Mathematical Modeling and Analysis of Atmospheric Carbon Dioxide (CO₂) Dynamics with Human Population Growth and Vehicle Services Age

Abstract

In this study, we propose a mathematical model for the dynamics of atmospheric carbon dioxide (CO₂) with human population growth and an age-structured vehicle population. The positivity and boundedness of the model’s solutions are verified, and the global stability of an interior equilibrium point is proved. We calibrate the model parameters using the least-squares method. Simulations of the model show good alignment with the reported atmospheric CO₂ data. Moreover, the sensitivity analysis shows that human-related emission factors and emissions from old vehicles are the main contributors to CO₂ accumulation. The results of this study recommend that reducing anthropogenic emissions, accelerating the retirement of old vehicles, and strengthening carbon sequestration initiatives be considered to mitigate future atmospheric CO₂ accumulation.

1. Introduction

The rapid escalation of atmospheric carbon dioxide (CO₂) over recent decades has emerged as the defining crisis of the modern era. The continuous accumulation of atmospheric CO₂ acts as the primary driver of global climate change, yielding profound and long-lasting adverse impacts on environmental stability, weather patterns, air quality, and public health [1,2,3]. This crisis is largely driven by human activities, including industrial expansion, fossil fuel combustion for energy production, large-scale deforestation, growing transportation demands, and intensive agricultural practices that weaken the Earth’s natural capacity to absorb carbon [4,5,6]. Among the various sectors contributing to CO₂ emissions, transportation is widely recognized as one of the largest and fastest-growing sources worldwide [7,8,9]. In particular, the transport sector accounts for approximately 15% of total anthropogenic greenhouse gas emissions and around 23% of all energy-related greenhouse gas releases globally [10].

Vehicle fleet characteristics such as service age, total mileage, and mechanical condition significantly influence CO₂ emission levels. Data from fleet-tracking systems highlight substantial variations in emissions related to vehicle deterioration over time. Korzeb et al. [11] demonstrated that vehicles with cumulative mileage up to 86,000 km emit substantially less CO₂ than those operating between 86,001 and 160,000 km. Correspondingly, Singam et al. [12] identified vehicle age, baseline fuel consumption, driving habits, and maintenance frequency as critical determinants affecting emissions. These observations are corroborated by engineering analyses and data-driven models, which reveal that component wear degrades engine combustion efficiency, thereby elevating carbon emissions [13].

In response to the growing concern over atmospheric CO₂ accumulation, various mathematical models have been developed to examine how socioeconomic, technological, and environmental factors govern CO₂ dynamics [5,14,15,16,17,18,19]. For instance, Misra et al. [14] proposed a compartment model that examined the relationship between atmospheric CO₂ accumulation and mitigation budget allocations, finding that rising CO₂ levels directly increase funding requirements for mitigation efforts. On the ecological side, Sundar et al. [15] showed that targeted green belt development and seaweed farming can substantially reduce ambient carbon levels by enhancing photosynthetic absorption near major urban emission sources. Addressing demographic drivers, Fors [16] developed a mathematical model demonstrating that human population growth accelerates deforestation, which in turn raises the equilibrium concentration of atmospheric CO₂. Additionally, nonlinear dynamic models have been used to assess carbon fluctuations under varying plant absorption capacities [19], as well as the feedback relationships between reforestation efforts and human population pressures [17,18].

Most existing mathematical models focus on broad interactions among human populations, industrial activities, and atmospheric CO₂ concentrations. Even when vehicular activity is explicitly incorporated, vehicles are often assumed to emit CO₂ at the same constant rate, regardless of their service age or condition. For example, Mehmood et al. [20] developed a model to investigate the effects of emissions from fossil-fueled vehicles on atmospheric CO₂ concentrations and global warming. However, their study assumed identical CO₂ emission rates for all vehicles and did not account for differences in emissions due to variations in vehicle age and condition.

Donald et al. [5] developed a nonlinear compartmental model to investigate the contribution of vehicular activities to atmospheric CO₂ accumulation. The model incorporated human population, vehicle population, forest biomass, and atmospheric carbon dioxide as interacting compartments. The authors analyzed the existence and stability of equilibrium points and estimated model parameters using available data. Sensitivity analysis and numerical simulations were also conducted to assess the influence of various parameters on CO₂ dynamics. The study demonstrated that increasing vehicular activities significantly elevates atmospheric CO₂ concentrations. However, the model assumed that all vehicles emit CO₂ at the same rate, without considering differences arising from vehicle age or condition. This limitation provides an opportunity for extending the model by incorporating age-structured vehicle classes.

Liu et al. [21] employed a system dynamics approach to evaluate the impact of transportation policies on carbon dioxide emissions. The study simulated the effects of measures such as improving public transportation, implementing vehicle restrictions, and promoting fuel efficiency. Their findings suggested that integrated transport policies can substantially reduce future emissions. However, the analysis focused primarily on policy scenarios and simulation outcomes without conducting rigorous mathematical investigations such as equilibrium analysis or stability assessment.

Chang et al. [22] proposed a real-time emission estimation framework using intelligent transportation system technologies. The model incorporated traffic characteristics such as vehicle speed, acceleration, and road conditions to estimate CO₂ emissions more accurately. The study demonstrated the usefulness of traffic monitoring systems for managing urban emissions and improving transport efficiency. Despite its practical significance, the work focused on short-term emission estimation and did not address the long-term dynamics of atmospheric CO₂ accumulation.

Cardona et al. [23] developed a system dynamics model to estimate and project CO₂ emissions from the transport sector under different development scenarios. The study assessed the long-term implications of transportation growth and explored strategies for emission mitigation. The results emphasized the importance of sustainable transport planning in controlling emissions. Nevertheless, the model did not explicitly describe atmospheric CO₂ accumulation mechanisms and lacked an analytical examination of the system’s dynamic properties.

Overall, these studies demonstrate substantial progress in modeling carbon dioxide dynamics and transportation-related emissions. However, they reveal important research gaps, particularly the absence of age-structured vehicle populations and the limited integration of transportation dynamics with rigorous mathematical analysis of atmospheric CO₂ accumulation. These gaps motivate the development of models that account for heterogeneous vehicle fleets and provide a deeper understanding of the long-term impacts of transportation on climate change.

2. Model Formulation

In this study, we develop a deterministic compartmental mathematical model to describe the dynamics of atmospheric CO₂ concentration influenced by human population growth, vehicle fleet expansion, and human activities. The model consists of five interacting state variables: the human population, $H\left(t\right)$; three vehicle age classes, $V_1\left(t\right)$, $V_2\left(t\right)$, and $V_3\left(t\right)$; and the accumulated atmospheric CO₂ concentration, $C\left(t\right)$.

Human population dynamics: Although the logistic growth model has some limitations, such as assuming a homogeneous population and neglecting socioeconomic, cultural, and policy factors, it has been widely used to describe human population growth [24,25,26]. Therefore, for simplicity and consistency with previous studies, we assume that the human population follows logistic growth.

Let $H\left(t\right)$ denote the human population at time t. The population grows at an intrinsic rate s, while environmental limitations restrict its growth through the carrying capacity K. Under these assumptions, the human population dynamics are given by

$${\displaystyle \frac{dH}{dt}}=sH\left(1-{\displaystyle \frac{H}{K}}\right).$$

Human population dynamics play an important role in the model because vehicle ownership and other human activities depend on population size. We assume that new vehicles enter the system at a rate proportional to the human population, given by $\mathsf{\Lambda }H$, where $\mathsf{\Lambda }$ is the per capita vehicle acquisition rate.

Vehicle fleet dynamics: The vehicle fleet is divided into three groups according to vehicle age and cumulative mileage. Based on fleet distribution data reported in [27,28,29], the vehicle classes are defined as follows:

• $V_1\left(t\right)$ represents new vehicles with less than three years of service and cumulative mileage below 30,000 miles.
• $V_2\left(t\right)$ represents mid-aged vehicles with 3–6 years of service and mileage between 30,000 and 100,000 miles.
• $V_3\left(t\right)$ represents old vehicles with more than seven years of service and mileage exceeding 100,000 miles.

Vehicles are assumed to move sequentially through these classes as they age and accumulate mileage. New vehicles progress to the mid-aged class at rate ${\alpha }_1$, while mid-aged vehicles move to the old vehicle class at rate ${\alpha }_2$. Vehicles may also leave each class because of retirement, scrappage, or mechanical failure. These removals occur at rates ${\mu }_1$, ${\mu }_2$, and ${\mu }_3$, respectively.

An important assumption of the model is that older vehicles emit more CO₂ due to reduced engine efficiency and technological deterioration. Consequently, the emission rates satisfy

$${\beta }_1<{\beta }_2<{\beta }_3,$$

where ${\beta }_i$ represents the CO₂ emission rate of the i-th vehicle class. The vehicle population dynamics are therefore given by

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{V}_1}{dt}}& =\mathsf{\Lambda }H-({\alpha }_1+{\mu }_1)V_1,\hfill \\ \hfill {\displaystyle \frac{d{V}_2}{dt}}& ={\alpha }_1{V}_1-({\alpha }_2+{\mu }_2)V_2,\hfill \\ \hfill {\displaystyle \frac{d{V}_3}{dt}}& ={\alpha }_2{V}_2-{\mu }_3{V}_3.\hfill \end{array}$$

Atmospheric CO₂ dynamics: The final component of the model describes the accumulated atmospheric carbon dioxide concentration, denoted by $C\left(t\right)$. Atmospheric CO₂ accumulates from three main sources:

• Natural processes and biological respiration contribute CO₂ at a constant rate $\delta$.
• Human activities, including energy consumption, industrial production, and land-use changes, contribute CO₂ at a rate proportional to the population size, given by $\theta H$.
• Vehicular emissions arise from the three vehicle classes and are represented by

  $${\beta }_1{V}_1+{\beta }_2{V}_2+{\beta }_3{V}_3.$$

To capture the major mechanisms of the carbon cycle, the model incorporates two pathways for CO₂ removal. The first pathway represents natural sequestration through forests, soils, and oceans, occurring at rate ${\gamma }_{1}C$. The second pathway represents human mitigation efforts, such as carbon capture technologies and reforestation programs, occurring at rate ${\gamma }_{2}C$. Therefore, the atmospheric CO₂ dynamics are described by

$${\displaystyle \frac{\mathit{dC}}{\mathit{dt}}}=\delta +\theta H+{\beta }_1{V}_1+{\beta }_2{V}_2+{\beta }_3{V}_3-({\gamma }_1+{\gamma }_2)C.$$

Combining the above components, we obtain the following system of nonlinear differential equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\mathit{dH}}{\mathit{dt}}}=\mathit{sH}\left(1-{\displaystyle \frac{H}{K}}\right),\hfill \\ {\displaystyle \frac{{\mathit{dV}}_1}{\mathit{dt}}}=\mathsf{\Lambda }H-({\alpha }_1+{\mu }_1)V_1,\hfill \\ {\displaystyle \frac{{\mathit{dV}}_2}{\mathit{dt}}}={\alpha }_1{V}_1-({\alpha }_2+{\mu }_2)V_2,\hfill \\ {\displaystyle \frac{{\mathit{dV}}_3}{\mathit{dt}}}={\alpha }_2{V}_2-{\mu }_3{V}_3,\hfill \\ {\displaystyle \frac{\mathit{dC}}{\mathit{dt}}}=\delta +\theta H+{\beta }_1{V}_1+{\beta }_2{V}_2+{\beta }_3{V}_3-({\gamma }_1+{\gamma }_2)C,\hfill \end{array}\right.\end{array}$$

(1)

where the initial data

$$H\left(0\right)\ge 0,V_1\left(0\right)\ge 0,V_2\left(0\right)\ge 0,V_3\left(0\right)\ge 0,C\left(0\right)\ge 0.$$

To ensure biological, physical, and environmental realism, all model parameters are assumed to be positive. The definitions of the state variables and model parameters are presented in

Table 1. Model state variables and their description.

State Variables	Description
$H\left(t\right)$	Human population size at time $t$
$V_1\left(t\right)$	New vehicle cohort (mileage $<30,000$ miles)
$V_2\left(t\right)$	Mid-aged vehicle cohort (mileage $30,000$–$100,000$ miles)
$V_3\left(t\right)$	Old vehicle cohort (mileage $>100,000$ miles)
$C\left(t\right)$	Cumulative atmospheric ${\mathrm{CO}}_2$ concentration

and

Table 2. Model parameters and their description.

Parameters	Description
$s$	Intrinsic growth rate of the human population
$K$	Environmental carrying capacity for the human population
$\mathsf{\Lambda }$	Per capita vehicle acquisition rate
${\alpha }_1,{\alpha }_2$	Transition rates between vehicle age cohorts (aging rates)
${\mu }_i$	Retirement/scrapping rates for vehicle class $i$ ($i=1,2,3$)
${\beta }_i$	${\mathrm{CO}}_2$ emission intensity of vehicle class $i$ ($i=1,2,3$)
$\theta$	Human activity emission coupling factor (energy consumption, industrial processes, land-use changes)
$\delta$	Constant emission rate from natural and biological respiration processes
${\gamma }_1$	Natural ${\mathrm{CO}}_2$ depletion coefficient (oceanic and terrestrial sinks)
${\gamma }_2$	Anthropogenic ${\mathrm{CO}}_2$ removal coefficient (sequestration and reforestation)

, respectively.

3. Qualitative Analysis

3.1. Positivity and Boundedness of a Solution

Since the proposed model describes the dynamics of the human population, vehicle fleets, and atmospheric CO₂ concentration, it is necessary to ensure that all state variables remain non-negative for all time $t>0$. This requirement guarantees the biological and physical feasibility of the model.

In this section, we establish two fundamental properties of system (1): positivity and boundedness of the solutions.

Theorem 1 Positivity of Solutions).

(Consider the system (1) with strictly positive parameters and initial conditions

$$H\left(0\right)>0,V_1\left(0\right)>0,V_2\left(0\right)>0,V_3\left(0\right)>0,C\left(0\right)>0.$$

Then, the solutions

$$H\left(t\right),V_1\left(t\right),V_2\left(t\right),V_3\left(t\right),C\left(t\right)$$

remain strictly positive for all $t\ge 0$.

Proof. 

The first equation of the system (1) is a differential equation of the logistic form:

$${\displaystyle \frac{\mathit{dH}}{\mathit{dt}}}=\mathit{sH}\left(1-{\displaystyle \frac{H}{K}}\right).$$

The solution for $H\left(t\right)$ is given by

$$H\left(t\right)={\displaystyle \frac{\mathit{KH}\left(0\right)e^{\mathit{st}}}{K+H\left(0\right)(e^{\mathit{st}}-1)}}.$$

Given that $s>0$ and $H\left(0\right)>0$, the exponential nature of the solution ensures that $H\left(t\right)>0$ for all $t\ge 0$.

To establish the positivity of $V_1\left(t\right)$, we consider the second equation of system (1):

$${\displaystyle \frac{{\mathit{dV}}_1}{\mathit{dt}}}=\mathsf{\Lambda }H-({\alpha }_1+{\mu }_1)V_1.$$

Using the integrating factor $e^{({\alpha }_1+{\mu }_1)t}$, the general solution is

$$V_1\left(t\right)=e^{-({\alpha }_1+{\mu }_1)t}\left[V_1\left(0\right)+\mathsf{\Lambda }{\int }_0^{t}H\left(\tau \right)e^{({\alpha }_1+{\mu }_1)\tau }d\tau \right].$$

Given that $H\left(\tau \right)>0$, all terms on the right-hand side are positive. Consequently, $V_1\left(t\right)>0$ for all $t\ge 0$.

Similarly, for the mid-aged vehicle cohort:

$${\displaystyle \frac{{\mathit{dV}}_2}{\mathit{dt}}}={\alpha }_1{V}_1-({\alpha }_2+{\mu }_2)V_2,$$

using the integrating factor $e^{({\alpha }_2+{\mu }_2)t}$ the general solution is

$$V_2\left(t\right)=e^{-({\alpha }_2+{\mu }_2)t}\left[V_2\left(0\right)+{\alpha }_1{\int }_0^t{V}_1\left(\tau \right)e^{({\alpha }_2+{\mu }_2)\tau }d\tau \right].$$

Since $V_1\left(\tau \right)>0$, it follows that $V_2\left(t\right)>0$ for all $t\ge 0$.

For the old vehicle cohort $V_3$:

$${\displaystyle \frac{{\mathit{dV}}_3}{\mathit{dt}}}={\alpha }_2{V}_2-{\mu }_3{V}_3,$$

and using the integrating factor $e^{{\mu }_{3}t}$, the general solution is

$$V_3\left(t\right)=e^{-{\mu }_{3}t}\left[V_3\left(0\right)+{\alpha }_2{\int }_0^t{V}_2\left(\tau \right)e^{{\mu }_3\tau }d\tau \right].$$

Since $V_2\left(\tau \right)>0$, we obtain $V_3\left(t\right)>0$ for all $t\ge 0$.

Finally, we consider the equation of atmospheric CO₂ concentration $C\left(t\right)$:

$${\displaystyle \frac{\mathit{dC}}{\mathit{dt}}}=\delta +\theta H+{\beta }_1{V}_1+{\beta }_2{V}_2+{\beta }_3{V}_3-({\gamma }_1+{\gamma }_2)C.$$

Using the integrating factor $e^{({\gamma }_1+{\gamma }_2)t}$, we have:

$$C\left(t\right)=e^{-({\gamma }_1+{\gamma }_2)t}\left[C\left(0\right)+{\int }_0^t\left(\delta +\theta H+{\beta }_1{V}_1+{\beta }_2{V}_2+{\beta }_3{V}_3\right)e^{({\gamma }_1+{\gamma }_2)\tau }d\tau \right].$$

Because all parameters and state variables are positive, the integrand is positive, ensuring $C\left(t\right)>0$ for all $t\ge 0$. Therefore, each state variable remains strictly positive for all $t\ge 0$ whenever the initial data are strictly positive. This proves that the region ${\mathbb{R}}_{+}^5$ is positively invariant for system (1). □

Theorem 2 (Boundedness of Solutions).

Consider the system (1) with strictly positive parameters and non-negative initial conditions in ${\mathbb{R}}_{+}^5$. Then all solutions of the system remain bounded for all $t>0$. Moreover, there exists a positively invariant bounded region

$$\mathsf{\Omega }=\left\{(H,V_1,V_2,V_3,C)\in {\mathbb{R}}_{+}^5:H\le K,V_1\le M_1,V_2\le M_2,V_3\le M_3,C\le M_4\right\},$$

which attracts all the trajectories.

Proof. 

Since the right-hand side of the system (1) is locally Lipschitz in ${\mathbb{R}}_{+}^5$, solutions exist and are unique for all $t>0$. From the first equation of the system (1):

$${\displaystyle \frac{\mathit{dH}}{\mathit{dt}}}=\mathit{sH}\left(1-{\displaystyle \frac{H}{K}}\right)$$

which is a standard logistic equation. This implies the following:

$$0\le H\left(t\right)\le max\left\{H\right(0),K\},\mathrm{for}\mathrm{all}t>0.$$

In particular, if $H\left(0\right)\le K$, then $0\le H\left(t\right)\le K$ for all $t>0$, ensuring $H\left(t\right)$ is uniformly bounded. Next, consider the dynamics of the new vehicle cohort $V_1\left(t\right)$:

$${\displaystyle \frac{{\mathit{dV}}_1}{\mathit{dt}}}=\mathsf{\Lambda }H-({\alpha }_1+{\mu }_1)V_1.$$

Given $H\left(t\right)\le K$, we obtain

$${\displaystyle \frac{{\mathit{dV}}_1}{\mathit{dt}}}\le \mathsf{\Lambda }K-({\alpha }_1+{\mu }_1)V_1.$$

By the comparison theorem, it follows that

$$0\le V_1\left(t\right)\le max\left\{V_1\left(0\right),{\displaystyle \frac{\mathsf{\Lambda }K}{{\alpha }_1+{\mu }_1}}\right\}=M_1,\mathrm{for}\mathrm{all}t>0.$$

For the mid-aged vehicle cohort $V_2\left(t\right)$:

$${\displaystyle \frac{{\mathit{dV}}_2}{\mathit{dt}}}={\alpha }_1{V}_1-({\alpha }_2+{\mu }_2)V_2.$$

The boundedness of $V_1\left(t\right)$ implies

$${\displaystyle \frac{{\mathit{dV}}_2}{\mathit{dt}}}\le {\alpha }_1{M}_1-({\alpha }_2+{\mu }_2)V_2.$$

Applying the comparison theorem again yields

$$0\le V_2\left(t\right)\le max\left\{V_2\left(0\right),{\displaystyle \frac{{\alpha }_1{M}_1}{{\alpha }_2+{\mu }_2}}\right\}=M_2,\mathrm{for}\mathrm{all}t>0.$$

For the old vehicle cohort $V_3\left(t\right)$:

$${\displaystyle \frac{{\mathit{dV}}_3}{\mathit{dt}}}={\alpha }_2{V}_2-{\mu }_3{V}_3.$$

Using the boundedness of $V_2\left(t\right)$, we obtain

$${\displaystyle \frac{{\mathit{dV}}_3}{\mathit{dt}}}\le {\alpha }_2{M}_2-{\mu }_3{V}_3.$$

By comparison, it follows that

$$0\le V_3\left(t\right)\le max\left\{V_3\left(0\right),{\displaystyle \frac{{\alpha }_2{M}_2}{{\mu }_3}}\right\}=M_3,\mathrm{for}\mathrm{all}t>0.$$

Finally, consider the carbon concentration equation

$${\displaystyle \frac{\mathit{dC}}{\mathit{dt}}}=\delta +\theta H+{\beta }_1{V}_1+{\beta }_2{V}_2+{\beta }_3{V}_3-({\gamma }_1+{\gamma }_2)C.$$

Since $V_1\left(t\right),V_2\left(t\right),$ and $V_3\left(t\right)$ are bounded, let

$$M=\delta +\theta K+{\beta }_1{M}_1+{\beta }_2{M}_2+{\beta }_3{M}_3.$$

Thus,

$${\displaystyle \frac{\mathit{dC}}{\mathit{dt}}}\le M-({\gamma }_1+{\gamma }_2)C.$$

By the comparison theorem,

$$0\le C\left(t\right)\le max\left\{C\left(0\right),{\displaystyle \frac{M}{{\gamma }_1+{\gamma }_2}}\right\}=M_4,\mathrm{for}\mathrm{all}t>0.$$

Hence, all solution sets of the system (1) remain bounded for all $t>0$. Consequently, there exists a positively invariant compact region $\mathsf{\Omega }\subset {\mathbb{R}}_{+}^5$ that attracts all the trajectories of the system (1). □

3.2. Equilibrium Points of the System

Equilibrium points of the system (1) correspond to steady states where all state variables remain constant over time. These points are obtained by setting the right-hand sides of the system to zero. That is, an equilibrium point

$$(H^{*},V_1^{*},V_2^{*},V_3^{*},C^{*})$$

satisfies

$$\mathit{dH}/\mathit{dt}=\left({\mathit{dV}}_1\right)/\mathit{dt}=\left({\mathit{dV}}_2\right)/\mathit{dt}=\left({\mathit{dV}}_3\right)/\mathit{dt}=\mathit{dC}/\mathit{dt}=0.$$

Using system (1), the equilibrium conditions are therefore given by

$$\left\{\begin{array}{c}0={\mathit{sH}}^{*}\left(1-{\displaystyle \frac{H^{*}}{K}}\right)\hfill \\ 0=\mathsf{\Lambda }{H}^{*}-({\alpha }_1+{\mu }_1)V_1^{*}\hfill \\ 0={\alpha }_1{V}_1^{*}-({\alpha }_2+{\mu }_2)V_2^{*}\hfill \\ 0={\alpha }_2{V}_2^{*}-{\mu }_3{V}_3^{*}\hfill \\ 0=\delta +\theta H^{*}+{\beta }_1{V}_1^{*}+{\beta }_2{V}_2^{*}+{\beta }_3{V}_3^{*}-({\gamma }_1+{\gamma }_2)C^{*}\hfill \end{array}\right.$$

The first equation satisfies either

$$H^{*}=0$$

or

$$H^{*}=K.$$

Thus, system (1) admits two equilibrium points.

If

$$H^{*}=0,$$

$$V_1^{*}=V_2^{*}=V_3^{*}=0,$$

and

$$C^{*}={\displaystyle \frac{\delta }{{\gamma }_1+{\gamma }_2}}.$$

Thus, the system admits the boundary equilibrium

$$E_0^{*}=\left(0,0,0,0,{\displaystyle \frac{\delta }{{\gamma }_1+{\gamma }_2}}\right).$$

This equilibrium corresponds to a hypothetical scenario in which no human population and no vehicles exist, and the atmospheric CO₂ concentration is determined solely by natural processes and environmental sequestration.

If

$$H^{*}=K,$$

we obtain the interior equilibrium

$$E_1^{*}=(H^{*},V_1^{*},V_2^{*},V_3^{*},C^{*}),$$

where

$$\begin{array}{cc}\hfill H^{*}& =K,\hfill \\ \hfill V_1^{*}& ={\displaystyle \frac{K\mathsf{\Lambda }}{{\alpha }_1+{\mu }_1}},\hfill \\ \hfill V_2^{*}& ={\displaystyle \frac{K\mathsf{\Lambda }{\alpha }_1}{({\alpha }_1+{\mu }_1)({\alpha }_2+{\mu }_2)}},\hfill \\ \hfill V_3^{*}& ={\displaystyle \frac{K\mathsf{\Lambda }{\alpha }_1{\alpha }_2}{({\alpha }_1+{\mu }_1)({\alpha }_2+{\mu }_2){\mu }_3}},\mathrm{and}\hfill \\ \hfill C^{*}& ={\displaystyle \frac{{\beta }_1{V}_1^{*}+{\beta }_2{V}_2^{*}+{\beta }_3{V}_3^{*}+\delta +K\theta }{{\gamma }_1+{\gamma }_2}}.\hfill \end{array}$$

Since all parameters are positive, the interior equilibrium $E_1^{*}$ is strictly positive and biologically feasible.

3.3. Local Stability Analysis of the Equilibrium Points

To analyze the local stability of the equilibrium points, we first compute the Jacobian matrix of system (1).

Let

$$E=(H,V_1,V_2,V_3,C),$$

then Jacobian matrix $J\left(E\right)$ of system (1) is

$$J\left(E\right)=\left(\begin{array}{ccccc}s(1-2H/K)& 0& 0& 0& 0\\ \mathsf{\Lambda }& -{\alpha }_1-{\mu }_1& 0& 0& 0\\ 0& {\alpha }_1& -{\alpha }_2-{\mu }_2& 0& 0\\ 0& 0& {\alpha }_2& -{\mu }_3& 0\\ \theta & {\beta }_1& {\beta }_2& {\beta }_3& -{\gamma }_1-{\gamma }_2\end{array}\right).$$

Local Stability of the boundary equilibrium point $E_0^{*}$:

To investigate the local stability of the boundary equilibrium point $E_0^{*}$, the Jacobian matrix is evaluated at this point, yielding

$$J\left(E_0^{*}\right)=\left(\begin{array}{ccccc}s& 0& 0& 0& 0\\ \mathsf{\Lambda }& -{\alpha }_1-{\mu }_1& 0& 0& 0\\ 0& {\alpha }_1& -{\alpha }_2-{\mu }_2& 0& 0\\ 0& 0& {\alpha }_2& -{\mu }_3& 0\\ \theta & {\beta }_1& {\beta }_2& {\beta }_3& -{\gamma }_1-{\gamma }_2\end{array}\right).$$

Since the matrix is lower triangular, its eigenvalues are given by the diagonal entries:

$${\lambda }_1=s,{\lambda }_2=-({\gamma }_1+{\gamma }_2),{\lambda }_3=-({\alpha }_1+{\mu }_1),{\lambda }_4=-({\alpha }_2+{\mu }_2),{\lambda }_5=-{\mu }_3.$$

Since $s>0$, one eigenvalue is positive. Therefore, the equilibrium point $E_0^{*}$ is unstable.

Theorem 3.

The boundary equilibrium point $E_0^{*}$ of system (1) is unstable.

This result indicates that a system with no human population and no vehicles cannot persist. Any small introduction of population will cause the system to move away from $E_0^{*}$.

Local Stability of the Interior Equilibrium $E_1^{*}$: Evaluating the Jacobian matrix at $E_1^{*}$ gives

$$J\left(E_1^{*}\right)=\left(\begin{array}{ccccc}-s& 0& 0& 0& 0\\ \mathsf{\Lambda }& -{\alpha }_1-{\mu }_1& 0& 0& 0\\ 0& {\alpha }_1& -{\alpha }_2-{\mu }_2& 0& 0\\ 0& 0& {\alpha }_2& -{\mu }_3& 0\\ \theta & {\beta }_1& {\beta }_2& {\beta }_3& -{\gamma }_1-{\gamma }_2\end{array}\right).$$

The eigenvalues of this matrix are

$${\lambda }_1=-s,{\lambda }_2=-({\gamma }_1+{\gamma }_2),{\lambda }_3=-({\alpha }_1+{\mu }_1),{\lambda }_4=-({\alpha }_2+{\mu }_2),{\lambda }_5=-{\mu }_3.$$

Since all parameters are positive, all eigenvalues are negative. Thus we have proved the following.

Theorem 4.

The interior equilibrium point $E_1^{*}$ of system (1) is locally asymptotically stable within the invariant region Ω.

4. Global Stability Analysis

Theorem 5.

The interior equilibrium point $E_1^{*}$ of system (1) is globally asymptotically stable in the invariant region Ω.

Proof. 

First, consider the human population equation

$${\displaystyle \frac{\mathit{dH}}{\mathit{dt}}}=\mathit{sH}\left(1-{\displaystyle \frac{H}{K}}\right).$$

The logistic equation admits the explicit solution

$$H\left(t\right)={\displaystyle \frac{{\mathit{KH}}_0}{H_0+(K-H_0)e^{-\mathit{st}}}},$$

where $H_0=H\left(0\right)>0$. Consequently,

$$\underset{t\to \infty }{lim}H\left(t\right)=K.$$

Since the interior equilibrium satisfies

$$H^{*}=K,$$

it follows that

$$\underset{t\to \infty }{lim}\left(H\left(t\right)-H^{*}\right)=0.$$

Next, define the error variable

$$e_1\left(t\right)=V_1\left(t\right)-V_1^{*}.$$

Using the equation

$${\displaystyle \frac{{\mathit{dV}}_1}{\mathit{dt}}}=\mathsf{\Lambda }H-({\alpha }_1+{\mu }_1)V_1,$$

together with the equilibrium relation

$$0=\mathsf{\Lambda }{H}^{*}-({\alpha }_1+{\mu }_1)V_1^{*},$$

we obtain

$${\dot{e}}_1=\mathsf{\Lambda }(H-H^{*})-({\alpha }_1+{\mu }_1)e_1.$$

Applying the variation-of-constants formula yields

$$e_1\left(t\right)=e^{-({\alpha }_1+{\mu }_1)t}{e}_1\left(0\right)+\mathsf{\Lambda }{\int }_0^t{e}^{-({\alpha }_1+{\mu }_1)(t-\tau )}\left(H\left(\tau \right)-H^{*}\right)d\tau .$$

Since ${\alpha }_1+{\mu }_1>0$ and

$$H\left(t\right)-H^{*}\to 0\mathrm{as}t\to \infty ,$$

the first term converges to zero exponentially, while the integral term also tends to zero. Therefore,

$$\underset{t\to \infty }{lim}{e}_1\left(t\right)=0,$$

which implies

$$\underset{t\to \infty }{lim}{V}_1\left(t\right)=V_1^{*}.$$

Similarly, define

$$e_2\left(t\right)=V_2\left(t\right)-V_2^{*}.$$

Using

$${\displaystyle \frac{{\mathit{dV}}_2}{\mathit{dt}}}={\alpha }_1{V}_1-({\alpha }_2+{\mu }_2)V_2$$

and the equilibrium identity

$$0={\alpha }_1{V}_1^{*}-({\alpha }_2+{\mu }_2)V_2^{*},$$

we obtain

$${\dot{e}}_2={\alpha }_1{e}_1-({\alpha }_2+{\mu }_2)e_2.$$

Applying the variation-of-constants formula gives

$$e_2\left(t\right)=e^{-({\alpha }_2+{\mu }_2)t}{e}_2\left(0\right)+{\alpha }_1{\int }_0^t{e}^{-({\alpha }_2+{\mu }_2)(t-\tau )}{e}_1\left(\tau \right)d\tau .$$

Since ${\alpha }_2+{\mu }_2>0$ and $e_1\left(t\right)\to 0$, it follows that

$$\underset{t\to \infty }{lim}{e}_2\left(t\right)=0.$$

Hence,

$$\underset{t\to \infty }{lim}{V}_2\left(t\right)=V_2^{*}.$$

Proceeding similarly, let

$$e_3\left(t\right)=V_3\left(t\right)-V_3^{*}.$$

Using

$${\displaystyle \frac{{\mathit{dV}}_3}{\mathit{dt}}}={\alpha }_2{V}_2-{\mu }_3{V}_3$$

and the equilibrium relation

$$0={\alpha }_2{V}_2^{*}-{\mu }_3{V}_3^{*},$$

we obtain

$${\dot{e}}_3={\alpha }_2{e}_2-{\mu }_3{e}_3.$$

Hence,

$$e_3\left(t\right)=e^{-{\mu }_{3}t}{e}_3\left(0\right)+{\alpha }_2{\int }_0^t{e}^{-{\mu }_3(t-\tau )}{e}_2\left(\tau \right)d\tau .$$

Since ${\mu }_3>0$ and $e_2\left(t\right)\to 0$, we conclude that

$$\underset{t\to \infty }{lim}{e}_3\left(t\right)=0,$$

which implies that

$$\underset{t\to \infty }{lim}{V}_3\left(t\right)=V_3^{*}.$$

Finally, define

$$e_4\left(t\right)=C\left(t\right)-C^{*}.$$

Using the carbon dynamics equation

$${\displaystyle \frac{\mathit{dC}}{\mathit{dt}}}=\delta +\theta H+{\beta }_1{V}_1+{\beta }_2{V}_2+{\beta }_3{V}_3-({\gamma }_1+{\gamma }_2)C,$$

together with the equilibrium identity

$$0=\delta +\theta H^{*}+{\beta }_1{V}_1^{*}+{\beta }_2{V}_2^{*}+{\beta }_3{V}_3^{*}-({\gamma }_1+{\gamma }_2)C^{*},$$

we obtain

$${\dot{e}}_4=\theta (H-H^{*})+{\beta }_1{e}_1+{\beta }_2{e}_2+{\beta }_3{e}_3-({\gamma }_1+{\gamma }_2)e_4.$$

Applying the variation-of-constants formula yields

$$e_4\left(t\right)=e^{-({\gamma }_1+{\gamma }_2)t}{e}_4\left(0\right)+{\int }_0^t{e}^{-({\gamma }_1+{\gamma }_2)(t-\tau )}\left[\theta \left(H\left(\tau \right)-H^{*}\right)+{\beta }_1{e}_1\left(\tau \right)+{\beta }_2{e}_2\left(\tau \right)+{\beta }_3{e}_3\left(\tau \right)\right]d\tau .$$

Since

$$H\left(t\right)-H^{*}\to 0,e_1\left(t\right)\to 0,e_2\left(t\right)\to 0,e_3\left(t\right)\to 0,$$

and ${\gamma }_1+{\gamma }_2>0$, it follows that

$$\underset{t\to \infty }{lim}{e}_4\left(t\right)=0.$$

Therefore,

$$\underset{t\to \infty }{lim}C\left(t\right)=C^{*}.$$

Combining the above limits, we obtain

$$\underset{t\to \infty }{lim}(H\left(t\right),V_1\left(t\right),V_2\left(t\right),V_3\left(t\right),C\left(t\right))=(H^{*},V_1^{*},V_2^{*},V_3^{*},C^{*}),$$

for every initial condition in $\mathsf{\Omega }$. Hence, the interior equilibrium point $E_1^{*}$ is globally asymptotically stable in the positively invariant region $\mathsf{\Omega }$. □

5. Quantitative Analysis

5.1. Data Set Introduction

To calibrate the model parameters, numerical simulations were initialized at an initial time $t_0=2010$. The initial atmospheric CO₂ concentration was set to $C\left(0\right)=390.1$ parts per million (ppm) according to data reported by the national oceanic and atmospheric administration [30]. The corresponding global human population was $H\left(0\right)=6.99\times {10}^9$, based on United Nations demographic data [31].

In 2010, approximately $1.015\times {10}^9$ vehicles were operating worldwide [32]. Based on the fleet distribution ratios reported in [27,28,29], the global vehicle population was partitioned into three mileage-based cohorts. Specifically, the initial conditions were set as

$$V_1\left(0\right)=2.5375\times {10}^8,$$

$$V_2\left(0\right)=4.5675\times {10}^8,$$

$$V_3\left(0\right)=3.405\times {10}^8,$$

corresponding to new, mid-aged, and old vehicles, respectively. Observed atmospheric CO₂ data for the period 2010–2025 were obtained from the National Oceanic and Atmospheric Administration [30] and are listed in

Table 3. Atmospheric CO₂ concentration (ppm) from 2010 to 2025.

Year	2010	2011	2012	2013	2014	2015	2016	2017
CO2 (ppm)	390.1	391.85	394.06	396.74	398.81	401.01	404.41	406.76
Year	2018	2019	2020	2021	2022	2023	2024	2025
CO2 (ppm)	408.72	411.65	414.21	416.41	418.53	421.08	423.61	425.31

.

The unknown parameters were estimated by fitting the model to observed CO₂ concentration data spanning 2010–2025 using nonlinear least squares optimization. Specifically, MATLAB, R2024a, lsqcurvefit function was employed, which minimizes the sum of squared residuals between the model output and observed data. Since box constraints were imposed on all parameters, the trust region reflective algorithm was automatically selected as the underlying solver. All parameters were bounded within the interval [0, 1] to ensure physically meaningful estimates. The optimization was configured with a maximum of 5000 function evaluations and a function tolerance of ${10}^{-8}$ to ensure adequate convergence. To assess the reliability of the estimated parameters, 95% confidence intervals were computed using MATLAB’s nlparci function, based on the residuals and Jacobian matrix returned at the optimal solution.

5.2. Model Fitting Performance Evaluation

To evaluate the predictive performance of the proposed model, the Normalized Root Mean Squared Error (RMSE) and the Nash–Sutcliffe Efficiency (NSE) are used.

The RMSE measures the percentage difference between simulated and observed values and indicates the model’s average prediction error. Based on established evaluation criteria [33,34,35], model performance is categorized as excellent (RMSE ≤ 10%), good (10% < RMSE ≤ 20%), reasonable (20% < RMSE ≤ 30%), or poor (RMSE > 30%).

The NSE is another widely used statistical measure for evaluating the predictive performance of simulation models. It compares the variance of model errors with the variance of the observed data. The NSE value ranges from $-\infty$ to 1, where a value closer to 1 indicates a better agreement between simulated and observed values. Following the classification criteria established in the literature [34,36,37], the model performance is categorized as very good (0.75 < NSE < 1), good (0.65 ≤ NSE ≤ 0.75), satisfactory (0.50 ≤ NSE < 0.65), or unsatisfactory (NSE < 0.50). The RMSE and NSE are computed, respectively, as

$$\mathrm{RMSE}={\displaystyle \frac{100}{\overline{O}}}\times \sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(P_i-O_i\right)}^2}$$

(2)

and

$$\mathrm{NSE}=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(P_i-O_i\right)}^2}{{\sum }_{i=1}^n{\left(O_i-\overline{O}\right)}^2}}$$

(3)

In these equations, $O_i$ and $P_i$ represent the observed and predicted values, respectively, while $\overline{O}$ denotes the mean of the observations and $n$ is the sample size.

Using Equation (2), the RMSE value was calculated as 0.286%, indicating excellent model accuracy. Similarly, applying Equation (3) produced an NSE value of 0.9962, which also falls within the very good performance category.

These statistical results indicate that the proposed model accurately reproduces the observed atmospheric CO₂ dynamics and demonstrates strong predictive capability. Figure 1 shows the model output (solid blue curve) closely follows the observed data points, demonstrating a high degree of agreement between the model and empirical observations. Therefore, the estimated parameters can be considered for further simulations and long-term projections. The parameter values, together with their corresponding units, are summarized in

Table 4. Parameter values and their corresponding units.

Parameters	Value	Unit	Source
$H\left(0\right)$	$6.96\times {10}^9$	Person	[31]
$V_1\left(0\right)$	$2.5375\times {10}^8$	Vehicle	[27,28,29,32]
$V_2\left(0\right)$	$4.5675\times {10}^8$	Vehicle	[27,28,29,32]
$V_3\left(0\right)$	$3.405\times {10}^8$	Vehicle	[27,28,29,32]
$C\left(0\right)$	$390.1$	ppm	[30]
$s$	$0.046$	$1/\mathrm{Year}$	Fitted
$K$	$11\times {10}^9$	Person	[25]
$\mathsf{\Lambda }$	$1.04\times {10}^{-4}$	Vehicle/(Person · Year)	Fitted
${\alpha }_1$	$0.164$	$1/\mathrm{Year}$	Fitted
${\alpha }_2$	$3.789\times {10}^{-2}$	$1/\mathrm{Year}$	Fitted
${\mu }_1$	$8.109\times {10}^{-3}$	$1/\mathrm{Year}$	Fitted
${\mu }_2$	$4.87\times {10}^{-2}$	$1/\mathrm{Year}$	Fitted
${\mu }_3$	$5.715\times {10}^{-2}$	$1/\mathrm{Year}$	Fitted
${\beta }_1$	$2.187\times {10}^{-10}$	ppm/(Vehicle · Year)	Fitted
${\beta }_2$	$6.836\times {10}^{-10}$	ppm/(Vehicle · Year)	Fitted
${\beta }_3$	$4.912\times {10}^{-9}$	ppm/(Vehicle · Year)	Fitted
$\theta$	$3.02\times {10}^{-10}$	ppm/(Person · Year)	Fitted
$\delta$	$0.035$	ppm/Year	Fitted
${\gamma }_1$	$0.0016$	ppm/Year	Fitted
${\gamma }_2$	$0.00276$	ppm/Year	Fitted

.

5.3. Numerical Simulations

In this section, we present numerical simulations to validate the global asymptotic stability of the interior equilibrium $E_1^{*}=(H^{*},V_1^{*},V_2^{*},V_3^{*},C^{*})$, as analytically established in Theorem 5. The simulations are conducted using the parameter values listed in Table 4. To confirm that the system consistently converges to $E_1^{*}$, the governing equations are integrated across multiple distinct initial points.

Figure 2a–d displays the phase space trajectories projected onto the $H–V_1–C$, $V_1–V_2–C$, $V_2–V_3–C$, and $V_1–V_2–V_3$ subspaces, respectively, simulated over the time span $t\in [0,3000]$.

As illustrated, regardless of their initial points, all trajectories converge to the interior equilibrium point $E_1^{*}$. These numerical trajectories verify that the interior equilibrium state is globally attractive and highly robust against variations in initial perturbations. Ultimately, these simulation results strongly align with and validate the formal analytical proof provided in Theorem 5.

5.4. Global Sensitivity Analysis

To quantify how individual model parameters influence atmospheric CO₂ concentrations, we employed Latin Hypercube Sampling (LHS) in conjunction with Partial Rank Correlation Coefficient (PRCC) analysis. This LHS–PRCC framework provides a robust global sensitivity measure, where the absolute magnitude of the PRCC (ranging from 0 to 1) identifies the parameters with the most significant impact on model outcomes [38,39]. Within this context, the sign of the coefficient dictates the direction of influence: positive PRCC values indicate a direct relationship, whereas negative values signify an inverse relationship [40].

For the LHS matrix generation, all parameters in model (1) were assigned uniform distributions. To ensure statistical convergence and the robustness of the resulting PRCCs, a sample size of $N=3000$ was generated. This global analysis was executed by perturbing each parameter within a $\pm 50\%$ range of its established baseline value, thereby providing a comprehensive view of the system’s sensitivity and capturing potential extreme real-world scenarios.

The ranking of parameters based on their absolute PRCC values is presented in

Table 5. Parameter ranking (most influential first).

Rank	Parameter	PRCC Value
1	$\theta$	$+0.789$
2	${\beta }_3$	$+0.386$
3	${\gamma }_2$	$-0.372$
4	${\mu }_3$	$-0.252$
5	${\alpha }_2$	$+0.0973$
6	${\mu }_2$	$-0.0951$
7	${\mu }_1$	$-0.0461$
8	${\beta }_2$	$+0.0346$
9	$\mathsf{\Lambda }$	$+0.018$
10	${\beta }_1$	$+0.0173$
11	${\alpha }_1$	$-0.00643$

. The results indicate that the human activity emission coupling factor ($\theta$) is the most influential parameter, with the highest PRCC value ($+0.789$). This strong influence suggests that human-driven activities such as energy consumption, industrial processes, and land-use changes play a dominant role in determining atmospheric CO₂ levels. Therefore, reductions in emissions associated with these activities could significantly decrease atmospheric CO₂ concentration.

The CO₂ emission rate from old-aged vehicles (${\beta }_3$) is identified as the second-most-influential parameter (PRCC $=+0.386$). This finding highlights the substantial contribution of older vehicles to atmospheric CO₂ emissions, mainly due to inefficient combustion systems and outdated emission-control technologies. Closely following this is the anthropogenic depletion coefficient (${\gamma }_2$), which ranks third with a strong negative sensitivity of $-0.372$. This inverse relationship highlights the critical role of mitigation strategies, such as reforestation and carbon sequestration, in actively counterbalancing atmospheric accumulation.

Finally, parameters governing vehicle lifecycles, specifically the retirement rate of old-aged vehicles (${\mu }_3$) demonstrate a moderate yet noticeable influence (PRCC $=-0.252$). This suggests that while sequestration and industrial regulations are primary levers, accelerating the removal of inefficient vehicles from the transport network provides a significant secondary benefit for climate stabilization.

The sensitivity analysis presented in Figure 3 identifies $\theta$ and ${\beta }_3$ as the parameters with the most significant positive PRCC values. This confirms that atmospheric CO₂ accumulation is most strongly driven by human activity emissions and the output from aging vehicle fleets.

Conversely, the parameter ${\gamma }_2$ shows a strong negative sensitivity, indicating that it plays an important role in reducing atmospheric CO₂ concentration. The simulation results show that increasing ${\gamma }_2$ leads to a decrease in CO₂ levels. Similarly, the parameter ${\mu }_3$, which represents the retirement rate of old vehicles, also has a negative effect on CO₂ levels.

Figure 4 presents a comparative sensitivity analysis of projected CO₂ concentrations over 30-years. As illustrated in Figure 4a, reducing the human activity emission factor ($\theta$) yields the most substantial decline in atmospheric CO₂ concentration, notably, a 25% reduction in $\theta$ lowers the projected CO₂ level from 454.88 ppm to 436.7 ppm. Similarly, Figure 4b reveals that decreasing the emission rate of old vehicles (${\beta }_3$) markedly reduces overall CO₂ concentrations. In contrast, Figure 4c demonstrates that increasing the anthropogenic CO₂ removal coefficient (${\gamma }_2$) enhances carbon sequestration capacity. Finally, Figure 4d indicates that accelerating the retirement rate of old vehicles (${\mu }_3$) further contributes to CO₂ mitigation by removing high-emitting vehicles from system.

6. Results and Discussion

Balancing atmospheric CO₂ mitigation with the continuously increasing demand for transportation remains one of the defining challenges of sustainable development. In this study, we developed and analyzed a nonlinear mathematical model of atmospheric CO₂ dynamics by coupling atmospheric carbon concentration with human population growth and an age-structured vehicle population. Unlike many existing mathematical models that treat transportation emissions as a homogeneous source, the proposed model explicitly accounts for the differential contributions of vehicle cohorts by age. This enables the model to capture the elevated emissions associated with aging vehicle fleets and their long-term influence on atmospheric carbon accumulation.

The qualitative analysis demonstrated that the solutions of the model remain positive and bounded for all biologically feasible initial conditions, confirming the physical relevance of the proposed framework. The stability analysis further showed that the boundary equilibrium point, $E_0^{*}$, is unstable, whereas the interior equilibrium point, $E_1^{*}$, is globally asymptotically stable. From an environmental perspective, these findings proved that the coupled human–vehicle–atmospheric system naturally evolves toward a persistent equilibrium characterized by non-zero atmospheric CO₂ concentrations.

This interpretation is consistent with the findings of large-scale climate assessments. According to the Intergovernmental Panel on Climate Change (IPCC) report [41], continued economic and population growth, together with ongoing dependence on fossil fuels, will continue to increase greenhouse gas emissions unless effective mitigation measures are implemented. Similarly, analyses of the global carbon budget show that current emission trends are not consistent with the pathways needed to limit global warming to internationally agreed targets [42]. Therefore, the mathematical stability results obtained in this study provide theoretical support for the empirical evidence that atmospheric carbon accumulation is unlikely to decline on its own without deliberate policy interventions.

Using historical data, model calibration, and parameter estimation, projected an atmospheric CO₂ concentration of approximately 455 ppm by the year 2040 if present trends persist. This estimate falls within the range suggested by previous climate investigations and stabilization studies. For example, den Elzen et al. [43] showed that delayed mitigation efforts substantially increase the likelihood of exceeding stabilization thresholds, while Mukherji et al. [44] highlighted that continued anthropogenic forcing under current development trajectories results in sustained increases in atmospheric carbon concentrations. Moreover, observational records reported by NASA [45] continue to document a persistent rise in atmospheric CO₂, reinforcing the plausibility of our model projections. The agreement between our model and these climate assessments suggests that such Mathematical models can effectively reproduce the dominant mechanisms governing long-term atmospheric carbon dynamics.

The global sensitivity analysis based on the Partial Rank Correlation Coefficient (PRCC) method identified human-related activities as the most influential determinants of atmospheric CO₂ accumulation. This result agrees with the broad scientific consensus that energy consumption, industrial activities, land-use change, and population-driven demand are the principal drivers of anthropogenic climate change [41,42].

More importantly, our analysis identified the emission rate associated with older vehicles as the second-most-influential parameter affecting future atmospheric carbon levels. This finding extends previous transportation studies by demonstrating that fleet composition may be as important as fleet size in determining long-term environmental outcomes. Transportation research has consistently shown that older vehicles emit substantially higher levels of pollutants because of technological obsolescence, engine deterioration, and the absence of modern emission-control systems. Studies by Anenberg et al. [46] and Barrett et al. [47], for example, reported that replacing aging vehicle fleets with cleaner technologies yields considerable environmental and public-health benefits. Similarly, assessments by the International Energy Agency have emphasized that improvements in vehicle efficiency can be offset by slow fleet turnover and increasing travel demand [48]. By explicitly incorporating vehicle age structure, the present model provides a mathematical explanation for these empirical observations and identifies aging vehicles as a critical leverage point for intervention.

7. Recommendations

The PRCC analysis identified the human activity emission coupling factor, $\theta$, as the most influential parameter governing atmospheric CO₂ accumulation. Consequently, emission-reduction policies should primarily target sectors associated with human activities, including energy production, industrial processes, and land-use practices. The sensitivity analysis further indicates that a 25% reduction in the human activity emission parameter could decrease the projected atmospheric CO₂ concentration by approximately 17 ppm by 2040 relative to the baseline projection. This finding suggests that interventions aimed at improving energy efficiency, expanding renewable energy deployment, and promoting low-carbon production practices would have the greatest impact on mitigating future atmospheric CO₂ accumulation.

The older-vehicle emission parameter, ${\beta }_3$, was identified as the second-most-influential factor affecting atmospheric CO₂ dynamics. This result demonstrates that aging vehicle fleets contribute disproportionately to long-term carbon accumulation. The model predicts that a 25% reduction in emissions from older vehicles could lower the projected atmospheric CO₂ concentration by approximately 9 ppm by 2040. Therefore, transportation policies should prioritize accelerated fleet renewal through stricter vehicle inspection and maintenance programs, vehicle scrappage incentives, the enforcement of emission standards for aging vehicles, and increased support for the adoption of cleaner and electric vehicle technologies.

Furthermore, the anthropogenic CO₂ depletion parameter, ${\gamma }_2$, was found to play an important role in mitigating atmospheric carbon accumulation. Sensitivity analysis indicates that increasing the effective anthropogenic CO₂ removal rate by 25% could reduce the projected atmospheric CO₂ concentration by approximately 8 ppm by 2040. Accordingly, governments and stakeholders should establish measurable targets for expanding reforestation and afforestation programs and invest in carbon capture, utilization, and storage technologies to enhance atmospheric carbon removal.

8. Future Work

In future research, the model can be extended to include additional factors, such as electric vehicles and the adoption of renewable energy.