Assume there is a set of age nodes \(0=x_0,x_1,…,x_n\) and migration probabilities \(\migprob_0, \migprob_1,\ldots,\migprob_{n-1}\) for age intervals \([x_i,x_{i+1})\). The first step in the HYBRID_INTENSITY build method is to calculate the constant instantaneous migration intensity \(\conintensity_i\) over interval \([x_i,x_{i+1})\) \begin{equation} \conintensity_i=\frac{-1}{x_{i+1}-x_i}\log(1-\migprob_i). \end{equation} Let \(\intensity_i\) be the instantaneous intensity of migration at exact age \(x_i\). For the interior nodes \(i=1,2,\ldots,n-1\) calculate \(\intensity_i\) by linearly interpolating \(\conintensity_{i-1}\) and \(\conintensity_i\) \begin{equation} \intensity_i=\frac{x_{i+1}-x_i}{x_{i+1}-x_{i-1}}\conintensity_{i-1} + \frac{x_i-x_{i-1}}{x_{i+1}-x_{i-1}}\conintensity_i \end{equation} In the interior intervals \([x_i,x_{i+1})\) where \(i=1,2,\ldots,n-2\) make \(\intensity(x)\) a quadratic function of age \begin{equation} \intensity(x) = \frac{x_{i+1}-x}{\dx_i}\intensity_i + \frac{x-x_i}{\dx_i}\intensity_{i+1} +\frac{\curvature_i}{2} (x_{i+1}-x)(x-x_i) \end{equation} where \(\dx_i=x_{i+1}-x_i\) and \begin{equation} \curvature_i=\frac{12}{\dx_i^2}\left(\frac{\intensity_i+\intensity_{i+1}}{2}-\conintensity_i\right), \quad i=1,\ldots,n-2. \end{equation} For the first interval \([x_0,x_1)\) and the last interval \([x_{n-1},x_n)\) make the force linear in age so that \begin{eqnarray} \intensity_0 &=&2\conintensity_0 - \intensity_1,\\ \intensity_n &=&2\conintensity_{n-1}-\intensity_{n-1}, \end{eqnarray} and \begin{eqnarray} \curvature_0&=&0,\\ \curvature_{n-1}&=&0. \end{eqnarray}