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\begin{document}

\title{\bf Supplemental Information}

\section{The Sisk Model}
In this section, we provide the details of the epidemiological model of violence proposed by Sisk and collaborators. Their model divides individuals into several compartments: Susceptible $(S)$, three Exposed states ($E_1$, $E_2$, $E_3$), and Infected $(V)$. Note that ``infected" is equivalent to `violent' in this context.  The model is described by the following system of ordinary differential equations:
 \begin{subequations} \label{eq:viral}
\begin{empheq}[left=\empheqlbrace]{align}
S' &= - (\mu_1  + \mu_2 + \mu_3) S V, \label{eq:viralS}\\
E_1' &= \mu_1 SV - \omega_1 E_1 V - \omega_2 E_1 V + \beta_1 E_2, \label{eq:viralE1}\\
E_2' &= \mu_2 SV - \omega_3 E_2V -  \beta_1 E_2 + \omega_1 E_1 V + \beta_2 E_3, \label{eq:viralE2}\\
E_3' &= \mu_3 SV - \omega_4 E_3 V - \beta_2 E_3 +\omega_2E_1V  + \omega_3 E_2 V + \beta_3V, \label{eq:viralE3}\\
V' &= \omega_4 E_3V -\beta_3 V, \label{eq:viralV}
\end{empheq}
\end{subequations}
%
The top component of 
Figure \ref{fig:sirtokin_infec} illustrates this compartment model and Table \ref{tab:violence_model} describes the variables and parameters in this system. 
\begin{table}[ht]
\centering
\begin{tabular}{|p{2cm}|p{5cm}|p{6.5cm}|}
\hline
\textbf{Symbol} & \textbf{Description} & \textbf{Role in the Model} \\ \hline
$S$ & Susceptible & Individuals who have not been exposed to violence but can become exposed through contact with violent individuals. \\ \hline
$E_1, E_2, E_3$ & Exposed states & Individuals at progressively higher levels of exposure to violence. $E_1$ = low exposure, $E_2$ = moderate exposure, $E_3$ = high exposure. \\ \hline
$V$ & Infected / Violent & Individuals actively exhibiting violent behavior. \\ \hline
$\mu_1, \mu_2, \mu_3$ & Susceptible-to-exposed transition rates & Rate at which susceptible individuals move to each exposed state through contact with violent individuals. \\ \hline
$\omega_1, \omega_2, \omega_3, \omega_4$ & Exposed-to-exposed or exposed-to-violent rates & Rate at which exposed individuals progress to higher exposure or violent state through interactions. \\ \hline
$\beta_1, \beta_2$ & Backward transition rates & Rate at which individuals move from a higher exposed state back to a lower exposed state. \\ \hline
$\beta_3$ & Exit/recovery rate & Rate at which violent individuals leave the violent state, representing recovery or disengagement. \\ \hline
\end{tabular}
\caption{Description of the compartments and parameters of the Sisk et al. epidemiological model of violence.}\label{tab:violence_model}
\end{table}

\begin{figure}[h]
    \centering
    \includegraphics[width=\linewidth]{sirkinetic2.pdf}
    \caption{}
    \label{fig:sirtokin_infec}
\end{figure}

\section{The Progressive-Risk Model}
In this section, we discuss the Progressive-Risk Model.  Let $S(t,u)$ denote the density of susceptible individuals with exposure level $u \in [0,1]$ at time $t$, and let $V(t)$ denote the total violent population at time $t$. The dynamics of PRM are governed by the following equations:


\begin{align*}%\label{eq:sirtokin_infec}
\frac{\partial S(t,u)}{\partial t} &= \mu \int_0^1 T_{\mu}(x,u)S(t,x)V dx - \mu S(t,u)V+ \beta \int_0^1 T_{\beta}(x,u)S(t,x) dx - \beta S(t,u)\\
&\quad-\omega T_{\omega}(u)S(t,u)\int_0^1 S(t,x)dx+  \frac{\alpha}{\epsilon}T_{\alpha}(u)V\\
\frac{dV(t)}{dt} &= -\alpha V(t) + \omega \int_0^1\int_0^1 T_{\omega}(u)S(t,u) S(t,x)dxdu
\end{align*}

with transition functions defined below in Table \ref{tab:transition_kernels}. 
\begin{table}[H]
\centering
\begin{tabular}{|p{4cm}|p{7cm}|}
\hline
\textbf{Transition Functions} & \textbf{Definition / Meaning} \\ \hline
$T_{\mu}(x,u)=\delta_{x + \mu^*(1-x) - u}$ & Moves an individual from exposure $x$ to $u$ after interacting with violent individuals. \\ \hline
$T_{\beta}(x,u)=\delta_{(1-\beta^*)x - u}$ & Moves an individual from exposure $x$ to $u$ when exposure decreases. \\ \hline
$T_{\omega}(u)=\chi_{[1-\epsilon,1]}(u)$ & Susceptibles with a high propensity towards violence transition to the violent state at rate $\omega$. \\ \hline
$T_{\alpha}(u)=\chi_{[1-\epsilon,1]}(u)$ & Movement from the violent state to a highly exposed state. \\ \hline
\end{tabular}
\caption{Transition functions used in the kinetic model.}
\label{tab:transition_kernels}
\end{table}
We describe the model terms below, followed by a summary of the variables and parameters, with their descriptions in Table \ref{tab:kinetic_parameters}.  

\begin{itemize}
    \item \textit{Exposure from violent interactions:} The term $\mu\int_0^1  T_{\mu}(x,u) S(t,x) V dx - \mu S(t,u)V$ models the interaction between susceptibles (with propensity towards violence $x$) with violent individuals.  This interaction increases the propensity towards violence to $u$.  The subtraction term ensures mass conservation.
    \item \textit{Recovery:} The term $\beta \int_0^1 T_{\beta}(x,u) S(t,x) dx - \beta S(t,u)$ captures the impact of interventions or societal influences that reduce violent behavior. It models how individuals can gradually lower their exposure to violence over time, reflecting disengagement from violent activity.

    \item \textit{Interactions among susceptibles:} The term $-\omega T_\omega(u) S(t,u) \int_0^1 S(t,x) dx$ models how highly exposed susceptibles influence others near maximal exposure, increasing their risk of becoming violent.
    \item \textit{Recruitment from violent population:} The term $ \frac{\alpha}{\epsilon} T_\alpha(u) V$ models the recruitment of highly exposed susceptibles by violent individuals, capturing how interactions with violent actors can push these individuals into the violent population.

    \item \textit{Violent population dynamics:} $dV/dt = -\alpha V(t) + \omega \int_0^1\int_0^1 T_\omega(u) S(t,u) S(t,x) dx du$. The first term represents recovery or exit from violence, and the second term accounts for new violent individuals arising from interactions among highly exposed susceptibles.
\end{itemize}


\begin{table}[H]
\centering
\begin{tabular}{|p{2.5cm}|p{10cm}|}
\hline
\textbf{Parameter} & \textbf{Interpretation} \\ \hline
%$S(t,u)$ & Density of susceptible individuals with exposure level $u \in [0,1]$ at time $t$. \\ \hline
%$V(t)$ & Total violent population at time $t$. \\ \hline
$\mu$ & Rate at which susceptibles increase exposure due to interactions with violent individuals. \\ \hline
$\beta$ & Rate of backward progression; describes reduction of exposure over time (disengagement). \\ \hline
$\omega$ & Rate of interactions among susceptibles that can push highly exposed individuals toward violence. \\ \hline
$\alpha$ & Rate of exit from violent population or recruitment of highly exposed susceptibles by violent individuals. \\ \hline
$\epsilon$ & Small parameter defining the interval of high exposure near 1 for the recruitment and interaction functions. \\ \hline
%$T_\mu(x,u)$ & Function governing upward transition in exposure due to violent interactions. \\ \hline
%$T_\beta(x,u)$ & Function governing backward transition in exposure (decreasing exposure). \\ \hline
%$T_\omega(u)$ & Function restricting susceptible-susceptible interactions to individuals with high exposure ($u \approx 1$). \\ \hline
%$T_\alpha(u)$ & Function restricting recruitment from violent individuals to highly exposed susceptibles ($u \approx 1$). \\ \hline
\end{tabular}
\caption{Parameters found in PRM and their interpretations.}\label{tab:kinetic_parameters}
\end{table}
\begin{comment}
\begin{table}[H]
\centering

\begin{tabular}{|p{4cm}|p{5cm}|p{5cm}|}
\hline
\textbf{Parameter / Term} & \textbf{Kinetic Model} & \textbf{Sisk et al. Model} \\ \hline
Susceptible population & $S(t,u)$ : density across continuous exposure $u \in [0,1]$ & $S$ : single compartment \\ \hline
Violent population & $V(t)$ : total violent individuals & $V$ : total violent individuals \\ \hline
Exposure increase & $\mu$ with kernel $T_\mu(x,u)$ : transition from exposure $x \to u$ via violent interactions & $\mu_i$ : transition rates from $S$ to discrete exposed compartments $E_1, E_2, E_3$ \\ \hline
Recovery & $\beta$ with kernel $T_\beta(x,u)$ : decrease in exposure & $\beta_i$ : transitions from higher to lower exposed states; $\beta_3$ exit from $V$ \\ \hline
Interactions among susceptibles & $\omega$ with kernel $T_\omega(u)$ : interactions among high-exposure susceptibles moving some toward $V$ & Not modeled; interactions only via contact with $V$ \\ \hline
Recruitment from violent population & $\alpha$ with kernel $T_\alpha(u)$ : highly exposed susceptibles recruited into $V$ & Not explicitly separate; captured by $\omega_i$ terms between $E_i$ and $V$ \\ \hline
High-exposure threshold & $\epsilon$ : defines near-maximal exposure for $T_\omega$ and $T_\alpha$ & Not needed; discrete compartments implicitly represent levels of exposure \\ \hline
\end{tabular}
\caption{Comparison of parameters in the kinetic model and the Sisk et al. (2022) compartmental model.}\label{tab:parameter_comparison}
\end{table}
\end{comment}

\subsection{Steady-state Analysis Via Violence Persistence Number}



\subsubsection{The Corresponding VPN for the Sisk Model}\label{sec:apprepsir}


We illustrate how to derive the correspoinding VRN of the Sisk model given by \eqref{eq:viral}. We begin by identifying the infected compartments in the system: $E_1$, $E_2$, $E_3$, and $I$. These correspond to stages in the progression of infection, with susceptible $S$ becoming exposed through three distinct channels parameterized by $\mu_1$, $\mu_2$, and $\mu_3$, and transitioning through the exposure compartments before eventually becoming fully infectious in the $I$ class. The equations governing this system are:

To derive $\mathcal{V}_0$, we express the system in the form $\dot{X} = \mathcal{F}(X) - \mathcal{V}(X)$, where $\mathcal{F}$ contains the terms corresponding to new infections, and $\mathcal{V}$ captures all other transitions (progression between compartments, recoveries, etc.). Focusing on the vector $X = [E_1, E_2, E_3, I]^T$, we identify the new infections entering each compartment at the disease-free equilibrium (DFE), where $S \approx 1$ and $E_1 = E_2 = E_3 = I = 0$. The new infection vector is thus:

$$
\mathcal{F}(X) =
\begin{bmatrix}
\mu_1 I \\
\mu_2 I \\
\mu_3 I \\
0
\end{bmatrix},
$$

reflecting that new exposures arise from interactions between susceptible individuals and the infectious class.

In contrast, the transition vector $\mathcal{V}(X)$ accounts for outflows and internal transfers in the infected subsystem. Linearizing around the DFE (where $I = 0$), and neglecting nonlinear terms involving $I$, we obtain the following simplified form:

$$
\mathcal{V}(X) =
\begin{bmatrix}
\beta_1 E_2 \\
\beta_1 E_2 + \beta_2 E_3 \\
\beta_2 E_3 + \beta_3 I \\
\beta_3 I
\end{bmatrix}.
$$

Next, we compute the Jacobians of $\mathcal{F}$ and $\mathcal{V}$ with respect to the infected state vector $X$. The Jacobian $F = D_X \mathcal{F}$ at the DFE is given by:

$$
F =
\begin{bmatrix}
0 & 0 & 0 & \mu_1 \\
0 & 0 & 0 & \mu_2 \\
0 & 0 & 0 & \mu_3 \\
0 & 0 & 0 & 0
\end{bmatrix},
$$

and the Jacobian $V = D_X \mathcal{V}$ at the DFE is:

$$
V =
\begin{bmatrix}
0 & -\beta_1 & 0 & 0 \\
0 & \beta_1 & -\beta_2 & 0 \\
0 & 0 & \beta_2 & -\beta_3 \\
0 & 0 & 0 & \beta_3
\end{bmatrix}.
$$

The next-generation matrix is given by $K = F V^{-1}$, and the VRN $\mathcal{V}_0$ is defined as the spectral radius (dominant eigenvalue) of $K$. Due to the structure of $F$ and $V$, one can exploit the fact that each pathway $\mu_i$ leads independently to infectiousness through a chain of transitions governed by the rates $\beta_i$. This cascade allows us to interpret $\mathcal{V}_0$ as the sum of contributions from each exposure route, resulting in:

$$
\mathcal{V}_0 = \frac{\mu_1}{\beta_1} + \frac{\mu_2}{\beta_2} + \frac{\mu_3}{\beta_3}.
$$

This expression encapsulates the average number of secondary infections generated via each pathway $E_1 \to I$, $E_2 \to I$, and $E_3 \to I$, assuming that each infected individual proceeds independently through a specific route with probability proportional to $\mu_i$. The result provides a clear threshold condition: if $\mathcal{V}_0 < 1$, the infection will die out, while if $\mathcal{V}_0 > 1$, an outbreak may occur.

\subsubsection{The Violence Persistence Number for PRM}\label{app:rep_meso} 
To compute the VPN $\mathcal{V}_0$ for PRM we follow the classical next-generation method by decomposing the linearized dynamics around the disease-free equilibrium into two parts: an infection operator $\mathcal{F}$, representing the generation of new infections, and a transition operator $\mathcal{V}$, encoding the movement and recovery of individuals within the system. The disease-free equilibrium is given by $S(u) = \delta(u)$ and $V = 0$, and we perturb around this equilibrium using $S(t,u) = \delta(u) + g(u)$ and $V(t) = v$, retaining only first-order terms in the perturbation variables $g$ and $v$.

The infection operator $\mathcal{F}$ is obtained from the terms in the PDE that create new violent actors or lead to higher propensities toward violence. Linearizing these yields

$$
\mathcal{F}[g, v](u) = 
\begin{bmatrix}
v \left[\mu(\delta(u - \mu^*) - \delta(u)) + \alpha \frac{1}{\epsilon} \chi_{[1 - \epsilon, 1]}(u) \right] \\
0
\end{bmatrix},
$$

as the generation of new infectious mass depends only on the viral load $v$. The linearized transition operator is

$$
\mathcal{V}[g, v](u) = -\begin{bmatrix}
\beta \left( \frac{1}{1 - \beta^*} g\left( \frac{u}{1 - \beta^*} \right) \chi_{[0, 1 - \beta^*]}(u) - g(u) \right) + \omega \chi_{[1 - \epsilon, 1]}(u) g(u) \\
- \alpha v + \omega \int_{1 - \epsilon}^1 g(x)\, dx
\end{bmatrix}.
$$

To invert $\mathcal{V}$, we solve $\mathcal{V}[g, v] = \phi = (\phi_1, \phi_2)$. The second component gives

$$
v = \frac{\omega}{\alpha} \int_{1 - \epsilon}^1 g(x)\, dx - \frac{1}{\alpha} \phi_2.
$$

The first component yields the piecewise definition:

$$
g(u) = 
\begin{cases}
\frac{1}{1 - \beta^*} g\left( \frac{u}{1 - \beta^*} \right) + \frac{1}{\beta} \phi_1(u), & u \in [0, 1 - \beta^*), \vspace{4pt}\\
\frac{1}{\beta} \phi_1(u), & u \in [1 - \beta^*, 1 - \epsilon), \vspace{4pt}\\
\frac{1}{\beta + \omega} \phi_1(u), & u \in [1 - \epsilon, 1].
\end{cases}
$$
This provides the inverse $\mathcal{V}^{-1}$, which we now apply in $\mathcal{K} = \mathcal{F} \mathcal{V}^{-1}$. Since $\mathcal{F}$ depends only on $v$, we substitute the above expression to obtain:
$$
\mathcal{K}[\phi](u) =
\left( \frac{\omega}{\alpha} \int_{1 - \epsilon}^1 g(x)\, dx - \frac{1}{\alpha} \phi_2 \right)
\left[ \mu(\delta(u - \mu^*) - \delta(u)) + \alpha \frac{1}{\epsilon} \chi_{[1 - \epsilon, 1]}(u) \right].
$$

This operator has the form $\mathcal{K}[\phi] = \langle \phi, \eta \rangle \psi(u)$. We notice that it is rank-one, with output aligned along a fixed function  $ \psi(u) = \mu(\delta(u - \mu^*) - \delta(u)) + \alpha \frac{1}{\epsilon} \chi_{[1 - \epsilon, 1]}(u) $, and scaling factor determined by the inner product with a dual function $\eta$. To compute its spectral radius, we evaluate $\mathcal{K}[\psi] = \mathcal{V}_0 \psi$, using $\phi = \psi$ and $\phi_2 = 0$. Then $g^* = \mathcal{V}^{-1}_1[\psi]$ is given over the interval $[1 - \epsilon, 1]$ by:
$$
g^*(x) = \frac{1}{\beta + \omega} \psi(x) = \frac{1}{\beta + \omega} \left[ \mu(\delta(x - \mu^*) - \delta(x)) + \alpha \frac{1}{\epsilon} \right].
$$

Thus, we compute:
$$
\int_{1 - \epsilon}^1 g^*(x)\, dx 
= \frac{1}{\beta + \omega} \left[ \mu \int_{1 - \epsilon}^1 (\delta(x - \mu^*) - \delta(x)) dx + \alpha \frac{1}{\epsilon} \cdot \epsilon \right].
$$

We have:
$$
\int_{1 - \epsilon}^1 \delta(x - \mu^*) dx = \chi_{[1-\epsilon,1](u)}, \quad \int_{1 - \epsilon}^1 \delta(x) dx = 0,
$$

and thus the delta contribution is $\mu$. Therefore:

$$
\int_{1 - \epsilon}^1 g^*(x) dx = \frac{1}{\beta + \omega} \left( \mu \chi_{[1-\epsilon,1](u)}+ \alpha \right).
$$

Substituting into the expression for the spectral radius:

$$
\mathcal{V}_0 = \frac{\omega}{\alpha} \int_{1 - \epsilon}^1 g^*(x) dx = \frac{\omega}{\alpha} \cdot \frac{\mu \chi_{[1-\epsilon,1](u)} + \alpha}{\beta + \omega}.
$$




\section{The Mesoscopic Violence–Response Model}
In this section we provide the details of MVPM.  
As before, define $S(t, u)$ as the general population with $u$ representing an individual's tendency toward violence.  In this model, we let $V(t, v)$ represent the violent subgroup, where $v$ indicates the level of violence.  We define a new variable $R(t,w)$ to denote the response population, where $w$ reflects the arrest rate. This fully mesoscopic approach allows us to capture more detailed feedback mechanisms, for example, how the level of violence affects response strategies and vice versa. It also highlights how variation within each group contributes to complex social dynamics. The kinetic system we study is given by:

%\begin{align*}
%\frac{\partial S(t,u)}{\partial t} =& \int_0^1 \mu T_{\mu}(x,u)S(t,x)V dx - \mu S(t,u)V+ \int_0^1 \beta T_{\beta}(x,u)S(t,x) dx - \beta S(t,u)\\
%&-\omega T_{\omega}(u)S(t,u)\int_0^1 S(t,x)dx+ \frac{\alpha }{\epsilon}T_{\alpha}(u)V\\
%\frac{dV(t)}{dt} &= -\alpha V(t) + \omega \int_0^1\int_0^1 T_{\omega}(u)S(t,u) S(t,x)dxdu
%\end{align*}


\begin{subequations}\label{eq:Fullkinetic}
\begin{align}
    \frac{\partial S(t,u)}{\partial t} &=  \mu\int_0^1\int_0^1 T_\mu(x,u) S(t,x)V(t,y) dxdy -\mu S(t,u)\int_0^1V(t,y)dy  \notag\\
    &\quad + \beta\int_0^1 T_{\beta}(x,y) S(t,x)dx -\beta S(t,u) -\omega T_\omega(u) S(t,u)\int_0^1 S(t,x)dx\notag\\
    &\quad  + \frac{\alpha }{\epsilon}\int_0^1\int_0^1 T_\alpha(u)V(t,y)R(t,z)dydz , \label{eq:fullyS}\\
    \frac{\partial V(t,v)}{\partial t} &= -\alpha \int_0^1\int_0^1 V(t,y)R(t,z)dydz + (1-\alpha)
    \int_0^1\int_0^1 T_R(x,y,u) V(t,y)R(t,z)dydz \notag\\
    &\quad + \gamma \int_0^1\int_0^1 T_{\alpha}(u) S(t,u)S(t,x)dxdu. \label{eq:fullyV}
\end{align}
\end{subequations}

The evolution of $S$ is driven by several key processes. We only discuss the new processes not included in PRM. 
\begin{comment}
    
The first term in equation \eqref{eq:fullyS},

$$
\mu\int_0^1\int_0^1 T_\mu(x,u) S(t,x)V(t,y) dxdy 
$$
represents how individuals in the general population become more prone to violence through interactions with violent individuals. The function $\delta_{x + \mu^*(1-x)y - u}$ ensures that when a person with baseline predisposition $x$ interacts with someone at violence level $y$, their new state becomes $u = x + \mu^*(1-x)y$. The parameter $\mu \in (0,1)$ determines how strongly the violent individual influences the other person.
To balance this increase, the model includes a loss term,
$$
-\mu S(t,u)\int_0^1V(t,y)dy,
$$
which subtracts individuals from state $u$ as they change due to these interactions, ensuring total population is conserved. The next pair of terms,

$$
\beta\int_0^1 T_{\beta}(x,y) S(t,x)dx -\beta S(t,u) ,
$$

models the natural decrease in an individual's tendency toward violence. In this process, someone at state $x$ moves to a lower state $u=(1-\beta^*)x$, representing a gradual reduction in aggression over time. This could be due to factors like fatigue, fear, or social pressure. The second term subtracts individuals from state $u$ to conserve the total population as these transitions occur.
\end{comment}
The first new terms model institutional interventions.  The term
$$
\frac{\alpha }{\epsilon}\int_0^1\int_0^1 T_\alpha(u)V(t,y)R(t,z)dydz ,
$$

describes how the response team redirects violent individuals back into the general population. The indicator function $\chi_{[1-\epsilon, 1]}(u)$ limits this effect to people whose violence level is near the highest point, reflecting focused interventions on the most violent individuals. The integral shows the combined effect of violence and response.  The following term,

$$
-\omega T_\omega(u) S(t,u)\int_0^1 S(t,x)dx,
$$

models how individuals with high violence predisposition are recruited into the violent group at a rate $\omega$. This recruitment depends on the overall size of the general population, representing social tension.


Equation \eqref{eq:fullyV} describes how the violent subgroup $V$ changes over time.  The first term,

$$
- \alpha \int_0^1 \int_0^1 V(t,y) R(t,z) \, dy \, dz,
$$

represents the reduction of violence through the response. A fixed portion $\alpha$ of violent activity is reduced, summarizing both deterrence and arrests. The next term,

$$
(1 - \alpha) \int_0^1 \int_0^1 T_R(x, y, u) \, V(t,y) R(t,z) \, dy \, dz,
$$

captures more complex outcomes where violence continues or even increases despite the response. The function $T_R$ models effects that response team interactions have on the propensity towards violence.  The final term,

$$
\gamma \int_0^1\int_0^1 T_{\alpha}(u) S(t,u)S(t,x)dxdu.,
$$

describes how people from the general population become violent. The indicator limits this to those already close to becoming violent. This process shows how violence can spread socially. People with a high tendency toward violence may be pushed over the edge when around others with similar tendencies.

\begin{table}[H]
\centering
\begin{tabular}{|p{6.5cm}|p{8cm}|}
\hline
\textbf{Term} & \textbf{Interpretation} \\ \hline
%$\small  
%\mu\int\int T_\mu(x,u) S(t,x)V(t,y)\,dxdy
$
%& Increase in exposure of susceptibles due to contact with violent individuals. \\ \hline

%$\small\displaystyle 
%-\mu S(t,u)\!\int_0^1\!V(t,y)\,dy
%$
%& Loss from state $u$ due to upward transitions induced by violent contact (mass balance). \\ \hline

%$\small\displaystyle 
%\beta\!\int_0^1\! T_\beta(x,u) S(t,x)\,dx
%$
%& Exposure reduction among susceptibles. \\ \hline

%$\small\displaystyle 
%-\beta S(t,u)
%$
%& Loss from state $u$ due to downward transitions (mass balance). \\ \hline
$\small\displaystyle 
-\omega T_\omega(u)\,S(t,u)\!\int_0^1\!S(t,x)\,dx
$
& Recruitment of highly exposed susceptibles into violent activity through susceptible–susceptible interactions. \\ \hline

$\small\displaystyle 
\frac{\alpha}{\epsilon}\!\int\!\!\int T_\alpha(u) V(t,y)P(t,z)\,dydz
$
& Response team diverts violent individuals back into the susceptible population, concentrated near the highest exposure levels. \\ \hline

$\small\displaystyle 
-\alpha\!\int\!\!\int V(t,y)P(t,z)\,dydz
$
& Reduction of violent behavior due to response strategies. \\ \hline

$\small\displaystyle 
\hspace{-3pt}(1-\alpha)\!\int\!\!\int T_R(x,y,u) V(t,y)P(t,z)\,dydz
$
& Response strategies that lead to persistence or escalation of violence (e.g., retaliation effects). \\ \hline

$\small\displaystyle 
\gamma\!\int\!\!\int T_\alpha(u) S(t,u)S(t,x)\,dxdu$
& Emergence of new violent individuals due to high‐exposure interactions among susceptibles. \\ \hline

\end{tabular}
\caption{Interpretation of key terms in MVRM.}
\label{tab:term_explanations}
\end{table}

\subsection{Response Team Functions}
In this section we discuss the details of the response function, which we take of the form $T_R(x,y,u)=\delta_{\phi(x,y)-u}.$ This function shows how interactions between violent individuals and police affect the level of violence.  The first function we consider is the {\bf escalation}, which has the form: 
\begin{equation*}%\label{eq:transright}
    \phi(x,y) = x - \theta xy,
\end{equation*}
which models a steady increase in violence after interacting with the response. For a violent person at level $x$ interacting with a response team member at intensity $y$, their violence level changes to $u = x + \theta(1 - x)y$. This means the response raises violence more when the person’s initial level $x$ is low. As $x$ gets closer to 1, the increase is reduced, modeling a saturation effect where very violent individuals become only slightly more violent, even with higher response levels.

The second function is the {\bf de-escalation function}, given by  
\begin{equation*}  
    \phi(x, y) = \delta_{x - \theta (1-x) y}.  
\end{equation*}  
where $\theta$ is a parameter that governs the strength of the de-escalation. This function represents a monotonic response. Individuals are shifted downward from their initial level $x$ by an amount proportional to both $x$ and the response intensity $y$. Thus, higher values of $y$ produce stronger reductions, and individuals with lower violence levels experience larger decreases. This reflects a uniformly suppressive effect of the response team.  

The third  function is the {\bf threshold function}, given by the function
\begin{equation*}%\label{eq:transtoward}
    \phi(x, y) =
\begin{cases}
x - \theta(x - a)(1-y), & \text{if } x > a, \\
x + \theta(a - x)y, & \text{if } x \leq a,
\end{cases}
\end{equation*}

where $a$ is a threshold level that changes the behavior of the function, and $\theta$ is a parameter that governs the severity of the transition. This function represents a non-monotonic response. Individuals below the threshold experience escalation shifting toward $a$. Those above the threshold are moderated, moving downward toward $a$. This reflects a context-dependent response effect: violent individuals are partially suppressed, while less violent individuals are provoked to incrase their propensity towards violence.



% We test two different transition functions under two policing scenarios: one with moderate policing centered at $p = 0.5$, and another with aggressive policing centered at $p = 0.9$. In both cases, the initial distributions of violence $V(t,v)$ and the general population $S(t,u)$ start centered at 0.5. All distributions are normal.

For the simulations in Figure 7 we used a normalized gaussian of the form

\begin{equation*}%\label{eq:PoliceDistribution}
    R(t,z) = 0.01\frac{\exp^{-(z-0.5)^2/0.1}}{\int_0^1 \exp^{-(x-0.5)^2/0.1}dx}.
\end{equation*}

We fix the parameter $\mu = 0.8$, $\mu^* = 0.2$, $\beta = 0.01$, $\beta^* = 0.1$, $\epsilon = 0.1$, $\omega = 0.07$, $\alpha = 0.01$, $a=0.6$, $\gamma = 0.4$ . The initial states for $V$ and $S$ are 
\begin{equation*}
    V(0,v) = 0.04\frac{\exp^{-(v-0.5)^2/0.01}}{\int_0^1 \exp^{-(x-0.5)^2/0.01}dx}\quad\text{and} \quad S(0,u) = 0.95\frac{\exp^{-(u)^2/0.5}}{\int_0^1 \exp^{-(x)^2/0.5}dx}.
\end{equation*}


\end{document}