## 2 Modeling Procedure
### Hypothetical Model
\[\tau_{4}\frac{d\eta_{4}}{dt}=\beta_{42}\eta_{2}(t-\theta_{6})+\beta_{43}\eta_{3}(t- \theta_{8})+\beta_{46}\eta_{6}(t-\theta_{17})+\beta_{45}\eta_{5}(t-\theta_{19})- \eta_{4}(t)+\zeta_{4}(t) \tag{3}\]  
\[\tau_{5}\frac{d\eta_{5}}{dt}=\gamma_{57}\tilde{\xi}_{7}(t-\theta_{15})+\gamma_{ 510}\tilde{\xi}_{10}(t-\theta_{20})+\beta_{54}\eta_{4}(t-\theta_{12})-\eta_{5 }(t)+\zeta_{5}(t) \tag{4}\]  
\[\tau_{6}\frac{d\eta_{6}}{dt}=\gamma_{64}\tilde{\xi}_{4}(t-\theta_{11})+\gamma_ {68}\tilde{\xi}_{8}(t-\theta_{18})-\eta_{6}(t)+\zeta_{6}(t). \tag{5}\]  
The extended model includes two additional components. The first one is incorporating an optimal range for step goals. This was accomplished by adding a goal attainment signal that affects self-efficacy based on the effectiveness of the required behavior. Secondly, a self-regulation loop was established by using internal cues and is represented by the \(C_{sr}\) block. The idea behind this block is derived from internal model control (IMC) [23]) and is mathematically defined in the equation outlined in [11]. The self-regulation process is represented through a controller that adjusts action cues based on the discrepancies between the set goal (\(u_{8}\)) and the measured outcome (\(y_{4}\)). The controller needs to allow for a partial set point tracking to permit other intervention components (such as points) to influence changes in the output. According to the IMC formulation, the self-regulator is a classical feedback controller. See Equation (6). \[c_{sr}=\frac{K_{sr}}{\gamma_{64}\beta_{46}}\frac{\tau_{4}s+1}{\lambda s-K_{sr} +1}. \tag{6}\]  
The performance of the self-regulator is determined by two parameters: \(\lambda\), which indicates the closed-loop speed of response, and \(K_{sr}\), which ranges between 0 and 1 and represents the level of integral action, with 1 indicating perfect integral action. Equations (1)-(5), without disturbances or delays (\(\zeta_{i}=0,\theta_{i}=0,\forall_{i}\)), and self-regulator \(C_{sr}\), with state \(\eta_{7}\), were considered. Assuming that the dominant effect on the output is obtained from the time constant of the behavior inventory (\(\eta_{4}\)), the state matrices are shown in Equations (7)-(9). \[A=\begin{bmatrix}\frac{-1}{\tau_{2}}&0&0&\frac{\beta_{25}}{\tau_{2}}&0&0\\ 0  
\[N_{sr}=\frac{1-K_{sr}}{\lambda}. \tag{11}\]  
A stimulus in the form of a reward mechanism was employed to encourage the desired behavior in the intervention, whereby users receive a specified number of points (\(\xi_{9}\)) upon reaching their daily step goals (\(u_{8}\)). These points can later be exchanged for tangible rewards. This is represented by an if/then block that is considered external to the simplified SCT model due to its non-linear nature, as depicted in \[\tau_{2}\frac{d\eta_{2}}{dt}=\gamma_{29}\xi_{9}(t-\theta_{21})+\beta_{25}\eta _{5}(t-\theta_{14})-\eta_{2}(t)+\zeta_{2}(t) \tag{1}\]0  
The obtained model can be described by means of the parameters shown in Table 1, which were partially obtained through Secondary Data Modeling in Section 2.1, using a semi-physical estimation procedure (combining physical knowledge and experimental data to create a mathematical model).