# Modelling Vector Transmission and Epidemiology of Co-Infecting Plant Viruses

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Modelling

#### 2.2. Invasion Thresholds

#### 2.3. Formal Reduction to a Model That Does Not Track Vectors Explicitly

## 3. Results

#### 3.1. Only One Virus Can Invade in Absence of the Other

#### 3.2. Comparison with Results of Model That Does Not Track Vectors Explicitly

#### 3.3. Neither Virus Can Invade in Absence of the Other

## 4. Discussion

- i
- Virus A is able to complete a full infection cycle, invade a disease-free population and settle at an endemic equilibrium in the absence of virus B. By contrast, virus B can systematically infect a plant but cannot be acquired by a vector in the absence of co-infection in the host plant. Successful invasion of virus B depends on a complex relation between vector acquisition and inoculation rates, the relative inoculation success of viruses A and B and the prevalence of infected hosts and infective vectors carrying virus A.
- ii
- Virus A and virus B can both complete their infection cycles but are unable to invade a disease-free population in the absence of the other virus. The only biologically feasible endemic equilibrium is a co-infection equilibrium. The system will approach the co-infection equilibrium if the initial prevalences of virus A and B are sufficiently large, i.e., if the initial condition is in the basin of attraction of the co-infection equilibrium. Otherwise, if the initial virus prevalences are so low that the initial condition is in the basin of attraction of the disease-free equilibrium, both viruses will disappear from the system. The separatrix between these two different outcomes represents a curve of tipping points. On either side of these tipping points, we have contrasting dynamics, namely a disease-free versus a co-infected system. We have seen that increased vector mortality rates (e.g., due to vector control programs) moves the separatrix by making the co-infection equilibrium less resilient. This could lead to an abrupt (rather than gradual) extinction of co-infection.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Basic Reproduction Number

## Appendix B. Invasion Reproduction Number

**Figure A1.**For the vector-explicit model, virus A equilibrium prevalences for the vector ${z}_{a}=z{a}_{eq}/\overline{V}$ and for the host ${i}_{a}=i{a}_{eq}/\overline{P}$ are graphed in (

**A**,

**C**,

**E**) and the corresponding invasion reproduction curves are graphed in (

**B**,

**D**,

**F**) as a function of acquisition and inoculation parameters. The parameter values are the same as in Figure 2 with the exception that in panels B, D and F parameters ${\gamma}_{A}=2$ and ${\gamma}_{B}=0.1$ (solid curve), 0.5 (dashed curve), 1 (dashed-dot curve).

**Figure A2.**The time courses of one successful and one unsuccessful invasion of virus B are graphed for the vector-explicit model. The virus A host prevalences ${I}_{A}/P$ are graphed in panels (

**A**,

**C**,

**E**) and the virus B host prevalences ${I}_{B}/P$ are graphed in panels (

**B**,

**D**,

**F**) for the two sets of parameter values identified by black circles in Figure 2. The initial condition is perturbed slightly from the virus A equilibrium, $(\overline{V}-z{a}_{eq}-0.1,v{a}_{eq},0.1,\overline{P}-i{a}_{eq}-30,i{a}_{eq}-10,10,20)$. Parameter values for the graphs in panels (

**A**–

**F**) are the same as in Figure 2A–F with ${\gamma}_{B}=0.9$.

## Appendix C. Reduction to a Vector-Implicit Model

**Figure A3.**For the vector-implicit model, the virus A host equilibrium prevalences ${i}_{a}=i{a}_{eq}/\overline{P}$ are graphed in panels (

**A**,

**C**,

**E**) and the corresponding invasion reproduction curves are graphed in panels (

**B**,

**D**,

**F**) as a function of acquisition and inoculation parameters. Parameter values are as in Figure 3A–F with the exception that in panels B, D and F the parameter values for ${\gamma}_{A}=2$ and for ${\gamma}_{B}=0.1$ (solid curve), 0.5 (dashed curve), 1 (dashed-dot curve).

**Figure A4.**The time courses of two successful invasions and one unsuccessful invasion of virus B are graphed for the vector-implicit model. The virus A host prevalences ${I}_{A}/P$ are graphed in panels (

**A**,

**C**,

**E**) and the virus B host prevalences ${I}_{B}/P$ are graphed in panels (

**B**,

**D**,

**F**) for the three sets of parameter values identified by the black circles in Figure 3. The initial condition is perturbed slightly from the virus A equilibrium, $(\overline{P}-30,{\overline{I}}_{A}-10,10,20)$. Parameter values are as in Figure 3A–F with ${\gamma}_{B}=0.9$.

**Figure A5.**Comparison of the virus A equilibrium prevalences and invasion reproduction numbers in the vector-explicit model in (

**A**,

**B**) to the vector-implicit model in (

**C**,

**D**) as a function of vector recovery rate ${\delta}_{A}$. Parameter values are ${\widehat{\theta}}_{A}={\theta}_{A}/(c+{\delta}_{A})=0.05$, ${\alpha}_{A}=0.3$, ${\gamma}_{B}=0.9$, ${\delta}_{A}\in [0,19]$, ${\theta}_{A}=0.05(c+{\delta}_{A})\in [0.05,1]$ and other parameter values as in Table 2.

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**Figure 1.**Compartmental diagrams (

**left**) for the vector model and (

**right**) the host plant model, described by the differential Equations (1) and (2), respectively. Meanings of the symbols for model parameters are summarised in Table 2.

**Figure 2.**For the vector-explicit model, the virus A equilibrium prevalences for the vector ${z}_{a}=z{a}_{eq}/\overline{V}$ and for the host ${i}_{a}=i{a}_{eq}/\overline{P}$ are graphed in panels (

**A**,

**C**,

**E**) and the corresponding invasion reproduction numbers ${\mathcal{R}}_{invB}$ are graphed in panels (

**B**,

**D**,

**F**) as a function of acquisition (${\alpha}_{A}$) or inoculation (${\theta}_{A}$ and ${\theta}_{B}$) parameters. Virus B may invade the virus A equilibrium if the invasion reproduction number is greater than the threshold value one. The parameters values that are not varied are fixed at their default values (see Table 2): $\Lambda =10$ per month per unit area, $\mathsf{\Phi}=30$ per month, $c=1$ per month, ${\alpha}_{A}=0.2$, ${\alpha}_{B}=0$, ${\alpha}_{AB}=0.5$, ${\u03f5}_{A}={\u03f5}_{B}=0.5$, ${\delta}_{A}={\delta}_{B}=3$ per month, $\mu =1/12$ per month, $\sigma =1000\mu $ (1000 per year per unit area), ${\theta}_{A}=0.8$, ${\theta}_{B}=0.5$ and $\omega ={\omega}_{A}={\omega}_{B}=0$. Parameter values for the two invasion reproduction curves in (

**B**,

**E**,

**F**) are ${\gamma}_{B}=0.9$ for the solid curve and ${\gamma}_{B}=0.25$ for the dashed curve. (

**B**) The two black circles on the solid curve are at values of ${\alpha}_{A}=0.1,0.8$; (

**D**) ${\theta}_{A}=0.3,0.8$; (

**F**) ${\theta}_{B}=0.2,0.8$. The invasion reproduction number is ${\mathcal{R}}_{invB}=\rho ({M}_{invB})$ as defined in (5).

**Figure 3.**For the vector-implicit model, the virus A host equilibrium prevalences $i{a}_{eq}/\overline{P}$ are graphed in panels (

**A**,

**C**,

**E**) and the corresponding invasion reproduction numbers ${\mathcal{R}}_{invB}$ are graphed in panels (

**B**,

**D**,

**F**) as a function of the acquisition (${\theta}_{A}$) and inoculation (${\theta}_{A}$ and ${\theta}_{B}$) parameters. The same parameter values are applied as in the vector-explicit model in Figure 2. The three black circles on the solid curve in (

**B**) are at values of ${\alpha}_{A}=0.05,0.1,0.8$; in (

**D**) ${\theta}_{A}=0.2,0.3,0.8$; (

**F**) ${\theta}_{B}=0.05,0.2,0.8$. The invasion reproduction number equals ${\mathcal{R}}_{invB}=\rho ({\mathcal{M}}_{invB})$ as defined in (8).

**Figure 4.**(

**A**) The basins of attraction of the disease-free equilibrium (DFE) (white region) and the co-infected equilibrium (shaded region) are graphed as a function of the initial frequency ${f}_{a}$ of virus A and frequency ${f}_{b}$ of virus B in the plant population. See Equations (9) and (10) for the initial conditions. The points mark the prevalences of virus A and B in the plant population at the DFE (at the origin) and the stable co-infection equilibrium (in the interior). The gray square marks the relative frequency of virus A and B in the vector population at the stable co-infection equilibrium. The blue cross and red asterisk indicate initial conditions in different basins of attraction, for which time plots show convergence either to the DFE (${I}_{AB}=0$) or to the co-infection equilibrium (${I}_{AB}/P>0$) in (

**B**). Parameter values are $\Lambda =1,\mathsf{\Phi}=250,c=1,{\alpha}_{A}=0.005,{\alpha}_{B}=0.0005,{\alpha}_{AB}=0.15,{\u03f5}_{A}={\u03f5}_{B}=0.5,{\delta}_{A}=20,{\delta}_{B}=40,\mu =1/12,\sigma =100\mu ,{\theta}_{A}=0.8,{\theta}_{B}=0.4,\omega =\mu /2={\omega}_{A}={\omega}_{B},{\gamma}_{A}={\gamma}_{B}=0.5$ such that ${\mathcal{R}}_{0A}=0.95$ and ${\mathcal{R}}_{0B}=0.02$.

**Figure 5.**(

**A**) The changes in the size of the basins of attraction for the DFE and the co-infected equilibrium are graphed as a function of the initial frequency ${f}_{a}$ of virus A and ${f}_{b}$ of virus B when the death rate of vectors decreases from $c=1.0$ to $c=0.7$, with values of $c=0.7,0.8,0.9,1.0$. The boundary separating the two basins of attraction at $c=0.7$ is closest to the origin = DFE and at $c=1.0$ is furthest from the origin. Other parameter values are as in Figure 4. The star symbol marks the initial condition for the disease progress curves in (

**B**), which shows the time series of the prevalence of co-infected plants for varying levels of per-capita plant mortality, c. Initial conditions are fixed at (9) and (10) with initial virus prevalences ${f}_{a}={f}_{b}=0.1$ in the plant population.

**Figure 6.**Basins of attraction for the DFE and the co-infection equilibrium for different values of ${\u03f5}_{A}$ and ${\u03f5}_{B}=1-{\u03f5}_{A}$. The blue curve with $\u03f5=0.5$ corresponds to the separatrix shown in Figure 4 with ${\mathcal{R}}_{0A}=0.95,{\mathcal{R}}_{0B}=0.02$. Decreasing ${\u03f5}_{A}$, i.e. increasing ${\u03f5}_{B}$, reverses the asymmetry in the basins of attraction.

**Table 1.**Summary of some representative experimental work on co-infection at different levels of complexity.

Pathosystem | Authors | Key Message | |
---|---|---|---|

Two virus strains | Potato virus Y (PYV)—Myzus persicae (host Capsicum annuum) | Moury et al. [15] | Two strains were equally transmissible and competition was studied to estimate the size of bottlenecks imposed by vector transmission. If there was a cost of virulence, modelling showed that virulent strains would go extinct. |

${\mathrm{PVY}}^{\mathrm{NTN}}$ compared with other strains—Myzus persicae | Srinivasan et al. [21] | Previous work had suggested some specificity in transmission of strains. The rate of infection for ${\mathrm{PVY}}^{\mathrm{NTN}}$ was higher than for other strains, a vector-related outcome as this was not observed with mechanical transmission. | |

${\mathrm{PVY}}^{\mathrm{NTN}}$ compared with ${\mathrm{PVY}}^{\mathrm{O}}$—Myzus persicae | Carroll et al. [22] | The necrotic strain was transmitted more efficiently than the wild-type. Co-infection would more likely result from inoculation by multiple aphids feeding on plants infected with the different strains rather than by single aphids feeding on multiple plants infected with the different strains. | |

Two virus species—common vector | Barley yellow dwarf virus/cereal yellow dwarf virus—Rhopalosiphum padi | Lacroix et al. [24] | The co-inoculation of BYDV-PAV lowered the CYDV-RPV infection rate but only at low nutrient supply rates. Broader environmental and nutritional factors can alter co-infection interactions and outcomes. |

Watermelon mosaic virus/zucchini yellow mosaic virus—Aphis gossypii | Salvaudon et al. [25] | ZYMV accumulated at similar rates in single and mixed infections, whereas WMV was much reduced in the presence of ZYMV. ZYMV also induced host changes that gave strong vector preference for infected plants; whereas WMV did not, although it was still readily acquired from mixed infections. | |

Rice tungro spherical virus/rice tungro bacilliform virus—Nephottetix virescens | Holt and Chancellor [26] | Infection by each virus alone results in less pronounced symptoms. RTBV is retained in the vector for a longer period. When a vector carries both viruses, co-inoculation is common. When inoculative with RSTV alone the infection probability is higher. | |

Two virus species—multiple vector species | Bean pod mottle virus—Epilachna varivestis/soybean mosaic virus—Aphis glycines | Penaflor et al. [27] | Singly-infected plants with either BPMV or SMV increased soybean palatability, potentially enhancing acquisition of BPMV from BPMV plants and secondary infection of BPMV from SMV plants. BPMV infection had little effect on A. glycines, whereas SMV infection reduced aphid population growth but increased the preference for infected plants. With co-infection, effects on population growth were reversed and aphids showed a preference for co-infected plants. |

Multiple virus species—multiple vector species | Grapevine leafroll-associated viruses (GLRaVs)—mealybugs/scale insects | Naidu et al. [28] | The exact role of GLRaVs in disease etiology remains unclear. With mealybugs, transmission is of a semi-persistent manner with a lack of vector-virus specificity. |

Blaisdell et al. [29] | Co-infections of GLRaVs are frequent in grapevines although with some spatial separation with implications for transmission and epidemiology. | ||

Sweet potato chlorotic stunt virus/sweet potato feathery mottle virus/multiple viruses—multiple vector species | Untiveros et al. [30] | Six viruses from the same or different virus families interacted synergistically with sweet potato virus disease, with increased disease symptoms, virus accumulation and movement in plants, and reduced yield of storage roots. All inoculations were made by grafting; no conclusions can be drawn on vector transmission effects. | |

Multiple virus species—multiple vector species, multiple hosts | Ecological networks formed by multiple co-infecting viruses in multiple hosts | McLeish et al. [31] | Co-infection networks were found to lead to strong non-random associations compared with single infections. Single infections were mostly related to habitat parameters, whereas co-infections were more related to ecological heterogeneity and ecosystem-level processes. |

Vector | Default Values | Default Values | |
---|---|---|---|

Parameters | Section 3.1 and Section 3.2, Appendix B and Appendix C | Section 3.3 | |

$\Lambda $ | vector birth rate | 10/month/area | 1/month/area |

c | per capita vector natural death rate | 1/month | 1/month |

$\mathsf{\Phi}$ | number of plants visited/time by a vector | 1/day | 8.33/day |

${\delta}_{A}$ | per capita infective vector recovery rate from virus A | 3/month | 0.66/day |

${\delta}_{B}$ | per capita infective vector recovery rate from virus B | 3/month | 1.33/day |

${\alpha}_{A}$ | probability non-infective vector acquires virus A from ${I}_{A}$ per plant visit | 0.2 | 0.005 |

${\alpha}_{B}$ | probability non-infective vector acquires virus B from ${I}_{B}$ per plant visit | 0 | 0.005 |

${\alpha}_{AB}$ | probability non-infective vector acquires a single virus, A or B, from ${I}_{AB}$ per plant visit | 0.5 | 0.15 |

${\u03f5}_{A}$ | conditional probability of acquiring virus A from a co-infected plant ${I}_{AB}$, given a successful acquisition | 0.5 | 0.5 |

${\u03f5}_{B}$ | conditional probability of acquiring virus B from co-infected plant ${I}_{AB}$, given a successful acquisition (${\u03f5}_{B}=1-{\u03f5}_{A}$) | 0.5 | 0.5 |

Plant | Default Values | Default Values | |

Parameters | Section 3.1 and Section 3.2, Appendix B and Appendix C | Section 3.3 | |

$\mu $ | per capita mortality and or harvest of plants | 1/year | 1/year |

$\sigma $ | seeding or planting rate | 1000/year/area | 100/year/area |

${\theta}_{A}$ | probability an infective vector with virus A inoculates a healthy plant per visit | 0.8 | 0.8 |

${\theta}_{B}$ | probability an infective vector with virus B inoculates a healthy plant per visit | 0.5 | 0.5 |

${\gamma}_{A}$ | relative inoculation success of virus A (as compared to a heathy plant) in a plant ${I}_{B}$, infected with a single virus B | 0.9 | 0.5 |

${\gamma}_{B}$ | relative inoculation success of virus B (as compared to a healthy plant) in a plant ${I}_{A}$, infected with a single virus A | 0.25, 0.9 | 0.5 |

$\omega $ | per-capita viral A or B loss rate in a plant infected with single virus | 0 | 0.001/day |

${\omega}_{A}$ | per-capita viral B loss rate (A is retained) from a co-infected plant ${I}_{AB}$ | 0 | 0.001/day |

${\omega}_{B}$ | per-capita viral A loss rate (B is retained) from a co-infected plant ${I}_{AB}$ | 0 | 0.001/day |

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Allen, L.J.S.; Bokil, V.A.; Cunniffe, N.J.; Hamelin, F.M.; Hilker, F.M.; Jeger, M.J.
Modelling Vector Transmission and Epidemiology of Co-Infecting Plant Viruses. *Viruses* **2019**, *11*, 1153.
https://doi.org/10.3390/v11121153

**AMA Style**

Allen LJS, Bokil VA, Cunniffe NJ, Hamelin FM, Hilker FM, Jeger MJ.
Modelling Vector Transmission and Epidemiology of Co-Infecting Plant Viruses. *Viruses*. 2019; 11(12):1153.
https://doi.org/10.3390/v11121153

**Chicago/Turabian Style**

Allen, Linda J. S., Vrushali A. Bokil, Nik J. Cunniffe, Frédéric M. Hamelin, Frank M. Hilker, and Michael J. Jeger.
2019. "Modelling Vector Transmission and Epidemiology of Co-Infecting Plant Viruses" *Viruses* 11, no. 12: 1153.
https://doi.org/10.3390/v11121153