Transmission is a key element in understanding the epidemiology of plant virus diseases, particularly those transmitted by arthropod vectors [1
]. In general, four modes of transmission, non-persistent, semi-persistent, persistent-circulative and persistent-propagative, can be distinguished. Each of these modes has a characteristic time period for acquisition from infected plants, retention in the vector, and inoculation to healthy plants [5
], although some virus groups such as the torradoviruses do not fit neatly into these categories [6
]. Other aspects important for arthropod transmission include transovarial and transtadial transmission, and the “helper strategy” [3
] in which a helper virus can be transmitted by the vector but the dependent virus can only be transmitted in the presence of the helper, a strategy modelled by Zhang et al. [7
Co-infection of hosts by two or more plant viruses is common in both agricultural crops [8
] and natural plant communities [10
]. Because of this, the literature on co-infection by plant viruses is extensive, although often not related to transmission. Indeed, extending epidemiological models to go beyond a single pathogen species was relatively-recently highlighted as a key challenge in modeling plant diseases in [12
] (challenge 4). Co-infection almost always leads to interactions between viruses during transmission and within-plant processes that can strongly influence disease development in individual plants and ultimately spread in a plant population. The strength and direction of interactions can vary with both negative and facilitating effects, involving within-cell processes, cell-to-cell movement, vector acquisition and inoculation, symptom development and virulence, and yield loss. Reports on cellular interactions have been the most prevalent, mostly for replication rates and virus titre, but some studies have shown clear interactions with vectors over short (epidemiological) and long-term (evolutionary) time scales [13
Most experimental studies on the relationship between co-infection and transmission have been done for viruses with non-persistent transmission by aphids. Syller [14
] reviewed the literature on “simultaneous” transmission of plant viruses by vectors, emphasizing the acquisition component of transmission with little consideration of inoculation. The use of the term “simultaneous” is ambiguous—what seems to be suggested is that two different virus particles can be acquired instantaneously by a vector during a single probe where there is spatial separation between two viruses. However, for non-persistent transmission, with a following probe [15
] on the same or different plant, one of the viruses can become detached and no longer be available for inoculation. Transmission can present a real bottleneck in the virus life cycle [16
]. In a subsequent review, Syller and Grupa [17
] differentiate between simultaneous inoculation (which they call co-infection) and sequential inoculation (which they call super-infection). They claim that synergistic interactions within-plants most often arise between unrelated viruses. Synergism is defined as a facilitative effect in which accumulation of one or both viruses in the host plant increases; in the case of the effect on just one virus, it has been called asymmetric synergism [18
]. Synergism has also been used to describe more severe disease symptoms than induced by either virus alone. Syller and Grupa [17
] concentrate more on antagonistic effects, such as cross protection [19
] or, as has been termed “super-infection exclusion” in which related viruses or virus strains are used preventively to exclude more virulent strains. Mascia and Gallitelli [20
] note the contributions that mathematical modeling could make “in forecasting challenges deriving from the great variety of pathways of synergistic and antagonistic interactions” (p. 176).
Co-infection can cover scenarios ranging from two viruses (or virus strains)/one vector through to many viruses/many vectors, but with some nuances. There is an extensive literature on co-infection across this range. Some representative but not exhaustive publications are noted in Table 1
, together with some key messages. Many publications acknowledge that there are several or many (in the case of aphids) vector species for a given plant virus, but the experiments reported only involve one vector. Similarly, the same virus and vector can infect more than one host (cucumber mosaic virus is an extreme example) and hence cause more than one disease. Co-infection with virus strains differing in virulence (or other characteristic) can lead to the same set of interactions and consequences as found with virus species. A good example of two strains of the same virus species with shared vector species is potato virus Y (PVY) [21
] on potato and other hosts [15
]. There are many examples of two co-infecting virus species with a shared vector species [24
]. Similarly, there are many cases where two co-infecting viruses have quite different vectors taxonomically [27
]. Co-infection is manifested in more complex situations with multiple viruses and vectors such as with grapevine leafroll disease [28
] and sweet potato virus disease (SPVD) [30
]. At an even higher level of complexity, the ecological networks formed by multiple co-infecting viruses and multiple hosts were analyzed by McLeish et al. [31
In this paper, we formulate a general epidemiological model for one vector species and one plant species that allows for co-infection of the host plant by two virus species or strains. The model is used to investigate the role of vector transmission on co-infection, specifically acquisition and inoculation, as well as antagonistic and synergistic interactions. The basic reproduction number provides a condition for invasion of a single virus infection. For co-infection, we derive a new invasion threshold. Given that a single virus can persist in the host plant, the invasion threshold highlights some key epidemiological parameters for successful co-infection. In addition, we investigate the roles of the vector acquisition and inoculation parameters when one virus depends on an autonomous virus for its successful transmission, or when both viruses are unable to invade alone but can facilitate co-infection if they occur in high enough prevalences. We also explicitly test when the simplifications—which are almost always left implicit—in models which do not explicitly include the infection status of vectors lead to potentially misleading results.
Co-infections are pervasive in plant virus epidemiology; yet, mathematical models keeping track of co-infections often leave vector dynamics implicit. This is understandable since (i) keeping track of co-infections makes the models less tractable mathematically, (ii) modeling vector dynamics explicitly may not be necessary relative to the research question addressed, and (iii) the biological knowledge or the data may be too scarce to reasonably increase the model’s complexity [38
]. Moreover, it is well known [10
] that vector-implicit models may reasonably approximate vector-explicit models when vector dynamics can be considered to be fast with respect to epidemiological dynamics in the plant host population (such as in non-persistent transmission). Although see [41
] for a different model of non-persistent transmission of viruses that goes beyond differential equations to track the epidemiological effects of vector dynamics.
However, we have showed that vector-implicit and vector-explicit models may yield qualitatively different results (Figure 2
and Figure 3
). The default parameter set we considered were appropriate for persistent-circulative transmission with vector infective periods of about 10 days (Table 2
), and the qualitative difference in the models’ outcomes highlights a key biological feature of the vector-explicit model that is not accounted for in its vector-implicit analogue. This key feature is competition for vectors [16
]. Competition occurs because we assumed one vector cannot be inoculative with two viruses at the same time at the point of inoculation. This assumption may reflect the fact that the first acquired virus saturates stylet receptor sites in non-persistent transmission, or that it precludes second inoculation of other viruses as long as the vector retains the first virus. The latter phenomenon would resemble the so-called “super-infection exclusion” usually associated with the host plant rather than with the vector [16
]. Nevertheless, competitive exclusion (or pre-emptive competition) of one virus by another can also occur within vectors [42
]. This way, infective vectors are only able to inoculate one virus at the point of inoculation. Therefore, viruses indirectly compete for vectors. However, the vector-implicit model is valid only if the vector recovery rate is sufficiently large, meaning that the vector loses the virus very quickly after acquisition (as is the case in non-persistent viruses). Therefore, it is very unlikely that a virus cannot be transmitted because the vector had already acquired another virus. These results highlight the importance of the mode of vector transmission in models of vectored plant viruses.
To illustrate the implications of our model, we considered two distinct special cases with two viruses (or virus strains) denoted A and B:
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.
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.
In this paper, we have formulated a general epidemiological model for potential co-infection in a host plant by two virus species or strains. We used the model to investigate the effects of vector transmission on co-infection, specifically acquisition and inoculation, as well as antagonistic and synergistic interactions between viruses. We showed that reducing the model to a vector-implicit model can lose some of the key features gained when the vector is included explicitly (such as competition for vector). We derived a new invasion threshold that determines whether or not a second virus can invade a host population in which a first virus had successfully established. The invasion threshold highlights the key epidemiological parameters that are important for successful co-infection. However, the invasion threshold only applies near the virus A equilibrium when the invading virus B species/strain is at low prevalence levels. The dynamics of the models away from the virus A equilibrium can be quite complex and exhibit bistability, where both the virus A equilibrium and the co-infection equilibrium are stable. In this case, the initial prevalences of viruses A and B determine whether virus B can successfully invade. We also investigated the potential for co-infection and the conditions that need to be satisfied when one virus depends on an autonomous virus for its successful transmission or when both viruses are unable to invade alone. For persistent-circulative transmission, competition between viruses/virus strains, either direct or indirect, as they move through the vector is identified as a key and challenging area for further research that would improve modeling attempts to predict the epidemiological consequences of co-infection.