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Problem 1. Designing a distributed personal-medical system
Designing a distributed personal-medical system
Presenter
Professor Jeff Sanders, African Institute for Mathematical Sciences, South Africa
Problem statement
Can we design a distributed (i.e. de-centralised) system to manage personal medical history?
People expect to be, and are in normal years, far more mobile than ever before. Mobile phones and the web provide the communications and information they need, but national border control, personal ID and health systems remain resiliently conventional in spite of being essential for travel. By comparison currency has been potentially liberated, with the advent of Bitcoin.
Can we do the same for personal health records?
If you're stung by a jellyfish on holiday in northern Queensland, break a leg skiing in the Alps, or catch a virus whilst working in Wuhan, immediate treatment requires knowledge of your medical history, allergies and insurance. Record books are unreliable, easy to lose and to keep with you at all times. Is there some distributed system which allows access to your personal medical information from anywhere in the world? What functionality should it support? What are the ethical requirements of such a design?
- This project begins with a brief survey of centralised versus distributed systems, emphasising the local thinking needed to design a distributed protocol which achieves some global property.
- It then considers the features and functionality desired of a distributed personal medical system, emphasising those not achievable in a conventional system (for instance exploitation of data analytics). Ethics, accountability, and intrusion are important considerations that participants may not be used to addressing.
- A system is designed which incorporates the desired features. Coding is not required, but reasoning about correctness of behaviour of our design is.
A distributed system will be thought of as a `data structure' i.e. as consisting of state on which certain operations are defined, like `Update a record', and `Query a record'. We must decide what `state' the system has in order to be able to express its operations accurately. The first step is to under stand what exactly is a `data structure' and how to describe it using discrete mathematics in the Z notation.
A typical industrial problem requires a solution which optimises some quantity. When an optimal solution is unrealistic an acceptably efficient one is sought. It might be: what standard length of aluminium minimises wasteage over various uses. Or what treatment of the residue of sugar cane, for use in generating electricity all-year round, minimises use of energy. The solution is typically described by differential equations and then constraints are imposed to optimise the solution.
This project is an example of an important family of contemporary problems whichfit that paradigm, but whose solution is, instead of a solution to a differential equation, a design (of optimal or acceptable efficiency). Ingenuity is required to invent the design and mathematics is needed to express it and to show that it functions as desired. The design might enable several robots to coordinate in exploring a mine. It might be for a secure electronic voting system. Or it might be for on-line sales featuring specials that benefit from market data. The mathematics is discrete and straightforward though perhaps unfamiliar.
An implementation (i.e. code) is not sought. Indeed our time will be spent on deciding then expressing features and reasoning about how they would be captured in a design.
Presentation
MISG 2021 Graduate Modelling Camp Problem 1 Presentation
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MISG 2021 Graduate Modelling Camp Problem 1 Report-back Presentation
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Problem 2. 网易体育 Protection Methods
Problem 2. 网易体育 Protection Methods: masks, social distancing
Presenter:
Professor Nevile Fowkes, University of Western Australia, Perth Australia
Problem Statement:
The aim here is to gain an understanding of the physical processes involved in the use of masks and in social distancing. We want to know how best to use these tools to reduce the spread of the disease.
We have only 2 days and limited interaction so we will simply aim to
- produce a report on the present state of knowledge of the processes and
- set up simple models.
Masks
The aim is to design the best mask consistent with personal comfort. Evidently an impermeable mask exactly fitting onto the individual's face will prevent the spread but will kill the wearer. Also a comfortable but very crude and poorly fitting mask will be useless. So the question is:
What permeability (fabric thickness and weave) and fitting parameters (size, shape) will be `best' for the wearer and for a potential victim.
Some observations:
- The droplets (containing the virus) are carried in the air stream and will either be absorbed in the mask fabric or expelled through or around the mask.
- The volume of space between the mask and face must be sufficient and/or the mask be flexible enough to allow for comfortable air exchange due to breathing.
Droplet Dispersal
One would expect sneezing to be the main transmission method but normal breathing may also be important because of the longer time span involved. (We will not consider contact transmission.)
A sneeze is essentially an air jet (carrying droplets) that spreads out due to entrainment(?). The concentration of droplets in the air stream thus reduces with distance from the `impulsive' sneeze source. How far is safe and how will mask design effect this.
It would be useful to look at recent work. A useful list of puplications is given below. Also we need to learn some physiology.
References
- Dbouk T, Drikakis D. On respiratory droplets and face masks. Physics of Fluids. 2020;32(6). doi.org/10.1063/5.0015044 (https://aip.scitation.org/doi/10.1063/5.0015044)
- Cummings C. P. Ajayi O. J., Mehendale F. M., Gabi R., Viola I. M. The dispersal of spherical droplets in source-sink flows and their relevance in the 网易体育 pandemic. Physics of Fluids 32, 083302 (2020). 3. Wang B., Wu H., Wan X. Transport and fate of human droplets- A modelling approach. Physics of Fluids 32, 083307 (2020).
Supporting Material
网易体育 Protection Methods: masks, social distancing
On respiratory droplets and face masks
How to design the perfect face mask – the effect of compressibility on filters
Masks: Modelling workshop Organisation
Presentation by Professor Neville Fowkes, University of Western Australia
Presentation
2021 MISG Study Group Problem 2 Presentation 1
2021 MISG Study Group Problem 2 Presentation 2
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Problem 3. Turbulent two-dimensional jet
Problem 3. Turbulent two-dimensional jet
Presenter:
Professor David Mason, University of the Witwatersrand, Johannesburg
Problem statement
In order to understand the spread of the corona virus and social distancing, an elementary model of a sneeze as a turbulent axisymmetric jet will be considered.
In a turbulent flow the fluid velocity is separated into a mean flow and a fluctuation. The derivation of the Reynolds averaged equations, the Reynolds stresses and the concept of eddy viscosity will first be reviewed. The turbulent axisymmetric jet will be modelled using Prandtl’s mixing length model for the eddy viscosity. A similarity solution for the turbulent jet will be investigated. The solution will be analysed to estimate a safe social distance.
Presentation
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2021 MISG Graduate Modelling Camp Problem 3 Report Back
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Problem 4: Predicting heart disease using logistic regression
Problem 4: Predicting heart disease using logistic regression
Presenter:
Professor Montaz Ali, School of Computer Science and Applied Mathematics, University of the Witwatersrand
Problem Statement
It is well known that a number of medical factors are the main cause of heart disease and heart attack. These factors are blood pressure level, sugar levels, body mass index and cholesterol. The problem will need a collection of data from people who had heart attacks and also from people who never had attacks. This data will then be used to build a mathematical model to predict probability of heart attack of any individual. The size of the collected data will be N where each i-th data will be a vector form x with components (x1,x2,x3,x4,x5). The variables x1 and x2 represent blood pressure levels which vary with person (Normal: x1<120, x2<80; Elevated: 120<x1<129, x2<80; Hypertension: 130<x1<139, 80<x2<90; Hypertension state2: x1>140, x2>90). The variable x3 represents body sugar levels (Normal: x3 varies between 70 to 99; Diabetes: x3 varies between 100 to 125). The variable x4 represents body mass index (Underweight: x4<18.5; Normal: 18.5<x4<24.9; Overweight: 25<x4<30; Obesity: x4>30). The variable x5 represents Total Cholesterol (Normal:x5<200; Borderline: 200<x5<239; High: x5>240). After the data collection students have to generate N nominal data each with value of all 5 variables of which 80% will be used to build a logistic model and once the model has been built the model will be tested on the remaining 20% data. An additional binary variable w will be used to denote if the person had a heart attack or not having a condition i.e. values of (x1,x2,x3,x4,x5). If yes then w=1, else w=-1.
Please note that one knows from the given data that if a person had a heart attack or not (w=1 or w=-1) but when the model predicts, it will be the probability of a heart attack of a person with given condition (x1,x2,x3,x4,x5).
The mathematical model will be a logistic regression (LR). LR incorporates two models: Linear Regression and a non-linear transformation. First a linear model will be created from the dataset, say Lm(x) which will then go through a nonlinear transformation to give a value between 0 and 1 (think of the sigmoid function). We want to predict if a person has a high risk (high probability) or low risk (low probability), or average risk given his/her medical condition. In effect we want to obtain Pr(w/x) i.e. the probability of w given x which has two values (each of which must add up to 1). The modelling approach will use the data set to construct a maximum likelihood function. This means that we will find the parameter values such that the model predicts the data set with maximum probability.
Supporting Materials
Mathematical model for logistic regression
Presentation
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Problem 5: Green roofs to mitigate the Urban Heat Island
Problem 5: Green roofs to mitigate the Urban Heat Island
Presenter:
Dr Adewunmi Gideon Fareo, University of the Witwatersrand, Johannesburg
Problem statement
In order to determine if greening of roofs in cities and urban areas mitigate the urban heat island effect, we will focus on the physics of heat diffusion by considering a simple one-dimensional heat equation.
Introduction
The Urban heat Island is a phenomenon where the temperature of urban areas and a region of a city (generally the inner city) is higher than the surrounding less dense outskirts.
Close to 50% of the world’s population is living in urban areas or cities. According to a united nations report, it is predicted that by the year 2050, 66% of the global population will live in urban areas, resulting in higher levels of densification. The continuous increase in urban inhabitants has led to the spread of urban areas into nearby farmlands causing farm reclamations and deforestation activities. When vegetative spaces, like farmlands are turned into urban structures, there is often an associated rise in temperatures, a phenomenon first noticed in London in the 19th century, and referred to as the Urban Heat Island effect (UHI).
It has been recognized that the temperature variation is primarily due to the absorption of radiation during daytime by dense materials like concrete, dark materials like bitumen and the roofs of buildings that make up the cities and urban areas. This heat from solar radiation is stored until the evening when it is released into the atmosphere by convection. Dark materials like bitumen tend to absorb much more heat than lighter colours that reflect most of the solar radiation back into the atmosphere. The combination of these materials causes the inner city areas to be several degrees warmer than the surrounding less dense areas.
The surrounding less dense areas or nearby countryside has a lot of vegetation and less of concretes, bitumen, etc, and therefore is not affected by the Heat Island effect.
There is a body of knowledge that suggests that the vegetation on green roofs can mitigate the urban heat island through shading, insulation of the soil layer and evapotranspiration.
References
- Takebayashi, H., Moriyama, M., Surface heat budget on green roof and high reflection roof for mitigation of urban heat island, Building and Environment vol 42(8), 2971-2979, (1997).
- Susca, T., Gaffin, S.R., Dell'Osso, G.R., Positive effects of vegetation: Urban heat island and green roofs. Environmental pollution, vol 159, 2011, 2119-2126.
- Fitchett, A., Govender, P., Vallabh, P., An exploration of green roofs for indoor andexterior temperature regulation in the South African interior. Environment Development and Sustainability 22, 5025–5044 (2020).
- Taha, H. Urban climates and heat islands: albedo, evapotranspiration, and anthropogenic heat. Energy and buildings, 25(2), 99-103, (1997).
Presentation
2021 MISG Study Group Problem 5 Green Roofs Presentation
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2021 MISG Study Group Problem 5 Green Roofs Report-back Presentation
