# SIR - A Model for Epidemiology

SIR stands for Susceptible, Infected and Recovered (or alternatively Removed) and indicates the three possible states of the members of a population afflicted by a contagious decease.

### An example model*

In order to demonstrate the possibilities of modeling the interactions between these three groups we make the following assumptions:

• The size of this population grows with immigration and declines with natural deaths.
• Infection depends on contact of infected with not yet infected and not yet immune, i.e. with susceptible members of the population.
• The decease is not really dangerous. It does not increase mortality.
• The number of infected has no influence on migration and on the healing rate of infected.
• Infected can be healed, and once they are healed they are immune against the decease.
• The risk of contagion is equal for all members of the population.
• Newly immigrated are neither infected nor immune.

*This version of a SIR-model follows the suggestions of Desch Gertrud W. / Propst Georg: Quantitative Systemwissenschaften.

A causal diagram for these assumptions could look as follows:

In this model, the variables called susceptible, infected and recovered are stocks. They are quantities with a certain size at time $$t$$. The functions $$S(t), I(t)$$ and $$R(t)$$ tell us how many individuals each class contains at time $$t$$.

Immigration $$\beta$$, fatalities $$\mu$$, infection $$\lambda$$ and healing $$\gamma$$ are flows or rates, measured as events per day.

Immigration rate is a constant $$\beta$$ which is independent of all other processes. It adds to the $$S$$-class.

The death rate defines a constant probability $$\mu$$ of fatalities per day which every individual is exposed to. The amount of daily fatalities in the $$S$$-, $$I$$- and $$R$$-class therefore is $$\mu*S$$, $$\mu*I$$ and $$\mu*R$$ respectively . The death rate diminishes these classes.

The infection rate defines a constant probability $$\lambda > 0$$ of infections per day which every susceptible is exposed to. If we assume the frequency of contacts between susceptible and infectious being proportional to their numbers, the amount of daily infections in the $$S$$-class could be calculated as $$\lambda*I*S$$. This factor diminishes the $$S$$-class and increases the $$I$$-class.

Finally, the healing rate defines a constant probability $$\gamma > 0$$ of healings per day which every infected is exposed to. The number of daily healings thus is $$\gamma*I$$. This factor diminishes the $$I$$-class and increases the $$R$$-class.

Using differential equations, the interaction of these dynamics can be captured in the following way:

$\frac{dS}{dt}=\beta – \mu * S_t – \lambda *S_t*I_t$ $\frac{dI}{dt}=– \mu*I_t+ \lambda*S_t*I_t- \gamma*I_t$ $\frac{dR}{dt}=– \mu*R_t + \gamma * I_t$

Or alternatively denoted as difference (or recursive) equations:

$S_{t+1}=S_t+(\beta – \mu*S_t – \lambda*S_t*I_t )*dt$ $I_{t+1}= I_t+(– \mu*I_t+ \lambda*S_t*I_t- \gamma*I_t)*dt$ $R_{t+1}=R_t+( – \mu*R_t + \gamma * I_t) *dt$

The following interactive model allows testing these assumptions with varying parameters and initial values of $$S = 100 000$$, $$I = 100$$ and $$R = 0$$.

$$10^{-6}$$

Implemented in Vensim, this SIR-model could look like follows:

Input values for the above VENSIM-model are:

### The steady state

As can be seen from the above plots, the dynamics of the model approach a steady state or fixed point.

As a steady state is a state in which dynamics do no longer grow nor decline, a steady state can be found by setting the differential equations to zero:

$\beta – \mu*S – \lambda*S*I = 0$ $– \mu*I+ \lambda*S*I- \gamma*I=0$ $– \mu*R + \gamma * I = 0$

which gives:

$S=\frac{\mu+\gamma}{\lambda}$ $I=(\frac{\beta}{\mu+\gamma})-\frac{\mu}{\lambda}$ $R=\beta*\frac{\gamma}{\mu*(\mu+\gamma)}-\frac{\lambda}{\gamma}$

In numbers:

$S=80200$ $I=49.377$ $R=19750.6$