MITIGATION OF CLIMATE CHANGE DUE TO EXCESSIVE CARBON DIOXIDE EMISSION AND ACCUMULATION: A MATHEMATICAL MODEL APPROACH

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INTRODUCTION
The global community is currently plagued with many environmental challenges. Most of these problems have been attributed to global climate change arising from over exploitation of our planet. Climate change is caused by excessive emission and accumulation of greenhouse gases into our environment.The most crucial contributor of all these gases to the current narrative of climate change issue is carbon dioxide. Excessive emission of this gas into our environment raises the global temperature beyond optimal level and the attendant consequences of this are manifested through various natural disasters such as aridity, flooding, drought, and many others. Human activities such as agriculture, industrialization, urbanization, deforestation, transportation, mining, electricity generation have contributed greatly to the excessive emission of the greenhouse gases, especially carbon dioxide into our environment. The devastating effect of climate change is felt by almost every country in the world. There is growing concern about climate change because of the rise in average global temperature. This has been attributed greatly to excessive emission and accumulation of greenhouse gases. Civilization and anthropogenic activities have prompted the hazards of global warming, climate change and bad impacts on our quality of life [1].
Six greenhouse gases are covered under the Kyoto Protocol. Carbon dioxide makes up the greatest part of greenhouse gas emissions and has become the most vital anthropogenic greenhouse gas. The greatest contributions to GHG emissions are from the electricity and petroleum industries, followed by the transportation sector and other industries. Decision makers formulate practicable and sustainable environmental policies, regulate energy structure of industrial production, and efficiently decrease carbon emissions and help in achieving a country's environmental stability through accurate carbon dioxide forecasting [2]. According to the National Plan for Adaption to Climate Change (NPACC), South America risks losing 23% of its species as a consequence of climate warming [3]. Brazil as the fifth largest country in the global geography, made a commitment under the Paris Agreement, that by 2025, greenhouse gases emissions would be at 37% and that by 2020, carbon dioxide emission will be around 43% [3]. [4] noted that there was a rise in temperature of water due to excessive emission of greenhouse gases, which consequently led to a decrease in the level of dissolved oxygen and also a rise in the rate of circulation of disintegrated oxygen by aquatic inhabitants, thereby leading to a decline in the density of these aquatic animals.
Different mathematical models on climate change have been developed by considering effects of one or two of deforestation, human population, population pressure, migration, urbanization, industrialization, awareness, to mention but a few. [5] in studying the problem of global warming due to increasing emission of greenhouse gases, 83% of which were from different human activities and the remaining 17% being from wild animals, developed and analyzed a four compartment mathematical model. Their findings revealed that human beings were solely responsible for the current increased emission of the greenhouse gases and global warming, and the attendant consequences of this was felt by both the human community and the forest ecosystem.
[6] studied and investigated a mathematical model applied to green building concept for sus- on the price of carbon abatement and the rate at which the expected CO 2 concentration in the atmosphere should be reduced. [9] developed a nonlinear mathematical model to study the effect of reforestation as well as the delay encountered between the measurement of forest data and implementation of reforestation efforts on the control of atmospheric concentration of carbon dioxide. From their analysis, concentration of atmospheric carbon dioxide decreased due to reforestation. [10] developed an ecological type nonlinear mathematical model for the removal of carbon dioxide from the regional atmosphere by externally introduced liquid species, which could react with this gas and remove it by gravity. He showed analytically and numerically that the concentration of carbon dioxide decreased as the rate of introduction of externally introduced liquid species increased. [11] developed a simplified mathematical model to study the effect of carbon dioxide on the mechanism of global warming by considering emissivity of the gases (water vapour and carbon dioxide) to be dependent on three variables which were temperature, gas concentration and beam length of the atmospheric layer defined through the model.
Their findings revealed that the temperature of the Earth rose enormously with CO 2 concentration. [12] provided an accessible mathematical approach used to design models that show smooth patterns of global carbon dioxide emissions which are in agreement with the UN climate targets. His findings through appropriate interpolations showed that the smooth pathways would overcome a global lack of no-carbon energy in the short run and tolerate low emissions that could disappear as adequately required from 2040s with the direct removal of CO 2 becoming immaterial in the long run. [13] used optimal control theory to obtain effective and useful distribution of investments in reforestation and encouragement of certain technology to attain a carbon dioxide emission schedule for 2020 in the Legal Brazilian Amazon (BA). Their simulation results showed that a forest area of about 3.7 × 10 6 km 2 was needed for CO 2 emission target of 3.76 × 10 8 tonnes in 2020, which could require about 4.5 × 10 5 km 2 reforestation out of this total land area. [14] in studying the grave problem of climate change caused by rise in the mean global temperature resulting from excessive emission of greenhouse gases, especially carbon dioxide, developed a nonlinear mathematical model made up of five compartments. Their results revealed that the concentration of CO 2 reduced as the rates of spray of liquid phase and solid particulate matters increase in the atmosphere and could be eliminated, if the rates of the spray of the external species were very large. In trying to reduce carbon dioxide emissions through carbon capture and storage in a saline aquifer, [15] developed and analyzed simple mathematical models for the trapping processes of CO 2 . His findings revealed that the emission of CO 2 can be greatly reduced using saline aquifer. [16] developed a mathematical model for removing carbon dioxide from the atmosphere by the introduction of external species which reacts with the CO 2 to render it harmless and also through plantation of leafy greenbelts (which also removes the CO 2 gas through the natural process of photosynthesis). From their findings from both the analytical and numerical simulation, as the rate of introduction of external species In the study we are proposing to undertake, we shall narrow our focus on three lead papers.
[17], [18], and [19] developed mathematical models by considering the effects of externally introduced liquid species; externally introduced liquid species and plant biomass density; and awareness of human population respectively, on reduction of consequences of global warming. No single mitigation measure is enough to combat the dangers posed by climate change.
Therefore, we need a combination of mitigation measures to produce a better result. Hence, we propose to incorporate photosynthetic biomass density, good conservation policies, enlightenment programmes, and direct air capture technology in the model dynamics of reducing the excessive concentration of carbon dioxide emission and accumulation into the atmosphere.

MODEL FORMULATION
The following assumptions were made in the formulation of the model: i. All parameters and variables involved in the model are non-negative since the system under consideration is biological; ii. Excessive emission and accumulation of carbon dioxide, a greenhouse gas, into the atmosphere is solely responsible for current climate change; iii. The concentration of carbon dioxide in the atmosphere constantly increases due to human activities such as rapid increase in industrialization, excessive combustion of fossil fuels, urbanization, deforestation, modern lifestyles, to mention but a few; iv. There is a threshold concentration level (maximum tolerant level) of carbon dioxide beyond which the model becomes irrelevant; v. The effect of climate change arising from excessive emission and accumulation of carbon dioxide into the atmosphere will be under check by incorporating some mitigation measures; vi. The emission and accumulation of carbon dioxide follows a form of logistic growth model pattern; vii. The growth of the photosynthetic biomass density also follows a logistic growth pattern; viii. Photosynthetic biomass density can be increased through plantation, afforestation, seed dispersal and pollination; ix. The rate of incorporation of each mitigation measure is directly proportional to the concentration of carbon dioxide; x. Awareness of the hazards of excessive concentration of carbon dioxide in the atmosphere and the relevance of discouraging deforestation activities as well as encouraging good conservation policies would contribute to a reduction of excessive emission of carbon dioxide into the atmosphere.
The variables considered in this model are: Excessive concentration of carbon dioxide, C(t); Photosynthetic biomass density, P(t); Good conservation policies density, R(t); Enlightenment programmes density, E(t) and Direct air capture technology density, T(t). The intrinsic rate of accumulation of carbon dioxide in the atmosphere is denoted by β and the term βC 1 − C C m represents the cumulative accumulation of carbon dioxide in the atmosphere, where C m represents the maximum tolerated concentration of CO 2 beyond which the model becomes nonmeaningful. Due to the interaction between carbon dioxide in the atmosphere and the photosynthetic biomass, the concentration of CO 2 reduces at a rate of d 1 . In trying to mitigate against excessive emission of this gas, d 2 represents rate of reduction of the gas from implementation of good conservation policies, d 3 represents rate of reduction of concentration of CO 2 in the atmosphere due to success of the enlightenment programmes and d 4 represents the rate of reduction in concentration of CO 2 due to use of technology that could capture this gas and convert it to a less harmful substance or even store it for other use. Finally, µ 0 represents the natural rate of depletion in concentration of CO 2 in the atmosphere. The intrinsic rate of growth of the photosynthetic biomass is denoted by ω and the cumulative growth of this photosynthetic biomass is assumed to follow a logistic growth pattern as seen in the term ωP 1 − P N , where N is the carrying capacity for this photosynthetic biomass. Effective awareness programmes lead to people engaging in good conservation policies such as reforestation, afforestation, avoidance of bush burning, reclaiming of desert lands through some modern methods and many others. This in turn is assumed to contribute to the growth of the photosynthetic biomass density at a rate of τ. µ 1 represents the rate of decrease in photosynthetic biomass density due to natural phenomena such as wildfires, disease outbreak, flooding, drought, overgrazing and many more. There is also a decrease in this biomass due to human activities such as cutting down of forests for food, shelter, industrialization, urbanization, woods, honey, different medicinal products and many others. This could also be referred to as the artificial depletion in the photosynthetic biomass density and this occurs at a rate of µ 2 .
In the third equation, we assumed that the rate of of implementation of good conservation policies is directly proportional to the concentration of CO 2 in the atmosphere and hence, the reason for the term a 1 C, where a 1 is the rate of reduction of carbon dioxide due to good conservation policies. Due to positive impact of enlightenment, we assumed there is an improvement in good conservation policies and knowledge about the dangers of unchecked emission of carbon dioxide. Negligence or evasion of good conservation policies is assumed to to occur at rate of a 0 . Same analogy goes for the fourth and fifth equations. Hence, the deterministic nonlinear compartmental climate model equations are: subject to the non-negative initial conditions: In our formulation, it is assumed that the functions C(t), P(t), R(t), E(t), T (t) and their corre-

Concentration Number.
Using the analogy of the basic reproduction number, R 0 , in epidemiological mathematical models [20], [21], [22]; predation number , P 0 , in prey-predator relationship mathematical model [23] ;consumption number, C 0 , as also used in ecological models [24], [25], [26], [27], which are all using the idea of the Next Generation matrix [20], we have come up with a similar threshold quantity which we call Concentration Number, C * .
The concentration number, C * , is a threshold quantity that determines how dangerous the concentration of CO 2 is in the atmosphere. If C * < 1 , then the emission and accumulation of CO 2 is not harmful to the ecosystem. However, if C * > 1, then we have excessive emission and accumulation of CO 2 in the atmosphere which could lead to climate change as time evolves and hence, mitigation measures have to be put in place to check this excessive emission. To get the concentration number, we split the model equations to form two matrices F i and V i respectively. F i matrix contains interaction terms of the model equations while V i contains a negation of the non-interactive terms of the model equations. The matrix got from F i is denoted F while the one got from V i is denoted V. C * is the spectral radius of the next generation matrix, FV −1 . That is, We obtain the mathematical expression for concentration number, C * , as follows: Considering equation (1) at equilibrium; Therefore, the required point for obtaining the Concentration Number is thus: P, R, E and T are zero because it is assumed that at this point, there is no mitigation measure so that we would be able to find out if there is excessive emission or not.
Obtaining matrices F i and V i , we have: The matrices F and V are obtained by evaluating the Jacobian matrices of F i and V i at the point given by equation (8) as follows: By evaluation, we have: The next generation matrix (NGM), FV −1 , is determined using equations (8) and (13) and the result is given thus: The characteristic equation corresponding to FV −1 is: From the above, the eigenvalues of FV −1 are: The concentration number, C * is thus the spectral radius of FV −1 . That is, Considering the above eigenvalues, the concentration number which is equivalent to the maximum eigenvalue is 3.2. Equilibrium Points. The equilibrium points of the model system given by equations (1), (4) and (5) can be obtained by equating each of these equations to zero and solving for the dependent variables. That is; A summary of the four equilibrium points obtained are: Where,

Stability Analysis.
To carry out the local stability analysis of the equilibrium points obtained, we linearize the model system given by equations (1) to (5) by generating a Jacobian (characteristic) matrix, J, and then evaluating this matrix at each of the equilibrium points.
Proof. At the first equilibrium point (trivial equilibrium point), ε 0 = (0, 0, 0, 0, 0), we have the Jacobian matrix given by equation (17) as: The characteristic equation for the Jacobian matrix, J 0 , is given by: Evaluating this determinant, we obtain: Since all the parameters are non-negative, then; An equilibrium point is locally asymptotically stable if all the eigenvalues are real and negative.

Parameters and their Values.
The following parameter values were used to calculate the concentration number, equilibrium points, sensitivity indices and numerical simulation of the formulated model.  it helps in experimental designs, data assimilation and reduction in complexity of nonlinear models [29]. Here, we performed sensitivity analysis on the concentration number, C * got in equation (16), to determine the influence of each parameter on the dynamics of the formulated model and those that have higher impact on climate change due to excessive CO 2 emission. The results revealed the parameters that would aid the choice of the best mitigation measure(s) to adopt in combating excessive emission and accumulation of CO 2 in the atmosphere.
The normalized forward sensitivity index (NFSI) denoted by say, Ψ , of a variable say, Θ , with respect to a parameter say, π, is the ratio of the relative change in Θ to the relative change in π [29]. If Θ is differentiable with respect to π, the NFSI is defined mathematically as Results of the sensitivity analysis on each parameter of the concentration number( C * ) using the above NSFI formula and the parameter values in Table (2) are presented in Table (3). From the sensitivity analysis carried out using the parameter values in Table (2), we obtained the indices to be either positive or negative (Table 3). From the results presented in Table (3

NUMERICAL SIMULATIONS
The numerical simulation is performed to support the results of theoretical analysis and also present further discussions of the dynamics of the model system given by equations (1) to (5).
Using the parameter values in Table (2), the four equilibrium points for the model were calculated as:  Table (2) and the inbuilt solver, ode45 (operates based on 5th order Runge-Kutta method), the results given by Table (4) and Figures (1) to (12) were obtained. We note that the infinity value recorded for time taken to reach minimum excessive concentration of CO 2 in Table (   The excessive concentration of CO 2 in the atmosphere is dangerous and could pose some threats which could manifest in form of environmental phenomena like flooding, drought, wildlife migration, disease outbreak and many others. In Figure (1), three different hypothetical scenarios between the accumulation rate of CO 2 , β , and the intrinsic growth rate of the photosynthetic biomass, ω are simulated. The least value of the maximum excessive concentration of CO 2 (1.3748) was obtained when the rate of accumulation,β , is less than the intrinsic growth rate of the photosynthetic biomass. This scenario could solely be because adequate attention was paid to conserving and increasing the growth of the photosynthetic biomass which in turn led to more removal of the excessive concentration of CO 2 from the atmosphere. If the rate of increase in the photosynthetic biomass could effectively be maintained and sustained, it would lead to a great reduction of the excess CO 2 and possibly its total removal from the atmosphere.
The second scenario shows a moderate value for the maximum excessive concentration of CO 2 (2.7461) when the rate of accumulation of CO 2 is equal to the intrinsic growth rate of the photosynthetic biomass. But the highest value for the maximum excessive concentration of CO 2 (4.4508) was obtained when the accumulation rate of CO 2 was more than the intrinsic growth rate of the photosynthetic biomass. This result is to be expected and aligns with a common reasoning that density of available photosynthetic biomass is overwhelmed in the job of sequestering CO 2 from the atmosphere. Hence, the reason for the highest value compared to the other two earlier scenarios. Therefore, if this be the case, other mitigation measures may be needed in conjunction with the photosynthetic biomass (the natural regulator of CO 2 ) to help in curbing this excessive concentration that could cause other environmental hazards.
In Figure (2), the results obtained showed that when the accumulation rate was less than the reduction rate of CO 2 by the photosynthetic biomass, there was barely no excessive concentration of carbon CO 2 in the atmosphere to be removed (as seen by the absence of the line for β < d 1 in Figure (2)). This could be because the photosynthetic biomass has to remove any surplus concentration of CO 2 in the atmosphere due to their high density. When the two rates were equal (accumulation and reduction by photosynthetic biomass rates), we obtained a lower value for the maximum excessive concentration of CO 2 (0.4054). However, when the accumulation rate was higher than the reduction rate of CO 2 by the photosynthetic biomass, we got a very high value for the maximum excessive concentration of CO 2 (2.7461). This is about 6 times higher than the previous value of 0.4054. This situation reflects reality as the density of photosynthetic biomass available may not be enough to remove the excess accumulated concentration of CO 2 from the atmosphere. Hence, other mitigation measures are required.
In Figure(3   The natural mechanism in place to remove CO 2 from the atmosphere is the anabolic process called photosynthesis, through which 'chlorophyllic' biomass (photosynthetic biomass) manufacture food to sustain the ecosystem. But due to increased human population that leads to further quest for survival, accommodation, industrialization, urbanization and others, the natural system in place is overwhelmed. Forests are destroyed in search of many products and this lead to a reduction in the photosynthetic biomass density. By engaging in good conservation practices like reforestation, afforestation and minimizing excessive and uncontrolled destruction of the photosynthetic biomass, we could increase greatly the photosynthetic biomass density which in turn would contribute to reducing this excessive emitted and accumulated concentration of CO 2 in the atmosphere. This is evident in Figure ( imply that if we can put in some effort in increasing the photosynthetic biomass density, we can reduce the excessive emission and accumulation of CO 2 in the atmosphere by 12.11% and 32.25% respectively as seen in Table (4).
The photosynthetic biomass alone may not have the capacity to effectively remove the excessive emitted and accumulated concentration of CO 2 from the atmosphere within a relatively short period of time. Hence, we need other mitigation measures such as good conservation policies, enlightenment programmes and direct air capture technology to support the natural mechanism of photosynthetic biomass.
In Figure ( In Figure (12 ),comparison of different cases between the accumulation rate of CO 2 , β , and the combined rates of photosynthetic biomass, good conservation policies, enlightenment programmes and direct air capture technology are shown. When the accumulation rate was less than these combined rates from all the mitigation measures, the excessive concentration was removed and hence, the reason there is no line in Figure ( 12) to depict this particular scenario.
When the accumulation rate of CO 2 was equal to the combined rates of the mitigation measures, we got a maximum excessive concentration value of about 0.1000. But for the situation where the accumulation rate was higher than the combined rates of all the measures, we got a higher maximum excessive concentration of 0.3830 which is about 3 times higher than the previous value of 0.1000.

CONCLUSION
In this research paper, we focused our attention on carbon dioxide which is the most vital greenhouse gas that contributes to the recent global warming. We developed a five compartment deterministic nonlinear mathematical model to study this problem of climate change due to excessive emission and accumulation of carbon dioxide. In the mathematical analysis of the model, four equilibrium points were obtained. The local stability analysis of each of the equilibrium points was done. Using the idea of the next generation matrix (NGM), a novel threshold quantity called Concentration Number was derived. For our developed model, we obtained the concentration number to be 2.4207 using the parameter values given in Table(2). Since this value is greater than one, it means the excessive concentration of this gas could pose a potential threat to the environment. Hence, other mitigation measures were needed. The sensitivity analysis on the concentration number was done using the normalized forward sensitivity index (NFSI) method. The results revealed the most crucial parameters that influenced the dynamics of the model such as the accumulation rate of carbon dioxide, intrinsic growth rate of the photosynthetic biomass and others. These results also influenced and informed the choice of parameters varied in the simulation of the model.
The numerical simulation of the model was done using MATLAB R2013b software. Many influential parameters identified by carrying out sensitivity analysis were varied during the simulation of the model. A summary of the findings from the simulation results showed that the excessive and accumulated concentration of CO 2 in the atmosphere can be reduced.

DATA AVAILABILITY
The data used in the numerical simulations and other calculations in this study were got from research materials whose sources are appropriately cited as seen in Table (2) while others were fixed or assumed.

ACKNOWLEDGEMENT
The corresponding author on behalf of the authors is grateful to the African Union Commission for her financial assistance and sponsorship to study under her Pan African University Scholarship Scheme in Pan African University Institute of Basic Sciences, Technology and Innovation (PAUISTI).

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.