### Hans H. Diebner's Research

## Epidemiological Standard SIR Model

The simplest epidemiological SIR system obeys the differential equations
dS/dt = -

dI/dt =

dR/dt =

*a*SIdI/dt =

*a*SI -*b*IdR/dt =

*b*I
The population under consideration consists of susceptibles (healthy individuals subject to infection),

*S*, infected individuals,*I*, and removed/recovered individuals,*R*, (here immune individuals). The graphics shows the time course of these three sub-populations: S(t) red, I(t) green, R(t) blue. Parameter values: b=0.02, a=0.02...0.2 (tuneable with slider).The dynamics starts with a relative population of

*S(t=0) = 0.99*, i.e., with 99% susceptible índividuals and with a very small infected fraction of 1%, i.e.,

*I(t=0) = 0.01*. The relative infection rate,

*a*, which can also be interpreted as successful contact rate between the individuals with respect to the infection determines the velocity at which the infection propagates. The spread of infection is proportional both to the number of susceptibles and the number of infected individuals. The susceptibles, therefore, decrease at a rate

*a*SI (negative sign), and the infected increase at the same rate (positive sign).

The infected individuals recover at a relative rate

*b*and become immune, i.e., the fraction of

*b*I is taken from the infected per time unit (negative sign) and shifted to the immune subgroup (positive sign). Towards the end of the epidemics the susceptible population fades out and the total population consists of immune individuals only. Inbetween the fraction of infected individuals increases initially and tends towards zero in the further time course, namely when all susceptibles run through the infection and eventually became immune.

The condition for the small initial infection not to increase reads:

dI/dt =

For an almost 100% susceptible polulation (S=1) it holds that *a*SI -*b*I < 0*a < b*has to be fulfilled. Please shift the slider to the leftmost position to have

*a = b = 0.02*, thus preventing an epidemic spread.