ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. MODELS FOR PREDICTING GLOBAL PLASTIC WASTE YUEXING HAO, GLENN SHAFER (FACULTY ADVISOR) ✵ ABSTRACT For more than half a century, plastic prod- ucts have been a part of people’s lives. When plastic waste is thrown into nature, it can cause a sequence of dangerous effects. Previous researchers esti- mated that global plastic waste in 2020 will be more than 400 million tons. To reduce plastic waste, they built scientific models to analyze the sources of plas- tic and provided solutions for regenerating these plastic wastes. However, their models are static and inaccurate, which may cause some false predictions. In this paper, we first observe the distribution of the real-world plastic waste data. Then, we build simple exponential growth model and logistics model to match these data. By testing different models on our plots, we discover that the SELF-ADAPTIVE MODEL is the best to describe and correctly predict our future plastic waste production, as this model combines the benefits of SIMPLE EXPONENTIAL GROWTH MODEL and the LOGISTIC MODEL. The self-Adaptive model has the potential to minimize the error rate and make the predictions more accurate. Based on this model, we can develop more accurate and informative solu- tions for the real-world plastic problems. 1 INTRODUCTION Plastic waste became a serious problem in the 21st century. According to data from 2017, sci- entists estimate that human beings produced around 8300 million metric tons (Mt) of virgin plas- tics. As of 2015, approximately 6300 Mt of plastic waste had been generated. Of this waste, only around 9% had been recycled[1]. Most plastic prod- ucts are sent directly to landfills or make their way to the ocean. It may take one hundred to thousands of years for plastic products to decompose, and the available places for landfill waste are becoming fewer and fewer[2,3]. Therefore, we should prevent this global crisis from becoming more serious. In this paper, we will introduce three models and find the model best suited for better prediction making. Based on the previous data, we will consider more factors and make the model more adaptive to the real-world situations. There are some research- ers who modeled past plastic consumption and pre- dicted the future. However, our model will be more accurate and have better data visualization. 2 METHODOLOGY To estimate the maximum levels of single- use or disposable plastic product waste, we should first know the constraints and extent. The constraints are the EFFICIENCY of the natural environment or the waste disposal companies at processing plastic pol- lution. If their efficiency of disposing plastic products is higher than the rate of people producing plastic waste, there will be no plastic pollution at all. There- fore, we consider the constraint to be the availability of processing plastic waste. The extent is the RANGE of our plastic waste, since our plastic waste cannot in- crease to infinity. We should set an upper bound and lower bound for the weight of our waste. The hard part of this problem is that if na- tions and governments do not accurately know their nation’s total production of plastic waste, it increases the difficulty of managing single-use plastic waste comprehensively. The composition of modern plas- tic waste is complex because the waste may contain both single-use and multi-use plastics. However, we can estimate the disposable plastics product usage by finding the characteristics of plastics waste’s his- torical change. Then, based on this trend, we can predict the quantity change of plastic waste in the fu- ture. In addition, there emerges another difficult aspect of this problem: there are many influencing factors regarding the generation of plastic waste. ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II There is no simple pattern or law that can be deter- mined by looking at the data. We should, therefore, develop a model which does not need a typical data distribution law and use less theoretical calculations that can still make an accurate prediction. That is to say, we may not need to add much mathematical functions or calculations inside the model, but the model can adaptive the different situations by itself. Therefore, in this paper, we employ the SELF-ADAPTIVE MODEL to estimate and analyze the data on plastic waste production over the past 5 to 10 years, and then predict its production in the next 30 years. It is called a self-adaptive model because this model does not have specific relationships among factors in a system. We cannot clearly indicate the relation- ships between the factors in the data, such as the in- fluences of countries’ rules, population density, lev- els of development. Thus, it is a good choice to use the self-adaptive model for this problem. 3 BUILDING PLASTIC POLLUTION MODELS In this part, we first define plastic pollution into seven kinds of waste: Textiles, Transportation, Packaging, Electronic, Consumer Products, Indus- trial Machinery, Building & Construction. These dif- ferent kinds of waste will be considered as different variables. Currently, plastic packaging is the greatest source of primary plastic production. Because of the rapid development of online shopping and delivery services, the demand for plastic has increased rap- idly in just a few years. Thus, we can find some con- nections between these variables and the total amount of plastic waste[2]. To estimate the maximum mitigation levels for plastic waste without destroying the environment, we take the seven kinds of waste from different kinds of single-use or disposable plastic waste into ac- count. Correspondingly, we consider each plastic waste factor’s weights, the resources to process, the total budget, and the cost of the environmental dam- age it causes. We also set the damage levels for the environment to make our models more accurate. Af- ter comparing each factor, we find the best combi- nation to maximally mitigate the dangerous effects that plastic waste brings about. i. The Simple Exponential Growth Model Before the 21st century, from the 1960s to the 1990s, the simple exponential growth model (also called as Malthusian Model) is a great model for de- scribing the exponential growth of plastic waste[5]. The model could perfectly match with the data and creates great visualization. However, after fitting the data, we discov- ered that plastic waste’s rate of increase is not as fast after 2000. This is because nations and governments, as intervention variables, have paid more attention on this global crisis and already taken some measures to prevent more plastic pollution. To show the intervention variables, we decided to switch our models to be more accurate. FIGURE 1: Global plastic production per year. As shown in the figure, plastic waste was largely static from 1950-1960. After this, plastic products became prevalent in humans’ lives, and plastic waste increased rapidly from 1960 on- ward. There were some significant increases between the 1970s and 2000s[4]. TO ESTIMATE THE MAXIMUM MITIGATION LEVELS FOR PLASTIC WASTE WITHOUT DESTROYING THE ENVIRONMENT, WE TAKE THE SEVEN KINDS OF WASTE FROM DIFFERENT KINDS OF SINGLE-USE OR DISPOSA- BLE PLASTIC WASTE INTO ACCOUNT. ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II ii. The Logistic Model The logistic model was developed by Bel- gian mathematician Pierre Verhulst[8]. He came up with this model’s idea by thinking about the rate of population increased may be limited. It may be af- fected by many other factors, which may not always increase. We agree with the thoughts of Varhulst: the global plastic waste may not increase after reaching some points. Therefore, we used Logistic Model here. However, after 2015, we observe that the model’s coefficient is not a constant anymore. The curve’s rate of increase will change because of inter- vention variables. That is to say, the model’s coeffi- cient should be a flexible number. To make the co- efficient adjustable, based on our Logistic model’s differential equation, we created another self-adap- tive differential equation. iii. The Self-Adaptive Model Based on the differential equation from last section, we develop a more advanced model called a SELF-ADAPTATION model, for which the coefficients are flexible and easy to change. Some researchers call this kind of model a grey model, since it can be flex- ible between black model or white model. Under this model, we create a self-adaptation differential equation, which can be easily adapted in different conditions. By employing this model, our predic- tions are close to reality and the error is small. Since the simple exponential growth model’s predictions between 1950-1970 and Lo- gistic model’s predictions before 2010 are accurate, we will employ both models in our new self-adaptive model. The coefficient of plastic waste before 2010 is nearly static[4]. However, after 2010, when more countries and nations began to ban or limit the use of single-use or disposable plastic products, the co- efficients of the plastic waste curve began to de- crease. After 5-10 years, the self-adaptive model will change the coefficients, which makes it more adapt- able to real-world data. After comparison of these models, we chose to employ a self-adaptive model as our main model to predict future plastic waste production. 4 SELECTION OF THE MODELS The criteria of model’s selection are based on the accuracy of the models. (See APPENDIX 1)  Firstly, we started by building a SIMPLE EXPONENTIAL GROWTH model and trying to find a valuable result. Before the 21st century, around 1960s to 1990s, the Malthusian model is considered a great model for describing the exponential growth since the function is proportional to the speed to which the function grows.  However, after fitting the data, we discovered that the plastic waste’s increasing rate is not that fast anymore. This is due to the fact that nations and governments, as intervention variables, paid more attention on this global crisis and already took some measurements to prevent more plas- tic pollutions. To show the intervention variables, we decided to switch our models to a more accu- rate model. FIGURE 2: The Logistic Model for fitting the real-world data. This plot shows that our logistic model is perfectly suited to the real data, where the x-axis represents the year, and the y-axis represents plastic production. Compared to our previous model, the logistic model is closer to the real- word data from 1960 to 2015. ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II  Secondly, we built a LOGISTICS model. The logistics model matches with our real data from 1960- 2015. However, after 2015, we found out that the model’s coefficient is not a constant anymore. The increasing rate will also change because the intervention variables. That is to say, the model’s coefficient should be a flexible number. To make the coefficient adjustable, based on our Logistics model’s differential equation, we created another self-adaptive differential equation.  Lastly, we move our eyes to the SELF-ADAPTIVE model, in which the coefficients are flexible and easy to change. Using this model, we created a self-adaptation differential equation. By employ- ing this model, our predictions and the reality are similar, and the error is small. Therefore, after comparisons, we chose to employ the self-adaptive model as our main model to pre- dict the future plastic waste productions. 5 RESULTS To examine the accuracy and efficiency of our models, we compared each year’s plastic waste and the data calculated from each model. All these three models' error rates are low in the beginning before the 21st century[4]. Then, depending on the different functions, each model has some deviation from the real-world data. By graphing all the models (FIGURE 3), we can easily see the differences between the models and the best model for future prediction. Therefore, we concluded that the self-adaptive model is the most accurate one, performing much better than the simple exponential growth model or logistic model. The strength of our model is that we take three models and use them to continue to improve the accuracy of our model. We built up the first sim- ple exponential growth model, which gave us a gen- eral idea about the trend of plastic waste. Based on the prediction model, we came up with another Lo- gistic model. By employing the Logistic model, the results perfectly match our data. We then tried to use Logistic model to predict our future data; however, the results deviated from the prediction data. We fig- ured out that the problem might be the coefficients of our Logistic model, which should not be a con- stant. Since the coefficients are flexibly changing, we switched our model to a self-adaptive model. 6 DISCUSSION There are a number of plastic waste preven- tion techniques, which can commonly be summa- rized as the 4 Rs: reduce, reuse, recycle, and recover. Before, people summarized this technique as 3Rs, FIGURE 3: The prediction of different models. In this figure, we can visually compare the prediction results of these three models. Blue dots represent real-world data[4], and while yellow dots represent prediction data made by pre- vious researchers[5]. We can clearly see that a blue linear line of a trend. This line just shows a trend of the future plastic waste but has no meaning to the predictions. The simple exponential growth model (RED) provides a great example of the maximum level of plastic pollution that the system can handle. It could perfectly suit current data, but not future data. The logistics model (PURPLE) shows the ex- ponential growth of the prediction. After 2030, the logis- tics model deviates with the real-world data. The self- adaptive model (GREEN) displays the flexible curve which suits not only the current data, but also future data. By us- ing a self-adaptive model, our predictions can be more de- tailed and precise. ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II which did not include recovery[6]. However, more and more people are realizing that it is not enough to recycle. They also need to reduce and reuse recy- cled plastics. The total population of the above 15 coun- tries are around 5 billion, including the development and developed countries. From figure 4, we find out that developing countries usually produce less plas- tic waste than developed countries. All nations are responsible for managing their waste production. For example, the countries above the trend should make more actions to prevent increasing plastic waste further. Governments can design laws to pre- vent highly plastic polluting companies from pro- ducing single-use or disposable plastics. They can also reduce taxes for those companies that have plastic recycling technology and produce multiple- use or biodegrade plastic products. These actions could limit single-use or disposable plastic produc- tion. Moreover, they would provide a cleaner and healthier plastic cycle. Since plastic pollution has become a global crisis, governments, companies, and human beings have realized the urgency of recycling plastic prod- ucts and reducing single-use or disposable plastic. In our self-adaptive model, we predict that the rate of increase of plastic waste in 2030 and 2040 will be much slower than the previous year’s rates∎ FIGURE 4: The relationship between GDP and Plastic Waste. We choose 15 countries from different regions and development levels[4]. The sizes of the circles represent the total plastic waste of that country. We can conclude from the graph that when GDP goes up, plastic waste per year also increases. In the other word, the total plastic wastes have some connections with the country’s degree of development. ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 7 REFERENCE [1] Geyer, R., Jambeck, J. R., & Law, K. L. (2017). Production, use, and fate of all plastics ever made. Science Advances, 3(7), e1700782. [2] Jambeck, J. R., Geyer, R., Wilcox, C., Siegler, T. R., Perryman, M., Andrady, A., … & Law, K. L. (2015). Science, 347(6223), 768-771. [3] Li, W. C., Tse, H. F., & Fok, L. (2016). Plastic waste in the ma- rine environment: A review of sources, occurrence and ef- fects. Science of the Total Environment, 566, 333-349. [4] Hannah Ritchie and Max Roser (2020). Plastic Pollution. Pub- lished online at OurWorldInData.org. [5] Eriksen, M., Lebreton, L. C., Carson, H. S., Thiel, M., Moore, C. J., Borerro, J. C., ... & Reisser, J. (2014). Plastic pollution in the world's oceans: more than 5 trillion plastic pieces weigh- ing over 250,000 tons afloat at sea. PloS one, 9(12), e111913. [6] Lebreton, L., Andrady, A. (2019). Future scenarios of global plastic waste generation and disposal. Palgrave Commun 5. [7] Cramer, J.S. (2002) The Origins of Logistics Regression. Chapter 9 of Logit Models from Economics and Other Fields, Cambridge University Press. ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 8 SUPPLEMENTARY TABLES 9 SEE ALSO HTTPS://PREZI.COM/VIEW/BJDEXJKYPKMAV4ZXQ2TZ/ YEAR PLASTIC WASTES (TONNES) SIMPLE EXPONENTIAL GROWTH MODEL ERROR (%) LOGISTICS MODEL ERROR (%) SELF-ADAPTIVE MODEL ERROR (%) 1960 1970 1980 1990 1995 2000 2005 8,000 35,000 70,027 120,383 152,068 213,209 263,002 - 35,110 70,069 120,443 151,099 213,028 263,799 - 0.3143 0.0599 0.0498 -0.6372 -0.0849 0.0303 - 35,069 70,090 120,409 152,430 213,790 262,997 - 0.1971 0.0899 0.0216 0.2381 0.2725 -0.0019 - 34,980 70,431 120,011 151,995 213,563 263,503 - -0.0571 0.0769 -0.0390 -0.0480 0.1717 0.1905 2010 2011 2012 2013 … 2017 2018 2019 313,000 325,089 338,797 352,642 … 368,782 398,603 442,488 320,263 329,994 380,863 423,569 … 457,592 482,905 540,000 2.3204 1.5088 12.4163 20.1130 … 24.0820 21.1494 22.0372 320,067 324,989 339,002 353,059 … 369,728 399,766 444,050 0.2257 -0.0308 0.0605 0.1183 … 0.2565 0.2918 0.3530 313,067 325,209 338,997 352,368 … 368,597 398,645 442,766 0.2578 0.03690.0 590 -0.0777 … -0.0501 0.0105 0.0628 2020 2021 2022 2023 2024 2025 … 486,323 510,360 546,085 584,311 625,213 668,978 … 589,670 600,678 634,335 669,895 700,000 724,330 … 21.2507 17.6970 16.1605 14.6470 11.9618 8.2741 … 487,982 513,480 546,999 582,994 624,785 670,012 … 0.3411 0.6113 0.1673 -0.0543 -0.6846 0.1546 … 486,373 510,299 546,327 583,994 625,869 668,321 … 0.0461 0.0623 0.0443 -0.0542 0.0105 -0.0982 … 2030 715,806 768,450 7.3545 715,949 0.1997 624,330 0.0529 2040 876,893 1000,785 14.1285 879,323 0.2771 823,780 0.0947 2050 1,275,997 1,868,544 46.4379 1,276,544 0.4264 1,276,544 0.0426 SUPPLEMENTAL TABLE 1: The Comparison of Total Plastic Wastes and its Simulation based on three models https://prezi.com/view/BJdeXjKyPkmaV4zXq2tz/