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Theory Versus Reality
I tried to do a more accurate version of Lotka's fit of US population growth. The data used came from Appendex A of the US Census publication Measuring America. Since the day of the year that the...
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SI Pandemic Model and the Logistic Equation
One can produce a very simple model for a pandemic involving just infecteds and susceptibles, the haves and the have nots, with the infection rate being proportional to rate of interaction between...
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Single Variable Equations for the SIR Model
One can also derive single variable equations for the SIR Model. We start with a short review of the model.We can eliminate I from the equation for the rate of change for S as follows.Note that the...
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A Closer Look at US Population Growth
Since undergoing open heart surgery last month I've been trying to analyze the US census data in hopes of being able to make a better prediction of future growth. If we assume the rate of change in...
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Monitoring CA Covid-19 Data
One can download California Covid-19 statistics which come in a Comma Separated Value (CSV) file format that one can open with a spreadsheet program or app and save.To get daily data for the entire...
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A Fit for the CA Covid-19 Data
I managed to get a fit for the CA coronavirus data presented in the last blog. The first step was fit a cubic polynomial of the known cases, K, to dK/dt.Which gave the following results. This...
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A Higher Order Fit for the CA Covid-19 Data
Fitting a fifth degree polynomial to the rate of change, dK/dt, gave an even better fit to the CA Covid-19 Data. The coefficients of the powers of K were,The resulting fits were,Unfortunately, the...
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Data Analysis of the CA Covid-19 Data
If one wants a fit that is useful for predictions the fit function for dK/dt should tend to zero as K becomes large. Since polynomials tend to infinity as K goes to infinity we can try fitting...
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Is It Time for Some Error Analysis?
It seems we've reached the point where it would be advantageous to look at possible errors. If one includes the number of deaths with the number of cases in K one gets similar results with the fit...
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Turning the Corner?
Here's another comparison of S* and I=dK/dt from the CA Covid-19 fit which might be interpreted as "turning the corner" on the coronavirus. The plot is versus time and again the initial date is 4/1/20.
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Using Relative Changes to Determine Rates
One can use the relative change in a quantity to determine a rate of change such as ΔI/I=Qdt. When one does this with the CA Covid-19 data and a calculation based on the fit polynomial one gets a...
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A Possible Formula for Q
If one calculates and plots Q/I one gets the following curve.This looks somewhat like an exponential decay function and if we try to fit one we get the following. The value for λ was chosen by...
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Checking Out the QI Model
There was a lot of interest in "curve flattening" but the fits don't appear to show much evidence of any. There does appear to be a little lowering near the peak but could it have been due to the...
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Beginning of a Second Wave in SoCal?
Theory can't always be trusted as can be seen in recent CA Covid-19 data.The current data is starting to deviate from the fitted curve in what appears to be the beginning of a second wave of the...
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Reading CA Covid-19 County Data & Preparing Plots
Reading the CA Covid-19 case data files is initially challenging but is easier than one might think. One starts by collecting information on the number of county entries with the same name.Next a...
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An Interpretation of the dI/dt Eqn
One might wonder why dI/dt is proportional to I squared and not just I. It could be that the number of potentially infected individuals in Q around the infected individuals is proportional to I as...
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When was the Jupiter-Saturn Conjunction?
There's been a lot of talk about The Great Conjunction lately but little information on when it occurs. Given ephemeris data from JPL's Horizons on the positions of planets Jupiter and Saturn we can...
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Doing Simulated Experiment to Verify a Fit Procedure
Suppose you want to do an experiment but you're not sure if the results will be as accurate as needed. One can do a simulation first to evaluate the results. The first thing we need is some simulated...
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Combining Multiple Experiments
If one repeats the same experiment a number of times the errors become more uniformly distributed.A plot of the data shows a more uniform spread.We would expect the error of the average value of y...
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Determining the Uncertainty in the Coefficients
One might wonder what the uncertainty in values of the coefficients in the last blog might have been. Repeated "experiments" shows there is some variation present in the results. Can we get a better...
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Validity of the Formula for the Uncertainty in the Fit Coefficients
My derivation of the uncertainty in the fit coefficients assumed that the expected value <δyjδyk>=0 for k≠j. It is not obvious that this is so but we can demonstrate it in the following...
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LS Fit Using Orthogonal Polynomials over the Data Interval
One can make the vectors or functions used to compute the fit coefficients more symmetrical if one defines a set of orthogonal polynomials on the interval covered by the data. The data for the 9...
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The Principle of Least Action
Maupertuis loosely introduced the Principle of Least Action in 1741. A few years later in 1744 in Methodus inveniendi Euler associated Action with an integral that was a minimum for the path a mass...
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Formulas for v·dr
The formulas for v·dr modified to contain the variations are,with t being the tangent unit vector to the trajectory gives,
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More on the Principle of Least Action
It appears that the way the variation is done affects the minimum of the action integral. If one arranges the variation so that it is normal to tangent of the curve the number of terms contributing...
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