# MA999 - Module Resources

**Term 2, Tu 10-11 in B3.01, Th 13-15 in B3.01**

Masters students for whom this is not advertised as a usual option (e.g. MASDOC, MASt) are welcome to contact the module leader R.S.MacKay@warwick.ac.uk if interested in taking it.

**Robert MacKay on Gaussian Processes: **Jan 9 10.00, Jan 11 13.00&14.00, Feb 20 10.00, Feb 22 13.00&14.00

A Gaussian Process (GP) is a random function such that for any finite set of evaluation points the values of the function have a joint Gaussian distribution. Given data, one can fit samples from the process. One can also fit parameters of the process and compare the evidence for different GPs. This is a type of modelling that does not directly use underlying physics or biology. We will examine its use in various contexts, the primary motivation being to detect inter-area oscillations in electricity transmission networks. These are oscillations in power flow which tend to occur when the power flow between two parts of the network is large.

Recommended reading:

An incomplete draft paper which will form the basis for at least the first 3 lectures (not for circulation)

Rasmussen C, Williams C, Gaussian Processes for machine learning (2006)

Roberts S et al. Gaussian processes for time-series modelling. Phil Trans Roy Soc A, 2013, vol. 371, no 1984, p. 20110550.

Lloyd JR et al. Automatic construction and natural-language description of nonparametric regression models. arXiv preprint arXiv:1402.4304, 2014.

Duvenaud D et al. Structure discovery in nonparametric regression through compositional kernel search. arXiv preprint arXiv:1302.4922, 2013.

http://www.automaticstatistician.com

**Ed Brambley on Aeroacoustics of thin boundary layers: **Jan 16 10.00, Jan 18 13.00, Jan 25 13.00, Jan 30 10.00, Feb 1 13.00, Feb 8 13.00

Aircraft engines are loud, but not as loud as they used to be. For a new aircraft to be allowed to fly, it must meet ever stricter noise limits during takeoff and landing. Manufacturers such as Boeing and Airbus are competing to make lighter, cheaper, more powerful engines that still meet these new noise limits. Research on aircraft engine noise (aeroacoustics) involves experimental, computational, and theoretical (mathematical) work, with the theory being the cheapest and** **(arguably) the most insightful. This topic explains recent theoretical research on the effects of thin boundary layers on sound absorbing liners in aircraft engines. The most important results covered are less than 10 years old, and this is a topic of active research.

Probable content:

Overview of sound in aircraft engines. Governing equations (LNSE, LEE) and derivatives (Pridmore-Brown). Acoustic modes in a cylinder. Impedance models of nonrigid surfaces. Impedance boundary conditions with flow (Ingard-Myers, Auregan, Brambley, viscous). Surface modes and instabilities. Putting it all together to make aircraft engines quieter. Current and future research.

Supplementary reading:

While this topic stands on its own, books that complement this topic include:

* Modern Methods in Analytical Acoustics, Crighton, Dowling, Ffowcs Williams, Heckl & Leppington (1992, Springer)

* Perturbation Methods, Hinch (1991, CUP)

* Fluid Mechanics, Landau & Lifshitz (1987, Elsevier)

**Matthew Turner on Collective motion: **Feb 13 10.00, Feb 15 13.00&14.00, Feb 27 10.00, Mar 1 13.00&14.00

Topics that will be covered: Review of collective motion in thermodynamic and animal systems. Physical, metric and metric-free interactions. Exercises with coding approaches.

Useful resources include:

Sample python code is available here. This code is tested on Python 2.7 so students should make sure that they have a compatible version of Python installed. Students would be advised to look through the code before the Thursday sessions (there are some questions at the bottom of each script).

Vicsek et al. (1995) Novel type of phase transition in a system of self-driven particles Phys Rev Lett

Giardina (2008) Collective behavior in animal groups: theoretical models and empirical studies HFSP J

Ballerini et al. (2008) Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence

from a field study PNAS

Lauga and Powers (2009) The hydrodynamics of swimming microorganisms Rep Prog Phys

Pearce et al. (2014) Role of projection in the control of bird flocks PNAS

Topaz and Bertozzi (2004) Swarming patters in a two dimensional kinematic model for biological groups SIAM J. Appl. Math.

Buhl et al. (2006) From Disorder to Order in Marching Locusts Science

Popkin (2016) The Physics of Life Nature 529, 16-18

T. Sanchez et al. (2012) Spontaneous motion in hierarchically assembled active matter Nature 491, 431

You may (perhaps) find the book by Frank Schweitzer interesting.

**Ron Crump **(Jan 18 14.00, Jan 25 14.00)**, Ben Collyer **(Jan 23 10.00, Feb 6 10.00)** and Samuel** **Brand **(Feb 1 14.00, Feb 8 14.00)** ** on** Mathematical modelling in Epidemiology**

**Sam Brand: Modelling vector-borne disease transmission
**The reproductive ratio for an epidemic is usually described as the average number of secondary infections due to a single primary infectious individual. If control efforts (like vaccination) can reduce the reproductive ratio below one then the disease will be eradicated. However, for diseases spread by insect biting this simple definition needs re-interpreting, and usually any control method targets either the human host (e.g. vaccination) or the insect vectors (e.g. insecticidal spraying). We’ll go through redefining the reproductive ratio to take into account vector-borne transmission, and calculate the different levels of control required for eradication. We will discuss how different assumptions about both the incubation period in the insects, and their bionomics, affect reproductive ratio prediction. If time allows we’ll also cover numerical solution of analytically intractable cases.

Reading: Diekmann, O., Heesterbeek, J. & Metz, J. A. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology 28, 365–382 (1990).

Keeling, M. J. & Rohani, P. Modeling Infectious Diseases in Humans and Animals. (Princeton University Press, 2008).

Brand, S. P. C. et al. The Interaction between Vector Life History and Short Vector Life in Vector-Borne Disease Transmission and Control. PLoS Comput Biol 12, e1004837 (2016).

**Ben Collyer: Modelling the dynamics of parasitic worms within a host population.**

When modelling the dynamics of parasitic worms within a human population, the burden of infection (how many worms each individual carries) as well the individuals' infectious status is an important factor in disease transmission. We'll go from modelling the distribution of worm burden in a population to simplified models that make use of empirical observations. If there's time we'll run some of these models numerically to see what control strategies may be effective.

Reading: Anderson, Roy M., Robert M. May, and B. Anderson. Infectious diseases of humans: dynamics and control. Vol. 28. Oxford: Oxford university press, 1992.

Hollingsworth, T. Déirdre, et al. "Quantitative analyses and modelling to support achievement of the 2020 goals for nine neglected tropical diseases." Parasites & Vectors 8.1 (2015)

Anderson, R. M., et al. "What is required in terms of mass drug administration to interrupt the transmission of schistosome parasites in regions of endemic infection?." Parasites & Vectors 8.1 (2015)

Stylianou, Andria, et al. "Developing a mathematical model for the evaluation of the potential impact of a partially efficacious vaccine on the transmission dynamics of Schistosoma mansoni in human communities." Parasites & Vectors 10.1 (2017)

Truscott, J. E., et al. "A comparison of two mathematical models of the impact of mass drug administration on the transmission and control of schistosomiasis." Epidemics 18 (2017)

Gurarie, David, and Charles H. King. "Population biology of Schistosoma mating, aggregation, and transmission breakpoints: more reliable model analysis for the end-game in communities at risk." PLoS One 9.12 (2014)

schistolecture2-solutions.ipynb

**Ron Crump: Modelling disease dynamics via back-calculation**

This means using knowledge of the incubation period distribution to make

inferences about past infection from time series of incidence data.

Work I've been doing on leprosy will be used as an example.

**Oral presentations by students (20 mins + 5 mins for questions)**

15 March, 13.00-15.00, B3.01

**Deadline for submission of written reports**

12.00, 23 April