This was a blog required for my Applied Calc. class where we had to blog on the topics that we were going to discuss in the next class, it helped with understanding it all.
Reading it myself now, I have no idea was I was on about. It must be all the big words that meant something at the time. Never thought anyone other than my prof. would be looking at it.
:/
Please carry on to: Bringing the Light. Now THAT I can explain… I think. lol. xoxo.
Thursday, July 30, 2009
Wednesday, April 23, 2008
CapStone Seminar (Math)
A Mathematical Model of Bisphenol A pharmocakinetics in the Mouse
By: Stephanie Abascal
Bisphenol A (BPA) is a chemical that is used in epoxy resins and a lot of plastic products ranging from baby bottles to the soda cans. This chemical is a threathening one when taken into the body as it causes health risks from sterilization to being a precursor to breast cancer.
There are mathematical equations namely, ODEs, that can model the levels of BPA in the body. These models are similar if not the same as those used in the SIR model that we studied in class. Abascal used Kawamoto's physiological based pharmacokinetic (PBPK) model which studies what the body of a human or mouse for example does to the BPA drug. This analysis inlcudes 27 equation which she cuts down to 3 and working with the equation that refers to the foetus and nondimensionalised it by dividing by some constant with the same units. This takes away the complexity making it easier to understand. BPa never leaves the body but accumulates.
This is bad as BPA can be found in baby bottles; directly affecting your baby.
Abascal made the final conclusion: BPA is bad! Keep the following in mind: The higher the number of the plastic, the harder it is and the harder it is the more likely is it to release BPA into whatever it holds. (The Macalester water bottles are made of a number 7 plastic!!!!)
By: Stephanie Abascal
Bisphenol A (BPA) is a chemical that is used in epoxy resins and a lot of plastic products ranging from baby bottles to the soda cans. This chemical is a threathening one when taken into the body as it causes health risks from sterilization to being a precursor to breast cancer.
There are mathematical equations namely, ODEs, that can model the levels of BPA in the body. These models are similar if not the same as those used in the SIR model that we studied in class. Abascal used Kawamoto's physiological based pharmacokinetic (PBPK) model which studies what the body of a human or mouse for example does to the BPA drug. This analysis inlcudes 27 equation which she cuts down to 3 and working with the equation that refers to the foetus and nondimensionalised it by dividing by some constant with the same units. This takes away the complexity making it easier to understand. BPa never leaves the body but accumulates.
This is bad as BPA can be found in baby bottles; directly affecting your baby.
Abascal made the final conclusion: BPA is bad! Keep the following in mind: The higher the number of the plastic, the harder it is and the harder it is the more likely is it to release BPA into whatever it holds. (The Macalester water bottles are made of a number 7 plastic!!!!)
Tuesday, April 15, 2008
10.7: SIR and the Spread of Disease
Main Points
Mathematics has an application in everything in everyday life, including to ability to predict an epidemic and therefore the amount of vaccine that would be needed with respect to a population which is of course growing exponentially; (however for the SIR model to work population is help constant).
The S-I-R model:
S - susceptibles (these are not sick yet but could be, future)
I - Infecteds (these are sick already, present)
R - Recovered (these have already been sick, past)
The 'S' reduce over time as they get sick when they come in contact with the 'I's. The two are thus proportional. (infections and contact) thus giving the formula: dS/dt = - (rate at which S gets sick) = -aSI. (the newly sick are then aSI). Those recovering are proportional to those infected: bI; therefore: dI/dt = rate ('S's get sick - 'I's cured) = aSI - bI.
The recovered are no longer susceptible and therefore also proportional to those infected: dR/dt = bI. ('a' is how infectious the disease is and 'b' is rate at which 'I's become 'R's.)
The threshold population value is computed by b/a.
A vaccination should be administered according to the threshold population value (TPV). The entire population save TPV should be vaccinated to avoid an epidemic.
Challenges
I dont understand how the 192 of the dI/dS equation was obtained. And why is there a peak only went the initial value is greater than 192? Shouldn't all initial values no matter what it is produce a peak with its trajectory? I do not exactly understand the significance of a threshold population value.
Mathematics has an application in everything in everyday life, including to ability to predict an epidemic and therefore the amount of vaccine that would be needed with respect to a population which is of course growing exponentially; (however for the SIR model to work population is help constant).
The S-I-R model:
S - susceptibles (these are not sick yet but could be, future)
I - Infecteds (these are sick already, present)
R - Recovered (these have already been sick, past)
The 'S' reduce over time as they get sick when they come in contact with the 'I's. The two are thus proportional. (infections and contact) thus giving the formula: dS/dt = - (rate at which S gets sick) = -aSI. (the newly sick are then aSI). Those recovering are proportional to those infected: bI; therefore: dI/dt = rate ('S's get sick - 'I's cured) = aSI - bI.
The recovered are no longer susceptible and therefore also proportional to those infected: dR/dt = bI. ('a' is how infectious the disease is and 'b' is rate at which 'I's become 'R's.)
The threshold population value is computed by b/a.
A vaccination should be administered according to the threshold population value (TPV). The entire population save TPV should be vaccinated to avoid an epidemic.
Challenges
I dont understand how the 192 of the dI/dS equation was obtained. And why is there a peak only went the initial value is greater than 192? Shouldn't all initial values no matter what it is produce a peak with its trajectory? I do not exactly understand the significance of a threshold population value.
Thursday, April 10, 2008
10.6: Modeling the interaction of two populations.
Main Points
In reality, entities such as populations tend to interact with others other thna themselves. An analysis of this situation would require two differential equations. Exampels are the Predator-Prey model and the Symbosis model. In the former model, Lotka-Volterra equations are used. In this model, in the absense of predators the prey's population increases and in the absense of prey the predators' population decreases. Multiple population functions can be plotted as a function of time and this information is gotten from the shape of the trajectories of the equations.
Challenges
Is this concept is like those before only with a simultaneous equation twist to it? I do not know how to interprete the slope field or the trajectories. I do not know how the trajectories are obtained . . .
In reality, entities such as populations tend to interact with others other thna themselves. An analysis of this situation would require two differential equations. Exampels are the Predator-Prey model and the Symbosis model. In the former model, Lotka-Volterra equations are used. In this model, in the absense of predators the prey's population increases and in the absense of prey the predators' population decreases. Multiple population functions can be plotted as a function of time and this information is gotten from the shape of the trajectories of the equations.
Challenges
Is this concept is like those before only with a simultaneous equation twist to it? I do not know how to interprete the slope field or the trajectories. I do not know how the trajectories are obtained . . .
Tuesday, April 8, 2008
10.5: Applications and modeling.
Main Points
For the differential equation: dy/dx = k(y - A) the general solution is y = A + Ce^kt for any constant C. An equilibrium solution is constant for all the values of the independent variable, at this point the graph of the solution is a horizontal line. This solution is stable when a change in the initial conditions produces a solution that heads towards the equilibrium as the independent variable heads towards positive infinity and is unstable when the solution of the initial conditions veeraway from the equilibrium. A differential equation may have more than one equilibrium.
Newton proposed that the temp of a hot object decreases at a rate proportional to the difference between it and the surroundings and the same with cold objects.
Challenges
I feel as though the book is too specific with its examples and thus understanding the topic outside the example is a little difficult. Obtaining the general solution is fairly easy when we are given the differential equation is given. How do we do anything else with it? And why are there a number of graphs for a solution? Do all differential equations have equilibrium solutions?
(I think I really need to go over that class I missed with you)
For the differential equation: dy/dx = k(y - A) the general solution is y = A + Ce^kt for any constant C. An equilibrium solution is constant for all the values of the independent variable, at this point the graph of the solution is a horizontal line. This solution is stable when a change in the initial conditions produces a solution that heads towards the equilibrium as the independent variable heads towards positive infinity and is unstable when the solution of the initial conditions veeraway from the equilibrium. A differential equation may have more than one equilibrium.
Newton proposed that the temp of a hot object decreases at a rate proportional to the difference between it and the surroundings and the same with cold objects.
Challenges
I feel as though the book is too specific with its examples and thus understanding the topic outside the example is a little difficult. Obtaining the general solution is fairly easy when we are given the differential equation is given. How do we do anything else with it? And why are there a number of graphs for a solution? Do all differential equations have equilibrium solutions?
(I think I really need to go over that class I missed with you)
Sunday, April 6, 2008
10.4: Exponential Growth and Decay
Main Points
For the differential equation dy/dx = ky, its general solution is y = Ce^kt for any constant C. When k > 0 this is known to be exponential growth and when k < 0, this is exponential decay. y = C when t = 0. Exponential growth solution curves become exponential decay curves when they are reflected over the y axis. A differential equation is only an approximation of a representation of a number of values, but for large values this approximation method is pretty accurate.
Challenges
In example 1, I do not understand how the solution for iii was obtained. I'm not sure I understand how to set up a differential equation.
For the differential equation dy/dx = ky, its general solution is y = Ce^kt for any constant C. When k > 0 this is known to be exponential growth and when k < 0, this is exponential decay. y = C when t = 0. Exponential growth solution curves become exponential decay curves when they are reflected over the y axis. A differential equation is only an approximation of a representation of a number of values, but for large values this approximation method is pretty accurate.
Challenges
In example 1, I do not understand how the solution for iii was obtained. I'm not sure I understand how to set up a differential equation.
Thursday, April 3, 2008
10.2: Solutions to differential equations.
Main Points
A differential equation (DE) is one involving the derivative of an unknown function. The unknown is a function rather than a number. This function is computed by finding the rate of change per some y-value for a number of x values for the differential equation and finding the formula that fits the pattern of values. This formula then becomes the solution of the differential equation. The general solution of a differential equation is the family of functions of the solution of that DE. A DE satisfied by both the solution and initial condition is a particular solution.
Challenges
I know I should probably understand this but how and why is the unknown of a differential equation a functino rather than a number? Does the solution of a differential equation always include a constant?
A differential equation (DE) is one involving the derivative of an unknown function. The unknown is a function rather than a number. This function is computed by finding the rate of change per some y-value for a number of x values for the differential equation and finding the formula that fits the pattern of values. This formula then becomes the solution of the differential equation. The general solution of a differential equation is the family of functions of the solution of that DE. A DE satisfied by both the solution and initial condition is a particular solution.
Challenges
I know I should probably understand this but how and why is the unknown of a differential equation a functino rather than a number? Does the solution of a differential equation always include a constant?
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