Capstone Presentation:
The topic being studied at the presentation on Friday was Bisphenol A (hereby referred to as BPA), the chemical described as synthetic estrogen, present in many hard plastics found in everyday life especially bottles. There are many health risks to this organic chemical if it enters the body. It can potentially lower sperm count, result in sterilization as well as breast cancer. The current Canadian and U.S. limit of non harmful BPA consumption is 25 mg/ day.
The student who conducted the experiment aimed to test how long BPA would last in a fetus if consumed. She compartmentalized the body and used an ODE per compartment to model the flow of BPA in it. She derived a number of equations testing the dosage of BPA and the time in lasts in first body and thereby the fetus. She also used a physiological based pharmacokinetic model to test what the body does to the drug once its enters.
To graph the equations, she used Kawamoto et Al's model. She combined BPA as glucogen and the actual amount of drug in the body to plot a single line. She questioned whether BPA would disappear from a fetus over a period of time but she ultimately found that it remained at a constant rate within the fetus.
The student then took all her 27 equations from the compartmentalized study of the body and used varying toy model of bodies and fetuses to arrive at a single equation through nondimensionalization(i.e., making all main variables dimensionless).
The implications of her study were that BPA products should now be outlawed as it has been in Canada. Surprisingly enough baby bottles still cotain a significant amount of BPA in other parts of the world.
Through mathermatical modeling, the equations she arrived at concluded that a contstant low level of BPA will always remain in a fetus if injected at any given point in time. Also, doubling the dose of BPA will double the level of it remaining in the fetus.
I thought the study was very interesting and had a lot of relevant knowledge. The extrapolation of her data on the graph plotted bothered me a little, but someone asked the question in class and she seemed to prove it made sense even though i'm not entirely convinced. That is mainly because I didn't really understand the Kawamoto et Al's model used in the study.
Thursday, April 24, 2008
Tuesday, April 15, 2008
SECTION 10.7: Modeling the spread of a disease
This section deals with the direct application of solving more than differential equation at the same time. It helps determine whether a disease is an epidemic or not and can help determined the level of vaccination necessary to prevent the disease.
Epidemics are studied based on an S-I-R model where S is the number of susceptible candidates for the sickness, I is the number of people already suffering from the disease and R is the number of people who have recovered from the disease. On the whole, in this model, it is assumed that the rate of change of susceptibles is related to th negative rate of change of the number of susceptibles who get sick. The rate of change of the number infected is the rate of change of susceptibles getting sick excluding the rate of change of those who have recovered.
( dS/dt= -aSI) and (dI/dt=aSI-bI) where a and b are constants. a measures how infectious the disease is and b represents the rate at which infected people are removed from infected population.
As a result of the above analysis, a threshold value is determined where the threshold population = b/a.
The section provides a very interesting study of exactly how calculus of this form can be directly applied in the field of science (especially biology). It could effectively help save lives when studied on a much larger scale.
This section deals with the direct application of solving more than differential equation at the same time. It helps determine whether a disease is an epidemic or not and can help determined the level of vaccination necessary to prevent the disease.
Epidemics are studied based on an S-I-R model where S is the number of susceptible candidates for the sickness, I is the number of people already suffering from the disease and R is the number of people who have recovered from the disease. On the whole, in this model, it is assumed that the rate of change of susceptibles is related to th negative rate of change of the number of susceptibles who get sick. The rate of change of the number infected is the rate of change of susceptibles getting sick excluding the rate of change of those who have recovered.
( dS/dt= -aSI) and (dI/dt=aSI-bI) where a and b are constants. a measures how infectious the disease is and b represents the rate at which infected people are removed from infected population.
As a result of the above analysis, a threshold value is determined where the threshold population = b/a.
The section provides a very interesting study of exactly how calculus of this form can be directly applied in the field of science (especially biology). It could effectively help save lives when studied on a much larger scale.
Thursday, April 10, 2008
SECTION 10.6: Modeling the Interaction of Two Populations
I understand that this section involves analyzing systems in which two differential equations are required in order to find a legitimate solution. But the reasoning for this has been using a phase plane of the two unknown variables studied in each differntial equations resulting in a slope field diagram and equillibrium points. As we have not yet studied slope fields in class, I was thoroughly confused in this reading. Beyond understand why the two equations are required in a situation such as the robin and the worm, I am unable to apply this concept to other real world situations because most of the solutions in this section don't make sense to me at this point. I apologize for the lack of detail in this blogpost but I geniunely don't understand the key points of this section!
I understand that this section involves analyzing systems in which two differential equations are required in order to find a legitimate solution. But the reasoning for this has been using a phase plane of the two unknown variables studied in each differntial equations resulting in a slope field diagram and equillibrium points. As we have not yet studied slope fields in class, I was thoroughly confused in this reading. Beyond understand why the two equations are required in a situation such as the robin and the worm, I am unable to apply this concept to other real world situations because most of the solutions in this section don't make sense to me at this point. I apologize for the lack of detail in this blogpost but I geniunely don't understand the key points of this section!
Tuesday, April 8, 2008
SECTION 10.5: Applications and Modeling
In this section, derivatives related to the original equation with a difference of a constant A are taken into consideration, i.e., dy/dt = k (y-A), where k and A are constants.
Just as the solution to derivatives in the earlier section were found, the solution to the above mentioned derivative can also be determined.
If (y-A) is considered to be an exponential equation, then it must be in the form of C(e^kt).
Earlier we found that the exponential equation dy/dt = ky impliest hat y = C(e^kt). Therefore, this time the solution will vary by a constant A.
dy/dt = k(y-A) implies that y = A + C(e^kt) for any constant C.
Equilibirum solutions:
1. It is constant for all values of the independent variable. The graph is a horizontal line. They can be identified by setting the derivative of the function to zero.
2. It is stable is a small change in the initial conditions gives a solution which tends toward the equillibrium as the independent variable tends to positive infinity.
3. An equilibrium solution is unstable if a small change in the initial conditions gives a solution curve which veers away from the equilibrium as the independent variable tends to positive infinity.
This same principle is applied to Newton's law of heating and cooling by which
Rate of change in temperature = constant * Temperature difference.
This section although not directly applicable to economics is useful in understanding other everyday phenomenon. It is useful in the field of biology in order to know levels of drug concentration in a person. It is useful when solving murder cases or in hospitals when estimating times of death.
In this section, derivatives related to the original equation with a difference of a constant A are taken into consideration, i.e., dy/dt = k (y-A), where k and A are constants.
Just as the solution to derivatives in the earlier section were found, the solution to the above mentioned derivative can also be determined.
If (y-A) is considered to be an exponential equation, then it must be in the form of C(e^kt).
Earlier we found that the exponential equation dy/dt = ky impliest hat y = C(e^kt). Therefore, this time the solution will vary by a constant A.
dy/dt = k(y-A) implies that y = A + C(e^kt) for any constant C.
Equilibirum solutions:
1. It is constant for all values of the independent variable. The graph is a horizontal line. They can be identified by setting the derivative of the function to zero.
2. It is stable is a small change in the initial conditions gives a solution which tends toward the equillibrium as the independent variable tends to positive infinity.
3. An equilibrium solution is unstable if a small change in the initial conditions gives a solution curve which veers away from the equilibrium as the independent variable tends to positive infinity.
This same principle is applied to Newton's law of heating and cooling by which
Rate of change in temperature = constant * Temperature difference.
This section although not directly applicable to economics is useful in understanding other everyday phenomenon. It is useful in the field of biology in order to know levels of drug concentration in a person. It is useful when solving murder cases or in hospitals when estimating times of death.
Sunday, April 6, 2008
SECTION 10.4: Exponential Growth and Decay
A solution is defined as a function that is the same as its derivative. For example, an exponential function of the form e^x has a derivative equal to e^x. For an exponential equation, if the function contains a constant (k), the derivative of the function is a constant times the function, i.e., essentially the same as the original function. It implies that the rate of change of e^x is proportional to e^x where the constant k is the constant of proportionality.
The general equation for an exponential function is y = C(e^kt) where k is defines exponential growth if more than 0 and exponential decay if less than 0. The constant C is the value of y when t is 0. In a growth function expressing population, C would represent the initial population.
Exponential growth and decay can occur in many every day circumstances. It is relevant in the study of population growth, marine biology, pollution in cities, drug levels in people etc. My intended major still stands to be economics and so i am definitely aware through the book and other instances that learning to understand rates of change of derivatives functions whether those of exponential growth or decay are important to understand many aspects of banking (beyond just compund interest).
A solution is defined as a function that is the same as its derivative. For example, an exponential function of the form e^x has a derivative equal to e^x. For an exponential equation, if the function contains a constant (k), the derivative of the function is a constant times the function, i.e., essentially the same as the original function. It implies that the rate of change of e^x is proportional to e^x where the constant k is the constant of proportionality.
The general equation for an exponential function is y = C(e^kt) where k is defines exponential growth if more than 0 and exponential decay if less than 0. The constant C is the value of y when t is 0. In a growth function expressing population, C would represent the initial population.
Exponential growth and decay can occur in many every day circumstances. It is relevant in the study of population growth, marine biology, pollution in cities, drug levels in people etc. My intended major still stands to be economics and so i am definitely aware through the book and other instances that learning to understand rates of change of derivatives functions whether those of exponential growth or decay are important to understand many aspects of banking (beyond just compund interest).
Thursday, April 3, 2008
SECTION 10.2: Solutions of differential equations
A differential equation is formed when including the derivative of an unknown function. Its solution is any function that can be reduced to the differential equation/ satisfies it. The solution to the differential equation will be of numerical form and sometimes (if not always), the solution can be written as a another formula including a constant. To find the value for this constant, the initial value of unknown variable being studied needs to be known. Then substituting that value in the formula, we can find the constant.
A solution that satisfies a differential equation with a set initial condition is called a particular solution. The differential equation and the initial condition together is called an inital value problem.
This kind of information is useful in economics for a situation in which a person is aware of the rate of change of economic growth but needs to estimate more future values by formulating a general formula from the available derivative of unknown growth variable. This kind of theory is very important in understanding general trends of population growth etc. It is like moving backwards to find a general function when known the derivative.
A differential equation is formed when including the derivative of an unknown function. Its solution is any function that can be reduced to the differential equation/ satisfies it. The solution to the differential equation will be of numerical form and sometimes (if not always), the solution can be written as a another formula including a constant. To find the value for this constant, the initial value of unknown variable being studied needs to be known. Then substituting that value in the formula, we can find the constant.
A solution that satisfies a differential equation with a set initial condition is called a particular solution. The differential equation and the initial condition together is called an inital value problem.
This kind of information is useful in economics for a situation in which a person is aware of the rate of change of economic growth but needs to estimate more future values by formulating a general formula from the available derivative of unknown growth variable. This kind of theory is very important in understanding general trends of population growth etc. It is like moving backwards to find a general function when known the derivative.
Tuesday, April 1, 2008
SECTION 10.1: Mathematical Modeling: Setting up a differential equation
A differential equation is one that is formed based on the information on rates of changes of available values in the equation. This equation can then be analyzed in order to obtain the original function.
From the examples given in the textbook, it was observed that the rate of change of a certain quantity being study was effectively adjustable to the continuous rate of change of an increase or decrease of that quantity and the constant rate at which it is being increased or decreased. Assuming the Quantity being studied is Q,
dQ/dt = aX-Y where a determined the continuous rate of change and Y determines the constant rate of change. X would refer to the original value of the quantity being studied.
The Logistic Model:
The magnitude of a certain quantity being studied (e.g., population) is proportional to its initial value times the difference between its carrying capacity(the limit to which population in a town can increase) and the current value.
In other words if carrying capacity is L, the original and th original value is P then:
dP/dt = kP(L-P) where k is the constant of proportionality. This is known as a 'logistic differential equation'.
I found it difficult to understand exactly how the carrying capacity of a quantity as large as population can really be determined.
It is apparent that known how to use models with differntial equations assessing rates of changes can be important in assessing a company's revenues in terms of the changing wages it has to pay its workers. This section of mathematics is again directly relatable to economics.
A differential equation is one that is formed based on the information on rates of changes of available values in the equation. This equation can then be analyzed in order to obtain the original function.
From the examples given in the textbook, it was observed that the rate of change of a certain quantity being study was effectively adjustable to the continuous rate of change of an increase or decrease of that quantity and the constant rate at which it is being increased or decreased. Assuming the Quantity being studied is Q,
dQ/dt = aX-Y where a determined the continuous rate of change and Y determines the constant rate of change. X would refer to the original value of the quantity being studied.
The Logistic Model:
The magnitude of a certain quantity being studied (e.g., population) is proportional to its initial value times the difference between its carrying capacity(the limit to which population in a town can increase) and the current value.
In other words if carrying capacity is L, the original and th original value is P then:
dP/dt = kP(L-P) where k is the constant of proportionality. This is known as a 'logistic differential equation'.
I found it difficult to understand exactly how the carrying capacity of a quantity as large as population can really be determined.
It is apparent that known how to use models with differntial equations assessing rates of changes can be important in assessing a company's revenues in terms of the changing wages it has to pay its workers. This section of mathematics is again directly relatable to economics.
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