\begin{equation}\label{eq:infectivity}
\beta(\tau)  =
 \begin{cases} 
     \frac{\beta_{\text{max}}}{V_{\text{max}}} \big[\exp(k_{1}\tau) - 1\big] & \tau\leq t_{\text{inc}} - T_{p} \\
      \beta_{\text{max}} & t_{\text{inc}} - T_p \leq \tau < t_{\text{inc}}\\
      \beta_{\text{max}}\big[\exp\big(k_{2}[\tau-t_{\text{inc}}]\big) - [V_{\text{max}}]^{-1}\big] & \tau \geq t_{\text{inc}}
   \end{cases}\,,
\end{equation}
