# Force of Infection

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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.