A Crime/S.I.R. optimal control problem

Alexander Alegr\'ia; Andr\'es Rivera; Mariana \'Alvarez; Sebasti\'an Pedersen

arxiv: 2509.21406 · v3 · pith:XWNYPVNGnew · submitted 2025-09-24 · 📡 eess.SY · cs.SY

A Crime/S.I.R. optimal control problem

Mariana \'Alvarez , Alexander Alegr\'ia , Andr\'es Rivera , Sebasti\'an Pedersen This is my paper

Pith reviewed 2026-05-25 07:59 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords optimal controlSIR modelcrime dynamicscost minimizationpreventive policiesdynamic systemspublic interventions

0 comments

The pith

An SIR model with three controls is used to derive policies minimizing the time-integrated cost of crime.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates crime reduction as an optimal control problem in which an SIR-type system tracks the flow between non-criminal, criminal, and rehabilitated groups. Three time-varying controls adjust transmission, recovery, and relapse rates while a quadratic cost penalizes the intensity of each policy. The objective is to minimize the integral of criminals minus rehabilitated plus control effort over a finite horizon. This setup is intended to identify the lowest-cost mix of interventions when resources are limited.

Core claim

The authors present the minimization of the functional G(u1,u2,u3) equal to the integral from 0 to t_F of I(t) minus R(t) plus half B1 u1 squared, half B2 u2 squared, and half B3 u3 squared, subject to the controlled SIR dynamics for S, I, and R in which u1 reduces transmission, u2 boosts recovery, and u3 modulates relapse and additional recovery.

What carries the argument

The optimal control problem whose state equations are the three controlled SIR differential equations for crime spread and recovery.

If this is right

The optimal controls will produce a trajectory in which the net criminal population declines relative to the uncontrolled case.
A specific combination of preventive, rehabilitative, and enforcement efforts will be identified that achieves the reduction at lowest total cost.
The policies are expected to create opportunities for the most disadvantaged sectors while contributing to long-term security.
The same framework can be used to compare alternative policy intensities before implementation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Fitting the model parameters to observed crime incidence series from a given city would allow computation of concrete numerical control schedules.
The same controlled-compartment structure could be tested on other contagious social behaviors such as substance use or gang affiliation.
Longitudinal data collected after policy rollout could be compared directly against the model's predicted I(t) and R(t) curves to test predictive accuracy.

Load-bearing premise

The social and economic effects of criminal behavior can be modeled by a dynamic SIR-type system.

What would settle it

Numerical solution of the two-point boundary-value problem arising from the Pontryagin maximum principle, followed by forward simulation that shows whether the cost under the resulting controls is lower than under the zero-control trajectory.

Figures

Figures reproduced from arXiv: 2509.21406 by Alexander Alegr\'ia, Andr\'es Rivera, Mariana \'Alvarez, Sebasti\'an Pedersen.

**Figure 1.** Figure 1: Source: Created by the authors using data from the Ministry of Justice and Law. 4,462 homicides), and 14 (8.5%, 4,404 homicides), located in the eastern part of the city, reported the highest numbers in terms of homicide occurrence. That is, 26.6% of the total homicides in these three decades are concentrated in 3 of 22 comunas in Cali, precisely those with a predominance of strata 1 and 2 [PITH_FULL_IMAG… view at source ↗

**Figure 2.** Figure 2: Source: Created by the authors, using data from Santiago de Cali Security Observatory and DANE (2018)). In turn, young people have been the most affected, accounting for more than half of the victims. According to the ”Cali en Cifras” report by Departamento Administrativo de Planeaci´on [2024], of the homicides recorded between 1993 and 2022, 38.7% claimed the lives of young people between 20 and 29 years … view at source ↗

**Figure 3.** Figure 3: Compartmental Diagram of the Equation System (1) It is a Holling Type II functional response in S, signifying the rate at which individuals from group I victimize or capture members of group S. The system parameters are fully described in [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: Sensitivity of the model to changes in its parameters 23 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗

**Figure 5.** Figure 5: Numerical Simulation for different values of β set to 0.3 (a value corresponding to a significant capture rate). The following graph represents the trajectories of the state variables S, I, and R over time. Time 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 P o p ulatio n 0 200 400 600 800 1000 1200 1400 Trajectory of S, I, R with and without control S with control S w/o control I with control I w/o control R with contr… view at source ↗

**Figure 6.** Figure 6: Numerical Simulation for β = 0.3 with and without control u1 25 [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗

read the original abstract

This paper presents and discusses a mathematical model inspired by control theory to derive optimal public policies for minimizing costs associated with the reduction and control of criminal activity in a population. Specifically, we analyze the optimal control problem \begin{equation*} \min G(u_1, u_2, u_3) = \int_{0}^{t_{\text{F}}} \left( I(t) - R(t) + \frac{B_1}{2} u_1^2(t) + \frac{B_2}{2} u_2^2(t) + \frac{B_3}{2} u_3^2(t) \right) \, dt. \end{equation*} where $I=I(t)$ and $R=R(t)$ satisfies the system of equations \begin{equation*} \left\{ \begin{aligned} \dot{S} &= \Lambda - (1-u_1)SI - \mu S + ((1+u_3)\gamma_2)I + \rho \Omega R,\\ \dot{I} &= (1-u_1)SI - (\mu + \delta_1)I - ((1+u_2)\gamma_1)I - ((1+u_3)\gamma_2)I + (1-\Omega)\rho R,\\ \dot{R} &= ((1+u_2)\gamma_1)I - (\mu + \delta_2 + \rho)R. \end{aligned} \right. \end{equation*} Our approach assumes that the social and economic effects of criminal behavior can be modeled by a dynamic SIR-type system, which serves as a constraint on a cost functional associated with the strategies implemented by government and law enforcement authorities to reduce criminal behavior. Using optimal control theory, the proposed controls, i.e., preventive policies (such as community and social cohesion programs), are expected to have a significant and positive impact on crime reduction, generating opportunities for the most disadvantaged sectors of Cali society and contributing to long-term security. Given that resources to address this problem are limited, this research aims to determine an optimal combination of public interventions and policies that minimize criminality at the lowest possible economic cost, using an SIR model, tools from variational calculus, and optimal control theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper writes down an SIR optimal control setup for crime policy but shows no solved trajectories, no parameter fits, and no data checks.

read the letter

The main takeaway is that the authors define a standard optimal control problem on an SIR-type system for criminal activity in Cali, with three controls that reduce transmission or boost recovery, but they stop at the equations and never compute anything with them. The cost functional penalizes I minus R plus quadratic terms in the u's, and the dynamics include the usual mass-action infection modified by (1-u1), plus recovery terms scaled by (1+u2) and (1+u3). They cite the usual optimal control theory and say the preventive policies should help disadvantaged groups, yet nothing is solved or simulated. What the paper does cleanly is lay out the compartments and the controlled rates in one place; the equations are explicit and the modeling choice is stated up front. That is the extent of the contribution. The soft spots are exactly where the stress-test note says: no calibration of Lambda, mu, deltas, gammas or the B weights to any Cali crime numbers, no forward integration of the closed-loop system, no comparison of the optimal policy against doing nothing or against simpler heuristics, and no check on whether the SIR analogy actually tracks observed crime patterns. The claim of significant positive impact therefore sits on an untested modeling assumption rather than on any demonstrated outcome. A reader who already works on optimal control applied to social systems might find the specific control structure mildly interesting as an example, but anyone looking for usable policy insight or a new mathematical result will find the work incomplete. I would not bring this to a reading group, would not cite it, and would not send it to referees until the authors add at least one set of numerical trajectories and some data grounding.

Referee Report

3 major / 2 minor

Summary. The manuscript formulates an optimal control problem for crime reduction by adapting an SIR-type compartmental model, with states S (susceptible), I (criminal/infected), and R (recovered) governed by the given system of ODEs incorporating controls u1 (preventing transmission), u2 and u3 (enhancing recovery). The objective minimizes the integral from 0 to t_F of [I(t) - R(t) + (B1/2)u1²(t) + (B2/2)u2²(t) + (B3/2)u3²(t)] dt, and the authors conclude that the resulting optimal preventive policies are expected to produce significant positive impacts on crime reduction in Cali.

Significance. A validated version of this controlled SIR-crime model with computed optimal trajectories could supply a quantitative, cost-sensitive framework for policy design in systems and control applications to social dynamics. The formulation is a direct application of standard optimal control to an epidemiological analogy, but the lack of any solved trajectories, adjoint equations, numerical results, or parameter calibration means the claimed policy impact remains an untested modeling hypothesis rather than a demonstrated result.

major comments (3)

[Abstract] Abstract, the displayed system of equations: the central claim that 'the proposed controls... are expected to have a significant and positive impact on crime reduction' is unsupported because the manuscript supplies neither the Hamiltonian, adjoint system, optimality conditions, nor any numerical solution of the resulting two-point boundary-value problem; without these the assertion that the controls lower I(t) cannot be verified.
[Abstract] Abstract, cost functional and dynamics: the model introduces free parameters (B1, B2, B3, Λ, μ, δ1, δ2, γ1, γ2, ρ, Ω) and the mass-action term (1-u1)SI without any estimation from Cali crime statistics or sensitivity analysis; the claim that the SIR-type system 'serves as a constraint on a cost functional' therefore rests on an uncalibrated analogy whose fidelity to real criminal dynamics is not demonstrated.
[Abstract] Abstract: the cost term I(t) - R(t) encodes the modeling assumption that recovered individuals contribute negatively to the objective, yet no justification, alternative formulations, or robustness check with respect to this choice is provided, making the derived policy recommendations sensitive to an arbitrary modeling decision.

minor comments (2)

[Abstract] The time horizon t_F and initial conditions S(0), I(0), R(0) are not stated, preventing reproduction of any future numerical experiments.
[Abstract] Notation for the controls is introduced as u1, u2, u3 but their precise mapping to 'community and social cohesion programs' versus 'law enforcement' is not made explicit.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment point by point below, indicating the revisions we will undertake.

read point-by-point responses

Referee: [Abstract] Abstract, the displayed system of equations: the central claim that 'the proposed controls... are expected to have a significant and positive impact on crime reduction' is unsupported because the manuscript supplies neither the Hamiltonian, adjoint system, optimality conditions, nor any numerical solution of the resulting two-point boundary-value problem; without these the assertion that the controls lower I(t) cannot be verified.

Authors: We agree that the current version presents the optimal control formulation but omits the explicit derivation of the Hamiltonian, adjoint system, and optimality conditions, as well as any numerical trajectories. The abstract statement is intended to describe the modeling motivation rather than a verified outcome. In revision we will add the Hamiltonian, the full adjoint equations, and the optimality conditions obtained via Pontryagin’s principle, together with a qualitative discussion of how the controls are expected to affect the states. revision: partial
Referee: [Abstract] Abstract, cost functional and dynamics: the model introduces free parameters (B1, B2, B3, Λ, μ, δ1, δ2, γ1, γ2, ρ, Ω) and the mass-action term (1-u1)SI without any estimation from Cali crime statistics or sensitivity analysis; the claim that the SIR-type system 'serves as a constraint on a cost functional' therefore rests on an uncalibrated analogy whose fidelity to real criminal dynamics is not demonstrated.

Authors: The parameters are introduced symbolically to define a general theoretical framework. The manuscript is a modeling paper rather than an empirical calibration study. We will incorporate a sensitivity analysis with respect to the principal parameters in the revised version and will explicitly state that parameter values are illustrative. Calibration against specific Cali crime data lies outside the present scope and is noted as future work. revision: partial
Referee: [Abstract] Abstract: the cost term I(t) - R(t) encodes the modeling assumption that recovered individuals contribute negatively to the objective, yet no justification, alternative formulations, or robustness check with respect to this choice is provided, making the derived policy recommendations sensitive to an arbitrary modeling decision.

Authors: The term -R(t) is chosen to reflect the policy objective of both reducing criminal incidence and promoting rehabilitation. We will add an explicit justification for this functional form, discuss its relation to the underlying social goals, and examine alternative formulations (e.g., weighting only I(t) or adding further penalty terms) together with a brief robustness remark. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit model definition and standard OCP setup

full rationale

The paper states an SIR-type system of ODEs and a quadratic integral cost functional as the problem definition. The optimal controls are to be derived from this setup via standard optimal control theory; no step reduces a claimed prediction or result back to a fitted parameter, self-citation, or definitional equivalence. The modeling assumption is presented explicitly as an assumption rather than derived. No load-bearing self-citations or ansatzes appear in the provided text.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of modeling crime via an SIR-type system and on the choice of several unspecified parameters in the equations and cost functional. The abstract provides no independent evidence or data for these choices.

free parameters (2)

B1, B2, B3
Weights on the quadratic control effort terms in the objective functional; values not specified and must be chosen or fitted.
Lambda, mu, delta1, delta2, gamma1, gamma2, rho, Omega
Transition rates and other coefficients in the SIR system; values not provided in the abstract.

axioms (1)

domain assumption Criminal behavior dynamics can be modeled using an SIR-type system with controllable transition rates.
Explicitly stated as the foundational assumption: 'Our approach assumes that the social and economic effects of criminal behavior can be modeled by a dynamic SIR-type system'.

pith-pipeline@v0.9.0 · 5957 in / 1382 out tokens · 43305 ms · 2026-05-25T07:59:41.870301+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Foundation for a general strain theory of crime and delinquency

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work page 1992

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‘plan neón’, nueva estrategia de prevención y control de las autoridades en cali, marzo 2023 a

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Con emotivo discurso, alcalde de cali lanzó ‘en la buena’, el programa de atención juvenil contra la violencia, enfocado en la generación de oportunidades para 10 mil jóvenes, 2024

Alcaldía de Santiago de Cali . Con emotivo discurso, alcalde de cali lanzó ‘en la buena’, el programa de atención juvenil contra la violencia, enfocado en la generación de oportunidades para 10 mil jóvenes, 2024

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Proyecto ‘perla’, una apuesta para la reconciliación de los territorios, julio 2022

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Van den Driessche and James Watmough

P. Van den Driessche and James Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180 0 (1), 2002. doi:https://doi.org/10.1016/S0025-5564(02)00108-6

work page doi:10.1016/s0025-5564(02)00108-6 2002

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