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

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.

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.

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.

[4] C. Guinat, G. Nicolas, T. Vergne, A. Bronner, B. Durand, A. Courcoul, M. Gilbert, J.-L. Guérin, and M. C. Paul. Spatio-temporal patterns of highly pathogenic avian influenza virus subtype H5N8 spread, France, 2016 to 2017. Eurosurveillance, 23(26):1700791, June 2018.

[5] L. Loth, L. T. Pham, and M. A. Stevenson. Spatio-temporal distribution of outbreaks of highly pathogenic avian influenza virus subtype H5N1 in Vietnam, 2015–2018. Transboundary and Emerging Diseases, 68(1):13–20, 2019.

[6] R. Métras, T. Porphyre, D. U. Pfeiffer, A. Kemp, P. N. Thompson, L. M. Collins, and R. G. White. Exploratory Space-Time Analyses of Rift Valley Fever in South Africa in 2008–2011. PLOS Neglected Tropical Diseases, 6(8):e1808, Aug. 2012.

[7] B. D. Ripley. The Second-Order Analysis of Stationary Point Processes. Journal of Applied Probability, 13(2):255–266, 1976.

[8] J. W. Wilesmith, M. A. Stevenson, C. B. King, and R. S. Morris. Spatio-temporal epidemiology of foot-and-mouth disease in two counties of Great Britain in 2001. Preventive Veterinary Medicine, 61(3):157–170, Nov. 2003.

## Mathematical modelling of Australian COVID-19 response: A PhD student perspective

It has been a little over 12 months since COVID-19 became a regular headline in the Australian media, but I would not be alone in saying it has definitely felt longer. At the time I was entering the third year of my PhD in mathematics and epidemiology, so when news broke of the new virus in late 2019, I was certainly paying attention. Little did I know it would affect not only my life as a researcher in the field, but everyone across the world.

Although COVID-19 restrictions have disrupted my study and research like many other HDR students, I have been fortunate that my work in modelling COVID-19 made progress towards my thesis.

In late February 2020 I was asked if I could help in the modelling effort, and at first this was supporting Dr Andrew Black and Dr James Walker in examining Australia’s pandemic preparedness and border closures. This modelling work by Andrew and James formed part of the advice on closing the international border. It was a frantic period of time, with a rapidly evolving situation.  Seeing this body of work influence policy was the first of many instances

It wasn’t long after that my supervisor Prof. Joshua V. Ross asked if I was interested in developing and providing a forecast of COVID-19 cases to the COVID-19 response. I would be lying if I didn’t say Imposter Syndrome didn’t tell me to run the other way. Fortunately, and with encouragement from my supervisors and the wider COVID-19 modelling group, I didn’t give in and dived into the work.

To better describe our model, I will briefly introduce some important epidemiological concepts. An important epidemiological parameter is the effective reproduction number Reff, which can be defined as the average secondary number of infections from an infectious individual. This can vary through time, as behaviour changes through the epidemic, through social distancing and public health policy changes.

Reff can be retrospectively estimated through examining the number of cases over time, but to forecast cases using a mechanistic model, it must incorporate some estimate of the future transmission potential and/or arrival of infected cases. The relatively low number of cases in Australia also creates difficulties in utilising methods that rely on historical case incidence. Measures of mobility of each Australian jurisdiction provided by Google and survey results of the public’s behaviour in adhering to personal distancing measures provides the ability to link these indicators to an estimate of the effective reproduction number. This allows for a mechanistic model to forecast cases.

We estimated Reff using historical case incidence and an established method from the literature. To forecast Reff forward, we calibrated a model that links social mobility and personal distancing measures to these estimates of Reff

Within Australia, there have been jurisdictional level differences in policy and response to social distancing, but the underlying culture and mobility patterns may have commonalities. As such, we employed a hierarchical model to partially pool information between jurisdictions, while allowing for inferred differences where they may occur.

After calibrating the model and using Bayesian inference to learn the parameters, we then forecast the social mobility and distancing metrics using a random walk with drift in each jurisdiction. The model then gives a posterior predictive distribution on the Reff over time.

The relatively few cases of local transmission in Australia, in conjunction with strict border control measures internationally and domestically, makes it natural to forecast the number of cases in each jurisdiction using a stochastic branching model. This generative model, using estimates from the literature for epidemiological parameters, can be paired with the time varying effective reproduction number to forecast COVID-19 cases in Australian jurisdictions. This framework adapts to changing public health policies and responses to the ongoing pandemic, particularly during small outbreaks and the irregular but frequent responses to outbreaks seen in Australia.

This forecasting model was run every week, and the results contributed to an ensemble forecast that was provided to various bodies in the Australian Government. This ensemble forecast was often considered by Chief Health Officers in determining the appropriate course of action, and was even shown a few times at media press conferences.

As mathematicians, it is rare that we get to personally observe the impacts of our research, let alone at my level as a PhD candidate. While the pressure and high stakes definitely gave me some sleepless nights, to see policy and action consider my work was incredibly fulfilling, and I highly recommend any HDR student take any opportunity to work on research with direct and immediate impacts like the COVID-19 response. Don’t let your Imposter Syndrome dissuade you from contributing, as every effort, however minor, helps. Your unique perspective will always be valuable in discussions, and you will almost certainly be supported by an amazing and dedicated team as well as your supervisors, as I did in my work!

## Contribution of mathematical modelling to COVID-19 response strategies in regional and remote Australian Aboriginal and Torres Strait Islander communities

(This is a guest post by Dr Rebecca Chisholm, Dr Ben Hui and Associate Professor David Regan 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.)

The health and science communities recognised early on in the SARS-CoV-2 pandemic that Aboriginal and Torres Strait Islander Australians were likely to be at high risk of COVID-19 infection and severe outcomes, due to high rates of comorbidities associated with severe outcomes [1,2], and multiple factors predisposing to increased SARS-CoV-2 transmission [2,3,4].  In March 2020, the Australian Government convened the Aboriginal and Torres Strait Islander Advisory Group on COVID-19 (IAG), co-chaired by the Department of Health and the National Aboriginal Community Controlled Health Organisation. The role of the IAG was to develop and deliver a National Management Plan to protect Aboriginal and Torres Strait Islander communities.  Our research groups—located at the Doherty Institute, the Kirby Institute and La Trobe University—were commissioned to carry out modelling, under the guidance of the IAG, to help inform aspects of this plan related to regional and remote communities.

Our prior research and existing modelling frameworks enabled us to quickly begin the process of responding to the questions of interest to the IAG which included:

• How important is a timely response to the first identified case of COVID-19?
• Who should be quarantined and/or tested in communities?
• How important is it to test people when they are in quarantine and prior to exit from quarantine?
• Is there a role for community-wide lockdown in initial containment?

Together, we repurposed a stochastic, individual-based modelling framework which had previously been developed at the Kirby Institute to examine the dynamics of sexually transmitted infections in remote communities [5].  Within this framework, we incorporated a model of population mobility and household structure relevant to disease spread via close contact in remote communities. This model was originally developed at La Trobe University and the Doherty Institute as part of a research program focused on understanding the drivers of high prevalence of Group A Streptococcus disease in these communities [4].  We also integrated a COVID-19-specific disease transmission model and the effects of various public health responses.  Throughout this model building process, we regularly engaged with the IAG and representatives from other peak bodies and public health units to iteratively refine details and assumptions (described in Box 1 and Figure 1).

To address the questions of interest to the IAG, we used the model to simulate and analyse a number of outbreak response scenarios. We designed the scenarios in consultation with public health service providers working closely with communities (with options varying by jurisdiction and community). These included:

• Case isolation, with or without an exit test, and with various expected delays between case identification and response;
• Case isolation and quarantining the contacts of a case (based on different definitions of contacts), with or without exit tests, and with or without tests on entry to quarantine;
• Case isolation and population lockdown (entire community quarantined), with or without exit tests, and with various levels of assumed compliance to lockdown.

Box 1. Brief model summary. The individual-based, computational model we designed simulated the “silent” introduction of SARS-CoV-2 into a remote community of either 100, 500, 1000 or 3500 people, the subsequent transmission of SARS-CoV-2 within the community, and the public health response. The model explicitly represented the infection status of each community member, as well as their age and place of residence within the community, which were tracked and updated daily.  Community members were assumed to have close family connections across multiple dwellings in the community (their so-called “extended household”), between which their time at home was distributed, and within which they were at higher transmission risk compared to individuals staying in different dwellings (Figure 1a).  Infected community members were further classified according to whether or not they would present to healthcare services for testing (if symptoms developed and were recognized, and fear/stigma did not prevent individuals from presenting, Figure 1b). At the time we developed our model, there had been no  SARS-CoV-2 transmission in Australian Aboriginal and Torres Strait Islander communities.  Therefore, our model was parameterized based on the experience of SARS-CoV-2 in other populations [6], but accounting for the expected increase in transmission due to enhanced mixing anticipated in interconnected and overcrowded households [2,3].

To gain an understanding of the range of possible epidemic outcomes, we used our model to run 100 simulations of each outbreak response scenario  (defined by a set of parameters controlling the transmission of SARS-CoV-2, the public health response, and the assumed response of community members to the response).  For different response scenarios, we compared and reported the median and interquartile range of several model outputs of interest, including the percentage of the community who were infected at the peak of the outbreak (peak infection prevalence) and by the end of the outbreak (the attack rate), the number of cases identified versus the number of cumulative infections over time, the total number of person-days community members were in quarantine for, and the number of tests performed.

We sought regular feedback on the response scenarios considered, and our interpretation and communication of model outputs.  This ensured we were always addressing relevant questions and faithfully relaying our findings (summarised in Box 2 and Figure 2).

Our work informed both the CDNA National Guidance for remote Aboriginal and Torres Strait Islander Communities for COVID-19 [7] and the Australian Health Sector Emergency Response Plan for Novel Coronavirus (COVID-19) [8].  We have since submitted a publication for peer-review describing our work, currently available as a pre-print [9].  We also worked together with the IAG to develop a plain-language document containing key messages for health services [10], and a plain-language presentation [11] containing key messages for Health service decision makers and community leaders to consider when deciding how a remote community will respond to a COVID‐19 outbreak.

To date, efforts to protect Australian Aboriginal and Torres Strait Islander peoples from COVID-19 are working – there have been no incursions of SARS-CoV-2 into remote Australian Aboriginal and Torres Strait Islander communities, and the incidence of locally-acquired cases among all Australian Aboriginal and Torres Strait Islander peoples is six-times lower than the Australia-wide incidence [12].

Box 2. Brief summary of findings. Our analysis indicated that without an effective public health response, an introduction of SARS-CoV-2 into a regional or remote Australian Aboriginal and Torres Strait Islander community would likely result in rapid spread.  Furthermore, multiple secondary cases would likely be present in a community by the time the first case is identified, indicating that capacity for early case detection and a prompt response would be crucial in constraining an outbreak.  A response involving case isolation and quarantining of close contacts of cases defined by extended household membership was found to significantly reduce peak infection prevalence compared to the non-response scenario, but subsequent waves of infection consistently led to unacceptably high attack rates in excess of 80% in modelled scenarios.  Rapidly initiating an additional 14-day, community-wide lockdown of non-quarantined households could reduce the attack rate to less than 10%, but only if compliance with the lockdown was at least 80% (Figure 2).

### References

[1] Chen T, Wu D, Chen H, Yan W, Yang D, Chen G, Ma K, Xu D, Yu H, Wang H: Clinical characteristics of 113 deceased patients with coronavirus disease 2019: retrospective study. Bmj 2020, 368. https://doi.org/10.1136/bmj.m1295

[2] Australian Institute of Health and Welfare: The health and welfare of Australia’s Aboriginal and Torres Strait Islander peoples: 2015. In. Canberra: AIHW; 2015. https://doi.org/10.25816/5ebcbd26fa7e4

[3] Koh D: Migrant workers and COVID-19. Occupational and Environmental Medicine 2020:oemed-2020-106626. https://doi.org/10.1136/oemed-2020-106626

[4] Chisholm RH, Crammond B, Wu Y, Bowen A, Campbell PT, Tong SY, McVernon J, Geard N: A model of population dynamics with complex household structure and mobility: implications for transmission and control of communicable diseases. PeerJ 2020, 8:e10203. https://doi.org/10.7717/peerj.10203

[5] Hui BB, Gray RT, Wilson DP, Ward JS, Smith AMA, Philp DJ, Law MG, Hocking JS, Regan DG: Population movement can sustain STI prevalence in remote Australian indigenous communities. BMC Infectious Diseases 2013, 13:188. https://doi.org/10.1186/1471-2334-13-188

[6] Sanche S, Lin YT, Xu C, Romero-Severson E, Hengartner N, Ke R: High Contagiousness and Rapid Spread of Severe Acute Respiratory Syndrome Coronavirus 2. Emerg Infect Dis 2020, 26(7). https://doi.org/10.3201/eid2607.200282

[7] Communicable Disease Network Australia: National Guidance for remote Aboriginal and Torres Strait Islander Communities for COVID-19. 2020, Department of Health, Commonwealth of Australia [https://www.health.gov.au/resources/publications/cdna-national-guidance-for-remote-aboriginal-and-torres-strait-islander-communities-for-covid-19]

[8] Department of Health, Commonwealth of Australia. Australian Health Sector Emergency Response Plan for Novel Coronavirus (COVID-19). 2020, Department of Health, Commonwealth of Australia [https://www.health.gov.au/resources/publications/australian-health-sector-emergency-response-plan-for-novel-coronavirus-covid-19]

[9] Hui BB, Brown D, Chisholm RH, Geard N, McVernon J, Regan DG: Modelling testing and response strategies for COVID-19 outbreaks in remote Australian Aboriginal communities. medRxiv 2020, 2020.10.07.20208819. https://doi.org/10.1101/2020.10.07.20208819

[10] Department of Health, Commonwealth of Australia. COVID-19 Testing and Response Strategies in Regional and Remote Indigenous Communities: Key Messages for Health Services. 2020, Department of Health, Commonwealth of Australia [https://www.health.gov.au/resources/publications/covid-19-testing-and-response-strategies-in-regional-and-remote-indigenous-communities-key-messages-for-health-services]

[11] Department of Health, Commonwealth of Australia. Impact of COVID-19 in remote and regional settings. 2020, Department of Health, Commonwealth of Australia [https://www.health.gov.au/resources/publications/impact-of-covid-19-in-remote-and-regional-settings]

[12] Aboriginal and Torres Strait Islander Advisory Group on COVID-19. Aboriginal and Torres Strait Islander Advisory Group on COVID-19 Communique Update: 14 December 2020. Department of Health, Commonwealth of Australia [https://www.health.gov.au/resources/publications/aboriginal-and-torres-strait-islander-advisory-group-on-covid-19-communiques]

## Informing the COVID-19 response: mathematicians’ contributions to pandemic planning and response

COVID-19 has changed how we live our lives, and will continue to do so for some time yet. Australia has been fortunate in many ways. We have clearly defined borders which are able to be managed effectively. We have a highly functional public health system. Despite the challenges in Victoria in mid-2020, ultimately we increased testing and contact tracing capacity enough to suppress transmission. We have, for the most part, seen coherent leadership from our state and Commonwealth political leaders.

The ‘science’—from clinical and lab-based research to mathematical modelling—has been listened to and, again for the most part, acted upon.

But world class research is clearly not sufficient to manage the pandemic. The United Kingdom—the ‘home’, and I would argue ‘intellectual powerhouse’, of mathematical epidemiology—has suffered greatly. As has the United States and much of Europe.

In Australia, the field of mathematical epidemiology is still in development. It was only 2005 when Australia’s National Health and Medical Research Council made its first major investment, funding a ‘Capacity Building Grant’ in infectious diseases modelling to support public health. I was fortunate to be appointed as a post-doctoral research fellow under the scheme, as were a number of other now well-known mathematical epidemiologists including Professor Jodie McVernon (Doherty Institute) and A/Prof James Wood (UNSW), who trained in infectious disease modelling and mathematical physics respectively. The grant was led by Professor Raina MacIntyre, a prominent epidemiologist and media figure in Australia’s COVID-19 response.

From the outset, we were engaged with the Australian Government Department of Health’s Office of Health Protection, the body responsible for preparedness and response to public health emergencies in Australia. At the time, the focus was on SARS and pandemic influenza.

As a mathematician, I have maintained an open dialogue with the Commonwealth for over 15 years. Through contractual research, we have developed stochastic models for border incursions, examined optimal distribution strategies for limited supplies of antivirals, estimated the volume of Personal Protection Equipment (PPE) required in a response and examined optimal strategies for vaccination. These analyses informed the development and multiple revisions of the Australian Health Management Plan for Pandemic Influenza (AHMPPI). In February 2020, with input from me and colleagues, the AHMPPI was rapidly adapted for COVID-19.

Throughout this 15 year period, we regularly visited Canberra to sit around the table with Health leadership, including four different Chief Medical Officers of Australia and their advisors. Both parties learnt a lot through that collaboration. As a mathematician I learnt how to communicate the purpose, limitations and relevance of our models. The government learnt to appreciate what models could and could not do. What policy decisions they could and could not inform. We gained a shared understanding that deeply quantitative work primarily delivered qualitative insights. And we learnt to trust each other.

Trust—not just in the science, but in the people conveying that science—is, in my view, the most fundamental requirement for the effective contribution of scientific knowledge to policy and response.

As a qualified and trusted advisor, I have contributed in two ways. With my team, we have undertaken mathematical analyses and delivered those results to government. But my responsibilities also include interpretation and evaluation of the (global and emerging) literature. Can Imperial College’s COVID-19 modelling on ‘lockdowns’ be applied to Australia? Are optimal vaccination strategies developed for other countries applicable here? Are real-time analysis methods for a well-established outbreak—like those developed at the London School of Hygiene and Tropical Medicine—applicable to Australian case data?

I believe that Australia benefited from the deep engagement and trust developed between academics and the Commonwealth over 15 years. The trust lies not just with the advisors, such as me. The trust extends through to a cultivated broader trust in the scientific research performed by others and interpreted and evaluated by those advisors.

And with that, where does mathematical analysis make a difference?

Often, it is in what we (as mathematicians) may perceive as surprisingly simple points.

Epidemic theory describes how a pathogen spreads through the community. Scaling out the average duration of infectiousness, and ignoring some biological subtleties, the rate of change in prevalence I (the proportion of individuals who are Infectious in the population) is described by a non-linear ordinary differential equation:

dI/dt = (R0S − 1)I

where R0 is the ‘basic Reproduction number’, the number of secondary cases arising from a single case in an otherwise fully susceptible population, and S is the proportion of the population that is Susceptible.

With R0 > 1 and S sufficiently large (as it is at the beginning of an epidemic), prevalence (that is, I) grows exponentially and S decreases (as dS/dt = −R0SI).

Eventually, depletion of the susceptible pool (S) modified the dynamics. The resultant non-linear dynamics are what make infectious diseases both mathematically interesting, and conceptually challenging for public health policy makers to respond to.

In early 2020, my team delivered a report to government which explored the possible change in the total number of infections over the course of an epidemic due to various percentage reductions in transmissibility. For our purposes here, this is as simple as considering how the size of an epidemic depends upon R0, although we did not report the analysis in this way to government.

A textbook analysis yields the ‘final size equation’, which relates R0 to the size (as t→∞) of the epidemic, Z = 1 − S(t→∞):

Z = 1 − eR0 Z

This is a non-linear relationship. By February 2020, we suspected the R0 for COVID-19 was in the range 1.5–3. At the upper end of this range, a 50% reduction (due to say, some level of physical distancing) has an important but fundamentally challenging effect—an epidemic with a modified reproduction number of 3/2 = 1.5 still spreads explosively, resulting in a vast number of infections. But if COVID-19’s reproduction number was at the lower end, a 50% reduction would prevent the virus from spreading entirely as 1.5/2 = 0.75 < 1 and transmission cannot be sustained.

Such results are entirely unsurprising to us as mathematicians. We understand the importance of non-linearities and of features such as bifurcations. It is natural for us to view the transmission of an infectious disease as a dynamical system. But these important points are anything but intuitive and easily missed by decision makers.

With trust and open communication channels, important findings, as well as viewing the entire pandemic and our response to it as a dynamical system, proved influential in Australia’s early response.

Simple analyses emphasised the value of mathematical reasoning. They highlighted the risks of ‘intuitive linear thinking’ but they also demonstrated how mathematical analysis can overcome that limitation. Models can be used to anticipate or reason on the (positive or negative) impacts of alternative response strategies.

Subsequent scenario analyses (with more ‘realistic’ and nuanced models calibrated to COVID-19 epidemiology) laid the groundwork for our initial response and for the monitoring and evaluation of the ‘effective reproduction number’ of COVID-19 throughout 2020 and into 2021. Collectively, these mathematical capabilities have contributed to the Australian government’s risk-assessment process for managing the pandemic.

National policy guidance relies upon in-depth mathematical modelling and analyses, conducted both in-house, nationally and around the world. But while necessary, the existence of that research is not sufficient to have impact. To have impact, to be effective, also requires relationships, ‘translators’ and, above all, (mutual) trust.

## Australian mathematicians rise to the challenge of COVID-19

(This is a guest post by Dr Joel Miller, introducing a miniseries of articles/essays by Australian mathematicians involved in the pandemic response)

Mathematics plays an integral role in our daily lives.  A smart phone that guides you to your destination relies on mathematical routines to calculate your position, other algorithms find the optimal route, and yet others ensure that the communications from your phone are secure.  The central role that mathematics plays throughout scientific disciplines comes largely because our mathematical models of the natural world, built on observation, give us remarkable predictive power and allow us to design systems that perform optimally.

At the outset of the COVID-19 pandemic, we did not have time to do careful experiments comparing different policies before implementing them.  What we had was information about how efficiently the disease spreads, some hints that it could spread through asymptomatic or presymptomatic infected individuals, and estimates of the distribution of severity in different age groups.  That knowledge grew as different countries began to experience outbreaks.

Armed with this knowledge policy makers were forced to make decisions about their response.  They needed a way to turn this limited information about the mechanisms underlying disease spread into projections of what the future would hold.  Mathematical modelling was the tool that let us rigorously determine what consequences could follow from different policy decisions and different plausible disease properties.  The modelling effort relied on a wide range of techniques and modellers from different backgrounds and career levels, ranging from student to senior academic, as well as researchers working within health departments.

Lives have been upended by the COVID-19 pandemic, and by our response to it.  In this series some of the mathematical modellers who played a role in advising Australia’s (thus far) stunningly successful response give their perspective on the role that they played, showing how mathematicians at many levels played a key role in the decisions that led to COVID-19’s effective elimination in Australia.