Reconciling contrasting effects of nitrogen on host immunity and pathogen transmission using stoichiometric models.

Hosts rely on the availability of nutrients for growth, and for defense against pathogens. At the same time, changes in host nutrition can alter the dynamics of pathogens that rely on their host for reproduction. For primary producer hosts, enhanced nutrient loads may increase host biomass or pathogen reproduction, promoting faster density-dependent pathogen transmission. However, the effect of elevated nutrients may be reduced if hosts allocate a growth-limiting nutrient to pathogen defense. In canonical disease models, transmission is not a function of nutrient availability. Yet, including nutrient availability is necessary to mechanistically understand the response of infection to changes in the environment. Here, we explore the implications of nutrient-mediated pathogen infectivity and host immunity on infection outcomes. We developed a stoichiometric disease model that explicitly integrates the contrasting dependencies of pathogen infectivity and host immunity on nitrogen (N) and parameterized it for an algal-host system. Our findings reveal dynamic shifts in host biomass build-up, pathogen prevalence, and the force of infection along N supply gradients with N-mediated host infectivity and immunity, compared with a model in which the transmission rate was fixed. We show contrasting responses in pathogen performance with increasing N supply between N-mediated infectivity and N-mediated immunity, revealing an optimum for pathogen transmission at intermediate N supply. This was caused by N limitation of the pathogen at a low N supply and by pathogen suppression via enhanced host immunity at a high N supply. By integrating both nutrient-mediated pathogen infectivity and host immunity into a stoichiometric model, we provide a theoretical framework that is a first step in reconciling the contrasting role nutrients can have on host-pathogen dynamics.

Data driven mixed effects modeling of the dual process framework of addiction among individuals with alcohol use disorder.

Alcohol use disorder (AUD) comprises a continuum of symptoms and associated problems that has led AUD to be a leading cause of morbidity and mortality across the globe. Given the heterogeneity of AUD from mild to severe, consideration is being given to providing a spectrum of interventions that offer goal choice to match this heterogeneity, including helping individuals with AUD to moderate or control their drinking at low-risk levels. Because so much remains unknown about the factors that contribute to successful moderated drinking, we use dynamical systems modeling to identify mechanisms of behavior change. Daily alcohol consumption and daily desire (i.e., craving) are modeled using a system of delayed difference equations. Employing a mixed effects implementation of this system allows us to garner information about these mechanisms at both the population and individual levels. Use of this mixed effects framework first requires a parameter set reduction via identifiability analysis. The model calibration is then performed using Bayesian parameter estimation techniques. Finally, we demonstrate how conducting a parameter sensitivity analysis can assist in identifying optimal targets of intervention at the patient-specific level. This proof-of-concept analysis provides a foundation for future modeling to describe mechanisms of behavior change and determine potential treatment strategies in patients with AUD.

Incorporating the mutational landscape of SARS-COV-2 variants and case-dependent vaccination rates into epidemic models.

Coronavirus Disease (COVID-19), which began as a small outbreak in Wuhan, China, in December 2019, became a global pandemic within months due to its high transmissibility. In the absence of pharmaceutical treatment, various non-pharmaceutical interventions (NPIs) to contain the spread of COVID-19 brought the entire world to a halt. After almost a year of seemingly returning to normalcy with the world's quickest vaccine development, the emergence of more infectious and vaccine resistant coronavirus variants is bringing the situation back to where it was a year ago. In the light of this new situation, we conducted a study to portray the possible scenarios based on the three key factors: impact of interventions (pharmaceutical and NPIs), vaccination rate, and vaccine efficacy. In our study, we assessed two of the most crucial factors, transmissibility and vaccination rate, in order to reduce the spreading of COVID-19 in a simple but effective manner. In order to incorporate the time-varying mutational landscape of COVID-19 variants, we estimated a weighted transmissibility composed of the proportion of existing strains that naturally vary over time. Additionally, we consider time varying vaccination rates based on the number of daily new cases. Our method for calculating the vaccination rate from past active cases is an effective approach in forecasting probable future scenarios as it actively tracks people's attitudes toward immunization as active case changes. Our simulations show that if a large number of individuals cannot be vaccinated by ensuring high efficacy in a short period of time, adopting NPIs is the best approach to manage disease transmission with the emergence of new vaccine breakthrough and more infectious variants.

Effects of environmental variability on superspreading transmission events in stochastic epidemic models.

Superspreaders (individuals with a high propensity for disease spread) have played a pivotal role in recent emerging and re-emerging diseases. In disease outbreak studies, host heterogeneity based on demographic (e.g. age, sex, vaccination status) and environmental (e.g. climate, urban/rural residence, clinics) factors are critical for the spread of infectious diseases, such as Ebola and Middle East Respiratory Syndrome (MERS). Transmission rates can vary as demographic and environmental factors are altered naturally or due to modified behaviors in response to the implementation of public health strategies. In this work, we develop stochastic models to explore the effects of demographic and environmental variability on human-to-human disease transmission rates among superspreaders in the case of Ebola and MERS. We show that the addition of environmental variability results in reduced probability of outbreak occurrence, however the severity of outbreaks that do occur increases. These observations have implications for public health strategies that aim to control environmental variables.

Elements of disease in a changing world: modelling feedbacks between infectious disease and ecosystems.

An overlooked effect of ecosystem eutrophication is the potential to alter disease dynamics in primary producers, inducing disease-mediated feedbacks that alter net primary productivity and elemental recycling. Models in disease ecology rarely track organisms past death, yet death from infection can alter important ecosystem processes including elemental recycling rates and nutrient supply to living hosts. In contrast, models in ecosystem ecology rarely track disease dynamics, yet elemental nutrient pools (e.g. nitrogen, phosphorus) can regulate important disease processes including pathogen reproduction and transmission. Thus, both disease and ecosystem ecology stand to grow as fields by exploring questions that arise at their intersection. However, we currently lack a framework explicitly linking these disciplines. We developed a stoichiometric model using elemental currencies to track primary producer biomass (carbon) in vegetation and soil pools, and to track prevalence and the basic reproduction number (R0 ) of a directly transmitted pathogen. This model, parameterised for a deciduous forest, demonstrates that anthropogenic nutrient supply can interact with disease to qualitatively alter both ecosystem and disease dynamics. Using this element-focused approach, we identify knowledge gaps and generate predictions about the impact of anthropogenic nutrient supply rates on infectious disease and feedbacks to ecosystem carbon and nutrient cycling.

A Time Scales Approach for Modeling Intermittent Hormone Therapy for Prostate Cancer.

Prostate cancer is a common cancer among males in the USA and is often treated by intermittent androgen deprivation therapy. This therapy requires a patient to alternate between periods of androgen suppression treatment and no treatment. Prostate-specific antigen levels are used to track relative changes in tumor volume of prostate cancer patients undergoing intermittent androgen deprivation therapy. During this therapy, there is a pause between treatment cycles. Traditionally, continuous ordinary differential equations are used to estimate prostate-specific antigen levels. In this paper, we use dynamic equations to estimate prostate-specific antigen levels and construct a novel time scale model to account for both continuous and discrete time simultaneously. This allows us to account for breaks between treatment cycles. Using empirical data sets of prostate-specific antigen levels, a known bio-marker of prostate cancer, across multiple patients, we fit our model and use least squares to estimate two parameter values. We then compare our model to the data and find a resemblance on treatment intervals similar to our time scale.

Orchard layout and plant traits influence fruit yield more strongly than pollinator behaviour and density in a dioecious crop.

Mutualistic plant-pollinator interactions are critical for the functioning of both non-managed and agricultural systems. Mathematical models of plant-pollinator interactions can help understand key determinants in pollination success. However, most previous models have not addressed pollinator behavior and plant biology combined. Information generated from such a model can inform optimal design of crop orchards and effective utilization of managed pollinators like western honey bees (Apis mellifera), and help generate hypotheses about the effects of management practices and cultivar selection. We expect that the number of honey bees per flower and male to female flower ratio will influence fruit yield. To test the relative importance of these effects, both singly and simultaneously, we utilized a delay differential equation model combined with Latin hypercube sampling for sensitivity analysis. Empirical data obtained from historical records and collected in kiwifruit (Actinidia chinensis) orchards in New Zealand were used to parameterize the model. We found that, at realistic bee densities, the optimal orchard had 65-75% female flowers, and the most benefit was gained from the first 6-8 bees/1000 flowers, with diminishing returns thereafter. While bee density significantly impacted fruit production, plant-based parameters-flower density and male:female flower ratio-were the most influential. The predictive model provides strategies for improving crop management, such as choosing cultivars which have their peak bloom on the same day, increasing the number of flowers with approximately 70% female flowers in the orchard, and placing enough hives to maintain more than 6 bees per 1000 flowers to optimize yield.

Nitrate-N-mediated toxicological responses of Scenedesmus acutus and Daphnia pulex to cadmium, arsenic and their binary mixture (Cd/Asmix) at environmentally relevant concentrations.

Several biomarkers used for ecological risk assessment have been established for single contaminant toxicity, many of which are less predictive of the influence of media and/or dietary nutrients on toxicity outcomes of contaminant mixtures. In this study, we investigate toxicological responses and life traits of Scenedesmus acutus and Daphnia pulex to heavy metals (cadmium-Cd, arsenic-As, binary mixture-Cd/Asmix) in media and diets with varied nutrient (nitrate-N) conditions (low-LN, median-MN, optimum-COMBO). Results showed that nitrate-N-mediated metal inhibitory effects on growth and productivity of primary producer (S. acutus) were significantly interactive (p < 0.05; effect size, ƞ2≤56 %). Cadmium toxicities (Cd-IC50s) in S. acutus were 1.2×, 5.3×, and 4.3× As-IC50s in LN, MN and COMBO media, respectively, while mixture (Cd/Asmix) toxicities were synergistic in MN medium and partial additivity in COMBO and LN media. Nitrate-N and metal exposure effects on S. acutus nutrient stoichiometry, metal uptake and bioaccumulation were significantly interactive (p < 0.05, ƞ2≤100 %). Moreover, survival of primary consumer (D. pulex) was significantly impaired by single and mixed dietary-metal exposures with greater effect under LN condition coupled with significant interactive effects on reproductive capacity (p < 0.05, ƞ2≤21.2 %) but not on swimming activity. We recommend that nitrate-N-mediated metal exposure effects/toxicity in bioindicator species should be considered during ecological risk assessments.

Disease-mediated ecosystem services: Pathogens, plants, and people.

Despite the ubiquity of pathogens in ecological systems, their roles in influencing ecosystem services are often overlooked. Pathogens that infect primary producers (i.e., plants, algae, cyanobacteria) can have particularly strong effects because autotrophs are responsible for a wide range of provisioning, regulating, and cultural services. We review the roles of pathogens in mediating ecosystem services provided by autotrophs and outline scenarios in which infection may lead to unexpected outcomes in response to global change. Our synthesis highlights a deficit of information on this topic, and we outline a vision for future research that includes integrative theory and cross-system empirical studies. Ultimately, knowledge about the mediating roles of pathogens on ecosystem services should inform environmental policy and practice.

Trophic-Level Interactive Effects of Phosphorus Availability on the Toxicities of Cadmium, Arsenic, and Their Binary Mixture in Media-Exposed Scenedesmus acutus and Media and Dietary-Exposed Daphnia pulex.

Various anthropogenic activities simultaneously alter essential mineral nutrients and contaminant content in the environment. Depending on essential nutrient conditions, the uptake and effects of contaminants in exposed organisms may be altered. The addressing of ecological risk assessment (ERA) of contaminant mixtures has proven difficult. Furthermore, most assessments involving single contaminant exposures do not consider the interaction of essential nutrients on toxicological end points. Hypotheses for toxicological effects of cadmium (Cd), arsenic (As), and their binary mixture (Cd/Asmix) include alteration under varying dietary and media phosphorus (P) conditions. However, interactive effects and effect size (η2) are largely unknown. Here, we investigated the toxicities of Cd-, As-, and Cd/Asmix-treated media and diets on Scenedesmus acutus (a primary producer) and Daphnia pulex (a primary consumer), under varied media and dietary P conditions [low (LP), median (MP), and optimum (COMBO)]. Our results showed significant (p < 0.05) interactive effects and concentration dependent growth inhibition of S. acutus. The toxicity (at day 7) of Cd against S. acutus was 2×, 11×, and 4× that of As in LP, MP, and COMBO conditions, respectively, while the joint toxicity effects of Cd/Asmix were partially additive in LP and COMBO, and synergistic in MP media. Furthermore, acute lethal toxicity (96 h) of Cd in D. pulex was ∼60× that of As, while Cd/Asmix joint toxicity was synergistic. Chronic toxicity (14 d) in D. pulex showed significant (p < 0.05) interaction of As and P-availability on survival, reproduction, and behavior (distance moved, velocity, acceleration and mobility), while Cd and P availability showed significant interactive effect on rotational behavior. Dose response effects of Cd, As, and Cd/Asmix in S. acutus and D. pulex were either monophasic or biphasic under varying nutrient conditions. This study provides empirical evidence of the interactive effects of media/dietary P and toxic metals (Cd, As, and Cd/Asmix) at environmentally relevant concentrations, emphasizing the need for consideration of such interactions during ERA.

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles.

In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations (FDEs) as approximations of some type of fractional nonlinear birth and death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While ordinary differential equations (ODEs) are commonly used to model epidemics, FDEs are more flexible in fitting empirical data and theoretically offer improved model predictions. The question arises whether, in practice, the benefits of using FDEs over ODEs outweigh the added computational complexities. While important differences in transient dynamics were observed, the FDE only outperformed the ODE in one of out three data sets. In general, FDE modeling approaches may be worth it in situations with large refined data sets and good numerical algorithms.

The effects of excess food nutrient content on a tritrophic food chain model in the aquatic ecosystem.

Ecological stoichiometry is an approach that focuses on the balance of energy and elements in environmental interactions, and it leads to new insights and a better understanding of ecological processes and outcomes. Modeling under this framework enables us to investigate the effects of nutrient content (i.e., food quality) on organisms, whether the imbalance involves insufficient or excess nutrient content. In this paper, we develop and analyze a tritrophic food chain model that captures the phenomenon known as the "stoichiometric knife-edge", where consumer growth is limited under conditions of excess nutrients. The model tracks two essential elements, carbon and phosphorus, in each species. The dynamics of the system such as boundedness and positivity of the solutions, existence and stability conditions of boundary and internal equilibria are analyzed. Through numerical simulations and bifurcation analyses, we observe the dynamics of the system switching between periodic oscillations and chaos. Our findings also show that nutrient-rich food consumption can cause adverse effects on species.

Mechanistically derived Toxicant-mediated predator-prey model under Stoichiometric constraints.

Studies in ecological stoichiometry highlight that grazer dynamics are affected by insufficient food nutrient content (low phosphorus (P)/carbon (C) ratio) as well as excess food nutrient content (high P:C). Contaminant stressors affect all levels of the biological hierarchy, from cells to organs to organisms to populations to entire ecosystems. Eco-toxicological modeling under the framework of ecological stoichiometry predicts the risk of bio-accumulation of a toxicant under stoichiometric constraints. In this paper, we developed and analyzed a Lotka-Volterra type predator- prey model which explicitly tracks the environmental toxicant as well as the toxicant in the populations under stoichiometric constraints. Analytic, numerical, slow-fast steady state and bifurcation theory are employed to predict the risk of toxicant bio-accumulation under varying food conditions. In some cases, our model predicts different population dynamics, including wide amplitude limit cycles where producer densities exhibit very low values and may be in danger of stochastic extinction.

Compensatory Foraging in Stoichiometric Producer-Grazer Models.

Nutritional constraints are common as food resources are rarely optimally suited for grazing species. Elemental mismatches between trophic levels can influence population growth and foraging behaviors. Grazing species, such as Daphnia, utilize optimal foraging techniques, such as compensatory feeding. Here, we develop two stoichiometric producer-grazer models, a base model that incorporates a fixed energetic foraging cost and an optimal foraging model where energetic foraging costs depend on food nutritional content. A variable energetic foraging cost results in cell quota-dependent predation behaviors. Analyzing and comparing these two models allows us to investigate the potential benefits of stoichiometric compensatory foraging behaviors on grazer populations. Optimal foraging strategies depend on environmental conditions, such as light and nutrient availability. In low-light conditions, fixed energetic foraging appears optimal regardless of the nutrient loads. However, in higher light conditions and intermediate nutrient loads, grazers utilizing compensatory foraging strategies gain an advantage. Overall, grazers can benefit from compensatory feeding behaviors when the food nutrient content of their prey becomes low or high.

Environmental seasonality on predator-prey systems under nutrient and toxicant constraints.

Bioaccumulation of toxicants in aquatic food webs can pose risks to ecosystem function and human health. Toxicant models of aquatic ecosystems can be improved by incorporating realistic environmental impacts such as nutrient availability and seasonality. It is well known that the carrying capacity of predator-prey systems can vary seasonally due to environmental cycles resulting from natural and human activities. As such, incorporating seasonal variation in the carrying capacity of a predator-prey system provides a better understanding of the underlying population dynamics bioaccumulation of toxicants. Here, we develop a seasonally varied predator-prey model subject to concurrent nutrient and toxicant stressors. We investigate the effects of seasonality on population dynamics to increase understanding of the complex governing processes of the trophic transfer of nutrients, energy, and toxicants. We observe that the strength of seasonality can shift solutions from periodic to quasi-periodic and models that neglect environmental seasonality may be under-predicting adverse effects of toxicity.

Seasonal Variation of Nutrient Loading in a Stoichiometric Producer-Consumer System.

Recent discoveries in ecological stoichiometry have indicated that food quality in terms of the phosphorus/carbon (P/C) ratio affects consumers whether the imbalance involves insufficient or excess nutrients. This phenomenon is called the "stoichiometric P/C knife-edge." In this study, we develop and analyze a producer-consumer model which captures this phenomenon. It assesses the effects of (external) nutrient (P) loading on consumer dynamics in an aquatic environment by mechanistically deriving and accounting for seasonal variation in nutrient loading. In the absence of seasonal effects, previous models suggest that the dynamics are Hopf bifurcation, saddle-node bifurcations, and limit cycles. However, seasonal effects can have major implications on the predicted solutions and enrich population dynamics. Bifurcation analyses demonstrate that seasonal forcing can cause both periodic and quasi-periodic solutions.

Modeling Approach Influences Dynamics of a Vector-Borne Pathogen System.

The choice of a modeling approach is a critical decision in the modeling process, as it determines the complexity of the model and the phenomena that the model captures. In this paper, we developed an individual-based model (IBM) and compared it to a previously published ordinary differential equation (ODE) model, both developed to describe the same biological system although with slightly different emphases given the underlying assumptions and processes of each modeling approach. We used both models to examine the effect of insect vector life history and behavior traits on the spread of a vector-borne plant virus, and determine how choice of approach affects the results and their biological interpretation. A non-random distribution of insect vectors across plant hosts emerged in the IBM version of the model and was not captured by the ODE. This distribution led simultaneously to a slower-growing vector population and a faster spread of the pathogen among hosts. The IBM model also enabled us to test the effect of potential control measures to slow down virus transmission. We found that removing virus-infected hosts was a more effective strategy for controlling infection than removing vector-infested hosts. Our findings highlight the need to carefully consider possible modeling approaches before constructing a model.

Special issue: Resource explicit population models.

Preface. Guest editors: Yang Kuang, Angela Peace and Hao Wang.

Genotypic Selection in Spatially Heterogeneous Producer-Grazer Systems Subject to Stoichiometric Constraints.

Various environmental conditions may exert selection pressures leading to adaptation of stoichiometrically important traits, such as organismal nutritional content or growth rate. We use theoretical approaches to explore the connections between genotypic selection and ecological stoichiometry in spatially heterogeneous environments. We present models of a producer and two grazing genotypes with different stoichiometric phosphorus/carbon ratios under spatially homogenous and heterogeneous conditions. Numerical experiments predict that selection of a single genotype, co-persistence of both genotypes, and extinction are possible outcomes depending on environmental conditions. Our results indicated that in spatially homogenous settings, co-persistence of both genotypes can occur when population dynamics oscillate on limit cycles near a key stoichiometric threshold on food quality. Under spatially heterogeneous settings, dynamics are more complex, where co-persistence is observed on limit cycles, as well as stable equilibria.

Dynamics of a Producer-Grazer Model Incorporating the Effects of Phosphorus Loading on Grazer's Growth.

Phosphorus is an essential element for all life forms, and it is also a limiting nutrient in many aquatic ecosystems. To keep track of the mismatch between the grazer's phosphorus requirement and producer phosphorus content, stoichiometric models have been developed to explicitly incorporate food quality and food quantity. Most stoichiometric models have suggested that the grazer dynamics heavily depends on the producer phosphorus content when the producer has insufficient nutrient content [low phosphorus (P):carbon (C) ratio]. However, recent laboratory experiments have shown that the grazer dynamics are also affected by excess producer nutrient content (extremely high P:C ratio). This phenomenon is known as the "stoichiometric knife edge." While the Peace et al. (Bull Math Biol 76(9):2175-2197, 2014) model has captured this phenomenon, it does not explicitly track P loading of the aquatic environment. Here, we extend the Peace et al. (2014) model by mechanistically deriving and tracking P loading in order to investigate the growth response of the grazer to the producer of varying P:C ratios. We analyze the dynamics of the system such as boundedness and positivity of the solutions, existence and stability conditions of boundary equilibria. Bifurcation diagram and simulations show that our model behaves qualitatively similar to the Peace et al. (2014) model. The model shows that the fate of the grazer population can be very sensitive to P loading. Furthermore, the structure of our model can easily be extended to incorporate seasonal P loading.