A cellular model of virus-host innate immune interactions

We present a comprehensive mathematical model that provides an integrated framework for studying virus-innate immune interactions at the single-cell level. Building on two foundational models that individually describe the viral dynamics and immune response, [26,43] respectively, our model describes the combined dynamics of the virus and host immune response in an infected immune cell (Fig 1A). Previously, we had developed a detailed mathematical representation of (+)RNA virus life cycle that describes the intracellular dynamics of (+)RNA viral constituents, including the effects of vesicular compartment formation, viral mutations, and host cell variability [26]. Although this original viral dynamics model provided a quantitative framework for examining viral replication and growth, it lacked the cellular innate immune response. To address this, we have now incorporated the major immune cascade dynamics (inspired from Burkart et al. [43]), which models the RIG-I mediated detection of viral dsRNA and downstream signaling, and the JAK-STAT pathways. Our model, consisting of 77 equations, captures the essential nonlinear interactions between viral replication and innate immune signaling while remaining computationally tractable. To achieve this, we introduce a set of strategic simplifications that retain the principal regulatory motifs without attempting to reconstruct the full biochemical intricacy of the system. To maintain model tractability, we consolidated upstream sensing of RNA viruses into a unified dsRNA–RIG-I module, modeling immune detection exclusively through this primary cytosolic axis [49,50]. Negative regulation was similarly abstracted into a uniform feedback term. While omitting cell-type-specific modulators, these simplifications preserve the essential feedforward activation and delayed negative regulation inherent to innate immune signaling, enabling a precise characterization of the kinetic interplay between viral replication and RIG-I-mediated induction.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1. Integrated mathematical model of the viral life cycle with innate immune pathways.

A Schematic representation of the intracellular model coupling the (+)RNA virus life cycle with host innate immune signaling. Viral infection begins with the extracellular virions (V₀) binding and entering the host cell, followed by uncoating and release of the viral RNA. This RNA undergoes translation to produce structural () and non-structural proteins (), and replication via a double-stranded RNA intermediate (), often sequestered within membrane compartments to evade immune detection. Leakage of replication intermediates into the cytoplasm can trigger recognition by pattern recognition receptors like RIG-I, activating the MAVS and downstream transcription factors such as IRF3/7 and NFB. This results in the production and extracellular release of type I interferons. Extracellular IFN-I binds to its receptor, activating the JAK-STAT signaling cascade via STAT phosphorylation, which induces the transcription of interferon-stimulated genes (ISGs). ISGs can either suppress the viral life cycle (ISGav, strength given by ; green repression T-shaped arrows) or auto-regulate the innate immune pathway (ISGn, strength parameterized by ; blue repression T-shaped arrows). Viral antagonism, particularly via non-structural proteins, can suppress the activity of antiviral ISGs (suppression strength given by ; red repression T-shaped arrows). Created in BioRender. K, R. (2026) https://BioRender.com/9v006fh B Viral (left) and ISG (right) dynamics under the influence of various virus-immune interactions, normalized to 96 h-point levels without virus-immune interactions (S₀: , blue). The red line depicts dynamics under the action of viral antagonism and immune suppression along with negative auto-regulation (: ). Dynamics with immune activation with antiviral response alone (: , , magenta) and host immune activation with negative autoregulation but no virus mediated immune suppression (: and , black) is shown for comparison. C Comparison of fold reduction in total viral load () between the case and various viruses with the entire virus-immune interaction network () is shown; HCV (black bar), JEV (white bar), and a few hypothetical viruses (grey bars). All the model parameter values of the hypothetical virus, , are based on HCV, except for , which is representative of JEV. virus variants were limited to since HCV and JEV primarily differ in these parameters.

https://doi.org/10.1371/journal.pcbi.1014395.g001

The integration of viral and immune dynamics was achieved by identifying the mechanisms that links them - (a) through viral dsRNA detection that facilitates RIG-I activation of the innate immune cascade, and (b) by detailing the reciprocal interactions between immune response effectors and the viral elements (Fig 1A). During infection, viral double-stranded RNA (dsRNA, the replication intermediate) accumulates in the cytoplasm as it leaks from the replication compartments [17,51,52] (equation 4). This cytoplasmic dsRNA becomes accessible to RIG-I, enabling its detection and subsequent activation of the innate immune response. In parallel, ISGs mount a robust and multifaceted defense by targeting viral entry, replication, translation, assembly, and release [9,12,53]. To account for these immune responses, we employ a Hill-function-based repression model to represent the ISG-mediated inhibition of these viral processes (Fig 1A, equations 1, 2, 3, 4; also S1 Text equations- S3, S4, S5, S6, S7, S9, S10, and S11). Apart from the effect of innate immune response on viral dynamics, we also account for how viral proteins actively interfere with the innate immune response by targeting key components of the signaling pathways. To model this interference, we introduce viral protein-mediated repression of signal propagation via MAVS, IKK-TBK1, and the STAT1/STAT2 receptor complexes [54–56] (Fig 1A, S1 Text equations- S14, S21, S23, S44, and S46). Finally, we incorporate the regulation of IRF7 (equations 5 and 6), a master transcriptional regulator of type I interferons (esp. IFN) previously overlooked in earlier models. IRF7 plays a critical role in amplifying IFN responses through a positive feedback loop mediated by JAK-STAT signaling, where initial IFN production (mediated by IRF3) induces IRF7 expression (details provided in Methods). To precisely capture the dynamics of the cellular immune response, innate immune parameters are adapted from prior studies that estimate them using experiments with synthetic 5’ppp-dsRNA or IFN to stimulate cells [15,43]. Virus-specific parameters for HCV and JEV are obtained from our earlier work that reconciles experimental data of viral protein and RNA dynamics with viral life cycle dynamics [26].

To investigate the broad impact of the bidirectional virus-host interactions on infection progression, we first employ an aggregated parameter approach to examine the viral-immune dynamics in our comprehensive integrated model. For this purpose, we introduced two lumped tuning parameters: , representing the strength of immune-mediated suppression of the viral processes (equations 1, 2, 3, 4; also S1 Text equations- S3, S4, S5, S6, S7, S9, S10, and S11; see Methods), and , capturing the extent of viral protein-mediated suppression of the immune response (S1 Text equations- S14, S21, S23, S44, and S46; see Methods). Additionally, serves as a tuning parameter to modulate the negative feedback effect of anti-inflammatory ISGs on the JAK-STAT pathway (S1 Text equations- S37, S39, S41, S42, S43, and S44; see Methods). At the molecular level, these interactions influence different processes such as immune sensor activation, signaling, viral replication, and protein translation to varying extents; however, for analytical simplicity, we will assume a uniform regulatory effect throughout for each of the immune-mediated suppression and the viral antagonism processes. This approach ensures that the model remains interpretable while allowing us to explore the broader implications of these interactions on the infection outcomes. We begin by defining a baseline scenario (S₀) devoid of counteracting interactions between the immune or viral components (, Fig 1B). Under this condition, the model successfully reproduces the dynamics of Hepatitis C virus (HCV) proteins, consistent with experimental observations (S1 Fig) [25,26]. The JEV-specific parameters were likewise estimated previously in [26] against the data of Uchil and Satchidanandam (2003) [57], and are used here without modification. In this scenario, immune pathways are gradually activated, as evidenced by the increase in levels of interferon-stimulated genes (ISG). Next, we consider the scenario where the antiviral effects of the innate immune response are included (: ). Here, the viral infection is substantially disrupted, and the ensuing reduction in immune activation reflects a resolution phase driven by the absence of sustained viral stimulus. In the subsequent scenario, we incorporate the negative self-regulation of the innate immune pathway (: ). This results in a dampened immune response, leading to a less pronounced reduction in viral load compared to . Finally, we examine the complete system by including the virus-mediated suppression of the innate immune pathways with the combined action of the immune pathways (: ). In this scenario, the immune response is further suppressed, as indicated by a significant reduction in ISG levels. In fact, 96 h-point ISG expression in is approximately five orders of magnitude lower than in the baseline scenario (S₀), suggesting strong repression of the immune response due to virus interference. However, there is only a modest change in the level of virus growth across the two scenarios (Fig 1B). Notably, compared to , the complete system exhibits a substantially higher viral load alongside a marked further suppression of ISG levels, underscoring the dominant role of virus-mediated immune antagonism in tipping the balance from viral clearance toward persistent infection. To further assess the robustness of the model, we examined how the qualitative dynamics of viral titre change across a range of multiplicities of infection MOI = 1.2 and MOI = 4.9 (S2 Fig). Across both conditions, the four scenarios (S₀, , , ) reproduce qualitatively similar ordering and trends in normalized viral titre trajectories, confirming that the interplay between viral replication and immune suppression is robust to the choice of initial inoculum.

By adjusting the model parameter values (S3 Table) to reflect the dynamics of Japanese Encephalitis Virus (JEV) [26], the system predicts kinetics that are qualitatively similar to those observed for HCV (S3 Fig). However, virus-specific differences emerge when comparing the values at the end of the simulation window across different scenarios and in the temporal dynamics of the immune response, particularly ISG induction, where displays distinct kinetics between HCV and JEV, with JEV showing a more rapid rise and sustained high ISG levels, in contrast to the attenuated ISG response observed in HCV. The immune-mediated suppression of viral infection, as indicated by the 96 h-point viral load ratio , is markedly higher for JEV (≈350) compared to HCV (≈30), highlighting a stronger suppression effect on JEV than HCV (Fig 1C). We hypothesize that this disparity arises from key differences in the viral life cycle parameters of JEV. Notably, JEV exhibits a faster viral translation rate (higher ) and larger number of replication complexes (higher ) compared to HCV. These factors drive a more rapid and substantial accumulation of replication intermediates, which in turn facilitates earlier and stronger activation of the immune response. Consequently, when the antiviral effects of the immune response are incorporated (), the reduction in viral load is significantly greater for JEV than for HCV. This highlights the critical role of viral replication dynamics in shaping the interplay between viral persistence and immune suppression.

To validate this hypothesis, we constructed a panel of hypothetical viral strains, each retaining the characteristics of HCV except for a single substituted feature that is derived from JEV (Fig 1C). For example, the parameter , parameterizing the timescale of replication-compartment formation, is 2-fold smaller in JEV than in HCV. Reducing to the JEV-like value in HCV yields a chimeric virus with faster compartment formation and earlier exposure of cytoplasmic dsRNA. In scenario (), the chimeric strain , incorporating JEV-like carrying capacity (), exhibits the strongest suppression of viral load relative to the baseline scenario (), with reductions ≈150-fold and closely recapitulating JEV-like behavior. The strain , incorporating a JEV-like translation rate (), also shows enhanced suppression, albeit to a lesser extent, indicating a secondary contribution of translation dynamics to immune-mediated control. In contrast, chimeras modifying other parameters, including the replication-compartment formation timescale (), compartmentalized replication intermediates formation rate (), and rate of egress (), exhibit comparatively modest changes in viral suppression and largely retain HCV-like behavior, with reductions typically ≈30-fold. Together, these results indicate that increasing the number of replication complexes is the dominant factor driving enhanced immune-mediated suppression, with faster viral translation providing an additional, but secondary, contribution. Mechanistically, this likely reflects more rapid accumulation of viral products, leading to earlier immune activation and more effective antiviral responses, consistent with observations in other systems [58].

Reciprocal negative feedback between viral and immune processes shapes the viral-immune transition boundary

Experimental studies have demonstrated that early and robust innate immune responses, particularly those mediated by interferon-stimulated genes (ISGs), are essential for limiting viral replication and achieving viral clearance [59,60]. In contrast, delayed or dysregulated immune responses, often resulting from viral immune evasion strategies, can lead to immune escape and prolonged infection [61–64]. We argue that infection outcomes are governed by the dynamic titration of viral antagonism against host immune strength, both of which fluctuate along a biological continuum shaped by viral load, replication kinetics, and host-specific cellular or genetic contexts [65–68]. For example, different viral strains can differ in their ability to produce immune antagonists or stimulate ISG expression, while host immune activity is modulated by polymorphisms in interferon signaling components, ISG expression profiles, and metabolic or stress conditions. Motivated by these biological complexities, we investigated whether our virus-immune model is capable of capturing the dichotomy in infection outcomes and identifying the key cellular and viral features that govern the balance between viral clearance and immune escape.

The strength of viral antagonism () determines distinct immune response dynamics. At low values, ISG levels remain high and the viral loads remain low, indicating robust immune activation and effective viral clearance. However, as increases, ISG levels are progressively suppressed, particularly at higher viral loads, reflecting the dominance of viral antagonism and the transition to immune escape (Fig 2A). For HCV, the infection outcome switches from a ‘virus-low, ISG-high’ state to a ‘virus-high, ISG-low’ state (Fig 2A) as surpasses 0.58, suggesting that HCV can overcome immune control above this critical level. Further, a threshold viral load is evident below which ISG levels remain low, regardless of , suggesting that such low viral loads are insufficient to trigger strong immune activation.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. Virus-immune transition boundary and sensitivity analysis reveal critical regimes in infection.

A The phase plot of viral load and antiviral ISG levels during hepatitis C virus infection is shown for various strengths of viral antagonism (). Each curve shows the trajectory of the system in the state space, with arrows indicating the direction of time from the initial state at 0 h.p.i to the state reached by 96 h.p.i (circles). The system shifts from a ‘virus-high, ISG-low’ state (pink region) to a ‘virus-low, ISG-high’ state (grey region) as increases, with the clear transition at = = 0.58 represented by the dashed line. B The viral load ( at 96 h.p.i) HCV is shown as a function of and . C Correlation coefficients between viral load () and individual parameters near the VITB boundary show how a parameter influences the viral load at the end of the simulation window, thus identifying antiviral and proviral parameters. D Parameter sensitivity indices for and evaluated at the infection threshold point: (, ) = (1, 0.58) are negatively correlated. Colored dots denote model parameters with threshold sensitivity score > 1 (see Methods). E The VITB boundary for HCV infection systems is shown for different values of (left) and (right). The solid line, dashed line, and the solid line with square symbols show the VITB boundary when the parameter is unperturbed, reduced by half, and doubled, respectively. F Fold change in the HCV viral load at 96 h-point when the corresponding parameter values were (top panel) halved (magenta-, green-, or (bottom panel) doubled (black-, blue-, and orange-) along the VITB boundary.

https://doi.org/10.1371/journal.pcbi.1014395.g002

We can now evaluate how virus infection is established as a function of the strength of the two key regulatory interactions in our model: the immune-mediated viral suppression (by varying ) and the viral antagonism of the immune response (by varying ). To delineate the transition between immune control and viral escape, we systematically varied the lumped parameters, and while maintaining the reciprocal parameter at a fixed value. By evaluating the total viral output (), at the end of the simulation window, we identified the critical thresholds, and , which represent the system’s points of maximal sensitivity. Specifically, these thresholds were defined by the peak of the first-order numerical derivative of . For instance, was determined by locating the sharpest transition in viral titre as a function of immune activity, while an analogous procedure was employed to derive from the maximal slope of the viral output curve with respect to viral antagonism. Mapping across the and parameter space reveals a transition boundary separating bimodal outcomes: (a) a low-virus regime, where high and low result in effective immune clearance, and (b) a high-virus regime, where low and high leads to immune escape (Fig 2B). The locus of critical points (, ) defines the Viral-Immune Transition Boundary (VITB) that captures the tipping point between innate immune control and virus immune escape and crossing precipitates a rapid switch between virus-low/ISG-high and virus-high/ISG-low states. Near this boundary, the infection outcome is characterized by extreme sensitivity, where minor variations in the virus-host balance fundamentally dictate whether an infection resolves or persists. There is a marked positive correlation between the critical values of (immune activity) and (viral antagonism), indicating a balancing interaction: as viral antagonism intensifies, enhanced immune activity is necessary to counteract it. Overall, the VITB boundary emphasizes the nonlinear dynamics of the virus-immune system, where robust immune activation (high ISG levels) can suppress viral replication, but strong viral antagonism can overwhelm the immune response, leading to persistent infection.

An analogous trend is detected for JEV (S5 Fig). However, with the same level of innate immune activity (), JEV requires a lower level of antagonism to evade immune control ( = 0.16) compared to HCV ( = 0.58). This is consistent with JEV being more strongly suppressed by the innate immune response than HCV in the absence of antagonism (Fig 1C), once we note that sets a per-protein efficacy while the net antagonistic burden scales with total antagonist abundance. JEV’s higher translation rate () – identified above as a primary driver of its enhanced dsRNA-mediated immune activation – also accelerates accumulation of viral antagonists, so a smaller suffices for immune evasion. HCV’s slower protein synthesis [26] correspondingly demands higher per-protein efficacy to reach a comparable burden.

Our initial approach involved broad perturbations across multiple interactions (varying and ) rather than specific molecular-level processes. However, such simultaneous perturbations are unlikely to occur in natural systems. A more plausible scenario is that the fitness of the virus or the immune response is influenced by specific changes in a single or a small subset of parameters. To explore this, we investigated whether the Virus-Immune Boundary (VITB) emerges naturally from the virus-immune interaction network under more realistic conditions, where only a limited set of parameters change for different viral strains or cell types. Notably, similar sharp transitions between viral clearance and escape states with distinct VITB boundaries were observed for HCV when we varied specific model parameters, such as , , and (S4 Fig). This sharp transition at the VITB boundary provides a unique opportunity to dissect the relative contributions of viral and host processes to infection outcomes without the need to simulate the full system dynamics under individual parameter variations.

By leveraging the system’s heightened sensitivity near the VITB boundary, we can identify key parameters that strongly influence infection outcome. The analysis in Fig 2C illustrates how this approach enables the identification of critical drivers of the infection threshold (at = 0.58, = 1 for HCV). Specifically, we simulate the infection outcomes using a set of 25,000 parameter combinations, where each parameter is sampled from a log-uniform range spanning two orders of magnitude (0.1× to 10× the nominal value) (see S2 and S4 Tables, and Methods). This approach allows us to recover a bimodal distribution in viral loads (S6 Fig) that highlights the pronounced dichotomy in infection states near the VITB boundary. Approximately 60% of the parameter combinations lead to viral clearance, while 40% result in high viral load (>1 nM). Notably, neither outcome strongly dominates the parameter ensemble, placing our model in a regime consistent with the cell-to-cell variability in infection outcome reported in isogenic cell populations infected with (+)RNA viruses [32,45,48,69,70]. Correlation between the sampled parameter values and the resulting viral load helps us in identifying proviral and antiviral factors (Fig 2C). We find that the degradation rate of viral RNA (), proteins (), internalized virus () or extracellular virus particles () is strongly correlated with reduced viral load, highlighting their obvious role as critical parameters controlling viral growth. We also find that time constant for development of replication compartments is also strongly correlated with reduced viral load. Similarly, parameters that enhance viral processes—such as viral translation (), replication (), export of viral (+)RNA from replication compartments (), fusion of viral and endosomal membranes (), or the carrying capacity of compartmentalized replication complexes () - are positively correlated with higher viral loads. On the other hand, enhancing immune detection, for example, via increasing the leakage of compartmentalized viral dsRNA into the cytoplasm (), leads to a reduction in viral load. Among the host cell properties, potent immune signaling and activation can similarly reduce virus production. For instance, enhanced phosphorylation of TBK1 or IKK (), or increased activation rates of MAVS () or upregulation of host protein () can considerably decrease viral load. Conversely, when immune factors undergo rapid removal due to enhanced degradation or deactivation, we observe increase in viral load suggesting diminished innate immunity. This is exemplified for the case of ISG mRNA () degradation, and dephosphorylation of MAVS (), or kinases ().

To further dissect the mechanisms governing the balance between immune and viral interplay, we performed an infection threshold sensitivity analysis to quantify how changes in crucial viral and host parameters affect the system’s tipping point (details in Methods). A distinct inverse correlation is evident in how various parameters affect and , underscoring the antagonistic nature of these two sets of interactions (Fig 2D). Interestingly, some molecular-level processes stand out as more proviral or antiviral than the others. Among the viral processes, parameters that directly influence protein production, such as translation rate , or indirectly by boosting RNA synthesis, like replication rate , exhibit the strongest negative influence on (S7 Fig). For instance in the case of HCV, a + 10% increase in , reduces the viral antagonism () required for virus escape by 9.4% (from 0.58 to 0.525) for a fixed (= 1). Thus, enhanced viral replication and translation aid in efficient immune escape. Regarding the immune response, mechanisms that enhance ISG activity ( and , ) reduce the required strength of the immune response to control the viral proliferation. Conversely, we observe that enhanced leakage of compartmentalized viral dsRNA into the cytoplasm () can decrease . For instance, at the VITB boundary, when the rate of leakage of compartmentalized dsRNA is is increased by +10%, the antiviral activity () required for immune control is decreased by 4.5% (from 1 to 0.95) for HCV. This provides additional evidence that elevated may trigger earlier and more abundant production of immune agonists (dsRNA), which support the immune response (Fig 2C). In case of JEV, swift formation of replication compartments (small ) result in rapid dsRNA accumulation. This triggers a potent immune activation and requiring only mild immune response to suppress it. Conversely, in HCV infection, slower dsRNA production postpones immune activation, necessitating stronger immune suppression to control infection.

We next extend our infection threshold sensitivity analysis to investigate such virus-immune tipping point dependence on various critical cellular and viral processes (identified in Fig 2D) along the entire VITB boundary. Figs 2E, S8, and S9 illustrates how a twofold increase or a 50% decrease in ISG translation efficiency or viral dsRNA leakage influences the VITB boundary across the entire range of viral antagonism () and immune activity () for HCV. Higher ISG RNA translation rate () requires lower immune activity to counteract viral antagonism (downward shift in VITB boundary) and achieving viral clearance, while reducing it leads to weakening of immune control (upward shift in VITB boundary) (Fig 2E left). We observe that a two-fold decrease in ISG translation () leads to a 3–16 fold increase in HCV viral load at different points on the transition boundary (Fig 2F top). Effectively, an increase in leads to a downward shift of the VITB boundary in the space, indicating that change in the host ISG RNA translation rate consistently impact the system’s behavior uniformly along the transition boundary. This trend is also observed for a two-fold higher ISG mRNA degradation rate that results in an ∼ 3–16 fold increase in viral load at different points along the VITB boundary.

Unlike host-mediated processes, perturbations in viral processes, such as replication (), or translation () influence system behavior in a regime-dependent manner across the transition boundary interface (Fig 2E, right; S8 Fig). At high levels of both viral antagonism and immune activity, increasing the rate of translation () shifts the VITB boundary upward, requiring stronger immune activity () to counteract viral antagonism () and reduced levels of viral proteins makes it easier for the immune system to suppress viral replication. Similar asymmetric changes in the VITB are also observed when other virus-mediated proviral processes are impeded (S9 Fig). Such dependence of these viral processes even influences the viral load levels in a regime-dependent manner along the VITB boundary (Fig 2F). When virus–immune interactions are weak (low and ), viral load () is largely insensitive to changes in , or (Fig 2F). However, in the strongly interacting regime of the VITB boundary, small changes in the proviral factors have disproportionately large effects. For instance, along the boundary at high (= 10), a two-fold increase in or yields ∼60-fold and ∼50-fold increases in , respectively (Fig 2F). Comparable trends for JEV (S8, S9, and S10 Figs) suggest this regime-dependent sensitivity is a general feature of (+)RNA virus-host immune interactions.

These regime-dependent effects of viral parameters can be explained by two factors: (a) the dual role of viral dsRNA as both a viral replication intermediate and an immune activator, and (b) the double-negative feedback loop between viral antagonism () and immune suppression of viral replication (). Given the dual role of viral dsRNA, changes in viral processes, like viral replication, translation, and replication compartmentalization, that alter its levels exert both pro-viral and antiviral influences. These counter-influences tend to balance each other at the VITB interface. When the negative feedback interaction strengths are high, strong mutual antagonism amplifies even minor imbalances, producing pronounced shifts in infection dynamics and outcome (Fig 2E and 2F).

Dynamic phases of immune response during infection progression

Deciphering the temporal dynamics of immune responses during infection is essential for identifying key regulatory mechanisms, critical immune checkpoints, and potential therapeutic targets. In complex biological systems like viral infections, immune factors and viral components interact over distinct temporal phases, each defined by specific molecular patterns. However, due to the vast array of immune components, monitoring the dynamics of all elements is impractical in experimental settings, and identifying the most informative markers poses a complex challenge. We can leverage our virus-immune response model to address this challenge.

Here, we investigate the temporal profiles of immune components and examine their temporal correlations to pinpoint effective indicators of cellular infection spread. To capture the immune system’s underlying dynamics, we examine its behavior under three distinct virus-immune regimes (Figs 3A, S11, S12, and S13): (a) at the VITB boundary (), (b) viral clearance regime (), and (c) immune escape regime (). Variables exhibiting substantial change in their dynamics (coefficient of variation ≥0.05) were retained (see Methods), and their trajectories across all regimes were concatenated and z-normalized, enabling a unified comparison of temporal patterns. Hierarchical clustering of the z-normalized trajectories of immune factor dynamics across these regimes revealed eight distinct groups of dynamically co-regulated immune factors (Fig 3A). Each cluster exhibits unique temporal patterns that can be delineated into three major temporal phases of immune response during infection: an ‘early activation (EA)’ phase (0-24h), an ‘intermediate response (IR)’ phase (24–72 h), and a ‘late resolution (LR)’ phase (72–96 h)(Fig 3B).

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. Dynamics of innate immune effectors under different infection outcomes.

A Clustered z-normalized dynamics of immune pathway components under three viral-immune regimes: low (; top), critical (; middle) and high (; bottom) are plotted. Distinct clusters are indicated by the color-coded dendrogram. B The panels show the dynamics of viral proteins (grey) and the mean temporal profiles of the immune pathway clusters identified in A, revealing their characteristic dynamic behavior under each regime. C The metric, quantifying the divergence between dynamics of immune factors across the low and the high regimes, is plotted (see Methods; bar edge colors reflect the cluster assignment). The inset shows the metric for the early phase (0–24 h). Dots indicate factors that show high divergence in both time periods.

https://doi.org/10.1371/journal.pcbi.1014395.g003

In the ‘early activation’ phase, the newly formed viral dsRNA is detected by the RIG-I, triggering a sudden but transient increase in the levels of the “Early responders” cluster (Fig 3A and 3B). This group is dominated by the activated forms of antiviral signal transducers (aMAVS, pIKK, pTBK1, pIKK, and pNF). Their levels reach a peak around 12–24 hours, followed by a sharp decay. Overall, “Early responders” display similar initial transient dynamics in all three infection regimes. However, their induction level is inversely proportional to the severity of the infection, i.e., strong activation of the “Early responders” is associated with the viral clearance regime. Additionally, “Early responders” increase further in the resolution phase, likely due to reinforcement of immune signaling when initial immune responses are effective. Concurrently, the “IRF3–IFN” cluster — comprising pIRF3, IFNmRNA, and IFNmRNA — undergoes a similarly transient early peak followed by sharp decay. The “RIG-I Sentinel” cluster (aRIGI) is also engaged in this phase, exhibiting rapid sigmoid-like activation within the first 24 h that reflects prompt detection of cytoplasmic dsRNA by the pattern recognition machinery. The late-phase behaviour of both clusters is regime-dependent and is described below.

During the ‘intermediate response’ phase, as the viral replication is established and viral dsRNA increases, RIG-I activation increases. This initiates the activation of the “IFN-ISG transcriptional hub” (Fig 3A and 3B), which encompasses phosphorylated STATs, IRFs, and IFN/ISG mRNAs. These immune components exhibit a progressive increase in their levels as the infection advances, reaching their highest concentration during the ‘intermediate response’ phase. However, the increase was substantially repressed under high (immune escape regime), which can be attributed to the suppression of the signaling pathways by the viral proteins (Fig 1A). Conversely, when considering viral clearance characterized by low , the elevation in levels during the resolution phase is significantly greater than in other regimes. This suggests robust reinforcement of pathways and effective antiviral transcriptional activity. In this phase, activation of the “Secondary responders”, comprising STAT-bound IFN receptor complexes (ARC, ARC-STAT2c, and ARC-STAT12c), plays a pivotal role in directing immune responses. This group exhibits a bell-shaped dynamic, rising to a peak around 36–48 h before gradually declining, potentially due to feedback inhibition and resource depletion. The peak was most pronounced under the viral clearance regime, and its attenuation at high is consistent with diminished upstream IFN-driven receptor engagement due to robust viral immune suppression. Activation of this complex thus represents a pivotal relay between IFN production in the IR phase and the transcriptional amplification that follows in late resolution.

In the ‘late resolution’ phase, the regime-dependent fates of the early-activated clusters become apparent. The “RIG-I Sentinel” (aRIGI) declines under low virus (), consistent with clearance of the viral dsRNA stimulus, but remains persistently elevated under high virus despite impaired downstream signaling. Similarly, the “IRF3-IFN” cluster shows a secondary rise specifically under low infection (low ), reflecting renewed IRF3-driven transcription as infection resolves. Distinct from these is the “IRF7 Amplifier” (pIRF7), which emerges as the dominant factor of this phase: it remains near baseline through most of the infection before rising steeply after 72 h exclusively under low infection (low ), driven by accumulated IFN-mediated induction of IRF7 and its positive feedback activation. This late amplification is entirely absent under the immune escape regime.

dynamics during virus infection can be understood from the behaviour of the “,” , and “–I” forms (Fig 3A and 3B). Cytoplasmic forms a standalone cluster and is quickly depleted due to phosphorylation, nuclear import, and its capture by . This rapid drop highlights its role as the input pool for downstream signaling. As translocates and phosphorylates into , it induces IFNs and feedback regulator . The transient peak of “” is concomitant with “Early Responders” group dynamics, peaking around 12–24 hours followed by a sharp decay. The “” is a regulatory module that constitutes a distinct cluster exhibiting a delayed but progressive monotone increase from approximately 24 h, consistent with its role as a feedback brake whose accumulation depends on prior -driven transcription. Under high , this increase is attenuated, reflecting a partially active and inefficient cycle caused by viral suppression of the MAVS–IKK signaling pathway. However, under low viral load, the rise is more pronounced owing to enhanced signaling activity.

Taken together, our analysis reveals a hierarchical temporal cascade in the host antiviral response, wherein identified clusters reflect functionally distinct stages of innate immune signaling and effector activation. The clusters show significant sensitivity to viral load, with their rapid modulation highlighting crucial points where host detection and response pathways may succeed or fail. Notably, the early divergence of the “IFN-ISG transcriptional hub” and “pIRF7” in various infection regimes suggests these molecules can serve as early indicators of infection outcomes, even before clear phenotypic signs appear.

We employ a metric to compare the difference between high and low viral load scenarios (Fig 3C, see Methods). Phosphorylated IRF7, nascent IFN transcripts, and early ISGs are key factors in viral infection outcomes, particularly in the initial stages of infection, indicating that the immediate-early transcriptional circuit plays a crucial role in initiating effective antiviral states (Fig 3C-inset). Throughout the infection period, the persistent impact of IFN levels, phosphorylated IRF7, highlights the critical roles of strong downstream amplification in immune regulation (Fig 3C). These findings map the dynamics of viral antagonism directly onto distinct regulatory layers of the host response, illustrating how viral interference at both early detection and later transcriptional amplification stages can rewire immune trajectories and shape disease outcomes. Our model facilitates the understanding of the hierarchical and temporal structure of the antiviral response, identifying specific molecular clusters as crucial indicators of the infection progression.

IFN Desensitization is Mediated by Receptor-Kinase Complex and TYK Dynamics

IFNs are widely used as therapeutic agents for the treatment of chronic viral infections, certain cancers, and autoimmune diseases, due to their potent antiviral and immunomodulatory effects [71,72]. However, their clinical efficacy is often limited by the phenomenon of desensitization, where repeated or prolonged IFN exposure leads to reduced cellular responsiveness [73]. This behavior is thought to arise from feedback inhibition and regulatory dynamics within the JAK-STAT signaling pathway [74–76].

Because our model incorporates key immune pathways that regulate the IFN response, it can capture the dose-dependent desensitization effect for a naive (uninfected) cell (see Methods). A high dose of IFN (100 nM) sharply induces ISGs responses with ‘ISGn mRNA’, an early transient negative regulator [15], and the ‘ISGav mRNA’, a late antiviral transcript [15], quickly rising to 0.75 nM and ≈ 140 nM, respectively (Fig 4A and 4B). This activation is brief, with both mRNA dropping significantly and stabilizing at approximately 0.01 nM and 60 nM within 24 hours. A second high (100 nM) IFN dose 24 hours later elicits no additional increase in ISGn or ISGav mRNA, signifying strong desensitization. Conversely, administering a high subsequent IFN dose (100 nM) following an initial low dose (1 nM) results in an increase in ISGn and ISGav, albeit it is modest compared to the substantial increase observed with a single high-dose (100 nM) of IFN. This signifies limited IFN desensitization consistent with experimental reports of dose- and time-dependent IFN desensitization [15,30].

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. Dose-dependent IFN-desensitization.

The kinetics of A an ISG transcript involved in immune suppression (early ISG, ISGn) and B an ISG transcript involved in viral suppression (late ISG, ISGav) are compared under distinct two-dose interferon (IFN) treatments. The second dose is administered 24 hours after the initial dose. The left panels represent Condition 1, where an initial low dose of 1 nM IFN is followed by either a low (1 nM, yellow line) or high (100 nM, orange line) second dose. The right panels represent Condition 2, where an initial high dose of 100 nM IFN is followed by either a low (1 nM, yellow line) or high (100 nM, orange line) second dose. C The values, representing deviations in the dynamics of JAK-STAT pathway components between the two IFN priming conditions, are plotted. The top-ranking species, which indicate model components most sensitive to the initial IFN dose, are highlighted. D Time-course dynamics of select model species with the two highest (: RTKC, TYK) and two moderate (: IFNARd, JAK) values are shown, illustrating IFN dose-dependent differences in system response.

https://doi.org/10.1371/journal.pcbi.1014395.g004

Comparing these scenarios, most downstream signaling components remain similar at re-exposure, but key upstream regulators, particularly TYK2 and the IFN receptor complex (IFNAR1/2), show marked depletion after high-dose exposure (Fig 4C and 4D). This suggests that desensitization arises from the destabilization or inhibition of receptor-proximal components, as reported in experimental studies [15,74–77].

To further test this hypothesis, we simulated a two-dose IFN regimen (100 nM followed by 100 nM). We selectively restored the RTKC, TYK, and JAK levels before the second dose. Restoring the receptor-TYK2 complex (RTKC) partially recovered both ISGs expression, while restoring TYK2 alone had no effect (S14 Fig), indicating that negative feedback primarily targets the RTKC. Reintroducing JAK, whose levels remained unchanged across conditions, also had no impact on ISG expression. These results confirm that desensitization is driven by the depletion or inhibition of the receptor-kinase complex, which serves as the key regulator of downstream IFN signaling.

Harnessing interferon and antiviral synergies to enhance viral suppression

Building on our insight about IFN-mediated immune response regulation, we examined how interferon administration can modulate the outcome of viral infection. The timing and dosage of IFN treatment are suggested as critical factors influencing its efficacy [78–80]. Our model shows that IFN treatment markedly decreases viral load in a dose-dependent fashion when given before or shortly after infection (Fig 5A). The timing of administration significantly impacts treatment efficacy. Specifically, a low IFN dose (1 nM) given 1–7 days before infection yields an antiviral effect over 100 times greater than a high dose (>10 nM) applied a few hours post-infection (Fig 5A). We observe that IFN potency declines rapidly once replication is established, with inhibition strongest during the earliest stages of infection. This is consistent with studies showing that early replication and translation events represent a vulnerable bottleneck for viral control [78], that dengue virus is sensitive to IFN only before productive replication [79], and that SARS-CoV-2 becomes largely refractory to IFN following initial replication due to antagonism of the JAK–STAT pathway [80].

Download:

• PNG
  larger image
• TIFF
  original image

Fig 5. Effect of IFN dosage and timing on viral infection.

A Heatmap showing the viral load (nM) as a function of the timing of IFN addition (hours post-infection, h.p.i.) and the IFN dosage (nM). Simulations were conducted under a virus-high regime, where the virus exhibits enhanced immune escape capabilities (virus-high regime; ). B Viral load dynamics under IFN treatment for two IFN dosages (blue: 1 nM, red: 3 nM) and two levels of viral countermeasure efficiency (dotted lines: ; solid lines: ) are shown. The prophylactic efficacy is quantified using the area over the curve (AOC), represented by the light copper shaded region. A higher AOC value corresponds to a stronger prophylactic effect, as it reflects a more substantial reduction in viral load. C Heatmaps illustrating the normalized AOC (nAOC) values as a function of IFN dosage (nM) and strength of key viral parameters: (top) viral countermeasure efficiency (), (middle) viral replication rate () and (bottom) viral dsRNA leakage rate (). Here, , , and represent the baseline or WT values and the ratios- , , and , represent the change w.r.t to the WT values. D Heatmaps illustrating the fold change in normalized AOC (nAOC) as a function of IFN dosage (nM) and key viral parameters: (top) viral countermeasure efficiency (), (middle) viral replication rate () and (bottom) dsRNA leakage rate . Warmer colors (pink) indicate higher fold changes in nAOC, signifying improved prophylactic efficacy.

https://doi.org/10.1371/journal.pcbi.1014395.g005

Prompted by this result, we examine the effectiveness of IFN as a prophylactic agent and its efficacy in combination with direct-acting antivirals (Fig 5C, 5D, and S15 Fig). To standardize comparisons across infection scenarios, we quantify antiviral efficacy by estimating the fold reduction in viral load integrated over the treatment duration (area-over-curve metric, AOC; Fig 5B). This area-over-curve metric, normalized against no IFN exposure (referred here as nAOC, see Methods), captures the time-of-addition-independent antiviral potency of the strategy, with higher nAOC scores indicating superior viral suppression. As anticipated, the AOC for low-dose IFN (1 nM; ≈60, Fig 5B, dashed blue) was significantly lower than that of a higher dose (3 nM; ≈89, Fig 5B, dashed red). Furthermore, the efficacy of 1 nM prophylactic IFN exposure was inversely correlated with viral antagonism (). Specifically, the AOC decreased from ≈60 at = (Fig 5B, dashed blue) to ≈56 when antagonism was doubled ( = 2; Fig 5B, solid blue). These trends confirm that IFN-mediated suppression scales positively with dosage but is fundamentally constrained by the strength of viral immune evasion.

We next examined how direct-acting antivirals (DAAs), acting on distinct steps of the viral life cycle, influence the prophylactic efficacy of IFN. Specifically, we focused on viral processes that emerged as key determinants of infection threshold: the efficiency of viral antagonism of innate immune signaling, the viral replication rate, and the rate of formation of compartmentalized replication complexes. Interferon prophylactic efficacy is significantly augmented by all three mechanisms: attenuating viral immune antagonism (), restricting viral replication kinetics (), or promoting dsRNA leakage into the cytosol (). By shifting the viral-host equilibrium toward immune-mediated control, these interventions enhance the potency of IFN. This synergistic effect is particularly pronounced at low to intermediate IFN doses where IFN only partially restrains the infection and viral fitness plays a dominant role in defining disease progression. Within this sensitive regime, IFN potency is significantly augmented by modest attenuation of viral antagonism (e.g., reduced to baseline), moderate inhibition of replication kinetics ( decreased to baseline), or accelerated dsRNA leakage ( increased to baseline) substantially increase IFN effectiveness by collectively tipping the viral-host equilibrium toward host-mediated control. To quantify the therapeutic benefit of DAA supplementation, we calculated the fold change in nAOC across varied viral parameters (Figs 5D and S15). These process-specific interventions markedly potentiated IFN-mediated suppression, with the most pronounced synergistic gains observed at low IFN concentrations. Conversely, at higher dosages, IFN monotherapy proved sufficient to achieve robust viral inhibition independently, suggesting a dose-dependent saturation point for the relative benefit conferred by combination therapy.

Together, these findings underscore the crucial role of early IFN intervention in prophylactic settings, particularly when combined with direct-acting antivirals. They further demonstrate how synergy between antiviral therapeutics and innate immune responses can be strategically exploited to enhance antiviral activity while minimizing IFN-associated toxicity [13,14].

Ethics statement

This is a computational study. No animals or humans and/or their tissue samples were used.

Models integration and extension

To create a detailed, mechanistic model of the virus–innate immune response, we integrated the detailed intracellular viral life cycle model by Chhajer et al. [26] with the RIG-I–IFN–ISG signaling model by Burkart et al. [43]. To achieve this integration, we introduced additional variables, parameters, and equations to capture extracellular viral dynamics, the leakage and sensing of double-stranded RNA (dsRNA), the regulation of Interferon Regulatory Factor 7 (IRF7), and multilayered feedback mechanisms involving both viral antagonism and host regulatory pathways.

The viral life cycle model was extended to include extracellular dynamics of viral entry and internalization. Viral entry was modeled as a first-order process governed by the rate constant k_en, leading to depletion of extracellular virus (V₀). Internalized virus () accumulates as a result and can either undergo uncoating at a rate or be degraded at a rate . Upon uncoating, viral RNA is released into the cytoplasm and contributes to the cytoplasmic viral RNA pool (R_cyt). To account for this, the original equation describing cytoplasmic viral RNA dynamics was modified to include an additional term reflecting input from uncoated virus. All other terms in this equation, as well as the ODEs describing the dynamics of structural and non-structural viral proteins (), replication compartment formation (), and replication complexes within compartments (), remain identical to the original framework by Chhajer et al. [26].

A critical extension of the viral model was the explicit inclusion of dsRNA leakage from replication compartments. While the original model did not accommodate this, cytosolic dsRNA serves as a key activator of the innate immune response and is central to coupling viral replication with RIG-I sensing. We therefore introduced a leakage term at rate , following an approach similar to Zitzmann et al. [35]. Once in the cytoplasm, dsRNA undergoes natural degradation at a rate , can be replenished through the deactivation of RIG-I at a rate , and is cleared through RIG-I–mediated sensing at a rate proportional to the product of RIG-I and dsRNA concentrations (). This formulation establishes the mechanistic link between the viral life cycle and the innate immune module.

The innate immune signaling model of Burkart et al. [43] was extended by incorporating the dynamics of Interferon Regulatory Factor 7 (IRF7). IRF7 is a transcription factor essential for the induction of type I interferon genes, particularly in plasmacytoid dendritic cells and during late-phase antiviral responses [102]. It is also responsible for its own induction via IFN-dependent transcription [103]. Given its high degree of homology with IRF3, we assumed IRF7 undergoes phosphorylation and nuclear translocation through a mechanism analogous to IRF3. Specifically, cytoplasmic IRF7 is phosphorylated by activated TBK1 and IKK, translocates into the nucleus, and regulates interferon gene expression. Once in the nucleus, phosphorylated IRF7 (pIRF7) can be dephosphorylated or exported back into the cytoplasm. Its synthesis is driven by translation from IFN-induced mRNA, while constitutive degradation occurs with a first-order decay rate.

In addition to coupling viral and immune processes mechanistically, we introduced multilayered feedback to capture the dynamic regulation that occurs at different levels. On the host side, interferon-stimulated genes (ISGs) target distinct stages of the viral life cycle. For example, CH25H and IFITMs inhibit viral entry [9,12], IFIT and TRIM family proteins disrupt RNA and protein synthesis [9,12], and viperin interferes with assembly and egress by disrupting viral envelope formation [9,12]. In contrast, viruses of the Flaviviridae family have evolved to counteract host defenses by antagonizing RIG-I signaling and JAK-STAT signaling cascades, thereby suppressing type I IFN production and signaling [54–56]. Finally, the host itself incorporates intrinsic negative regulatory mechanisms to limit excessive immune activation. In particular, molecules such as SOCS proteins and USP18 dampen IFN-induced signaling, ensuring resolution of the antiviral state [15,74,75].

To quantitatively capture these interacting layers of feedback, we introduced three lumped pseudo-variables representing distinct functional modules: (efficacy of viral antagonism), (potency of ISG-mediated antiviral activity), and (efficiency of host negative regulation of immune signaling). These pseudo-variables act by modulating either the intrinsic process rates or the degradation rates of relevant molecular species.

A complete description of the model, along with equations (S1 Text-Model descriptions), detailed variable definitions, initial conditions (S1 Table), parameter values, and sources (S2 and S4 Tables), is provided in the Supplementary Information. Altogether, the integrated model comprises 77 ordinary differential equations and 129 parameters. No parameters were re-estimated in the present work; all values were taken from the source models, where they had been estimated by approximate Bayesian computation (viral life cycle, Chhajer et al. [26] and Zitzmann et al. [35]), negative log-likelihood minimisation with multi-start initialisation (RIG-I/IFN signalling, Burkart et al. [43]; JAK-STAT signalling, Maiwald et al. [29] and Kok et al. [15]). Direct quantitative measurements for several parameters were included when available (entry/uncoating rates, Zitzmann et al. [35]; ISG mRNA kinetics from Maier et al. [70]; IRF7 dynamics from Lopacinski et al. [38] and Zou et al. [104]). Practical identifiability was assessed in those source studies via profile-likelihood analysis and pairwise posterior correlation. A complete listing of parameters with values, units, sources, and the estimation method used in each source is provided in S2 and S4 Tables. The three lumped pseudo-variables - (ISG-mediated antiviral suppression), (viral antagonism), and (auto negative feedback) - were incorporated as systematic tuning parameters varied across their full biological range to explore the model’s behaviour space. The governing equations and parameter values for each module are documented in full in the SI. Key modifications introduced during integration to mechanistically couple the two (viral and immune) subsystems are described below.

Viral life cycle and dsRNA detection.

We added the following variables and equations to the viral life cycle model proposed by Chhajer et al. [26] to incorporate the extracellular viral dynamics of viral entry and internalization:

(1)(2)

Subsequently, the equation monitoring the levels of cytoplasmic viral RNA is changed:

(3)(4)

Innate immune signaling pathway.

The dynamics of Interferon Regulatory Factor 7 (IRF7) are modelled as shown below:

(5)(6)

Incorporation of feedback at multiple levels by both the virus and innate immune system.

Feedback was modeled as a generic modulation of rates:

(7)

and the degradation rates are changed as

(8)

where denotes rate constants of viral entry (), fusion (), replication (), translation (), egress (), and represents degradation rates of viral RNA (), and proteins (), is and [A] is levels of antiviral ISG (ISGav). For immune signaling processes, , with viral proteins () attenuating MAVS activation (k_MAVS), IRF3/IRF7 activation (), and STAT complex activation (k₁₃). In addition, host-driven negative feedback is incorporated by setting and [A]= ISGn, which downregulates activated receptor complex (ARC) formation (k₇).

Calculation of critical values and infection threshold sensitivity analysis

To identify the critical value of (or ), we systematically vary one of or keeping the other fixed, and record the extracellular viral concentration at the end of the simulation (). Next, we evaluate the first-order numerical derivative of the logarithm of viral output () with respect to the variable varied and identify the point of maximum change as the critical value. and were varied in the logarithmic scale to span a large dynamic range.

To evaluate how different model parameters influence the critical values of immune activity and viral antagonism, we performed infection threshold Sensitivity Analysis. To evaluate the sensitivity of at the infection threshold point, ‘(, )’, we fix as . When we look at the ‘’ model parameter, we alter the value of the parameter from its nominal value () to and re-evaluate the critical value of viral antagonism (). Finally, we evaluate the sensitivity index as the ratio of the relative change in threshold value with respect to the relative change in the parameter value, using the following equation:

(9)

As sensitivity captures the slope of the threshold function with respect to the model parameter, we use a five-point central difference formula instead of the naive single-point difference formula, to calculate the sensitivity index more accurately. The threshold values were evaluated at .

(10)

determines the influence of the ‘’ model parameter on the critical viral antagonism required for immune escape. Analogously, we can calculate ; here we fix as , and assess how changes in the model parameters affect the immune activity threshold () necessary for viral suppression. To identify the most important parameters that influence both the thresholds, we rank the parameters based on threshold sensitivity score, calculated as the sum of squares of the corresponding sensitivity indices, i.e.,

Latin Hypercube Sampling and correlation with viral load

To evaluate the influence of host and viral parameters on infection outcomes, we performed Latin Hypercube Sampling (LHS). Each model parameter was varied within a log-uniform range

(11)

where denotes the original value of parameter i. A total of N = 25000 parameter sets were generated, each comprising the full parameter vector,

(12)

To quantify the relationship between individual parameters and infection outcome, we computed the Pearson correlation coefficient between each parameter and the viral load across all LHS-generated parameter sets:

(13)

where and denote the standard deviations of parameter and viral load , respectively. The resulting correlation coefficients were then ranked to identify the top 10 positively correlated and top 10 negatively correlated parameters. These parameters were visualized in a bar plot to highlight the most influential drivers of viral load variability (Fig 2C).

Normalized dynamics based clustering

In our analysis, we perform hierarchical clustering on normalized time-series data for each model variable under three conditions - A: low (when = 0.5), B: critical (when = ), and C: high (when = 2). Variables with zero standard deviation or with a low coefficient of variation () were excluded, ensuring retention of only dynamically informative variables. We then concatenate the dynamics of the selected variables in the three regimes and normalize them using Z-score normalization. The mean and standard deviation used for the normalization is based on the concatenated dynamics.

A pairwise distance matrix was computed across all retained variables, followed by hierarchical clustering based on the average linkage criterion. The resulting dendrogram was cut to yield k = 8 clusters. For each cluster, variable memberships and their average correlation to the cluster mean trajectory were computed. Variable-wise dynamics were additionally annotated with marker sizes inversely proportional to the distance to the centroid.

This analysis was based on our recently published model-based experiment design algorithm, PARSEC [99]. Like the analysis here, the PARSEC groups model variables/observables, but the clustering is based on differences in parameter sensitivity signatures instead of dynamical correlations.

Quantification of dynamic differences using a -like trajectory score

To quantify dynamic differences in temporal responses between conditions (specify the conditions), we defined a log-transformed mean squared deviation (-like) score for each variable. For a given variable , the score between two conditions c₁ and c₂ was calculated as:

(14)

where denotes the trajectory of variable k at time t under condition c, T is the number of time points. This measure captures the average squared deviation between log-scaled trajectories and is mathematically equivalent to a -like score. A small positive number = 10⁻⁸ is added to the variables so that the resulting logarithmic value is defined and finite.

We applied this metric in two distinct contexts. First, in the clustering analysis, it was used to compare responses between conditions A and C (Fig 3C). Second, in the interferon (IFN) stimulation experiments, the same metric was employed to compare trajectories between two IFN priming protocols: one in which cells received 100 nM IFN at 24 h following a 1 nM priming dose, and another in which the same 100 nM boost followed a 100 nM priming dose (Fig 4C).

Normalized area over the curve (nAOC) calculation - estimating prophylactic efficiency of IFN pretreatment

To evaluate the prophylactic action, we simulate the effect of various dosages of IFN (as shown in Fig 5) administered k days before the start of infection, where . We report the resulting viral load at 48 hours post-infection (). In the logarithmic scale, we compute the area between this curve () and a horizontal line at (a high viral load threshold reference) between k = 0 to k = 30 days using the trapezoidal rule. Specifically, the AOC was calculated using:

(15)