the population.

(of 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 a population is affected by the disease.

(of English. Life expectancy ) denotes the average life expectancy in the population.

Assumptions

• It is assumed that a rectangular age pyramid, as is typically found in developed countries with low child mortality and frequent reaching the life expectancy.
• It is a homogeneous mixture of the population provided. This means that the studied individuals socialize random and not be confined mainly to a smaller group. This condition is rarely justified, however, it is necessary to simplify the mathematics.

The endemic status

An infectious disease is endemic, if it exists continuously without external influences within a population. This means that on average each sick person infects another exactly. If this value is low, the disease would die, he would be bigger it would develop into an epidemic due to exponential growth. Mathematically, this means:

For a disease with high basic reproduction rate remains endemic (assuming no existing immunity), so the actual number of susceptibles must necessarily be low.

With the assumption made ​​above about the age pyramid can be assumed that each individual of the population reached exactly the life expectancy and then dies. If the average age of infection is, younger individuals are in the middle susceptible, while older individuals have been immunized by previous infection ( or are still infectious). Consequently, the proportion of susceptibles for the disease is:

In endemic case, however, is also true:

This is true

Which allows an estimation of the basic reproduction rate by easily identifiable data.

For a population with exponential age pyramid shows that

The mathematics used in this case is more complex and thus outside the scope of this analysis.

The mathematics of vaccinations

When the immunized proportion of the population (or " immunization " ) is above the level required for herd immunity, a disease can not remain in endemic condition within this population. An example of a global success in this way is the eradication of smallpox, the last case was documented in Somalia in 1977. Currently, the WHO operates a similar vaccination strategy to eradicate polio.

The degree of joint is referred to as immunity. As for an endemic state

Must be satisfied, because the immune proportion of the population and ( as in this simplified model, each individual is either susceptible or immune). Then:

This is the threshold of collective immunity, it must be exceeded in order for the disease dies out. The calculated value here is the critical immunization threshold. It is the minimum proportion of the population ( or shortly thereafter ) must be immunized by vaccination to birth, so that the disease dies out in the given population.

Vaccination program below the critical immunization threshold

Are antisera is not sufficiently effective or can not be applied to sufficiently broad front, for example due to social resistance (see, for example, MMR vaccine), so the vaccination program is not to excel in the situation. However, such a program can disrupt the balance of infection and cause unforeseen problems.

Suppose the immunized at birth proportion of the population amounts to (where ) and the disease have the basic reproduction rate. Then the vaccination program changed to, where

This change takes place simply because of the decline in the number of potentially susceptibles. is nothing else than without the individuals which would become infected under normal circumstances, but due to the vaccination will not.

Due to this lower basic reproduction rate is also the average age among the non- vaccinated changed to a value.

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

However true, therefore:

Thus, the vaccination program increased the average age of infection. Unvaccinated individuals are now by the presence of the vaccinated group a reduced infection rate.

This effect is, however, disadvantageous in diseases whose course is more severe with increasing age. With a high probability of fatal evolution can not apt vaccination program in extreme cases more victims among the unvaccinated demand, as it would have been without the vaccination program.

Vaccination programs above the critical immunization threshold

The excess of a vaccination program, the critical immunization threshold of a population for a significant period, the disease is stopped within this population. If this elimination carried out worldwide, it leads ultimately to eradicate the disease.