Impact of ontogenic changes on the dynamics of a fungal crop disease model motivated by coffee leaf rust.

Ontogenic resistance has been described for many plant-pathogen systems. Conversely, coffee leaf rust, a major fungal disease that drastically reduces coffee production, exhibits a form of ontogenic susceptibility, with a higher infection risk for mature leaves. To take into account stage-dependent crop response to phytopathogenic fungi, we developed an SEIR-U epidemiological model, where U stands for spores, which differentiates between young and mature leaves. Based on this model, we also explored the impact of ontogenic resistance on the sporulation rate. We computed the basic reproduction number [Formula: see text], which classically determines the stability of the disease-free equilibrium. We identified forward and backward bifurcation cases. The backward bifurcation is generated by the high sporulation of young leaves compared to mature ones. In this case, when the basic reproduction number is less than one, the disease can persist. These results provide useful insights on the disease dynamics and its control. In particular, ontogenic resistance may require higher control efforts to eradicate the disease.

Improving sustainable crop protection using population genetics concepts.

Growing genetically resistant plants allows pathogen populations to be controlled and reduces the use of pesticides. However, pathogens can quickly overcome such resistance. In this context, how can we achieve sustainable crop protection? This crucial question has remained largely unanswered despite decades of intense debate and research effort. In this study, we used a bibliographic analysis to show that the research field of resistance durability has evolved into three subfields: (1) "plant breeding" (generating new genetic material), (2) "molecular interactions" (exploring the molecular dialogue governing plant-pathogen interactions) and (3) "epidemiology and evolution" (explaining and forecasting of pathogen population dynamics resulting from selection pressure[s] exerted by resistant plants). We argue that this triple split of the field impedes integrated research progress and ultimately compromises the sustainable management of genetic resistance. After identifying a gap among the three subfields, we argue that the theoretical framework of population genetics could bridge this gap. Indeed, population genetics formally explains the evolution of all heritable traits, and allows genetic changes to be tracked along with variation in population dynamics. This provides an integrated view of pathogen adaptation, in particular via evolutionary-epidemiological feedbacks. In this Opinion Note, we detail examples illustrating how such a framework can better inform best practices for developing and managing genetically resistant cultivars.

A thermal trade-off between viral production and degradation drives virus-phytoplankton population dynamics.

Marine viruses interact with microbial hosts in dynamic environments shaped by variation in abiotic factors, including temperature. However, the impacts of temperature on viral infection of phytoplankton are not well understood. Here we coupled mathematical modelling with experiments to explore the effect of temperature on virus-phytoplankton interactions. Our model shows the negative consequences of high temperatures on infection and suggests a temperature-dependent threshold between viral production and degradation. Modelling long-term dynamics in environments with different average temperatures revealed the potential for long-term host-virus coexistence, epidemic free or habitat loss states. We generalised our model to variation in global sea surface temperatures corresponding to present and future seas and show that climate change may differentially influence virus-host dynamics depending on the virus-host pair. Temperature-dependent changes in the infectivity of virus particles may lead to shifts in virus-host habitats in warmer oceans, analogous to projected changes in the habitats of macro-, microorganisms and pathogens.

Self-financing model for cabbage crops with pest management.

Smallholder farmers rely on their farm earnings to cover operating costs and generate income. That is not an easy task because of the pests, which reduce yields and generate plant protection costs. The farm yield and plant protection depend on the budget capacity of the farmer. In this work, we want to explore conditions for a sustainable and self-financing cabbage farm. We propose then a non-linear mathematical model for cabbage crops by considering the current account of the plantation as a dynamic variable. We assume that this variable increases due to the sale of cabbages, and provides for the seedling purchase, the plant protection costs, and the grower's income. In the first part, we analyze the model without pest management. We determine how the budget must be spent and we show the existence of a double transcritical bifurcation. We quantify the seasonal yield and income, and estimate the damage due to pest herbivory. In the second part, we analyze a slightly simplified version of our model and obtain the existence of a backward bifurcation. Furthermore, we show that botanical pesticides can be used to prevent pest spread with relatively low plant protection costs.

Impulsive modelling of rust dynamics and predator releases for biocontrol.

Fungal diseases cause serious damages in crop worldwide. In particular, coffee leaf rust (CLR), caused by fungus Hemileia vastatrix attacks coffee leaves and reduces coffee yield. This paper presents a multi-seasonal model of the CLR development in the coffee plantation with continuous dynamics during the rainy season and a discrete event to represent the simpler dynamics during the dry season. Biological control using predators through one or more discrete introduction events over the year is then added. Analytical and semi-numerical studies are performed to identify how much and how frequently predators need to be introduced through the definition of a threshold value, as a function of various parameters. We show that the best strategy to efficiently control the disease depends on the predator mortality: low mortality parasites need be released only once a year, while high mortality parasites should be released more frequently to ensure their persistence in the plantation. This work hence provides qualitative and quantitative bases for the deployment of predator-based biocontrol, a promising alternative to fungicides for rust control.

Modelling and control of a banana soilborne pest in a multi-seasonal framework.

We study the infestation dynamics of banana or plantain plants by Radopholus similis, a plant-parasitic nematode that causes severe damages. Two control strategies are implemented in our model: pesticides, which are widely used, and fallows, which are more environmentally friendly. To represent the host-parasite dynamics, two semi-discrete models are proposed. During each cropping season, free nematodes enter the plant roots, on which they feed and reproduce. At the end of the cropping season, fruits are harvested. In the first model, the parent plant is cut down to be replaced by one of its suckers and pesticides are applied. In the second model, the parent plant is uprooted and a fallow period is introduced, inducing the decay of the free pest populations; at the beginning of the next cropping season, a pest-free vitroplant is planted. For both models, the effective reproduction number of pests is computed, assuming that the infestation dynamics are fast compared to the other processes, which leads to the model order reduction. Conditions on the pesticide load or the fallow duration are then derived to ensure the stability of the periodic pest free solution. Finally, numerical simulations illustrate these theoretical results.

Multi-seasonal modelling of plant-nematode interactions reveals efficient plant resistance deployment strategies.

Root-knot nematodes, Meloidogyne spp., are soil-borne polyphagous pests with major impact on crop yield worldwide. Resistant crops efficiently control avirulent root-knot nematodes, but favour the emergence of virulent forms. Since virulence is associated with fitness costs, susceptible crops counter-select virulent root-knot nematodes. In this study, we identify optimal rotation strategies between susceptible and resistant crops to control root-knot nematodes and maximize crop yield. We developed an epidemiological model describing the within-season dynamics of avirulent and virulent root-knot nematodes on susceptible or resistant plant root-systems, and their between-season survival. The model was fitted to experimental data and used to predict yield-maximizing rotation strategies, with special attention to the impact of epidemic severity and genetic parameters. Crop rotations were found to be efficient under realistic parameter ranges. They were characterized by low ratios of resistant plants and were robust to parameter uncertainty. Rotations provide significant gain over resistant-only strategies, especially under intermediate fitness costs and severe epidemic contexts. Switching from the current general deployment of resistant crops to custom rotation strategies could not only maintain or increase crop yield, but also preserve the few and valuable R-genes available.

Sensitivity analysis to identify key parameters influencing Salmonella infection dynamics in a pig batch.

In the context of managed herds, epidemiological models usually take into account relatively complex interactions involving a high number of parameters. Some parameters may be uncertain and/or highly variable, especially epidemiological parameters. Their impact on the model outputs must then be assessed by a sensitivity analysis, allowing to identify key parameters. The prevalence over time is an output of particular interest in epidemiological models, so sensitivity analysis methods adapted to such dynamic output are needed. In this paper, such a sensitivity analysis method, based on a principal component analysis and on analysis of variance, is presented. It allows to compute a generalised sensitivity index for each parameter of a model representing Salmonella spread within a pig batch. The model is a stochastic discrete-time model describing the batch dynamics and movements between rearing rooms, from birth to slaughterhouse delivery. Four health states were introduced: Salmonella-free, seronegative shedder, seropositive shedder and seropositive carrier. The indirect transmission was modelled via an infection probability function depending on the quantity of Salmonella in the rearing room. Simulations were run according to a fractional factorial design enabling the estimation of main effects and two-factor interactions. For each of the 18 epidemiological parameters, four values were chosen, leading to 4096 scenarios. For each scenario, 15 replications were performed, leading to 61440 simulations. The sensitivity analysis was then conducted on the seroprevalence output. The parameters governing the infection probability function and residual room contaminations were identified as key parameters. To control the Salmonella seroprevalence, efficient measures should therefore aim at these parameters. Moreover, the shedding rate and maternal protective factor also had a major impact. Therefore, further investigation on the protective effect of maternal or post-infection antibodies would be needed.

Using singular perturbations to reduce an epidemiological model: application to bovine viral diarrhoea virus within-herd spread.

Studying the spread of a pathogen in a managed metapopulation such as cattle herds in a geographical region often requires to take into account both the within- and between-herd transmission dynamics. This can lead to high-dimensional metapopulation systems resulting from the coupling of several within-herd transmission models. To tackle this problem, we aim in this paper at reducing the dimension of a tractable but realistic dynamical system reproducing the within-herd spread. The context chosen to illustrate our purpose is bovine viral diarrhoea virus (BVDV) transmission in a cattle herd structured in two age classes and several epidemiological states, including two infectious states (transiently and persistently infected). Different time scales, corresponding to the epidemiological and demographic processes, are identified which allow to build a reduced model. Singular perturbation technique is used to prove that, under some non-restrictive conditions on parameter values, the behaviour of the original system is quite accurately approximated by that of the reduced system. Simulations are also performed to corroborate the approximation quality. Our study illustrates the methodological interest of using singular perturbations to reduce model complexity. It also rigorously proves the biologically intuitive assumption that transiently infected individuals can be neglected in a homogeneous population, when capturing the global dynamics of BVDV spread.

Why, when and how should exposure be considered at the within-host scale? A modelling contribution to PRRSv infection.

Understanding the impact of pathogen exposure on the within-host dynamics and its outcome in terms of infectiousness is a key issue to better understand and control the infection spread. Most experimental and modelling studies tackling this issue looked at the impact of the exposure dose on the infection probability and pathogen load, very few on the within-host immune response. Our aim was to explore the impact on the within-host response not only of the exposure dose, but also of its duration and peak, for contrasted virulence levels. We used an integrative modelling approach of the within-host dynamics at the between-cell level. We focused on the porcine reproductive and respiratory syndrome virus, a major concern for the swine industry. We quantified the impact of exposure and virulence on the viral dynamics and immune response by global sensitivity analyses and descriptive statistics. We found that the area under the viral curve, an indicator of the infection severity, was fully determined by the exposure intensity. The infection duration increased with the strain virulence and, for a given strain, exhibited a positive linear correlation with the exposure intensity logarithm and the exposure duration. Taking into account the exposure intensity is hence necessary. Besides, representing the exposure due to contacts by a single punctual dose would tend to underestimate the infection duration. As the infection severity and duration both contribute to the pig infectiousness, a prolonged exposure of the adequate intensity would be recommended in an immuno-epidemiological context.

How to prevent viremia rebound? Evidence from a PRRSv data-supported model of immune response.

BACKGROUND: Understanding what determines the between-host variability in infection dynamics is a key issue to better control the infection spread. In particular, pathogen clearance is desirable over rebounds for the health of the infected individual and its contact group. In this context, the Porcine Respiratory and Reproductive Syndrome virus (PRRSv) is of particular interest. Numerous studies have shown that pigs similarly infected with this highly ubiquitous virus elicit diverse response profiles. Whilst some manage to clear the virus within a few weeks, others experience prolonged infection with a rebound. Despite much speculation, the underlying mechanisms responsible for this undesirable rebound phenomenon remain unclear. RESULTS: We aimed at identifying immune mechanisms that can reproduce and explain the rebound patterns observed in PRRSv infection using a mathematical modelling approach of the within-host dynamics. As diverse mechanisms were found to influence PRRSv infection, we established a model that details the major mechanisms and their regulations at the between-cell scale. We developed an ABC-like optimisation method to fit our model to an extensive set of experimental data, consisting of non-rebounder and rebounder viremia profiles. We compared, between both profiles, the estimated parameter values, the resulting immune dynamics and the efficacies of the underlying immune mechanisms. Exploring the influence of these mechanisms, we showed that rebound was promoted by high apoptosis, high cell infection and low cytolysis by Cytotoxic T Lymphocytes, while increasing neutralisation was very efficient to prevent rebounds. CONCLUSIONS: Our paper provides an original model of the immune response and an appropriate systematic fitting method, whose interest extends beyond PRRS infection. It gives the first mechanistic explanation for emergence of rebounds during PRRSv infection. Moreover, results suggest that vaccines or genetic selection promoting strong neutralising and cytolytic responses, ideally associated with low apoptotic activity and cell permissiveness, would prevent rebound.

Using a climate-dependent model to predict mosquito abundance: application to Aedes (Stegomyia) africanus and Aedes (Diceromyia) furcifer (Diptera: Culicidae).

Mosquitoes, acting as vectors, are involved in the transmission of viruses. Thus, their abundances, which strongly depend on the weather and environment, are closely linked to major disease outbreaks. The aim of this paper is to provide a tool to predict vector abundance. In order to describe the dynamics of mosquito populations, we developed a matrix model integrating climate fluctuations. The population is structured in five stages: two egg stages (immature and mature), one larval stage and two female flying stages (nulliparous and parous). The water availability in breeding sites was considered as the main environmental factor affecting the mosquito life-cycle. Thus, the model represents the evolution of the mosquito abundance in each stage over time, in connection with water availability. The model was used to simulate the abundance trends over 3 years of two mosquito species, Aedes africanus (Theobald) and Aedes furcifer (Edwards), vectors of the yellow fever virus in Ivory Coast. As both these species breed in tree holes, the water dynamics in the tree hole was reproduced from daily rainfall data. The results we obtained showed a good match between the simulated populations and the field data over the time period considered.

Principal process analysis of biological models.

BACKGROUND: Understanding the dynamical behaviour of biological systems is challenged by their large number of components and interactions. While efforts have been made in this direction to reduce model complexity, they often prove insufficient to grasp which and when model processes play a crucial role. Answering these questions is fundamental to unravel the functioning of living organisms. RESULTS: We design a method for dealing with model complexity, based on the analysis of dynamical models by means of Principal Process Analysis. We apply the method to a well-known model of circadian rhythms in mammals. The knowledge of the system trajectories allows us to decompose the system dynamics into processes that are active or inactive with respect to a certain threshold value. Process activities are graphically represented by Boolean and Dynamical Process Maps. We detect model processes that are always inactive, or inactive on some time interval. Eliminating these processes reduces the complex dynamics of the original model to the much simpler dynamics of the core processes, in a succession of sub-models that are easier to analyse. We quantify by means of global relative errors the extent to which the simplified models reproduce the main features of the original system dynamics and apply global sensitivity analysis to test the influence of model parameters on the errors. CONCLUSION: The results obtained prove the robustness of the method. The analysis of the sub-model dynamics allows us to identify the source of circadian oscillations. We find that the negative feedback loop involving proteins PER, CRY, CLOCK-BMAL1 is the main oscillator, in agreement with previous modelling and experimental studies. In conclusion, Principal Process Analysis is a simple-to-use method, which constitutes an additional and useful tool for analysing the complex dynamical behaviour of biological systems.

Modelling Salmonella transmission among pigs from farm to slaughterhouse: Interplay between management variability and epidemiological uncertainty.

Salmonella carriage and cutaneous contamination of pigs at slaughter are a major risk for carcass contamination. They depend on Salmonella prevalence at farm, but also on transmission and skin soiling among pigs during their journey from farm to slaughterhouse. To better understand and potentially control what influences Salmonella transmission within a pig batch during this transport and lairage step, we proposed a compartmental, discrete-time and stochastic model. We calibrated the model using pork chain data from Brittany. We carried out a sensitivity analysis to evaluate the impact of the variability in management protocols and of the uncertainty in epidemiological parameters on three model outcomes: prevalence of infection, average cutaneous contamination and number of new infections at slaughter. Each outcome is mainly influenced by a single management factor: prevalence at slaughter mainly depends on the prevalence at farm, cutaneous contamination on the contamination of lairage pens and new infections on the total duration of transport and lairage. However, these results are strongly affected by the uncertainty in epidemiological parameters. Re-excretion of carriers due to stress does not have a major impact on the number of new infections.

Integrative model of the immune response to a pulmonary macrophage infection: what determines the infection duration?

The immune mechanisms which determine the infection duration induced by pathogens targeting pulmonary macrophages are poorly known. To explore the impact of such pathogens, it is indispensable to integrate the various immune mechanisms and to take into account the variability in pathogen virulence and host susceptibility. In this context, mathematical models complement experimentation and are powerful tools to represent and explore the complex mechanisms involved in the infection and immune dynamics. We developed an original mathematical model in which we detailed the interactions between the macrophages and the pathogen, the orientation of the adaptive response and the cytokine regulations. We applied our model to the Porcine Respiratory and Reproductive Syndrome virus (PRRSv), a major concern for the swine industry. We extracted value ranges for the model parameters from modelling and experimental studies on respiratory pathogens. We identified the most influential parameters through a sensitivity analysis. We defined a parameter set, the reference scenario, resulting in a realistic and representative immune response to PRRSv infection. We then defined scenarios corresponding to graduated levels of strain virulence and host susceptibility around the reference scenario. We observed that high levels of antiviral cytokines and a dominant cellular response were associated with either short, the usual assumption, or long infection durations, depending on the immune mechanisms involved. To identify these mechanisms, we need to combine the levels of antiviral cytokines, including IFNγ, and IL10. The latter is a good indicator of the infected macrophage level, both combined provide the adaptive response orientation. Available PRRSv vaccines lack efficiency. By integrating the main interactions between the complex immune mechanisms, this modelling framework could be used to help designing more efficient vaccination strategies.

Pathogen population dynamics in agricultural landscapes: the Ddal modelling framework.

Modelling processes that occur at the landscape scale is gaining more and more attention from theoretical ecologists to agricultural managers. Most of the approaches found in the literature lack applicability for managers or, on the opposite, lack a sound theoretical basis. Based on the metapopulation concept, we propose here a modelling approach for landscape epidemiology that takes advantage of theoretical results developed in the metapopulation context while considering realistic landscapes structures. A landscape simulator makes it possible to represent both the field pattern and the spatial distribution of crops. The pathogen population dynamics are then described through a matrix population model both stage- and space-structured. In addition to a classical invasion analysis we present a stochastic simulation experiment and provide a complete framework for performing a sensitivity analysis integrating the landscape as an input factor. We illustrate our approach using an example to evaluate whether the agricultural landscape composition and structure may prevent and mitigate the development of an epidemic. Although designed for a fungal foliar disease, our modelling approach is easily adaptable to other organisms.

The role of mathematical modelling in understanding the epidemiology and control of sheep transmissible spongiform encephalopathies: a review.

To deal with the incompleteness of observations and disentangle the complexities of transmission much use has been made of mathematical modelling when investigating the epidemiology of sheep transmissible spongiform encephalopathies (TSE) and, in particular, scrapie. Importantly, these modelling approaches allow the incidence of clinical disease to be related to the underlying prevalence of infection, thereby overcoming one of the major difficulties when studying these diseases. Models have been used to investigate the epidemiology of scrapie within individual flocks and at a regional level; to assess the efficacy of different control strategies, especially selective breeding programmes based on prion protein (PrP) genotype; to interpret the results of scrapie surveillance; and to inform the design of surveillance programmes. Furthermore, mathematical modelling has played an important role when assessing the risk to human health posed by the possible presence of bovine spongiform encephalopathy in sheep. Here, we review the various approaches that have been taken when developing and analysing mathematical models for the epidemiology and control of sheep TSE and assess their impact on our understanding of these diseases. We also identify areas that require further work, discuss future challenges and identify data gaps.

Modelling Salmonella spread within a farrow-to-finish pig herd.

Delivery of infected pigs to the slaughterhouse is a major source of pork meat contamination by bacterial hazards to humans. We propose a model of Salmonella spread within a farrow-to-finish pig herd, assuming the prevalence in infected delivered pigs depends on the whole pig life-time and growing process. This stochastic discrete-time model represents both the population dynamics in a farrow-to-finish pig herd using batch management, and Salmonella spread. Four mutually exclusive individual health states were considered: Salmonella-free, seronegative shedder, seropositive shedder and seropositive not shedding carrier, making the distinction between seropositive animals and shedders. Since indirect transmission is the main route of transmission, the probability of infection depends on the quantity of Salmonella in the pigs' environment (Q). A dose effect function is used with two thresholds, assuming saturation in exposure for high Q vs. a minimum exposure for low Q. Salmonella is introduced in an initially Salmonella-free 150-sow herd. Prevalence of shedders and seroprevalence are calculated over time in batches of sows and pigs, and in groups of delivered pigs, composed of pigs from different batches. The model shows very variable seroprevalence over time within a herd among delivered groups, as well as among replications. The mean seroprevalence and the mean shedding prevalence are 19.3% and 13.8% respectively. A sensitivity analysis shows that the Salmonella quantity shed and the maternal protective factor are the most influential parameters on Salmonella prevalence in delivered pigs.

Genotype-level variation in lifetime breeding success, litter size and survival of sheep in scrapie-affected flocks.

Five different sheep flocks with natural outbreaks of scrapie were examined to determine associations between individual performance (lifetime breeding success, litter size and survival) and scrapie infection or PrP genotype. Despite different breed composition and forces of infection, consistent patterns were found among the flocks. Regardless of the flock, scrapie-infected sheep produced on average 34 % fewer offspring than non-scrapie-infected sheep. The effect of scrapie on lifetime breeding success appears to be a function of lifespan as opposed to fecundity. Analysis of litter size revealed no overall or genotype differences among the five sheep flocks. Survival, however, depends on the individual's scrapie status (infected or not) and its PrP genotype. Susceptible genotypes appear to perform less well in lifetime breeding success and life expectancy even if they are never affected with clinical scrapie. One possible explanation for these results is the effect of pre-clinical scrapie. Additional evidence supporting this hypothesis is discussed.