SIS model

In mathematical epidemiology , a branch of theoretical biology , the SIS model is a semi-realistic approach to describing the spread of infectious diseases without building up immunity. This article uses the differential equations. For an introductory article on elementary mathematics, see Mathematical Modeling of Epidemiology .

requirements

Infected people go back to the group of healthy after recovery.

When SIS model two groups of individuals can be distinguished: at the time   referred to the number (the healthy s usceptible individuals) and the number of infected ( i nfectious individuals). Furthermore, let  the total number of individuals be. The SIS model can then be used for diseases that have the following characteristics: ${\displaystyle t}$ ${\displaystyle S(t)}$ ${\displaystyle I(t)}$ ${\displaystyle N}$

• After the disease is cured, each individual immediately returns to the group of healthy people and can be infected again.
• Infected people are immediately contagious.
• Healthy people get sick at the linear rate . ${\displaystyle c}$
• Infected people recover at the linear rate . ${\displaystyle \omega }$
• Each group has the same probability of interacting with each other. This justifies the assumption of linear relationships.
• All parameters remain in the biologically meaningful range, that is . ${\displaystyle S(t),I(t)\in [0,\infty )}$

Differential equations of the SIS model

The spread of the disease under consideration is usually formulated in the form of ordinary differential equations:

Course of the number of infected and healthy people.

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} S}{\mathrm {d} t}}&=-cIS+\omega I\\{\frac {\mathrm {d} I}{\mathrm {d} t}}&=cIS-\omega I.\end{aligned}}}$

The conservation of population size follows from the equations:

${\displaystyle {\frac {\mathrm {d} N}{\mathrm {d} t}}={\frac {\mathrm {d} I}{\mathrm {d} t}}+{\frac {\mathrm {d} S}{\mathrm {d} t}}=0\Rightarrow N=I(t)+S(t)={\text{const.}}}$

Because of this, the SIS model can be fully implemented ${\displaystyle S(t)=N-I(t)}$

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} t}}=cI(N-I)-\omega I=(cN-\omega )I-cI^{2}}$

describe. Define how the DGL can be written as . ${\displaystyle A=cN-\omega }$ ${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} t}}=(A-cI)I}$

Solutions of the differential equation

By separating the variables it follows: from which the function with the initial condition follows through a simple partial fraction decomposition and integration : ${\displaystyle {\frac {\mathrm {d} I}{(A-cI)I}}=\mathrm {d} t}$ ${\displaystyle I(t)}$ ${\displaystyle I(0)=I_{0}>0}$

${\displaystyle I(t)={\frac {AI_{0}e^{At}}{A+cI_{0}\left(e^{At}-1\right)}}.}$

The number of healthy follows from the solution for . ${\displaystyle S(t)}$ ${\displaystyle S(t)=N-I(t)}$ ${\displaystyle I(t)}$

Analysis of the DGLs using dimensionless quantities

To simplify the analysis one goes over to dimensionless quantities: ${\displaystyle u_{1}:={\frac {S}{N}},u_{2}:={\frac {I}{N}},\theta =\omega t,r:={\frac {cN}{\omega }}}$

${\displaystyle {\begin{aligned}{\frac {du_{1}}{d\theta }}&=-ru_{1}u_{2}+u_{2}\\{\frac {du_{2}}{d\theta }}&=ru_{1}u_{2}-u_{2}\end{aligned}}}$

The change can be estimated up through: . ${\displaystyle {\frac {du_{2}}{d\theta }}}$ ${\displaystyle {\frac {du_{2}}{d\theta }}=ru_{2}-u_{2}=(r-1)u_{2}}$

This simplified differential equation leads to an exponential decrease for r <1, ​​so that the disease disappears completely from the population. For r> 1, the fixed point is aimed for in the long term . The disease remains widespread. ${\displaystyle ({\frac {1}{r}},1-{\frac {1}{r}})}$

Differentiation from other models

In addition to the SIS model, there are other simple models in epidemiology that can be described with ordinary differential equations. These are in particular the following:

• The SIS model is an extension of the SI model , in which individuals cannot recover.
• An alternative extension is the SIR model , in which individuals become immune to the disease.

See also

• SI model (contagion without recovery)
• 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)

literature

• Nicholas F. Britton: Essential Mathematical Biology . Jumper
• Sebastian Möhler: Spread of Infectious Diseases. ( tu-Freiburg [PDF; accessed on March 12, 2020]).