(Mooney and Swift project 5.2, p. 310)
Described in more detail by M&S, but: this is a differential equation model for the rates of armament expenditure of two countries locked in an arms rate. The state variables \(x\) and \(y\) are the annual rates of expenditure, and we suppose the rates of expenditure depend positively on the expenditure of the other country (with coefficients \(a\), \(b\)) and negatively on one’s own expenditure (with coefficients \(-m\), \(-n\), so that your expenditure would decay exponentially if the other country stopped spending). If we further assume there are “underlying grievances” \(r\) and \(s\) that drive an increase in expenditure even in the absence of spending by the other country: then we get
\[ \begin{split} \frac{dx}{dt} & = ay - mx +r \\ \frac{dy}{dt} & = bx - ny + s \end{split} \]
See the analytical questions in sections 1 and 2 of the project, which describe analyses that are definitely within your capabilities
Carbon is a key element in terrestrial ecosystems. It enters the soil when plants die, or shed leaves and branches (called “litterfall”); it leaves the system by being turned into carbon dioxide by bacterial metabolism or other chemical processes. It makes sense to model this process via one ODE (for the amount of carbon in the litter compartment) or multiple ODEs (considering carbon moving through various compartments in the soil).
See part 1 of Mooney and Swift’s project.
See Mooney and Swift or B. M. Bolker, Pacala, and Parton (1998).
Mooney and Swift project 5.10
It shouldn’t be too hard to find pharmacokinetic parameters such as the rate of metabolism/loss of various drugs: for example Remien et al. (2012) (this is not necessarily the best reference; I found it following links from a quick Google search for “drug metabolism ODE parameters” (a Google Scholar search might have been better).
This is a model from Keen (1995), later elaborated by Grasselli and Costa Lima (2012). In the equations below \(\lambda\) represents employment rate (fraction of the work force employed); \(\omega\) represents the wages share of national income; \(w\) represents workers’ wages as a function of the unemployment rate \(\lambda\). (This is the simplest version of the model!)
\[ \begin{split} \frac{d\lambda}{dt} & = \lambda \left(\frac{1-\omega}{\nu} - \alpha-\beta-\gamma\right) \\ \frac{d\omega}{dt} & = \omega(w(\lambda)-\alpha) \\ w(\lambda) & = A/((B-C \lambda)^2 - D) \end{split} \]
Read Keen (1995) and Grasselli and Costa Lima (2012) and hope your head doesn’t explode.
We have seen a discrete-time version of the SIR (susceptible/infected/recovered) model in class; the continuous-time variant is even better known. There are hundreds of variants of the SIR model dealing with various complexities of disease biology and human society (Hethcote 1994). One recent variation is the SHERIF model (Champredon et al. 2017), 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). To make things simpler, consider the SIFR model, which includes transmission caused by contact occurring at funerals.
\[ \begin{split} \frac{dS}{dt} = -\beta S I - \beta_F S F \\ \frac{dI}{dt} = \beta S I + \beta_F S F - \gamma I \\ \frac{dF}{dt} = \gamma I - \sigma F \\ \frac{dR}{dt} = \sigma F \end{split} \]
Changes in credit risk of a company can be modeled as a Markov model where companies move among categories until they default. In general we expect credit risks to be ordered: in general, considering movement from compartment \(i\) to \(j\), we expect that \(m_{ij}\) decreases monotonically with \(|i-j|\) (the bigger distance between credit risks, the less likely it is to make that transition in one step). We could take a leap and imagine some special cases such as
\[ \mathbf M = \left( \begin{array}{cccc} \rho & 1-\rho/2 & 0 & \dots \\ 1-\rho & \rho & 1-\rho/2 & \dots \\ 0 & 1-\rho/2 & \rho & \dots \\ 0 & 0 & 1 -\rho/2 & \dots \\ 0 & 0 & 0 & \dots \\ \vdots & \vdots & \vdots & \ddots \end{array} \right) \] (companies can only move one step up or down in a time period).
Given the transition matrix, we want to know some of the standard things for a Markov matrix with absorbing states, e.g.: what is the expected time to end up in default, starting from various states? (Since there is only a single absorbing state in this case, we don’t need to calculate which absorbing state we end up in.)
Reduce the matrix until it’s very small (2x2? 3x3?), and/or simplify the structure of the matrix to a point where the linear algebra can be done analytically.
See notes from Lozinski and Grasselli (to be posted)
Bolker, B. M., S. W. Pacala, and W. J. Parton Jr. 1998. “Linear Analysis of Soil Decomposition: Insights from the Century Model.” Ecological Applications 8 (2): 425–39.
Champredon, David, Michael Li, Benjamin M. Bolker, and Jonathan Dushoff. 2017. “Two Approaches to Forecast Ebola Synthetic Epidemics.” Epidemics, February. doi:10.1016/j.epidem.2017.02.011.
Grasselli, M. R., and B. Costa Lima. 2012. “An Analysis of the Keen Model for Credit Expansion, Asset Price Bubbles and Financial Fragility.” Mathematics and Financial Economics 6 (3): 191–210. doi:10.1007/s11579-012-0071-8.
Hethcote, Herbert W. 1994. “A Thousand and One Epidemic Models.” In Frontiers in Mathematical Biology, 504–15. Lecture Notes in Biomathematics. Springer, Berlin, Heidelberg. https://link.springer.com/chapter/10.1007/978-3-642-50124-1_29.
Keen, Steve. 1995. “Finance and Economic Breakdown: Modeling Minsky’s ‘Financial Instability Hypothesis’.” Journal of Post Keynesian Economics 17 (4): 607–35.
Li, Shou-Li, Ottar N. Bjørnstad, Matthew J. Ferrari, Riley Mummah, Michael C. Runge, Christopher J. Fonnesbeck, Michael J. Tildesley, William J. M. Probert, and Katriona Shea. 2017. “Essential Information: Uncertainty and Optimal Control of Ebola Outbreaks.” Proceedings of the National Academy of Sciences 114 (22): 5659–64. doi:10.1073/pnas.1617482114.
Remien, Christopher H., Frederick R. Adler, Lindsey Waddoups, Terry D. Box, and Norman L. Sussman. 2012. “Mathematical Modeling of Liver Injury and Dysfunction After Acetaminophen Overdose: Early Discrimination Between Survival and Death.” Hepatology 56 (2): 727–34. doi:10.1002/hep.25656.