# Mathematical modeling of carbon dioxide emissions with GDP linkage: sensitivity analysis and optimal control strategy

## TL;DR

This paper uses a mathematical model to study how GDP, forests, and population affect CO₂ levels, finding that forests have a bigger impact than GDP on reducing emissions.

## Contribution

The novelty lies in using a mathematical model with sensitivity analysis to compare the influence of GDP and forests on CO₂ emissions.

## Key findings

- GDP-driven CO₂ emissions and atmospheric CO₂ concentration have limited sensitivity in the model.
- Forests show higher sensitivity and greater influence on CO₂ levels compared to GDP.
- Optimal control strategies suggest prioritizing forestation for effective CO₂ mitigation.

## Abstract

Climate change and global warming are among the most significant issues that humanity is currently facing, and also among the issues that pose the greatest threats to all mankind. These issues are primarily driven by abnormal increases in greenhouse gas concentrations. Mathematical modeling serves as a powerful approach to analyze the dynamic patterns of atmospheric carbon dioxide. In this paper, we established a mathematical model with four state variables to investigate the dynamic behavior of the interaction between atmospheric carbon dioxide, GDP, forest area and human population. Relevant theories were employed to analyze the system’s boundedness and the stability of equilibrium points. The parameter values were estimated with the help of the actual data in China and numerical fitting was carried out to verify the results of the theoretical analysis. The Partial Rank Correlation Coefficient (PRCC) determines the sensitivity ofan input parameter to the output by measuring the correlation between a single input parameter and the model output. The sensitivity analysis of the compartments with respect to the model parameters was analyzed by using the PRCCand the Latin Hypercube Sampling test.The results indicate that the sensitivity of GDP-driven CO₂ emissions and GDP-governed atmospheric CO₂ concentration to the system is not significant. This implies that within the GDP-driven mitigation framework, the regulatory effect of GDP on atmospheric CO₂ concentration is relatively limited, and its significance is less pronounced than that of forests. Therefore, future relevant strategies should prioritize parameters with higher sensitivity (e.g., forestation). Apply the optimal control theory to regulate the atmospheric carbon dioxide level and provide the corresponding numerical fitting. Finally, corresponding discussions and suggestions were put forward with the help of the results of the theoretical analysis and numerical fitting.