Three mechanisms for the IFN-induced control of viral infection

Three possible antiviral mechanisms of IFN are allowed for in our model: 1) induction of a virus-resistant state for target cells; 2) a reduction in the viral production rate from infected cells; and 3) activation of NK cells to induce apoptosis in infected cells (Fig 4). With the additional inclusion of a strain-specific antibody response, the following equations describe the single virus-strain system: (1) (2) (3) (4) (5) (6) (7)

The change in viral load (dV/dt) includes four components, the production term (pI/(1+sF)) in which virions are produced by infected cells (I) at a rate p subject to an IFN-dependent scaling factor of (1/(1+sF)) [17, 23, 28], the viral natural decay/clearance (cV) with a decay rate of c, the neutralisation term (μVA) by antibody (A), and a consumption term (βVT) due to binding to and infection of target cells (T). s indicates the sensitivity of the production rate to IFN. The term g(T + R)(1 − (T + R + I)/C_t) models target cell (re-)growth by both target cells and resistant cells (those protected by the IFN) but limited by a maximum cell number C_t (e.g. due to the spatial capacity, [18]). Target cells (T) are consumed by virus (V) due to binding (β′ VT), the same process as βVT, where β ≠ β′ allows for different measurement units of assays used to detect virus. IFN (F) induces the protective transition from T to R at rate ϕFT and resistant cells (R) lose protection, reverting to susceptible target cells at a rate ρ [28].

Infected cells (I) increase due to the infection of target cells by virus (β′ VT) and die at a (base) rate δ. The term κIF models the killing of infected cells by IFN-activated NK cells [28]. IFN (F) is modelled using simple dynamics that only include production (qI) and natural decay (dF) [36]. Antibodies (A) are produced by activated B cells. We model the proportion of activated B cells by state B. The activation of B cells is induced by an increase in V. Parameter values and their justification are given in Table 1.

For a clearer comparison between the different hypothesised mechanisms by which the innate response contributes to viral control, we consider three models of single virus, each of which includes only one of the mechanisms shown in Fig 4:

• Model 1: including an IFN-induced virus-resistant state of the target cells (by letting s = 0 and κ = 0),
• Model 2: including an IFN-induced diminished viral production rate (by letting ϕ = ρ = ξ = 0 and κ = 0),
• Model 3: including killing of infected cells by IFN-activated NK cells (by letting s = 0 and ϕ = ρ = ξ = 0).

Most importantly, for these three models, the terms of antiviral action appear in different equations. The virus-resistant terms appear directly in the equation for dT/dt and thus modulates the viral load (V) in an indirect way (Model 1). Similarly, killing of infected cells by NK cells (κIF) exerts an indirect control on viral production by changing the infected cell kinetics (Model 3). In contrast, Model 2 assumes direct control of viral production by IFN (due to the term pI/(1+sF)). Thus, we capture the diversity of plausible viral control mechanisms, in particular both indirect and direct pathways.

Models for re-exposure viral kinetics

In order to capture the kinetics of primary–challenge infection experiments, we introduce a re-exposure model in which we assume that the two different viruses share the same source of target cells and IFN, but induce distinct and non-crossreactive antibody responses: (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) An additional subscript (1 or 2, following the existing ones if there is already a subscript like m₁, m₂ and m₃) has been introduced for all the relevant variables and parameters to indicate the primary and challenge viruses. Due to a paucity of experimental data, all parameters for the two viruses are assumed to be equal unless otherwise specified.

As for the single-virus model, we also extend the re-exposure model to three models, each of which includes only one of the innate immune response mechanisms:

• Model R1: including an IFN-induced virus-resistant state of the target cells (by letting s₁ = s₂ = 0 and κ₁ = κ₂ = 0),
• Model R2: including an IFN-induced diminished viral production rate (by letting ϕ = ρ = ξ = 0 and κ₁ = κ₂ = 0),
• Model R3: including killing of infected cells by IFN-activated NK cells (by letting ϕ = ρ = ξ = 0 and s₁ = s₂ = 0).

Methods for numerical simulations and statistical analyses

The ordinary differential equation (ODE) models were solved using MATLAB’s ode15s ODE solver (The MathWorks, Natick, MA). We set an absolute tolerance of 10⁻¹² on all variables for accuracy. For the single-exposure models (e.g. Eqs 1—7), initial conditions were V = 1, T = C_t with all other variables set to zero at t = 0. For the re-exposure models (e.g. Eqs 8—18), initial conditions were V₁ = 1, T = C_t with all other variables set to zero at t = 0. V₂ = 1 was then introduced at the time of challenge. The resolution of the simulated time series shown in the figures was set to be one hundred points per day. When analysing the re-exposure model results, we introduced an indicator, the moving-correlation (MC) coefficient, defined to be the correlation coefficient of a subset of the time series within a moving window, to indicate the periods where the rates of change of the two viral loads were either synchronised or desynchronised. The moving-window was set to be 0.2 days (corresponding to 20 points based on the time series resolution), which we found was sufficient to correctly capture both the relationships and the turning points (smaller values do not further improve the determination of phase-transition points). Determination of these critical phase-transition times was also confirmed by observation based on the time course of solutions. The first peak of the secondary viral infection separating Phase 1 and 2 (defined in the Results) was determined by finding the points where dV₂/dt = 0. MATLAB code is provided in the Supporting Information.

IFN-mediated induction of a resistive state for target cells still results in viral control via target-cell depletion

The level of IFN should significantly influence the kinetics of viral infection based on the (three) model formulations. The control of the level of IFN is achieved by using different IFN production rates (q). Here we examine how the behaviour of the three models changes for different rates of IFN production and how the models compare to one another.

For Model 1, a higher rate of IFN production (and thus a higher attained IFN level) is able to maintain a considerable level of healthy cells in the virus-resistant state, which in turn facilitates a relatively rapid replenishment of target cells immediately following the control of viral infection. However Model 1 fails to prevent the occurrence of a temporary depletion of target cells (see S1 Fig in the Supporting Information). Indeed, target-cell depletion remains the underlying mechanism for control. This is not a surprising result, as in Model 1 increasing IFN leads to a decrease of the term −ϕFT in Eq 2, which facilitates the consumption of target cells.

A detailed study of the change in viral load for both low and higher levels of IFN production (q) is shown in Fig 5A and 5B, where the change of viral load (dV/dt in Eq 1) is decomposed into its four components (appearing on the right-hand side of Eq 1), whose relative contributions to the change in viral load vary by the stage of infection. Both figures show that the single virus infection may be deconstructed into three distinct stages. In the first (“early”) stage of infection (0–2 days) virions are primarily consumed by binding to the target cells (due to an almost full target cell pool) and natural decay, whereas the contribution from antibody is negligible. The second stage (3–5 days) features a significant drop in the term for binding to target cells, which confirms that high IFN levels do not prevent a temporary depletion of the target cells for Model 1. In the last stage, starting around day 5, antibodies begin to dominate the removal of virions.

In contrast to Model 1 in which target-cell depletion is the primary mechanism of viral control, both Model 2 and Model 3 are able to maintain a relatively high level of target cells when a sufficiently high IFN production rate is assumed (S2B and S3B Figs). The conservation of a high level of target cells is also clearly reflected by Fig 5D and 5F wherein the curve representing binding to target cells (βVT) does not show the quick drop evident in Model 1’s dynamics during the second stage post infection. This implies that the decrease in viral load in the second stage for Models 2 and 3 with a larger IFN production rate is driven by mechanisms other than effective limitation in the number of target cells.

When the IFN production rate is small (q = 10⁻⁷), as shown in Fig 5A, 5C and 5E, all three models converge (as expected) to generate qualitatively the same dynamical behaviours as from the simplest TIV model lacking an explicit, time-dependent innate immune response (see S4 Fig).

To study how target-cell depletion varies with the IFN production rate (q) in greater detail, we now explore model behaviour as q increases from 10⁻⁸ to 5 × 10⁻⁵. With increasing q, both Model 2 and Model 3 gradually prevent a temporary depletion of target cells (measured by the minimum of target cell number within the first 7 days post-infection) whereas Model 1 fails to do so (Fig 6; S5–S7 Figs show examples of full time courses for relevant model compartments). Even when allowing the transition rates for the production (ϕ) and decay (ρ) of IFN to be sampled from the space {(ϕ, ρ) ∈ [0, 10] × [0, 100]}, we find the minimum target cell number for Model 1 is restricted to lie within the grey region in Fig 6. Note that for some intermediate values of q the models may lose their ability to completely clear virus (see S6 and S7 Figs), likely due to a lack of immune components or incorporation of only one innate immune mechanism for each case (see Discussion for further comments). These results confirm that Model 1 primarily utilises target cell depletion for viral control and demonstrate that Models 2 and 3 may also have different dynamical properties depending on the IFN production rate.

Challenge virus kinetics are strongly influenced by the inter-exposure interval

Having established the mechanisms Models 1–3 use to control viral infection, we now move onto an examination of the behaviours of the re-exposure models, in which two viruses (the primary and challenge viruses) are introduced consecutively with an inter-exposure interval (IEI). We first study Model R1 in detail, focusing on how the model recaptures the clear dependence upon the IEI shown in the experimental data (Figs 1 and 2). We then present the results of the other two re-exposure models based on that analysis, and through a comparison evaluate the differences between the three models.

Fig 7 shows that the solutions of Model R1, in particular the viral kinetics of the second virus (red curves), change dramatically as the IEI increases from 1 day (A) to 14 days (F). These changes are summarised and can be explained as follows:

• For a 1 day interval (Fig 7A), the two viruses undergo an initially synchronised increase followed by a synchronised decrease (i.e. co-infection). The synchronised increase occurs in the very early stage of infection when target cell numbers remain sufficiently high, corresponding to the first stage of single-virus infection as examined in the previous section. Following target cell depletion, both viruses decrease and are eventually cleared by strain-specific antibody. The dynamics are akin to those for a single virus infection.
• For a 2 day interval (Fig 7B), a short period of synchronised increase is followed by a synchronised decrease (indicated by the MC coefficient of 1). However, this is then followed by a desynchronised period (MC coefficient of -1) wherein the primary virus is cleared while the challenge virus grows once more. During the initial period of synchronised growth the challenge virus’ load is two orders of magnitude smaller than that of the primary virus and the challenge virus in Fig 7A. This may be understood by the strong depletion of target cells by this time. Such a low viral load does not effectively activate antibody production for that virus so that following the brief drop (synchronised with the first virus due to temporary depletion of target cells) the challenge virus increases again with the replenishment of the target cell population.
• For a 3 day interval (Fig 7C) there is no period of synchronised growth. Rather, a synchronised decrease appears immediately following challenge with the second virus due to the temporary depletion of target cells, which is the key feature of the second stage for the single-virus infection (recall Fig 5). Challenge with the second virus during this second stage of the primary virus infection (around 3–5 days) leads to qualitatively the same behaviours (shown later). Similar to Fig 7B, after the initial drop, the second virus experiences a full cycle of infection.
• For an IEI over 6 days (Fig 7E and 7F) the primary virus has essentially been cleared at the time of challenge (stage three of the primary virus infection). During this stage target cell numbers are increasing, enabling immediate infection by the second virus.

Fig 8 summarises all the observed behaviours of Model R1 and indicates the different phases including productive co-infection (Phase 1, grey), an early synchronised decrease (Phase 2, red), a desynchronised phase (Phase 3, green) and final removal of the challenge virus (Phase 4, blue). Importantly, all four phases can be easily mapped to experimental data (Figs 1–3), and always appear in the described order. We will see later that the other two re-exposure models, although exhibiting qualitative differences from Model R1, do not alter the order established here.

As we have analysed, the infection dynamics are closely related to the stage of the primary virus infection at the time of challenge with the second virus, explaining the strong influence of the exposure interval in both the experimental observations and model outputs. All of our observations can be summarised in a single figure (Fig 9A). Reading horizontally, we see that the four phases (separated by colours) appear in order through time. Viewed vertically, the figure shows that the choice of the IEI strongly affects the qualitative behaviours of the re-exposure model (distinguished by dashed lines). Importantly, all regions of this plane may be classified as one of the four phases, suggesting a complete picture has been obtained through this classification procedure. Because of the concise nature of this method of showing re-exposure results, this type of figure will be used to show further results for the alternative re-exposure models (Models R2 and R3).

The hierarchy of viral infection is reproduced by the re-exposure models and provides new insight into model selection

Continuing with the study of Model R1, we now examine whether it can generate qualitatively different re-exposure behaviours (i.e. generate patterns different from that in Fig 9A) by only changing the IFN production rates of the two viruses, q₁ and q₂. As shown in the previous section on single infection, different IFN production rates drive different model behaviours. A very small IFN production (e.g. 10⁻⁷) resembles a model that lacks any IFN-induced protective effect whereas a model with a relatively large IFN production rate (e.g. 5 × 10⁻⁶) shows a significant conversion of target cells to the virus-resistant state. Results are given in Fig 9 where four different combinations of values for q₁ and q₂ are examined. S10 Fig presents further combinations of values for q₁ and q₂ drawn from a wider range. All sub-figures (Fig 9; S10 Fig) show a qualitatively similar pattern. In particular, regardless of the level of IFN production (and thus regardless of whether the resistant state is introduced or not), all show the existence of Phase 2 (red) which characterises the inhibitory effect of the primary virus infection on the challenge virus. Based on our previous analyses of a single viral infection, this is a result of target cell depletion which cannot be avoided by solely introducing the virus-resistant state. Thus, Model R1 (with all other parameters fixed and equal for the two viruses) fails to reproduce the hierarchy of viral infection shown in the data; e.g. primary infection with A(H1N1)pdm09 strongly inhibited influenza B virus challenge dynamics (Fig 1), whereas the latter showed a very limited ability to inhibit the former (Fig 2). The subtle quantitative differences visible in Fig 9 and S10 Fig are understood by considering that a larger q₁ (regardless of q₂) induces a larger virus-resistant cell population (R) and so more rapid replenishment of the target cell pool. Consequently, an increased q₁ leads to a shorter duration of Phase 2 (IEI of day 3–5) as also shown in S1 Fig.

Moving on to consider Models R2 and R3, two different patterns emerge when varying the IFN production rate, as demonstrated by comparing the top row to the bottom row in Figs 10 and 11 for Models R2 and R3 respectively (also see S8 and S9 Figs for examples of time courses of the solutions). These patterns successfully reproduce the hierarchy of viral infection observed from the experimental data (Figs 1 and 2), as we now illustrate. Consider the case that A(H1N1)pdm09 strongly stimulates the immune response (high q) and influenza B provides weaker stimulation (low q). Then if we take the primary virus to be A(H1N1)pdm09 (q₁ = 5 × 10⁻⁶) and the challenge virus to be influenza B (q₂ = 10⁻⁷) we observe that A(H1N1)pdm09 delays infection with influenza B for short IEIs (Fig 10B and S8 Fig). Conversely, if influenza B is administered first (q₁ = 10⁻⁷), then challenge with A(H1N1)pdm09 (q₂ = 5 × 10⁻⁶) results in co-infection for short IEIs (Fig 10C). Results for more combinations of values for q₁ and q₂ drawn from a wider range are provided in S11 Fig (for Model R2) and S12 Fig (for Model R3). In all scenarios for Models R2 and R3, high q₁ prevents depletion of the target cell pool during the primary virus infection (see Fig 6). While exerting a weak delay on the challenge virus for an IEI of 1–3 days (seen in both data and simulation results), the continued availability of target cells allows for productive replication, preventing the system from displaying Phase 2 dynamics. Similar to Model R1, we observe that the patterns are primarily dominated by the IFN production rate of the first virus but nearly independent of that of the second virus (S10–S12 Figs).

In summary, our three models—with their alternative hypothesised mechanisms for the action of the innate response leading to viral control—are each capable of capturing the dynamics of a single virus infection and the main features of primary–challenge experiments. However, Model R1, with its reliance upon the virus-resistant state fails to reproduce the hierarchy of viral infection (i.e. it always produces Phase 2 dynamics). For the other two mechanisms—a decreasing viral production rate (Model R2) or an induced killing of infected cells by NK cells (Model R3)—we have shown that both are able to reproduce all the behaviours including the hierarchy of viral infection observed in the experimental data. We have made these observations based on the assumption that the viruses’ underlying kinetic properties are the same and that their differing ability to induce IFN production is the mediator of observed difference. In the Supporting Information (S13 and S14 Figs), we extend our study by exploring some alternative models in which other virus-immunity parameters are allowed to vary (in addition to the IFN production rate), and demonstrate that Models R2 and R3 can still reproduce the observed hierarchies, while Model R1 remains reliant upon target-cell depletion and so is less capable of capturing the observations.

Stochasticity may explain the complete blocking of the challenge virus at short inter-exposure intervals

We have shown that the re-exposure model can successfully reproduce the phenomena of co-infection and delay (Fig 3). However, we also observe complete blocking of the challenge virus following primary infection in some circumstances (Fig 1, IEIs of 3 and 7 days, and Fig 3). Although the reason for complete inhibition remains unclear, that it only occurs in some of the experimental replicates [10] suggests that stochastic effects in terms of viral dynamics may be important. We hypothesise that failure of the challenge virus may occur when the number of virions (V_2num) drops to a sufficiently low value such that stochastic effects become dominant [37]. For example, the case of an initially synchronised decrease (see Fig 3 and Fig 7C) makes our hypothesis possible. Here we examine this by using a stochastic model derived from the deterministic model used to generate Fig 6C (see the Supporting Information for details on the stochastic model, its implementation and parameterisation). Fig 12 shows that, although the solution of the deterministic model shows a rebound in viral load, the stochastic model results in two possible classes of solution: delayed infection with the second virus (“success”) or blocked infection with the second virus (“failure”).

To quantify this stochastic phenomenon, we investigate the dependence of the success/failure rate on the initial number of virions (see Fig 12C). Results show that the failure rate increases as the initial number of virions decreases in the presence of an innate immune response. Although the control case shows a 0% chance (0 out of 1000 simulations) of failure for the primary virus infection with an initial number of 40 virions, the same number of virions in the challenge inoculum is insufficient to generate any re-infection events. However, as the initial number of virions (for both the primary and challenge viruses) increases from 40 to 5000, the chance of generating successful re-infection events increases to 100%, with a half chance of success when approximately 700 virions are present in the inoculum (as used to generate Fig 12A and 12B). This implies that failure could be due to an insufficient number of successfully infecting virions in the challenge virus inoculum, even if this number is sufficient to reliably induce infection with the primary virus.