Foundations of Applied Mathematics

MATH 4700 – Fall 2024

Practice Final

1 Hungry, Hungry Lions (20 points)

Suppose some ecosystem contains some lions and a rather small number of zebras. Zebras eat grass, which is in abundant supply, while there is nothing for the lions to eat except for zebras. Here you will be asked to model the interactions between three populations:

• well-fed lions,

• hungry lions,

• zebras.

Well-fed lions are simply those that have recently had occasion to eat zebra, while hungry lions are those which have gone without zebra meat for a while.

Assume that, in the absence of any external influences, each of the subpopulations would have a constant per capita birth rate and death rate. Of course these birth rates and death rates will be different for the different populations.

a. (7 points) Define variables and parameters for a population model to describe this ecosystem, indicating dimensions. Preferably, define your constant param-eters so that they are all positive, but if you have a negative parameter, state explicitly that it is negative.

b. (3 points) What kinds of inequalities are suggested by the information given above? (No assumption is made that this is a long-lived, stable ecosystem; the species could be either flourishing or extinct.)

c. (10 points) Write down a set of differential equations to model the dynamics of the populations of the ecosystem.

2 Perturbation Theory (30 points)

Consider the following nondimensional system of equations where ε is a small nondi-mensional parameter (|ε| ≪ 1):

a. (10 points) Develop a regular perturbation theory to construct a good approx-imation to the solution of the initial value problem. You should explicitly solve for the leading order nontrivial main term, that is, the most important term in the perturbation expansion for x and y which is not zero. You should also obtain the differential equations including initial conditions for the next most important nonzero term in the perturbation series expansion for the solutions x and y. But you do not need to solve these differential equations. Just convince yourself that the solutions are not zero. If a solution is zero, it means you should derive the differential equation for the next term in the perturbation series expansion of the solution, until you get a differential equation which has a nonzero solution. Your answer is complete if you can write an approximations for x(t) and y(t) each as a sum of:

• a nonzero leading order term for which you provide an explicit formula, plus

• a nonzero correction term for which you provide a differential equation with initial conditions.

• an estimate of the error of your approximation.

b. (10 points) Sketch the solution for (x(t), y(t)) on the phase plane over a time long enough that x varies significantly. Note you do not need to obtain a full analytical approximation to sketch the solution graphically.

c. (10 points) Use a suitable approximation method to obtain an analytical de-scription for how x(t) varies with time, to leading order, as it moves significantly away from its initial value.

3 Statistical Analysis of SEIR Model (20 points plus bonus points)

In class, we discussed the SEIR model for susceptible (S), exposed (E), infective (I), and recovered (R) populations:

where N = S +E +I +R is the total population. There are many kinetic coefficients (all positive) here whose meaning aren’t important for the problem. The initial conditions are: S(0) = 500, E(0) = 0, I(0) = 27, and R(0) = 0.

Suppose we take the following two parameters as uncertain:

• the average time of infectiousness τI , which is given a prior distribution with a symmetric triangular shape on the values 10 ≤ τI ≤ 18 with the peak value (mode) at τI = 14,

• the infectivity parameter β is given a prior distribution of a normal distribution with a peak at µ = 2.25, a width σ = 1, truncated to reject negative values.

The ground truth values are τI = 14 and β = 2.25.

a. (5 points) Here are the predictions of the dynamical model based purely on prior beliefs, using the same convention as in class (solid line for ground truth, dashed line for the mean of the simulations sampled from the prior, with in-dividual samples from the prior parameter distribution plotted in faint lines). The dashed lines and solid lines are essentially coincident, meaning the mean of the prior predictions is very close to the ground truth. What might explain this?

Figure 1: Prior predictions of SEIR model.

b. (5 points) The infective population I(t) and recovered population R(t) were taken as observables, with random counting errors of magnitudes σI = 75 and σR = 20, respectively. Furthermore, the recovered population was observed with a bias of -10 on average. The data scientist’s observation model was the same as the true observation model.

The posterior distribution obtained from observations at 100 time points equidis-tributed over the time interval 0 ≤ t ≤ 30 can be summarized (via the reported posterior mean and standard deviation):

• τI = 14 ± 1,

• β = 2.42 ± 0.22.

How do the posterior and prior distributions of the parameters τI and β compare in terms of their accuracy and precision? Explain.

c. (10 points plus bonus points) Here are the predictions of the dynamical model based purely on the posterior distribution of the parameters, using the same convention as in class (solid line for ground truth, dashed line for the mean of the simulations sampled from the posterior, with individual samples from the posterior parameter distribution plotted in faint lines). The observed data is plotted by discrete circles. As not much is happening after the end of the data collection period, let’s focus on understanding the behavior. of the various populations during the time interval 0 ≤ t ≤ 30. How did the data collected affect the inference of how the various populations behaved on the time interval 0 ≤ t ≤ 30, relative to the prior beliefs? The bonus points are for good thoughts on why the data affected this inference. Feel free to continue your answer on the next page.

Figure 2: Posterior predictions of SEIR model.

d. (5 bonus points) What would be the point of even using a model to predict what happened over a time interval where we already have collected observa-tional data?

4 Influential Mathematics (25 points plus 5 bonus points)

Imagine a nation (Lineland) which is represented spatially in a one-dimensional way as an interval [0, L] where L = 3000 km is the distance between the east and the west coast. Each point x represents the north-south cross-section of Lineland which is a distance x from the west coast.

a. (7 points) Write a partial differential equation model with initial conditions to describe the evolution of the number density ρ(x, t) of individuals in Lineland as a function of east-west location x and time t subject to the following require-ments:

• The population density of the nation starts as (2 + cos(2πx/L)) × 105 people/km, i.e., there is a higher population density near the coasts than the midlands.

• People are moving and traveling, with individual variations but with a bias to move away from the coasts (I would have said the opposite 5 years ago). But no one enters or leaves Lineland, and we neglect births and deaths on the time scale of interest.

You should write down a specific partial differential equation, but it can involve positive parameters that you do not specify numerical values for. Explain as precisely as you can the meaning of any parameters you introduce. And do use the quantitative specifications from the problem in your model.

b. (5 points) Sketch the initial population density ρ(x, 0) and an approximate representation of ρ(x, t) after t = 1 year.

c. (8 points) Consider a social practice (say a certain way of wearing clothing or a way of verbal communication) which people find interesting to adopt when they see others perform. it in real life, but for some reason is not influential via social media. Write a partial differential equation model to describe the number density ρS(x, t) of individuals in Lineland adopting the new social practice as a function of east-west location x and time t which has the following features:

• At the present time t = 0, the social practice has been adopted by 5% of the population living within 100 km of the west coast, and by nobody anywhere else.

• The motion of people in Lineland from part a is not correlated with (related to) whether or not the individuals adopt the social practice

• A given individual who has not adopted the new social practice is more likely to adopt this practice when they see others in their (real-world) vicinity doing the new social practice.

• People who are doing the new social practice find it lame and stop doing it after about 2 months on average.

You should write down a specific partial differential equation, but it can involve positive parameters that you do not specify numerical values for. Explain as precisely as you can the meaning of any parameters you introduce. And do use the quantitative specifications from the problem in your model.

d. (5 bonus points) How would your model in part c change if the social practice could be spread by social media?

e. (5 points) How would you use the solution to your model to part c to count the number of people adopting the new social practice in the middle third of the country one year from now?

5 Standing, Walking, or Running (30 points plus 15 bonus points)

Consider a random walker moving on a one-dimensional lattice with spacing ∆x and time step ∆t. At each time step, the walker independently makes one of three choices with the indicated probabilities:

• stands still with probability ps,

• moves one lattice site to the left with probability pℓ , or

• moves two lattice sites to the left with probability p2.

The random walker starts at position m = 0. Define w(m, N) to be the probability that the random walker is at position m after N time steps.

a. (10 points) Derive a master equation to relate w(m, N + 1) to its value at the previous time step N, for N ≥ 0. Explain your reasoning.

b. (10 points) Now suppose that the lattice sites are separated by a small dis-tance ∆x and the time steps take a short time ∆t, and that Np random walkers are moving independently according to the above rules, all starting from lattice site m = 0 at time step N = 0. Take a continuum limit of the master equa-tion to obtain a macroscale partial differential equation for the number density ρ(x, t) of Np random walkers each moving independently according to the model stated above. Relate all coefficients in the partial differential equation to the microscale parameters. Assume that ps, pℓ , and p2 are not close to the values of 0 or 1.

c. (10 points) Sketch the number density ρ(x, t) as a function of x for t = 0, t = 1, and t = 10.

d. (15 bonus points) What other macroscale equations can be achieved by taking ps, pℓ , and/or p2 to be O(∆x) close to 0 or 1 as the continuum limit is taken?