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Dynamical Model for the Sustainable Development Goals

This paper presents a mathematical model that simulates and predicts the dynamics of the UN Sustainable Development Goals by incorporating resource distribution, international cooperation, and goal correlations, demonstrating its ability to reproduce real data and evaluate hypothetical scenarios for optimizing global progress.

Original authors: Alberto García-Rodríguez, Tzipe Govezensky, Julia Tagüeña, Kimmo K. Kaski, Rafael A. Barrio

Published 2026-04-24

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CC BY 4.0

Original authors: Alberto García-Rodríguez, Tzipe Govezensky, Julia Tagüeña, Kimmo K. Kaski, Rafael A. Barrio

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world's 17 Sustainable Development Goals (SDGs) as 17 different rooms in a massive, crumbling mansion that we all need to fix by the year 2030. These rooms represent things like ending poverty, cleaning the oceans, and ensuring good health.

The problem is that despite our best intentions, we aren't fixing the rooms fast enough. In fact, if we keep doing exactly what we're doing, we'll miss the deadline.

This paper introduces a mathematical "flight simulator" for the world. Instead of just guessing what will happen, the authors built a computer model to see how different actions might change the outcome. Think of it as a video game where you can tweak the settings to see if the world can actually win the game.

Here is how their model works, explained through simple analogies:

1. The Three Main Ingredients

The authors realized that fixing the mansion depends on three main things:

The Wallet (Resources): How much money and effort each country spends. In the model, this is like how much fuel a car has. If you have a tiny tank, you can't drive very far.
The Teamwork (Cooperation): Countries are neighbors. In the model, they are connected by invisible strings. If one neighbor fixes their roof, it might help the neighbor next door. If they ignore each other, everyone struggles alone.
The Domino Effect (Correlations): The 17 goals are not isolated. Fixing one room often helps (or hurts) another.
- The Good Domino: Cleaning the water (Goal 6) helps people stay healthy (Goal 3).
- The Bad Domino: Sometimes, trying to build a factory to create jobs (Goal 8) might accidentally pollute the air (Goal 13), making the air goal harder to reach.

2. How the Model Simulates the Future

The authors fed real data from the last 20 years into their computer. They treated the world like a giant network of friends.

The "Memory" Factor: The model remembers what happened in the past. You can't just snap your fingers and fix a problem; progress takes time and builds on previous steps.
The "Neighbor" Factor: The model checks if Country A is helping Country B. If they cooperate, they move faster. If they are isolated, they move slowly and chaotically.

3. What the Simulations Revealed

The authors ran the simulator with different settings to see what would happen in the future. Here are the surprising results:

Scenario A: The "Lone Wolf" Approach
If countries stop talking to each other and try to fix their own problems alone, the progress is incredibly slow and messy. Some countries get stuck, while others barely move. It's like trying to fix a house while your neighbors are building a wall between you.
Scenario B: The "Too Much Connection" Trap
They tried to make the countries cooperate too much. While this helped speed things up initially, the model showed that because the goals are so tangled (some help each other, some fight each other), the system eventually hit a wall and got stuck. You can't just "cooperate" your way out of a bad design.
Scenario C: The "Magic Wand" (No Correlations)
They imagined a world where the goals had no connection to each other. Surprisingly, this was actually slower! It turns out that the "domino effect" is necessary. When you fix one thing, it should help fix the others. If they are totally disconnected, you lose that free boost.
Scenario D: The "Deep Pockets" Solution
This was the most dramatic result. The authors asked: "What if every country decided to spend 30% of their resources on these goals instead of the usual 10%?"
The Result: The simulation showed that if everyone stepped up their spending, the world could actually finish the job by 2040. It's not a magic trick; it's just a matter of putting more fuel in the tank.

The Big Takeaway

The paper concludes with a mix of realism and hope.

The Bad News: If we keep doing exactly what we are doing now (spending a little bit, not cooperating enough, and ignoring how goals affect each other), we will miss the 2030 deadline. The "mansion" will remain broken.
The Good News: The model proves that it is possible to fix it. We just need to change the variables. We need to:
1. Spend more money (increase the resource allocation).
2. Talk to each other (increase cooperation).
3. Understand the connections (make sure our actions on one goal don't accidentally break another).

In short: The world has the blueprint to fix the house. The math says we can do it, but only if we stop playing the game on "easy mode" and start working together with more resources and better strategy.

1. Problem Statement

The United Nations' 2030 Agenda outlines 17 Sustainable Development Goals (SDGs) as global challenges. However, current progress is significantly slower than required to meet the 2030 targets. Existing models (e.g., iSDG, FeliX) often suffer from data gaps, regional limitations, or oversimplification of the complex interdependencies between social, economic, environmental, and institutional factors. There is a critical need for a global, dynamic mathematical model that:

Replicates historical SDG progress data.
Accounts for the intricate correlations between different goals.
Incorporates resource allocation and international cooperation.
Allows for "what-if" scenario testing to optimize policy interventions.

2. Methodology

The authors propose a continuous-time dynamical system model based on three primary factors: resource distribution, international cooperation, and goal correlations.

2.1 Mathematical Formulation

The core of the model is a differential equation describing the progress $x(i, \alpha, t)$ of country $i$ toward goal $\alpha$ at time $t$ . The equation is:

$\frac{dx(i, \alpha, t)}{dt} = \underbrace{r(i) \cdot r_0 \cdot (1 - e(i, \alpha)) \cdot [1 - x(i, \alpha)] / g_l}_{\text{Resource-driven progress}} + \underbrace{g_k \cdot nn(i, \alpha, t) + \frac{g_k}{2} \cdot nnn(i, \alpha, t)}_{\text{Network cooperation}} + \underbrace{g_{int}(i, \alpha, t)}_{\text{Goal correlations}} + \underbrace{\frac{1}{m} \int_{t-m}^{t} x(i, \alpha, t') dt'}_{\text{Memory effect}}$

Key Components:

Resource Term: Depends on total resources $r(i)$ , spending rate $r_0$ , and the normalized spending matrix $e(i, \alpha)$ . The term $[1-x]$ ensures progress slows as the goal is approached (saturation), limited by $g_l$ .
Network Cooperation: Modeled via an adjacency matrix $A$ $A$ .
- $nn$: Influence from immediate neighbors (weighted by degree $k_i$ ).
- $nnn$: Influence from next-nearest neighbors.
- $g_k$ : Weighting parameter for neighbor influence.
Goal Correlations ( $g_{int}$ ): A coupled term where progress in one goal affects others.
- Defined by a matrix $g_{links}$ (size $17 \times 17$ ) representing correlations between goals (positive for synergy, negative for trade-offs).
- Weighted by a country-specific vector $g_i$ .
Memory: An integral term representing the influence of past progress over a time window $m$ .

2.2 Data and Calibration

Data Source: Sustainable Development Report 2025 (scores for 107 countries from 2000–2024) and Worldometer GDP data (2022).
Clustering: Countries were grouped into 6 clusters based on similar SDG trajectories using unsupervised machine learning (from a previous study). This reduces complexity and handles data noise.
Numerical Integration: Solved using the Euler method with a time step $dt = 0.001$.
Parameter Tuning: Parameters ( $g_k, g_i, m, r$ ) were adjusted to fit the historical data (2000–2024) before projecting into the future.

3. Key Contributions

Novel Dynamical Framework: Introduction of a differential equation model that explicitly couples resource allocation, network topology (cooperation), and goal interdependencies (synergies/trade-offs).
Empirical Validation: The model successfully reproduces historical SDG trajectories for 88 countries (grouped into 6 clusters) using real-world data, validating its predictive capability.
Scenario Simulation: The model serves as a tool for "what-if" analysis, allowing researchers to isolate specific variables (e.g., increasing cooperation or changing correlation strengths) to observe their impact on global SDG attainment.
Identification of Bottlenecks: The study quantitatively identifies that current progress is hindered not just by lack of resources, but specifically by negative correlations between goals and insufficient international cooperation.

4. Key Results and Findings

4.1 Reproduction of Real Data

The model accurately fits the historical data (2000–2024). It correctly predicts the stagnation of specific goals, notably SDG 12 (Responsible Consumption) and SDG 13 (Climate Action). The model attributes this to these goals being highly positively correlated with each other but strongly anti-correlated with nearly all other goals, creating a systemic drag.

4.2 Impact of Cooperation ( $g_k$ )

Complete Cooperation: When the adjacency matrix is fully connected (all countries cooperate), progress is uniform and rapid.
No Cooperation: When the matrix is diagonal (no cooperation), progress is slow, and dispersion between countries increases significantly.
Conclusion: International cooperation reduces disparities and accelerates goal attainment.

4.3 Impact of Goal Correlations ( $g_i$ )

High Correlation (Current Reality): Increasing the weight of correlations accelerates initial progress but leads to stagnation around 2080 due to the negative trade-offs inherent in the correlation matrix.
Zero Correlation ( $g_i = 0$ ): Surprisingly, removing correlations slows dynamics drastically, preventing goal attainment even over 100 years. This suggests that positive synergies are essential for momentum.
Positive Correlation Only: Simulations with a random matrix of only positive correlations show that if all goals were synergistic, success would be inevitable.

4.4 Impact of Resource Allocation

Baseline: Current annual spending is modeled at 10% of resources.
Scenario: Increasing annual spending to 30% of resources results in a dramatic shift, with all SDGs approaching success by approximately 2040.

5. Significance and Implications

Policy Guidance: The model provides a quantitative basis for policymakers to understand that simply increasing resources is insufficient without addressing the structural trade-offs between goals (e.g., economic growth vs. climate action) and enhancing international cooperation.
Strategic Optimization: It suggests that to meet the 2030 agenda, a more detailed program is required that accounts for specific national socio-economic contexts and cultural differences, rather than a "one-size-fits-all" approach.
Methodological Advance: By combining machine learning (for clustering) with dynamical systems theory, the study offers a scalable framework for analyzing complex socio-ecological systems.
Warning on Predictions: The authors caution that while the model captures long-term trends, predictions rely on assumptions (e.g., random matrices for resource distribution) due to data gaps. However, the model effectively highlights the direction of necessary interventions.

In summary, the paper demonstrates that achieving the SDGs is mathematically feasible but currently blocked by the complex interplay of anti-correlated goals and insufficient resource allocation. The model suggests that a tripling of resource investment combined with optimized cooperation strategies could bring the world back on track for the 2030 targets.