The simulation model used here is one of a suite of epidemiological models developed to support the study and management of the COVID-19 outbreak in Australia. The suite builds on a stochastic individual-based or agent-based model (IBM), which follows infected individuals through multiple stages and alternative fates as the disease progresses. Individual-based models are flexible but are computationally expensive to run when the number of individuals becomes large. This makes IBM more difficult to undertake rigorous statistical calibration, or to run large ensembles to quantify stochastic variation. Thus, we developed a more computationally efficient analogue of the IBM: a stochastic compartment model (SCM) which followed daily cohorts of infected individuals through the same stages and fates. For extremely demanding computations, including Monte Carlo-based Bayesian inference, the SCM was approximated by a simpler deterministic compartment model (DCM) that follows daily cohorts through the same stages and fates, except that probabilistic transitions are replaced by proportional allocations.
The model suite represents the public health measures implemented in Australia in an attempt to prevent or limit outbreaks. These measures include: testing; contact tracing; self-quarantine/self-isolation of detected cases and contacts; border controls (self-quarantine and then hotel quarantine of overseas arrivals); and social distancing directives. Our model suite predicts the number of new infected cases each day as a result of transmission from current infectious cases, taking into account the effects of self-isolation and self-quarantine.
This section provides an overview of the model structure and function. Further details are provided in Appendix 1.
The model suite represents the key stages and possible outcomes in the evolution of COVID-19 infection, as portrayed in Fig. 1. Some asymptomatic COVID-19 cases never develop symptoms, but do become infectious, although they are believed to be less infectious than symptomatic cases. Newly infected cases in the model are immediately assigned to either an asymptomatic category (with probability PA) or a symptomatic category with probability (1-PA). Those in the symptomatic category do not develop symptoms until TS days post-infection. The timing of the onset of symptoms matters because, at least during the period considered in this study, testing (and therefore detection, reporting and contact tracing) in Australia were mostly confined to those displaying symptoms.
Both asymptomatic and symptomatic categories become infectious after TI days. Importantly, transmission can occur at least 1 to 2 days before symptoms became apparent (TI < TS), hindering attempts to control COVID-19 outbreaks. In our model suite, the infectious period ends at TF days, chosen so as to yield appropriate values for the reproductive number R0 and the growth rate of uncontrolled outbreaks.
A proportion of COVID-19 patients develop severe symptoms requiring hospital admission. In our model suite, this occurs after TH days, with probability PH. Of those admitted to hospital, a further proportion will develop fatal complications, with probability PM. Those with fatal complications die with daily probability PD, starting TD days after infection. All other patients start to recover with daily probability PR, beginning TR days after infection. Our model suite only tracks cases for a maximum period of TM days after infection after which cases have either recovered or died. Transitions in status related to onset of symptoms, onset and cessation of infectivity, and hospital admission occur simultaneously across all individuals in the same cohort. We developed versions of the models which allow these transitions to be spread over windows of multiple days noting that this made a negligible difference to our simulations.
Testing and detection of symptomatic cases occur in the model suite either through community testing, or through testing of those in self-quarantine or self-isolation. Community testing of cases self-reporting with symptoms was less effective in the first wave in Australia (March to May), as tests were in short supply and only contacts of overseas arrivals or known cases were tested. In Victoria’s second wave, attendance at community testing clinics was strongly encouraged, although some studies suggest only a modest proportion of those with COVID-19-like symptoms volunteered to be tested. In the model suite, the daily probability for a symptomatic case being tested and detected as positive at a community testing centre were allowed to increase from a low value of 0.2 in the first wave, to a higher value PDC after June.
Testing and detection of symptomatic cases already in self-quarantine is assumed to be more effective, and their daily probability of detection PDSQ is set to 0.8. The model assumes that all severe cases are tested and detected on admission to hospital.
Contact tracing has been a key and controversial public health measure against COVID-19 in Australia, and its representation in the model has been given particular attention. For each new infected case, the IBM keeps track of the ID of the responsible source case. If and when its source case is detected, an (undetected) case becomes eligible to be traced, with a daily probability PT. Once traced, cases are placed in self-quarantine noting that all detected cases are required to self-isolate.
The representation of contact tracing in the SCM and DCM is more challenging. In these models, only the numbers of cases in subcategories within a daily cohort are known. Our model suite tracks the proportions of detected cases within source cohorts, weighted by their relative contributions to each new cohort. They use this to calculate the proportion of members in each daily cohort which become subject to contact tracing. This calculation cannot replicate exactly the tracking of individual sources in the IBM, but provides a good approximation, allowing the SCM to closely replicate ensemble output from the IBM.
A highly effective contact tracing program would be expected to have values of the 24 h tracing efficiency PT close to 1. Simulations show that contact tracing can still be highly effective with much lower values of PT because the proportion of detected cases builds cumulatively over time. In the Victorian second wave, contact tracing by itself was insufficient to stop growth in infections and a severe lockdown was eventually imposed. We reproduced this fact in the model suite by assuming that a proportion PU of infected cases were permanently undetected and untraceable.
Contact tracing is resource intensive, especially as the number of cases increases. For each detected case, a number of downstream contacts of order 10 must be identified, contacted, asked to self-quarantine, and monitored for development of symptoms, and to check compliance, for 14 days. Thus, for 100 daily detected cases, contact tracers could have up to 14,000 contacts under management. The Australia-wide tracing capacity TCAP is assumed to be between 100 and 500 daily detected cases but we note that in specific jurisdictions, such as Victoria during its second wave, the capacity could have been below 100 new daily cases, in the absence of end-to-end automated process for enabling and recording contact tracing (Legal and Social Issues Committee 2020).
Australian border controls are implemented in the model as quarantine requirements on overseas arrivals. After 17 March 2020, overseas arrivals were required to self-quarantine at home. After 28 March they were required to enter hotel quarantine. The model simulations provided here are driven by reported numbers of daily detected COVID-19 cases among overseas arrivals in Australia. Positive overseas arrivals in quarantine are assumed to be detected immediately upon displaying symptoms and are represented in the model as new infectives appearing TS days before being reported. Reported overseas cases are assumed to be accompanied by additional undetected asymptomatic cases, in the ratio PA:1–PA.
Social distancing (SD) or lockdown measures have played a key role in controlling both the first and second waves in Australia. The model suite represents SD implicitly as changes in the effective daily transmission rate G (new infections per infectious case per day). The maximum transmission rate in the absence of social distancing is denoted by G0. The minimum transmission rate achieved during the severe nation-wide lockdown in April 2020, which ended the first wave, is denoted by GLD. Movement data suggest there was a gradual relaxation following the first wave, up until early July. The extent of this relaxation of SD in early July is measured by the parameter RSD given by (G–GLD)/(G0–GLD). In the control scenarios described later, social distancing stringency is characterised by the control variable SD = (G0–G)/(G0–GLD).
In the IBM, the number of daily new infections produced by an infected individual is assumed to be a random variable drawn from an over-dispersed negative binomial distribution with mean G and dispersion coefficient k’. In the SCM, the number of daily new infections from a pool of X infectious individuals is then also negative binomial, with adjusted parameters G. X and k’.X. The choice of the dispersion parameter k’ is discussed further in the Appendix. In the DCM, the number of new daily infections is just the expected value G.X.
Individuals have different transmission rates, depending on their status. Cases in the asymptomatic category are assumed to have a mean transmission rate equal to FA times that of those in the symptomatic category. While there should be zero transmission from cases in self-isolation or self-quarantine, in practice some transmission occurs. The model suite assumes the transmission rate from those in self-quarantine or self-isolation is reduced by the ‘leakage’ factor PL.
Hotel quarantine was initially assumed to be 100% effective in preventing transmission. However, cases of transmission from within hotel quarantine contributed to Victoria’s second wave, and multiple other cases of transmission from within hotel quarantine have been observed since in Australia. Accordingly, the model allows the generation of community infected cases from within hotel quarantine with a (very low) daily probability PQ.
All the models used in this study assume homogeneous mixing of infected with a susceptible pool of size SUS. The number of susceptible people, POP, is initially set to 20 million, assuming approximately 80% of the Australian population was initially susceptible. New infected cases are subtracted daily from SUS, and the daily transmission rate is multiplied by the fraction (1–SUS/POP). Given the small size of Australian outbreaks to date, the reduction in the susceptible pool size has negligible effect on transmission rates in the simulations presented here.
Parameter uncertainty and Bayesian inference
The model suite has 24 model parameters (Table 1). In July–August 2020, when this study was undertaken, many of these parameters, particularly those related to the natural history of the disease, were considered to be well-constrained by prior knowledge. But others, particularly those defining the effectiveness of Australian control measures, were poorly constrained. We wanted to understand the capacity of the model to reproduce Australian observations of daily cases prior to that time, and the extent to which those observations could constrain the uncertain parameters, so as to reduce uncertainty in model simulations of responses to future control measures.
A Monte Carlo Bayesian inference procedure was, therefore, undertaken to fit the model to Australian observations obtained from https://www.covid19data.com.au/ and https://www.worldometers.info/coronavirus/#countries for the period 20 February to 5 July 2020, noting that estimation was completed 6 August.
A simple sample importance resample (SIR) procedure was used to obtain the posterior distribution of parameters. An ensemble of 200,000 simulations was generated, using independent random samples from the prior parameter distribution (the prior was treated as uniform on the parameter ranges specified in Table 1, and parameters were treated as independent in the prior). Because of the large ensemble size, the DCM was used as a fast approximation to the SCM in this procedure. Comparison of posterior ensembles from the SCM and DCM suggested this provided a good approximation to the posterior parameter distribution for both models. Both SCM and DCM were able to reproduce the observed time series well. (See Appendix 2 for detailed methods and results.)
Table 1 gives the maximum likelihood values of the uncertain parameters. The inference procedure yielded a large (10,000 member) random sample from the posterior parameter distribution. In the model scenarios described below, parameters for each simulation were drawn randomly from this posterior sample. This allowed us to represent the uncertainty in model predictions due to residual parameter uncertainty, as well as the uncertainty arising from stochastic events within the model itself.
Mandated social distancing measures were reintroduced on 9 July 2020 in Victoria, as rapidly growing daily case numbers reached a weekly average of 100. Model simulation scenarios (Table 2) were designed to assess the effectiveness of implementing different levels of social distancing at that trigger level. Social distancing levels in these scenarios are defined by the control variable SD = (G0–G)/(G0–GLD), so SD = 0 corresponds to no social distancing, and SD = 1 corresponds to the lockdown obtaining in April 2020.
For each simulated SD level, from 0.5 to 1.0, mandated measures remain in place for a minimum 40-day period and then social distancing is relaxed in a linear fashion over 60 days. These scenarios assume highly effective border controls and quarantine for all new arrivals into Australia and PQ is set to zero. Social distancing is not relaxed until there is no recorded community transmission. Thus, each of the six scenarios in Table 2 assumes the goal is to achieve no community transmission.
Two suppression scenarios were also simulated (see Table 3). In each suppression scenario, stringent social distancing measures (SD = 1.0) are imposed when the weekly average of new daily recorded cases is 100, but relaxation is triggered by a weekly average of 20 daily recorded cases. In suppression scenario A, social distancing is imposed for a minimum of 40 days before the relaxation criteria is assessed, while in suppression scenario B, there is no minimum period. For both scenarios, once the relaxation criteria are met, gradual relaxation to zero social distancing occurs over a 60-day period. In each scenario, border quarantine leakage (failure) occurs with a daily probability of 0.2% per infected arrival from overseas (PQ = 0.002).
For each of these scenarios, the SCM was used to generate an ensemble of 1000 runs, drawing parameter sets randomly from the posterior distribution produced by the Bayesian inference procedure described above. The simulated ensemble outputs were statistically analysed and daily percentiles calculated.
Economy-wide costs of the national and high stringency social distancing that began in March 2020 are based on Australian Bureau of Statistics (ABS) data at a Victorian level equivalent to approximately $210 million per lockdown day (Kompas et al. 2020). Economy costs of a lockdown were assumed to be linear in the different levels of mandated social distancing, noting that greater social distancing and an increased frequency of cycles of high stringency social distancing followed by relaxation are likely to more than proportionally increase economy costs. COVID-19 related fatalities are valued at $4.9 million per value of statistical life (VSL), sourced from Prime Minister and Cabinet (Prime Minister and Cabinet 2020).