**3. Basic Model of a Toxic Relationship**

In this section, we devise a modified version of the model described in [1] where the level of love an individual puts in a relationship varies over time, depending both on oblivion, on the partner's love and on the perception one has of the appeal of the partner.

We denote by 1, 2 the submitted and the toxic partner respectively.

We assume for simplicity that the partner 2's appeal perceived by partner 1, *A*1, decreases at the same rate of love *x*<sup>1</sup> (i.e., oblivion's rate is the same), it increases as individual 1 loves more, and it decreases as fast as the difference between the amount of love partner 2 puts into in the relationship, *x*2, and the desired amount of love that partner 1 would expect from partner 2, *x*ˆ2.

We also assume that *A*<sup>1</sup> depends on a constant factor *M*<sup>2</sup> of appeal i.e., income, personality, physical appearance, etc.

We study the stability of the following system of dynamic equations:

$$\begin{cases}
\dot{\mathbf{x}}\_1(t) = -a\mathbf{x}\_1(t) + \beta \overline{\mathbf{x}\_2} + \gamma\_1 A\_1(t) \\
\dot{A}\_1(t) = -aA\_1(t) + k[\mathbf{x}\_1(t) + (\overline{\mathbf{x}\_2} - \mathbf{x}\_2)] + M\_2
\end{cases} \tag{1}$$

where


**Definition 1.** *If there exists a minimum level of desired love x*ˆ2 *for individual 1 with x*<sup>2</sup> ≤ *x*ˆ2 *and if x*<sup>1</sup> > 0 *then we say that the relationship between 1 and 2 is a toxic relationship.*

In a toxic relationship, individual 1 loves partner 2 even if the partner 2 does not reciprocate (neither at the minimum possible level normally requested *x*ˆ2), since the second partner only provides *x*<sup>2</sup> and *x*<sup>2</sup> < *x*ˆ2 .This happens because attraction *A*<sup>1</sup> of partner 1 towards partner 2 is very high.

We are going to prove that system (1) has a steady state stable equilibrium.

**Proposition 1.** *Assume kγ*<sup>1</sup> < *<sup>α</sup>* < *<sup>β</sup> and M*<sup>2</sup> ≥ *kx*ˆ2 *, then state steady* (*x*<sup>∗</sup> <sup>1</sup>, *A*<sup>∗</sup> 1 )

$$x\_1^\* = \frac{\alpha\beta\overline{x\_2} + k\gamma\_1(\overline{x\_2} - \mathfrak{x}\_2)}{\alpha^2 - k\gamma\_1} + \frac{\gamma\_1M\_2}{\alpha^2 - k\gamma\_1}$$

$$A\_1^\* = \frac{\beta k\overline{x\_2} + \alpha k(\overline{x\_2} - \mathfrak{x}\_2)}{\alpha^2 - k\gamma\_1} + \frac{\alpha M\_2}{\alpha^2 - k\gamma\_1}$$

*is an asympotically stable equilibrium for system* (1)*.*

**Proof.** Let *A* be the coefficient matrix of the system

$$A = \begin{pmatrix} -\alpha & \gamma\_1 \\ k & -\alpha \end{pmatrix}.$$

since *detA* > 0 and *trA* < 0 the conclusion follows.

**Remark 1.** *Notice that in the steady state equilibrium, the love for the partner is always positive and*

*it grows as the depreciation rate decreases with respect to the sensitivity towards the partner's appeal. If partner 1 has x*ˆ2 = 1 *and partner 2 provides x*<sup>2</sup> = 0*, the first partner still loves the second one since*

$$x\_1^\* = -\frac{k\gamma\_1}{a^2 - k\gamma\_1} + \frac{\gamma\_1 M\_2}{a^2 - k\gamma\_1}$$

*We notice that if the amount of love partner 2 puts in the relationship is equal to the desired love of partner 1, i.e., x*<sup>2</sup> = *x*ˆ2*, then*

$$x\_1^\* = \frac{\alpha \beta \overline{x\_2}}{a^2 - k \gamma\_1} + \frac{\gamma\_1 M\_2}{a^2 - k \gamma\_1}$$

*and*

$$A\_1^\* = \frac{\beta k \overline{\chi\_2}}{a^2 - k \gamma\_1} + \frac{\kappa M\_2}{a^2 - k \gamma\_1}$$

*This means that the first partner's love is always positive if the second partner's perceived appeal overcomes the negative impact of the unreciprocated love.*

*In particular, if x*<sup>2</sup> = *x*ˆ2 = 0*, then*

$$x\_1^\* = \frac{\gamma\_1 M\_2}{a^2 - k\gamma\_1}$$

*and*

$$A\_1^\* = \frac{aM\_2}{a^2 - k\gamma\_1}$$

*So we see that the partner 1 still loves partner 2 only depending upon the factor M*<sup>2</sup> *which represents wealth, or other benefits.*

If individual 1 starts loving individual 2 at time *t* = 0, then 1 will love 2 forever. This is true unless the love desired by 1 *x*ˆ2 attains a specific value

**Proposition 2.** *If partner 1 has an amount of desired love*

$$
\hat{x}\_2 = \frac{(\alpha\beta + \gamma\_1 k)\overline{x\_2} + \gamma\_1 M\_2}{k\gamma\_1}
$$

*then partner 1 stops loving partner 2.*

**Proof.** From the previous Proposition we see that *x*∗ <sup>1</sup> = 0 with this choice of *x*ˆ2.

**Definition 2.** *Individual 1 overcomes a toxic relationship, i.e., 1 is healed, if x*∗ <sup>1</sup> = 0*.*

**Remark 2.** *Hence we see that from the above Proposition, individual 1 is healed only if the love they desire equals a specific amount.*

In practice, this is hard to achieve because it is not straightforward for a person to select a specific value of love to put in a relationship. This is why in the next Section we investigate if there are other ways to get out of such a relationship.

#### **4. Healing**

In this section, we look for conditions to help the submitted partner 1 to heal from toxicity, since in the previous Section this condition could be attained only if the submitted partner has a specific value of desired love *x*ˆ2.

We are going to consider two possible ways: the first one is by reducing the toxic partner's appeal 2 via a subsidy *S* > 0, while the second one is going to be the presence of a third partner denoted by 3.

We begin introducing a subsidy *S* = *sM*2. This represents a payment from the government or another organization to the abused partner to cover personal living expenses they might incur when leaving the relationship, such as rent. Here *s* is the factor of proportionality chosen by the government.

In system (1) we introduce the subsidy *S* in the dynamic of *A*<sup>1</sup>

$$\begin{cases}
\dot{x}\_1(t) = -a\mathbf{x}\_1(t) + \beta \overline{\mathbf{x}\_2} + \gamma\_1 A\_1(t) \\
\dot{A}\_1(t) = -aA\_1(t) + k[\mathbf{x}\_1(t) + (\overline{\mathbf{x}\_2} - \mathbf{x}\_2)] + M\_2(1-s)
\end{cases} \tag{2}$$

**Proposition 3.** *The subsidy S* = *sM*<sup>2</sup> *with*

$$s = \frac{\alpha \beta \overline{\mathbf{x}\_2} + k \gamma\_1 (\overline{\mathbf{x}\_2} - \mathbf{x}\_2)}{\gamma\_1 M\_2} + 1$$

*is healing individual 1.*

**Proof.** It is easy to see that in this case the steady state *x*∗ <sup>1</sup> *<sup>s</sup>* of system (2) is

$$\mathbf{x}\_{1s}^{\*} = \frac{\mathfrak{a}\beta\overline{\mathbf{x}\_{2}} + k\gamma\_{1}(\overline{\mathbf{x}\_{2}} - \mathbf{x}\_{2}^{\*})}{\mathfrak{a}^{2} - k\gamma\_{1}} + \frac{\gamma\_{1}M\_{2}(1-s)}{\mathfrak{a}^{2} - k\gamma\_{1}}.$$

hence the result follows from the above assumptions.

**Remark 3.** *Notice that*

$$x\_{1\_s}^\* = x\_1^\* - \frac{\gamma\_1 M\_2}{\varkappa^2 - k\gamma\_1}s$$

*so the subsidy reduces the love of partner one towards partner two.*

Another way to heal might be the presence of a third person, denoted by 3, we want to see if 1 can heal from the second partner. System (1) becomes

$$\begin{cases} \begin{aligned} \dot{\mathbf{x}}\_1(t) &= -a\mathbf{x}\_1(t) + \beta(\overline{\mathbf{x}\_2} - \mathbf{x}\_3) + \gamma\_1 A\_1(t) \\ \dot{A}\_1(t) &= -aA\_1(t) + k[\mathbf{x}\_1(t) + (\overline{\mathbf{x}\_2} - \mathbf{x}\_2^\*) - \mathbf{x}\_3] + M\_2 - M\_3 \end{aligned} \end{cases} \tag{3}$$

where *x*<sup>3</sup> is the amount of love of individual 3 towards individual 1 and *M*<sup>3</sup> with *M*<sup>3</sup> < *M*<sup>2</sup> is the appeal of the third partner.

**Proposition 4.** *If x*<sup>3</sup> *is such that*

$$\mathfrak{x}\_{\mathfrak{I}} = \overline{\mathfrak{x}\_{\mathfrak{I}}} + \frac{\gamma\_1 \left[ (M\_2 - M\_3) - k \mathfrak{x}\_2 \right]}{\mathfrak{a}\mathfrak{f} + k \gamma\_1}$$

*then 3 is healing individual 1.*

**Proof.** It is enough to see that in this case the steady state *x*∗ <sup>13</sup> of system (3) is

$$\mathbf{x}\_{13}^\* = \frac{a\beta(\overline{\mathbf{x}\_2} - \mathbf{x}\_3) + k\gamma\_1[(\overline{\mathbf{x}\_2} - \mathbf{x}\_2) - \mathbf{x}\_3]}{a^2 - k\gamma\_1} + \frac{\gamma\_1(M\_2 - M\_3)}{a^2 - k\gamma\_1}.$$

**Remark 4.** *Notice that*

$$\mathbf{x}\_{1\cdot 3}^{\*} = \mathbf{x}\_1^{\*} - \frac{\alpha\beta + k\gamma\_1}{\alpha^2 - k\gamma\_1}\mathbf{x}\_3 - \frac{\gamma\_1}{\alpha^2 - k\gamma\_1}M\_3.$$

*Hence, the third partner reduces the love of partner one towards partner two, similarly to the subsidy's role in the previous case.*

If in addition to partner 3 offering a certain fixed amount of love *x*<sup>3</sup> to partner 1 there is also a subsidy *S* = *sM*2, then proceeding as before we are going to consider the system

$$\begin{cases}
\dot{x}\_1(t) = -a\mathbf{x}\_1(t) + \beta(\overline{\mathbf{x}\_2} - \overline{\mathbf{x}}\_3) + \gamma\_1 A\_1(t) \\
\dot{A}\_1(t) = -aA\_1(t) + k[\mathbf{x}\_1(t) + (\overline{\mathbf{x}\_2} - \mathbf{x}\_2) - \overline{\mathbf{x}}\_3] + M\_2(1 - \overline{\mathbf{s}}) - M\_3
\end{cases} \tag{4}$$

**Proposition 5.** *The subsidy S* = *sM*<sup>2</sup> *with*

$$\overline{s} = \frac{\alpha\beta(\overline{\overline{\mathbf{x}\_2}} - \overline{\mathbf{x}\_3}) + k\gamma\_1[(\overline{\overline{\mathbf{x}\_2}} - \mathbf{x}\_2) - \overline{\overline{\mathbf{x}}\_3}]}{\gamma\_1M\_2} + 1 - \frac{M\_3}{M\_2}$$

*is healing individual 1.*

**Proof.**

$$x\_{1.35}^{\*} = \frac{a\beta(\overline{x\_2} - \overline{x}\_3) + k\gamma\_1[(\overline{x\_2} - \overline{x}\_2) - \overline{x}\_3]}{a^2 - k\gamma\_1} + \gamma\_1 \frac{M\_2(1 - \overline{s}) - M\_3}{a^2 - k\gamma\_1}$$

and the result follows.

**Remark 5.** *Notice that in the previous case we found that if individual 1 loves 2 without a correspondence, then the only way to get x*<sup>1</sup> = 0 *is by introducing a subsidy that counterbalances the addiction due to income M*2*.*

*Instead, when there is also a third partner there is no need to take s* > 1 *because the love term x*<sup>3</sup> *also acts as a counterweight. This results in a lower minimum subsidy required to heal partner 1 since*

$$\overline{s} = s - \frac{(\alpha\beta + k\gamma\_1)\overline{x}\_3 + \gamma\_1M\_3}{\gamma\_1M\_2}$$

#### **5. Discussion**

A toxic relationship can be defined as a relationship characterized by one partner displaying behaviours that are emotionally and often physically damaging the other partner.

A healthy relationship involves mutual caring, respect, compassion and a strong interest in the partner's happiness. In the couple both individuals share control and decision-making. On the contrary, a toxic relationship is characterized by insecurity, selfcenteredness, dominance and control.

When two individuals have a toxic relationship, we usually look at the toxic partner's behaviour. We must also observe the individual who is the recipient of the toxic behaviour. In fact, according to psychologists, the reasons that push adults into remaining in toxic reationships, which will almost inevitably damage them emotionally or physically, need to be thoroughly investigated. We think that this often happens because addiction may play a very important role: the partner's appeal grows over time regardless of the amount of love the they invest in the relationship.

In this paper, we approach these issues from a theoretical point of view with the purpose of devising an analytical model which can highlight the main points of this problem and can shed light on useful policy solutions. To this purpose we use the model of Rinaldi [1], adding the possibility of the toxic partner's appeal evolving dynamically according to a specific law of motion. Our model assumes the law's dependence on oblivion, on the excess of love with respect to the partner and on a constant parameter that measures the values of the source of addiction(for instance income or wealth, etc.). Our model shows that in the most simple case, where the partner's behaviour is given (exogenous) and hopelessly toxic, there is an asympotically stable equilibrium for high values of addiction, with a submitted partner always in love.

Nevertheless, an opportune measure of correction based on subsidies can be introduced. On the other hand, the lack of government help is often one of the main reasons for the low levels of immediately reported domestic abuses. For instance, according to

GROVIO( Group of Experts on Action against Violence against Women and Domestic Violence and EstremeConseguences) of the European Council, about 80% of abuse in Italy happens at home, and there are not enough dormitories to host over 5000 women who have left their houses to escape abusive partners. Alongside dedicated laws, penalties and high compensation for the monetary, biological and moral damage, public funds are essential. Furthermore, dedicated public funds play a key role in reducing any dependence on employers or co-workers, and in incentivizing reports from victims. Nevertheless, union organizations offering cheap or free legal help, supporting in searching for other jobs and reintegration are very useful.

To take this problem into account , we look for an alternative policy and we study how the results of our theorethical model change when an alternative third part enters the scene and competes for the submitted parter's love with the toxic partner. In this case lower or zero subsidies can be necessary as the third part works like a substitute to decrease the toxic dependance. Anyway a mix of policy intervention based both on subsidies and on help from third part can be fashioned. Therefore, when a policy based only on subsidies is not sufficient, other factors can be useful to rescue from toxicity.

Often, subsidies are not high enough or people need also to preserve their dignity,security, counselling, etc. In our model for instance a high source of addiction may also come from a very low self esteem which can otherwise be raised offering victims other opportunities like alternative jobs or legal/counseling support. This is why we believe that the best solutions relies in a mix of policies where alonside with the government, private organizations offer support to the victims of abuse as well placement offices in finding new jobs.

With this work we aimed to give a contribution to the literature on Economics of Love and in particular we hope to incentive more theoretical and empirical studies on relationships and to devise better policy solutions. In particular we think that a mix of policies, directly or indirectly created to fight this phenomenon based on monetary help and supportive institutions and organizations could be very effective.

A future development of this paper would be to consider people with low self esteem who are most subject to this kind of addiction and where psychoterapy for couples is strongly needed. In the extreme cases when toxicity externates in physical violence also objective policy solutions are likely to be necessary.

An interesting follow-up investigation we would like pursue is the study of the alltoo-common instances of bullying at the workplace. In these instances, too often abuse and harassment are considered normal events, so that the workers who suffer them prefer to remain silent. This occurs especially when the victim's job position is precarious and when the employer can affect the victim's financial condition on a whim. Istat data for Italy tell us that 9 out of 100 women, during their working life, have suffered harassment or blackmail with a sexual background at work (1 million 403 thousand), but that only 20% talk about it with someone (usually office colleagues) and only 0.7% complain, for fear of retaliation, shame, or for a distorted sense of guilt, slander. The company is damaged by these occurrances as well as the workers as the effects are a lower productivity, increased risk of accidents and conflict. The inevitable repercussions then fall on the health service (treatments, drugs) and on the social security system (illnesses, injuries, etc.).

**Author Contributions:** Both authors equally contributed in every aspect of the writing of this article. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
