This usage is in keeping with use of vec in matrix theory and extended to cover scalars and nested arrays.
Problem Statement
I am dealing with percoaltion theory for disease spread. My programme currently randomly immusies a percentage of the population and then infects one of them and the final programme lets the disease spread.
However I want to be able to call different stages of the disease spread from the final programme. I thought of doing this by calling each matrix created after every run in the loop called I_pop and put them into a matrix so i can easily call each partition of that matrix. however I dont know how to return all matrices from my loop and not just the one created at the end.
Also, I need to plot the population at the start and see the progress at the end.
Any help would be appreciated.
if there is any other easier way to handle it, please comment or suggest!
Where I've added variables or functions as an aside, I've set them to Math Style "User 7", which I have renamed to "Comments". Mathcad treats variables with different Math Styles as different
To set the Math Style of a variable/function simply click on it and select "Comments" from the Style drop down in the Edit Toolbar.
I also use Text Styles (such as the Navy Bold style of this note) to distinguish my comments from those contained within the original document.
There are aslo a few quite useful functions included in the collapsed Area "Utilities" above.
The Worksheet
Create a mask for the Final step in order to cover the out of bounds areas.
As the array is simply all zeros, the simplest way to create is to set the last element to zero and Mathcad will fill in all the rest as zero by default.
Now the population under investigation lies from mask[1,1] to mask[10,10]
Initialise the randomly immunisd fraction of the population
x is the fraction of immunised population
Total population is given by
Thus
If the population lies within the mask from mask1,1 to mask10,10, ie 100 cells, why does P = 11 (ie, 121 cells)?
Created the initial matrix of inunfected population
Where 1 is the data value assigned to immunised population.
Now to randomly select an infected person and implement them into the population.
As an aside, it's normally good practice to define a global parameter, such as P, and then provide a local redefinition of it.
It's probably good enough for development work, but you might want to look at how you generate the infection source locations.
The function
Where Two Is the data value assigned to an infected person
No need to include both indices when dealing with a vector; just the one will suffice.
Number of infected people =
Just to show you can do it, I've put the definitions of ip and p_out as embedded Math regions inside the Text regions.
Previous Number of Infected =
Counter =
Now to implement the programme function to show the disease spread to anyone near them who is not immmune.
Your function Impop may not be doing what you intended it to do.
This means that as the loop progresses, it will almost certainly encounter cells that weren't infected at the start of the loop but were infected by one of its "predecessors" in the loop.
One way round this is to make of copy of the initial grid and then update that.
I'm not quite sure what ip is supposed to represent, but if it's meant to give the number of newly infected cells, then it should be incremented each time a vulnerable cell is infected.
I've added some simple functions
Quick check that the boundary test works and only increments ip for non-boundary cells.
useful code fragment to replace the ip+... expression when debugging.
Original Impop function: note how the infection has spread to all of the cells downward and to the right in one cycle rather than spreading evenly up and down, left and right with only the immediate, vulnerable neighbours of the infected cells being affected.
Alternative Impop function: the infection has only affected the immediate, vulnerable neighbours of the infected cells, which is possibly what was intended.
Visually assess the changed cells ...
In the next iteration, note how the original function works slowly back and to the left, but propagates rapidly below and to the right, whereas the alternative version spreads more slowly and only to neighbours of infected cells.
The Propagation of the Disease for the Population and Random Source of Infection Shown Above.
You're on the right tracks for collecting all of the infection data.
The program below returns the infection array for each cycle.
a vector of index numbers for each cycle
the change in infection numbers between each cycle
A plot of how the infection spreads at each cycle, showing the number of infected cells at each and the number of new infections since the last cycle.
Now to invesitgate how the spread of infection depends upon the level of immunisation.
... changed to <x
I've modified the way that the immunized-infected initial populations are created. I've created a vector of immunization rates and then called your pop function to create each of the (5) initial populations.
Create a number of randomly immunized populations
number of populations
corresponding range variable.
generate a evenly spread range of immunization rates
linspace is in the Utilities Area
generate the corresponding populations
check the immunization rate vector
examine the initial populations
check the stats look OK for each pop
Infect each population
index of the cell to infect
To infect each population, we need to get at each population within U_pop and then set the source element in each one.
Unfortunately, whilst this is an obvious, if somewhat unwieldy, way of setting an element in a nested matrix, it doesn't work at worksheet level (ie, using the := definition operator.
Inspect the initial populations to make sure they are consistently and correctly infected.
I'm going to bypass this part of the original code ...
And reuse some of the new code, making it into function to ease its application to each population ...
Infect all of the initial populations
population development (note use of vectorization operator)
the result is nested array of the (history,IP) pair for each population.
choose one of the populations
immunization density
Inspect the differences each cycle
Calculate the infection difference between each cycle