# A Mathematical Model for Intimate Partner Violence

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## Abstract

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## 1. Introduction

## 2. Mathematical Modeling

#### 2.1. Non-influenced Intimate Partner Interaction

**A0.**In a similar way and following [10], we assume the existence of non-influenced steady states $\tilde{v}$ and $\tilde{i}$ for each individual, man and woman indexes, respectively. Those states are independent of each other intimate partner. Actually, they depend on their own personal history with respect to the IPV problem. For this reason, we assume that the non-influenced steady states are directly proportional to two causal factors: Violence seen in childhood and acceptance of violence. So, we define the non-influenced man’s violence steady state as $\tilde{v}={\alpha}_{1}{\beta}_{1}$ and the non-influenced woman’s independence steady state as $\tilde{i}=(1-{\alpha}_{2})(1-{\beta}_{2}),$ where ${\alpha}_{1},{\alpha}_{2}$ are the parameters that quantify the violence watched in childhood for man and woman, respectively, and ${\beta}_{1},{\beta}_{2}$ are the parameters that quantify the acceptance of machismo for the man and woman, respectively. Notice that the non-influenced independence index of the woman decreases according to the level of violence watched during her infancy and her acceptance of IPV [18]. We will discuss these parameters in more detail in the next section.**A1.**These non-influenced steady states correspond to an equilibrium point $(\tilde{v},\tilde{i})$ of the following differential equation system:$$\begin{array}{cc}\frac{dv\left(t\right)}{dt}& ={\delta}_{1}(\tilde{v}-v\left(t\right)),\phantom{\rule{1.em}{0ex}}v\left(0\right)={v}_{0},\\ \frac{di\left(t\right)}{dt}& ={\delta}_{2}(\tilde{i}-i\left(t\right)),\phantom{\rule{1.em}{0ex}}i\left(0\right)={i}_{0},\end{array}$$

#### 2.2. Influenced Intimate Partner Interactions

**A2.**The rate at which the man’s violence index increases depends on the man’s need to control the woman which is caused by a low man’s self-esteem [14]. In particular, we assume that it is inversely proportional to the man self-esteem, $1/\gamma $ in ${k}_{1}(1-{\rho}_{1})\frac{{\beta}_{1}}{\gamma}i\left(t\right)$. Besides, we assume that the variation rate of $v\left(t\right)$ depends linearly on the woman’s independence index. This is because the man will have more need to control the woman when woman becomes more independent. This is supported by patriarchal theory as described in [13]. Moreover, the coefficients in the modeling term ${k}_{1}(1-{\rho}_{1})\frac{{\beta}_{1}}{\gamma}i\left(t\right)$ were set considering the influence of the man’ s acceptance of machismo ${\beta}_{1}$, as well as, possible self-regulatory failures ${\rho}_{1}$. It is important to remark, that the self-regulatory factor of a person is something difficult to quantify because depends on several random factors. For these reason and as a first approximation, we assume that these coefficients are constant.**A3.**In an analogous way for the modeling term ${k}_{2}(1-{\rho}_{2})\mu v\left(t\right)$, we assume that the degree of violence that the man exerts against the woman, in addition with the influence of the social environment, are factors that can change the woman’s independence index. This might cause an increase in the level of the woman submission ($\mu $ negative) caused by the social pressure of the woman’s family or friends to remain in the relationship. Or it can increase the level of the woman’s independence index ($\mu $ positive) when the woman is enrolled in empowerment programs. Let us remark the importance of the sign change of $\mu $ which implies a turning point in the IPV dynamics of the couple. On the other hand, notice that ${\rho}_{2}$ is included in this term and is the woman’s self-regulatory coefficient. So, we propose that $v\left(t\right)$ influences linearly the change rate of $i\left(t\right)$ and vice versa.**A4.**According to the latest National Survey on Violence against Women in Mexico (ENVIM) [2], the frequency of alcohol consumption by a violent man is proportional to the number of aggressions perpetrated by a battering man. Also, that the alcohol consumption events are generally periodic with usually weekly or monthly periodicity. In order to model this behavior, we use the modeling term $h\left(t\right)$. We assume that this situation can be properly described by a sine function with parameters that allow us to control its amplitude ($\rho $) and its frequency ($\omega $): $h\left(t\right)=\rho (sin(\omega t)+\eta )$, where $\eta $ is a positive constant.

#### 2.3. Variable and Parameter Scales

## 3. Analysis of the Model without Alcohol Consumption

**Case 1.**If ${\tau}^{2}-4\Delta <0,$ we have that ${({\delta}_{1}-{\delta}_{2})}^{2}<-4{k}_{1}{k}_{2}(1-{\rho}_{1})(1-{\rho}_{2})({\beta}_{1}/\gamma )\mu $, then the equilibrium point $({v}^{*},{i}^{*})$ is asymptotically stable, but the solutions are oscillating. A necessary condition is that $\mu <0$ which means that external factors make the woman’s independence index decrease. This may be caused by family or social pressure on the woman, or by her beliefs that lead her to a highly dependent state. In these circumstances, the IPV for this couple tends to the stable steady state which indicates woman submission and a constant level of IPV. It is important to remark that the left-hand side of the inequality is the difference between ${\delta}_{1}$ and ${\delta}_{2}$. Then, we can have this case when the inertia coefficients are very similar or equal, and there is social pressure on the woman. Another way in which this condition may hold corresponds to a high social pressure, low self-regulatory conditions, high acceptance of machismo or very low man’s self-esteem.**Case 2.**If ${\tau}^{2}-4\Delta \ge 0$ and $\Delta \ge 0$, or equivalently ${({\delta}_{1}-{\delta}_{2})}^{2}\ge -4{k}_{1}{k}_{2}(1-{\rho}_{1})(1-{\rho}_{2})({\beta}_{1}/\gamma )\mu \ge -4{\delta}_{1}{\delta}_{2}$, then the equilibrium point is asymptotically stable and the solutions are not oscillatory. Notice that $\mu $ could be positive or negative. For this case, we can find a lot of possible parameter combinations which could lead us to this situation where the intimate partner interaction stabilizes rapidly. That means that the man’s violence and the woman’s independence indexes reach a stationary level and remain there for a long time. Observe that in this case, the inertias for both of the members of the couple could be similar or different, one from each other. However, independently of this, the level of IPV will be high when there exists high acceptance of machismo, low self-esteem or low self-regulation for the man. Otherwise, the model predicts a moderated level of IPV. Another fact is that depending on the $\mu $ value, the level of the woman’s independence will stabilize on a level of independence or dependency.**Case 3.**If ${\tau}^{2}-4\Delta >0$ and $\Delta <0$ or equivalently, ${\delta}_{1}{\delta}_{2}<{k}_{1}{k}_{2}(1-{\rho}_{1})(1-{\rho}_{2})({\beta}_{1}/\gamma )\mu $, then the equilibrium point is unstable and it is necessary that $\mu >0$ for the condition holds. Notice that the condition $\mu >0$ means that external factors are leading to an increment in the woman’s independence index. In particular, this could be achieved through empowerment programs for the woman. Then, our model reproduces the reported fact that this type of programs can lead to break the IPV cycle. But, it is important to remark that other situations must also occur. For example, if the coefficients ${\delta}_{1}$ and ${\delta}_{2}$ are relatively high, the coefficients ${\rho}_{1}$ and ${\rho}_{2}$ have to be close to zero, and ${\beta}_{1}$ high, or $\gamma $ small. In any case, we will have a steep climb on the IPV or the woman’s independence.

## 4. Numerical Simulations

#### 4.1. Model without Alcohol Consumption

**Scenario 1**Couple with a profile of moderated IPV. In this case, we consider a moderated violent man and a non-independent woman. We are assuming that both of the members of the couple have observed violence during their childhood, and they accept machismo as a lifestyle. This determines the non-influenced steady state. The rest of the parameters are such that the man’s self-esteem is medium as well as the self-regulatory level. With respect to the woman, its self-regulatory level is a medium level, and she is experiencing social pressure to continue in the relationship. Let us recall that for this case $h\left(t\right)=0$ because the man does not consume alcohol regularly. In Figure 2, we can observe that the solutions stabilize after an oscillatory behavior which corresponds to case 1, previously discussed. Also, that under these conditions the man’s violence index increases while the woman’s independence index reduces when it is compared with non-influenced steady state. This reduction usually is caused by a decrease in the woman’s self-esteem which could be explained as a consequence of the woman suffering IPV. On the other hand, let us remark that the maximum value of the violence index is $0.6$. If this value is extrapolated to the original scale, it corresponds to aggressions described as shaking and pulling to mistreat to the woman. For the level at which the violence index stabilize, we can say a daily man’s behavior corresponds to violent acts such as rough touching or destruction of personal belongings.

**Scenario 2**Couple with a profile of moderated IPV and empowered woman. For this case, the parameters were set with values similar to the values of Scenario 1. Both of the members of the couple have observed a slightly medium level violence during their childhood, and they have a medium acceptance level of machismo. The man’s self-esteem is medium while that his self-regulatory parameter is lesser than the corresponding woman’s parameter value. The main change is that the woman empowerment is high as a consequence of being involved in empowerment programs. In Figure 3, we can observe that the solutions for both of the indexes increase rapidly. This is because the man responds violently to the increment in the woman’s independence. Let us recall that the man’s self-esteem is related to the man’s need to control the woman. For this case, violence reaches values so high that they escape of any real situation of IPV. Then, we can interpret this situation as a scenario where the woman faces severe violence perpetration (eventually the woman’s death when the violence index exceeds one). A more optimistic interpretation is that the relationship breaks because the woman’s independence increases faster than the IPV. It is important to remark that when the machismo acceptance or the man’s self-esteem is higher, the IPV grows even faster than the woman’s independence. Notice that influenced steady state cannot be reached by the solutions because it is non-stable.

#### 4.2. Model with Alcohol Consumption

**Scenario 3**Couple with a moderated profile of IPV and man’s alcohol consumption. In this case, we take the same parameter set of Scenario 1 and include the effect of the periodic alcohol consumption by the man ($h\left(t\right)\ne 0$), i.e., we are considering $\rho =0.25,\omega =\pi /2,\eta =0.35.$ In Figure 4, we can observe that the model with man’s periodic alcohol consumption follows the qualitative model behavior without alcohol consumption. In this case, the violence index increases and the independence index decreases, but the solution oscillates. Let us observe that, there is a time interval where the oscillations are changing in amplitude (transient behavior) to finally reach a periodic solution. Note the that maximum value of the violence index almost reach one, which corresponds to assassination attempt according to the IPN scale. Finally, the solution for $v\left(t\right)$ oscillates in an approximate range of $[0.5,0.8]$. When these values are interpreted according to IPV scale, they correspond to psychological and physical violent acts ranging from controlling the woman action, destroying personal belongings and death threats among others. We also note that there is the displacement of the steady state point, when it is compared with the case without alcohol consumption. This means that the man’s violence index and the woman’s dependency index increases, confirming that alcohol worsens the IPV perpetration. This agrees with the behavior reported one by the literature [2].

**Scenario 4**Couple with a profile of empowered woman and man’s alcohol consumption. In this case, we take the parameter set of Scenario 2 and include the effect of the periodic alcohol consumption by the man ($h\left(t\right)\ne 0$), i.e., we add the parameters values $\rho =0.15,\omega =\pi /2,\eta =0.35.$ The results of this simulation are an oscillating increment in both of the indexes which is the same qualitative behavior found in the scenario 2, but with oscillations. See Figure 5. This even occurs more rapidly than in the case with a non-alcoholic man showing the emphasizing effect of periodic alcohol consumption.

## 5. Discussion of the Model and Its Application

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Diagram of model (2) showing the variables and their interaction terms.

**Figure 2.**Violent man without alcohol consumption and submissive woman. ${\alpha}_{1}=0.7,{\alpha}_{2}=0.6,{\beta}_{1}=0.6,{\beta}_{2}=0.5,\gamma =0.5,{\delta}_{1}=0.2,{\delta}_{2}=0.25,{\rho}_{1}=0.5,{\rho}_{2}=0.5,\mu =-0.2,(v\left(0\right),i\left(0\right))=(0.4,0.2)$.

**Figure 3.**Violent man without alcohol consumption, empowerment woman. ${\alpha}_{1}=0.5,{\alpha}_{2}=0.6,{\beta}_{1}=0.3,{\beta}_{2}=0.3,\gamma =0.6,{\delta}_{1}=0.25,{\delta}_{2}=0.25,{\rho}_{1}=0.5,{\rho}_{2}=0.5,\mu =0.6$, $\left(v\right(0),i(0\left)\right)=(0.4,0.2)$.

**Figure 4.**Violent man with alcohol consumption, non-independent woman. ${\alpha}_{1}=0.7,{\alpha}_{2}=0.6,{\beta}_{1}=0.6,{\beta}_{2}=0.5,\gamma =0.5,{\delta}_{1}=0.2,{\delta}_{2}=0.25,{\rho}_{1}=0.5,{\rho}_{2}=0.5,\mu =-0.2,\rho =0.25,\omega =\pi /2,\eta =0.35.$

**Figure 5.**Violent man with alcohol consumption, empowered woman. ${\alpha}_{1}=0.5,{\alpha}_{2}=0.6,{\beta}_{1}=0.3,{\beta}_{2}=0.3,\gamma =0.6,{\delta}_{1}=0.25,{\delta}_{2}=0.25,{\rho}_{1}=0.5,{\rho}_{2}=0.5,\mu =0.6,\rho =0.25,\omega =\pi /2,\eta =0.35.$

Variable or Coefficient | Description | Scale | Reference |
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$v\left(t\right)$ | violence index | $[0,30]$ | [20,21] |

$i\left(t\right)$ | independence index | $[0,1]$ | [22] |

${\alpha}_{1}$, ${\alpha}_{2}$ | violence in childhood | $[0,12]$ | [17,23] |

${\beta}_{1}$, ${\beta}_{2}$ | acceptance of machismo | $[0,1]$ | [24,25] |

${\delta}_{1}$, ${\delta}_{2}$ | coping styles | several | [27,28] |

${\rho}_{1}$, ${\rho}_{2}$ | self-regulatory coefficients | $[0,8]$ | [16] |

$\gamma $ | man’s self-esteem | Rosenberg $[0,40]$ | [26] |

$\mu $ | external factors | risk ratio | [15] |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Delgadillo-Aleman, S.; Ku-Carrillo, R.; Perez-Amezcua, B.; Chen-Charpentier, B.
A Mathematical Model for Intimate Partner Violence. *Math. Comput. Appl.* **2019**, *24*, 29.
https://doi.org/10.3390/mca24010029

**AMA Style**

Delgadillo-Aleman S, Ku-Carrillo R, Perez-Amezcua B, Chen-Charpentier B.
A Mathematical Model for Intimate Partner Violence. *Mathematical and Computational Applications*. 2019; 24(1):29.
https://doi.org/10.3390/mca24010029

**Chicago/Turabian Style**

Delgadillo-Aleman, Sandra, Roberto Ku-Carrillo, Brenda Perez-Amezcua, and Benito Chen-Charpentier.
2019. "A Mathematical Model for Intimate Partner Violence" *Mathematical and Computational Applications* 24, no. 1: 29.
https://doi.org/10.3390/mca24010029