## 4 The Stimulus Response Model (SRM)
About the inhibitor effect, _fi(t)_, assuming that it actuates with some delay, it tends to decrease happiness proportionally to the happiness delay, _z(t)_, per unit of drug amount \(M\):  
\[m(t)=m(t-\Delta t)+\Delta t\cdot(-fm(t)) \tag{1}\]1  
Parameter \(q\) in (11) represents the strength or power of the inhibitor effect, and its value also depends on the kind of drug and on the individual characteristics. The only restriction for \(q\) is that its value must be greater or equal to zero. Note that the greater the value of \(q\) the greater the inhibitor effect of the considered drug on the individual happiness [21; 22]. That is, the greater the values of \(p\) and \(q\) are the more intense is, respectively, the excitation effect and the inhibitor effect. By substituting (9), (10) and (11) in (8), and adding the initial happiness value at the initial time \(t=t_{0}\):  
\[ \begin{array}{c}\frac{dy(t)}{dt}=a(b-y(t))+\frac{p}{M}\cdot s(t)\cdot y(t)- \frac{q}{M}\cdot z(t)\\ y(t_{0})=y_{0}\end{array}  \tag{12}\]  
Considering the balance equation in the infinitesimal limit form for the happiness delay level variable, \(z(t)\), of the HD:  
\[\frac{dz(t)}{dt}=-fh(t)+fl(t) \tag{13}\]  
According to the literature about the subject [21; 22], the hypotheses about the mathematical patterns of the two flows _fh(t)_ and _fl(t)_ that appear in (13) are the following. About the happiness inhibitor delay, _fh(t)_, it is a decreasing control about the happiness delay with a time parameter \(\tau\) as follows:  
\[fh(t)=\frac{z(t)}{\tau} \tag{14}\]  
The only restriction for parameter \(\tau\) value in (14) is that it has to be positive. About the happiness excitation delay, _fl(t)_, it tends to increase the happiness delay as the product of the present stimulus and happiness:  
\[fl(t)=s(t)\cdot y(t) \tag{15}\]The initial value of \(z(t)\) in \(t=t_{0}\) is zero due to it just starts to act when the stimulus arises from this instant. Therefore, considering this initial value and substituting (14) and (15) in (13):  
\[\frac{dz(t)}{dt}=-\frac{z(t)}{\tau}+s(t)\cdot y(t)\\ z(t_{0})=0 \tag{16}\]  
Equations (6), (7), (12), and (16) define the stimulus response model (SRM) to study the short-term happiness dynamical response to a stimulus provided by a drug single dose, such as the here considered ones: alcohol and caffeine. However, the SRM can be presented in a briefer way. On the one hand, the drug dynamics is given by (6) and (7); nevertheless, they define an independent subsystem of the SRM from which the drug in blood variable \(s(t)\) can be analytically found:  
\[s(t)=s_{0}e^{-\beta\cdot t}+\left\{ \begin{array}{c}\frac{\alpha\cdot M}{ \beta-\alpha}\big{(}e^{-\alpha\cdot t}-e^{-\beta\cdot t}\big{)}\ :\ \alpha\
eq\beta\\ \alpha\cdot M\cdot t\cdot e^{-\alpha\cdot t}\ :\ \alpha=\beta\end{array} \right. \tag{17}\]  
On the other hand, Equation (16) is a linear differential equation in \(z(t)\); therefore, it can be integrated and substituted into (12):  
\[\frac{dy(t)}{dt}=a(b-y(t))+\frac{p}{M}\cdot s(t)\cdot y(t)-\frac{q}{M}\cdot \int\limits_{t_{0}}^{t}e^{\frac{t-t}{t}}\cdot s(r)\cdot y(r)\ dr\\ y(t_{0})=y_{0} \tag{18}\]  
Equation (18) is an integrodifferential equation where the inhibitor effect arises explicitly with a continuous delay (the integral term). Both Equations (17) and (18) also define the SRM in a much more concise way. However, the complexity of (18) forces the researcher to find its solutions numerically. Consequently, it is more advantageous to find numerical solutions for happiness \(y(t)\) by using Equations (12) and (16), while the solutions for the drug in blood variable \(s(t)\) are found directly by (17) (see following section).