Force of Infection


Consider the SIR model again.

\[ \def\arraystretch{1.5} \left\{ \begin{array}{ll} S_{t+1} = S_{t} – b S_t I_t \\
I_{t+1} = I_t +b S_t I_t-gI_t  \\
R_{t+1} = R_t+gI_t \end{array} \right. \]

The rate at which new infections are made is $b S I$. The number $bI$ now represents \[ \frac{ \text{the number of new infections}}{\text{the number of susceptibles} \times \text{time}}. \]
We denote this number by $\lambda$ and call it the force of infection (FOI). In general $\lambda$ is time-dependent. If we plug it in the SIR model we get
\[ S_{t+1} = S_{t} –  \lambda_t S_t. \]
In the continuous case we get the differential equation
\[ \frac{dS}{dt} = – \lambda S. \]
This can be solved for a general $\lambda(t)$, which gives
\[ S(t) = S(0) e^{-\int_0^t \lambda(u)du}. \]

The previous equations shows that when $\lambda$ is known, the evolution of the disease can be found quickly. However gaining the data to calculate $\lambda$ can be difficult.
In the following we will first introduce morphologic events such as death and births. Then we shall apply the FOI method to population in classes.

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