The basic reproduction number of vector-borne plant virus epidemics

Van den Bosch, Frank.

Computational and Systems Biology, Rothamsted Research, Harpenden AL5 2JQ, United Kingdom

Jeger, Michael J.*

Centre for Environmental Policy, Imperial College London, Silwood Park, Ascot SL5 7PY, United Kingdom

*Corresponding author: m.jeger@imperial.ac.uk

Abstract

The basic reproduction number R0 is a key parameter in plant disease epidemiology, which largely determines whether or not an epidemic will occur in a plant population. The next generation matrix approach to deriving and calculating the basic reproduction number of a plant virus epidemic is described. The approach is illustrated through a series of examples of increasing complexity, ranging from the simplest case of one vector transmitting one virus to a single host, to the case of multiple vectors, to combined horizontal (vector) and vertical (seed) transmission, and where vector control using insecticides is practised. The importance of parameters representing host and vector population dynamics and their interaction in the absence of disease is stressed, and the constraints these place on the calculation of the basic reproduction number. Finally, mention is made of further elaborations to the approach that could prove useful in plant virus epidemiology.

1. Introduction

The basic reproduction number is defined as the total number of infections arising from one newly infected individual introduced into a healthy population. On the population level the basic reproduction ratio is the ratio between subsequent generations. Originally developed with an emphasis on infectious diseases of humans and animals, the generality of the approach is recognised as equally applicable for plant diseases. In that context, specification of R0 gives a calculation tool that enables the estimation of the number of secondary infections that will occur in a healthy and susceptible plant population following the introduction of the pathogen, in the case considered here a virus. For disease to persist and for an epidemic to develop the value of R0 must be greater than one. Hence R0 is also the invasion threshold for an epidemic to occur. We note first that in epidemics modelled by systems of non-linear differential or difference equations it is possible to derive criteria that also determine whether invasion will occur or not (e.g. Jeger et al., 2015). However,

although these criteria are often helpful and biologically interpretable, they may not provide a numerical value for the number of secondary infections as described for R0 above.

A systematic approach to calculating R0 is through the Next Generation Matrix (Diekmann et al., 1990; Heesterbeek, 2002) approach, as interpreted and described for plant diseases by van den Bosch et al. (2008); and most recently, with an emphasis on the mathematical derivation and analysis, for vector-borne plant diseases by Shi et al. (2014). Although virus diseases were analysed in van den Bosch et al. (2008), the involvement of vectors in transmission was not considered explicitly. This is because the R0 of these systems have some specific properties when calculated with the next generation matrix approach.

This paper discusses these properties through the use of four examples: 1) the simplest case of one vector and one virus, 2) multiple vectors and one virus, 3) combined vector and seed transmission, and 4) vector control through the use of insecticides, where we calculate the ‘time in stage’ resulting from exposure to the insecticide. We will also introduce an additional topic in the use of the next generation matrix approach, the internal steady-state of the disease free system in the absence of the virus, in which the importance of vector population dynamics parameters is shown. In so doing, our aims are to illustrate the applicability and flexibility of the next generation matrix approach, to discuss the limitations of interpretation and use of the method, and to outline briefly extensions that thus far have had limited application in vector-borne plant virus diseases.

2. Examples2.1 The simplest case: one plant-host, one virus, one vector

We consider first the case where there is one plant host and one vector (Table 1). In this case the host population is healthy (uninfected) and the vector does not carry the virus (non-viruliferous). Suppose now that one viruliferous vector Y and/or one infected plant I is introduced into the population. We assume that both the total plant and insect population are very large so that no density dependence effects need to be considered. In this simple case there are no latent periods in either the host or vector populations: i.e. healthy plants become infectious instantaneously when inoculated and vectors become viruliferous and capable of inoculating healthy plants as soon as they acquire the virus.

The dynamics are shown schematically in Fig.1 Example 1. In this Example and for the others considered in Fig.1 the intention is to show a general framework for the considering the virus dynamics, not a formal Markov chain representation. The equations describing the dynamics can be written as:

I n+1=R11 I n+R21Y n

Y n+1=R12 I n+R22Y n

Where R11 is the number of infected hosts that one infected host individual causes during its lifetime (which for vector-host relations is obviously zero), R21 is the number of infected

hosts that one infected vector causes during its lifetime, R12 is the number of infected vectors that one infected host causes during its lifetime and finally R22 is the number of infected vectors that one infected vector causes during its lifetime (which is 0). This equation can be written in the form of the next-generation matrix as:

( I n+1

Y n+1)=(R11 R21

R12 R22)( I nY n)

where R11=¿ R22=0 , R21=βH τY ,R12=αX τ I . Where H and X are the healthy host population density and the non-viruliferous vector population density, respectively. The parameters are described in Table 2. So, in the generation n + 1, plants will be inoculated at a rate βH per infectious vector for the period of time τY the vector remains infective, assumed to be constant; non-viruliferous vectors will acquire virus at a rate αX per infected plant for the period of time τ Ian infected plant remains infectious, also assumed to be constant.

Following the assumptions and procedures outlined above, the next generation matrix is

A=( 0 βH τYαX τ I 0 )

The dominant eigenvalue of the next generation matrix is interpreted as R0 (Heesterbeek, 2002), where the time-step of the matrix is the generation time.

We note that this matrix can also be written as the product of two, biologically intuitive to interpret, matrices of infection rates and lifetimes:

A=( 0 βHαX 0 )(τ I 0

0 τY )From this formula the following characteristic equation can be derived R0

2−βH τY αX τ I=0

By solving this quadratic equation the basic reproduction number is found to be R0=√βH τY αX τ I

or equivalently,

R02=βH τY αX τ I

A heuristic argument for the derivation of the basic reproduction number could start with the introduction of one viruliferous vector. As discussed above, this vector infects H healthy hosts per time unit, and does so for Y time units. After its infectious period the viruliferous vector thus leaves HY infected plants behind. Each of these infected plants cause X vectors to become viruliferous per time unit. Each plant is infectious for I time units so leaves XI viruliferous vectors behind. In total, one viruliferous vector thus produces βH τY αX τ I viruliferous vectors. Thus it could be concluded that R0=βH τY αX τ I .

However, if we follow the definition of R0 strictly, the reasoning above has taken two cyclically alternating generations, one from vector to host plant and one from host plant to vector, and the definition of R0 is per [cycle] generation. The next generation matrix approach does correspond exactly to the definition, as it takes one, either host or vector, generation as its reference time step. The expression so derived thus gives the [true] average value for the numbers of infections that arise per infection in one generation step.

We agree, however, that in some circumstances the assumption of a two-step sequential process makes perfect biological sense. For example, when a virus can be transmitted through infected seed it makes most agronomic sense to discuss the multiplication factor of the fraction of seeds infected, which includes the two step process from seed to vector back to seed. Similarly, even when a completely healthy crop population is planted, the subsequent arrival of a very few viruliferous vectors in an otherwise non-viruliferous immigrant population can lead to the inoculation of plants. The sequence is then vector-plant-vector. The respective residence times of the virus in the plant and vector are important aspects that affect the interpretation of the next generation approach.

2.2 Example 2. One host, one virus with multiple vectors

For many plant virus diseases there may be more than one vector species involved in transmission, usually closely-related species sharing many biological characteristics. The analysis from the previous section is readily extended to the case where there are multiple vectors transmitting virus to the host plant (Fig. 1 Example 2). A similar extension can also be made for the case of one vector affecting multiple hosts (not shown).

Following the same procedures, and using the subscript n to denote the nth vector species, the next generation matrix is then

A = (0 β1H τY 1

α1 X1 τ I 0⋯ βnH τYn⋯ 0

⋮ ⋮α nX n τ I 0

⋯ 0⋯ 0 )

with the characteristic equation

R0n−1(R0

2−√∑i=1

n

βiH τYiα i X i τ I )=0

where the summation is made over all n vector species.

By solving this quadratic equation the basic reproduction number is found to be

R0=√∑i=1

n

βiH τYiα i X i τ I

This example shows the use of the next generation matrix approach as compared to the method of constructing R0 through a heuristic argument. It is possible, for each vector, to

reason through what the R0 for this species would be, but the question then is ‘how to average all these’ across the multiple vectors to arrive at a basic reproductive number for the virus? The next generation matrix does exactly that.

As an extension to the multiple vector case, we can consider competition between vector species; in the simplest case using Lotka-Volterra competition equations. We take this up in a later section where we show how the steady-state population densities in the absence of disease (the ‘disease-free equilibrium’) enter into the calculation of R0.

2.3 Example 3. Vector transmission combined with seed transmission.

Vector transmission is one means of ensuring horizontal transmission from an infected to a healthy plant within the same cohort. Seed transmission by contrast ensures vertical transmission from an infected plant to its progeny. For many plant virus diseases there is often a combination of both vector and seed transmission which has important epidemiological consequences.

If we assume that the plant population density is stable, this implies that for each host there is on average one progeny plant. Let q be proportion of the progeny of infected plants infected due to seed transmission (Fig. 1 Example 3).

The next generation matrix, combining both vector and seed transmission, is then

A=( q βH τYαX τ I 0 )

with characteristic equation

R02−q R0−βH τY αX τ I=0

so that

R0=−q±√q2+4 βH τY αX τ I

2

This equation for R0 corresponds to the general equation for dominant eigenvalue of a 2×2 matrix derived on van den Bosch et al. (2008, equation 7), with the elements Rij filled in with the parameters for seed transmission. With seed transmission alone, β = 0 and hence R0 = q < 1 (the proportion of infected progeny), so the virus will not persist. As a consequence, always some level of horizontal transfer is needed , such as provided by vector transmission, for R0 to be greater than one, and hence for the virus to persist/invade. This conclusion also applies when less restrictive assumptions on plant fecundity are made (Hamelin et al., 2016). Pollen transmission from an infected donor plant enables both horizontal transmission (directly to the receptor plant vegetative tissues) and indirect vertical transmission in which the seed of the receptor plant is infected. In this way, pollen combined with seed transmission enables virus persistence in the absence of a vector (Hamelin et al., 2016).

The derived R0 expression above is difficult to interpret intuitively. As also pointed out in van den Bosch et al. (2008), there may be pitfalls in interpreting threshold criteria from non-linear differential equations used to describe the system. With such equations the infectious periods are exponentially distributed rather than constant, and it is possible to show that with seed transmission the invasion criterion is given by

q+ βH τY αX τ I>1

This criterion, which is equivalent to the R02 expression for the single vector case in 2.1 with

the addition of the seed infection probability q, can be interpreted retrospectively, is a simpler expression than R0 derived from the next generation matrix, and corresponds to how plant disease epidemiologists have viewed the problem in the past. So there is much to say for other approaches than the next generation matrix, but it is important to realise that in most cases the next-generation-matrix will guarantee that what is derived is an invasion threshold and the correct R0.

2.4 Example 4. Vector control through insecticides.

In normal agronomic practices, insecticides are used to control populations of insect pests which can have severe effects on crop yield and performance. Application of insecticides is known to affect the behaviour of the insects at sub-lethal doses; and if the insect is also a virus vector, this may affect virus transmission by increased insect activity and dispersal.

Here we concentrate on how control using insecticides can affect calculation of the basic reproduction number. The example is a case of type change (see van den Bosch et al 2008). A vector can become viruliferous (i.e. acquire the virus) when it is or is not exposed to the insecticide. A viruliferous vector not exposed to the insecticide can become a viruliferous vector exposed to the insecticide (Fig. 1 Example 4). For the purposes here we ignore sub-lethal behavioural effects in which vectors may revert from the exposed to the non-exposed state.

This situation can be modelled by means of two matrices: an infection rate matrix, and a time in stage transition matrix, as discussed below.

The ‘Infection rate’ matrix is R=[ 0 βH β sHαX 0 0α s X 0 0 ]

where the subscript s indicates acquisition and inoculation when exposed to the insecticide.

The ‘Time in stage’ matrix is T=[τ I 0 00 τY 00 τY →YS τYS ]

Where I is the mean time an infected host remains in the infectious state, Y is the mean time an insect individual that is not hit by the insecticide remains in the viruliferous state, YS

is the time an insect individual hit by the insecticide remains in the viruliferous state and τY →YS represents the mean time a viruliferous individual remains not hit by the insecticicde.. The product of these two matrices is then the next generation matrix (van den Bosch et al 2008) and can be used to derive the basic reproduction number:

The next generation matrix is then A = RT, with characteristic equation

λ {λ2−αX τ I ( βH τY+βsH τY →YS )+α s X τ I βsH τYS}=0

So,

R0=√βH τY αX τ I+β sH τYSα s X τ I+ βsH τY →YSαX τ I

The terms in the expression can each be interpreted as the R0 values related to each type change that occurs as a consequence of insecticide application. The parameters τY and τY →YS are dependent. The larger is τY →YS, the smaller is τY . If there is a simple relation between insecticide application rate and the values of τY and τY →YS, then R0 could be plotted against application rate to evaluate the effectiveness of control.

The terms in the transition ‘Time in stage’ matrix depend on a range of processes and are closely related to the background system. For example the time a vector spends as not exposed to the insecticides depends on its intrinsic death rate as well as on the insecticide application rate. Constructing the time-in-stage matrices can be challenging. However, if the rate constants with which vectors move between stages and out of the system can be specified, then putting these in matrix form and inverting the matrix gives the time-in-stage matrix T.

It is also known that biological control through the use of predators or parasitoids can have similar effects on vector behaviour and movement and increase the rate of transmission in some systems (Jeger et al., 2011b). Operations such as the roguing of diseased plants or vegetation management practices which disrupt vector populations may also have a similar effect and could be considered in a way similar to that of insecticide control.

3. The disease free system

3.1 Healthy plants and non-viruliferous vector

In van den Bosch et al. (2007), it was shown that the steady-state density of the host population is an important parameter in the R0 expression. In most of the discussed examples this quantity could be found by a simple calculation. However, in their one virus example used, a linked plantation-nursery system, it was necessary to define how tree densities in propagator trees and nurseries were related. In the examples discussed in the present paper, there are two populations, the host plant and the vector, interacting with the virus and it becomes apparent that the steady-state size of each

population in the disease free system (healthy plants and non-viruliferous vectors) is an important issue.

In the simplest case, the host plant and vector density in the disease free system, H and X, appear in the equation for Ro. These densities are set by the agronomic/ecological system being considered. In cases where the interest lies in the effects of changes in parameters of this disease free system on the probability of disease invasion, and hence on R0, a model for the dynamics of the disease free system needs to be build.

For example, assume that the host plant and vector may be related in the absence of disease, because the vector is an herbivore. A simple system for the linked dynamics of host plant and herbivore is

dH (t)dt

=rH (t)(1−H (t )K )−γH (t)X ( t)

dX (t)dt

=θγH (t)X (t)−ρX (t)

where the herbivore parameters are defined in Table 3.

The densities in steady-state are then derived as

H = ρ /θγ

X= rγ−1γrρθγK

In the simplest model (Section 2.1, Fig. 1 Example 1), we obtain by substitution

R0=√β τY α τ I r ( ρθ γ 2 )(1− ρθγK )

R0 can then be plotted as function of the parameters representing vector population dynamics, such as vector death rate ρ or feeding rate γ (Figure 2). This gives a non-monotonic curve showing for example that for very inefficient and very efficient herbivores the virus dies out (R0 < 1). As the feeding rate of the vector increases, the basic reproduction number increases rapidly to a peak, the magnitude and position of which is inversely related to the vector death rate, i.e. the maximum basic reproduction number is higher (and is positioned earlier) with lower vector death rates. However, at higher vector feeding rates, although the basic reproduction number is decreasing, there is a direct relationship with vector death rate, i.e. the basic reproduction number is higher with higher vector death rates.

In general, the effects of control measures/agronomic measures that for example affect the vector feeding rate or death rate, but do not directly affect the virus or its transmission, will

substantially affect the host and vector populations and can be evaluated by developing models for the disease free system.

We also note that all methods introduced so far only work when the background system converges to a stable steady state. It is far from trivial to study R0 in systems where this background system is not in steady state. For crop plants, where there is a fixed planting rate, the steady state population size, given information on poor emergence or mortality due to environmental factors, can normally be estimated. For vectors the assumption of a steady-state population size is more problematic unless the population is migratory and subsequently immigrants and emigrants cancel out. Compared to the crop the intrinsic birth and death processes of the vector population in general will happen on a faster time scale.

3.2 Competition between two vector species

As noted when considering multiple vector species transmitting the same virus, competition may affect the calculation of the basic reproduction number. In determining steady-state population densities, we can calculate internal steady states for two competing vector species, where these exist (depending on the size and direction of the competition), and show how competition enters into the R0 expression.

Competition between species can be described by the Lotka-Volterra competition equations. First we assume that each vector species reach the same steady-state values, the X above, when only one is present, i.e. they are herbivores with the same feeding characteristics on the host. Then in the absence of disease, and for simplicity setting X to 1, it can be shown (Appendix) that with both vector species present the internal steady state for each species is given by

X1=(1−a12)

1−a12 a21

X2=(1−a21)

1−a12 a21

The parameters a12and a21are known as the competition coefficients. They represent the extent to which species 1 competes with species 2. However this internal steady state is only stable (i.e. both vector species co-exist) if a12 and a21 are each less than 1. If both coefficients are equal to 1, there is no distinction between the two species in their density-dependent effect. If one of the competition coefficients is greater than 1 and the other less than 1, then one of the two species goes extinct. If both competition coefficients are greater than 1 (Figure 3), then one of the species still goes extinct but which one depends on initial conditions, effectively which one got there first.

For stable coexistence to occur, the inter-species competition must be less than the within-species competition that occurs within normal density-dependence for each species alone (its carrying capacity). Hence, in the absence of disease, the steady-state density for each co-existing species X1 and X2 will be less than the steady state value for that species when present aloneX . Hence substituting the steady state values in the next generation matrix

for multiple vectors allows investigation of the effect of competition between two co-existing vector species. However, the expressions above represent the simplest case where the two coexisting species are herbivores with the same feeding characteristics (encounter and feeding rates) in the absence of disease.

As in the case of a single vector the derivation of steady-state vector densities in the absence of virus permits an analysis of the effects of competition between vectors on the basic reproduction number and hence disease dynamics. In a different context, within-plant competitive interactions between virus strains using the Lotka-Volterra equations have been analysed for conditions under which strain co-existence is possible (Jeger et al., 2011a).

4. Further elaborations and applications

This paper is intended as a general introduction to the calculation and use of the basic reproduction number in plant virus epidemiology. Further elaborations have been made with some applications for plant virus-vector systems. A further threshold quantity, the type reproduction number (Roberts and Heesterbeek, 2003), gives an indication of the effort to control infection in the host or the vector and can be used as a guide as to how best to direct control measures (Hebert and Allen, 2016). Examples of its application for vector-borne plant virus diseases are for optimal control using insecticides and roguing (Allen et al., 2017).Thus far, the derivation of the basic reproduction number has been entirely deterministic. Stochastic approaches can give probabilistic representations of R0 (Cao and Denu, 2016) in cases where realistic variation is included in models. Such models can lead to disease extinction even when the R0 values derived from deterministic models are greater than one (Hebert and Allen, 2016). Climatic variation has been included in R0 calculations but for human not plant vector-borne disease (Cordovez et al., 2014). Variation also arises from vegetational/topographical features and in these cases spatial formulation of models gives spatial reproduction numbers (Cisse et al., 2016).For plant viruses, a full picture of their epidemiology would range from their within-cell multiplication to vector acquisition and inoculation, to vector dispersal at the landscape scale, and their interaction with other biota such as natural enemies (Jeger et al., 2011b). A challenge for future work would be to define multi-scale reproductive numbers in which derivation at a given hierarchical scale is dependent on parameters determined at the lower scale (Jesse et al., 2011).

5. Conclusion

The next-generation matrix method is a structured way of calculating an expression that is both an invasion threshold and definitely the basic reproduction number. The next generation matrix approach will always result in the basic reproduction number defined as the total number of new infections cause by one infection. This has the definite advantage that it allows a direct comparison of the basic reproduction number between different systems. However, the next-generation matrix method does not always lead to expressions that agree with how epidemiologists have viewed the problem in the past, and do not

always lead to expressions that can be interpreted retrospectively. Other ways of developing invasion thresholds do often give interpretable expressions that can prove useful in determining whether invasion will occur but do not enable calculating a value for the number of secondary infections following invasion.

Appendix

Lotka-Volterra competition between two herbivores sharing the same plant host

A simple system for the linked dynamics of host plant and herbivore is

dHdt

=rH (1−HK )−γHX

dXdt

=θγH X− ρX

with steady state values

H = ρ /θγ

and

X= rγ−1γrρθγK

Suppose now that the two herbivores X1 and X2 are present and feed on the same host plant. We can then write

d H c

dt=r H c(1−H c

K )−γ H c (X1+X2)

where the subscript c indicates that two herbivores feed on the host plant. We assume that the life history parameters of the two herbivoresγ ,θ and ρ have the same values. However, interspecific competition may occur between the two herbivores. In general interspecific competition will only occur when there is intraspecific (density-dependent) competition so, based on the Lotka-Volterra competition equations, denoting the interspecific effect of X1 on X2 as a12and of X2 onX1 as a21we can write

d X1

dt= (θγ H c−ρ ) X1(1−

X 1

X−a12

X2

X )d X2

dt=(θγ H c−ρ)X2(1− X2

X−a21

X1

X )

where X represents the steady state density of each herbivore when present alone, considered to be the maximum density for that species on the host.

We now set all three equations simultaneously to zero and solve for the steady state densities.

Setting the terms (1−X 1

X−a12

X2

X )and (1−X 2

X−a12

X1

X ) equal to zero and solving gives

X1

X=

1−a12

1−a12a21

X2

X=

1−a21

1−a12 a21

There then remains the steady state value for Hc, when competition occurs between the two herbivores. Substituting the values for the two herbivore densities into the equation for the host population and setting to zero implies

r H c(1−H c

K )=γ H c X (1−a12+1−a21

1−a12a21)

Substituting for X and simplifying gives

H c

K=1−(1− ρ

θγ K )( 2−a12−a21

1−a12a21)

Substituting back H=ρ /θγ , the steady state host density when only one herbivore is present, gives

H c=K (1−(1−HK )( 2−a12−a21

1−a12a21 ))It can be shown that the steady states H c, X1, and X2, are positive if and only if a12 and a21 are each less than 1. We note that this condition also applies for two competing species competing for a finite resource in the standard Lotka-Volterra competition equations. We can also show that H c<H , and that X1+X2>X when the two competing herbivores co-exist. In other words, two competing but co-existing herbivores will lead to a lower plant host density than when only one is present, and that the total density of the two herbivores will be greater than with one alone.

The difference from the standard competition model is that we are modelling a herbivore-plant host (more generally a predator-prey) system but with two competing herbivores. Such systems have been analysed previously in the mathematical literature (Alebraheem and Abu-Hasan, 2012; Dubey and Upadhyaya, 2004). The interaction between predation and competition has been analysed at higher trophic levels by Chesson and Kuang (2008). None of these consider the case where a ‘predator’ (herbivore) is also a vector and transmits a pathogen to its ‘prey’ (plant).

References

Alebraheem, J., Abu-Hasan, Y. 2012. Persistence of Predators in a Two Predators-One Prey Model with Non-Periodic Solution. Applied Mathematical Sciences 6, 943 - 956.

Allen, L. J. S., Bokil, V. A., Jeger, M. J., Lenhart, S. 2017. Optimal control of a vector transmitted viral disease of crops. (Submitted)

Cao, Y., Denu, D. 2016. Analysis of stochastic vector-host stochastic epidemic model with direct transmission. Discrete and Continuous Dynamical Systems Series B. 21, 2109-2127.

Chesson, P., Kuang J. J. 2008. The interaction between predation and competition. Nature 456, 235-238.

Cisse, B., El Yacoubi, S., Goubiere, S. 2016. The spatial reproduction number in a cellular automaton model for vector-borne diseases applied to the transmission of Chagas disease. Simulation: Transactions of the Society for Modeling and Simulation International 92, 141-152.

Cordovez, J. M., Rendon, L. M., Gonzalez, C., Guhl, F. 2014. Using the basic reproduction number to assess the effects of climate change in the risk of Chagas disease transmission in Colombia. Acta Tropica. 129, 74-82.

Diekmann, O., Heesterbeek, J. A. P., Metz, J. A. J. 1990. On the definition and the computation of the basic reproduction ratio R0 in models for infectious-diseases in heterogeneous populations. Journal of Mathematical Biology 28, 365-382.

Dubey, B., Upadhyay, R. K. 2004. Persistence and Extinction of One-Prey and Two-Predators System. Nonlinear Analysis: Modelling and Control 9, 307–329.

Hamelin, F. M., Allen, L. J. S., Prendeville, H. R., Reza Hajimorad, M., Jeger, M. J. 2016. The evolution of plant virus transmission pathways. Journal of Theoretical Biology.

Hamelin, F. M., Allen, L. J. S., Prendeville, H. R., Reza Hajimorad, M., Jeger, M. J. 2017. The evolution of mutualistic plant-virus symbioses through trade-offs between virulence and transmission. Virus Research (Submitted, this issue).

Hebert, M. P., Allen, L. J. S. 2016. Disease outbreaks in plant-vector-virus models with vector aggregation and dispersal. Discrete and Continuous Dynamical Systems Series B. 21, 2169-2191.

Heesterbeek, J. A. P. 2002. A brief history of R-0 and a recipe for its calculation. Acta Biotheoretica 50, 189-204 (with erratum 375-376).

Jeger, M. J., van den Bosch, F., Madden, L. V. 2011a. Modelling virus- and host-limitation in vectored plant disease epidemics. Virus Research 159, 215-222.

Jeger, M. J., Chen, Z., Powell, G., Hodge, S., van den Bosch, F. 2011b. Interactions in a host plant-virus-vector-parasitoid system: modelling the consequences for virus transmission and disease dynamics. Virus Research 159, 183-193.

Jeger, M. J., van den Bosch, F., McRoberts, N. 2015. Modelling transmission characteristics and epidemic development of the tospovirus-thrip interaction. Arthropod-Plant Interactions 9, 107-120.

Roberts, M. G., Heesterbeek, J. A. P. 2003. A new method to estimate the effort required to control an infectious disease. Proceedings of the Royal Society London B 270, 1359-1364.

Shi, R., Zhao, H., Tang, S. 2014. Global dynamic analysis of a vector-borne plant disease model. Advances in Difference Equations 2014: 59.

Van den Bosch, F., McRoberts N., van den Berg, F., Madden, L. V. 2008. The basic reproduction number of plant pathogens: matrix approaches to complex dynamics. Phytopathology 98, 239-249.

Table 1 List of model variables

Variable

Description

H Density of healthy host plants per unit areaI Density of infected host plants per unit areaX Density of non-viruliferous vectors per unit areaY Density of viruliferous vectors per unit area

Table 2 List of transmission parameters

Parameter Descriptionβ Inoculation coefficient (time-1): the product βH gives the rate at which healthy plants

are inoculated per viruliferous vectorα Acquisition coefficient (time-1): the product αX gives the rate at which non-

viruliferous vectors acquire virus per infected plantτY Vector infective period (time): the (constant) period of time a viruliferous

vector remains infectiveτI Plant infectious period (time); the (constant) period of time an infected plant

remains infectiousq Proportion of infected progeny from seed of an infected plant

Table 3 List of herbivore population parameters

parameter descriptionr Intrinsic growth rate of herbivore population (time-1)γ Herbivore feeding rate (time-1)ρ Herbivore death rate (time-1)ϴ Conversion factor from herbivore feeding to population growtha12 Interspecific competitive effect of herbivore 1 on herbivore 2a21 Interspecific competitive effect of herbivore 2 on herbivore 1

Figure legends

1. Schematics illustrating disease dynamics for the four examples in the text illustrating the use of the next generation matrix approach for deriving and calculating the basic reproduction number

Example 1. One vector species transmitting one virus to a single plant species. Variables I (infected host) and Y (viruliferous vector) are shown in circles, with arrows representing flows labelled with the acquisition (α) and inoculation (β ¿ parameters (Table 1).

Example 2. Multiple (n) vector species transmitting the same virus to a single plant species. Variables I (infected host) and Y1 …. Yn (vector species) are shown in circles, with arrows representing flows labelled with the acquisition (α) and inoculation (β ¿ parameters labelled 1, 2, …n (Table 1).

Example 3. One vector species transmitting one virus to a single plant species and seed transmission to progeny also occurs. Variables I (infected host) and Y (viruliferous vector) are shown in circles, with arrows representing flows labelled with the acquisition (α) and inoculation (β ¿ parameters and the probability of seed infection (q) (Table 1).

Example 4. One vector species transmitting one virus to a single plant species, and where disease control using insecticides is applied. Variables I (infected host), Y (viruliferous vector not exposed to insecticide), and Ys (viruliferous vector exposed to insecticide) are shown in circles, with arrows representing flows labelled with transmission parameters (Table 1). Exposure to insecticide is represented by the dotted line. The subscripts s indicate acquisition (α) and inoculation (β ) when exposed to the insecticide

2. The basic reproduction number R0 calculated from equation Y in relation to the vector feeding rate (γ) for three values of the vector death rate (ρ). Default values used for other parameters are shown in Tables 1 and 2.

3. The basic reproductive number R0 where there is competition between two vectors transmitting the same virus. The interspecific effect of X1 on X2 is denoted by a12and of X2 onX1 by a21Both competition coefficients α12 and α21 are ≥ 1, and one of the vector species will go to extinction (which one depends on initial conditions). The maximum R0 value (3.8) is obtained when α12 = α21 = 1 (i.e. when intra-species competition equals inter-species competition) but declines linearly as one of the vector species outcompetes the other

Figure 1

Figure 2

Figure 3