SIR Modelle

One of the frst infectious disease spread models was developed in 1760 by Daniel Bernoulli. The aim of the model was to determine the effectiveness of variolation of healthy people with the smallpox virus. Variolation is an outdated immunization method against smallpox. Real modern deterministic epidemic disease spread modeling started in the 20th century. Especially since the 1950s epidemiological modeling is widespread.
When analyzing spreading of infectious diseases, mostly dynamic epidemic models are used. The time component must be linked to the speci c stages of the infection life-cycle. Thereby the considered population is divided into groups which are then assigned to the respective infectious disease life-cycle phase. Hence, such models are denoted as compartmental models which were developed by Kermack and McKendrick in 1927.

Most epidemic models are based on this approach.
In the simplest form, the population is considered being constant and that some infected individuals are inserted in the population. In order to analyze the progress of the infection, the population is divided into the following three classes:

S: number of susceptibles (individuals who have the ability to get sick)
I: number of infectives (individuals who are infected and have the ability to transmit the disease)
R: number of recovered people with immunity or who are isolated or death

Using of differential equations, the time evolution of this model can be described by:

S(t), I(t) and R(t) are the numbers of individuals of the respective class at time point t. Hence, in this simpli ed model N = S(t) + I(t) + R(t), with N is the total number of individuals in the model. is the contact rate and
is the average infectious period. A typical simulation result of a SIR epidemic model is visualized.

A further model which is also often used for infectious disease spread modeling is an extension of the SIR model. It is known as MSEIR model and forms the basis for this work (MSEIRS).

Class M includes newborns with passive immunity. After passive or temporary immunity (MSEIRS) disappears these individuals move to the class of susceptibles S. If a susceptible individual has adequate contact with an infected individual, then the susceptible individual moves to the exposed class. The individual is in the latent period. After the latent period, the individual moves to class I of infectives. Now the individual can transmit the disease to susceptible individuals. After the infective period, the individual moves to the recovered class R. These individuals are permanently or temporarily immune. A further feature of this model is that, coming from any class, an individual can die, naturally or due to the disease.

Besides the presented two models, there exist several other compartmental model types but all of them are based on the classic SIR model. The compartments are arranged in an order which is most adequate for the disease. Some common models are: SI, SIS, SEI, SIRS, SEIR, SEIRS and