Estimating unreported cases of Covid-19 in a region or country presents a significant challenge, specially when they do not coincide with asymptomatic cases. This paper introduces a mathematical extension to deterministic models of total case growth, designed to describe the temporal evolution of unreported cases for each reported case of Covid-19. The proposed expression accounts for the growth of unreported cases in flat regions and their decline towards the end of an epidemic wave. We apply this extension to both a Gompertz growth model and a generalized Richards model, fitting these models to data from 30 countries/regions using Bayesian methods. The results show that fits from the extended models outperform those from the standalone Gompertz and generalized Richards models, as evidenced by superior values in the Bayesian Information Criterion (BIC) and R 2 metrics. In conclusion, the extended models offer a more accurate depiction of Covid-19 epidemic wave progression and provide short-term predictions of epidemic trends and criteria for identifying the end of an epidemic wave.

Environmental systems suffer from multiple interacting stressors. Each stressor can act on different parts of the system and at different time scales. This hampers measuring and predicting the stressors’ impacts on ecosystems. We propose a conceptual method that integrates available data with physical constraints over relevant time scales to predict management outcomes in data-scarce systems affected by multiple stressors. We first predict the combined stressor levels that threaten a management target and then define stressor equivalents to to convert between. These “ball-park” estimates of critical stressor levels help to identify how the threat posed by interacting stressors responds to its management. Our approach assists managers in the decision-making process regarding when to manage a system and how to monitor. We illustrate our concept with a case study of an invaded island ecosystem, yet our approach is useful for other data-poor environmental systems that suffer from multiple cumulative stressors.