Your browser (Internet Explorer 8 or lower) is out of date. It has known security flaws and may not display all features of this and other websites. Learn how to update your browser.

X

Menu

Mat375

Markov Chain SIR Model 3/26

Sally Dufek
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....
Andy Long

Next up: an SIR.

Markov Chain Mat375 Olinick

  • 2 years 2 months ago

Hoffmann's Galla Example

Andrew E Long
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

The Galla people have a most unusual set of age classes for their males (five of them). In this model, we look at the ages at which fathers enter, by comparison with their sons.

Thanks Mike! Interesting examples, as always....
Andy Long

Markov Chain Mat375 Olinick

  • 2 years 1 month ago

Galla Example

Sally Dufek
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

Galla Age Distribution Model.

Thanks Mike! Interesting examples, as always....
Andy Long

Markov Chain Mat375 Olinick

  • 2 years 1 month ago

Galla example

Leah Gillespie
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....
Andy Long

Next up: an SIR.

Markov Chain Mat375 Olinick

  • 2 years 1 month ago

Clone of A Simple Infection-only SIR (Susceptible, Infected, Recovered) Example

Sally Dufek
This is a simple example of (part of a) simple SIR (Susceptible, Infected, Recovered) model, suggested by De Vries, et al. in A Course in Mathematical Biology.

They wanted to illustrate the comparative behavior of differential equations and discrete difference equations. We know that differential equations are generally solved numerically by discretizing them, so that the comparison is a little bit rigged....

A comparable model in Mathematica is available at
http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-w-discrete-version.nb

SIR Math Modeling Mat375

  • 2 years 3 months ago

Driehaus English Mother/Daughter Birth Weights

Rachel Driehaus
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....
Andy Long

Next up: an SIR.

Markov Chain Mat375 Olinick

  • 2 years 2 months ago

Clone of English Mother/Daughter Birth Weights

Clay Frink
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....
Andy Long

Next up: an SIR.

Markov Chain Mat375 Olinick

  • 2 years 1 month ago

Galla Example

Lizzy Compton
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....
Andy Long

Next up: an SIR.

Markov Chain Mat375 Olinick

  • 2 years 1 month ago

Newton's Method

Andrew E Long
MAT375: Non-linear Exam....

This insight implements Newton's method as an InsightMaker model.

It is important to use Euler's method, with step-size of 1. That's what allows us to get away with this!:)

Fun to try a couple of different cases, so I have built four choices into this example. You can choose the function ("Function Choice" of 0, 1, 2, or 3) using the slider.

Andy LongSpring, 2020



Mat375 Mathematics Numerical Analysis

  • 1 month 2 weeks ago

Driehaus SIR (poisoning would be better) Markov Model

Rachel Driehaus
This is an example I thought of after reading Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

It's an SIR-type model, but one where the equilibrium (ws,wi,wr) is always the same, even as the weights in the transition matrix change.

Actually it might be better to think of this as a poisoning model: the rate of infection is constant, and independent of the existence of an infected population. That's more like disease due to an environmental effect (e.g. lead-poisoning from smelters, or mercury poisoning from the burning of coal). So infected would mean "effected", and "recovered" might be "treated" -- and ultimately released, to be exposed again.

This shows that the equilibrium does not determine the transition probabilities: two different transition matrices can have the same ultimate equilibrium.

There is a constraint on the infection rate that I haven't figured out how to build in:

InfectionRate < Min[1,wi/ws, wr/ws]

I can allow InfectionRate to vary up to 1 if I take
ws < wi
and
ws < wr
However if you violate that, you'll get interesting solutions with negative values of populations. The dynamics are pretty interesting in that case, however! If you want to see them, you'll have to remove the constraints that I put on the parameters in the Recover and LossOfImmunity parameters.

Thanks Mike! Interesting examples, as always....
Andy Long

Markov Chain Mat375

  • 2 years 1 month ago

Galla Example

Terra Ficke
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

Galla age distribution model.
Thanks Mike! Interesting examples, as always....
Andy Long

Next up: an SIR.

Markov Chain Mat375 Olinick

  • 2 years 1 month ago

Clone of English Mother/Daughter Birth Weights

Austin Hardesty
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....
Andy Long

Next up: an SIR.

Markov Chain Mat375 Olinick

  • 2 years 2 months ago

Non-dimensionalized Logistic Growth

Andrew E Long
This (simplest!) model demonstrates logistic growth.The original differential equation looks like

y'(t) = b y(t) (1 - y(t)/K)

where K is the carrying capacity of the quantity y.

But if we divide each side of the equation by K, we obtain
d(y/K)/dt = b (y/K) (1-y/K)
Defining a new variable w, the population relative to its carrying capacity, we obtain

dw/dt = b w (1 - w)
Finally we divide both sides by b, to write
dw/d(bt) = w (1 - w)
So if we work in dimensionless time units of bt, we have
w' = w (1 - w)
where the derivative is with respect to the variable bt=τ. .
τ=τ
This       This equation, as simple as possible, contains all the dynamics (all the ways the population can behave), while masking the "trivialities"; but it kind of hides the physical aspects of the problem. So it's easy to study, but harder to interpret: alas, you can't have it all!:) 

τ=1 when t=1b: so if b=.5/year, then τ=1 when t=2.

So the larger b (the greater the birthrate), the shorter the real time t to give τ=1. τ=τ=
τ=

Math Modeling Mat375 Logistic Non-dimensionalize

  • 2 months 1 week ago

Clone of English Mother/Daughter Birth Weights

Donna Odhiambo
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....
Andy Long

Next up: an SIR.

Markov Chain Mat375 Olinick

  • 2 years 1 month ago

Clone of Cannibalistic and Chaotic Flour Beetles

Terra Ficke
This is an example from Cushing's book An Introduction to Structured Population Dynamics. ​

The parameters initially included reproduce the bifurcation results on p. 39 of Cushing's manuscript.
The tuning parameter is b, the birthrate.

p. 37: The LPA flour beetle model.

The bifurcation diagram for parameter b is on page 39;
The bifurcation diagram for mu adult is on p. 59;
The bifurcation diagram for C pa is on p. 60.

Andy Long

Leslie Matrices Mat375 Cushing

  • 2 years 1 month ago

Clone of MAT 375 Midterm file: Model of Isle Royale: Predator Prey Interactions

Matthew Gall
This model illustrates predator prey interactions using real-life data of wolf and moose populations on the Isle Royale.

We incorporate logistic growth into the moose dynamics, and we replace the death flow of the moose with a kill rate modeled from the kill rate data found on the Isle Royale website.

Thanks to Jacob Englert for the model if-then-else structure.

I start with these parameters:
Wolf Death Rate = 0.15
Wolf Birth Rate = 0.0187963
Moose Birth Rate = 0.4
Carrying Capacity = 2000
Initial Moose: 563
Initial Wolves: 20

I used RK-4 with step-size 0.1, from 1959 for 60 years.

The moose birth flow is logistic, MBR*M*(1-M/K)
Moose death flow is Kill Rate (in Moose/Year)
Wolf birth flow is WBR*Kill Rate (in Wolves/Year)
Wolf death flow is WDR*W

Environment Ecology Populations Midterm Mat375

  • 2 years 2 months ago

Clone of Cannibalistic and Chaotic Flour Beetles

Jacob Englert
This is an example from Cushing's book An Introduction to Structured Population Dynamics. ​

The parameters initially included reproduce the bifurcation results on p. 39 of Cushing's manuscript.
The tuning parameter is b, the birthrate.

p. 37: The LPA flour beetle model.

The bifurcation diagram for parameter b is on page 39;
The bifurcation diagram for mu adult is on p. 59;
The bifurcation diagram for C pa is on p. 60.

Andy Long

Leslie Matrices Mat375 Cushing

  • 2 years 1 month ago

Clone of English Mother/Daughter Birth Weights

Lizzy Compton
This is an introductory example from Olinick's book An Introduction to Mathematical Models in the Social and Life Sciences. ​

"A recent study focused on the relationship between the birth weights of English women and the birth weights of their daughters. The weights were split into three categories: low (below 6 pounds), average (between 6 and 8 pounds), and high (above 8 pounds). Among women whose own birth weights were low, 50 percent of the daughters had low birth weights, 45 percent had average weights, and 5 percent had high weights. Women with average birth weights had daughters with average weights half of the time, while the half was split evenly between low and high categories. Women with high birth weights had female babies with high weights 40 percent of the time, with low and average weights each occuring 30 percent of the time." p. 274-275.

For the Markov chain, you should make sure that you're taking time steps of length 1 in the settings, and Euler. RK-4 effectively looks beyond a single previous step, so it has a sort of memory!

Thanks Mike! Interesting examples, as always....
Andy Long

Next up: an SIR.

Markov Chain Mat375 Olinick

  • 2 years 2 months ago

Clone of A Sleek, non-dimensionalized SIR (Susceptible, Infected, Recovered) model

Adam May
This is an example of an SIR (Susceptible, Infected, Recovered) model that has been re-parameterized down to the bare minimum, to illustrated the dynamics possible with the fewest number of parameters.

We're rescaled this SIR model, so that time is given in infection rate-appropriate time units, "rates" are now ratios of rates (with infectivity rate in the denominator), and populations are considered proportions (unfortunately InsightMaker doesn't function properly if I give them all values from 0 to 1, which sum to 1 -- so, at the moment, I give them values that sum to 100, and consider the results percentages).

The new display includes the asymptotics: the three sub-populations will tend to fixed values as time goes to infinity; the infected population goes to zero if the recovery rate is greater than the infectivity rate -- i.e., the disease dies out.

Note the use of a "ghost" stock (for Total Population), which I think is a pretty cool idea. It cuts down on the number of arcs in the model graph.

A comparable model in Mathematica is available at
http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/SIRModel-rescaled.nb

SIR Math Modeling Mat375 Non-dimensionalize

  • 2 years 3 months ago

Pages