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

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

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

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

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

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

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

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

Next up: and SIR, and his interesting model of female birth weights.

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

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

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

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

Next up: and SIR, and his interesting model of female birth weights.

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

Galla with the different classes. 

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

Next up: an SIR.

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

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.

Galla age distribution model

This model illustrates predator prey interactions using real-life data of wolf and moose populations on the Isle Royale. It was "cloned" from a model that InsightMaker provides to its users, at
https://insightmaker.com/insight/2068/Isle-Royale-Predator-Prey-Interactions
Thanks Scott Fortmann-Roe.

I've created a Mathematica file that replicates the model, at
http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/Moose-n-Wolf-InsightMaker.nb

It allows one to experiment with adjusting the initial number of moose and wolves on the island.

I used steepest descent in Mathematica to optimize the parameters, with my objective data being the ratio of wolves to moose. You can try my (admittedly) kludgy code, at
http://www.nku.edu/~longa/classes/2018spring/mat375/mathematica/Moose-n-Wolf-InsightMaker-BestFit.nb

{WolfBirthRateFactorStart,
WolfDeathRateStart,
MooseBirthRateStart,
MooseDeathRateFactorStart,
moStart,
woStart} =
{0.000267409,
0.239821,
0.269755,
0.0113679,
591,
23.};

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

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

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

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

Next up: and SIR, and his interesting model of female birth weights.