A practical guide to mathematical methods for estimating infectious disease outbreak risks.

Mathematical models are increasingly used throughout infectious disease outbreaks to guide control measures. In this review article, we focus on the initial stages of an outbreak, when a pathogen has just been observed in a new location (e.g., a town, region or country). We provide a beginner's guide to two methods for estimating the risk that introduced cases lead to sustained local transmission (i.e., the probability of a major outbreak), as opposed to the outbreak fading out with only a small number of cases. We discuss how these simple methods can be extended for epidemiological models with any level of complexity, facilitating their wider use, and describe how estimates of the probability of a major outbreak can be used to guide pathogen surveillance and control strategies. We also give an overview of previous applications of these approaches. This guide is intended to help quantitative researchers develop their own epidemiological models and use them to estimate the risks associated with pathogens arriving in new host populations. The development of these models is crucial for future outbreak preparedness. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

A direct comparison of methods for assessing the threat from emerging infectious diseases in seasonally varying environments.

Seasonal variations in environmental conditions lead to changing infectious disease epidemic risks at different times of year. The probability that early cases initiate a major epidemic depends on the season in which the pathogen enters the population. The instantaneous epidemic risk (IER) can be tracked. This quantity is straightforward to calculate, and corresponds to the probability of a major epidemic starting from a single case introduced at time t=t0, assuming that environmental conditions remain identical from that time onwards (i.e. for all t≥t0). However, the threat when a pathogen enters the population in fact depends on changes in environmental conditions occurring within the timescale of the initial phase of the outbreak. For that reason, we compare the IER with a different metric: the case epidemic risk (CER). The CER corresponds to the probability of a major epidemic starting from a single case entering the population at time t=t0, accounting for changes in environmental conditions after that time. We show how the IER and CER can be calculated using different epidemiological models (the stochastic Susceptible-Infectious-Removed model and a stochastic host-vector model that is parameterised using temperature data for Miami) in which transmission parameter values vary temporally. While the IER is always easy to calculate numerically, the adaptable method we provide for calculating the CER for the host-vector model can also be applied easily and solved using widely available software tools. In line with previous research, we demonstrate that, if a pathogen is likely to either invade the population or fade out on a fast timescale compared to changes in environmental conditions, the IER closely matches the CER. However, if this is not the case, the IER and the CER can be significantly different, and so the CER should be used. This demonstrates the need to consider future changes in environmental conditions carefully when assessing the risk posed by emerging pathogens.

A simulation-based approach for estimating the time-dependent reproduction number from temporally aggregated disease incidence time series data.

Tracking pathogen transmissibility during infectious disease outbreaks is essential for assessing the effectiveness of public health measures and planning future control strategies. A key measure of transmissibility is the time-dependent reproduction number, which has been estimated in real-time during outbreaks of a range of pathogens from disease incidence time series data. While commonly used approaches for estimating the time-dependent reproduction number can be reliable when disease incidence is recorded frequently, such incidence data are often aggregated temporally (for example, numbers of cases may be reported weekly rather than daily). As we show, commonly used methods for estimating transmissibility can be unreliable when the timescale of transmission is shorter than the timescale of data recording. To address this, here we develop a simulation-based approach involving Approximate Bayesian Computation for estimating the time-dependent reproduction number from temporally aggregated disease incidence time series data. We first use a simulated dataset representative of a situation in which daily disease incidence data are unavailable and only weekly summary values are reported, demonstrating that our method provides accurate estimates of the time-dependent reproduction number under such circumstances. We then apply our method to two outbreak datasets consisting of weekly influenza case numbers in 2019-20 and 2022-23 in Wales (in the United Kingdom). Our simple-to-use approach will allow accurate estimates of time-dependent reproduction numbers to be obtained from temporally aggregated data during future infectious disease outbreaks.

Exact calculation of end-of-outbreak probabilities using contact tracing data.

A key challenge for public health policymakers is determining when an infectious disease outbreak has finished. Following a period without cases, an estimate of the probability that no further cases will occur in future (the end-of-outbreak probability) can be used to inform whether or not to declare an outbreak over. An existing quantitative approach (the Nishiura method), based on a branching process transmission model, allows the end-of-outbreak probability to be approximated from disease incidence time series, the offspring distribution and the serial interval distribution. Here, we show how the end-of-outbreak probability under the same transmission model can be calculated exactly if data describing who-infected-whom (the transmission tree) are also available (e.g. from contact tracing studies). In that scenario, our novel approach (the traced transmission method) is straightforward to use. We demonstrate this by applying the method to data from previous outbreaks of Ebola virus disease and Nipah virus infection. For both outbreaks, the traced transmission method would have determined that the outbreak was over earlier than the Nishiura method. This highlights that collection of contact tracing data and application of the traced transmission method may allow stringent control interventions to be relaxed quickly at the end of an outbreak, with only a limited risk of outbreak resurgence.

A theoretical framework for transitioning from patient-level to population-scale epidemiological dynamics: influenza A as a case study.

Multi-scale epidemic forecasting models have been used to inform population-scale predictions with within-host models and/or infection data collected in longitudinal cohort studies. However, most multi-scale models are complex and require significant modelling expertise to run. We formulate an alternative multi-scale modelling framework using a compartmental model with multiple infected stages. In the large-compartment limit, our easy-to-use framework generates identical results compared to previous more complicated approaches. We apply our framework to the case study of influenza A in humans. By using a viral dynamics model to generate synthetic patient-level data, we explore the effects of limited and inaccurate patient data on the accuracy of population-scale forecasts. If infection data are collected daily, we find that a cohort of at least 40 patients is required for a mean population-scale forecasting error below 10%. Forecasting errors may be reduced by including more patients in future cohort studies or by increasing the frequency of observations for each patient. Our work, therefore, provides not only an accessible epidemiological modelling framework but also an insight into the data required for accurate forecasting using multi-scale models.

Accurate forecasts of the effectiveness of interventions against Ebola may require models that account for variations in symptoms during infection.

Epidemiological models are routinely used to predict the effects of interventions aimed at reducing the impacts of Ebola epidemics. Most models of interventions targeting symptomatic hosts, such as isolation or treatment, assume that all symptomatic hosts are equally likely to be detected. In other words, following an incubation period, the level of symptoms displayed by an individual host is assumed to remain constant throughout an infection. In reality, however, symptoms vary between different stages of infection. During an Ebola infection, individuals progress from initial non-specific symptoms through to more severe phases of infection. Here we compare predictions of a model in which a constant symptoms level is assumed to those generated by a more epidemiologically realistic model that accounts for varying symptoms during infection. Both models can reproduce observed epidemic data, as we show by fitting the models to data from the ongoing epidemic in the Democratic Republic of the Congo and the 2014-16 epidemic in Liberia. However, for both of these epidemics, when interventions are altered identically in the models with and without levels of symptoms that depend on the time since first infection, predictions from the models differ. Our work highlights the need to consider whether or not varying symptoms should be accounted for in models used by decision makers to assess the likely efficacy of Ebola interventions.

A study of the transfer of tetracycline resistance genes between Escherichia coli in the intestinal tract of a mouse and a chicken model.

Experiments to demonstrate the transfer of genes within a natural environment are technically difficult because of the unknown numbers and strains of bacteria present, as well as difficulties designing adequate control experiments. The results of such studies should be viewed within the limits of the experimental design. Most experiments to date have been based on artificial models, which only give approximations of the real-life situation. The current study uses more natural models and provides information about tetracycline resistance as it occurs in wild-type bacteria within the environment of the normal intestinal tract of an animal. Tetracycline sensitive, nalidixic acid resistant Escherichia coli isolates of human origin were administered to mice and chicken animal models. They were monitored for acquisition of tetracycline resistance from indigenous or administered donor E. coli. Five sets of in vivo experiments demonstrated unequivocal transfer of tetracycline resistance to tetracycline sensitive recipients. The addition of tetracycline in the drinking water of the animals increased the probability of transfer between E. coli strains originating from the same animal species. The co-transfer of unselected antibiotic resistance in animal models was also demonstrated.

Antimicrobial resistance in Campylobacter spp., Escherichia coli and enterococci associated with pigs in Australia.

The major influences on the amplification and spread of antibiotic-resistant bacteria are the therapeutic use of antibiotics in human medicine and their use in livestock for therapy, prophylaxis and growth promotion. The use of veterinary antibiotics has many benefits to the livestock industries ensuring animal health and welfare, but use at subtherapeutic levels also exerts great selective pressure on emergence of resistant bacteria. The possible effect on human health is a problem of current debate. This study involved sampling pig carcasses, pig meat and assessing the level of resistance in zoonotic enteric bacteria of concern to human health. In South Australian pigs, thermophilic Campylobacter species showed widespread resistance (60-100%) to tylosin, erythromycin, lincomycin, ampicillin and tetracycline. No resistance was seen to ciprofloxacin. The enterococci demonstrated little resistance (0-30%) to vancomycin or virginiamycin, but the overall results from the antibiotic sensitivity testing of the enterococci have demonstrated how widespread their resistance has become. Escherichia coli strains showed widespread resistance to tetracycline and moderately common resistance (30-60%) to ampicillin and sulphadiazine. Resistance to more than one antibiotic was common. Pigs from New South Wales were also sampled and differences in resistance patterns were noted, perhaps reflecting different antibiotic use regimens in that state.

Long-term histologic results of vein grafting with the telescoping anastomotic technique.

We studied the long-term histologic results of a new method for autogenous vein grafting to examine whether stenosis at the anastomosis is maintained over time. Nineteen rat inferior epigastric veins were grafted into the femoral artery using a telescoping sleeve technique at both the proximal and the distal anastomoses. Specimens were studied macroscopically and histologically three months later. Stenosis at the anastomosis was located near the tip of the inserted vessel. The smallest inner diameters of the proximal and distal anastomoses were about 80% of the corresponding femoral artery diameter; no statistically significant difference was found between the two anastomoses. The grafts had a thickened wall due to intimal hypertrophy and fibrosis of the media. The inner diameter of the graft was, however, about twice that of the femoral artery, and these graft changes did not create any apparent constriction within the graft.

New method for autogenous vein grafting with the telescoping anastomotic technique.

An experimental study was performed to examine whether a microsurgical telescoping anastomotic technique could be applied twice in one vessel, for use in autogenous vein grafting. The rat inferior epigastric vein was grafted into a defect created in the femoral artery. The original telescoping method of Lauritzen was used, with two additional suture placements, to allow anastomosis at both proximal and distal sites. By placing the four sutures symmetrically and carefully timing the removal of the proximal and distal clamps, we achieved a patency rate of 77.3%. This is a new method for autogenous vein grafting that may serve as a prototype for an easier and possibly faster vein grafting technique. Our results appear to indicate that complete coaptation of the severed vessel ends is not necessarily required for patency in microvascular repair.