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 practical 'safe zone' technique for lag screw fixation of the fibula.

Introduction During ankle fracture fixation, iatrogenic trauma to retro fibula structures can result in morbidity and reoperation. We describe a safe zone for lag screw insertion. Materials and methods This study was completed in three sections. We identified the average entry and exit points for the lag screw using 45 Weber B ankle fractures identified from our trauma database. We then analysed 26 sequentially presented ankle magnetic resonance images, concentrating on axial sections at 4, 8, 12 and 16 mm above the ankle joint. Finally, we used 63 sequentially performed magnetic resonance scans to confirm the safe zone from these consistent structures. Results The typical lag screw exit point was 14.2 mm above the ankle joint (95% confidence Interval 11.3-17.1 mm). A safe zone trajectory occurred between 31 and 45 degrees taken from the anterior aspect of the flat fibular surface at this level. The obvious palpable landmark to direct screw trajectory and avoid 'at risk' structures was found to be the medial edge of the Achilles tendon. Our final dataset confirmed in 63 scans, the medial aspect of the Achilles tendon to be a consistent safe zone with a minimum distance of at risk structures of 4 mm. Conclusion This simple method of directing the fibula lag screw towards the palpable medial edge of the Achilles tendon is practical, easy to teach and directs the screw on a safe trajectory away from the most commonly injured structures around the back of the fibula.