In the presence of observational noise, the prevalence of cell–to–cell infection spread cannot be determined from fluorescence time series data alone

We sought to investigate whether an ODE model incorporating both cell–free viral infection and cell–to–cell infection, could be used to infer the balance of the two modes of spread, given a time series of observations of the fluorescent proportion of the cell population as in Kongsomros et al. [6]. We exhibit the basic properties of the ODE model in Fig 1 (the model is fully described in Methods “An ODE model for dual–spread dynamics”). Fig 1A shows the basic structure of the model and the parameters governing the model. We apply a standard target cell–limited model framework with a latent compartment and two modes of infection. That is, initially susceptible cells may become infected either through cell–to–cell infection—at a rate proportional to the infected proportion of the cell population—or through infection by cell–free virus—at a rate proportional to the quantity of extracellular virus in the system. Once initially infected, cells enter the first of K eclipse sub-stages (such that the duration of the eclipse stage is gamma–distributed, instead of exponentially–distributed, see [19, 20]), before becoming productively infected, at which stage they begin producing extracellular virus. Productively infected cells then die. We assume that cells become detectably fluorescent once they become productively infected, but that they remain fluorescent after death over the time scale of simulations, as observed in Kongsomros et al. [6].

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Fig 1. Schematics for the ODE model and proportion of infections from the cell–to–cell route.

(A) Schematic of the ODE model. (B) Proportion of infected cells over time as predicted by the ODE model for an array of values of α and β between zero and 2.5 and 2 × 10⁻⁶, respectively. The parameter values sampled to generate the plot are shown in the inset. (C) Calculation of P_CC. We keep track of the proportion of the cell population which has been infected by the cell–to–cell (CC) and cell–free (CF) infection over the course of infection. We define P_CC as the proportion of infections arising from the CC route at long time. (D) P_CC contour map on α–β space for the dual–spread ODE model. α and β have units of h^-1 and (TCID₅₀/ml)^-1h^-1, respectively.

https://doi.org/10.1371/journal.pcbi.1012264.g001

Throughout this work we will take the majority of the model parameters to be fixed (which is discussed in Methods “An ODE model for dual–spread dynamics”), aside from the two parameters governing the rates of cell–to–cell and cell–free infection, α and β, respectively. Fig 1B shows the dynamics of the infected cell proportion over time using the ODE model with a range of α and β values (throughout this work, α and β have units of h^-1 and (TCID₅₀/ml)^-1h^-1, respectively). We can quantify and describe the overall rate of infection progression by the exponential growth rate r (units of h^-1). This quantity, well established in the theory of both between–host and within–host infection dynamics, describes the initial rate of exponential expansion of the infected (or fluorescent) population [21, 22]. For further details refer to Methods “Exponential growth rate—r”.

We applied simulation–estimation techniques to investigate whether α and β could be inferred from the fluorescent cell time series of the model. We first selected three sets of (α, β) pairs resulting in different proportions of infections arising from each mechanism. Specifically, if we label the final fraction of infections arising from the cell–to–cell route as P_CC, we construct lookup tables on α–β space for this quantity, and use this to compute (α, β) pairs corresponding to P_CC values of approximately 0.1, 0.5, and 0.9, with a fixed exponential growth rate r of 0.52 in each case to ensure the overall dynamics progressed at a comparable rate. We show a graphic of the computation of P_CC in Fig 1C, and a contour map on α–β space for the ODE model in Fig 1D. For further details on P_CC, refer to Methods “Proportion of infections from the cell–to–cell route—P_CC”.

For each of the specified values of (α, β), we simulated the ODE model and, following Kongsomros and colleagues, we computed the fluorescent cell proportion F(t)—that is, the cumulative proportion of the initially susceptible population that has become infected—at t = {3, 6, 9, …, 30}h [6]. We then applied an observational model to this data to simulate the experimental process, by assuming a cell population size N_sample, and overdispersed noise modelled by a negative binomial distribution. We take N_sample = 2 × 10⁵ as in Kongsomros et al. [6] and set the dispersion parameter ϕ = 10², selected to impose a modest amount of noise on our observations, leading to the observed data vector . We specify the observation model in full in Methods “Simulation–estimation”, and explore the role of observational noise in more detail in S1 Text and S1 Fig.

Having obtained our observed data , we run a No U–Turn Sampling (NUTS) Markov Chain Monte Carlo (MCMC) algorithm [23] to obtain posterior density estimates for α and β. For each (α, β) pair to estimate, we run ten replicates of the simulation–estimation process. That is, for each replicate we apply random observational noise to the true fluorescence data and then re–estimate α and β using four independent and randomly seeded chains. We draw 2000 samples from each chain and discard the first 200 samples as a burn–in. We assume uniform priors for α and β on [0, 2.5]h^-1 and [0, 2 × 10⁻⁶](TCID₅₀/ml)^-1h^-1respectively, and assume a negative binomial likelihood. Further details of this simulation–estimation process are specified in Methods “Simulation–estimation”.

In Fig 2, we show the results of this fitting process. In Fig 2A–2C, we show heat maps of the density of posterior samples in (α, β) space from each chain of a single replicate fit. We do so for each of the three target parameter pairs. As a visual aid, we also plot the (α, β) contours corresponding to the true P_CC value and the true r value in each case. These plots show that the posterior samples for each pair of target parameters are spread out along the true r contour. While some samples are close to the target parameter pair, the chains do not appear to converge at this point. In S3 Fig, we show an equivalent plot to Fig 2A–2C as a scatter plot of accepted samples, which confirms that the chains are indeed well–mixed. In Fig 2D, for each target parameter pair, we show violin plots of the posterior distributions of P_CC and r for four replicate fits, along with a box plot of the posterior medians across all ten replicates. We also show the prior density of both of these quantities in grey. Fig 2D shows that while r is well estimated compared to its prior distribution—regardless of the choice of target parameters—P_CC cannot be practically identified even when a conservative amount of observational noise is present. While, at least for the case where the target P_CC = 0.1, the distribution of posterior medians can be somewhat accurate, posterior distributions from individual replicates are frequently far from the true value. Importantly, some of these posterior distributions have a high degree of precision, yet are inaccurate, for instance, Replicate 4 for the case where the target P_CC = 0.5. The individual posterior distributions for α and β, which we show in S4 Fig, show a similar practical unidentifiability. While the mode of the distributions roughly follows the true values of these parameters, the sample densities are dispersed widely, hence confidence intervals on α and β are wide. Overall, this experiment indicates that, when even a modest degree of observational noise is applied to the fluorescence data, only the exponential growth rate r can be accurately estimated: the proportion of infections arising from each mode of spread is lost in the observational process.

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Fig 2. Fitting fluorescence data with the ODE model does not permit inference of the prevalence of cell–to–cell spread.

(A)–(C) Posterior density as a contour plot in α–β space for a fit to fluorescence data with the ODE model where the true P_CC ≈ 0.1, 0.5, 0.9 and the value of r is held fixed. Density is shown for the 1800 samples from each chain after burn–in for a single replicate. We only show densities above a threshold value of 10⁻⁴. (D) Prior density and posterior densities from individual replicates for r and P_CC both with typical observational noise. We repeat this for three sets of parameters resulting in P_CC values of 0.1, 0.5 and 0.9 with a fixed r value. Dashed and solid horizontal lines mark the mean and median values respectively. We also show a box plot of the distribution of posterior medians across all replicates. There are ten replicates in total at each value of P_CC, of which we display four. The marginal posterior densities of α and β are shown in S4 Fig. α and β have units of h^-1 and (TCID₅₀/ml)^-1h^-1, respectively.

https://doi.org/10.1371/journal.pcbi.1012264.g002

We investigated the role of the level of observational noise in determining the quality of estimates of P_CC and r using the ODE model (for full details, see S1 Text). We found that for higher values of the dispersion parameter ϕ than we show here (that is, with less observational noise), estimates of P_CC were overall closer to the true value, however, the distribution of estimate medians still showed not insignificant variance, even when virtually all observational noise was removed. Subject to a higher level of observational noise, estimates of P_CC were almost entirely random. We show these results in full in S1 Fig.

Using a spatial model with spatial data, the balance of the modes of infection spread can be accurately inferred

We sought to apply a similar simulation–estimation procedure to a spatially–structured model of infection, to investigate whether a model capable of describing the actual structure of infection would provide better estimates of the proportion of each infection mechanism. We constructed an agent–based spatial model with an equivalent structure to the ODE model used in the previous result, where transitions between compartments of the model are replaced by probabilities of discrete cells, occupying specific positions in space, changing between states analogous to those in the ODE model. The notable difference in this construction is that while we still model cell–free infection based on a global extracellular viral reservoir, we now model cell–to–cell infection as a spatially local process. Specifically, we assume that the probability of cell–to–cell infection of a given cell is based on the infected proportion of its neighbours, instead of the global infected cell population as in the ODE model. This reflects the assumption—based on current biological understanding—that cell–free virions spread rapidly over the size of tissue we seek to model, whereas cell–to–cell infection is possible only between adjacent cells [3]. This process is illustrated in Fig 3A. Fig 3A shows a schematic of the spatial model, and illustrates the alternate formulation of the cell–to–cell infection mode. Note that, as illustrated in the schematic, cells are packed in a hexagonal lattice, which reflects the biological reality of epithelial monolayers and moreover ensures that adjacency between cells is well–defined. Full details of the spatial model can be found in Methods “A multicellular spatial model for dual–spread dynamics”.

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Fig 3. Dynamics and metrics of the spatial model.

(A) Schematic of the spatial model. The model follows the same structure as the ODE model with the exception that cell–to–cell infection is based on the proportion of a cells neighbours which are infected. (B) Cartoon of the calculation of infected neighbour proportion. (C) κ(t) is our clustering metric, computed as the mean proportion of neighbours of the fluorescent cells which are also fluorescent. (D) Typical time evolution of the cell grid using the spatial model under three α–β combinations, resulting in P_CC values of approximately 0.1, 0.5, and 0.9. Parameters were chosen such that the peak infected cell population is reached at approximately the same time in each instance. Initially infected cells are flagged with a unique colour and infections resulting from that lineage of cells are assigned the same colour. Target cells are marked in grey and dead cells in black. (E), (F), (G) Proportion of cell sheet infected, proportion of susceptible cells which are fluorescent over time, and the clustering metric κ(t) respectively. We show eight simulations for each of the α–β parameter pairs described above. α and β have units of h^-1 and (TCID₅₀/ml)^-1h^-1, respectively.

https://doi.org/10.1371/journal.pcbi.1012264.g003

In addition to the fluorescent proportion metric we introduced in the previous result, we developed an additional metric for the spatial model to describe the extent to which infected cells were clustered together. This metric, which we term κ(t), describes the mean proportion of neighbours of the fluorescent cells which are also fluorescent at time t. In Fig 3C we show a schematic which illustrates the computation of the fluorescent neighbour fraction at a number of fluorescent cells in a cell sheet. We define κ(t) explicitly in Methods “Clustering metric—κ(t)”. κ(t) has the property that when it is large, fluorescent cells tend to be clustered together and the infection is highly localised, whereas if it is small, the infection is diffuse.

In Fig 3D–3G, we demonstrate the behaviour of the spatial model under three (α, β) parameter pairs, chosen to result in a P_CC of approximately 0.1, 0.5, and 0.9, and to reach a peak infected cell fraction at approximately 18h. In Fig 3D, we visualise a section of the cell grid at a series of time points. We do so by assigning a unique index j = {1, 2, …,N_init} to each of the N_init initially infected cells and the extracellular virus they produce. Then, every time a susceptible cell is marked for infection during a simulation, we compute the probability that it was caused by each of the N_init viral lineages, and determine the lineage assigned to that cell. Infected cells are then coloured by their lineage. Once a cell dies, we change its colour to black. This construction allows us to visualise the spread of infection in space. Fig 3D shows that when cell–to–cell infection dominates, infection plaques are tightly clustered and infected cells of the same lineage tend to be found closer together. When cell–free infection dominates, there is no particular structure to the colouring of the cell sheet. In Fig 3E–3G, we show time series for the spatial model under the same three parameter schemes as discussed above: the proportion of the cell population which is infected over time, the fluorescent cell curve as discussed in the previous section, and the clustering metric κ(t). These time series indicate that even though the different parameter regime lead to vastly differently-structured infections—as can be seen in Fig 3D—their infected and fluorescent cell count dynamics as a time series are relatively similar, although there is some variation in the initial uptick of infection in the case where P_CC is large. By contrast, the time series for κ(t) shows substantial variation between the parameter values corresponding to low, roughly equal and high values of P_CC.

Since in the spatial model, cell–to–cell infection is constrained to act locally, infections that spread mainly through cell–to–cell infection are forced to spread radially. The size of the resulting infected cell population, therefore, grows in a non-exponential manner. For this reason, the exponential growth rate r is not well–defined in the case of the spatial model. As an alternative metric of the rate of growth of the infected cell population, we simply use the time of the peak infected cell population, which we label as t_peak. Since this, like P_CC, cannot be well-estimated a priori, we again resort to computing a lookup table of mean t_peak values on α–β space. For full details on the construction of these lookup tables and their corresponding surface plots, refer to Methods “Proportion of infections from the cell–to–cell route—P_CC”.

We computed (α, β) pairs for the spatial model which result in P_CC values of approximately 0.1, 0.5, and 0.9 and a common value of t_peak of approximately 18h, analogous to the values selected for the ODE model in our previous fitting experiment. For each of these parameter pairs, we ran simulations of the spatial model and reported the fluorescent proportion of the susceptible cells as well as the clustering metric κ(t) at times t = {3, 6, 9, …, 30}h, one time point per simulation. This model reflects the destructive experimental observation process. We provide full details of the observational model in Methods “Simulation–estimation”. The resulting observations collectively form our observed data vectors and . We then used Population Monte Carlo (PMC) methods to re–estimate α and β (full details in Methods “Simulation–estimation”) given this synthetic observational data. For each of the three target (α, β) pairs, we ran four replicates of the data generation and fitting process.

We show the results of this experiment in Fig 4. Fig 4, which follows a similar layout to Fig 2, shows that with the addition of clustering metric data, P_CC can now be robustly inferred using the spatial model. In Fig 4A–4C, we plot heat maps of the density of the final accepted posterior samples for α and β in α–β space for the three target parameter pairs, resulting in P_CC ≈ 0.1, 0.5, 0.9. These plots show posterior density distributed compactly around the true values of (α, β), instead of being dispersed along a t_peak contour as in the previous simulation–estimation. In Fig 4D, we show the weighted posterior distributions of P_CC and t_peak for individual replicates along with the distribution of weighted posterior means across replicates. As before, t_peak is still extremely well estimated in each case, however, now the posterior distributions for P_CC are also very accurate to the true value. Moreover, the posterior distributions for individual replicates are concentrated on the true values of P_CC with only modest confidence intervals, and the distributions of weighted mean estimates across replicates are extremely precise to the true values, meaning that carrying out inference with only a single data stream (as opposed to aggregating across multiple observations) was sufficient to estimate both P_CC and t_peak. This was not the case with the ODE model. We also show the individual posterior distributions for α and β in S5 Fig. S5 Fig shows a sharp peak of probability density around the true value of both α and β for each value of P_CC, especially when that mode of infection is minimal. We note that estimates for P_CC are especially sharp when the true value of P_CC is higher, suggesting that the dynamics in this high cell–to–cell scheme are particularly distinguishable.

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Fig 4. Fitting fluorescence and clustering data with the spatial model allows the prevalence of cell–to–cell spread to be determined.

(A)–(C) Posterior density as a contour plot in α–β space for a fit to fluorescence and clustering data where the true P_CC ≈ 0.1, 0.5, 0.9 and the infected cell peak time is held fixed at approximately 18h. We only show densities above a threshold value of 10⁻⁴. (D) Prior density and posterior densities from individual replicates for infected peak time (t_peak) and P_CC with target parameters as specified in (A)–(C). Dashed and solid horizontal lines mark the weighted mean and median values respectively. We also show a box plot of the distribution of posterior weighted means across all four replicates in each case. The marginal posterior densities of α and β are shown in S5 Fig. The replicates in bold are those plotted in (A)–(C). α and β have units of h^-1 and (TCID₅₀/ml)^-1h^-1, respectively.

https://doi.org/10.1371/journal.pcbi.1012264.g004

To test whether our results were dependent on the inclusion of the secondary data source, the clustering metric κ(t), we performed another set of simulation–estimations using the same methods as above, this time using only the fluorescence data (full details in S7 Text). We show the results of this fitting experiment in S6 Fig. This figure shows that, without the use of the clustering metric, estimates for P_CC are again very poor, while estimates for t_peak remain reasonably precise. This result, which mirrors what we observed with the ODE model, suggests that fluorescence data alone is not sufficient to imply the balance of the two modes of viral spread, even for the spatial model. We provide more discussion on this point in S4 Text.

The observational model used in our analysis here aims to recreate the noise incurred in an experimental setting. As such, we obtain our observational time series data by sampling one observation from each of a set of independent stochastic runs of the spatial model. This reflects the destruction of the cell culture in the observation process. However, it is certain that in experimental settings the observational process will incur additional noise than we have explicitly accounted for in this model. As such, we repeated the fitting process shown here after applying an additional negative binomial observational noise layer (the same as used for the ODE model) to both the fluorescence and clustering data. We discuss our results in S2 Text and S2 Fig. S2 Fig shows that estimates for both P_CC and t_peak with the spatial model are robust to a substantial degree of observational noise, especially when compared to applying the same levels of noise to the data and performing inference under the ODE model (shown in S1 Text). This finding suggests that estimates of P_CC using this approach is resilient to additional noise which may be incurred in an experimental setting.

The proportion of cell–to–cell spread can be inferred from diffusion–limited observational data within reasonable limits

So far, we have relied on the assumption that the diffusion of extracellular virions across the model tissue is sufficiently fast that the density of free virus in the system can be approximated as uniform. Clearly, this is a simplification of the biological reality. While the true value of the diffusion coefficient for free virions in media of differing properties is difficult to estimate [17, 24, 25], it is reasonable to assume that extracellular virions are to some extent constrained in the rate at which they spread across the tissue. At very slow diffusion, it may be that cell–free infection is indistinguishable from cell–to–cell infection. It is as yet unclear how well the approach we discuss here might apply to data collected from a diffusion–limited system.

To explore this, we developed an extended spatial model to include a spatially–structured viral density. We assume viral density is secreted continuously by infected cells uniformly in space across their surface, and free virus diffuses across the tissue according to linear diffusion with coefficient D. Throughout this work, we use units of CD²h^-1 for D, where CD is a cell diameter, taken here to be approximately 10μm for a typical respiratory epithelial cell [26]. For full details of the extended model, refer to Methods “A multicellular spatial model for dual–spread dynamics”, and S5 Text for implementation.

We investigated the behaviour of the extended model for varying values of the diffusion coefficient, and the proportion of cell–to–cell infection. The time of the peak infected proportion was held fixed at 18h as in the previous result. Again, α and β values corresponding to specified values of P_CC and t_peak were obtained using lookup tables, however, since these metrics are influenced by the choice of diffusion coefficient D, we constructed new lookup tables for each value of D tested. In Fig 5A, we show a visualisation of the cell grid at the completion of infection for a range of values of the diffusion coefficient and the P_CC. Here, we follow a similar approach to Fig 3D, where we assign each initially infected cell a unique colour and colour each newly infected cell by the lineage that infected it. In Fig 5A, we colour each cell—including dead cells—by the lineage with which they were infected. Fig 5A shows that when diffusion is very small () the difference in the final grid state is almost imperceptible between different cell–to–cell infection fractions. In each case, the grid is divided into large, single–colour foci, indicating that cell–free infections under this scheme are all extremely close to the infecting cell. As the diffusion coefficient increases to around , the edges of single–colour foci become frayed in low P_CC cases and the grid structure is more distinct from the high P_CC cases. For larger diffusion coefficients , the grid states cannot be distinguished from that of the infinite diffusion (uniform virus) case.

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Fig 5. Dynamics of the spatial model with diffusion–limited spread of extracellular virus.

(A) Final grid state following infection with the specified parameters. Initially infected cells are flagged with a unique colour and infections resulting from that lineage of cells are assigned the same colour. Here we show the final state of the cell grid, with each cell coloured by the lineage which infected it. *α and β values computed from lookup table for relevant diffusion coefficient, ensuring a time of peak infected cell proportion at approximately 18h and the indicated proportion of cell–to–cell infection. α and β values for each value of D, t_peak and P_CC used are specified in Table A in S5 Text. (B)–(F) The clustering metric, κ(t) for the same diffusion coefficients and α and β values as in (A). We show results from eight simulations in each case. (G) Maximum vertical distance between the mean κ(t) curves for P_CC = 0.9 and P_CC = 0.1 for varying diffusion coefficients. D has units of CD²h^-1, where CD is a cell diameter.

https://doi.org/10.1371/journal.pcbi.1012264.g005

In the previous result, we found that, while the time series for the proportion of infected cells in the sheet could not practically be distinguished for varying values of P_CC (provided the infected peak time was held fixed), the corresponding time series for the clustering metric κ(t) were clearly separated. Including this metric in our observational data therefore enabled the P_CC to be inferred. As such, we computed κ(t) time series for the same range of diffusion and P_CC values as in Fig 5A to test if such a distinction would be preserved. We ran eight simulations of the extended spatial model for each D–P_CC combination, and show the resulting κ(t) time series in Fig 5B–5F. Fig 5B–5F show that for diffusion coefficients of D = 10CD²h^-1 and above, the wide variation between time series for varying P_CC values is retained. Even for diffusion coefficients as low as D = 1CD²h^-1, there is still a noticeable distinction between the curves, however, at D = 0.1CD²h^-1there is very little variation. We quantify the variation between the curves by computing the maximum vertical variation between the low and high P_CC curves (P_CC = 0.1 and P_CC = 0.9, respectively) for each diffusion coefficient. We plot these in Fig 5G. Fig 5G confirms that for D ≥ 10CD²h^-1, there is as much distance between the curves as for the infinite diffusion case, but that this distance is lost rapidly for D < 1CD²h^-1. These results suggest that it is reasonable to expect that P_CC should be recoverable for a wide range of diffusion coefficients, including biologically likely values [24, 25].

We next carried out another round of simulation–estimations, where we generated diffusion–limited synthetic observational data using the extended spatial model under a range of values for the extracellular viral diffusion coefficient. We then use the (basic) spatial model—with diffusion misspecified as infinite—to re–fit the generated data. We computed target (α, β) parameter pairs corresponding to P_CC = 0.1, 0.5, 0.9 and t_peak = 18h separately at each value of D (these are specified in Table A in S5 Text). Moreover, since when diffusion is small the (α, β) pairs corresponding to this peak time exceed the support of the prior distributions for α and β as defined for the previous results, we conduct this series of simulation–estimations using wider prior distributions. Specifically, we take α_max = 40h^-1 and β_max = 5 × 10^-5(TCID₅₀/ml)^-1h^-1, following the definition in Methods “Simulation–Estimation”. Aside from these adjustments, these simulation–estimations were otherwise conducted using the same methods as in the previous result (summarised in Fig 4). We show the results of these simulation–estimations in Fig 6. Here we plot, as in previous figures, weighted posterior distributions for P_CC for each value of the diffusion coefficient. For each of these values, we show the weighted posterior distributions for each replicate as well as a box plot of the weighted means across the replicates. As a reference, we also include our previously–discussed results for the case where the observational data is generated with uniform virus (infinite diffusion). We show the analogous plot for the time to the peak infected proportion, t_peak, in S8 Fig, which shows, as in previous results, that t_peak is again well–estimated across each replicate, regardless of the value of the diffusion coefficient. By contrast, Fig 6 shows that the quality of estimation of P_CC is highly dependent on the value of the diffusion coefficient. In general, the quality of estimates dramatically decreases for smaller diffusion coefficients. Results for D ≥ 10CD²h^-1 approach the quality of fit obtained for the infinite diffusion case, however, there is a radical departure from the true values of P_CC for estimates where D = 0.1CD²h^-1 or 1CD²h^-1.

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Fig 6. Fitting data from diffusion–limited viral spread with the spatial model recovers the proportion of cell–to–cell infection for a realistic range of diffusion coefficients.

Prior density and posterior densities from individual replicates for t_peak for different values of D, the value of the extracellular viral diffusion coefficient used in the extended spatial model to generate observational data. We re–fit using the basic spatial model. For each value of D we also show a boxplot of the distribution of posterior weighted means across all four replicates. We show results for the case where the target values of α and β give rise to P_CC values of approximately 0.1, 0.5, and 0.9 and t_peak of approximately 18h for the specified value of D. α and β have units of h^-1 and (TCID₅₀/ml)^-1h^-1, respectively. *α and β values computed from lookup table for relevant diffusion coefficient, ensuring a time of peak infected cell proportion at approximately 18h and the indicated proportion of cell–to–cell infection. α and β values for each value of D, t_peak and P_CC used are specified in Table A in S5 Text. D has units of CD²h^-1, where CD is a cell diameter.

https://doi.org/10.1371/journal.pcbi.1012264.g006

While it is difficult to quantify the true value of the extracellular viral diffusion coefficient, Stokes–Einstein estimates for D for influenza or SARS–CoV–2 virions in water at room temperature or plasma at body temperature have been computed to be approximately 216CD²h^-1and 144CD²h^-1 respectively, assuming a cell diameter of approximately 10μm [25–27]. These diffusion coefficients are certainly sufficiently large to enable our approach here to apply, however, we note that several authors have assumed viral diffusion coefficients in various media to be orders of magnitude lower than these values (around ) [24, 25], in which case our approach may offer less precision in estimates of P_CC.

The P_CC = 0.1 case is estimated extremely poorly for the smaller diffusion coefficients, and even for D ≥ 10CD²h^-1, the centre of density for the posteriors still sits substantially above the true value (around 0.26). This is one instance of an overall systematic bias in these estimates which tends to predict higher values of P_CC than is actually present, especially when diffusion is small. This is because, when extracellular virus diffuses slowly, it is more likely to result in cell–free infections near infection foci which are then mistaken for cell–to–cell infections. When true cell–to–cell infection is rare this effect is exacerbated. While this systemic bias limits the ability of the inference to deduce precise estimates of the actual P_CC value in cases where the data is generated using a small diffusion coefficient, it may still be useful in providing an upper bound for this quantity, for instance in the case where D = 1CD²h^-1 and P_CC = 0.1, where our estimates would at least indicate that cell–to–cell infections are at least not the predominant mode of infection.

The posterior estimates in Fig 6 also have the striking feature that even when accuracy is very low, precision remains very high, and with consistent means across replicates. This property is a consequence of the misspecification of the model used to fit the data, which does not include finite diffusion. In S7 Fig we plot the same κ(t) trajectories as in Fig 5B–5F, but grouped by P_CC. S7 Fig demonstrates that, for small and equal cell–to–cell infection proportions (P_CC = 0.1 or 0.5), the κ(t) curve varies substantially for varying values of the diffusion coefficient. Thus, even if, as we predicted in Fig 5G, there is a significant difference between the κ(t) curves for different P_CC values and a given diffusion coefficient, those curves might be notably different to those for the infinite diffusion case. As such, we might obtain a better fit to the observed κ(t) values for an incorrect P_CC value. This might explain why the fits in Fig 6 appear to underperform compared to the predicted variation between curves in Fig 5G. Better estimates could potentially be obtained by refitting the observational data with a model which incorporated finite viral diffusion. However, fitting with such a model would require also fitting the diffusion coefficient D, and it is not clear a priori the accuracy with which this parameter can be inferred.

Inference on the prevalence of cell–to–cell infection is robust to smaller samples of the cell sheet

The clustering metric κ(t), as we have defined it, relies on sampling every fluorescent cell in the tissue at each observation time and calculating the proportion of its neighbours which are also fluorescent. However, in an experimental setting, it may be impractical if not impossible to observe the fluorescent state of every cell in the target population, especially in vivo. We sought to investigate whether approximations of κ(t) generated by sampling from subsets of the cell population would be sufficient to allow α and β—and therefore P_CC—to be inferred. We did so by carrying out simulation–estimations as in the previous result, but where the clustering metric is now approximated by κ_S(t), which is computed by randomly sampling S cells instead of sampling the entire grid. Full details of this adjusted simulation–estimation process are given in Methods “Clustering metric—κ(t)”.

To test the influence of the sample size S on estimation of P_CC, we performed a series of simulation–estimations on the spatial model using both fluorescence and approximate clustering data for varying sample sizes and target values of P_CC. These simulation–estimations were conducted using the same methods as in the previous results. We show the results of these simulation–estimations in Fig 7. Here we plot, as in previous figures, weighted posterior distributions for P_CC for each combination of target parameters and sample size, as well as box plots of the posterior weighted means across replicates in each case. Estimates for t_peak are again very precise across all replicates, as is shown in S9 Fig. Fig 7 shows that as the size of the sample becomes smaller and the approximation of κ(t) becomes coarser, posterior distributions for P_CC become wider and less confident, however, the centre of these distributions is still accurate, as can be seen in the box plots of posterior weighted means, which remain very compact and close to the true value of P_CC. This is true even for the smallest sample sizes and for any target value of P_CC. We see that increasing noise due to a reduction in sample size when approximating κ(t) does not result in biased estimates of P_CC, instead, merely a reduction of confidence. By contrast, as we mentioned in the previous result and S1 Text, while an increase in observational noise did lead to an increase in posterior distribution width, it also resulted in individual replicates where P_CC estimates were found in reasonably tight, inaccurate distributions. Finally, we also note that, as seen in the previous result, estimation of P_CC is far more precise in the case where the target value was higher. Even with the coarsest approximation of the clustering metric, the algorithm correctly identified the P_CC in this case with a high degree of precision. This suggests both that high P_CC dynamics of the spatial model are particularly distinctive—at least as far as the fluorescence and clustering time series are concerned—but also that only tiny samples of the cell sheet need to be measured in order to precisely infer the value of P_CC in this case.

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Fig 7. The prevalence of cell–to–cell infection can be recovered when the fluorescence clustering metric is computed on small samples of the tissue.

Prior density and posterior densities from individual replicates for P_CC for different values of S, the number of cells sampled to calculate the approximation κ_S(t) in fitting. Dashed and solid horizontal lines mark the weighted mean and median values respectively. For each value of S we also show a boxplot of the distribution of posterior weighted means across all four replicates. We show results for the case where the target values of α and β give rise to P_CC values of approximately 0.1, 0.5, and 0.9 and t_peak of approximately 18h. α and β have units of h^-1 and (TCID₅₀/ml)^-1h^-1, respectively.

https://doi.org/10.1371/journal.pcbi.1012264.g007

An ODE model for dual–spread dynamics

We employ an ODE model which is adapted from a typical model of viral dynamics with two modes of spread [18], which is in turn based on the standard model of viral dynamics [28]. We make the additional inclusion of a latent phase of infection, based on observations from data published by Kongsomros and colleagues [6]. We noticed a delay in the initial uptick of the fluorescent cell time series curve, indicating that cells only become detectably fluorescent once they are productively infected, that is, following the eclipse phase of infection. We tested having both single and multiple latent stages in the model—or equivalently, exponentially and gamma–distributed durations for the eclipse phase—and obtained dramatically improved agreement with the data when we assumed multiple latent stages before cells become detectably fluorescent. This approach is common in representing the eclipse phase of infection in the literature [19, 20]. We arrived at the following form of the model, in ODE form: (1) (2) (3) (4) (5) where T is the fraction of cells susceptible to infection, is the fraction of cells in the eclipse phase of infection, I is the fraction of cells in the productively infected state, and V is the quantity of extracellular virus. Since we wish to keep track of whether infections come from the cell–to–cell or cell–free infection routes, we incorporate the following subsystem which keeps track of the cumulative proportion of the target population which has become infected via the cell–to–cell mechanism (F_CC) or the cell–free mechanism (F_CF). We have (6) (7) (8) (9) (10) (11) where T₀ = T(0) is the initial target cell proportion. The sum of these two quantities, (12) is the cumulative proportion of the cell population which has become infected through either mechanism, which we take to be equivalent to the proportion of fluorescent cells as observed in Kongsomros et al. [6]. The assumption that cells remain fluorescent even after they die (over the time scale of interest) is justified by the observation that in Kongsomros et al. fluorescent proportions were observed to saturate at 100% at later times in their experiments.

Throughout this work, we will assume fixed values of the parameters K, γ, δ, p, and c, as specified in Table 1. These parameters were obtained by running a Bayesian parameter estimation for the form of the ODE model as defined above against fluorescent cell time series data in Kongsomros et al. [6], and selecting one particular posterior sample at random. We sketch this parameter estimation process in S6 Text. These values were selected simply to be indicative of the realistic range of values for these parameters and are sufficiently realistic for the purposes of this work. In each case we initiate the infection by setting T(0) = 0.99, I(0) = 0.01 and the remaining compartments to zero.

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Table 1. Fixed parameters used in our simulations.

https://doi.org/10.1371/journal.pcbi.1012264.t001

A multicellular spatial model for dual–spread dynamics

It is straightforward to adapt this system of ODEs into a spatially–structured multicellular model, that is, a model which tracks the dynamics of a finite number of discrete cells which each occupy some specified region of space and at any given point in time, may be in one of a set of cell states [32, 33]. Suppose we model the dynamics of a population of N cells. We associate with each of these cells an index i ∈ {1, 2, …, N}, and a cell state at time t given by σ_i(t), where the possible cell states correspond to the compartments of the ODE system, including the implicit dead cell compartment. That is, for any cell i, σ_i(t) ∈ {T, E, I, I^†}, representing the target, eclipse, infected and dead state respectively.

We consider a two–dimensional sheet of cells with hexagonal packing of cells and periodic boundary conditions in both the x and y directions, such that each cell has precisely six neighbours. This packing reflects the arrangement of cells in real epithelial monolayers and has the practical benefit that all adjacent cells are joined via a shared edge, avoiding any complications associated with corner–neighbours. Throughout this work we use a 50 × 50 grid of cells.

Below, we define both a basic and an extended spatial model. Throughout this work, we use the basic model for inference. The extended model is used only in specified instances for the generation of observational data. For the basic spatial model, following other authors [15, 17, 18, 34], we make the simplifying assumption that the dispersal of free virions over the computational domain is fast, and that the extracellular viral distribution can therefore be considered approximately uniform. As such, the equation for V in our spatial model changes only in notation from Eq (5): (13)

As such, cell–free infection is considered a spatially global mode of spread in our spatial model. By contrast, following results from the biological literature, we assume that cell–to–cell spread is a spatially local mechanism [3, 6]. As such, we assume that the probability of cell–to–cell infection in the spatial model depends not on the global proportion of infected cells as in Eq (1), but rather the proportion of a cell’s neighbours which are infected. Specifically, if we denote by ν(i) the set of indices of the cells neighbouring cell i, and by n_neighbours = 6 the fixed number of neighbours a cell can have, the probability of cell i becoming infected by cell–to–cell infection over a given time period depends on the term . Combining these two mechanisms, we obtain the following transition probability for target cell i to become (latently) infected over some time interval Δt: (14)

We also define an extended spatial model which relaxes the assumption that extracellular viral transport is approximately instantaneous. To do so, we assume that extracellular viral density obeys linear diffusion in the environment with diffusion coefficient D CD²h^-1(where CD is a cell diameter, defined as the constant distance between cell centres). If we denote by S_i the region of space (in ) occupied by cell i, we assume that extracellular virus is secreted by each productively infectious cell j uniformly over S_j, and that any susceptible cell k can become infected by the extracellular viral density in S_k. Specifically, we have for the virus equation (15) and, correspondingly, the transition probability for infection becomes (16)

We numerical solve the virus PDE using a implicit–explicit Finite Difference Method using nodes at each of the cell centres. For further details, refer to S5 Text.

For the eclipse phase, instead of implementing transition probabilities for each E^(k), for computational simplicity we instead sample a latent phase duration from its probability distribution at the time a cell first enters the eclipse state. That is, if we write for the time at which cell i enters the eclipse state, and for the time at which cell i enters the productively infected state, we have (17) where (18)

The remaining compartments are easily described by simple transition probabilities. (19) (20)

Together with appropriate initial and boundary conditions, Eqs (13), (14) and (17)–(20) define the basic spatial model, and Eqs (15)–(20) define the extended spatial model. Following equivalent initial conditions as for the ODE model, in both the basic and extended case we initiate infection by randomly selecting 1% of the cell sheet to be initially infected, and the remainder of the sheet to be susceptible to infection. We use periodic boundary conditions in x and y. In Fig 3A we show a schematic of the model as well as the layout of the cell grid. This is not a novel model: this model structure, or slight variations thereof, has been used in a number of recent publications describing infection dynamics with two modes of viral spread and has become somewhat of a standard approach in the field in recent years [7, 14, 15, 20].

As with the ODE model, we can additionally keep track of the cumulative proportion of infections arising from each mode of infection individually in the spatial model. In addition to the overall probability of infection in Eq (14), we can compute a probability of infection by each mode of spread individually as follows. Using the same Poisson process argument as above, the probability of cell–to–cell infection of cell i not taking place over the time interval [t, t + Δt) is given by (21) and the probability of cell–free infection of cell i not occurring over the same time interval is given by (22) for the basic model, and (23) for the extended model, where and are the events of a cell–to–cell infection and a cell–free infection occurring at cell i respectively. Note that we have to account for the fact that while, mathematically, both events may occur in the time interval [t, t + Δt), we need to assign a unique mode of transmission to each infection. We do so as follows. The following calculation is also derived in work by Blahut and colleagues [15]. If we write m(i) ∈ {CC, CF} for the mode of infection of cell i, then at the time of infection of cell i—that is, when —we compute the probability of each individual mode of transmission as follows: (24) and (25)

In our model, therefore, when an infection is detected, we draw a random number , and if p < P(m(i) = CC), we designate the infection a cell–to–cell infection, otherwise, it is considered a cell–free infection. We use a similar calculation to assign the viral lineage associated with an infection, which we used to construct the colouring of cells in Fig 3D, which we stipulate in full in S3 Text. The quantities F_CC and F_CF can easily be calculated for the spatial model as (26) (27) which allow us to keep track of the count of each type of infection event throughout simulations of the spatial model. As before, we define (28)

Metrics

Proportion of infections from the cell–to–cell route—P_CC.

We introduce the quantity P_CC to denote the proportion of infections arising from the cell–to–cell route. This is calculated by keeping track of the cumulative proportion of the target cell population which becomes infected by either infection mechanism over time. At long time—once the infection has essentially run its course—we compute P_CC as the fraction of the total infections which occurred via cell–to–cell infection. Using F_CC and F_CF as we have defined them, we have (29)

In Fig 1C we show an illustration of this calculation more generally.

P_CC quantifies the relative weight of the cell–to–cell route of infection and is therefore our target for estimation in this work. Its definition is general and is not specific to any particular model structure. P_CC cannot be directly calculated in closed form directly from the model parameters. Instead, we repeatedly simulate our model using parameters sampled from α–β space and compute P_CC in order to construct lookup tables. In Fig 1D, we plot a contour map of P_CC values for the ODE model in α–β space. In the case of the spatial model, we accounted for the inherent stochasticity of the model by running 20 simulations of the model at each (α, β) pair in the lookup table and kept track of mean P_CC values. The associated contour map for the spatial model is shown in Fig 8A. Contour plots were generated by computing contours over the lookup table using MATLAB’s contourf function. We interpolate between values on the lookup table by constructing spline fits along α and β contours.

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Fig 8. P_CC and t_peak contour maps on α–β space for the spatial model.

Contour maps for (A) P_CC, and (B) t_peak. α and β have units of h^-1 and (TCID₅₀/ml)^-1h^-1, respectively.

https://doi.org/10.1371/journal.pcbi.1012264.g008

Simulation–estimation

Throughout this work we conduct a series of simulation–estimation experiments to explore what can be learned about the roles of the two modes of viral spread based on observed model outputs. We outline here the general framework of this process.

For both the ODE and the spatial model, we begin by drawing a set of target values for the infection parameters α and β. As mentioned above, the values of the other model parameters are considered fixed and known. We then simulate the chosen model using these parameter values, and apply an observational model f(⋅) to its output to generate a set of observed data . The observational model f is designed to simulate the noise incurred in actual experiments. Throughout this work, we focus especially on the observed fluorescent cell proportion over time, since this is the main source of data reported by Kongsomros and colleagues [6].

For the ODE model, we obtain the observed fluorescent cell proportion by computing the true fluorescent cell time series F(t) (defined in Eq (12)) at each of a series of observation times, converting this proportion to a count of fluorescent cells and applying negative binomial noise. The negative binomial distribution reflects the observed error structure in [6], which is constructed from overdispersed count data. We assume some vector of observation times t = {t₁, t₂, …t_m} and define (33) where for i = 1, 2, …, m, where N_sampleF(t_i) and ϕ are the mean and dispersion parameter respectively of . N_sample is the number of cells measured for fluorescence, in a sense the size of the cell population. for all i = 1, 2, …m. In Fig 9B, we show an illustration of this observation process. The curve shown in blue is the true fluorescent proportion curve F(t). At each of the observation times, indicated with dots, we apply noise about the true value.

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Fig 9. Schematics of the fitting process and observational models.

(A) Schematic of the simulation–estimation process. (B)–(C) Observation model for fluorescent proportion of susceptible cells for (B) the ODE model and (C) the spatial model (first five points shown). In the spatial case we also show the observation model for the clustering metric κ(t). For the ODE model, we sample the true fluorescent proportion curve at a series of time points (shown in blue), then observe a value based on a negative binomial distribution centred on the true value (box plot of the distribution shown in orange). Here, ϕ = 10². For the spatial model, we run independent iterations of the stochastic model and observe one point from each. Note that additional observational noise can be applied to these data points, as we explore in S2 Text.

https://doi.org/10.1371/journal.pcbi.1012264.g009

After obtaining observed data , we then re–estimate α and β using Bayesian methods. We assume uniform prior distributions (34) (35) with α_max = 2.5h^-1, β_max = 2 × 10^-6(TCID₅₀/ml)^-1h^-1. We re–estimate α and β using No U–Turn Sampling (NUTS) Markov Chain Monte Carlo (MCMC) methods with a negative binomial likelihood (36) for i = 1, 2, …, m, where is the fluorescent proportion time series estimated by simulating the ODE model using samples and . For each estimation we use four chains seeded with random initial values and draw 2000 samples for each, including 200 burn–in samples. We assume N_sample = 2 × 10⁵, which was the number of cells used in the experiments in [6].

For the spatial model, we have two sources of observational data: both the fluorescent proportion time series and the clustering metric κ(t). Since the system is inherently stochastic, we do not add additional external noise, instead, we aim to emulate the experimental process whereby the fluorescent proportion of a cell population (and consequently the clustering metric) cannot be observed without destroying, or at least disrupting, the cell sheet. We implement this by sampling our observations from m independent simulations of the model. That is, if we have observation times t = {t₁, t₂, …, t_m} and m true fluorescence and clustering time series from independent simulations of the spatial model, F(t) = {F₁(t), F₂(t), …, F_m(t)} and K(t) = {κ₁(t), κ₂(t), …, κ_mt} respectively, we generate the two sets of observational data, (37) (38)

We show a demonstration of this observation process in Fig 9C.

Due to the stochasticity of the system, we use Approximate Bayesian Computation (ABC) to re–estimate α and β for the spatial model. In particular, we adapt the Population Monte Carlo (PMC) method introduced by Toni and collaborators [37] and revised by others [38, 39]. We sketch this method in pseudocode in Algorithm 1.

Algorithm 1 PMC algorithm for parameter estimation using the spatial model—fluorescence and clustering data

Input: Model , prior distributions for target parameters π_α(α) and π_β(β), target number of particles N_P, number of generations G, reference data and , distance metrics and , perturbation kernel K(⋅|⋅), initial acceptance proportion , threshold tightening parameter q.

Output: Weighted samples from the posterior distributions , .

Rejection sampling

for do

 Randomly draw and from π_α(α) and π_β(β), respectively.

 Obtain the model output using these parameters, .

 Compute the distance between model output and reference data and .

end for

n_{opt found} ← 0, T ← 0

while n_{opt found} < N_P do

 T ← T + 1, define as the indices i in the smallest T values of the and the smallest T values of the .

end while

for j = 1, 2, …, N_P do

 Set .

 Set w_j = 1/N_P.

end for

is the initial particle population. is the initial weight vector. Set the distance thresholds and as the q^th quantile of the and respectively.

Importance sampling

for g = 1, 2, …, G do

 Set number of accepted particles

 While N_accepted < N_P do

  Randomly draw a particle with probability w_j.

  Perturb particle by the kernel to obtain a new sample .

  Obtain the model output using these parameters, .

  Compute the distance between model output and reference data and .

  if and then

   Set N_accepted ← N_accepted + 1 and .

  else

   Return to start of while.

  end if

 end while

 for i = 1, 2, …, N_P do

  Set

 end for

 Set ,

 Set the distance thresholds and as the q^th quantile of the and respectively.

end for

In our case, the model is simply the time series F(t) obtained by a single simulation of the spatial model with parameters and , and evaluated at time points t, the vector of time points at which the reference data is obtained. We again use the uniform prior distributions in Eqs (34) and (35), although now with α_max = 10h^-1, β_max = 1.5 × 10⁻⁶(TCID₅₀/ml)^-1h^-1. For the perturbation kernel, we use the following definition proposed by Beaumont and colleagues: [38] (39) where Φ(x; μ, σ²) is a multivariate normal and Σ is the empirical covariance matrix of the particle population , using their weights . For the other parameters of the algorithm, we set N_P = 500, G = 5, p_0,accept = 0.3, and q = 0.5. We use euclidean distance for the distance metric d. For the case where we attempt only to estimate α and β using the spatial model and fluorescence data only, we slightly simplify the fitting process. We apply the same observational model, outlined in Eq (37), to the fluorescence data, and use a slightly simplified version of the PMC method to refit α and β. We provide full details in S7 Text.