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Deciding when to replace dairy bulls presents a complex challenge for artificial insemination (AI) companies. These decisions encompass multiple factors, including a bull’s age, predicted semen production, and estimated genetic merit. This study’s purpose was to provide a practical, objective tool to assist in these decisions. We utilized a Markov Chain model to calculate the economic valuation of dairy bulls, incorporating key factors such as housing costs, collection and marketing expenses, and the bull’s probable tenure in the herd. Data from a leading AI company were used to establish baseline values. The model further compared a bull’s net present value to that of a potential young replacement, establishing a relative valuation (BullVal $). The range of BullVal$observed spanned from −USD 316,748 to USD 497,710. Interestingly, the model recommended culling for 49% of the bulls based on negative BullVal$. It was found that a bull’s net present value was primarily influenced by market allocation and pricing, coupled with the interaction of semen production and genetic merit. This study offers a robust, data-driven model to guide bull replacement decisions in AI companies. Key determinants of a bull’s valuation included market dynamics, semen production rates, and genetic merit.

Although most of the genetic progress in production efficiency is achieved through selection at a global scale, locally, farm managers can also influence the selection process to better match genotypes and their varying herd environment. This study focused on the influence of a particular management decision-the use of extended lactation (EL) in dairy goat production systems- as it affects the survival and reproduction rates at the herd level, which may then shape different long-term selection responses. The objective was to understand and quantify the influences of EL and variability in achieved intake level on the responses to selection for production, reproduction, and longevity. An animal model of resource allocation between life functions was applied to the dairy goat. It predicts the trajectory of change in the herd genetic composition as affected by the feeding level and the selection pressure applied by the manager. During 40 yr, goats were selected for milk yield, reproduction, and, with a different selection weight for age (W-AGE), for longevity. Under varying achieved intake levels, increasing W-AGE improved the survival rate but a nonlinear effect was observed for the average milk yield and BCS. When moderately increasing W-AGE from 0, resources were reallocated from lactation towards body reserves and survival, which led to a trade-off at the herd level between improving survival and BCS and increasing milk yield. When further increasing W-AGE, old females became systematically preferred regardless of their reproductive status and the proportion of EL in the herd increased. Females undergoing EL had reduced energetic costs of reproduction, which improved their probability of survival. Across generations, an increased herd incidence of EL led to a relaxation of the selection pressure on the resource allocation to body reserves, which is normally imposed by the manager's priority to achieve successful reproduction at each mating. As selection for longevity progressed, the incidence of high-producing females increased within the herd, driving a long-term trend in increased milk production. Thus, the use of EL as a management tool led to an alleviation of the trade-off between milk yield progress and survival improvement. Although the model simplifies the underlying physiology of nutrient allocation, it provides insights into how farm manager strategies can influence the development of genotype x environment interactions and promote herd robustness.

Finding ways of increasing the efficiency of production systems is a key issue of sustainability. System efficiency is based on long-term individual efficiency, which is highly variable and management driven. To study the effects of management on herd and individual efficiency, we developed the model simulation of goat herd management (SIGHMA). This dynamic model is individual-based and represents the interactions between technical operations (relative to replacement, reproduction and feeding) and individual biological processes (performance dynamics based on energy partitioning and production potential). It simulates outputs at both herd and goat levels over 20 years. A farmer’s production project (i.e. a targeted milk production pattern) is represented by configuring the herd into female groups reflecting the organisation of kidding periods. Each group is managed by discrete events applying decision rules to simulate the carrying out of technical operations. The animal level is represented by a set of individual goat models. Each model simulates a goat’s biological dynamics through its productive life. It integrates the variability of biological responses driven by genetic scaling parameters (milk production potential and mature body weight), by the regulations of energy partitioning among physiological functions and by responses to diet energy defined by the feeding strategy. A sensitivity analysis shows that herd efficiency was mainly affected by feeding management and to a lesser extent by the herd production potential. The same effects were observed on herd milk feed costs with an even lower difference between production potential and feeding management. SIGHMA was used in a virtual experiment to observe the effects of feeding strategies on herd and individual performances. We found that overfeeding led to a herd production increase and a feed cost decrease. However, this apparent increase in efficiency at the herd level (as feed cost decreased) was related to goats that had directed energy towards body reserves. Such a process is not efficient as far as feed conversion is concerned. The underfeeding strategy led to production decrease and to a slight feed cost decrease. This apparent increase in efficiency was related to goats that had mobilised their reserves to sustain production. Our results highlight the interest of using SIGHMA to study the underlying processes affecting herd performance and analyse the role of individual variability regarding herd response to management. It opens perspectives to further quantify the link between individual variability, herd performance and management and thus further our understanding of livestock farming systems.

The rapid growth rate of COVID-19 continues to threaten to overwhelm healthcare systems in multiple countries. In response, severely affected countries have had to impose a range of public health strategies achieved via nonpharmaceutical interventions. Broadly, these strategies have fallen into two categories: 1) “mitigation,” which aims to achieve herd immunity by allowing the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus to spread through the population while mitigating disease burden, and 2) “suppression,” aiming to drastically reduce SARS-CoV-2 transmission rates and halt endogenous transmission in the target population. Using an age-structured transmission model, parameterized to simulate SARS-CoV-2 transmission in the United Kingdom, we assessed the long-term prospects of success using both of these approaches. We simulated a range of different nonpharmaceutical intervention scenarios incorporating social distancing applied to differing age groups. Our modeling confirmed that suppression of SARS-CoV-2 transmission is possible with plausible levels of social distancing over a period of months, consistent with observed trends. Notably, our modeling did not support achieving herd immunity as a practical objective, requiring an unlikely balancing of multiple poorly defined forces. Specifically, we found that 1) social distancing must initially reduce the transmission rate to within a narrow range, 2) to compensate for susceptible depletion, the extent of social distancing must be adaptive over time in a precise yet unfeasible way, and 3) social distancing must be maintained for an extended period to ensure the healthcare system is not overwhelmed.

I construct a dynamic social-network model of the COVID-19 epidemic which embeds the SIR epidemiological model onto a graph of person-to-person interactions. The standard SIR framework assumes uniform mixing of infectious persons in the population. This abstracts from important elements of realism and locality: (i) people are more likely to interact with members of their social networks and (ii) health and economic policies can affect differentially the rate of viral transmission via a person's social network vs. the population as a whole. The proposed network-augmented (NSIR) model allows the evaluation, via simulations, of (i) health and economic policies and outcomes for all or subset of the population: lockdown/distancing, herd immunity, testing, contact tracing; (ii) behavioral responses and/or imposing or lifting policies at specific times or conditional on observed states. I find that viral transmission over a network-connected population can proceed slower and reach lower peak than transmission via uniform mixing. Network connections introduce uncertainty and path dependence in the epidemic dynamics, with a significant role for bridge links and superspreaders. Testing and contact tracing are more effective in the network model. If lifted early, distancing policies mostly shift the infection peak into the future, with associated economic costs. Delayed or intermittent interventions or endogenous behavioral responses generate a multi-peaked infection curve, a form of 'curve flattening', but may have costlier economic consequences by prolonging the epidemic duration.

In this paper, we consider the infection-age-dependent Kermack-McKendrick model, where host individuals are distributed in a continuous state space. To provide a mathematical foundation for the heterogeneous model, we develop a -framework to formulate basic epidemiological concepts. First, we show the mathematical well-posedness of the basic model under appropriate conditions allowing for unbounded structural variables in an unbounded domain. Next, we define the basic reproduction number and prove pandemic threshold results. We then present a systematic procedure to compute the effective reproduction number and the herd immunity threshold. Finally, we give some illustrative examples and concrete results by using the separable mixing assumption.

The COVID-19 outbreak has highlighted our vulnerability to novel infections. Faced with this threat and no effective treatment, in line with many other countries, the UK adopted enforced social distancing (lockdown) to reduce transmission—successfully reducing the reproductive number R below one. However, given the large pool of susceptible individuals that remain, complete relaxation of controls is likely to generate a substantial further outbreak. Vaccination remains the only foreseeable means of both containing the infection and returning to normal interactions and behaviour. Here, we consider the optimal targeting of vaccination within the UK, with the aim of minimising future deaths or quality adjusted life year (QALY) losses. We show that, for a range of assumptions on the action and efficacy of the vaccine, targeting older age groups first is optimal and may be sufficient to stem the epidemic if the vaccine prevents transmission as well as disease.

Background More than half of the global population is under strict forms of social distancing. Estimating the expected impact of lockdown and exit strategies is critical to inform decision makers on the management of the COVID-19 health crisis. Methods We use a stochastic age-structured transmission model integrating data on age profile and social contacts in Île-de-France to (i) assess the epidemic in the region, (ii) evaluate the impact of lockdown, and (iii) propose possible exit strategies and estimate their effectiveness. The model is calibrated to hospital admission data before lockdown. Interventions are modeled by reconstructing the associated changes in the contact matrices and informed by mobility reductions during lockdown evaluated from mobile phone data. Different types and durations of social distancing are simulated, including progressive and targeted strategies, with large-scale testing. Results We estimate the reproductive number at 3.18 [3.09, 3.24] (95% confidence interval) prior to lockdown and at 0.68 [0.66, 0.69] during lockdown, thanks to an 81% reduction of the average number of contacts. Model predictions capture the disease dynamics during lockdown, showing the epidemic curve reaching ICU system capacity, largely strengthened during the emergency, and slowly decreasing. Results suggest that physical contacts outside households were largely avoided during lockdown. Lifting the lockdown with no exit strategy would lead to a second wave overwhelming the healthcare system, if conditions return to normal. Extensive case finding and isolation are required for social distancing strategies to gradually relax lockdown constraints. Conclusions As France experiences the first wave of COVID-19 pandemic in lockdown, intensive forms of social distancing are required in the upcoming months due to the currently low population immunity. Extensive case finding and isolation would allow the partial release of the socio-economic pressure caused by extreme measures, while avoiding healthcare demand exceeding capacity. Response planning needs to urgently prioritize the logistics and capacity for these interventions.

To address the urgent need to accurately predict the spreading trend of the COVID-19 epidemic, a continuous Markov-chain model was, for the first time, developed in this work to predict the spread of COVID-19 infection. A probability matrix of infection was first developed in this model based upon the contact frequency of individuals within the population, the individual’s characteristics, and other factors that can effectively reflect the epidemic’s temporal and spatial variation characteristics. The Markov-chain model was then extended to incorporate both the mutation effect of COVID-19 and the decaying effect of antibodies. The developed comprehensive Markov-chain model that integrates the aforementioned factors was finally tested by real data to predict the trend of the COVID-19 epidemic. The result shows that our model can effectively avoid the prediction dilemma that may exist with traditional ordinary differential equations model, such as the susceptible–infectious–recovered (SIR) model. Meanwhile, it can forecast the epidemic distribution and predict the epidemic hotspots geographically at different times. It is also demonstrated in our result that the influence of the population’s spatial and geographic distribution in a herd infection event is needed in the model for a better prediction of the epidemic trend. At the same time, our result indicates that no simple derivative relationship exists between the threshold of herd immunity and the virus basic reproduction number R0. The threshold of herd immunity achieved through natural immunity is significantly higher than 1 − 1/R0. These not only explain the theoretical misconceptions of herd immunity thresholds in herd immunity theory but also provide a guidance for predicting the optimal vaccination coverage. In addition, our model can predict the temporal and spatial distribution of infections in different epidemic waves. It is implied from our model that it is challenging to eradicate COVID-19 in the short term for a large population size and a wide spatial distribution. It is predicted that COVID-19 is likely to coexist with humans for a long time and that it will exhibit multipoint epidemic effects at a later stage. The statistical evidence is consistent with our prediction and strongly supports our modeling results.

We study a susceptible-exposed-infected-recovered (SEIR) model considered by Aguas et al. (In: Herd immunity thresholds for SARS-CoV-2 estimated from unfolding epidemics, 2021), Gomes et al. (In: J Theor Biol. 540:111063, 2022) where individuals are assumed to differ in their susceptibility or exposure to infection. Under this heterogeneity assumption, epidemic growth is effectively suppressed when the percentage of the population having acquired immunity surpasses a critical level - the herd immunity threshold - that is lower than in homogeneous populations. We derive explicit formulas to calculate herd immunity thresholds and stable configurations, especially when susceptibility or exposure are gamma distributed, and explore extensions of the model.