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Math 3MB3 Final Project - Topic Descriptions
Each project includes a description of a base model and some possible extensions. Your group will need to fill in the details of the base model and analyze that, then choose ONE of the suggested extensions and modify your model to explore that extension. Unless otherwise specified, you may construct either a discrete or continuous model.
Project 1: Carbon Cycle
Carbon is a key element in terrestrial ecosystems. It enters the soil when plants die, or shed leaves and branches (called “litterfall”, https://en.wikipedia.org/wiki/Plant_litt er); it leaves the system by being turned into carbon dioxide by bacterial metabolism or other chemical processes. In recent history, human industrial activity has begun to interfere substantially with the biological carbon cycle in various ways, leading to climate change (https://en.wikipedia.org/wiki/Climate_change).
In this project, you will model carbon cycling (https://en.wikipedia.org/wiki/Carbon cycle) from the atmosphere and back through various components of an ecosystem (e.g., plants, litterfall, humus [https://en.wikipedia.org/wiki/Humus]) and examine the effects of human activity on this cycle.
Base model
Start simply by tracking carbon levels in the litter on a forest floor alone (litter being naturally-occurring debris like leaves, branches, and deadfalls, that decay over time, not human-made trash). Let ct be the density of carbon in the litter at time t (measured in grams of carbon per square metre, or gC/m2 ).
Assume carbon enters the litter through litterfall continuously at a constant rate, z, and leaves at a rate proportional to the amount of litter currently in the system via humification (the conversion of litter into humus). Set up a simple discrete-time model for this scenario and analyze it.
Now account for atmospheric carbon by assuming that humus and litter both respirate carbon dioxide in to the atmosphere at rates proportion to the amount of each substance in the system, and that atmospheric carbon gets converted back into litter via plant growth, which produce litter at some rate proportional to the number of trees.
Set up a model for this scenario. You may find it easiest to set up one state variable for the density of carbon in each component of this system: litter, humus, atmosphere, and plants.
Finally, add some human activity to the model, and explore the repercussions. For instance, what if humans were to plant trees each year to draw out more atmospheric carbon? What if humans were to cut trees down to use this natural resource to produce goods? How might this affect model predictions?
Possible extensions
Expanded ecosystem
Model more components of the carbon cycle: plants (subdivided into leaves, branches, stems, and roots), litter, humus, and stable humus charcoal, denoted by x1 , x2 , ...x7 , respectively. Atmospheric carbon flows into plants via photosynthesis. Leaves, branches, and stems in- crease the carbon in litter, which then increases the carbon in the humus via humification. Roots increase the carbon directly in the humus (not via litterfall). Humus increases the carbon in the stable humus charcoal via carbonization. Litter, humus, and stable humus charcoal all increase carbon in the atmosphere via respiration.
As a simplification, assume that the atmosphere has a constant carbon content (un- changed either by giving carbon to plants or by absorbing carbon from the litter, humus, or stable humus charcoal), since it contains so much more carbon compared to the other components of the system. You can therefore imagine that the atmosphere is outside of the system and only model the other components. The atmosphere simply introduces a con- stant rate of carbon, z, into the system, and proportions p1 , ..., p4 indicate how much of this constant influx of carbon gets allocated to leaves, branches, stems, and roots, respectively. Parameters kij give the rate of carbon flow from xi to xj. Any carbon going back into the atmosphere can be thought of as simply leaving the system, which occurs at rate ki0 for carbon flowing from xi to the atmosphere.
A compartmental diagram of this model is as follows:
Model parameters for various ecosystems can be taken from the following table, which comes from ”A Simulation Study for the Global Carbon Cycle, Including Man’s Impact on the Biosphere” by Goudriaan and Ketner:
(The unit Gt is Gigatonnes, or a billion tonnes, where 1 tonne (metric ton) = 1000 kg = 1 Mg.)
Note that the “leaving litter” flow does not distinguish between carbon leaving litter by humification or respiration. The humification factor h gives the proportion of the “leaving litter” flow that goes into the humus, which leaves 1−h of the flow togo into the atmosphere via respiration. Similarly, the “leaving humus” flow must be divided into the proportion that goes into the stable humus charcoal via carbonization, c, and the proportion that goes into the atmosphere, 1 − c.
You could consider one or several ecosystems based on the parameters in the above table, and compare the predicted carbon-cycling behaviour between ecosystems.
Seasonal parameters
Many parts of the world experience seasonal climate variation, which affects the growth and decay of plants. Consider some seasonal variation in plant-related flows (e.g. by assuming that some or all of per-capita plant-related flow rates are not constant but vary sinusoidally with a period of a year). How does this change your model predictions?
Project 2: Drugs in the Body
The way in which drugs are administered to individuals and then metabolized by the body is of great concern in pharmacology and medicine: drug dosage over time must be high enough for some period to have a medicinal effect on the patient, but not so high that they may overdose. There are various drug delivery methods available to those designing patient therapies. Metabolic pathways also vary depending on drug administration and type.
In this project, you will compare the effects of various drug delivery methods and metabolic pathways on a patient’s dosage over time.
Base model
The rate at which the body processes drugs depends on two factors: the rate at which the drug is administered and the drug processing rate. Let A(t) be the amount of a drug in the body (in milligrams) at time t.
Then we can denote the dosing rate D(t), which is independent of the current amount of drug in the body A(t), and the processing rate P(A), which does depend on A. Then the model for A(t) can be cast as an “inflow minus outflow” relationship:
As a first pass, assume that a total of Dtot mg of the drug is given intravenously at a constant rate of r mg/hour for the first h hours, at which point drug administration ceases:
We also assume that drug processing occurs linearly, proportionate to the amount of drug currently in the system: P(A) = cA, where c is the clearance rate.
Note that, while D(t) is piecewise, you can split the domain into t ∈ [0, h] and t ∈ (h,∞), and then study what is happening in each piece of the domain to get a sense of the overall dynamics.
Assume the drug is given as a pill that is designed to dissolve slowly. As the pill dissolves, less and less of the medication is released. At any instant, the rate of release of the drug can be modelled by D(t) = Dmaxe−t/h.
Calculate how much of the drug is released by the pill in the time interval [0, h]. Compare this dosage plan to the intravenous plans you previously explored.
Possible extensions
Logistic drug metabolism
Assume that instead of P(A) = cA (exponential drug clearance), the drug is metabolized logistically, i.e., P(A) = cA(1 − A/K). Explain how one might interpret the new logistic parameter K in this context. How does this change the amount of drug in the bloodstream over time, compared to the intravenous treatment plans explored previously?
Intermediate absorption compartment
Consider a compartmental model where the drug needs to enter another compartment before becoming bio-available (e.g., it is injected in the blood but needs to diffuse into organs before it can be used, or it is swallowed into the digestive system and needs to diffuse into the blood stream). Model the amount of drug in each part of the body over time (blood, digestive system, organs—whichever apply in your context) as well as the diffusion processes between these body parts. Make a variety of assumptions for the diffusion mechanisms and explore how these affect model results.
Project 3: Conservation and Wildlife Management
It is important for ecosystems that the organisms within it remain in balance. If there are too few members of species X, it could go extinct, which may cause problems for other species that rely on X as a food source.
In this case, humans may undertake conservation efforts to save species X . On the other hand, if species X is too abundant, it may deplete its main food source, which could pose a risk to other species that feed on the same organism. Here, humans may consider wildlife management strategies, like hunting, to keep species X from growing too abundant.
In this project, you will model an organism’s population and the effects of various con- servation and wildlife management strategies on the that population.
Base model
Many species of wildcats are endangered, including the bobcat. To better understand bobcat population dynamics, we will construct an age-structured model for a bobcat population with 16 age classes. This model should include survival and reproduction, as in the age-structured model explored in lecture. Use the following parameters in your model:
Note that, just as in lecture, we are only tracking bobcats that can give birth, and these parameters reflect that assumption.
Draw a compartmental model diagram and use it to derive a general matrix model for this context (“general” meaning with symbols for parameter values) of the form P(⃗)(t+1) = M P(⃗)(t).
Use the best and worst case parameters in the model to determine the long-term model behaviour under each scenario. What is the long-term growth rate and distribution of the population in each case?
Now consider an intervention strategy where constant numbers of young bobcats (aged 0-2) are added to the population each year through breeding and conservation programs. How does this change your model predictions?
Possible extensions
Hunting
Assume now that instead bobcats are overly abundant and the local conservation authority is worried that they will overconsume their prey and potentially endanger them. Model the scenario where adult bobcats are now instead hunted at a constant rate each year (bobcats aged 3 and older). What does the model predict will happen in the long term? Suppose instead that the hunting rate is proportional to the current bobcat population. How much hunting can the bobcat population sustain before it too becomes endangered (i.e., its long- term growth rate predicts a decline)?
Catastrophes
Model the occurrence of sudden catastrophes to the population every n years. Catastrophes lower each age’s reproduction rate by some proportion p. Explore various parameter values for n and p and discuss the resilience of the population.
Project 4: Ebola
The way in which an infectious disease spreads in a human population depends on many factors, some of which are specific to the disease and population being studied. For instance, in the 2014 West African Ebola epidemic, some disease transmission occurred at funerals, as a result of contact between susceptible individuals and individuals who had recently died of Ebola.
In this project, you will extend a simple disease model to account for transmission at funerals. You may also consider other extensions appropriate for modeling the 2014 West African Ebola epidemic.
Base model
We explored the SIR (susceptible/infected/recovered) model in lecture (we didn’t explicitly discuss the R compartment though).
There are hundreds of variants of the SIR model dealing with various complexities of disease biology and human society. One variation is the SHERIF model http://www.sciencedirect.com/science/article/pii/S17554 36517300233, developed to analyze the recent West African Ebola outbreak, which adds Hospitalized, Exposed, and Funeral compartments to the SIR model (the order is chosen for pronounceability). Note that this paper discusses a continuous version of the model, instead of the discrete models we discussed in class.
To make things simpler, consider the SIFR model, which includes transmission caused by contact occurring at funerals. A discrete version of this model can be encoded in the following system of equations.
Explain what each term in the model equations denotes, and what the parameters rep- resent. Explore the long-term behaviour of this model. What does it predict for various parameter values?
Add vital dynamics, i.e., births and deaths, to the model: St+1 gains a +µN −µSt term; the other compartments use a loss term with a per capita rate µ (e.g. −µIt for the It+1 compartment)) and explore the long-term behaviour of the model. (If necessary, drop the F compartment and analyze the SIR equations.)
Possible extensions
Controlling funeral spread
Model public health interventions that reduce transmission at funerals by diverting some people from I directly to R without going through F. How does this change model predic- tions?
Add exposed and hospitalized individuals
Add the E and H terms to produce a discrete version of the full SHERIF model. Compare this model’s results to that of the SIFR model analyzed previously (with and without vital dynamics).
Project 5: Formation of Spots on Animals
This project will have you explore how we can use cellular automata to model pattern formation. One example of this is the formation of spots or stripes on animals.
Base model
Start by reading the “Turing patterns” subsection in chapter 11.5 of our course textbook. The base model for this project involves making sense of the model setup described here and successfully implementing a simulation of this model to produce a figure similar to what is shown in Figure 11.10. You should use periodic boundary conditions in this model.
In your final paper, you should explain in your own words what is happening in this model and what all the pieces of it mean. You should also explore multiple simulations of this model with different initial configuration patterns.
Possible extensions
Exploring the effects of different boundary conditions
In the base model, you used periodic boundary conditions in your simulations. Try rerunning the model using fixed boundary conditions instead. Compare and contrast your results with the two different boundary conditions.
Exploring the effects of different neighbourhoods
In the base model, you used Moore neighbourhoods of different radii. Try rerunning the model using von Neumann neighbourhoods (still with the same radii but now only taking the cells in the four main compass directions and omitting the cells on the diagonals). Compare and contrast your results with the two different neighbourhoods.