## 4 The Stimulus Response Model (SRM)
\begin{table}
\begin{tabular}{c c c c} \hline \hline
\begin{tabular}{c} su\(\cdot\)mL (Phase B) \\ su\(\cdot\)mg (Phase C) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Output variables of the SRM: symbols and dimensions. Let us deduce the model equations from this diagram. Take into account for this deduction the variable dimensions and units presented in Tables 4 and 5. First, to compute the drug in blood, \(s(t)\), and the drug in digestive organs, \(m(t)\), the following balance equations provided by the HD must be considered in finite differences form (\(\Delta t\) is a finite time increment):  
\[m(t)=m(t-\Delta t)+\Delta t\cdot(-fm(t)) \tag{1}\]  
\[s(t)=s(t-\Delta t)+\Delta t\cdot(fm(t)-fs(t)) \tag{2}\]  
However, Equations (1) and (2) can be written in the infinitesimal limit form (\(\Delta t\to 0\)):  
\[\frac{dm(t)}{dt}=-fm(t) \tag{3}\]  
\[\frac{ds(t)}{dt}=fm(t)-fs(t) \tag{4}\]  
From now onwards, the balance equations for the level variables of the HD will be written in the infinitesimal limit form. Note from the HD the causal influences on the flows \(fm(t)\) and \(fs(t)\). The hypothesis about how to compute them is provided by the drug chemical kinetics [21; 22]:  
\[fm(t)=\alpha\cdot m(t)\cdot f(s)=\beta\cdot s(t) \tag{5}\]  
Then, from (5) in (3) and (4), their initial values are added in the initial time \(t=t_{0}\):  
\[\frac{dm(t)}{dt}=-\alpha\cdot m(t) \tag{6}\]  
\[\frac{ds(t)}{dt}=\alpha\cdot m(t)-\beta\cdot s(t) \tag{7}\]  
Equations (6) and (7) provide the drug dynamics in organism, being \(\alpha\) and \(\beta\) two parameters to be calibrated for both participants and phases. Both parameters depend on the individual personality and on the kind of drug. For alcohol in Phase B, they are calibrated in the intervals \(\alpha\in[0.00118,\,0.02050]\) and \(\beta\in[0.00462,\,0.00533]\) following the literature about alcohol kinetics [22], and for caffeine in Phase C the corresponding intervals are \(\alpha\in[0.00137,\,0.091138]\) and \(\beta\in[0.00144,\,0.00304]\) following the literature about caffeine kinetics [21]. Considering the balance equation in the infinitesimal limit form for the happiness level variable, \(y(t)\) in the HD:  
\[\frac{dy(t)}{dt}=fr(t)+fe(t)-fi(t) \tag{8}\]  
Taking the literature about the subject as a base, the hypotheses about the mathematical patterns of the three flows considered in (8) are explained in the following paragraphs. About the homeostatic control, _fr(t)_, it is a control around the tonic or basal happiness \(b\). It is assumed that the happiness state, \(y(t)\), tends to return to the tonic happiness when it is modified by a stimulus:  
\[fr(t)=a(b-y(t)) \tag{9}\]  
Parameter \(b\) in (9) represents the value to which the \(y(t)\) variable would tend asymptotically under the ideal case that no other stimulus would influence the individual happiness. The restriction to be considered in the calibration of parameter \(b\) is that its value must belong to the interval of the corresponding happiness scale: \(b\in\ (0,20]\) for the ES and \(b\in(0,7]\) for the SFS. Parameter \(a\) represents the strength or power of the above-mentioned asymptotic dynamics and its only restriction is that its value must be greater or equal to zero. Both parameter values also depend on the kind of drug and on the individual characteristics. About the excitation effect, _fe(t)_, it tends to increase happiness proportionally to the stimulus, _s(t)_, and the present happiness, _y(t)_, per unit of drug amount \(M\):  
\[m(t)=m(t-\Delta t)+\Delta t\cdot(-fm(t)) \tag{1}\]0  
Parameter \(p\) in (10) represents the strength or power of the excitation effect, and its value also depends on the kind of drug and on the individual characteristics. The only restriction for \(p\) is that its value must be greater or equal to zero. Note that the greater the value of \(p\) the greater the excitation effect of the considered drug on the individual happiness [21; 22].