Mathematical model for the biocontrol of vector-borne viral diseases in solanaceous vegetable plants

TABLE OF CONTENTS PAGE

COVER PAGE

TITLE PAGE

DECLARATION

DEDICATION

APPROVAL PAGE

ACKNOWLEDGEMENTS

ABSTRACT

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF APPENDICES

CHAPTER ONE: INTRODUCTION
1.1 Background of the Study
1.2 Statement of the Problem
1.3 Aim and Objectives of the Study
1.4 Scope of the Study
1.5 Justification of the Study
1.6 Definitions of Basic Concepts/Terms

CHAPTER TWO: LITERATURE REVIEW
2.1 The Concept of Biological Control
2.2 Biological Control of Pests
2.3 Modelling of Interactions within Biological Spheres
2.4 Stability of Biological Systems
2.5 The Basic Reproduction Number and Disease Control
2.6 Sensitivity Analysis of Biological Models
2.7 Reviewed Literatures on Mathematical Models of Plants Diseases

CHAPTER THREE: METHODOLOGY
3.1 Introduction
3.2 The existing model
3.3 The Modified Model
3.4 Method of Model Analysis

CHAPTER FOUR: RESULTS
4.1 Introduction
4.2 Analytical Analysis of the modified model
4.2 Sensitivity Analysis
4.3 Analysis of Sub Model for Interspecific Competition
4.4 Analysis of Sub Model for Intra Specific Competition
4.5 Analysis of Predator – Prey Sub Model
4.6 Stability Analysis of the Predator – Prey Sub Model

CHAPTER FIVE: NUMERICAL SIMULATION AND DISCUSSION
5.1 Introduction
5.2 The Numerical Values used for Simulation
5.3 Numerical Simulation on the Modified Model
5.4 Discussion of the Results

CHAPTER SIX: SUMMARY, CONCLUSION AND RECOMMENDATIONS
6.1 Summary
6.2 Conclusion
6.3 Recommendations
6.4 Contribution to Knowledge

REFERENCES

APPENDICES

LIST OF TABLES

3.1a State Variables of the Existing Model

3.1b Parameters of the Existing Model

3.2 State Variables of the Modified Model

3.3 Parameters of the Modified Model and their Meaning

3.4 The Parameters used for the Predator – prey Sub model

4.1 Result of Sensitivity Analysis

5.1 The Parameter Values

5.2 Initial Values for the State Variables

LIST OF FIGURES

3.1 Flow Diagram of the Existing Model

3.2 Flow Diagram of the Modified Model

5.1a Effect of 8,375 Lady Beetle on the Population of Aphids, Thrips and Whiteflies

5.1b Effect of 16,875 Lady Beetle on the Population of Aphids, Thrips and Whiteflies

5.1c Effect of 33,750 Lady Beetle on the Population of Aphids, Thrips and Whiteflies

5.1d Effect of 50,625 Lady Beetle on the Population of Aphids, Thrips and Whiteflies

5.1e Effect of 67,500 Lady Beetle on the Population of Aphids, Thrips and Whiteflies

5.2 Effect of Initial Number of Lady Beetle on Infected Plants

5.3a Effect of Predation on Aphids, Thrips and Whiteflies at Abbildung in dieser Leseprobe nicht enthalten

5.3b Effect of Predation on Aphids, Thrips and Whiteflies at Abbildung in dieser Leseprobe nicht enthalten

5.3c Effect of Predation on Aphids, Thrips and Whiteflies at Abbildung in dieser Leseprobe nicht enthalten

5.3d Effect of Predation on Aphids, Thrips and Whiteflies at Abbildung in dieser Leseprobe nicht enthalten

5.3e Effect of Predation on Aphids, Thrips and Whiteflies at Abbildung in dieser Leseprobe nicht enthalten

5.4a Effect of Interspecific Competition among Vectors in the Absence of Lady Beetle

5.4b Effect of Interspecific Competition among Vectors in the Presence of Lady Beetle

5.5a Effect of Intraspecific Competition on Aphids

5.5b Effect of Intraspecific Competition on Thrips

5.5c Effect of Intraspecific Competition on Whiteflies

DECLARATION

I hereby declare that this thesis was written by me and it is a record of my own research work. It has not been presented before in any previous application for a higher degree. All references cited have been duly acknowledged.

SHAMAKI, Timothy Ado Date

DEDICATION

This thesis is dedicated to my mother, Mrs. Angito A. Shamaki, for all she has done to care and guide me through the most difficult path in my early life.

APPROVAL PAGE

This thesis entitled “Mathematical Model for the Biocontrol of Vector-Borne Viral Diseases (VBVDs) in Solanaceous Vegetable Plants” meets the regulations governing the award of the Doctor of Philosophy of the Modibbo Adama University, Yola and is approved for its contribution to knowledge and literary presentation.

Abbildung in dieser Leseprobe nicht enthalten

ACKNOWLEDGMENTS

First of all, with eternal gratitude, I acknowledge the grace of God upon my life. Therefore, my deepest thanks goes to Almighty God, He is the reason for everything that concerns me. Certainly, all that I am, have and will become, comes from Him and for Him; and to Him alone be the glory, honour and adoration for ever! Amen.

I would like to thank my supervisor for the much-needed guidance and support. Indeed, he has been a great mentor to me. And I am particularly grateful to him for his insightful ideas, knowledge, broad experience, and lots of countless useful suggestions that shaped my understanding of mathematical modelling. This thesis would not have been possible without his supervisory role, scrutiny and patience. Indeed, Prof. S. Musa is the best supervisor that a PhD student could ever have. He has always made time for me, and I will forever cherish our conversations about mathematical modelling and life as a whole. Surely, I owe him many thanks for contributing not only to my academic development but also life in general.

I must also thank my co supervisor, Dr. A. O. Adesanya, for his knowledge of mathematics. Equally, without his tireless support and advice, my efforts would have been fruitless. His demand for academic rigour has made me a better researcher. Thus, I am grateful to him for guiding me in the course of my PhD research.

Also, I would like to thank the Head of Department, Dr. A. M Alkali, the PG Coordinator, Dr. Y. B. Chukkol, for their leadership roles in spearheading the affairs of the PG programme. In a like manner, I would like to express my sincere gratitude to other members of staff in the department of mathematics for contributing to my academic development, specifically Prof. M. R. Odekunle, Prof. I. I. Adamu, Dr. A. Tahir, Dr. S. O. Adee, Dr. A. A. Momoh, Dr. A. M. Ayinde, Mr. P. A. Atsen, Mal. S. A. Abdullahi, Dr. A. Umar and Dr. H. B Aliyu.

I wish to thank my family, especially my wife, Tani Amina Ibrahim, for her love, support and sacrifice in the pursuit of my dreams. She has been a wonderful source of inspiration through thick and thin. Likewise my children – Tiara, Tabitha and Tryphena they have been so patient with me over the years. I salute your resoluteness. Also, special gratitude to my father, late Mr. Ado S. Gadaka, my mother, Angito Ado Shamaki and my siblings, the closeness we share, just as our parents taught us, makes life more fulfilling. I say thank you all.

Finally, I thank Mr. A. Augustine, a significant individual who has influenced my life and helped me succeed academically. I am grateful for his invaluable contributions. And last to be acknowledged herewith, but certainly not the least, is the Trinity Chapel - MAU, Yola. The Church has provided the spiritual fortress for my growth and development while here on campus. I say a big thank you and God bless you all. Amen.

ABSTRACT

This thesis treats the issue of Vector-Borne Virus Diseases (VBVDs) that are transmitted in solanaceous vegetable plants by incorporating three species of vectors (aphids, thrips and whiteflies). A mathematical model was developed that used lady beetles as biological control agents for controlling the spread of diseases in solanaceous vegetable plants through predation. The research adopted the linearization method; thus, it was proven that the system of ordinary differential equations (ODEs) of the model are well-represented epidemiologically and mathematically. The existence of the disease-free equilibrium (DFE) state and the disease-endemic equilibrium (DEE) state have been established by obtaining the values of the state variables at fixed points. Furthermore, Lyapunov direct and indirect method of stability analysis were used for proving the local and global stability of the DFE as well as the DEE respectively. Also, the basic reproduction number Abbildung in dieser Leseprobe nicht enthalten for the control of VBVDs was obtained using the operation of the next generation matrix (NGM), and a sensitivity analysis was performed on the Abbildung in dieser Leseprobe nicht enthalten using normalized sensitivity index. The result of sensitivity analysis indicated that the biting rate due to aphids, thrips and whiteflies can increase the disease burden in solanaceous vegetable plants if their rates are increased, among others. From the numerical experiments, is concluded that the biocontrol of VBVDs can be ascertained using lady beetles as biological control agents in tackling aphids, thrips and whiteflies in solanaceous vegetable plants. Simulation results have indicated that, vector pests were eradicated within four weeks after the introduction of 67,500 lady beetles on 13.5 hectare of farmland which resulted in complete elimination of infected plants between 4 to 6 weeks. Accordingly, the findings suggest that predation rate at Abbildung in dieser Leseprobe nicht enthalten is required for successful biocontrol. Therefore, it is recommended that biological control should be adopted as an integral part of solanaceous vegetable farming in order to control the spread of VBVDs.

LIST OF APPENDICES

i 1st Publication of Some of the Results of the Study

ii 2nd Publication of Some of the Results of the Study

iii 3rd Publication of Some of the Results of the Study

iv Data for Numerical Simulations

v Farmland Data and Initial Conditions

CHAPTER ONE INTRODUCTION

1.1 Background of the Study

Plants are essential to life. They are sources of food, medicines, clothes and are important for a healthy environment (Jackson and Chen-Charpentier, 2018). Unfortunately, plants can become infected with a disease and this has continued to ravage one-third of food quantity produced worldwide (Ciancio and Mukerji, 2008). According to Agrios (2005), “an estimated average of fourteen percent (14%) of world crops is lost to diseases due to pests”. Furthermore, there are a number of pests that have developed resistance to certain pesticides. Accordingly, not all parasites and diseases can be adequately controlled by the application of synthetic chemicals due to financial implication or/and harmful side effects on human life and the environment. To this end, Ciancio and Mukerji (2008) assert, “air, land and water are alarmingly polluted to the extent that several sensitive species are becoming extinct at the rate never experienced before on earth”. Hence, the need for a safer, economical and better means of safeguarding the planet.

With a projected population of 9.7 billion by 2050, the world’s population and its continued increase depends heavily on stable agricultural production in order to achieve sustainable food needs (Neofytou, Kyrychko and Blyuss, 2016). Thus, the issue of food security is central to the UN Sustainable Development Goals (SDGs). For this reason, it is desirable that applied mathematics works out a modality for the solution to the looming acute food shortage, since conventional methods of chemical control, as well as Integrated Pest Management (IPM) are associated with many harmful and detrimental challenges. In addition, mathematical epidemiology research is particularly necessary in the area of biological control, especially in view of the fact that there are many mathematical models describing the interaction between vectors and humans, but there are not so many that describe the relationship between plants and vectors (Su and Zhang, 2014).

A key area where research is required is in the field of solanaceous plants. This is due to the fact that solanaceous plants are extremely of high economic importance since they are widely used as ornamentals (petunia), sources of pharmaceutical compounds (tobacco, mandrake, belladonna, jimsonweed, etc.), and as food. In Africa, the four main solanaceous vegetable crops, used as food, are potato (Solanum tuberosum), tomato (Solanum lycopersicum), eggplant/aubergine (Solanum melongena), and pepper (Capsicum ssp.) with 365 million tons produced annually in Nigeria (Morris and Taylor, 2017). However, with all the economic benefits derived from the solanaceous vegetables plants, farming them is often characterized by severe tussle between local farmers and many pest varieties as well as diseases (Ghatak et al., 2017). Undoubtedly, the issue of diseases being spread by pests has already affected negatively not only on the welfare of the farmers but also on the Gross Domestic Product (GDP) of Nigeria. This indeed has negatively affected agricultural practices and has remained unaddressed for a long period of time (Wokoma, 2008).

In the tomato sector alone, previous research has shown that Nigeria ranks 14th largest tomato producer with 2.3 million tons per year (Borisade, Kolawole, Adebo and Uwaidem, 2017). Nevertheless, in Africa, Nigeria is the second largest producer, producing about 10.8% of fresh tomatoes in the West African Sub-region. This has caused Nigeria to be ranked the 13th importer of tomato paste in the World (PUNCH, 2017). This is evident by the fact that Nigeria's tomato yield is very low at an average of 5.47 tons per hectare compared to global average yields of 38.1 tons per hectare (Tomato Jos, 2017). That is why Borisade et al. (2017) stressed that, “in 2016 and 2017 alone, Nigeria has imported tomato paste estimated at about $720m”. Thus, Nigeria is ranked the largest consumers of vegetables in Sub-Saharan Africa with an average per capita consumption of 12 kg (Food and Agricultural Organization (FAO), 2019).

Vanguard Newspaper (2019) cites a 2016 scenario in which the Nigerian government claimed that a disease came into the country from the Republic of Niger through some flying insects, which destroys solanaceous vegetable plants under 48 hours. The five states affected by the outbreak include Kastina, Kano, Kaduna, Jigawa, and Nasarawa. The disease has also spread to other states in the South such as Oyo and Ogun States. Therefore, this study is an attempt to address these kinds of problems using mathematical modeling techniques. The study examined how biological control could be applied in solanaceous vegetable farming by determining how natural enemies (predators) can be used in order to reduce the incidence of disease and also to control the pests’ (vectors) population. Thus, this study modifies a generic model by Jackson and Chen-Charpentier (2018) by considering biological control against pests for annihilating Vector-Borne Viral Diseases (VBCDs) in solanaceous vegetable plants.

1.2 Statement of the Problem

For a long time, the issue of VBVDs being spread by pests has been a major setback in agricultural practices (Goh, Leitmann and Vincent, 1974). Thus, pests and diseases are viewed as constant threats to global agriculture (Moodley, Gubba and Mafongoya, 2019). Therefore, there is a need for new and adaptive strategies in order to reduce their economic impact on agricultural systems. In response to this, Jackson and Chen-Charpentier (2018) proposed a mathematical model on plant virus propagation with seasonality, but their model did not acknowledge the reality associated with insect vectors, as there is hardly a case where only one class of vector patronizes some certain plants. Moreover, the issue of intra specie and interspecies competition among insect vectors were not considered.

Mainly, Jackson and Chen-Charpentier (2018) stressed that insects are mostly active in the warm months and almost dormant in the cooler ones, and analyzed their model with periodic coefficients. As an observation, therefore, there is an exceptional case where plants are cultivated within one season (be it the cool or warm months). Therefore, this reserch incoperates three types of vector pests in solanaceous vegetable plants within a season.

1.3 Aim and Objectives of the Study

This research studied the biological control of VBVDs in solanaceous vegetable plants with the following objectives, to:

i) modify a mathematical model for biological control of VBVDs in solanaceous plants by incorporating three classes of vectors,
ii) establish the positivity of solution of the modified model,
iii) prove the existence and uniqueness of solution of the modified model,
iv) obtain the equilibrium points of the modified model,
v) derive a threshold condition (basic reproductive number, R₀) for the control of VBVDs of the modified model,
vi) establish the local and global stability of the modified model,
vii) obtain and analyze sub models that can be used to understand the dynamics of interactions among the species under consideration,
viii) carry out sensitivity analysis on the basic reproduction number so as to determine the range of parameters that have high impact on the modified model, and
ix) carry out numerical experiments (simulations) in order to determine the deployment of biological control strategy on the modified model.

1.4 Scope of the Study

This research is restricted to a biological control of VBVDs of solanaceous vegetable plants using compartmental modeling approach. The model is a system of first order nonlinear ODEs. Additionally, the study is limited to three solanaceous plants (i.e. tomato, pepper and eggplant). This is because solanaceous plants are among the world's most cultivated crops and given proper conditions and regular maintenance, they are relatively easy to grow (Johnny's Selected Seeds, 2018). We focused on viral diseases that affect solanaceous vegetable plants especially Yellow Leaf Curl Virus (YLCV), Spotted Wilt Virus (SWV) and Cucumber Mosaic Virus (CMV).

It is a common knowledge that these plant viruses also require some sort of carrier, known as vectors to transmit the pathogen from plant to plant. Therefore, the study is demarcated to a class of aphids (green peach aphids), thrips (T. tabasi) and whiteflies (Bemisia tabasi) because these ones are reported as the common problem associated with solanaceous plants which can be controlled by natural predatory enemies - ladybugs (hippodamia convergens) (Johnny's Selected Seeds, 2018).

1.5 Justification of the Study

Records have shown that, Nigeria, due to its large population, consume about 2.3 million tons of solanaceous vegetables per year with about 22kg per capita consumption (Idowu-agida, Nwaguma and Adeoye, 2010). This is frightening! It means Nigeria is likely to be in severe vegetable shortage if appropriate measures are not taken within a short period of time (Vanguard Newspaper, 2019). Therefore, it is unacceptable to continue overemphasizing the uses of old varieties of seedlings, diseases, pests and parasites, animal invasion and low soil fertility, among others, without proffering a scientifically workable solution. Hence, the need to address the problem using a mathematical modelling approaches.

1.6 Definitions of Basic Concepts/Terms

In this section, we defined the basic concepts/terms to be encountered in this work.

1.6.1 Solanaceous vegetable plants, ( ScienceDaily, 2019 )

These are nightshade plants, which belong to family of flowering plants that ranges from annual and perennial herbs to vines, liannas, epiphytes, shrubs and trees, and include a number of agricultural crops, medicinal plants, spices, weeds, and ornamentals. Many of the members of the family contain potent alkaloids, and some are highly toxic, but many including tomatoes, eggplant, bell and chili peppers, and tobacco are used as food.

1.6.2 Predator – prey interactions, ( Martcheva, 2015)

Predation, or predator - prey interactions, refers to the feeding on the individuals of a prey species by the individuals of a predator species for the survival and growth of the predator. The predator itself can be of two types - a specialist predator, whose entire feeding choice is restricted to a single prey species, and a generalist predator that feeds on many different prey species.

1.6.3 Infection, ( ScienceDaily, 2019 )

This refers to the appearance of disease symptoms accompanied by the establishment and spread of the pathogen. Normally, a pathogenic organism such as virus, fungus, bacterium, nematode, or parasitic flowering plant causes infectious plant diseases.

1.6.4 Biological control, (Ciancio and Mukerji, 2008)

The use of nature and/or modified organisms, genes or gene products to regulate or reduce pests in favour of human and animal populations, and agricultural crops besides protecting other beneficial organisms.

1.6.5 Plant disease, ( ScienceDaily, 2019 )

An impairment of the normal state of a plant that interrupts or modifies its vital functions. Plant diseases can be broadly classified according to the nature of their primary causal agent, either infectious or noninfectious.

1.6.6 Pathogen, ( ScienceDaily, 2019 )

A pathogen or infectious agent is a biological agent that causes disease or illness to its host. The term is most often used for agents that disrupt the normal physiology of a multi cellular animal or plant.

1.6.7 Mathematical model, (McGraw-Hill, 2003)

A mathematical representation of a process, system, device, or concept by means of a number of variables which are defined to represent the inputs, outputs, and internal states of the device or process, and a set of parameters, equations and inequalities describing the interaction of these variables.

1.6.8 Equilibrium point, (Boyce and DiPrima, 2008)

Given a system of differential equations, a solution to it in an interval is a differentiable function such that a solution is called an equilibrium point, critical point or steady state, if it is a constant for all , namely, for . In this case, satisfies since

1.6.9 Basic reproduction number (R ₀ ), (Research Nuggets, 2015)

The R0 referred to as the threshold quantity in epidemiology, it is the average number of secondary infections caused by a typical infected individual in a susceptible population. In most cases, when the reproduction number is less than one, the system has only the "disease-free" equilibrium, and the disease is expected to die out. As the number increases through one, a stable "endemic" equilibrium emerges, that is, the disease is maintained in the population without the need for external inputs.

1.6.10 Lyapunov function, (McGraw-Hill, 2003)

A function of a vector and of time which is positive-definite and has a negative-definite derivative with respect to time for nonzero vectors, is identically zero for the zero vector, and approaches infinity as the norm of the vector approaches infinity; used in determining the stability of control systems.

1.6.11 Globally asymptotically stable, (Khalil, 2002)

An equilibrium point of is stable (Lyapunov Stable (LS)) if for each there exist such that . And it is said to be globally asymptotically stable, if stable and .

1.6.12 Sensitivity analysis

Sensitivity analysis of a system of ordinary differential equation is defined as the derivatives of the solution with respect to the parameter(s) of interest. Thus, the sensitivity analysis is useful in understanding how a system of ordinary differential equation will change in accordance with change in the value of the parameter(s). To obtain a sensitivity index of any disease model, simply take the partial derivatives of with respect to each parameter.

CHAPTER TWO LITERATURE REVIEW

2.1 The Concept of Biological Control

Biological control is most commonly defined as the use of natural enemies to reduce the population size of pest species (McKimmie, 2000). It entails the suppression of damaging activities of one organism by one or more different organisms, often referred to as natural enemies (Coppel and Mertins, 2012). However, in terms of plant pathology, the definition of biological control refers to the purposeful utilization of introduced or resident living organisms, other than disease resistant host plants, to suppress the activities and populations of one or more plant pathogens (Pal and Gardener, 2006). Although not new to agricultural practice, modern efforts at biocontrol have taken place for about 100 years (Jackson and Chen-Charpentier, 2018). Biological control of pests has received great interest recently as an alternative to conventional pesticides. According to Naranjo, Ellsworth and Frisvold (2015), biological control is due to both environmental and economic concerns.

Biocontrol may be considered as a multi-trait phenomenon whose success depends on ability to compete for nutrients, adaptation to the changes in environmental conditions and above all protection of the host plant against pathogens (Stirling, 2011). In order to interact, plant beneficial organisms need to have some form of direct or indirect contact with the plant pathogens (Naranjo et al., 2015). The different types of interactions were named as mutualism, commensalism, neutralism, competition, parasitism and predation (Arthur and Mitchell, 1989). These terminologies originated in macro ecology but all of these types of interactions exist in the natural world (Coppel and Mertins, 2012). In plant science, the development of plant diseases involves both plants and microbes, the interactions that lead to biological control take place at different levels and rates (Pal and Gardener, 2006). Thus, biological control is a bio effect or-method of controlling pests (including insects, mites, weeds and plant diseases) using other living organisms (Jackson and Chen-Charpentier, 2018). It relies on predation, parasitism, herbivory, or other natural mechanisms, but typically also involves an active human management role.

Bio control is considered an important component of Integrated Pest Management (IPM) programmes (Baker, Green, and Loker, 2020.). Practitioners of biological control can be thought of as applied ecologists. Their view of agricultural systems considers the roles of all of the plant, animal, and microbial organisms present in the ecosystem. This view is in contrast to traditional insect control, for example, that targets one or more insect species while ignoring the effects of pesticides on the other organisms present. Murdoch, Chesson, and Chesson (1985) assert that there are several different approaches to biological control strategies: importation (sometimes called classical biological control) which involves searching for natural enemies in the geographic area(s) where the pest originated, then "augmentation" which involves adding natural enemies to the local population and "conservation" of natural enemies which seeks to promote those conditions that preserve or increase local populations .

Van Driesche, Hoddle and Center (2009) summited that these natural enemies can be introduced to the place(s) where the pest has become a problem. The term “beneficial insect(s)” or “beneficial(s)” are also used to refer to natural enemies or the insect in question (Gareth, 2006). The term beneficial insect, however, is a general one that refers to insects that are of value because of economic or other considerations. This includes bees, for example, which are pollinators as well as producers of honey. Natural enemies of insect pests, also known as biological control agents, include predators, parasitoids, and pathogens. Biological control agents of plant diseases are most often referred to as antagonists (Gareth, 2006). Biological control is one of the several strategies used to control pests to avoid economic damage on crop plants, in husbandry, or on recreation areas. It is also used against nuisance pests.

In solanaceous plants, the effectiveness of natural enemies (predators) on the solanaceous crop to prevent primary spread of diseases is somewhat questionable, given the rapidity of virus transmission once the vector lands on the plant (Jackson and Chen-Charpentier, 2018). However, biological control agents do help to reduce the overall pests’ densities and thereby reduce secondary spread in the crop (Yusuf and Benya, 2012). However, despite the widely recognized importance of biological control as a tool for managing and controlling invasive species, sourcing biological control agents for a particular application is still generally done in a rather ad hoc fashion (Gareth, 2006). What, then, are the factors that need to be taken into account when one is interested in biocontrol?

In particular, as part of developing a more general conceptual framework that would have predictive value for biocontrol management, Autry, Bayliss and Volpert (2018) asserts that “it is important to know for a host with a given set of life-history features, what types of pathogens are likely to be most effective with respect to long-term reductions in host population size”. This involves identifying important host (e.g. longevity, dispersal mode) and pathogen (e.g. transmission mode, stage-specificity of attack) life history features that can distinguish qualitative classes of weeds and biocontrol agents, and that are likely to significantly impact on initial rates of agent spread (Gareth, 2006).

Although, other schools of thought contended that the success of biocontrol programmes depends not only on what happens within single populations, but also on regional (metapopulation) persistence. In such situations, how the agent is deployed (e.g. the number and distribution of initial release sites) may influence overall dynamics and rates of spread, and therefore affects the host. In turn, these may determine the rate at which resistance evolves in the target host (Autry et al., 2018).

2.2 Biological Control of Pests

One of the common methods of biological control of pests is called importation, which involves the introduction of a pest's natural enemies to a new location where they do not occur naturally. This is usually done by government authorities (McKimmie, 2000). In many instances the complex of nature of the natural enemies associated with a pest may be inadequate, a situation that can occur when a biocontrol agent is accidentally introduced into a new geographic area, without its associated natural enemies (Van Driesche et al., 2009).

The process of importation involves determining the origin of the introduced pest and then collecting appropriate natural enemies associated with the pest or closely related species. Selected natural enemies are then passed through a rigorous assessment, testing and quarantine process, to ensure that they will work and that no unwanted organisms (such as hyper parasitoids) are introduced (Gareth, 2006). If these procedures are passed, the selected natural enemies are mass-produced and then released. Follow-up studies are conducted to determine if the natural enemy becomes successfully established at the site of release, and to assess the long-term benefit of its presence (Lenhart and Wortman, 2007).

To be most effective at controlling a pest, a biological control agent requires a colonizing ability, which will allow it to keep pace with the spatial and temporal disruption of the habitat. Its control of the pest will also be greatest if it has temporal persistence, so that it can maintain its population even in the temporary absence of the target species, and if it is an opportunistic forager, enabling it to rapidly exploit a pest population (Autry et al. 2018). However an agent with such attributes is likely to be non-host specific, which is not ideal when considering its overall ecological impact, as it may have unintended effects on non-target organisms. There are many examples of successful importation programmes, a predatory insect Vedalia Beetle (Rodoliacardinal) is a parasitoid fly introduced from Australia by Charles Valentine Riley in the 19thCentury. Within a few years, the cottony cushion scale was completely controlled by these introduced natural enemies (Gareth, 2006).

Thus, the classical biological control is long lasting and inexpensive. Other than the initial costs of collection, importation, and rearing, little expense is incurred. When a natural enemy is successfully established, it rarely requires additional input and it continues to kill the pest with no direct help from humans and at no cost (Wilson and Tisdell, 2001). However importation does not always work. It is usually most effective against exotic pests and less so against native insect pests. The reasons for failure are not often known but may include the release of too few individuals, poor adaptation of the natural enemy to environmental conditions at the release location, and lack of synchrony between the life cycle of the natural enemy and host pest.

Augmentation, which is another method, involves the supplemental release of natural enemies, boosting the naturally occurring population. Relatively few natural enemies may be released at a critical time of the season (inoculate release) or millions may be released (inundate release) (Lenhart and Wortman, 2007). An example of inoculate release occurs in greenhouse production of several crops. Periodic releases of the parasitoid, Encarsia formosa, are used to control greenhouse whitefly, and the predatory mite Phytoseiulus persimilis is used for control of the two-spotted spider mite (Lenhart and Wortman, 2007). Lady beetles, lacewings, or parasitoids such as those from the genus Trichogramma are frequently released in large numbers (inundate release). Recommended release rates for Trichogramma in vegetable or field crops range from 5,000 to 200,000 per acre (1 to 50 per square metre) per week depending on level of pest infestation (Gareth, 2006).

The conservation of existing natural enemies in an environment is the third method of biological pest control. Natural enemies are already adapted to the habitat and to the target pest, and their conservation can be simple and cost-effective. Predators are mainly free-living species that directly consume a large number of prey during their whole lifetime. Ladybugs, and in particular their larvae which are active between May and July in the northern hemisphere (Gareth, 2006), are voracious predators of aphids, and will consume mites, scale insects and small caterpillars. Lacewings, lady beetles, hoverfly larvae, and parasitized aphid mummies are usually present in aphid colonies. Cropping systems can be modified to favor the natural enemies, a practice sometimes referred to as habitat manipulation. Providing a suitable habitat, such as a shelterbelt, hedgerow, or beetle bank where beneficial insects can live and reproduce, can help ensure the survival of populations of natural enemies.

Biological control can potentially have positive and negative effects on biodiversity. The most common problems with biological control occur via predation, parasitism, pathogenicity, competition, or other attacks on non-target species (Van Driesche et al., 2009). Often a biological control agent is imported into an area to reduce the competitive advantage of an exotic species that has previously invaded or been introduced there, the aim being to thereby protect the existing native species and ecology. However, the introduced control does not always target only the intended species; it can also target native species. In Hawaii during the 1940s parasitic wasps were introduced to control a lepidopteron pest and the wasps are still found there today (Gareth, 2006). This may have a negative impact on the native ecosystem, however, host range and impacts need to be studied before declaring their impact on the environment.

Over the past 15 years with the rise in biological control, interest there has become a greater focus on the non-target impacts that could occur. In the past, many biological control releases were not thoroughly examined and agents of biological control were released without any consideration. When introducing a biological control agent to a new area, a primary concern is its host specificity (Autry et al., 2018). Generalist feeders (control agents that are not restricted to preying on a single species or a small range of species) often make poor biological control agents, and may become invasive species themselves. For this reason, potential biological control agents should be subject to extensive testing and quarantine before release into any new environment. If a species is introduced and attacks a native species, the biodiversity in that area can change dramatically. When one native species is removed from an area, it may have filled an essential ecological niche. When this niche is absent, it may directly affect the entire ecosystem.

2.2.1 Control of pathogens by plant resistance

Resistance refers to the ability of the host plant to overcome, either completely or in part, the effect of a pathogen (Cooke et al., 2006). This ability may vary from small scale, where there may only be a slight suppression of disease development, to large extent, where pathogenesis is incomplete. Incomplete pathogenesis can adequately suppress disease. If the effects are large enough, pathogen reproduction rates are slowed to the extent that the pathogen population merely replaces lost individuals but fails to increase in size (Van Driesche et al., 2009). If the resistance effects are small, disease increases more or less rapidly and, consequently, other control techniques will be required. However, the use of resistance is not without its own problems. Genetic uniformity within a plant population can lead to pressure for new virulence’s to develop within the pathogen population, or minor pathogens to become increasingly important in some agricultural production systems.

Resistance in plants can take two possible, though not mutually exclusive, forms; termed these horizontal and vertical resistance (Gareth, 2006). Horizontal resistance is expressed against all races of the pathogen, i.e. it is race, path type or biotype non-specific. Vertical, race-specific or specific resistance can be described as resistance that is race or biotype specific. In addition, consideration needs to be given to the concepts of induced resistance, non-host immunity and tolerance. Other terms have been used in some cases more or less synonymously, to describe horizontal resistance including race non-specific resistance, general resistance, durable resistance, polygenic resistance and partial resistance. However, Cooke et al., 2006) provides compelling reasons why the term horizontal is the most appropriate.

Horizontal resistance slows down the rate at which disease increases within a single plant or population of identical plants. It can, therefore, usefully be described as rate-reducing resistance. The mechanisms by which horizontal resistance slows the rate of epidemic development may be active or passive and may be expressed through reduced infectivity, lower levels of sporulation (in both cases resulting in a reduction in the basic infection rate), lengthened latent periods, or increased rate of removal of infectious tissue (reducing the infectious period). Frequently, experimenters have examined the nature of individual components of horizontal resistance to specified pathogens within selected varieties and have correlated these with the established reactions of the same varieties when infected by that same pathogen under field conditions.

2.2.2 The pathogen population dynamics

Population dynamics is the study of how the size and structure of populations respond to the forces that act on them (Turchin, 2003). Before it is possible to study changes in populations, it is first necessary to decide, what the individuals that are being counted. A population is composed of individuals at various stages in their life cycles. A measure of the population will have to include several numbers, describing the number of each type of individual per square metre (or per square metre of host). Changes in population size may come from birth, death, immigration or emigration (Begon, Mortimer and Thompson, 1996).

Immigration and emigration will usually only occur at certain stages. If a population is described by specifying the number of individuals in each of a series of stages, matrices can be used to model the population and the ways in which it will change (Caswell, 2000). The data describing the population can be set out in a vector or list of numbers in each stage. The changes to this list from one time-period to the next can be set out in a matrix or table showing two things: the proportion of each stage which will have advanced to the next stage (for example, from latent infection to sporulation infection, or from sporulation to over-seasoning stage), and the number of newborn individuals. The number of newborn will be proportional, among other things, to the total number of propagates produced by all the infectious stages of the pathogen. Emigration will usually be implicit in such a representation, causing a reduction in the number of births, since a live pathogen individual is usually associated with a plant host that cannot move far.

2.2.3 Parasitism and predation

The degree to which two species compete is determined by the similarity with which they use available resources (Schoener, 1982). Two species that use resources identically will not be able to coexist indefinitely in a stable environment. One will inevitably be very slightly more efficient and will grow more rapidly than the other, supplanting it slowly grows or rapidly will. This is the competitive exclusion principle. At first sight, this conclusion is incompatible with the regular occurrence, on the same plant organ, of many similar pathogens: for example, in the UK on wheat, it is easy to find leaves simultaneously infected with Mycosphaerella graminicola, Didymella exitialis and Phaeosphaeria nodorum (Begon et al., 1996).

In fact, several of the assumptions underlying the competitive exclusion principle are violated (Schoener, 1982). Hosts vary dramatically in abundance and quality during the year, so that the environment for a plant pathogen is never stable; pathogens and hosts are spatially aggregated, so that the strength of competition between species is reduced relative to that between related individuals within a dense patch of disease and there may well be subtle differences in resource use by apparently similar pathogen individuals, related to temperature and wetness requirements for infection, preferred age of host tissue, etc. (Shaw, 2006).

2.3 Modelling of Interactions within Biological Spheres

Mathematical population models have been used to study the dynamics of prey- predator systems. Lotka in 1925 and Volterra in 1927 proposed a simple model known as Lotka-Volterra model of prey-predator interactions (Hoppensteadt, 2006). Since then, many mathematical models have been constructed based on more realistic explicit and implicit biological assumptions. Modeling is a frequently evolving process, to gain a deep understanding of the mathematical aspects of the problem and to yield non-trivial biological insights; one must carefully construct biologically meaningful and mathematically tractable population models. Some of the aspects that need to be critically considered in a realistic and plausible mathematical model include; carrying capacity which is the maximum number of prey that the ecosystem can sustain in the absence of predator, competition among prey and predators which can be intraspecific or interspecific, harvesting of prey or predators and functional responses of predators (Fryxell and Lundberg, 1994).

Continuous predator - prey models have been studied mathematically (Martin and Ruan, 2001). The principles of Lotka - Volterra model, conservation of mass and decomposition of the rates of change into birth and death processes, remain valid until today and many theoretical ecologists adhere to these principles. Modifications were limited to replacing the Malthusian growth function, the predator per capita consumption of prey or the predator mortality by more complex functions such as the logistic growth. The functional responses all depend on prey-abundance only, but soon it became clear that predator abundance could influence this function. However, these models usually require more parameters and their analysis is complex. Therefore, they are, on one side, rarely used in applied ecology and, on the other side, have received little attention in the mathematical literature.

A simple way of incorporating predator dependence into the functional response was proposed by (Shi, Zhao and Tang, 2014) who considered this response as a function of the ratio. Interesting properties of this approach have emerged that are in contrast with predictions of models where the functional response only depends on prey abundance. The interaction of multiple predators, predation rate and prey selection are also modified by the habitat complexity. For aquatic predatory insects, rice fields, ponds, and temporary pools are habitats with ample heterogeneity with regard to spatial structures and prey species abundance (Kot, 2001). When ample prey species are available, structural complexity is more important in determining predator success.

Khalil (2002) studied the stability of three species population models consisting of an endemic prey (bird), an alien prey (rabbit) and an alien predator (cat). It may be pointed out here that all the above studies are based on the traditional prey dependent models. Recently, it has been observed that in some situations, especially when a predator have to search for food and have two different choice of food, a more suitable predator-prey theory should be based on the so-called ratio-dependent theory, in which the per-capita growth rate should be function of the ratio of prey to predator abundance, and should be the so-called predator functional response.

Karban, Hougen-Eitzman and English-Loeb (1994) proposed and analyzed a mathematical model of two competing prey and one predator species where the prey species follow Lotka-Volterra dynamics and predator uptake functions are ratio dependent. They derived conditions for the existence of different boundary equilibrium and discussed their global stability. They also obtain sufficient conditions for the permanence of the system. Khalil (2002) further studied the qualitative properties of a ratio dependent predator-prey model. They showed that the dynamic outcome of interactions depends upon parameter values and initial data. Three general forms of functional response are commonly used in ecological models: linear, hyperbolic, and sigmoidal.

How predators respond to changes in prey availability (functional response) is an issue of particular importance. There is evidence from several models that the type of functional response specified can greatly affect a model. That is why the models are simplified to a linear relationship between predator and prey population. Hence, this simplification is justified for many arthropod predators because they can rely on plant-provided alternative food sources such as pollen or nectar, the availability of which is unlikely to be influenced by the predator's consumption.

The theory of harvesting is important in natural resource management and bio economics. Most species have a growth rate, which more or less maintains a constant population equal to the carrying capacity of the environment (this of course depends on the population). In this case, the growth and death rates are nearly equal. The harvesting of species affects their mortality rates and if the harvesting is not too much the population will adjust to a new equilibrium. It has been evident that there is a need to develop ecologically acceptable strategies for harvesting any renewable resources such as fish, plants, animals etc.

It is interesting to note that even if the excess harvest does not threaten extinction, it can cause damage to the resource in the end. Massive fruit collection from the forest has an adverse effect on regeneration. The problem then is to determine a strategy, which ensures steady harvest year after year without a progressive decline in the abundance of the resource. We consider a logistic population growth model in which the mortality rate is enhanced by harvesting; by a term that is proportional to the existing population.

There are numerous studies on the effects of harvesting on population growth. Shi et al., (2014) observed that, in the context of predator-prey interaction, some studies that treat the populations being harvested as a homogeneous resource. Theoretical ecology remained silent about the astonishing array of dynamical behaviors of three species models for a long time. Of course, the increasing number of differential equations and the increasing dimensionality raise considerable additional problems for both the experimentalist and theoretician. Kot (2001) considered three level food webs -- two competing predators feeding on a single prey and a single predator feeding on two competing prey species. They obtain criteria for the system to be persistent.

Sabelis (1986) considered a two-prey one-predator harvesting model with interference. The model is based on Lotka-Volterra dynamics with two competing species, which are affected not only by harvesting but also by the presence of a predator, the third species. Optimal harvesting policy and the possibility of existence of a bioeconomic equilibrium is discussed. Karban et al., (1994) additionally proposed a two-predator one-prey system with ratio dependent predator growth rate. Criteria for local stability, instability and global stability of the non-negative equilibrium are obtained. They also discussed the permanent co-existence of the three species. Kot (2001) still considered a two predator; one prey model in which one predator interferes significantly with the other predator is analyzed. The analysis centers on bifurcation diagrams for various levels of interference in which the harvesting is the primary bifurcation parameter.

In most predator - prey models considered in the ecological literature, the predator response to prey density is assumed monotonic increasing; this inherent assumption means that the more prey population in the environment the better for the predator population. The problem of harvesting of competitive species has drawn the attention of bio-economists. Sabelis (1990) obtained the feasible dynamic equilibrium to obtain a limit of cost per unit effort and maximum value of effort for the joint harvesting of a logistic growth model of two competitive systems. Khalil (2002) has deliberated on the bio-economic model of a ratio - dependent prey - predator system with optimal harvesting. They proved that optimal equilibrium populations lead to a situation where total user's cost of harvest per unit effort equals the discounted value of the future profit. Jackson and Chen-Charpentier (2017) proposed a biological economic model based on a prey-predator dynamics based on delay differential equations where prey species are continually harvested and predation is considered type II functional response. Sabelis (1990) have further studied a predator-prey model with Allee effect and sigmoid functional response.

However, these models are less realistic. To make a model realistic one should include some of the past states of these systems; ideally, a real system may be modeled by differential equations with time-delays. Time-delays occur so often in almost every situation, that to ignore them is to ignore reality. The time delay is the inherent property of the dynamical systems and plays an important role in almost all branches of science and particularly in the biological sciences (Karban et al., 1994). The importance derives from the fact that many of the phenomena around us do not act instantaneously from the moment of their occurrence. For example, a change in the resources or environment does not affect the survival of existing populations immediately.

There is always a time lag between the moment an action takes place and its effect is observed. In ecology, more realistic models should include some of the past states, i.e., a real system should be modeled by differential equations with time delays (Jackson and Chen-Charpentier, 2017) mentioned that animals take some time to digest their food before further activities and responses take place and hence any model of species dynamics without delays is an approximation at best. Now it is beyond doubt that in an improved analysis, the effect of time-delay due to the time required in going from egg stage to the adult stage, gestation period, et cetera, has to be taken into account. Detailed arguments on the importance and usefulness of time-delays in realistic models may be found in the classical books of (Sabelis, 1986).

Recently, it is of interest to investigate the possible existence of chaos in the biological population. The subjects of chaos and chaos control are growing rapidly in many different fields such biological systems, structural engineering, ecological models, aerospace science, and economics. Food chain modeling provides challenges in the fields of both theoretical ecology and applied mathematics. Determining the equilibrium states and bifurcations of equilibria in a nonlinear system is also an important problem in mathematical models. Two-species continuous time models have only two basic patterns: approach to equilibrium or to a limit cycle.

In contrast, simple discrete time models of even a single species can exhibit chaotic behavior. But research of the last 25 years (Karban et al., 1994) demonstrates the very complex dynamics that can arise in model systems (in continuous time) with three or more species. Several of the early mathematical investigations of chaos were of ecological models (Sabelis, 1986). An investigation by Shi, et al., (2014) showed that a system of one predator and two competing prey can exhibit chaotic behavior. Nurul (2014) have been especially persuasive in his view that chaos may be a much more important phenomenon than ecologists had earlier believed. Kot (2001) argued that the reason why field ecologists have not been able to get any reliable proof of the existence of chaos is that they have preoccupied notions and that they are observing what they want.

It was argued by many authors that poor data quality (short and noisy character of the time-series) makes the techniques of nonlinear dynamics an unsuitable tool for the analysis of ecological data, and thus, leads to failure of such attempts. One common hindrance (in the study of nature populations) to understand the underlying dynamical process has been the non-availability of data of suitable length and precision. Another difficulty is that there is no method which can fix these parameters a priori .On the other hand, the existence of chaos in almost all the physical systems motivates one to critically study the same in natural populations.

2.4 Stability of Biological Systems

Stability is a very important issue in the study of nonlinear systems (Chen, 2005). Concisely, stability answers that question whether a small change (a small disturbance) of a physical system at some instant time changes the behavior of the system only slightly at all future time. There are undoubtedly a number of stability theories frequently associated with dynamical systems such as local stability, global stability, absolute stability and Lyapunov stability of equilibrium points (Martynyuk, Radziszewski and Szadkowski, 2020). Overall, an equilibrium point is said to be locally stable if small perturbations remain close to the equilibrium, and locally asymptotically stable if small perturbations eventually return to the equilibrium. As such, it does not predict the overall behavior of the system.

On the other hand, when small perturbations continue to move away from the equilibrium, the equilibrium point is said to be unstable. Equilibrium is globally stable if a system approaches the equilibrium regardless of its initial position. In most biological models, the Routh-Hurwitz criterion is used to determine asymptotic stability of equilibrium for nonlinear systems of ordinary differential equations (ODEs). For local stability, the Routh-Hurwitz criterion gives the necessary and sufficient conditions for all roots of the characteristic polynomial to have negative parts, thus implying asymptotic stability. In the mathematical theory of dynamical systems, the Lyapunov theory is commonly considered the most useful general theory for global stability (Lyapunov, 1992), and it is a mainstay in the engineering discipline of control theory (Giesl and Hafstein, 2015).

Many scholars detail that the Russian mathematician - Lyapunov developed a method, in 1982, for the analysis of the stability of ODEs. The method is used to establish global stability of a system of nonlinear ODEs (De Le’on, 2009). Moreover, in honor of his work, such functions are habitually referred to as Lyapunov functions (Korobeinikov, 2004). The method of Lyapunov functions has been used extensively in mathematical biology. For instance, Korobeinikov and Wake (2002) studied a Lyapunov function for SIR, SIRS and SIS epidemiological models which was later extended to a wider range of epidemic models SEIR and SEIS.

Lyapunov's direct method (also known as Lyapunov second method) provides a way of analysing the stability of nonlinear systems without actually solving the differential equations. The idea behind Lyapunov's direct method is that the system is stable if there exists some Lyapunov function in the neighbourhood of the equilibrium point. Thus, it can be shown that Lyapunov's direct method is a sufficient condition for the stability of nonlinear systems (Martynyuk et al., 2020). This method is one of the most powerful techniques for qualitative analysis of a dynamical system. Since dynamical systems are a main tool for modeling in the applied sciences. Therefore, Lyapunov functions appear in various branches of science such as meteorology, biology, computer science, and physics.

The technique employs an appropriate auxiliary function, called a Lyapunov function with specific properties, which is not easy to find, especially for multidimensional systems. But a general form of Lyapunov functions used in the literature of mathematical biology is originally from the first integral of a Lotka - Volterra system (Shuai and Van Den Driessche, 2013). Usually, when applied to disease models, some suitable coefficients have to be determined such that the derivative of along solutions of the model is non-positive, and such a determination becomes very challenging for models with high dimension.

2.5 The Basic Reproduction Number and Disease Control

In epidemiology, stability theory is used to understand the dynamics of infectious diseases as well as predicting their transmission pattern. In depicting the transmission of infectious diseases, the population is commonly divided into susceptibleAbbildung in dieser Leseprobe nicht enthalten, infectiousAbbildung in dieser Leseprobe nicht enthalten, and recovered Abbildung in dieser Leseprobe nicht enthalten individuals; see (Safi and Garba, 2012). The target is to checkmate the disease spread. In addition, the key concept that will help us achieve that is the basic reproduction number. The basic reproduction number, usually denoted by R₀, is defined as the average number of secondary infections produced by a single individual during his or her entire infectious period, in a fully susceptible population. Mostly, the basic reproduction number plays the role of a threshold parameter that predicts whether the disease will spread or die out.

In stability of infectious disease model, when R0 > 1, then introducing an infected individual into a population results in an epidemic (the disease will spread throughout the population). Nevertheless, when R0 < 1, then it means introducing a few infected individuals into a fully susceptible population will cause the disease to die out. Thus, basic reproduction number plays an important role in helping to quantify possible disease control strategies by focusing on the important aspects of a disease, determining threshold quantities for disease survival, and evaluating the effect of particular control strategies (van den Driessche, 2017). Such a quantity is relative, because the value of R0 for a specific disease depends on many variables, such as location and density of population. Furthermore, the larger the magnitude of R0, the faster the disease will spread, and presumably the more difficult it will be to control.

While R0 is a great concept in mathematical epidemiology and has been widely used since its first application in 1952 by George MacDonald, the mathematical definition of R₀ is problematic and in some cases ambiguous. The idea is derived from the idea of a reproductive number in population dynamics, which is defined as the expected number of offspring that one organism, will produce over its lifespan. Hartemink, Randolph, Davis and Heesterbeek (2008) conducted a review on the history of R₀, which opined, “the root of the basic reproduction concept can be traced through the work of Alfred Lotka, Ronald Ross, and others, but its first modern application in epidemiology was by George MacDonald in 1952, who constructed population models of the spread of malaria”. Hence, the basic reproductive number, R₀, is a key concept in epidemiology, and is inarguably one of the foremost and most valuable ideas that mathematics has brought to epidemic theory (Heesterbeek and Dietz, 1996).

If the basic reproduction number R0<1, then the disease free equilibrium is Locally Asymptotically Stable (LAS), and the disease cannot invade the population. On the other hand, if R0>1, then the disease will eventually invade the population (Hethcote, 1975). For simple homogeneous models, the reproductive number can be defined as the product of the number of contacts that one individual has per unit time, the probability of transmission per contact and the duration of the infectious period. However, for more complicated models with several infected compartments this simple heuristic definition of R0 is insufficient. A more general basic reproduction number can be defined as the number of new infections produced by a typical infective individual in a population at a Disease Free Equilibrium (DFE) state.

There are several different methods in which R0 can be calculated (Mitchell and Kribs, 2017). Some common methods of constructing R₀ are the survival function method, the Next Generation method, existence of an endemic equilibrium, final size equation, constant term of the characteristic polynomial, etc. Many of them yield different values of R0 for the same model, and many methods produce different values of R0 based on what the modeler considers appropriate. Each method derives its conditions from the threshold nature of R₀, yet many of these methods produce a value that is not consistent with the biological definition (Mitchell and Kribs, 2017). It is important to understand that employing one of the methods at random does not guarantee the calculation represents the number of secondary infections arising from a single infected individual. Many methods produce different values for R0 even in the same system. How can two different values simultaneously represent the number of secondary infections from a single infected individual? If more than the threshold capability of R0 is of concern, careful consideration should be taken when using a method to calculate R0 (Hethcote, 1975).

Perhaps the most common method of calculating R0 is the Next Generation method (Yang, 2014). This approach places appropriate terms from the infected class equations into the vectors and. terms that describe appearances of new infections belong in, and terms that describe a transfer of existing infections belong in and should be negated. The Jacobian matrices obtained by differentiating and with respect to the relevant subset of variables are computed and evaluated at a nontrivial disease-free equilibrium (DFE), resulting in the matrices and, respectively. Thus, most biologists will claim that there is only one value of R0 for any model. While that may be true, many indices exhibit the same threshold behavior.

2.6 Sensitivity Analysis of Biological Models

Mathematical modeling has become an important tool for assessing the effect of different intervention strategies on the spread of infectious diseases within a population (Cettl and Dvofiik, 1982). The majority of existing work falls into one of two categories. In the first category, intervention strategies are modeled by a constant parameter and the goal is to understand how changing the value of the parameter changes the dynamics of the system (Frank, 1978). Often the aim is to determine the best parameter value for a given performance measure. In the second category, intervention strategies are allowed to vary as a function of time and the goal is to determine the best function for a given performance measure (Frank, 1978). Due to the complexity of the models used, these studies are often numerical and are of varying generality. In some studies only a handful of specifically chosen functions are compared and in other studies a wide range of functions are considered (Cettl and Dvofiik, 1982).

Sensitivity analysis is extremely important for mathematical models. Sensitivity analysis studies the variation of the outputs of a model caused by variations in the inputs. In essence, sensitivity analysis determines which parameters and initial conditions (inputs) affect the quantities of interest (outputs) of the model the most. The first reason why this analysis is important is that it tells the researcher which parameters deserve the most numerical attention. A highly sensitive parameter should be carefully estimated as a small variation in that parameter will lead to large quantitative changes to the quantity of interest and may even produce qualitatively different results.

An insensitive parameter, on the other hand, does not require as much effort to estimate as a small variation in that parameter will not produce large changes to a quantity of interest. Many at times in model analysis, the most sensitive parameters are also the best established in the sense that the values do not change much from one time to the next. If this is the case, the second reason for sensitivity analysis becomes more pronounced. That is, sensitivity analysis highlights which parameters should be attacked in management strategies. One goal of mathematical modeling is to determine what the current outcome of a system may be, and if necessary, discover how to change any negative outcomes. Changing the values of the most sensitive parameters will be the most effective strategy in changing the results of the model (Cettl and Dvofiik, 1982). The modeler will then implement any applicable real-world scenarios that will change the value of the most sensitive parameter to obtain the most control over the outcome.

The sensitivity is computed by finding the derivatives of each variable with respect to each parameter at any time. To determine best control measures, knowledge of the relative importance of the different factors responsible for transmission is useful. Van den Driessche and Watmough (2002) asserted that, “initially disease transmission is related to R0 and sensitivity predicts which parameters have a high impact on R0”. The sensitivity index of R0 with respect to a parameter ω is. Another measure is the elasticity index (normalized sensitivity index) that measures the relative change of R0 with respect to ω, denoted byAbbildung in dieser Leseprobe nicht enthalten, and defined as:

Abbildung in dieser Leseprobe nicht enthalten (2.1)

The sign of the elasticity index tells whether R0 increases (positive sign) or decreases (negative sign) with the parameter; whereas the magnitude determines the relative importance of the parameter.

This index (2.1) can guide control by indicating the most important parameters to target, although feasibility and cost play a role in practical control strategy. If R0 is known explicitly, then the elasticity index for each parameter can be computed explicitly, and evaluated for a given set of parameters. The magnitude of the elasticity indices depends on these parameter values, which are probably only estimates. Latin Hypercube Sampling maximum is used to help identify the robustness of R0 to the parameters. Another technique to investigate this is to compute R0 over the feasible region of a given parameter while keeping the other parameters fixed at baseline values.

2.7 Re viewed Literatures on Mathematical Models of Plants Diseases

Xiao and Van Den Bosch (2003) formulated and analyzed a model concerning the effect of a wild host plant species on biologically based technologies for pest control (BBTs). The pest species considered do not only feed on the crop but also have a wild host species. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibrium, permanence and global stability are analyzed. It was said to be the first approach to the dynamics of bio-control programs when not only a crop is a resource for the pest but also a wild host plant species is present. The results however show that the presence of a wild host species can have a profound effect on the dynamics and effectiveness of a bio-control program.

Maiti, Pal and Samanta (2008) studied a tritrophic model consisting of tea plant, pest and predator to analyze different field observations. They considered the fact that growing consciousness among the tea industry to reduce the use of the chemical pesticides for pest control. They concentrated on predators that are beneficial insects that feed on harmful insects and mites, which incur considerable loss of production of tea. In essence, the effect of discrete time-delay on the tritrophic model is studied critically (Maiti et at., 2008). The dynamical behaviors of the model were studied both analytically using stability analysis and numerically by computer simulation. The results suggest some theoretical measures, which may be implemented for successful biological control.

Jeger, Jeffries, Elad and Xu (2009) developed a generic modelling framework to understand the dynamics of foliar pathogen and biocontrol agent (BCA) populations in order to predict the likelihood of successful biocontrol in relation to the mechanisms involved. The model considers biocontrol systems for foliar pathogens only and, although it is most applicable to fungal BCA systems, does not address a specific biocontrol system. Four biocontrol mechanisms (competition, antibiosis, mycoparasitism and induced resistance) were included within the model rubric. Qualitative analysis of the model showed that the rates of a BCA colonizing diseased and/or healthy plant tissues and the time that the BCA remains active are two of the more important factors in determining the final outcome of a biocontrol system. Further evaluation of the model indicated that the dynamic path to the steady-state population levels also depends critically on other parameters such as the host–pathogen infection rate.

Mailleret and Grognard (2009) studied a meta-model, augmented by a discrete part describing periodic introductions of predators as an impulsive model of augmentative biological control consisting of a general continuous predator–prey model in ordinary differential equations. The model assumed that the per time unit budget of biological control (i.e. the number of predators to be released) is fixed and the best deployment of this budget is sought in terms of release frequency. Furthermore, Mailleret et al., (2009) established the existence of an invariant periodic solution that corresponds to prey eradication is shown and a condition ensuring its global asymptotic stability. An optimization problem related to the preemptive use of augmentative biological control is then considered. The cost function to be minimised is the time needed to reduce an unforeseen prey (pest) invasion occurring at a worst time instant under some harmless level. The analysis shows that the optimization problem admits a countable infinite number of solutions. Thus, it is shown that the cost function is decreasing in the predator release frequency so that the best deployment of the biocontrol agents is to carry out as frequent introductions as possible.

Shi and Chen (2010) investigated an impulsive predator–prey model with disease in the prey for the purpose of integrated pest management. In the first part of the main results, they got the sufficient condition for the global stability of the susceptible pest-eradication periodic solution. This means if the release amount of infective prey and predator satisfy the condition, then the pest and the natural enemy will coexist for all time. Shi and Chen (2010) concludes, therefore, that integrated pest management strategy is superior to those who only release infective prey (pest) or only release predator (natural enemy). Hence, the mathematical results presented are considered as a prior strategy for pest management. Nevertheless, their model, supposed to release infective pests and natural enemies at the same time, which is not always true in real situations.

Rafikov and Holanda Limeira (2012) proposed a simple mathematical model of interaction between the sugarcane borer (Diatraea saccharalis) and its egg parasitoid (Trichogramma galloi). Based on Rafikov and Holanda Limeira model (2012), sugarcane borer is represented by the egg and larval stages, and the parasitoid is considered in terms of the parasitized eggs. Linear feedback control strategy is proposed to indicate how the natural enemies should be introduced in the environment. Their result shows that a great amount of parasitoids have to be introduced in initial days. This fact suggests that the proposed feedback control strategy can be integrated into existing biological control technologies, combining the feedback control with the traditional inundate pest control. Conclusively, this control strategy directs the ecosystem to the stable equilibrium point, which is reached at 40 days. After this period, according Rafikov and Holanda Limeira (2012) proposed control strategy, it is necessary to apply the constant control u * = 12.15 parasitized egg/day. It is not economically advantageous to use this constant control. In agricultural practice this control can be substituted by periodic releases of a small population of natural enemies.

Abdul Latif (2014) proposed and analyzed a mathematical modelling of induced resistance to plant disease. The underlying theory of induced resistance (IR) is concerned with the situation when there is an increase in plant resistance to herbivore or pathogen attack that results from a plant’s response triggered by an agent such as elicitors (also known as “plant activators”). The model studied by Abdul Latif (2014) adapted the traditional Susceptible – Infected - Removed (SIR) approach but is characterized by three main compartments, namely: susceptible, resistant and diseased. The model is generic and will be applicable to a range of plant-pathogen-elicitor scenarios. The outcome of model’s analysis predicted the relative proportion of plants in each compartment and quantitatively estimates the elicitor effectiveness. The numerical evaluations of this IR model provide a potential support tool for the development of elicitors that are more potent and its application strategies. Finally, an application of optimal control theory is derived to determine the best strategy for a continuous elicitor application.

Shi, Zhao and Tang (2014) developed an epidemic model which describes vector-borne plant diseases is proposed with the aim to investigate the effect of insect vectors on the spread of plant diseases. Firstly, the analytical formula for the basic reproduction number Abbildung in dieser Leseprobe nicht enthalten was obtained by using the next generation matrix method, and then the existence of disease-free equilibrium and endemic equilibrium is discussed. Secondly, by constructing a suitable Lyapunov function and employing the theory of additive compound matrices, the threshold for the dynamics is obtained. Thus, if , then the disease-free equilibrium is globally asymptotically stable, which means that the plant disease will disappear eventually; if Abbildung in dieser Leseprobe nicht enthalten > 1, then the endemic equilibrium is globally asymptotically stable, which indicates that the plant disease will persist for all time. Finally some numerical investigations are provided to verify our theoretical results, and the biological implications of the main results are that, the total number of the host plant, birth rate of the vector, and incidence rate of the disease can positively affect the value of Abbildung in dieser Leseprobe nicht enthalten; while the death rate of the host plant, the death rate of the vector, and the disease-induced death rate can negatively affect the value of Abbildung in dieser Leseprobe nicht enthalten; but the saturation rate of the incidence has no relation to the value of Abbildung in dieser Leseprobe nicht enthalten.

Murwayi, Onyango and Owour (2017) formulates and analyzes a dynamical nonlinear plant vector borne dispersion disease model that incorporates insect and plant population. The main point of concern was that insect vector borne plant diseases were considered a major concern due to abundance of insects in the tropics, which affects negatively on food security, human health and world economies. Hence, Murwayi et. al., (2017) assert that, “the plant diseases caused by pathogens like bacteria, viruses, fungi, protozoa and pathogenic nematodes are propagated through media such as water, wind and other intermediary carries called vectors, and are therefore referred to as vector borne plant diseases''. Consequently, the plants' yields are reduced significantly if they are infected by vector borne diseases. The main task is, therefore, the elimination or control of vectors, which can be achieved through understanding the process of propagation via Mathematical modeling (Murwayi et. al., 2017). To improve on the accuracy of their result, they incorporate climate change parameters. Their model also studied the presence of insects and plant population at equilibrium as well as the effect of wind as a parameter of climate change. They carried out local and global stability of the model in addition to sensitivity analysis of the basic reproduction number.

Jackson (2018) presented a PhD dissertation on modeling plant virus propagation and an optimal control. The whole dissertation comprised of four independent but interrelated published papers. Firstly, Jackson and Chen-Charpenier (2016) addressed the issue of modeling plant virus propagation with delays but since there were no data collected about a specific plant, virus, and insect vector; the authors assumed a general model. The major limitation of their work is that, it fails to acknowledge the reality associated with insect vectors, as there is hardly a case where only one class of vector patronizes a certain plant. Along with that, the issue of intra specie and interspecies competition among insect vectors was not investigated which this study will take into account.

Jackson and Chen-Charpenier (2017) presented a model of biological control of plant virus propagation with delays. The paper presented two plant virus propagation models, one with no delays and the other with two delays. In that case, the aspect of plant virus propagation with delays has been extensively studied. However, an anomaly occurred in the model formulation, a parameterAbbildung in dieser Leseprobe nicht enthalten, which was defined as plant disease induced death rate, was erroneously used as replanting rate of plant. In addition, the issue of biting rate of vectors due to plants has to be modified, as plants do not bite. Therefore, this study will amend these faults. Lastly, Jackson (2018) applied the model to a specific example and recommends that the parameters be modified to fit a particular situation. Consequently, this study is an attempt in that direction as it considers a particular situation.

Jackson and Chen-Charpentier (2018) proposed a model on plant virus propagation with seasonality. They considered six populations: susceptible plants infected plants recovered plants susceptible insect vectors infected insect vectors and predators . Each variable describes its respective population at time . In the words of Jackson and Chen-Charpentier (2018), the biting rate of insect vectors due to plants as well as the biting rate of plants due to insects is denoted by Abbildung in dieser Leseprobe nicht enthaltenandAbbildung in dieser Leseprobe nicht enthalten respectively, accounting for the periodic nature of the insects. They focused on modeling plant virus propagation with seasonality and argued that because of the seasonality, the insects are mostly active in the warm months and almost dormant in the cooler ones, and the model was analyzed with periodic coefficients. Finally, the fourth paper looked at an optimal control of plant virus propagation with seasonality and delays. Majorly, Jackson (2018) studied an optimal control problem. The aim was to minimize the cost of the insecticide, number of pests and forcers of infection. The result was well established, but it is outside the scope of this study.

Chowdhury, Basir, Takeuchi, Ghosh and Roy (2019) presented the formulation and analysis of a mathematical model for Jatropha curcas plantation with a view to control its natural pests using application of integrated pesticides. Bio-pesticides are costly; require a long term process and expensive to impose. However, if chemical pesticides are introduced in the farming system along with bio-pesticide, the process will be faster as well as cost effective. Thus, Chowdhury et al., (2019) studied the control of pests using an integrated approach i.e. using a combination of bio-pesticides and chemical pesticides. They identified the parameter for which stability switches may occur. Finally, optimal concentration profiles of both pesticides have been determined using optimal control theory to minimize its negative effects and to make the process cost effective. This has provided a new insight to the marginal farmers those who practice this in the real world system.

Amelia, Anggriani, Istifadah and Supriatna (2020) developed a mathematical model of the spread of the yellow virus in red chili plants with the growth of insects as vector carriers of disease following the logistical function. They show the value of the basic reproduction number from the model by determining the dominant eigenvalue of the next generation matrix. The results show that when this is smaller than one, non-endemic equilibrium points will be stable. Also, they provided examples of numerical simulations that describe the population of models that have been developed. The simulation results provided show that if the use of V. lecanii is more than 30%, the population of infected plants in the vegetative and generative phases will experience extinction, as will the infected population of B. tabaci.

Suryaningrat, Anggriani, Supriatna and Istifadah (2020) proposed a mathematical model of the vector-borne rice tungro disease considering predator-prey interaction between green leafhoppers and frogs. Their purpose was to investigate the effect of biological agents in the spread of diseases. But they also considered insecticide usage and its effect on the system. Using the theory of the dynamical system, Suryaningrat et al., (2020) examined the existence of equilibrium points. Furthermore, an optimal control strategy that suppresses the spread is proposed. By using the optimal control theory and the Pontryagin maximum principle, the optimal conditions of the disease spread model with natural predators are added to the control variable in the form of insecticide administration. Numerical simulation of this model shows that combining the control with natural predators is able to reduce or even eliminate the number of infected rice plants and vectors significantly.

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