2.1. Superinfection model

The deterministic model presented here is an extension of the basic coinfection model from our previous work (^{Pinky and Dobrovolny, 2016}) where now we introduce possible infection of single cells by both types of virus; we call this superinfection mechanism. The model presented here is the most general form of all the events we are interested in investigating. The model is represented by the following ordinary differential equations,

where i,j=1,2 and βij=pij=0 when i=j, and all the derivatives are taken with respect to time, t.

A schematic of the model is shown in Fig. 1 . Susceptible target cells, T, are infected by virus, V_i, at a rate β_iV_i. In addition, target cells regenerate at a constant rate, r, and decay naturally at a rate, a. Both kinds of infected target cells undergo eclipse phases where viruses exhibit intra-cellular activities to take over the host’s cellular mechanisms to produce viral genomes (RNA). We assume that the second virus can still infect the cells in eclipse phases such that E ₁ get infected with the second virus, V ₂, at a rate β ₂₁ and E ₂ get infected by the first virus, V ₁, at a rate β ₁₂. Thus cells in the eclipse phases become functional target cells for the other virus and these dually infected eclipse cells are called superinfected eclipse cells, E ₃. We further assume that once cells are infectious, they can no longer be infected by another virus. As soon as the eclipse phases are ready to produce viruses after the time durations of 1k1, 1k2 and 1k3, they become infectious cells, I ₁, I ₂ and I ₃. I ₁ and I ₂ produce viruses at rates, p ₁ and p ₂ while the superinfected infectious cells, I ₃, produce viruses of both types V ₁ and V ₂, at rates p ₁₂ and p ₂₁ respectively. These infectious cells produce viruses throughout their lifespans of 1δ1, 1δ2 and 1δ3. Viruses of both types, V ₁ and V ₂, decay at rates c ₁ and c ₂ respectively. Table 1 describes the model variables and parameters with values used for our simulations. All parameters are positive.

Model diagram for virus-virus coinfection with superinfection mechanism. Two different viruses infect the susceptible target cells simultaneously. Target cells regenerate at a constant rate and decay according to available target cells. Once infected by one type of virus, some of the infected cells are additionally infected by the other type of virus. Infection is established once the viruses successfully hack the cellular mechanisms for their own replication during the eclipse phase, eventually becoming infectious. Viruses replicate by infecting not only susceptible target cells but also the singly infected cells with a different virus type.

Definition of model variables and parameter values for model simulations.

We have not included an explicit immune response in our model for a number of reasons. We are interested in studying the mechanisms behind chronic coinfections, which occur more often in immunocompromised patients. Thus a complete lack of explicit immune response makes sense. Second, since we do not have any experimental data that explains immune response with respect to coinfection of two different viruses, our model does not explicitly account for any immune responses, however, this can be incorporated implicitly in the superinfection parameters, i.e. death rates of dually infected cells, 1δ3, superinfectivity rates, β_ij, and viral production rates from superinfected cells, p_ij, in the model. Moreover, studies show superinfection may cause both enhanced (^{Mosquera, Adler, 1998, Nowak, May, 1994}) and reduced (^{Brown et al., 2002}) production of viruses; we assume that superinfected infectious cells produce each type of virus at different rates, p ₁₂ and p ₂₁, than that of the singly infected cells so that we can explore the pathological consequences due to the different replication rates. Also it is reported that the susceptibility of target cells changes due to the occurrence of earlier infection (^{Anestad, Vainio, Hungnes, 2007, Laurie, Guarnaccia, Carolan, AWC, Aban, Petrie, et al., 2015, Simeonov, Gong, Kim, Poss, Chiaromonte, Fricks, 2010}). For example, infection by one virus may increase the probability of being subsequently infected by other types since the first infection may weaken a host’s immunity or resistance (^{Soares et al., 2016}). Another study showed that primary infection evokes an immune response which reduces the chance for secondary infection (^{Devevey, Dang, Graves, Murray, Brisson, 2015, Klemme, Louhi, Karvonen, 2016, Susi, Barrès, Vale, Laine, 2015}). So we consider different infection rates for superinfection to explore the possible outcomes. Lastly, we also allow the transition rate of superinfected eclipse phases and death rate of superinfected infectious cells to vary across a range of acceptable parameter values for both viruses based. While there is no direct experimental examination of eclipse phases or infectious cell lifespans during coinfection, since viruses are sharing the cell’s resources (^{Shinjoh et al., 2000}), it seems possible that the speed at which virions are produced is altered. On a larger scale, the observation by ^{Laurie et al. (2015)} that in a ferret model of human influenza, subsequent influenza infections with different strains (H1N1, H3N2, IBV) limits the time duration of virus replication also suggests that these transition rates are altered.

The terms of disease severity, superinfection and coinfection have ambiguous meanings in literature as there are no standard definitions for them in general (^{Sofonea et al., 2017}). In this paper, coinfection refers to infection caused by two different viruses in the respiratory tract at the same time, though not necessarily sharing the same target cells, while superinfection refers to infection of a single cell with two different viruses. In our case, we define disease severity by the two factors defined as viral load and duration of infection where two viruses coexist. Other than these two, order of inoculation, initial viral inoculum, number of coinfecting pathogens, virus specific interactions, host defense mechanisms can also influence disease severity.