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Those who have never wanted to predict how their love story could evolve throw the first stone! As a computer science nerd, I have always been amazed by the unpredictable side of love or even friendship.

As Kartik said in a previous post:

How best friends become enemies, atheists become fanatics, and true lovers start drifting apart for factors no one is to be blamed of…

In the following, I will only deal with love stories and restrict them to two individuals (\(I_1\)
and \(I_2\) respectively).

1. Introduction
   Love stories always start from an initial state that can be either of total indifference (\(I_1\) and \(I_2\) ignore each other), non reciprocal (\(I_1\) is interested (resp. indifferent) and \(I_2\) is indifferent (resp. interested)) or of mutual interest. During the first following weeks, the dynamics of feelings are in general quite turbulent, going from breakdown to euphoria in a cyclic way, cluttering one’s mind with a lot of unanswered questions.
   After this transient period, feelings may fade until the separation of \(I_1\) and \(I_2\) is amplified or remain the same, reaches a stable state or produces a limit cycle. A lot of parameters can vary these feelings: \(I_1\) and \(I_2\)‘s visual appearance could decrease over time, as well as libido. Consequently, if \(I_1\)‘s main interest was sex, it is likely that the system will reach an unavoidable separation, unless \(I_1\) changes her/his mind (Rinaldi et al., Modeling Love Dynamics.). This being said, mathematics, especially dynamic systems could be a interesting tool to better untangle love dynamics, characterize bifurcations (change in the stability/number of equilibria). (Rinaldi et al., Physica A, 392: 2013)
2. Mathematical Background
   Because I previously assumed a love story involving only two partners :), I will consider the following system composed of two ordinary differential equations (ODEs):
   \(
   \begin{equation}
   \label{eq:love_system}
   \left\{
   \begin{array}{l c}
   \dfrac{dx_1}{dt} = – \alpha_1 x_1(t) + \rho_1 A_2 + R_1(x_2), (1)\\
   \dfrac{dx_2}{dt} = – \alpha_2 x_2(t) + \rho_2 A_1 + R_2(x_1). (2)
   \end{array}
   \right.
   \end{equation}
   \)

    

   \(x_1\) and \(x_2\) correspond, respectively, to the love of \(I_1\) and \(I_2\). They can be negative, which means that \(I_1\) and/or \(I_2\) hate each other. Let’s have look at Equation (1):

   1. The first right-hand side term, namely \(-\alpha_1 x_1(t)\), is the term of oblivion. The higher \(\alpha_1\), the faster the forgetting processes.
   2. The second term is the appeal that \(I_2\) has on \(I_1\), and \(\rho_1\) is the sensitivity of \(I_1\) to \(A_2\) (see below).
   3. The third term is the reaction of \(I_1\) to \(I_2\)‘s love. For instance, some people like to be loved (they are called secure individuals). Conversely, others may feel overwhelmed as soon as their partner’s love is to high.

   Equation (2) follows the same rules.

   The appeal of a person depends on a lot of components such as beauty, intelligence, richness, generosity, openness, hygiene, honesty, humour and libido (of course one cannot define a person only be 10 characteristics, but we have to start somewhere). We define \(\lambda_j^h\) as the importance that the person \(j\) places on the \(h^{th}\) characteristic of i (\(j \neq i\)). Therefore the appeal can be defined as:

   \(
   A_i = \sum_h{\lambda_j^h A_i^h}.
   \)

    

   Importantly, we specify that

   \( 0 \leq \lambda_j^h \leq 1. \)

   In pratice, to set \(A_i^h\), we usually ask how the person considers her beauty, intelligence on a scale from -10 to 10. Thus if she is very rich, the corresponding \(A_i^h\) will be close to 10 and inversely. Interestingly, if the person is depressed, her global appeal \(A_i\) can take negative value.

   The reaction to the love term \(R_i (x_j)\) is often described by a sigmoidal function as shown below. This function is defined on \( ]-\infty; +\infty[ \) and takes values in \( [R_i^-, R_i^+] \) that respectively correspond to the minimum and maximum reactions (their value tightly spend on the people profil. For instance, a romantic person could have a very high \( R_i^+ \)). When \(x_1\) and \(x_2\) are negative (resp. positive), both blue and red curve tend to \(R_1^-\) (resp. \(R_1^+\)) and \(R_2^-\) (resp. \(R_2^+\)). The slope of the sigmoid could also be controlled. A high positive slope would mean that the corresponding individual is very sensitive to her/his partner’s love, and inversely. A negative slope would represent the case of people who hate to be loved. Based on these properties, one can create a lot of different psychological profiles. We could choose non bounded functions to model a very romantic person, using \(R(x) = x^3\) function. Additionally, some people like to be loved but at some point start to run away, when their partner’s love becomes too invasive. This kind of effect is observed in pharmacokinetics, when a substance that has health benefits in low doses becomes dangerous past a critical threshold. This is called hormesis effect, and can be model with the following function extracted from the Brain-Cousens model in ecotoxicology:

   \(
   R(x) = c + \frac{d-c+fx}{1+\exp{(b(\log(x)-\log(e)))}}
   \)

    

   

   Reaction function for secure individuals

   At \(t = 0\), if we consider that \(I_1\) and \(I_2\) are indifferent, \(x_1(0) = x_2(0) = 0\), \(R(x_1(0)) = R(x_2(0)) = 0\) and equations (1) and (2) become:

   \(
   \begin{equation}
   \label{eq:love_system_0}
   \left\{
   \begin{array}{l c}
   \dfrac{dx_1}{dt}(0) = \rho_1 A2, (1)\\
   \dfrac{dx_2}{dt}(0) = \rho_2 A1. (2)
   \end{array}
   \right.
   \end{equation}
   \)

    

   Therefore, the dynamics of the system only depends on the respective appeal of \(I_1\) and \(I_2\). For example, if \(I_2\)‘s characteristics totally match the expectations of \(I_1\), \(\dfrac{dx_1}{dt}(0)\) will be positive. The problem is that if a person is non-appealing, \(\dfrac{dx_1}{dt}(0)\) will be very close to 0, meaning that she could not be loved, which is not the case in practice. Actually, as the definition of appeal is quite subjective, an individual can be considered as non-appealing in its home country and very appealing in another one.

3. Model improvements
   The model described here is very basic and could be substantially improved. For instance, one could add the effect of family and friends, who may play a more or less significant role.
   Actually, some love stories involve more than two partners, either at the beginning, in the middle of the relationship or just before the end: one could introduce a third equation to investigate the effect of \(I_3\) on \(I_1\) and \(I_2\), for example.
4. External Resources
   This project is carried out by my previous master’s internship supervisor, Laurent Pujo Menjouet, when I was in France in 2011.
   A very nice presentation about love affairs was done by Laurent Pujo Menjouet (Université Claude Bernard Lyon 1, Institut Camille Jordan/INRIA Dracula).The ultimate goal of his project would be to provide an interface where people would have to enter their love profile, namely if they are married, single, describe their ideal partner, describe themselves as well. Based on these parameters, the simulator would predict all possible love trajectories depending on initial conditions. The user would also have to enter its goal allowing the simulator to provide advice to reach his wish or what to do in order to avoid having problems.The last step would be to collect datas from users, if they want to :), and use them to do statistical analysis and refine parameters with real life cases.You can access an experimental love simulator by clicking on the image below (only in French, unfortunately).
   

   The love simulator