Abstract

A mathematical model attempts to describe a system with mathematical language and concepts. Mathematical modeling is useful when it comes to predicting natural disasters. Today, mathematical modeling is used to forecast tornadoes and to predict the damage associated with tornadoes, earthquakes, hurricanes, and floods. Mathematical modeling has proven to be successful when calculating the evolution and occurrence of F5 tornados and in predicting the approximate location of a tornado. Mathematical modeling can also be used to predict the amount of damage an earthquake will have. It can provide people with earthquake warnings. Mathematical modeling has made it possible for researchers to create earthquake warning systems that provide regional notifications of earthquakes currently in progress. Furthermore, mathematical modeling can be used to predict hurricanes and flood mitigation. The following explores how mathematical modeling is used to assess and predict natural disasters.

**Table of Content**

Abstract…………………………………………………………………………………………………………………………………… 2

Introduction…………………………………………………………………………………………………………………………….. 4

Literature Review……………………………………………………………………………………………………………………. 4

Discussion and Conclusion……………………………………..…………………………………………………………….. 13

Bibliography………………………………………………………………………………………………………………………….. 15

**Mathematical Modeling to Predict and Assess Natural Disasters**

Mathematical modeling is an effective tool in several disciplines that attempts to describe a system or event with numbers. Mathematical modeling has become a common method of predicting and evaluating the potential damage of natural disasters. Natural disasters threaten both human life and property, resulting in a need for mathematical modeling. Mathematical modeling is used by scientists and researchers for weather forecasting, storm predictions, and predicting possible future damage. Mathematical modeling is currently used to assess and predict tornadoes, earthquakes, hurricanes, and floods. It is also used to assess the potential damage associated with these natural disasters as well. Mathematical modeling is used to warn people of tornadoes, earthquakes, hurricanes, and floods; helping them protect both their property and their life. The following explores how mathematical modeling is used to assess and predict natural disasters.

**Literature Review**

**Mathematical Modeling**

A mathematical model attempts to describe a system with mathematical language and concepts. Mathematical modeling refers to the process of creating a mathematical model. Mathematical models are utilized in several disciplines, such as earth science, biology, physics, meteorology, computer science, sociology, psychology, economics, and political science. There are several forms of mathematical modeling, such as differential equations, dynamical systems, game theoretic models, and statistical models (Varshney).

Professionals use mathematical modeling to describe the world with numbers. Mathematical models can be used to: 1) make future predictions, 2) relate quantities in real-life situations, and 3) data visualization. They can also be used to help people learn about a current system. Equations are one of the most common forms of mathematical modeling and take several factors into considerations when modeling a specific event. The numerical aspects of an equation make is possible for an individual to understand how various parts of one system are working together (De Wrachien).

**Predicting Natural Disasters**

Mathematical modeling is useful when it comes to predicting natural disasters. While natural disasters cannot be prevented, mathematical modeling can be used to help limit the damage associated with them. Today, mathematical modeling is used to forecast tornadoes and to predict the damage associated with tornadoes, earthquakes, hurricanes, and floods (Varshney). According to research, natural disasters intensify with each day that passes, threatening the safety and stability of people and buildings. Both cyclones and earthquakes are considered the most destructive out of all natural events, claiming the lives of many. There are also material and economic damages associated with all types of natural disasters, especially tornadoes, earthquakes, hurricanes, and floods. Thus, forecasting these natural events is of utmost importance for scientists in order to preserve human life and property during these events (Rodrigez et al).

There are five different classifications of natural disasters: meteorological, hydrological, climatological, geophysical, and biological. Meteorological natural disasters consist of storms, cyclones, hurricanes, and tornadoes. Hydrological natural disasters consist of floods and avalanches, while climatological natural disasters consist of wildfires and droughts. Lastly, geophysical natural disasters consist of landslides, tsunamis, volcanic eruptions, and earthquakes, while biological natural disasters consist of famine and disease outbreaks. The Center for Research on Epidemiology of Disasters reported there were 332 natural disasters in 2011, impacting 101 countries. These natural disasters were responsible for 30,770 deaths and impacted over 244 million individuals across the globe. These disasters are typically unpredictable and have a substantial impact on public health. They can also have both long-term and short-term heath impacts (Varshney).

There are two groups of mathematical modeling used for disaster mitigation. The first group of mathematical modeling is used to identify risk, and the second group is to forecast future extreme events. Currently, mathematical modeling plays a large role in forecasting weather, cyclone tracking, and hazard identification (Varshney). Researchers and scientists focus on mathematical modeling to both predict natural disasters as well as the damage they will leave behind. These mathematical models are used to give people warning of the natural disaster and helps authorities create adequate plans to keep populations safe. The following explores how mathematical modeling is used to predict and assess tornadoes, earthquakes, hurricanes, and floods.

*Tornadoes*

A tornado is a rotating air column that’s in contact with both a cumulonimbus cloud and the earth’s surface. Tornadoes typically vary in size and shape but are all observable funnels. While tornadoes travel various distances, they generally dissipate after a few miles. Most tornadoes travel at under 100 miles per hour. However, there have been reports of tornadoes traveling over 300 miles per hour. There have also been tornadoes that were over two miles wide and traveled over 62 miles. Thus, there is not a set of rules tornadoes follow when it comes to formation and dissipation. Given their size and shape, tornadoes are known to be destructive forces, threatening both human life and property (Arsen’yev).

Tornadoes are able create substantial damage. Currently, tornado warnings are given within minutes of an actual tornado touching down. Mathematical modeling is essential for tornado warnings and storm formation. Due to their rapidly developing subsystems, tornadoes are extremely difficult to forecast. Predicting the exact location a tornado is going to touch down is an exceptionally difficult task. However, recent advances in mathematical modeling for tornadoes has helped researchers determine the evolution and occurrence of F5 tornadoes. These tornadoes, the highest on the Fujita scale, can arise out of turbulent whirlwind supercells (Arsen’yev).

Mathematical modeling has been known to be successful when calculating the evolution and occurrence of F5 tornados. The Fujita scale is used to classify tornadoes based on their speed and size. Tornadoes classified as F5 are considered violent tornadoes with wind velocity from 400 to 514 km/h These tornadoes are the most damaging. In 1999, an F5 tornado in Oklahoma destroyed 4000 houses and killed 48 people. The tornado also left several people seriously injured. While F5-class tornadoes only consist of one percent of all tornadoes, they are the deadliest. Between 1950 and 2000, roughly 67 percent of tornado fatalities were caused by a F5-class tornado. Furthermore, approximately 5 percent of buildings are able to withstand the winds of a F5 tornado (Arsen’yev).

Arsen’yev explored mathematical modeling in predicting F5-class tornadoes. Researchers used asymmetric turbulence theory to calculate a F5 tornado. Researchers also took into consideration time, velocity of wind, and compressibility of air. The researcher was able to adequately predict both the evolution and occurrence of F5 tornadoes using time, velocity of wind, and compressibility of air. Researchers used mathematical modeling to calculate the tornado’s velocity through equations and formulas. Through this process, they were able to determine whether a storm cell had the ability to turn into a F5-class tornado. This mathematical model has been shown to be successful when determining the potential strength of a storm cell and allows scientists with the ability of measuring the evolution and occurrence of F5 tornadoes (Arsen’yev).

Mathematical modeling can also help predict the approximate location of a tornado. Tornado modeling has become necessary when it comes to hazard mitigation and risk analysis. Mathematical modeling has been known to help determine approximately where a F2 and above tornado will touch down and its probable intensity; helping businesses and individuals better prepare for the disaster. Advanced forecasting has made it possible to significantly reduce deaths, injuries, and damages caused by a tornado. Overall, mathematical modeling can help researchers forecast tornadoes with structure and timing. This modeling has become a valuable tool when forecasting and assessing tornado risk. On the other hand, due to the fact tornadoes are ever-changing and rapidly dynamic systems, forecasting exactly where they will touch down is extremely difficult (Arsen’yev).

*Earthquake*

An earthquake “is the result of a sudden release of energy in the Earth’s crust that creates seismic waves” (Varshney 205). There have been several methods used to predict the place and time an earthquake will occur. Currently, however, predictions are not able to specify the day or month. On the other hand, mathematical modeling can be used to estimate whether a segment of a fault may rupture in a specific decade. Mathematical modeling has made it possible for researchers to create earthquake warning systems that provide regional notifications of earthquakes currently in progress, but prior to the ground moving, allowing people in the area the ability to seek shelter prior to feeling the impact of the earthquake. Both regional and global databanks have been created through strong motion instrument networks. With this data, researchers are better able to predict strong earthquakes based on “application of empirical mathematical models fitted to the databanks” (Varshney 205). These types of mathematical models are called attenuation laws or ground motion models. These models “define the relationship between ground motion parameters and factors that affect the amplitudes of ground motion as are the released energy, the regional characteristics, the local soil characteristics, the type of fault, the radiation pattern, etc.” (Varshney 205).

Furthermore, some individuals have been able to predict and model global earthquakes using the Regressive Objective Regression (ROR) methodology. Rodriguez et al assessed ROR methodology by looking at an 11-year cycle of earthquakes in the Caribbean. Researchers were able to forecast the longitude, latitude, date, time, depth, and magnitude of an earthquake using ROR methodology. By forecasting atmospheric pressure, researchers were able to use ROR modeling to forecast a hurricane one year in advance (Rodriguez et al).

Mathematical modeling can also be used to predict the amount of damage an earthquake will have. Bagai et al used mathematical modeling to predict the damage an earthquake can have on both four-story and seven-story buildings. Researchers used the length of time of an earthquake to determine damage. Using mathematical modeling, researchers were able to model a building’s structural dynamic by applying Newton’s second law and by making the following assumptions: 1) all floors have masses, 2) a stiffness factor is incorporated in the linear restoring force of each floor, 3) a damping force exists between each floor, and 4) “horizontal earthquake oscillation, 𝐸𝑐𝑜𝑠𝜔𝑡 of the ground with amplitude 𝐸 and acceleration 𝑎 = −𝐸𝜔 2 𝑐𝑜𝑠𝜔𝑡, produces a force 𝐹 = 𝑚𝑎 = 𝑚 𝐸𝜔 2 𝑐𝑜𝑠𝜔𝑡 on each floor of the building” (Bagai et al 181). By making these assumptions, researchers were able to create a mathematical model “based on forced spring oscillation” (Bagai et al) to predict the damage of earthquakes on four-story and seven-story buildings.

Researchers took into consideration the mass of each floor, the stiffness parameters, and the damping parameters. For the floor’s mass, researchers calculated each floor as 1000 units. The stiffness parameter was set at 10000 units, while the damping parameters was 500 units. According to the mathematical model selected, researchers were able to determine that a two second earthquake will resonant with both four-story and seven-story buildings. However, an earthquake that lasts roughly 5.7 seconds will be more damaging to a four-story building than an earthquake lasting 3.54 seconds on a seven-story building. This study provides insight on why earthquakes of different frequencies have adverse effects on buildings with various number of floors (Bagai et al).

*Hurricanes*

Hurricanes, also called tropical cyclones, are low-pressure, rotating weather systems. While hurricanes have organized thunderstorms, they do not have fronts. Storms that have surface winds less than 40 miles per hour are referred to as tropical depressions. Tropical storms, on the other hand, have surface winds greater than 40 miles per hour. Once a storm’s winds reach 75 miles per hours, the storm is considered a hurricane. Hurricanes are categorized by the Saffir-Simpson Hurricane Wind Scale. This scale classifies hurricanes based on wind speeds and rates them from 1 to 5. The higher a hurricane’s category is, the larger potential the hurricane has to cause property damage. Hurricanes typically form in the Atlantic basin. This area includes the Gulf of Mexico, Caribbean Sea and the Atlantic Ocean. While hurricanes can occur anytime of the year, they are most common during June 1^{st} and November 30^{th}. According to the National Hurricane Center, there are approximately 12 hurricanes on average each year (Shmakin).

Hurricanes threaten both property and life. Mathematical modeling, however, provides authorities with proper tools to mitigate the damage associated with them. Mathematical modeling has become essential when it comes to protecting people and property from natural disasters, such as hurricanes. Hurricanes have been known to cause billions in damages and can result in the death of several individuals. Sometimes, hurricanes hit land with so much force, the impact lasts several years. Mathematical modeling has been successful in predicting the location and impact of hurricanes in order to help limit the damage associated with these natural disasters (Meriem et al).

Mathematical modeling is used to predict hurricanes. When predicting hurricanes, mathematical modeling looks at historical data of past storms, the intensity of the hurricane, and the trajectory the hurricane will take. Mathematical forecasting also looks how various climate changes can impact hurricane development. As this type of modeling improves, numerical modeling will help relief agencies better respond to the individuals impacted by hurricanes. For instance, mathematical modeling was used for Hurricane Sandy. Mathematical modeling was able to provide authorities with timely and specific information regarding the hurricane, enabling them to direct people to safety. As a result, mathematical modeling was able to save several lives before the hurricane hit (Meriem et al).

Mathematical modeling was effective during Hurricane Sandy because it was able to analyze two separate weather systems as well as understand the interaction between them. By understanding these interactions, scientists were able to identify the intensity and trajectory of the storm, providing authorities with proper time to notify the individuals who were located in the affected areas. These individuals were then able to take necessary steps in order to save their own lives and protect their property. The impact associated with Hurricane Sandy was horrible, however, due to the advances in mathematical modeling, authorities were able to lessen the impact of the storm (Meriem et al).

*Floods*

Mathematical modeling has also been successful in flood mitigation. Floods account for one-third of the natural disasters each year. They also account for one-third of the economic losses associated with natural disasters each year. It is expected that floods will only become more relevant and frequent in the future, due to urbanization, increase in population, and climate change. Mathematical models play an important role in flood mitigation and can be used for reducing and assessing the vulnerability of urban and rural flood prone areas. Mathematical models are considered the best tools when it comes to effective flood protection strategies (De Wrachien).

There are two types of mathematical modeling that can be used for floods: stochastic models and deterministic models. Stochastic modeling is used to determine the magnitude of the flood. By using climate variabilities, stochastic modeling helps researchers determine the magnitude of a specific flood and its risk. Deterministic modeling, on the other hand, is “defined as a mathematical procedure for predicting the changing magnitude, speed, and shape” (De Wrachien 63) of a flood. Mathematical modeling has proven to be the most successful way to understand the complexities of a flood. These models have replaced other strategies that were based on trial and error. Thus, mathematical modeling is a useful tool when it comes to designing efficient protection strategies and measures (De Wrachien).

**Discussion and Conclusion**

Overall, mathematical models are used to make future predictions relating quantities in real-life situations and data visualization. Mathematical modeling is useful when it comes to predicting natural disasters. While natural disasters cannot be prevented, mathematical modeling can be used to limit the damage associated with them. There are also material and economic damages associated with all types of natural disasters, especially tornadoes, earthquakes, hurricanes, and floods. Mathematical modeling has been shown to be increasingly effective at predicting and mitigating the damage associated with these natural disasters. Mathematical modeling will only be as effective as the chosen parameters, equations, numbers, and information considered in creating the model. There could be variables not included because they were deemed not significant, or variables may be included that were not important and should have been omitted. Consider a mathematical model to be like a machine: there is an input that undergoes a set of rules and then an output is generated based on the interrelationship of the input and set of rules. The machine can only do what it was designed to do and will only be effective or accurate if the set of rules are logically and carefully created. This is in part why weather forecasts are made only for seven to ten days out. Consider the infamous butterfly effect: something as small and seemingly insignificant as a butterfly flapping its wings in another part of the world could arguably have a domino effect on weather patterns across the world resulting in a weather event like strong tornadic winds.

Mathematical modeling seems to be the most effective when predicting and assessing hurricanes. Scientists and researchers are properly able to analyze cyclones and give people adequate advance warning. Tornadoes and earthquakes, on the other hand, are a little more difficult to predict. Researchers are able to use mathematical modeling to determine the approximate location of a tornado but are unable to use mathematical modeling to determine the exact location. While mathematical modeling is not perfect and has limitations when it comes to tornadoes, researchers and scientists are able to give people approximate locations which allows enough warning to take shelter. Earthquakes are similar. While researchers are unable to utilize mathematical modeling to predict the exact day of an earthquake, they can give people warning before the earth starts moving. While not perfect, mathematical modeling does help give people some warning for all four types of natural disasters.

Mathematical modeling is the most successful at predicting the damage associated with natural disasters. This is seen with both hurricanes and floods. Mathematical modeling is used with both hurricanes and floods to help mitigate the damage associated with them. Mathematical modeling has become essential when it comes to protecting people and property from natural disasters. Mathematical modeling was especially helpful during Hurricane Sandy. Because researchers and scientists were able to analyze the weather systems, they were able to identify the intensity and trajectory of the storm quickly and accurately. This provided authorities with sufficient time to notify the individuals who would be affected. In all the mathematical models, note that the past has no influence on the future and thus there is no mathematical model involving or accounting for past weather events. Thus, mathematical modeling is effective when it comes to predicting natural disasters and mitigating the damage associated with them.

Works Cited

Arsen’yev, Sergey. “Mathematical modeling of tornadoes and squall storms.” *Geoscience Frontiers* 2.2 (2011): 215-221. Print.

Bagai, Shobha and Parul Madaan. “A Mathematical Model for the Effect of Earthquake on High Rise Buildings of Different Shapes.” *Journal of Undergraduate Research and Innovation* 2.1 (2016): 180-188.

De Wrachien, Daniele. “Mathematical models in flood management: Overview and challenges.” *ResearchGate*, May 2010, https://www.researchgate.net/publication/271440272_Mathematical_models_in_flood_ anagement_Overview_and_challenges.

Meriem, Aouaouda and A. Ayadi. “Mathematical Modeling of Tropical Cyclones on the Basis of Wind Trajectories.” *Computational Mathematics and Mathematical Physics* 59.9 (2019): 1493-1507.

Rodriguez, Ricardo and David del Valle. “Mathematical Modeling and Its Applicability from Natural Disaster to Human Health.” *Research Journal of Medical Sciences* 2.2 (2022): 32-40.

Shmakin, A.B. “Cyclone, hurricanes, typhoons and tornadoes.” *Natural Disasters* 2 (N.D.): 1-6.

Varshney, Gauray. “Role of Mathematical Modeling in Preventing Natural Disaster.” *Journal Global Values* 7 (2016): 203-214.

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