Professor or A/Prof. @ Curtin University

School of Electrical Engineering, Computing, and Mathematical Sciences

$151,317 – $194, 454 (ALD – ALE) plus 17% superannuation

Fixed-term full-time research position (5 years)

Teaching and research appointment

Application closing date 10:00pm AWST, Friday 15th October 2021

Job Reference: 780919
  

The School of Electrical Engineering, Computing, and Mathematical Sciences at Curtin University brings together the university’s expertise in data science, artificial intelligence, and digital technologies. The school includes several major collaborative ventures with industry and government, including the Optus-Curtin Centre of Excellence in Artificial Intelligence, the ARC Industrial Training Centre for Transforming Maintenance through Data Science, the Cisco Centre for Networks, and the Curtin Institute for Computation.
  
Optimisation is one of the school’s core research themes and the optimisation group in the school (part of the mathematics and statistics discipline) is a key pillar in Curtin University’s data science portfolio. The university is now seeking a new professor or associate professor to provide academic leadership within the optimisation group, expand the group’s collaborations with industry and government, and foster and lead new demand-driven research funding opportunities. This is a research-focused position reporting to the group director. The successful candidate will be a senior member of the mathematics and statistics discipline at Curtin and will play a leading role in shaping the future of applied mathematics and data science at the university.

Position description

Application link

If you have a query in relation to the application process please contact our careers team on curtincareers@curtin.edu.au

If you have any further queries regarding the role, please contact:
Professor Ryan Loxton, Discipline Lead – Mathematics and Statistics, r.loxton@curtin.edu.au

Tutor/Senior Tutor in Statistics@University of Waikato

University of Waikato, Hamilton, New Zealand
Department of Mathematics

Closing date: 17 September 2021

Ko wai mātou? – Who are we? 

The Department of Mathematics within the School of Computing and 
Mathematics Sciences, offers a broad programme in pure, applied, and 
computational mathematics which gives opportunity for specialist and research 
preparation, but also serves the needs in mathematics and statistics of other 
disciplines in the University. 

Ngā kōrero mō te tūranga – About the role 

We have an opportunity for a qualified and passionate person to support 
teaching and learning for our popular statistics papers primarily through 
lectures and tutorials. 

You would be part of an expert team and involved in all aspects of teaching 
and support for undergraduate students.  In particular, you would deliver 
lectures, coordinate a team of sessional assistants, assist with the 
development of paper content and teaching resources, carry out day-to-day 
administration and run some computer laboratory sessions. 

For more information about the School of Computing and Mathematical 
Sciences, please visit https://www.cms.waikato.ac.nz/ 

Ko wai koe? – Who are you? 

To be successful in this role you will have a background in statistics, with a 
demonstrated teaching ability. You will also have a strong interest in 
supporting student learning in the transition from secondary school to 
university study, and inspiring students into considering further studies in our 
discipline. 

In addition, it is essential that you have an undergraduate degree in statistics 
or mathematics, and it is strongly preferred that you have a postgraduate 
degree, i.e. a Masters degree or PhD.

He aha ngā take me tono mai ai koe? – Why should you apply for this 
position? 


Our University provides a unique and satisfying work environment with many 
benefits to our staff including study opportunities and leave provisions. 

The salary will be commensurate with the level of appointment. Current salary 
range for Tutors is $45,624 to $66,830 per year and for Senior Tutors is 
$69,124 to $84,062 per year. 

This position is fixed term for two years and will commence in January 2022. 

Enquiries of an academic nature should directed to Associate Professor 
Daniel Delbourgo
, Head of Mathematics

For more information: position description, full advertisement.

Waikato Vacancies page (see Vacancy Number 410264)

Open letter to CEO of the Australian Research Council: Concerns about new ARC “no preprint rule”

(A pdf version of this letter is available here)

24 August 2021

Professor Sue Thomas
Chief Executive Officer
Australian Research Council

Dear Professor Thomas,

The Australian Physics, Astronomy, Chemistry, Mathematics and Statistics communities express grave concern about a recent change to Australian Research Council (ARC) rules to forbid reference to preprints anywhere in a grant application. We are particularly concerned about the impact on early career researchers whose ARC fellowship applications have recently been ruled ineligible because of a violation of this new rule.

We are not aware of any consultation with our scientific communities about this change. We urge the ARC to rescind this rule, as it is unworkable and inconsistent with standard practice in our disciplines.

Preprints are vital for the rapid dissemination of knowledge in physics, astronomy, chemistry, mathematics and statistics. This is particularly important in fields where there is a long lead-time between journal submission and publication. Citing preprints in publications, reports, or grant applications is an entrenched disciplinary norm in these fields. Experts and referees who encounter such citations know that preprints are not peer reviewed and are experienced in assigning them appropriate weight.

Preprint servers are also used to store other important scientific documents including white papers, PhD theses, software and instruction manuals, experimental design reports, and other technical documents. Although never intended for publication in a regular journal, it is common for such documents to be definitive references on certain topics and cited many hundreds of times.

Forbidding references to preprints prevents applicants from giving appropriate credit to the authors of ideas that informed their proposal. This constitutes academic misconduct. Doing so is contrary to the Australian Code for the Responsible Conduct of Research 2018, which requires researchers to both “Present information truthfully and accurately in proposing … research” (Principle 1) and “Appropriately reference and cite the work of others” (Principle 4).

Preprint servers, such as the physical sciences arXiv server, pioneered the development of open access publishing. They are an established part of the publishing landscape. Their use is fully consistent with the ARC Open Access Policy.

Major science funding agencies around the world permit or encourage preprints to be cited in grant proposals and funding reports. This includes the US funding agencies such as the Department of Energy (DOE), the National Science Foundation (NSF), NASA and the National Institutes of Health (NIH), the European Research Council (ERC), the French National Research Agency (ANR) and the UK funding agencies for Engineering and Physical Sciences (EPSRC), Biological Sciences (BBSRC) and Medical Sciences (MRC).

We are dismayed that promising research careers have been impacted and perhaps even ended because fellowship applicants cited preprints and other documents housed on preprint servers. We encourage the ARC to explore avenues to support the researchers affected.

We strongly recommend the ARC reverse its rule change as a matter of urgency, and permit authors to cite any relevant material in accordance with disciplinary conventions. We further recommend that any future proposed changes that represent a significant departure from disciplinary norms be subject to wider consultation with researchers and peak scientific bodies.

Yours sincerely,


Professor Sven Rogge, President, Australian Institute of Physics (AIP)
Professor Steven Bottle, President, Royal Australian Chemical Institute (RACI)
Professor John Lattanzio, President, Astronomical Society of Australia (ASA)
Professor Ole Warnaar, President, Australian Mathematical Society (AustMS)
A/Professor Jessica Kasza, President, Statistical Society of Australia (SSA)
A/Professor John Holdsworth, President, Australian and New Zealand Optical Society (ANZOS)
Professor Anthony Dooley, Chair, Australian Council of Heads of Mathematical Sciences (ACHMS)
Professor Tim Marchant, Director, Australian Mathematical Sciences Institute (AMSI)

Professor Brian P Schmidt AC FAA FRS, ANU Distinguished Professor, 2011 Nobel Laureate

Professor Harry Quiney, Head, School of Physics, The University of Melbourne
Professor Celine Boehm, Head, School of Physics, The University of Sydney
Professor Tim Senden, Director, Research School of Physics, The Australian National University
Professor Michael Morgan, Head, School of Physics and Astronomy, Monash University
Professor Susan Coppersmith, Head, School of Physics, University of New South Wales Sydney
Professor Peter Veitch, Head, School of Physical Sciences, The University of Adelaide
Professor Jingbo Wang, Head, Department of Physics, The University of Western Australia
Professor Igor Bray, Head, Physics and Astronomy, Curtin University
Professor Geoff Pryde, Head, Applied Maths and Physics, Griffith University
Professor David Spence, Interim Head, Department of Physics and Astronomy, Macquarie University
Professor Jamie Quinton, Head of Physics and Dean of Science, Flinders University
Professor Gary Bryant, Associate Dean (Physics), RMIT University
Professor John-David Dewsbury, Head, School of Science, UNSW Canberra
Dr Brenton Hall, Chair of Department of Physics and Astronomy, Swinburne University

Professor Scott Kable, Head, School of Chemistry, University of New South Wales Sydney
Professor Philip Gale, Head, School of Chemistry, The University of Sydney
Professor Richard O’Hair, Head, School of Chemistry, The University of Melbourne
Professor Phil Andrews, Head, School of Chemistry, Monash University
Professor Chris Sumby, Head of Chemistry, The University of Adelaide
Professor Alison Rodger, Head, School of Chemistry, Macquarie University
A/Professor David Wilson, Head, Department of Chemistry and Physics, La Trobe University
A/Professor Jennifer MacLeod, Head, School of Chemistry and Physics, Queensland U. of Technology
Professor Catherine Yule, Head, School of Science, Technology & Engineering, U. of the Sunshine Coast
A/Professor Andrew Seen, Head of Chemistry, University of Tasmania
Professor Richard John, Head of Chemistry, Griffith University

Professor Howard Bondell, Head, School of Mathematics and Statistics, The University of Melbourne
Professor Joseph Grotowski, Head, School of Mathematics and Physics, The University of Queensland
Professor Adelle Coster, Head, School of Mathematics and Statistics, University of New South Wales
Professor Warwick Tucker, Head, School of Mathematics, Monash University
Professor Andrew Hassell, Interim Director, Mathematical Sciences Institute, ANU
Professor Andrew Bassom, Head of Discipline, Mathematics, University of Tasmania
Dr Maureen Edwards, Head, School of Mathematics and Applied Statistics, University of Wollongong
Dr Christopher Lenard, Head of Department of Mathematics and Statistics, La Trobe University
Professor Alan Welsh, Chair, National Committee for Mathematical Sciences

Professor Dragomir Neshev, ANU, Director, ARC Centre of Excellence for Transformative Meta-Optical Systems (TMOS)
Professor Matthew Bailes, Swinburne University of Technology, Director, ARC Centre of Excellence for Gravitational Wave Discovery (OzGrav)
Professor Elisabetta Barberio, The University of Melbourne, Director, ARC Centre of Excellence for Dark Matter Particle Physics
Professor Lisa Kewley, Australian National University, Director, ARC Centre for Excellence in All-Sky Astrophysics in 3D (ASTRO 3D)
Professor Paul Mulvaney, The University of Melbourne, Director, ARC Centre of Excellence in Exciton Science
Professor Peter Taylor, University of Melbourne, Director, ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS)
Professor Michael Fuhrer, Monash University, Director, ARC Centre of Excellence in Future Low-Energy Electronics Technology (FLEET)
Professor Andrew White, The University of Queensland, Director, ARC Centre of Excellence for Engineered Quantum Systems (EQUS)

COVID-19: vaccines and variants

(This is a guest post by Dr Freya Shearer, as part of our miniseries of articles/essays by Australian mathematicians involved in the pandemic response. A pdf version of this article is available here.)

In the latter part of 2020, two key events shifted the course of the COVID-19 pandemic: the availability of vaccines and the emergence of variants of concern. 

A key question that rose up the agenda for policy makers — to what extent can vaccines mitigate the impacts of COVID-19, especially in the context of variants of concern? 

Due to the many nuances and uncertainties in estimating the future impacts of vaccination, ongoing monitoring of SARS-CoV-2 epidemiology in Australia while vaccines continue to roll out is critical to adaptive policy making. 

One of my roles in the COVID-19 response is to help provide epidemic situational assessment to Australian states and territories. As part of this, I have been coordinating (with Dr David Price) weekly (at least) situation reports submitted to the Communicable Disease Network of Australia (CDNA) and the Australian Health Protection Principal Committee (AHPPC) since April 2020. Each week the content of these reports represents a huge team effort – led by Professor James McCaw, an expert member of AHPPC (see the first essay in this series). A large nationally distributed group of mathematical and statistical modellers conduct analyses of trends in population behaviours (using mobility data from technology companies and anonymous population surveys), the effective reproduction number (using case data) and forecasts of daily case counts and hospital occupancy. We also estimate the propensity for the virus to spread based on behavioural data and the biological characteristics of SARS-CoV-2 [1]. A weekly summary is published in the government’s Common Operating Picture.  

It has been an enormous privilege to develop new analyses and metrics alongside policymakers to meet information needs specific to our context — for example, understanding the risk of epidemic activity during sustained periods of zero cases and an ongoing risk of importation(s). 

Over the past six months, the situational assessment teams have been continually adapting and extending their models to account for the differential impacts of new variants and vaccine products, and multiple co-circulating variants. 

Vaccines can act on multiple elements of transmission and/or disease. They can reduce:

  • Susceptibility to infection
  • The probability of onward transmission from immunised infected individuals
  • The probability of developing symptoms given infection
  • The probability of developing severe disease and death given infection

The COVID-19 vaccines registered for use in Australia are highly effective at reducing susceptibility to infection and onward transmission from immunised infected individuals for the Alpha variant. The resulting overall reductions in transmissibility are estimated to be > 90%. 

These vaccines also dramatically reduce the probability that immunised infected individuals will develop severe disease (> 85% estimated reduction) or die from their infection (also > 85% estimated reduction). However, vaccine efficacy is specific to each vaccine product and pathogen strain. At the time of writing, the Delta variant was set to become the dominant circulating SARS-CoV-2 strain globally. Early data shows decreased efficacy of our vaccines against the Delta variant (more on this later). 

Above and beyond the direct benefits of vaccination, everyone—vaccinated and unvaccinated—indirectly benefits from reduced exposure because others are vaccinated. This protection is a consequence of the reduction in disease transmission brought by the depletion of fully susceptible people (see the first essay in this series by Professor James McCaw for an explanation on the role of susceptible depletion). 

This indirect protection is important because even highly effective vaccines are imperfect; there is still a chance of severe disease and death for fully vaccinated people. Further, some people are unable to be vaccinated, due to underlying health issues for example.  

The level of vaccination coverage required to prevent sustained disease transmission is the critical vaccination or ‘herd immunity’ threshold. This threshold will vary according to pathogen transmissibility, vaccine efficacy, the population groups prioritised for vaccination, and levels of population mixing, among other factors. Higher pathogen transmissibility will increase the required level of coverage, while higher vaccine efficacy will decrease the required level of coverage. 

More specifically, the proportion of a homogeneously mixing population that would need to be vaccinated to prevent sustained transmission is given by (1–1/R0)/ε, where R0 is the “basic reproduction number”, the number of secondary cases arising from an index case in an otherwise fully susceptible population, and ε is the proportional vaccine efficacy at reducing transmission.  

Given an R0 value in many countries for wildtype SARS-CoV-2 of between 3 and 4, if we assume that vaccine efficacy is 90%, then the critical vaccination coverage becomes 75-85%. The Alpha variant is estimated to be around 50% more transmissible than wild type. This increases the critical vaccination threshold to 85–95%.

Early evidence suggests that the Delta variant is between 30 and 60% more transmissible than Alpha and may partially evade vaccine- and naturally-derived immunity. The critical vaccination threshold for a strain exhibiting such characteristics is nearing 100%, though we would expect significant population protection at lower vaccination levels and effective management of transmission. Due to the relatively short circulation period of Delta, very limited evidence is available on clinical severity and vaccine effectiveness against clinical outcomes. Early available data suggests lower vaccine effectiveness against the Delta variant compared to Alpha for most outcomes. 

The above simple calculations ignore important population heterogeneities that would make estimates higher or lower in specific settings. There are uncertainties in the transmissibility of Delta, indeed of any SARS-CoV-2 variant, including how it might vary geographically (even within Australia).  

An important part of our research is to adapt the above calculations to include a range of age-specific heterogeneities. The level of transmission reduction and protection against clinical outcomes achievable for a given population level of coverage depends on which sub-populations are prioritised for vaccination and differences in disease characteristics and social behaviours across different groups. 

The probability of hospitalisation, ICU admission and death given SARS-CoV-2 infection increases sharply with age, which is why vaccination programs in many countries have initially prioritised the oldest age groups.     

Groups at the highest risk of severe outcomes receive indirect protection through the vaccination of key transmitting population groups. Not all age groups contribute equally to transmission. Contributions vary because of different social contact rates, susceptibilities to infection and symptomatic fractions (which affect infectiousness) by age. For SARS-CoV-2, evidence suggests that susceptibility to infection and propensity to develop symptoms increases with age. Younger people and working age groups typically have more social contacts than older people.  

After accounting for all these factors, key transmitting ages for SARS-CoV-2 are estimated to be those aged 20–60 years. Whilst people under 20 have the highest numbers of contacts, they are less likely to spread SARS-CoV-2 to those contacts. People over 60 are more likely to spread SARS-CoV-2 to their contacts but typically have fewer contacts. People aged 20–60 have both relatively high numbers of contacts and ability to spread the disease. Thus, high coverage in these age groups is important for mitigating transmission and clinical outcomes. 

Future emerging variants are a key uncertainty in our ongoing management of COVID-19. With the emergence of each novel variant of concern, an early priority is to gather information on critical epidemiological indicators. What is its relative transmissibility compared to existing variants? What is the probability of severe illness given infection with the new variant? Does this vary by age group? What is the effectiveness of vaccines against the new variant? Does this differ by vaccine product? 

Estimating these quantities is extremely difficult in the early stages of variant emergence when information is scarce and detection efforts are highly varied and rapidly changing. Yet response planning/decisions are required before complete information is available. To support these decisions, analyses are made using the limited available data and must be continually reviewed and updated as evidence emerges.  

Ongoing global circulation of COVID-19, and global vaccine inequity mean that new variants will continue to emerge. Some of these variants will likely exhibit some combination of higher transmissibility, higher clinical severity and/or immune evasion. New vaccines/boosters will likely be needed to protect against such future variants. Response planning is designed to be adaptable to this continually evolving situation and many possible futures. For now, vaccination programs globally will continue to reduce transmission and harms related to COVID-19, together with public health and social measures.

Reference

[1] Nick Golding et al, Situational assessment of COVID-19 in Australia Technical Report 15 March 2021 (released 28 May 2021), https://www.doherty.edu.au/uploads/content_doc/Technical_Report_15_March_2021_RELEASED_VERSION.pdf

Research Fellow in Harmonic Analysis

Monash University, Melbourne, Australia
School of Mathematics

Closing date: Wednesday, 1 September 2021, 11:55 pm AEDT

The School of Mathematics is seeking an excellent Research Fellow to work as a part of a research project in analysis and PDE – investigating harmonic analysis (e.g. resolvent, Strichartz estimates, spectral multiplier) associated to Schrodinger operators and their application to dispersive equations (Well-posedness, asymptotic behaviour).

You would ideally have a doctoral qualification in Mathematics with extensive knowledge of harmonic analysis, functional analysis and PDE, with experience proving novel results in at least one of these areas.

Monash University seeks applicants for this opportunity who are able to demonstrate and present valid and current Australian work rights.

This role is a full-time position; however, flexible working arrangements may be negotiated.

Applications must be made via the Monash website:
https://careers.pageuppeople.com/513/cw/en/job/623542/research-fellow-harmonic-analysis

Scholarship opportunity: Kerry Landman Scholarship at the University of Melbourne

This scholarship will be available again in 2022, with applications opening on 1 October 2021, and I hope you may be able to assist in sharing this valuable opportunity through any appropriate channels you may have.

The Kerry Landman Scholarship supports high-achieving mathematics graduates that demonstrate a passion for education and wish to train as mathematics teachers. This scholarship is valued at up to $60,000 in total and students with a mathematics major who are enrolling in a Master of Teaching (Secondary) at the Melbourne Graduate School of Education are encouraged to apply. This scholarship aims to address the shortage of mathematically trained teachers who can passionately communicate the beauty and applications of mathematics to encourage, inspire and empower students to succeed. A successful candidate will have the potential to become a leader in a mathematics education.

Further detail regarding the scholarship can be found here.

National Science Week with MATRIX – Maths Documentaries

MATRIX is excited to host free online screenings of the following documentaries that celebrate the wonders of the mathematical sciences during National Science Week. We’d be extremely grateful if you could help us promote these events.  We would love for you, your teams and those in your networks to join us too. 

  • Maths Circles Around The World – 16 August 2021, 16.30 AEST 
  • Secrets of The Surface – The Mathematical Vision of Maryam Mirzakhani – 16 August 2021, 19.30 AEST 
  • Colors of Math – 17 August 2021, 19.30 AEST 
  • The Discrete Charm of Geometry – 18 August 2021, 19.30 AEST 

See the MATRIX webpage, the pdf flyer or the National Science Week website for more.

Applied Mathematician

Boeing

Closing date: 6th Aug 2021

Location:Brisbane
Category:Engineering / Technical
Position Type:Permanent
Job Reference:BOE/1550763

An exciting new role has opened up within Boeing Research & Technology Australia’s Brisbane Research Centre, located within the University of Queensland, St Lucia.

The successful candidate will work within a team of passionate mathematicians and engineers to develop state of the art GPU based multi-physics solvers. Essential skills include a familiarity with a range of computational methods for solving partial differential equations as well as experience in the implementation of such methods in a compiled language, particularly C, C++ or FORTRAN.  Experience in developing computational fluid dynamics codes, structural finite element codes or multi-physics codes is considered an advantage. Experience with high performance computing frameworks such as CUDA, OpenACC, OpenMP & MPI will also be considered beneficial.  

Required Qualifications:

  • Research higher degree in Mathematics (PhD or masters)
  • Bachelor degree in Math, Physics or Engineering

Applicants must be Australian Citizens to meet defence security requirements.

More information and application at this page.

Mathematical Modeller

Burnet Institute, Melbourne

Closing date: Tue, 17 Aug 2021 11:59pm Australian Eastern Time

The Opportunity 

Are you an early career researcher eager to undertake real-world mathematical modelling projects in infectious diseases and global health? This could be the role for you!

A unique opportunity exists for a Mathematical Modeller to join Burnet’s Modelling & Biostatistics team. Join an innovative team that is developing and applying epidemiological and costing models around the world across a range of disease areas including COVID-19, HIV, TB, malaria, viral hepatitis, maternal and child health.  

This is an exciting chance for you to:

  • Contribute to the development and application of epidemic and economic models to address major public health issues
  • Work with a globally based team of expert modellers, epidemiologists, economists, health and policy specialists, and computer scientists to improve models
  • Undertake modelling and analytical support that will be provided to governments, policy makers, and other partners including the World Bank, Global Fund, and international health agencies, to improve health budget investment towards better decision-making and impact around the world
  • Apply innovations and analytical skills to ensure high quality deliverables in a fast-paced environment
  • Contribute to and produce policy briefs, presentations, scientific papers and technical reports 

Refer to the attached position description for full details.

This is a full-time position, initially for a 2-year period with the possibility of extension. Work from the comfort of your own home or join us in the office, location is flexible!

Full details and application form at this link.

Mathematical models to support Victoria’s COVID-19 response: a blunt instrument to a complex problem

(This is a guest post by Dr Michael Lydeamore as part of our miniseries of articles/essays by Australian mathematicians involved in the pandemic response. A pdf version of this article is available here.)

Throughout Victoria’s COVID-19 response, a suite of mathematical and statistical models have been used to understand the spread and subsequent control of the pandemic. High-profile mathematical outputs, such as case forecasts, give a good picture of the general epidemic activity in a given region. However, when case numbers are small, forecasts can be very sensitive. Moreso, when there is only a handful of cases, more detailed factors such as geographic distribution of cases, or at-risk industries, are possibly more informative.

As part of my role in Victoria’s COVID-19 response, my team applied a number of statistical techniques to case data. These techniques have varying levels of sophistication, but one of the most used was Diggle’s space-time K-function [2]. This model is relatively unrefined, but the advantage of that is that very little information is required to compute it: just the date of infection of cases, and their geographical location. Both of these are collected almost immediately once the case is notified, meaning that this function can be calculated regularly and quickly, two factors that are critical in informing epidemic response.

The D_0 function can be interpreted as the proportional increase in case events at a given space-time arising from interactions at that space-time. In an infectious diseases context, this is a proxy measure of disease contagiousness. The lower this increase, the stronger the indication of successful intervention measures.

To calculate the D0 function, we start with the K-function (For some reason, spatial science has some of the most non-transparent function naming). The K-function is defined as the cumulative number of expected case events, K, as a function of the (straight-line) distance from an arbitrarily selected case [7]. That is,

Kd (s) = N -1i  ∑j≠i I [dij<s],      (1)

where N is the total number of cases, dij is the distance between case i and case j, and I is an indicator function. Eq (1) is sometimes termed the ‘spatial’ K-function, but by swapping out the dij term, it is possible to calculate a K-function across any attribute. For COVID-19, we used time, denoted Kt(t), as well as space. The time between two cases was the number of days between their onset dates (which we assumed were a proxy for infection dates) as opposed to the notification dates, for which there was sometimes a long delay. Figure 1 gives an illustration of calculating the K-function in (a) space and (b) time. For the arbitrarily chosen case—case B here—the K-function in space at 1 unit, KdB(1)=2/6, and at 2 units, KdB(2)=5/6. Comparatively, in time, KtB(1) = 1/6, and KtB(3) = 5/6. To estimate the K-function in it’s entirety, we would repeat this process for each of the other cases A–F.

So far, we have considered space and time completely separately. As the last example has shown, often the clustering of cases in space and time can be different. If space and time were completely independent, then the space-time K-function, denoted K(s,t), would be the product of the space and time K-functions. That is,

K(s,t) = Kd (s) × Kt(t).

However, that is rarely the case, particularly in infectious diseases, where disease spreads from one individual to another. In the example visualised in Figure 1, K(1,1) = 1/36 as cases A and B are within 1 unit and 1 day of each other (Note that we have N 2 here as we are considering the two dimensions). However, Kd (1) × Kt(1) = 2/6 × 1/6 = 2/36. (Note that we have labelled the cases here for the purposes of the example, but when actually calculating these functions the cases are considered unlabelled.) Because of this dependence between space and time, we are interested in estimating how many times greater K(s,t) is compared to the product of Kd (s) and Kt(t). Thus, we arrive at the definition of D0 function,

D0(s,t) = K(s,t)/(Kd(s) × Kt(t)).      (2)

The actual value of the D0 function is not particularly important. Rather, it is how this function changes over time or geographically that is the most important. The absence of space-time interaction (i.e. a relatively flat D0 function) is a sign of control success.

(a)
(b)

Figure 1: Graphical illustration of the K-function in (a) space and (b) time.

From March until the end of 2020, we estimated the D0 function as part of the routine reporting framework. It was regularly one of the fastest measures to show control success, but perhaps one of the most valuable moments was the reporting in mid-June. The period from May through to the start of June was one of relative calm, with 218 cases diagnosed over the 31 days, an average of roughly 7 notifications per 24 hours. A similar story was true at the start of June, with 114 community acquired or unknown source cases over the first 20 days: an average of under 6, relatively the same as what we had so far. In our routine report, using data up to June 20, 2020, something seemed amiss with the D0 function, reproduced in Figure 2. What was once a nicely clustered mass near 0 days and 0 kilometres seemed to have spread out, particularly over distance. There was, unusually, relatively little clustering to be seen.

After digesting this figure, we got together with the epidemiology team who had been compiling notes on all the cases and their contacts across Victoria. When we pooled all the evidence together—statistical models, case notes, forecasts of incoming numbers, genomic information, geographic risk profiles, and so much more—we arrived at a hypothesis we hoped would not be true: infection had been scatter-gunned across the greater Melbourne area. The more we looked, the more it seemed like it could be true. In the 10 days that followed, a further 369 community acquired or unknown cases would be notified, more than 6 times as many cases per day than that of the previous 3 weeks. Victoria’s second wave had arrived.

Figure 2: Estimate of the D0 function based on COVID-19 data up to June 20, 2020 in Victoria. The contagiousness estimated by this function is very low, despite signs of increasing case numbers.

The D0 function is not a new concept to infectious diseases. Although not typically applied in human diseases (as thankfully we don’t have many pandemics), it has been used in Rift Valley fever [6], highly pathogenic avian influenza [4,5] as well as a handful of others [1,3,8]. The technique is well-known, and in an environment where time is of the essence, it’s quick compute time was proven valuable. COVID-19 was, and is, an infection that we knew little about, and the ability to apply a tried and tested model when it was needed most meant that our ability to respond was as strong as it could be.

There is no crystal ball when it comes to pandemic predictions. No matter how complex the model we develop and apply, no one can ever accurately predict the future. Much of my and my team’s role was to synthesise the information from these relatively unsophisticated models and communicate them to the people who needed to know. In the example discussed here, it was the discussion of information with our epidemiologists that led to the conclusion, not a piece of data or a model. No-one knows when exactly the peak of infections will be, but between the epidemiology and the modelled data, we can come together and give an idea of whether we’re likely to see increases or if the control measures being applied are working.

Victoria has since reached a state of elimination for COVID-19, along with the rest of Australia. A feat shared only by a few globally. Here, we have seen one example of a response coming together to solve an issue, but it is far from the only example. If there’s one thing to take away from Victoria’s COVID-19 response, it’s that the pieces of the puzzle are always stronger together.

Acknowledgements: The author would like to thank Mark A. Stevenson and Kira Leeb for their comments on this article, as well as the Victorian Department of Health Analytics Team (COVID-19 Intelligence) for their work throughout the pandemic.

[1] E. Delmelle, I. Casas, J. H. Rojas, and A. Varela. Spatio-Temporal Patterns of Dengue Fever in Cali, Colombia. International Journal of Applied Geospatial Research (IJAGR), 4(4):58–75, Oct. 2013.

[2] P. Diggle, A. Chetwynd, R. Häggkvist, and S. Morris. Second-order analysis of space- time clustering. Statistical Methods in Medical Research, 4(2):124–136, June 1995.

[3] A. C. Gatrell, T. C. Bailey, P. J. Diggle, and B. S. Rowlingson. Spatial Point Pattern Analysis and Its Application in Geographical Epidemiology. Transactions of the Institute of British Geographers, 21(1):256–274, 1996.

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