**1. Introduction**

In this paper, we aim to give an additional contribution to the literature in Economics of Love devising a modified version of the model in [1]. We analyse the dynamics leading to toxic relationships, i.e., we focus our attention on situations where love is transformed into a negative dependence and the relationship produces a dangerous addiction.

Moreover, we investigate what intervention policy people could advocate to protect themselves and to reduce suffocating addiction in order to escape the trap of physical or psychological abuse either in family or at work. The dynamics of couples' relationships are analyzed through a system of differential equations representing the laws of motions of the amount of love that two individuals put in a relationship which also depends on any source of partner's appeal (financial, physical, intellectual, co-parenthood, etc.).

In our work, the appeal is assumed to change over time proportionally to both the effect of the other partner's love and to a variable representing the main source of addiction (for instance, wealth, status, physical appearence, etc.). The aim of our analysis is to build a model which takes into account the case where toxic relationships are at work, i.e., when one partner chooses to stay in a relationship despite the low or null amount of love received, for the sole reason that their partner's appeal makes it harder for them to terminate the relationship.

In 1995, Dr. Lillian Glass defines a toxic relationship as any relationship [between people who] do not support each other, where there's conflict and one seeks to undermine the other, where there's competition, where there's disrespect and a lack of cohesiveness [2].

By toxic relationship, we mean a relationship disorder characterized by a disparity, a non-egalitarian situation in which one of the two partners is dependent on the other one, triggering a mechanism of dominance and subjection. In these relationships one partner puts into the relationship much more effort than the other one, who, sometimes, could even put no effort at all into it. In an healthy relationship partners can still cut out their

**Citation:** Solferino, N.; Tessitore, M.E. Human Networks and Toxic Relationships. *Mathematics* **2021**, *9*, 2258. https://doi.org/10.3390/ math9182258

Academic Editor: David Carfì

Received: 27 July 2021 Accepted: 9 September 2021 Published: 14 September 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

spaces, maintaining a capacity for self-determination and taking benefit from reciprocity. In a toxic relationship the emotional dependance enters into play, making the partner our exclusive interlocutor, so that being happy and enthusiastic depends exclusively on the other person, just like drug addiction. In order to avoid the abandonment and the consequent lack of affection, the addicted partner cancels himself or herself out, while the counterpart exploits the relationship to feel admired and to exercise control. A toxic relationship implies psychological, and at times physical, violence, and can develop into a tragic eposide of murder. The majority of the victims of these murders are women

Another typical example of toxic relationship is bullying at a workplace, very often put into action by the boss. The unwillingness to call out such behaviour by the victims and spectators results in to favoring it. The reason for this behaviour is obviously the fear of being involved, of retaliation of some kind or even of losing the job.

In our work, we focus on the conditions which make a toxic relationshio to arise. Due to the exploited parter's addiction, the abusing parter's behaviour seems appealing regardless low affection. In Section 2 we rewiev the literature on this topic. In Section 3 we give an analytical definition of a toxic relationship and we solve an intertemporal dynamic model, where the toxic partner's behaviour is assumed to be exogenous. Assuming that the main source of addiction is income or wealth, we find that an asymptotically stable equilibrium with positive love is always possibile for a high enough level of appeal if subsidies that reduce addiction are introduced. In Section 4, we compare two alternative policies that can be adopted to heal from addiction. One policy consits in healing through a subsidy that can reduce the toxic partner's appeal. The alternative policy is to heal through a third, unconditionally reciprocating, part as a benchmark which represents an alternative, but less attractive, to the partner's love. This third part plays an important role in reducing the toxic partner's appeal. It substantially mimics not only the real presence of another partner but also the support from family, friends and overall private organizations in helping victims of domestic abuses recover their life fully. It may also represent private organizations that offer economic and psychological support as well as legal counselling to victims of bullying at workplace and placement offices which effectively help to find another job. Section 5 contains our discussion and future implications and research.
