How Differential Equations Reveal the Hidden Scale of Mexico’s Drug Cartels

A Mathematician’s Choice

Rafael Prieto‑Curiel, a Mexican applied‑mathematics graduate, left a well‑paid finance job to join the Mexico City police analytics unit, where he automated reporting, mapped crime hotspots, and helped cut average police response times from 17–20 minutes to about four minutes.

After five years he pursued a master’s in statistics and a PhD in mathematics and criminology at UCL, followed by a post‑doc at Oxford and a faculty position at the Complexity Science Hub in Vienna, consulting for the OECD and the World Bank.

Modeling a Criminal Ecosystem with ODEs

Data Foundations

Weekly homicide counts from INEGI (Mexico’s national statistics agency).

Missing‑person records from RNPDNO (national missing‑persons database).

Arrest and incarceration figures from the national prison census.

The authors assumed that a fixed proportion of homicides and arrests involved cartel members, estimating roughly 50,000 deaths or disappearances and 55,000 arrests linked to cartels between 2012 and 2022. They also reconstructed a network of alliances and rivalries among 2020’s active drug‑trafficking groups.

Core Equations

The study builds a 150‑equation coupled ordinary differential‑equation system, one for each cartel, of the form:

dN_i/dt = a N_i – b \frac{N_i}{\sum_j N_j} – c \sum_j A_{ij} N_i N_j – d N_i^2

where N_i is the size of cartel i , a is the recruitment rate (exponential growth), b scales national repression intensity, c captures bilateral conflict intensity (non‑zero only for allied or rival pairs A_{ij} ), and d represents saturation/exit effects that increase with the square of the cartel’s size.

Parameter Estimation and Calibration

Using observed weekly casualties and arrests, together with assumed heavy‑tailed initial cartel sizes, the authors minimized the squared error between model trajectories and the data to estimate the four parameters. Sensitivity analysis varied casualty and arrest parameters within realistic bounds, yielding a 2022 total cartel size confidence interval of 160,000–185,000 members. Adding 10 % more groups to the network increased the estimate by about 3.2 %.

What the Model Reveals

Cartel Size Estimates

The calibrated model predicts that the total number of cartel members grew from roughly 115,000 in 2012 to about 175,000 in 2022 – a net increase of 60,000. Maintaining this scale requires a minimum weekly recruitment of 350–370 individuals; otherwise the ecosystem would collapse under conflict, arrests, and internal attrition.

In 2021 the model estimates 19,300 new recruits, 6,500 deaths from inter‑cartel fighting, and 5,700 arrests, yielding a net weekly gain of about 7,000 members. Over the decade, 285,000 person‑years participated in cartels, with 18 % dying and 20 % imprisoned – a 38 % chance of death or incarceration for any entrant.

Policy Scenario Simulations

Four policy scenarios were projected to 2027:

Business‑as‑usual : weekly casualties rise +40 %, total cartel size +26 %.

Doubling arrests : casualties still increase +8 % (though more slowly), cartel size continues to grow.

Halving recruitment : casualties drop –25 %, cartel size shrinks –11 %.

Eliminating recruitment : it would take three years for cartel size to revert to 2012 levels, with a marked decline thereafter.

The counter‑intuitive result is that simply increasing arrests raises violence because it creates power vacuums that intensify inter‑cartel conflict and trigger faster recruitment. Reducing recruitment, by contrast, cuts casualties by 25 % and cartel size by 11 %.

Why Modeling Matters

The study shows that mathematical modeling can expose non‑linear dynamics that defy intuition, provide a common quantitative reference for policy debates, and estimate hidden variables such as total cartel membership from observable outcomes. It also warns that models can be mis‑used if assumptions are oversimplified.

Rafael has presented the work in dozens of interviews (CNN, The Guardian, etc.) and stresses that the model is a first quantitative attempt, not a definitive solution; structural socioeconomic factors still dominate the drug‑trade problem.