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# Mathematically Modeling the Dynamics of COVID-19 and Domestic Violence During Lockdown

The lockdown strategy that intended to slow COVID-19’s initial transmission caused many disruptions at home, especially for people who are at risk of domestic violence (DV). Multiple international reports indicate an increase in DV since the pandemic’s onset in late 2019 [3]. According to the exposure reduction theory for DV, extended periods of confinement at home provide ample opportunities for abusive behavior towards vulnerable individuals. Possible aggressions include verbal/emotional, physical, sexual, digital, and financial abuse [1, 2].

In addition, the presence of other lockdown-related stressors compounds the health threat of DV, which is already prevalent in the U.S. [5]. For instance, an increase in DV exacerbates pandemic-induced family conflicts, economic distress, and tension among family members; inadequate support for victims of family violence during the pandemic corelates with surges in China as well [6]. We can best represent these other stressors—which can manifest as alcohol consumption at home; limited access to social support and help due to confinement; and fears related to health, finances, and loss of schooling—through a deterministic model of DV growth during lockdown. Moreover, the disruptive nature of the pandemic and strict lockdowns created an opportunity for a synchronized DV cycle for the population of susceptible homes that would otherwise remain in different stages of violence escalation [4].

Here, we utilize mathematical modeling to better understand the lockdown dynamics of COVID-19 and DV; specifically, we use deterministic equations of growth to model our assumption that DV is increasing. Methods that incorporate data from DV police reports and surveys are limited because DV is highly underreported and depends on the victim’s perception of violence and capacity to inform others. This scenario motivates the development of an expanded compartmental epidemiological model that describes COVID-19 and DV interactions across multiple lockdown periods.

Figure 1. A COVID-19 compartmental model with susceptible $$(S)$$, exposed $$(E)$$, infectious $$(I)$$, quarantined $$(Q)$$, hospitalized $$(H)$$, deceased $$(D)$$, and recovered $$(R)$$ compartments. A lockdown-induced confinement compartment (blue background) includes COVID-19 dynamics (blue circles) and corresponding domestic violence victims (red circles, which are denoted by parameters $$V_S$$, $$V_E$$, $$V_I$$, $$V_D$$, and $$V_R$$). Figure courtesy of [2].

### The Mathematical Model

We derive our COVID-19 lockdown and DV model from the classical Kermack-McKendrick compartmental model with two states: lockdown-induced confinement and lockdown lift (see Figure 1). The COVID-19 contagion and DV both continue during home confinement; for the sake of our model, we assume a strict lockdown wherein 80 percent of the population has only limited interactions at home and the remaining 20 percent continues to interact normally. Figure 2 defines the parameters that we incorporated into our simulations.

Figure 2. Parameters in our COVID-19 lockdown and domestic violence model. Figure courtesy of the authors.

### Results

We performed simulations for three different lockdown scenarios to compare the effects of long-term confinement versus multiple shorter lockdown intervals that are separated by periods of lifted restrictions. Our three simulation scenarios are as follows:

• Scenario 1: One 60-day lockdown
• Scenario 2: Two 20-day lockdowns separated by a 20-day lift
• Scenario 3: Three 12-day lockdowns separated by two 12-day lifts.

We employed various COVID-19 parameters from the literature and assumed that the DV index parameter $$\alpha^V$$ increases five percent over a 60-day period of confinement [1]. Figure 3 presents the resulting COVID-19 and DV curves for the three lockdown simulations in a fully susceptible population of 3.3 million people. Although Scenario 3 produced 40,000 more COVID-19 infections than Scenario 1, it also yielded 105,000 fewer DV victims.

Furthermore, we obtained the basic reproduction number $$(R_0)$$ for our model and analyzed it for sensitivity of the two main parameters: the COVID-19 transmission rate $$\beta^C$$ and DV violence index $$\alpha^V$$. As expected, the result in Figure 4 corroborates $$R_0$$'s sensitivity to the COVID-19 transmission rate; changes in the DV violence index have minimal effect on the obtained confinement $$R_0$$.

Figure 3. Dynamics for a scenario with three separate lockdowns at time intervals that are measured as days since the lockdown’s onset: 120-132 days, 144-156 days, and 168-180 days. 3a. Graph of various domestic violence (DV) variables: Non-susceptible to DV $$(S1)$$, susceptible to DV $$(S2)$$, abusers $$(A)$$, victims $$(V)$$, and removed from home $$(R_C)$$. 3b. Graph of COVID-19 variables: Susceptible $$(S)$$, exposed $$(E)$$, infectious $$(I)$$, quarantined $$(Q)$$, hospitalized $$(H)$$, recovered $$(R)$$, and deceased $$(D)$$. The infectious category is subdivided into asymptomatic $$(I^A)$$ and symptomatic $$(I^S)$$. Figure courtesy of [2].

### Conclusions

As we learn more about the mechanisms behind DV’s connection to lockdown, many questions remain. Further exploring DV will improve mathematical modeling’s contributions to this costly social problem that has severe health implications beyond the COVID-19 pandemic. In addition to dangerously increasing the incidence of DV, preliminary studies indicate that lockdown measures have also escalated other problems that pertain to mental health. Our results suggest that shorter lockdown periods may help to alleviate DV, although they do not optimally limit the transmission of COVID-19.

Figure 4. The basic reproduction number $$(R_0)$$ as a function of domestic violence (DV) index $$\alpha^V$$ and transmission rates $$\beta^C$$. Figure courtesy of [2].
The development of programs that can facilitate communication with victims and provide support from a distance might aid in mitigating the reduction of access to assistance during lockdown. Biotechnology and wearable devices can also alert outsiders of a greater risk of violence, particularly physical aggression. Finally, continuous education campaigns that use new distance-learning skills may improve overall community education on healthy relationships, conflict management, and ways to recognize and report DV. With new surges in COVID-19 evolving yet again, both governmental and non-governmental organizations—as well as agencies and service sectors—must be aware of the needs of DV victims and provide them with appropriate and immediate assistance during the pandemic.

Carmen Caiseda presented this research during a minisymposium presentation at the 2021 SIAM Annual Meeting, which took place virtually in July 2021.

References
[1] Hsu, L.-C., & Henke, A. (2021). COVID-19, staying at home, and domestic violence. Rev. Econ. Household, 19, 145-155.
[2] Ohajunwa, C., Caiseda, C., & Seshaiyer, P. (2022). Computational modeling, analysis and simulation for lockdown dynamics of COVID-19 and domestic violence. Electron. Res. Arch., 30(7), 2446-2464.
[3] Piquero, A.R., Jennings, W.G., Jemison, E., Kaukinen, C., & Knaul, F.M. (2021). Domestic violence during the COVID-19 pandemic – Evidence from a systematic review and meta-analysis. J. Crim. Justice, 74, 101806.
[4] Walker, L.E. (1979). The battered woman. New York, NY: Harper Perennial.
[5] Walters, M.L., Chen, J., & Breiding, M.J. (2013). The National Intimate Partner and Sexual Violence Survey (NISVS): 2010 findings on victimization by sexual orientation. Atlanta, GA: National Center for Injury Prevention and Control, Centers for Disease Control and Prevention. Retrieved from https://www.cdc.gov/violenceprevention/pdf/nisvs_sofindings.pdf
[6] Zhang, H. (2022). The influence of the ongoing COVID-19 pandemic on family violence in China. J. Fam. Violence, 37(5), 733-743.

Comfort Ohajunwa is an undergraduate student at the College of William & Mary who is interested in mathematics and computer science. She has authored several journal papers on mathematical modeling research that examine the influence of human behaviors on COVID-19’s spread. Ohajunwa also participated in the 2021 Research Science Institute at the Massachusetts Institute of Technology, where she conducted research about the Boolean maximum 2-satisfiability problem. Carmen Caiseda is a professor and coordinator of the mathematics group at Inter American University of Puerto Rico (IAUPR) – Bayamon. She is co-principal investigator of the Data Science at IAUPR project, which is building a data science community of practice that impacts faculty, students, and professionals. As an undergraduate research mentor, Caiseda engages students in STEM via mathematical models of real-world challenges with sociocultural contexts. Padmanabhan Seshaiyer is a professor of mathematical sciences at George Mason University who previously served as chair of the SIAM Diversity Advisory Committee. He works in the broad area of computational mathematics, data science, biomechanics, design and systems thinking, and STEM education. Seshaiyer is also chair of the U.S. National Academies Commission on Mathematics Instruction and Associate Director for Applied Mathematics of the Math Alliance.