Fig 1.
Schematic of the innate and adaptive immune response during primary acute viral infection.
Exposure to the virus occurs at time 0 and extracellular viral load begins to rise (shaded green curve). Initial innate immune responses include phagocytosis of virus by neutrophils and macrophages, Type I interferon-induced antiviral resistance (IFN) (dark blue) and killing of infected cells by Natural Killer (NK) cells and other cell types (red). The black vertical dashed line denotes the transition between innate and adaptive immune responses. The adaptive immune response is triggered both by cytokine signaling to the lymph nodes and the migration of antigen-presenting cells from the tissue to the lymph nodes (not shown). In the later phases of infection, innate immune responses continue, but additional adaptive immune components come into play, including virus-specific cytotoxic T-cells (light blue) kill infected cells directly and also kill nearby cells through a variety of mechanisms. The orange vertical dashed line denotes the onset of the humoral adaptive immune response. B-cells produce virus-specific antibodies (orange line) which bind and inactivate virus directly and also allow its clearance and clearance of infected cells by other cell types. Tissue damage (shaded purple curve) accumulates due to cell death from direct responses to virus and from immune-cell killing by contact-mediated, diffusible factor-mediated and bystander-mediated mechanisms and eventually dissipates as cells proliferate to repair the damage (Adapted from [50,51]). The specific time course of all components varies among viruses, host tissues and host species, but the general sequence of events and immune response components are generally preserved.
Fig 2.
(A) Model objects, processes and interactions: Conceptual model of an epithelial tissue and lymph node. Schematic representation of the model objects, processes and interactions. Epithelial and immune cells refer to the two main classes of cells. Interactions occur within an extracellular environment, and a compartmental model of a lymph node controls immune-cell recruitment to the tissue. Together the epithelial-cell, extracellular-environment and immune-cell components represent the epithelial tissue. Each model object has associated processes that dictate its states and behaviors. Epithelial-cell processes include viral internalization (E1), viral replication (E2), viral release (E3) and cell death (E4). Immune cell processes include activation (I1), chemotaxis (I2), contact cytotoxicity (I3) and oxidative cytotoxicity (I4). Activated immune cells participate in oxidative cytotoxicity (I4) and secrete oxidative agents into the oxidizing-agent field (T3). Activated cells become inactive after 1 hour. The virus field (T1), cytokine field (T2) and oxidizing-agent field (T3) describe spatially-varying densities of extracellular components. Field processes describe diffusive transport and clearance of material in the extracellular environment and activated transport to the lymph nodes. The lymph node is a single-compartment model whose pro- or anti-inflammatory state specifies the recruitment or removal rate (L1) of immune cells in the epithelial tissue. The transport of cytokines to the lymph node promotes its proinflammatory state. (B) Viral Life Cycle: Interactions in the viral internalisation, replication and release models. Schematic representation of inputs, outputs and interactions between stages of the viral replication model. Extracellular viral particles (represented as continuous fields) are internalized by the viral internalization model and initiate the viral replication model. The main stages of the viral replication model are: unpacking, viral genome replication, protein synthesis and viral assembly and packaging. The output of the viral replication model passes to the viral release model, which transfers newly assembled viral particles from the cells into the extracellular environment. (C) Cell types and transitions. Epithelial cells are of type uninfected if they have not yet internalized any virus (E1). They are of type infected if they have internalized virus, but are not releasing virus into the virus field (viral release E3 is inactive). They are of type virus releasing if they are releasing virus into the extracellular virus field (i.e., viral release E3 is activated). Immune cells are initially inactive and do not participate in the oxidative cytotoxicity (I4) or chemotax towards higher concentrations of the cytokine field (I2). They become activated when they experience activation (I1). In all panels, dashed arrows with barbed heads represent transformations, solid arrows with barbed heads represent transport and solid arrows with lollipop heads represent regulation.
Fig 3.
Simulation of the progression of infection in a patch of epithelial tissue of size 360 μm x 360 μm starting from a single infected cell for a representative simulation using the baseline parameters given in Table 1.
(A) Snapshots of spatial configuration vs time, showing progression of a simulated infection. Columns, left to right: 0 minutes (time of initial infection), 4000 minutes (67 hours, 2 ¾ days) after infection, 8000 minutes (133 hours, 5 ½ days), 12000 minutes (200 hours, 8 ⅓ days), 16000 minutes (267 hours, 11 days), and 20000 (333 hours, 14 days) minutes. First row: epithelial-cell layer composed of uninfected (blue), infected (green), virus-releasing (red) and dead epithelial cells (black). Second row: position of immune cells in the extracellular environment layer. Third row: concentration of extracellular virus field. Fourth row: concentration of extracellular cytokine field. Fifth row: concentration of extracellular oxidative agent field. Fields are shaded on a logarithmic scale: red corresponds to the chosen maximum value specified in the first panel and blue corresponds to six orders of magnitude lower than the maximum value; colors saturate for values outside this range. (B-D) Simulation time series. (B) Number of uninfected (orange), infected (green), virus-releasing (red) and dead (purple) epithelial cells vs time on a logarithmic scale (0 values are overlaid at a non-logarithmic tick for clarity). (C) Total extracellular cytokine (magenta) and total extracellular virus (brown) vs time on a logarithmic scale. (D) Value of the immune recruitment signal S (yellow) and number of immune cells (grey) vs time on a linear scale. Simulations use periodic boundary conditions in the plane of the epithelial sheet, and Neumann conditions [57] normal to the epithelial sheet.
Table 1.
Parameter values in baseline parameter set.
Fig 4.
Patterns (classes) of spatiotemporal infection dynamics.
First row: snapshots of spatial configurations of the epithelial cells. Color coded: uninfected (blue), infected (green), virus releasing (red), dead (black). TImes from left to right 4000 minutes (67 hours, 2 ¾ days), 8000 minutes (133 hours, 5 ½ days), 12000 minutes (200 hours, 8 ⅓ days), 16000 (267 hours, 11 days) and 20000 minutes (333 hours, 14 days). The right border of each snapshot aligns with the corresponding time in the time series. Second row: number of uninfected (orange), infected (green), virus-releasing (red) and dead (purple) epithelial cells vs time on a logarithmic scale (with 0 included for clarity). Third row: total extracellular cytokine (magenta) and total extracellular virus (brown) vs time on a logarithmic scale. Fourth row: value of the immune recruitment signal S (yellow) and number of immune cells (grey) vs time on a linear scale. (A) No immune response: infection propagates unopposed until all epithelial cells have died from intracellular virus. (B) Widespread infection: weak immune response slows propagation of the infection, but no uninfected cells survive at the end of the simulation. (C) Slowed infection: uninfected and infected epithelial cells coexist at the end of the simulation. (D) Containment of infection: no infected or virus-releasing epithelial cells remain, uninfected cells survive and virus remains in the extracellular environment at the end of the simulation. (E) Recurrence: the number of infected and virus releasing epithelial cells goes to zero, but persistent extracellular virus infects new epithelial cells later on. (F) Clearance: the number of infected and virus-releasing epithelial cells goes to zero and the level of extracellular virus is negligible at the end of the simulation. The model in (A) omits the immune response (components L1, I1-I4). All parameter values are as in Table 1 and Fig 3 except for kon and βdelay (S1 Table).
Fig 5.
Sensitivity analysis of the number of uninfected epithelial cells vs time for variations in virus-receptor association affinity kon and immune response delay coefficient βdelay, showing regions with distinct infection dynamics.
Logarithmic pairwise parameter sweep of the virus-receptor association affinity kon and the immune response delay βdelay (×0.01,× 0.1,× 1,× 10,× 100) around their baseline values, with ten simulation replicas per parameter set (all other parameters have their baseline values as given in Table 1). The number of uninfected epithelial cells for each simulation replica for each parameter set, plotted on a logarithmic scale, vs time displayed in minutes. See Fig 4 for the definitions of the classes of infection dynamics.
Fig 6.
Sensitivity analysis of the number of infected epithelial cells vs time for variations in virus-receptor association affinity kon and immune response delay coefficient βdelay, showing regions with distinct infection dynamics.
Same parameter sweep as Fig 5. The number of infected epithelial cells for each simulation replica for each parameter set, plotted on a logarithmic scale, vs time displayed in minutes. See Fig 4 for the definitions of the classes of infection dynamics.
Fig 7.
Sensitivity analysis of the total amount of extracellular virus vs time for variations in virus-receptor association affinity kon and immune response delay coefficient βdelay, showing regions with distinct infection dynamics. Same parameter sweep as Fig 5.
The total amount of extracellular virus for each simulation replica for each parameter set, plotted on a logarithmic scale, vs time displayed in minutes. See Fig 4 for the definitions of the classes of infection dynamics.
Fig 8.
Number of uninfected cells vs time in simulations of a hypothetical drug treatment reducing the viral genome (e.g. RNA for SARS-CoV-2) replication rate (rmax) as a function of treatment potency and time of initiation of treatment.
Drug therapy is administered at a fixed time after infection and remains activated for the duration of the simulation. (A) Sample treatment, showing the time course of rmax. rmax is reduced by a multiplier which is one minus the potency of the drug at the given dose, 75% in (A), at a particular time of initiation of treatment (time delay of application), 12000 minutes (200 hours, 8 ⅓ days) in (A). (B) A parameter sweep of the potency of treatment (reduction in baseline viral replication rate rmax, vertical) and the time of treatment (dashed lines, horizontal) shows parameter regions where the majority of simulation replicas produce positive outcomes (green-shaded subplots), negative outcomes (orange-shaded subplots) and intermediate cases (intermediate shading or unshaded). Intensity of green and orange indicates the number of positive and negative outcome replicas for each parameter combination (treatment effectiveness). Green regions show that early intervention leads to positive outcomes (is effective) for most ranges of treatment potency, with high numbers of uninfected epithelial cells at the end of the simulation for almost all simulation replicas. Orange regions show that late interventions result in mostly negative outcomes (ineffective treatment) regardless of the potency, and that outcomes are more variable between replicas, with both positive and negative outcomes for most parameter sets. The number of uninfected epithelial cells for each simulation replica for each parameter set, plotted on a logarithmic scale, vs time displayed in minutes.
Fig 9.
Number of virus releasing cells vs time in simulations of a hypothetical drug treatment reducing the viral genome (e.g. RNA for SARS-CoV-2) replication rate (rmax) as a function of treatment potency (one minus the viral replication rate multiplier) and time of initiation of treatment.
The number of virus-releasing epithelial cells stays low when the intervention occurs early during infection (when the amount of extracellular virus is increasing rapidly), but continues to increase when the intervention occurs later (when the level of extracellular virus is at or near its maximum in the untreated case). Parameter values, axis types and time-scale and shading as in Fig 8.
Fig 10.
Levels of extracellular virus vs time in simulations of a hypothetical drug treatment reducing the viral genome (e.g. RNA for SARS-CoV-2) replication rate (rmax) as a function of drug potency (one minus the viral replication rate multiplier) and time of initiation of treatment.
Extracellular virus is cleared or near-cleared when intervention occurs soon after infection. Parameter values, axis types and time-scale and shading as in Fig 8.
Fig 11.
Difference in treatment effectiveness for different simulation replicas for perfect treatment potency (0 viral replication rate multiplier) near the time when the extracellular virus amount would reach its maximum in the untreated case (10000 minutes, 167 hours, 7 days).
(A) Select simulation replicas for this parameter set showing the variety of possible outcomes (treatment effectiveness). Spatial results show the epithelial and immune cell layers, and the extracellular virus field, at 4000, 8000, 12000, 16000, and 20000 minutes (67, 133, 200, 267 and 333 hours, 2 ¾, 5 ½, 8 ⅓, 11 and 14 days). Cell type colors are the same as in Fig 3A. Virus field values are scaled as in Fig 3A. (B) Time series for all simulation replicas for the selected parameter set: Right column, from top to bottom, number of uninfected epithelial cells, number of infected epithelial cells, number of virus-releasing epithelial cells, number of dead cells. Left column, from top to bottom: total amount of extracellular virus, total amount of cytokine, number of immune cells and immune response state variable. All variables except the immune signal plotted on a logarithmic scale vs time.
Fig 12.
Randomly distributed unsusceptible cells inhibit but do not stop spread of infection throughout the tissue.
(A) Distributions of epithelial cells during simulation time (columns) for varying fraction of unsusceptible cells (rows). From top to bottom, replicas were simulated with 10%, 20%, 30%, 40% and 50% of epithelial cells unsusceptible to viral internalization. Cell type colors are the same as in Fig 3A. (B) Number of uninfected cells during simulation time for ten replicas of each fraction of epithelial cells unsusceptible.
Fig 13.
Basic integration of an explicit model of hepatitis C virus subgenomic replication in a spatial context presents stochastic outcomes in ten simulation replicas.
(A) Schematic of implementing the integrated HCV model using the available modules described in Models and methods (“Main modules”) and add-on modules libraries hosted in the framework public repository. All main modules are imported except Viral Replication, which is replaced with Integrated HCV. Modules used in a simulation are specified in model specification. The integrated HCV model module is available in the add-on modules library. (B) Distributions of epithelial cells during simulation time for select replicas (labeled “a”, “b” and “c”) using the integrated HCV model. Cell type colors are the same as in Fig 3A. (C) Number of uninfected cells (left) and total extracellular virus (right) during simulation time for ten replicas using the integrated HCV model. Select results shown in (B) are annotated according to replica labels “a”, “b” and “c”.
Fig 14.
Even simple model extensions and extensions of extensions can produce notably different spatiotemporal emergent features.
(A) Schematic of implementing a modified epithelial cell model using the available modules described in Models and methods (“Main modules”) and add-on modules libraries hosted in the framework public repository. Modules used in a simulation are specified in model specification. The Simple Recovery and Neighbor Recovery model modules are available in the add-on modules library. (B) Distribution of epithelial cells during simulation time using the Simple Recovery (top) and Neighbor Recovery (bottom) models. Cell type colors are the same as in Fig 3A. (C) Number of uninfected cells during simulation time for ten simulation replicas using the Simple Recovery (left) and Neighbor Recovery (right) models.
Fig 15.
State diagram and interactions of epithelial cells.Epithelial cells can have one of four “cell types”: uninfected, infected, virus-releasing and dead.
Uninfected cells become infected cells when the viral uptake model (E1) internalizes viruses from the extracellular virus field (T1). Infected cells continue internalizing viruses from the extracellular virus field and become virus-releasing cells when the viral replication model (E2) produces sufficient newly assembled virions. Virus-releasing cells secrete viruses into the extracellular virus field (T1) according to the viral release module (E3) and secrete cytokines directly into the extracellular cytokine field (T2). Virus-releasing cells can die if the conditions of the virally induced cell-death model (E4) are met.
Fig 16.
Representative time series of viral internalization, replication and release models from Fig 2B.
A sample simulation of the viral replication model in a single epithelial cell. The model is initialized with one unit of internalized virus (U = 1), and the rest of the state variables set to zero (R = 0, P = 0, A = 0). No additional virus internalization occurs during this sample simulation. Dashed line indicates the time of the cell’s transition from the infected to virus-releasing cell type.
Fig 17.
State diagram and interactions of Immune cells.
Immune cells can adopt two different generalized types: inactive and activated. Inactive immune cells are recruited by the cytokine levels according to the immune recruitment module (L1).Transition from inactive to activated immune cells is determined by the immune activation module (I1) when cells are exposed to cytokines in the tissue. Activated immune cells amplify the cytokine signal by secreting cytokines to the extracellular environment. Activated immune cells chemotax towards virus-releasing cells (I2). Immune cells induce death of epithelial cells by direct cytotoxicity when coming into contact with infected cells (I3), bystander effect by killing neighbors of infected cells (I3) and through oxidative cytotoxicity (I4) by releasing cytotoxic oxidizing agents (T3) into the extracellular environment.