Mathematical models of cross protection in the epidemiology of plant-virus diseases

Cacao sustainability: The case of cacao swollen-shoot virus co-infection

The cacao swollen shoot virus disease (CSSVD) is among the most economically damaging diseases of cacao trees and accounts for almost 15-50% of harvest losses in Ghana. This virus is transmitted by several species of mealybugs (Pseudococcidae, Homoptera) when they feed on cacao plants. One of the mitigation strategies for CSSVD investigated at the Cocoa Research Institute of Ghana (CRIG) is the use of mild-strain cross-protection of cacao trees against the effects of severe strains. In this study, simple deterministic, delay, and stochastic ordinary differential equation-based models to describe the dynamic of the disease and spread of the virus are suggested. Model parameters are estimated using detailed empirical data from CRIG. The modeling outcomes demonstrate a remarkable resemblance between real and simulated dynamics. We have found that models with delay approximate the data better and this agrees with the knowledge that CSSVD epidemics develop slowly. Also, since there are large variations in the data, stochastic models lead to better results. We show that these models can be used to gain useful informative insights about the nature of disease spread.

The Epidemiology of Plant Virus Disease: Towards a New Synthesis

Epidemiology is the science of how disease develops in populations, with applications in human, animal and plant diseases. For plant diseases, epidemiology has developed as a quantitative science with the aims of describing, understanding and predicting epidemics, and intervening to mitigate their consequences in plant populations. Although the central focus of epidemiology is at the population level, it is often necessary to recognise the system hierarchies present by scaling down to the individual plant/cellular level and scaling up to the community/landscape level. This is particularly important for diseases caused by plant viruses, which in most cases are transmitted by arthropod vectors. This leads to range of virus-plant, virus-vector and vector-plant interactions giving a distinctive character to plant virus epidemiology (whilst recognising that some fungal, oomycete and bacterial pathogens are also vector-borne). These interactions have epidemiological, ecological and evolutionary consequences with implications for agronomic practices, pest and disease management, host resistance deployment, and the health of wild plant communities. Over the last two decades, there have been attempts to bring together these differing standpoints into a new synthesis, although this is more apparent for evolutionary and ecological approaches, perhaps reflecting the greater emphasis on shorter often annual time scales in epidemiological studies. It is argued here that incorporating an epidemiological perspective, specifically quantitative, into this developing synthesis will lead to new directions in plant virus research and disease management. This synthesis can serve to further consolidate and transform epidemiology as a key element in plant virus research.

Modelling Vector Transmission and Epidemiology of Co-Infecting Plant Viruses

Co-infection of plant hosts by two or more viruses is common in agricultural crops and natural plant communities. A variety of models have been used to investigate the dynamics of co-infection which track only the disease status of infected and co-infected plants, and which do not explicitly track the density of inoculative vectors. Much less attention has been paid to the role of vector transmission in co-infection, that is, acquisition and inoculation and their synergistic and antagonistic interactions. In this investigation, a general epidemiological model is formulated for one vector species and one plant species with potential co-infection in the host plant by two viruses. The basic reproduction number provides conditions for successful invasion of a single virus. We derive a new invasion threshold which provides conditions for successful invasion of a second virus. These two thresholds highlight some key epidemiological parameters important in vector transmission. To illustrate the flexibility of our model, we examine numerically two special cases of viral invasion. In the first case, one virus species depends on an autonomous virus for its successful transmission and in the second case, both viruses are unable to invade alone but can co-infect the host plant when prevalence is high.

Plant Virus Epidemiology: Applications and Prospects for Mathematical Modeling and Analysis to Improve Understanding and Disease Control

In recent years, mathematical modeling has increasingly been used to complement experimental and observational studies of biological phenomena across different levels of organization. In this article, we consider the contribution of mathematical models developed using a wide range of techniques and uses to the study of plant virus disease epidemics. Our emphasis is on the extent to which models have contributed to answering biological questions and indeed raised questions related to the epidemiology and ecology of plant viruses and the diseases caused. In some cases, models have led to direct applications in disease control, but arguably their impact is better judged through their influence in guiding research direction and improving understanding across the characteristic spatiotemporal scales of plant virus epidemics. We restrict this article to plant virus diseases for reasons of length and to maintain focus even though we recognize that modeling has played a major and perhaps greater part in the epidemiology of other plant pathogen taxa, including vector-borne bacteria and phytoplasmas.

Modeling Virus Coinfection to Inform Management of Maize Lethal Necrosis in Kenya

Maize lethal necrosis (MLN) has emerged as a serious threat to food security in sub-Saharan Africa. MLN is caused by coinfection with two viruses, Maize chlorotic mottle virus and a potyvirus, often Sugarcane mosaic virus. To better understand the dynamics of MLN and to provide insight into disease management, we modeled the spread of the viruses causing MLN within and between growing seasons. The model allows for transmission via vectors, soil, and seed, as well as exogenous sources of infection. Following model parameterization, we predict how management affects disease prevalence and crop performance over multiple seasons. Resource-rich farmers with large holdings can achieve good control by combining clean seed and insect control. However, crop rotation is often required to effect full control. Resource-poor farmers with smaller holdings must rely on rotation and roguing, and achieve more limited control. For both types of farmer, unless management is synchronized over large areas, exogenous sources of infection can thwart control. As well as providing practical guidance, our modeling framework is potentially informative for other cropping systems in which coinfection has devastating effects. Our work also emphasizes how mathematical modeling can inform management of an emerging disease even when epidemiological information remains scanty. [Formula: see text] Copyright © 2017 The Author(s). This is an open access article distributed under the CC BY-NC-ND 4.0 International license .

Modeling the effects of cross-protection control in plant disease with seasonality

Cross-protection in plants has been widely used to control losses caused by virus diseases in the world. Here, a non-autonomous plant-virus disease model was developed including cross-protection. Global dynamics of the model was discussed. Under the quite weak assumptions, integral form conditions were resolved for permanence of the system and extinction of diseases. Furthermore, we looked into the sufficient conditions that plants could be protected against the detrimental effects of infection by an infection with the mild virus isolates. Last, we performed numerical simulations. Our investigations suggested that cross-protection played an important role in controlling the spread of the challenging virus in plants.

Mathematical model of plant-virus interactions mediated by RNA interference

Cross-protection, which refers to a process whereby artificially inoculating a plant with a mild strain provides protection against a more aggressive isolate of the virus, is known to be an effective tool of disease control in plants. In this paper we derive and analyse a new mathematical model of the interactions between two competing viruses with particular account for RNA interference. Our results show that co-infection of the host can either increase or decrease the potency of individual infections depending on the levels of cross-protection or cross-enhancement between different viruses. Analytical and numerical bifurcation analyses are employed to investigate the stability of all steady states of the model in order to identify parameter regions where the system exhibits synergistic or antagonistic behaviour between viral strains, as well as different types of host recovery. We show that not only viral attributes but also the propagating component of RNA-interference in plants can play an important role in determining the dynamics.

Using contaminated tools fuels outbreaks of BananaXanthomonas wilt: An optimal control study within plantations using Runge–Kutta fourth-order algorithms

Optimal control theory is applied to a system of ordinary differential equations modeling banana Xanthomonas wilt within plantations. The objective is to reduce the proportion of infected plants by use of controls representing two types of preventive methods: vector and contaminated tool prevention. The optimal controls are characterized in terms of the optimality system, which is solved analytically and numerically for several scenarios.

Facilitative and antagonistic interactions between plant viruses in mixed infections

Mixed infections of plant viruses are common in nature, and a number of important virus diseases of plants are the outcomes of interactions between causative agents. Multiple infections lead to a variety of intrahost virus-virus interactions, many of which may result in the generation of variants showing novel genetic features, and thus change the genetic structure of the viral population. Hence, virus-virus interactions in plants may be of crucial significance for the understanding of viral pathogenesis and evolution, and consequently for the development of efficient and stable control strategies. The interactions between plant viruses in mixed infections are generally categorized as synergistic or antagonistic. Moreover, mixtures of synergistic and antagonistic interactions, creating usually unpredictable biological and epidemiological consequences, are likely to occur in plants. The mechanisms of some of these are still unknown. This review aims to bring together the current knowledge on the most commonly occurring facilitative and antagonistic interactions between related or unrelated viruses infecting the same host plant. The best characterized implications of these interactions for virus-vector-host relationships are included. The terms 'synergism' and 'helper dependence' for facilitative virus-virus interactions, and 'cross-protection' and 'mutual exclusion' for antagonistic interactions, are applied in this article.

Modelling virus- and host-limitation in vectored plant disease epidemics

Models of plant virus epidemics have received less attention than those caused by fungal pathogens. Intuitively, the fact that virus diseases are systemic means that the individual diseased plant can be considered as the population unit which simplifies modelling. However, the fact that a vector is required in the vast majority of cases for virus transmission, means that explicit consideration must be taken of the vector, or, the involvement of the vector in the transmission process must be considered implicitly. In the latter case it is also important that within-plant processes, such as virus multiplication and systemic movement, are taken into account. In this paper we propose an approach based on the linking of transmission at the population level with virus multiplication within plants. The resulting models are parameter-sparse and hence simplistic. However, the range of model outcomes is representative of field observations relating to the apparent limitation of epidemic development in populations of healthy susceptible plants. We propose that epidemic development can be constrained by virus limitation in the early stages of an epidemic when the availability of healthy susceptible hosts is not limiting. There is an inverse relationship between levels of transmission in the population and the mean virus titre/infected plant. In the case of competition between viruses, both virus and host limitation are likely to be important in determining whether one virus can displace another or whether both viruses can co-exist in a plant population. Lotka-Volterra type equations are derived to describe density-dependent competition between two viruses multiplying within plants, embedded within a population level epidemiological model. Explicit expressions determining displacement or co-existence of the viruses are obtained. Unlike the classical Lotka-Volterra competition equations, the co-existence requirement for the competition coefficients to be both less than 1 can be relaxed.

Spatiotemporal model of barley and cereal yellow dwarf virus transmission dynamics with seasonality and plant competition

Many generalist pathogens are influenced by the spatial distributions and relative abundances of susceptible host species. The spatial structure of host populations can influence patterns of infection incidence (or disease outbreaks), and the effects of a generalist pathogen on host community dynamics in a spatially heterogeneous community may differ from predictions derived via simple models. In this paper, we model the transmission of a generalist pathogen within a patch framework that incorporates the movement of vectors between discrete host patches to investigate the effects of local host community composition and vector movement rates on disease dynamics.We use barley and cereal yellow dwarf viruses (B/CYDV), a suite of generalist, aphid-vectored pathogens of grasses, and their interactions with a range of host species as our case study. We examine whether B/CYDV can persist locally or in a patch framework across a range of host community configurations. We then determine how pathogen-mediated interactions between perennial and annual competitors are altered at the local and regional scale when the host populations are spatially structured. We find that the spatial configuration of the patch system, host composition within patches, and patch connectivity affect not only the ability of the pathogen to invade a fragmented system, but also determine whether the pathogen facilitates the invasion of a non-native host species. Further, our results suggest that connectivity can interact with arrival time and host infection tolerance to determine the success or failure of establishment for newly arriving species.

Modeling the competition between viruses in a complex plant-pathogen system

In this article, we propose a mathematical model that describes the competition between two plant virus strains (MAV and PAV) for both the host plant (oat) and their aphid vectors. We found that although PAV is transmitted by two aphids and MAV by only one, this fact, by itself, does not explain the complete replacement of MAV by PAV in New York State during the period from 1961 through 1976; an interpretation that is in agreement with the theories of A. G. Power. Also, although MAV wins the competition within aphids, we assumed that, in 1961, PAV mutated into a new variant such that this new variant was able to overcome MAV within the plants during a latent period. As shown below, this is sufficient to explain the swap of strains; that is, the dominant MAV was replaced by PAV, also in agreement with Power's expectations.

Cross-protection: a century of mystery

Cross-protection is a phenomenon in which infection of a plant with a mild virus or viroid strain protects it from disease resulting from a subsequent encounter with a severe strain of the same virus or viroid. In this chapter, we review the history of cross-protection with regard to the development of ideas concerning its likely mechanisms, including RNA silencing and exclusion, and its influence on the early development of genetically engineered virus resistance. We also examine examples of the practical use of cross-protection in averting crop losses due to viruses, as well as the use of satellite RNAs to ameliorate the impact of virus-induced diseases. We also discuss the potential of cross-protection to contribute in future to the maintenance of crop health in the face of emerging virus diseases and related threats to agricultural production.

Modeling the dynamics of viral evolution considering competition within individual hosts and at population level: the effects of treatment

We consider two viral strains competing against each other within individual hosts (at cellular level) and at population level (for infecting hosts) by studying two cases. In the first case, the strains do not mutate into each other. In this case, we found that each individual in the population can be infected by only one strain and that co-existence in the population is possible only when the strain that has the greater basic intracellular reproduction number, R (0c ), has the smaller population number R (0p ). Treatment against the one strain shifts the population equilibrium toward the other strain in a complicated way (see Appendix B). In the second case, we assume that the strain that has the greater intracellular number R (0c ) can mutate into the other strain. In this case, individual hosts can be simultaneously infected by both strains (co-existence within the host). Treatment shifts the prevalence of the two strains within the hosts, depending on the mortality induced by the treatment, which is, in turn, dependent upon the doses given to each individual. The relative proportions of the strains at the population level, under treatment, depend both on the relative proportions within the hosts (which is determined by the dosage of treatment) and on the number of individuals treated per unit time, that is, the rate of treatment. Implications for cases of real diseases are briefly discussed.

A note on semi-discrete modelling in the life sciences

Semi-discrete models are a particular class of hybrid dynamical systems that undergo continuous dynamics most of the time but repeatedly experience discrete changes at some given moments. In the life sciences, since the first semi-discrete model was derived to describe population dynamics by Beverton & Holt (Beverton & Holt 1957 In Fisheries investigations, series 2, vol. 19), a large body of literature has been concerned with such modelling approaches. The aim of the present contribution is twofold. On the one hand, it provides a comprehensive introduction to semi-discrete modelling through two illustrative examples: the classical work by Beverton and Holt is recalled and an original example on immigration in a population model affected by a strong Allee effect is worked out. On the other hand, a short overview of the different applications of semi-discrete models in the life sciences is proposed.

The relationship of host-mediated induced resistance to polymorphism in gene-for-gene relationships

Gene-for-gene relationships are a common feature of plant-parasite interactions. Polymorphism at host resistance and parasite avirulence loci is maintained if there is negative, direct frequency-dependent selection on alleles of either gene. More specifically, selection of this kind is generated when the disease is polycyclic with frequent auto-infection. When an incompatible interaction occurs between a resistant host and an avirulent parasite, systemic defenses are triggered, rendering the plant more resistant to a later attack by another parasite. However, induced resistance (IR) incurs a fitness cost to the plant. Here, the effect of IR on polymorphism in gene-for-gene interactions is investigated. First, in an infinite population model in which parasites have two generations per host generation, increasing the fitness cost of IR increases selection for susceptible plants at low disease severity, while increasing the effectiveness of IR against further parasite attacks enhances selection for resistant plants at high disease severity. This reduces the possibility of polymorphism being maintained in host and parasite populations. In finite population models, the number of plants varies over time as a function of the disease burden of the population. Polymorphism in gene-for-gene relationships is then more stable at high disease prevalence and severity if IR reactions are more costly when there is competition for resources between plants.

Analysis of disease progress as a basis for evaluating disease management practices

The relationship between epidemiology and disease management is long-standing but sometimes tenuous. It may seem self-evident that improved understanding of epidemic processes will lead to more effective control practices but this remains a testable proposition rather than demonstrated reality. A wide range of models differing in mathematical sophistication and computational complexity has been proposed as a means of achieving a greater understanding of epidemiology and carrying this through to improved management. The potential exists to align these modeling approaches to evaluation of control practices and prediction of the consequent epidemic outcomes, but these have yet to make a major impact on practical disease management. For the immediate future simpler pragmatic approaches for analysis of disease progress, using nonlinear growth functions and/or integrated measures such as area under disease progress curves, will play a key role in informing tactical and strategic decisions on control treatments. These approaches have proved useful in describing control effectiveness and, in some cases, optimizing or changing control practices.