The hidden math of teen drug use: why some schools spiral and others stay clean

The hidden math of teen drug use: why some schools spiral and others stay clean

In one high school, 30% of students end up with sustained substance use. In another, nearly all stay clean. Identical policies, similar neighborhoods, same state laws. What explains the difference? According to a new mathematical model, the answer lies not in any single factor—but in the interplay of peer influence, school disengagement, and a hidden threshold: whether students who drop out due to drug use ever come back. When re-entry is rare (ϕ < 1), the school population shrinks with each wave of addiction, making recovery impossible no matter the intervention. But when reintegration is possible (ϕ = 1), the same school can stabilize—even if it passes through periods of high use. This isn’t just about drugs. It’s about how systems tip.

The Science

The model, called SCAR (Susceptible–Casual–Addicted–Resistant), was developed by Tamantha Pizarro, Jinni Su, Yixuan He, and Yun Kang at Arizona State University (Pizarro et al., 2026). It’s a compartmental dynamical system—like those used for infectious diseases—but adapted for adolescent behavior in a school setting. The population is divided into four groups:

• S: Susceptible non-users, at risk of initiation
• C: Casual or experimental users
• A: Students with sustained, problematic, or substance-use-disorder (SUD)-level involvement
• R: Resistant students embedded in protective, anti-use environments

The model tracks how students move between these states through peer influence, protective social norms, school turnover, disengagement (dropout), and re-entry after rehabilitation. Crucially, it does not assume a fixed school population. Instead, students in the A class may disengage at rate $\delta$, and only a fraction $\varphi$ of them return. This makes the total population $N(t)=S(t)+C(t)+A(t)+R(t)$ dynamic—not static.

The model is governed by a system of ordinary differential equations (ODEs) that capture:

• Peer-driven initiation: $S\to C$ via contact with casual ($\beta_{CS}$) or addicted ($\beta_{AS}$) users
• Escalation: $C\to A$ via contact with addicted peers ($\beta_{AC}$)
• Protective influence: $S\to R$ ($\beta_{RS}), $C \to S ($\beta_{RC}), $A \to C ($\beta_{RA}$)
• School turnover: new students enter at rate $\mu$, a fraction $\rho$ directly into R
• Disengagement: A students leave at rate $\delta$, a fraction $\varphi$ return to S

The key innovation is the return parameter $\varphi$. When $\varphi =1$, every student who disengages eventually returns—population is conserved. When $\varphi <1$, disengagement leads to net population loss. This seemingly small difference creates two entirely distinct dynamical regimes.

What They Found

The analysis reveals three major insights.

1. The Return Parameter $\varphi$ Splits Reality in Two

When $\varphi =1$, the total school population is constant: $\frac{dN}{dt}=0$. In this regime, the model behaves like a classical epidemic system. Biologically meaningful equilibria can exist—even with high levels of substance use.

But when $\varphi <1$, $\frac{dN}{dt}=-(1-\varphi )\delta A(t)<0$ whenever $A(t)>0$. That is, every student in sustained use reduces the total school population. This has a profound consequence: no true endemic equilibrium can exist. Even if the proportions of students in each class stabilize, the absolute numbers keep declining. The school is slowly emptying.

This means that in systems where dropout due to substance use is irreversible—where students don’t come back—no amount of treatment or prevention can restore equilibrium. The system is on a one-way path to collapse.

2. Initiation and Escalation Are Governed by Different Thresholds

The model identifies two distinct thresholds:

• The initiation threshold depends on ${\beta }_{CS},{\beta }_{AS},{\beta }_{RC},{\beta }_{RS}$, and $w$, the proportion of resistant students
• The escalation threshold depends on ${\beta }_{AC},{\beta }_{RA},\delta$, and $y$, the proportion of casual users

These are not the same. A school can be below the initiation threshold (few new users) but above the escalation threshold (casual users rapidly progressing to addiction). Or vice versa.

For example,

shows how ${\beta }_{CS}$, the influence of casual users on susceptible peers, drives initiation. But

shows that ${\beta }_{AC}$, the influence of addicted students on casual users, controls escalation. These are separate levers.

This implies that preventing first use is not the same as preventing addiction. A school might succeed at keeping kids from trying drugs but fail at stopping those who do from spiraling. Or it might allow experimentation but contain escalation through strong protective norms.

3. The System Can Be Bistable—Outcomes Depend on Starting Conditions

Perhaps most strikingly, the model exhibits bistability. For the same set of parameters, two different long-term outcomes are possible:

• A substance-free equilibrium, where use dies out
• A high-use equilibrium, where addiction becomes entrenched

Which one the system reaches depends on the initial conditions—how many students are using when the school year begins.

This is not just theoretical. Consider two schools with identical policies, peer influence rates, and re-entry pathways. One starts the year with 5% of students in casual use. The other starts with 15%. The first might converge to near-zero use. The second might tip into a persistent high-use state. The difference isn’t policy—it’s history.

Equilibrium Proportion of Addicted Students vs. Return Rate ϕ

Label	Value
ϕ = 1	0.85
ϕ = 0.6	0.4
ϕ = 0.3	0.1

Why This Changes Things

These findings challenge conventional wisdom in school-based substance use prevention.

Policy Is Not One-Size-Fits-All

Most school programs focus on universal prevention: D.A.R.E., peer education, awareness campaigns. These aim to reduce initiation by increasing ${\beta }_{RS}$ (protective influence) or decreasing ${\beta }_{CS}$ (peer pressure to try drugs).

But the SCAR model shows this is only half the battle. Escalation is a separate process, governed by different parameters. A student who experiments isn’t inevitably doomed—but they’re in a new risk category. Without targeted support, they may be pulled into sustained use by contact with addicted peers.

This suggests a layered strategy:

• Universal prevention to keep initiation low
• Early intervention for casual users to prevent escalation
• Recovery support for addicted students to enable de-escalation ($\beta_{RA}$)
• Re-engagement pathways to ensure $\varphi \approx 1$

The Hidden Cost of Dropout

The role of $\varphi$ is especially profound. When $\varphi <1$, the school loses students permanently. This isn’t just a tragedy for those individuals—it destabilizes the entire system. The model shows that in this regime, even if treatment reduces $A(t)$, the population keeps shrinking. There’s no recovery, only managed decline.

But when $\varphi =1$, the system can rebound. Students who disengage return, reintegrate, and can even become part of the resistant class over time. The school becomes a restorative ecosystem, not just a containment zone.

This reframes dropout not as a personal failure, but as a systemic feedback loop. Schools that lack re-entry programs aren’t just losing students—they’re ensuring future waves of addiction will be harder to contain.

Bistability Explains Persistent Disparities

Why do some schools remain clean while others struggle, even with similar resources? Bistability offers an answer: initial conditions matter.

A school that historically had high use may be trapped in a high-use equilibrium, even after improving policies. Conversely, a clean school might stay clean not because it’s doing more, but because it never crossed the tipping point.

This has equity implications. Schools in under-resourced communities often start with higher baseline use due to structural factors—poverty, trauma, lack of access to care. The model suggests they may be systemically disadvantaged, not because of weaker willpower or worse parenting, but because they’re already on the wrong side of a bifurcation.

What’s Next

The SCAR model is theoretical—but its implications are practical.

Open Questions

• Can we estimate $\varphi$ in real schools? What policies increase it? (e.g., re-enrollment programs, recovery high schools)
• How do ${\beta }_{AC}$ and ${\beta }_{RA}$ vary across substances? Is escalation faster for vaping than cannabis?
• Can schools shift from $\varphi <1$ to $\varphi =1$ through policy? What’s the cost?

Next Steps

The model calls for:

1. Measuring re-entry rates: Schools should track not just dropout, but whether students return—and under what conditions.
2. Targeting escalation: Interventions should distinguish between first-time users and those at risk of SUD.
3. Building restorative systems: Recovery isn’t complete until students are back in school, supported, and contributing to protective norms.

Caveats

The model is simplified. It assumes homogeneous mixing—every student interacts equally. In reality, peer networks are structured. It also doesn’t model relapse from R or direct entry into C or A. And it’s not fitted to data—yet.

But its power lies in clarity. By isolating mechanisms, it shows that substance use isn’t just a behavior problem—it’s a system problem. And systems can be redesigned.

The most hopeful insight? Reversibility changes everything. When schools become places where recovery is possible and re-entry is expected, the dynamics shift. The system isn’t doomed. It can heal.

That’s not just math. It’s a mandate.