5740-001 Math Modeling
Fall 2017

Class meets: W,   04:35PM-07:05PM    JTB 120
Office Hours: F, 02 PM-03PM, JWB 225
Instructor: Andrej Cherkaev, JWB 225, ph. 801-5816822, email: cherk@math.utah.edu (subject line must start with 5740)

A mathematical model is a formal simplified description of essential features of a system that allows for simulation and prediction of its output. Mathematical models include dynamical systems,  differential equations, game theoretic models, etc.
Mathematical models are widely used in the natural and social sciences and engineering providing a language for communications between mathematicians and physicists, engineers, computer scientists,  economists,  etc. Mathematical modeling usually refers to relatively new areas of applications of mathematical ideas.

In this course, we investigate models of various processes, such as species growth and interaction, epidemics, traffic flow, fair split, evolution, dynamics of unstable processes, a discrete chain of events,  metamaterials. The needed references must be found on the Internet or in the library. Example are computed using either Maple or any other platform such as Mathematica, MatLab, etc.

The class contains lectures and workshop-type presentations of projects by student groups. The grade is based on the presentations and reports, and on a couple of home work assignments.

Syllabus:  (preliminary list)

Introduction (1 week)
Introduction to Math. Modeling. Non-uniqueness of math model.
The Greatest model -  Universe (from Ptolemy to Big Bang)
Problem: Divide the class into working groups
Populations dynamics 1. Single species (2 weeks)
Classical models, Equations with delay
Leslie model of reproduction
Populations dynamics 2. Interacting species. (2 weeks)
Lotka-Volterra model, variations, several species
Dynamics of epidemics
Fairness and evolution (2 weeks)
Cooperative games: Fair shares
Evolutionary games
Traffic flow (3 weeks)
Continuum model. Shock waves. Dissipation
Discrete model. Cellular automata. Rule 184
Unstable motion. (2 weeks)
Bouncing box
Repeated Motion. (2 weeks)
Discrete Waves: Domino chain
Chain with bistable links
Material models: Metamaterials (2 weeks)
Auxetic materials
Morphing structures


1. Suggest an algorithm of forming groups of three for each project so that each student works with maximal number of classmates.\

2. Read from
A Mathematical Introduction to Population Dynamics
by Howie Weiss (Georgia Tech)
ch 2    2.1-2.6

3. Simulate and plot using Maple or any other platform
a. Logistic and Allee, Holling Type III growth, ( continuous  model)
b. Logistic, Logistic with delay (discrete model)

4. Show that in Holling Type III functional response (2.26). [p 24]
 point N=0 is always unstable.

Example of Maple code for plotting solutions to DE and discrete systems.

  Solve and plot solutions to Difference eqns:
x[1] = 0.01;
for i to 100 do
 x[i+1] := (1+(1-(1/10)*x[i])*.15)*x[i]
end do;
pointplot({seq([n, x[n]], n = 1 .. 50)})

x[1] := 1;  x[2] := 1;
for i from 2 to 100 do
x[i+1] := (1+(1-(1/2)*x[i-1])*.5)*x[i]
end do;
pointplot({seq([n, x[n]], n = 1 .. 20)});

Plot solutions to Differential equations

LG := diff(y(t), t) = k*y(t)*(K-y(t));
K := 40; k := 0.01;
ivs := [y(0) = 1, y(0) = 10, y(0) = 50];

DEplot(LG, y(t), t = 0 .. 50, ivs);
DEplot(LG, y(t), t = 0 .. 50, y = 0 .. 60, ivs, arrows = medium, linecolor = black);

Or, you may first solve the differential equation
and then plot the solution:

de := dsolve({diff(z(t), t) = a*(1-c*z(t))*z(t), z(0) = zo});
p := eval(z(t), de); p0 := eval(p, {a = 1.3, c = .2, zo = .1});
plot(p0, t = 0 .. 8);

HW2.  Project.
Model population growth using Leslie model. Modify model accounting for habitat limiting capacity.
Suggested sources:

HW3. From "Mathematical Introduction to Population Dynamics by Howie Weiss

1. Normalize the Lotka-Volterra model by scaling population sizes and time scale. Simulate dynamics of the normalized system for several values of the remaining parameter, graph the population density vs. time.

2.  Consider the normalized Lotka-Volterra model as in (3.41) (3.42)) with Holling-type responses of type II and type III, see section 2.4.3.  Find equilibrium points and comment, simulate and graph the population density vs. time. 

3. Problem 25. p. 54.  Consider the predator-prey system with a simple prey refuge  (eqs. (3.24) . (3.25))   How the equilibrium point is changed?  Linearize the problem near the equilibrium and comment on the change in the period depending on refuge size. Simulate the dynamics for several values of V, graph the population density vs. time.

HW4. Problems 17, 18, 19 p. 49

Yesterday we discussed
Transition of Infectious Diseases 

I reviewed chapter 4 from the book by Howie Weiss http://www.math.epn.edu.ec/emalca2010/files/material/Biomatematica/CUP101310.pdf 

and the article from Wikipedia: https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_MSEIRS_model

Using SIR model, we discussed
Qualitative behavior of solutions
steady-state solutions
critical parameter R_0 for starting the epidemics
integral curves
estimation  of S(\infty)
estimation of   R_0 knowing S(0) and S(\infty)

We mentioned modifications (other compartment models)
Including of vital  dynamics
SEIR model that includes a compartment of Exposed but not yet Infected individuals

Homework 5:

1. Conduct a similar analysis of epidemics using SEIR and SIRC models
SIRC Model (see the above article in Wikipedia) states that a part of recovered individual become Carriers and may infect Susceptible

2. Simulate the SIR, SEIR, and SIRC models, graph the solutions. 


We also started a discussion of the next project:   
Modeling Traffic Flow, Waves. 

The recommended text is: http://www.math.nyu.edu/faculty/childres/traffic3.pdf

Note: Due to our schedule, when you miss the class, you miss a week of classes! Try to attend every session. 

HW4. Problems 17, 18, 19 p. 49

Model A change in traffic due to
1. Opening a new line at the point x=rn
2. An exit at the point x=re

3. Sudden stop (red light)

We discussed

1) the shock waves in the traffic model and modification of the model that avoids shocks.
Additional reference:
see also

2) Cellular automata approach. Particularly

2.1 Simplest model: Rule 184:  https://en.wikipedia.org/wiki/Rule_184

2.2. more sophisticated models:(available through sciencedirect, login from a University computer)
 Cellular automata for one-lane traffic flow modeling by M.E. Larraga, J.A. del Rıo, L. Alvarez-lcaza,
 https://ac.els-cdn.com/S0968090X0400066X/1-s2.0-S0968090X0400066X-Cellular automata for one-lane traffic flow modelingmain.pdf?_tid=5eb0f636-a945-11e7-bf5f-00000aab0f01&acdnat=1507150286_ab90c816bf09aa1b1e3a43d4eb5e4f7f
A cell automation traffic flow model for mixed traffic
by Guangjiao Chena, Fankun Menga. Guolong Fua, Mingyang Dengb, Ling Lic

Homework: Prepare a presentation of one of these models at your choice. Pretend that this a job interview and you want to show that you are familiar with the subject.

Homework: Read the material below. Prepare a discussion of the differences and similarities between swarm behavior and gas behavior. Can shock waves develop in swarms?

Gas dynamics

http://www.mpia.de/homes/dullemon/lectures/hydrodynamicsII/chap_1.pdf (pp 9-22)

Swarm (flocking) behavior as an extension of the traffic model
http://seb-motsch.com/wp-content/uploads/2013/01/presentation_luminy.pdf (presentation)

HW Due 11/1/2017

Games. Cooperative games. Core. Shapley vector

Solve these cooperative games (due next week 11/1/2017)

Game 1.
- Player 1 owns six right gloves
- Players two and three own two left gloves each
- Player 4 owns four left gloves
Only a pair of gloves has a market value. 
Compute Shapley vector for each player

Game 2. 
Two large political parties control 48 percent of the votes each.
Three small political parties control 1.33 percent of the votes each.
Parties can form coalitions; The payload equals one if the coalition has the majority of votes, and equals zero otherwise.
Compute Shapley vector for each player

Evolutionary games (model discussion due to next week 11/1/2017, simulation due in 2weeks 11/8)

(a) https://en.wikipedia.org/w/index.php?title=Evolutionary_game_theory
(b) https://web.stanford.edu/~jhj1/teachingdocs/Jones-ess-notes.pdf
(c) https://www.math.psu.edu/belmonte/PaperFile/PhysRevE_TransitionMatrixEgames.pdf

Using (b) model the following game:  The population of G, C, R may fall ill with a fixed probability and die unless they are rescued. The rescuer loses some its fitness and the rescued gains more fitness than the rescuer loses. 
Species G always help everyone who is ill and receive help.
Species C cheat: they receive help, but do not help anyone.
Species R receive help and help only species R and G. 
1. Model the evolution of G
2. Model the co-evolution of G and C
3 Model the co-evolution of G, C and R

Compute and plot the evolutionary trajectories.

1. Plot the evolutionary trajectories for the evolutionary game above.

2. Write the equation and simulate motion of the rolling rectangular (or or cuboid) on the flat horizontal plane without friction. Simulate.
Hint:  Combine the pendulum motion and the continuity conditions at the switching  point

Newton's derivation of angular momentum: https://en.wikipedia.org/wiki/Angular_momentum#Newton.27s_derivation

HW: Domino problems

1. Magnifying dominoes
What is the relation between magnification factor, the height and thickness of the domino piece, and distance between them to make the cascade of falling dominoes possible?

2. Domino train
Find the references online. Build a model, assuming that dominoes are infinitely thin
How to compute the wave speed of falling dominoes?
What is the relation between  the height of the domino piece, distance between them, and the speed? 

HW: Nonlinear and unstable motion

1. Model oscillations of a mass-spring with dry friction https://en.wikipedia.org/wiki/Friction#Dry_friction
Model the discontinuous friction force with a continuous fast varying function.

2. Model oscillations of bystable mass-spring system.

HW: Design of materials

See the pdf file