Dynamical model and geometric insights in the discontinuity theory of immunity

(Eqs 2). We chose parameters and interactions to approximate dynamics typically observed during an immune response (Table I), but we mostly use this model as an entry point to explore dynamical aspects and fundamental geomet- ric properties of the immune response. We define an immune response as acute when infected cells are eliminated (typically following an exponential increase in T cell numbers, followed by a decrease in both immune chal- lenges and T cells), Fig. 1B. Conversely, following the discon- tinuity theory, for some other parameters, a regime might be reached where the immune challenge is not fully eliminated, coexisting with some (low-level) immune response: this situ- ation defines a chronic response, Fig. 1C. Fig. 2B illustrates the dynamics of the model for several values of the parameters defining the immune challenge, i.e. its growth rate r and initial value I(0) (other parameters we use are given in Table I). For graphical representation, we fo- cus on the dynamics of the immune challenge and effector T cells (see Supplement, Fig. 7 for the full dynamics of the other variables). The model presents a broad variety of possible im- mune responses. For a high value of r, i.e. a fast-growing im- mune challenge, the immune response is always acute: both immune challenge and T cells grow exponentially fast (with an increase in IL-2 and of activated Tregs) (Fig. 2B, top). In this situation, the infected cells are quickly eliminated and the number of effector T cells slowly decreases until it becomes zero; meanwhile, both IL-2 and Treg number relax to their initial values. For lower r and low value of the initial immune challenge I(0), the dynamics are initially qualitatively simi- lar, with a slower exponential increase in the number of both infected cells and effector T cells. The number of regulatory T cells and concentration of IL-2 also increase, but much less rapidly. Then, after some decrease and damped oscillations, all variables reach a non-zero, chronic equilibrium, character- ized by a balance between infection of cells and effector T cell response (Fig. 2B, bottom). For even lower values of r, for our initial parameter choices, we observe another chronic regime where both T cells and infected cells oscillate with time. The phase diagram in Fig. 2B illustrates the extent of chronic versus acute responses. Strikingly, there is a sharp boundary between chronic and acute regimes, around r = 4, with a vertical slope. This indicates that the nature of the re- sponse is based on the parameter r only. We contrast such a boundary with a hypothetical process where an immune re- sponse is triggered only when the immune challenge passes a pre-defined threshold after a fixed time (red dotted lines in .CC-BY 4.0 International licenseavailable under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint 4 Fig. 2B, see details in Supplement, section C). Hence, our simple yet realistic model of immune response implements the main tenant of discontinuity theory: the rate of change in the immune challenge, r, decides whether ef- fector T cells establish either an acute or a chronic response, irrespective of the precise initial immune challenge. We no- tice, however, that the chronic regime can only be reached for a range of initial challenge I(0). In particular, for high enough initial challenge I(0), the response is always acute, irrespec- tive of r (“Acute bolus” in Fig. 2B). This is not inconsistent with discontinuity theory: mathematically, putting a high level of initial challenge at t = 0 is similar to having an extremely fast growing immune challenge challenge from the onset, so that we would indeed expect the immune system to yield an acute reaction if I(0) is high enough. B. Discontinuity theory : coexistence and disappearances of immune trajectories through bifurcations To better understand the difference between chronic and acute regimes, we further reduce our model to two variables, I and T , by performing quasi-static approximations for [IL-2] and the number of Treg cells [33] (see details in Supplement F). This reduced model still displays similar properties of ab- solute sensitivity to the growth rate r, corresponding to the discontinuity theory (see comparisons in Supplement B), Fig. 3A. We derive an even simpler model, taking asymptotic lim- its for Tregs and IL-2 and rescaling variables and time (see

and methods) such that we consider the following reduced 2D system: { dTr d! = I0 {∀rIr + Tr (#rIr → 1)} dIr d! = I0 {1 + Ir (rr → Tr)} , (1) We explicitly introduce an operator, I0, indicating that the 2D field is interpolated towards the origin below the rescaled value I 0, corresponding to the minimal quantity of immune challenge necessary to its survival (see Supplement H). rr is the rescaled growth rate of the (rescaled) immune challenge, I r, which is depleted by (rescaled) T cells, Tr. Ir in turn acti- vates Tr with the (rescaled) rate ∀r, and the growth rate of Tr itself depends on Ir (#r term). Depletion/exhaustion of Tr oc- cur with a rescaled rate of 1. This minimal 3-parameter model still displays exquisite sensitivity to the growth rate of the im- mune challenge, as illustrated in the phase diagram of Fig. 3B, top, while only keeping the acute and chronic regimes, as shown in the bottom panel (we represent trajectories in log- scale). To interpret this model and to compare it to biological observables, we rescale T r and Ir back to T and I appropriately in all relevant figures and discussion. The rr parameter is also rescaled to r for easier comparison. This reduced 3-parameter dynamical model, Eq. 1, com- bines features of different existing models. Its basic dynam- ics are close to previously proposed models of immune re- sponses [34, 35] but it assumes additional nonlinearities for T cell growth. Those nonlinearities render the model more sim- ilar to generalized Lotka-V olterra models [36] with additional FIG. 3. A) Network of interactions of the reduced system. This cor- responds to equation 1. B) The different regimes in the parameter space of the reduced system. We show time series and phase portrait trajectories for the simulated systems. Examples of chronic (slow growth rate, pale blue region) and acute (fast growth rate, gray re- gion) infections are presented. See supplement A for more details. C) Bifurcation diagram of the system as the growth rate (r ) is varied. Snapshots of the phase portrait at different points along the bifurca- tion are shown. The shaded gray area corresponds to the basin of attraction of the chronic stable fixed point (FP). In the bifurcation diagram, we only shaded the region inside the unstable limit-cycle (LC). We use the same default parameters as in Fig. 2, Table I. Ini- tial conditions: [T-Cells]= 0, [I]= 1.0 linear rates ∀rIr and 1 in equation 1. If we integrate the model defined in Eq. 1 without the interpolation I0 towards an at- tractive origin, we get damped solutions, similar to chronic ones, irrespective of the value of rr (see Supplementary Fig- ure 16 and Appendix section I). This points towards an im- portant role of the nonlinear dynamics of immune challenge elimination when few of them are left. Indeed, such 2D reduction of the dynamics allows us to per- form phase-plane analysis in the T, I plane Fig. 3C. First, we observe that for all values of r (with scaling r r = r→∀ T ↑ ∃ ), there exists a stable fixed point (solid black point in Fig. 3B-C bot- tom), which corresponds to the chronic state at lowest value of r (light blue trajectory in Fig. 3B-C, bottom left). Con- versely, for very high r, the typical trajectory gets high in the I, T plane and circles back to the origin Fig. 3B, bottom right, as expected from an acute response. Crucially, this is true even if the initial challenge I(0) is very small (but non-zero). In other words, such acute response corresponds to an almost .CC-BY 4.0 International licenseavailable under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint 5 homoclinic trajectory [37] originating just above and circling back to the origin (light blue trajectory in 3B-C, bottom right). Notice that this acute trajectory circles around the stable fixed (solid black) point. We can then represent the basin of at- traction of the fixed point (grey, Fig. 3C bottom middle and right), which is then limited by an unstable limit cycle sepa- rating the acute response trajectories from the (chronic) fixed point (dashed purple lines in Fig. 3C middle and right). The unstable limit cycle is a direct mathematical conse- quence of the co-existence of a stable limit fixed point circled by a (almost) homoclinic trajectory. A homoclinic trajectory is an orbit in phase space starting from one point (technically a saddle point) and eventually converging to the same point after a long excursion. Such orbits are well known in neuroscience (e.g. neurons [38]). In an immune context, homoclinic trajec- tories are natural since, from a situation with no immune chal- lenge, no T cells, a small injection of strong immune challenge should trigger an immune response, eventually going back to the initial (cleared) state. The existence of such dynamics is the fundamental reason explaining the extreme sensitivity in response to r. Indeed, as another consequence of the coexis- tence between an unstable limit cycle and a stable origin, there is a saddle point, close to the origin (black cross in Fig. 3C). As r decreases, the unstable limit cycle grows, until it collides with this saddle point close to 0, Fig. 3C bottom middle, and thus disappears (through a subcritical homoclinic orbit bifur- cation [38], thus defining a true homoclinic orbit). Because of this bifurcation, acute trajectories such as the one in Fig. 3 C right, where low amount of initial challenges circle back to the origin, are no longer mathematically possible. Thus, one is left with the stable nonzero fixed point as the only stable attractor for many initial conditions I(0) in the absence of T cells above a threshold (gray zone represents the basin of at- traction in Fig. 3C), ensuring chronicity for lower values of r. So r acts as a control parameter for the bifurcation can- celing out the unstable limit cycle, explaining the sharp tran- sition between acute and chronic regime, and summarized in the bifurcation diagram at the top of Fig. 3C. Sensitivity to the growth rate only associated to discontinuity theory arises from the sudden change in the basin of attraction: the acute trajec- tories originating close to the origin for high r disappear, and are attracted to the chronic fixed point for low r for a broad range of I(0), compare Fig. 3C left with Fig. 3C right. III. LANDSCAPE GEOMETRIES FOR THE DISCONTINUITY THEORY OF IMMUNOLOGY. The properties described in the previous section suggest that the discrimination properties of the model are direct con- sequences of geometric features of the dynamical trajectories, in particular, the coexistence of acute and chronic trajecto- ries in phase space. To confirm this intuition, we reverse en- gineer a minimal geometric model for discontinuity theory, using the Evoscape approach [28]. This approach, inspired by geometric modelling in biology [39] and kernel-based ma- chine learning, relies on the combinations of simple dynami- cal modules to directly build “landscape-like” descriptions of biological systems in 2D, here the Immune challenge/effector T cell plane. We build landscapes based on two conditions, Fig. 4A : 1. there is an acute response for immune challenges achieving high growth rates r, even for very low initial values. This imposes a near-homoclinic acute trajectory from and to the origin in the limit of very high r, fur- ther associated to a stable fixed point at the origin. A corresponding landscape is shown in Fig. 4A, left. 2. there is a stable fixed point, topologically inside the acute homoclinic trajectory. A corresponding landscape is shown in Fig. 4A, middle Equations for each landscape are given in Supplement. The first condition is very natural for a functioning immune response: the systems should trigger an immune response even with a low number of immune challenges, then return to homeostasis after resolving the infection, analogous to an excitable-type trajectory [12, 28, 38]. Typically for such ex- citable/acute response, we expect the trajectory to circle an unstable attractor as seen in the Evoscape description in Fig. 4A, left. This first seems incompatible with the stability of a fixed point associated with the chronic response, Fig. 4A, middle. Y et, one can build a simple phase space encompassing those two constraints simply using a linear combination of those two landscapes, Fig. 4A, right. As expected by design, we can get an almost homoclinic/excitable trajectory arising from the combination of a repeller with flow (acute trajectory, condi- tion 1), itself around a stable fixed point in the center (chronic trajectory, condition 2). As a consequence, an unstable limit cycle naturally emerges. By varying the relative strength of modules (see details in Supplement), Fig 4B, the limit cycle can appear or disappear through a homoclinic bifurcation, and one can get either acute or chronic responses. To fully match discontinuity theory, one needs to relate the control parameter to growth rate, which requires some extra condition. We thus chose to add an extra “growth” module close to the y axis for the “acute” landscape only (light or- ange disk in Fig. 4A left). The resulting model has all the hallmarks of discontinuity theory: fast-growing immune chal- lenges reach the bottom attractor (corresponding to elimina- tion), blue trajectories in Fig. 4C while slow-growing im- mune challenges reach the chronic attractor, red trajectories in Fig. 4C. One can then reconstruct a phase diagram delin- eating chronic from acute regimes, Fig. 4D, which is very similar to the ones derived from the models in Figs. 2-3B. The fact that we can reproduce properties of the more ex- plicit models using a reverse-engineered landscape suggest that the observed geometry and sequence of bifurcations are very natural, and should be shared by models with similar properties irrespective of their precise mathematical formu- lation, as long as they satisfy conditions 1-2 above. .CC-BY 4.0 International licenseavailable under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint 6 FIG. 4. Reverse engineering a landscape model for the discontinuity theory of immunity A) Flows and potentials built using Evoscape[28]. The top row shows the flows, with the local Gaussian modules used indicated by circles. Green disks correspond to local attractors modules, purple rings to repelled, and orange modules to rotating flows. We also indicate examples of both acute and chronic trajectories. Landscapes corresponding to the flows are shown on the bottom row, with isolevels of the potential projected underneath the surface. The left column illustrates how to combine modules to get almost homoclinic trajectories corresponding to acute responses, circling to a bottom left attractor that would correspond to immune challenge elimination. The middle column is a simple attractor corresponding to the chronic response. Adding the two landscapes give a dynamic similar to what is observed in the explicit model, with acute/chronic trajectories going to two different attractors (green disks) separated by an unstable limit cycle. Equations can be found in Supplement B) Bifurcation diagram and phase plane analysis as the control (speed) parameter is varied, recapitulating the behaviours showing in Fig. 2C. C) Possible dynamics of the Immune Challenges as we vary the control parameter. D) Corresponding phase diagram. IV . DYNAMICAL ANTAGONISM AND INVERSION The transition from acute to chronic regime, with a bound- ary controlled by a single parameter r is very reminiscent of specific and sensitive discrimination by T cells, which is sen- sitive to the kinetics of interaction of ligands to T cell receptor, almost irrespective of their concentration. Such behaviour has been well characterized experimentally [4, 8, 40, 41] and fur- ther studied theoretically [9, 42]. In particular, it was proved that such “absolute discrimination” mechanisms always dis- play ligand antagonism [8, 43] as a phenotypic spandrel [44]. In those contexts, antagonistic properties occur at steady state. This motivated us to study antagonistic properties in the con- text of the discontinuity theory controlled by a dynamical bi- furcation like here. To do so, we expanded the simplified 3-parameter model of Eq. 1 to simulate a co-infection [45](we checked that the properties we describe below also hold for the expanded ver- sions of the models). We consider two distinct immune chal- lenges (I 1 and I2) with different rates (r 1 and r2) Fig. 5A- B. Dynamics of each of those immune challenges individu- ally are presented in Fig. 5C-D. We assume that those chal- lenges activate the same effector cells Fig. 5E. This situation is immunologically–realistic when immune challenges mutate to alter their infectivity and/or growth rate, while maintain- ing their antigenicity, which is now well established in multi- ple contexts from persistent bacteria [46] to tumor cells [47]. Equation 5 describes the dynamics of immune responses in these coinfections for the reduced system. The dynamics of those co-infections both as a function of time and in phase space are shown in Fig. 5F. For this exam- ple, we observe a counter-intuitive inversion in the behaviour of both immune challenges Fig. 5F left : while the slow grow- ing r challenge I 1 is eliminated, the fast growing challenge I2 oscillates then stabilizes at the stable chronic fixed point. In particular, this means that a chronic immune challenge can an- tagonize an acute one, to give rise to a chronic regime for the fast-growing challenge (while the slow-growing challenge is eliminated). Focusing on the phase plane trajectories, Fig. 5F right, we see that the trajectory of I 2 is very similar to the acute trajectory, but converges to the chronic fixed point. To fur- ther understand what happens, in Supplement (Fig. 18D), we show the basin contours of the chronic fixed point for I 2 as a function of the initial quantity of the chronically infected cells I1(0) : we see that addition of I1(0) modifies the unstable limit cycle, so that I2 crosses the boundary of the basin of attraction of the chronic fixed point very early during the simulated dy- namics. As I 1 increases, the basin contour retracts closer and closer to the chronic fixed point. Those counter-intuitive effects depend on the respective values of growth rates and initial immune challenges, as shown in Fig. 5 G . We see four regimes, where each im- mune challenge can be either chronic or acute. The “inver- .CC-BY 4.0 International licenseavailable under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint 7 FIG. 5. Dynamical antagonism and inversion A) Illustration of the dynamics for a single chronic response considered I1(0)=10.0, r1 = 0.2 and B) for an acute response I2(0)= 10.0, r2 = 0.5, with phase plane dynamics and time courses displayed in C-D. Other pa- rameters are the same as in Fig. 3E) Given the same two species presented in A and B, we can observe different outcomes to their isolated outcomes by combining them into a single infection (given by equation 5). F) Left: time series of two species combined, I1 and I2, undergoing, respectively, acute and chronic infections (the opposite of their original isolated outcomes, compare with dynamics in panels C-D). Right: trajectory in (T, I) space for each infection in the presence of the other one. G) Left: Infection outcomes for a combination of two species for different initial I 2 and r2. The val- ues of I1(0) and r1, leading to a chronic state for I1 when it is the only infectious species in the system, are fixed. The dotted red line represents the boundary between the chronic (inside) and acute (out- side) regions when I 1 is the only infectious species. Right: Cross- section of left plot, along the dotted black line. The growth rate for I2 (r2) is now fixed and the growth rate for I1 (r1) is allowed to move. I1(0)=100.0, T (0)=0.0 FIG. 6. Illustration of the process of initial infection and subse- quent “treatment”. A) The first immune challenge initially causes a chronic infection that reaches a steady state with I1 ↓= 0. Once this state is reached, a second immune challenge (I 2) is added. Differ- ent outcomes can occur where each species can be either completely eliminated (acute) or remain active (chronic). Each case maps to cases presented in B-F. B) Infection outcomes for a system with ini- tially a single species with r1 = 0.2. It evolves towards a chronic state. Once in this state, a second species is added, with its own r2, in amount I (ttreatment). Four possible outcomes are possible and are color-coded. Examples of these processes and their outcomes are presented in subplots C,D,E,F. Parameters are similar to Fig. 3. I1(0)= 10.0, T (0)= 0.0 sion” regime corresponds to the green region where, when put together, a challenge I1 that would be chronic alone is becom- ing acute in co-infection, while a challenge I2 acute alone be- comes chronic in co-infection. This regime thus occurs on the right of the normal chronic/acute boundary for I 2 (dotted red line in 5 G). We also observe a regime where both I1 and I2 are chronic although I2 alone would be acute, which also ex- tends on the right of the chronic/acute boundary for I2, again indicating antagonism on I2 by I1. Because antagonism here is associated with the dynamics of immune challenges, there are in fact multiple possibilities depending on the timing when an acute or chronic challenge is added. We also consider the opposite limit, where one (chronic) challenge is already established, and another one is added later, Fig. 6A. This corresponds to a two-tier infection, sequentially stimulating the same immune cells, which could happen for instance if a mutation occurs for an immune chal- lenge stabilized in the chronic regime. Fig. 6B-F illustrates the behavior of our 3-parameter model for this staggered co-infection, depending on the quantity and growth rate of the added immune challenge parameters I 2(0), r2 (see e.g. [48] for an example of such coinfection .CC-BY 4.0 International licenseavailable under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint 8 mixing helminth and bacteria). Compared to Fig. 5 when both challenges are added initially, we see on the phase dia- gram Fig. 6B that both the antagonistic (light blue) and in- version (green) regions largely extend towards higher r, over more than half an order of magnitude in r. This means that the slow-growing immune challenge I 1 in this region would typically drive a fast-growing one even more strongly towards the chronic regime. It is worth pointing out two other bio- logically relevant regimes. Panel E illustrates the inversion dynamics : while the initially chronic infection I 1 is elimi- nated, the acute infection I2 is becoming chronic. Since I2 is growing about ten times faster than I1, this indicates a strong worsening of the chronic infection. Conversely, in panel C, if enough I 2 is suddenly added, one can exit the chronic regime so that both challenges are eliminated. This is akin to a post- infection “vaccination” against the chronically established im- mune challenge I 1. V . DISCUSSION The discontinuity theory of immunity, proposed by Pradeu and collaborators, posits that the immune system is sensitive to the (quantitative) “speed of change” of immune challenges, rather than more qualitative features such as molecular signa- tures. It was used as a holistic framework to explain phe- nomena such as immune tolerance or autoimmune disease. Starting from a simple model of an immune response, dis- playing either acute/chronic activity depending on the growth rate r of the immune challenge, we derived a 3-parameter bi- dimensional model, recapitulating properties predicted by the discontinuity theory. In particular, the transition line sepa- rating acute and chronic regimes is purely controlled by the growth rate r over a broad range of initial sizes of immune challenges. Our model presents common features to previously pro- posed models, for instance the co-existence of a chronic (“per- sistence”) state with a clearance state was proposed in [49], or an excitable model for auto-immunity proposed in [12]. However, those models did not consider the discontinuity the- ory framework. Our model presents unique, generic geom- etry, with excitable trajectories going around the persistence state in phase space. Such geometric constraints ensure that an unstable limit cycle separates the acute and chronic trajec- tories, and as a consequence the discontinuity detection occurs through a subcritical homoclinic orbit bifurcation depending on the growth rate of immune challenge r, canceling the un- stable limit cycle. The properties described above would not depend on the particular details of the models, as long as the orbits can be reduced to 2D and a parameter such as r controls the acute to chronic regime. In particular, immunologists have documented with very high degree of granularity, how diverse immune responses take place depending on the type and size of infections (so-called Th1, Th2, Th17 etc. regimes for T cell responses). The generic aspect is only expected close to the bifurca- tion when the acute and chronic trajectories co-exist. We can not exclude that different geometries and bifurcations are ob- served for more complex dynamics, e.g. effectively living in much higher dimensions. However, we notice that, if the tran- sition from chronic to acute regime comes with a change of topology of the orbits, then by definition a global bifurcation is expected, and if acute immune responses indeed correspond to excitable dynamics [38], homoclinic bifurcations are a very natural scenario. Antagonism is a generic property of multiple decision- making or discrimination pathways, from multiple immune recognition processes to olfaction [43, 50, 51], leading to practical application for cancer immunotherapy [52]. In [31], it was mathematically demonstrated that absolute discrimina- tion, defined as ligand-based cellular decision-making based on one kinetic parameter irrespective of ligand concentrations, is necessarily associated with ligand antagonism, explain- ing why it is observed for many immune decisions [8]. We demonstrated here that discontinuity in the decision-making between acute and chronic response also displays antagonism, thus generalizing the results from [31] to a decision-making process based on a completely different mechanism (mathe- matical speaking), namely a global bifurcation. This leads to important biological predictions. In a disease context, it raises the possibility of dynamical “adversarial” strategies [53] for a pathogen or a tumor to leverage dynam- ical antagonism to escape immune responses. For instance, a slow-growing tumor (like I 1 in Fig. 6F) could first stabi- lize in the chronic regime of immunological response then later mutate into a faster-growing one (like I 2) while keeping the immunological response at bay within a chronic regime. One could even imagine a slow ramp-up of immune escape of such increasingly growing tumors so that a fast-growing tu- mor that would normally trigger an immune response could slowly evolve and remain “undetected” by the immune sys- tem. It is especially important to point out that in our model, slow growing challenges are not passive and actively antag- onize the detection of faster growing ones. Many observa- tions about the differential growth rates of primary tumors vs metastasis, cancer dormancy as well differential immune re- sponses (“immune privilege”), are consistent with such sce- nario [54, 55]. Of note, it has been recently observed that upon CAR-T treatment, tumors with slow growth tend to bet- ter survive [56]. This is consistent with both discontinuity theory and the observation in other contexts that persistor cells tend to escape treatment because of their slowed metabolism [46]. Conversely, the counterintuitive effects due to the im- mune challenge cross interactions could be used to tailor bet- ter treatments. For vaccine designs, bolus delivery is the nor- mal regime (top left, acute regime in Figure 2B) but it may be tolerizing and inducing a chronic immune response if the inital challenge is too small/not antigenic enough. Such effect could further be leveraged to rather induce tolerance. More generally, the coexistence of acute and chronic tra- jectories for the same immune challenge provides a simple, dynamical mechanism for an immune system to learn over time, thus allowing for a dynamic redefinition of the self [22]. Those predictions of 1. active antagonism of growing im- mune challenges and 2. mechanism of immune tolerance learning, naturally come from the geometric constraints im- .CC-BY 4.0 International licenseavailable under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint 9 TABLE I. Parameters used in the full model of 2 with their biological signification and value used in simulations. These are the values used for all figures, unless otherwise indicated. The time unit is “day”, i.e. all rates are implicitly per day. Symbol Biological signification V alue g Linear growth rate 101 r Exponential proliferation rate 0.2 ∀ Binding rate of T-cells to infected cells 10→4 T ↑ Concentration of naive T-cells at equilibrium 102 T ↑reg Concentration of regulatory T cells at equilibrium 104 # T-cell proliferation rate 104 % Consumption rate of IL2 by regulatory T- cells 10 &1 Inhibition rate of T-cells by regulatory T- cells 10→3 &2 Self-regulation rate of regulatory T-cells 10→3 ∃ Exhaustion rate of activated T-cells 5.0 µ Production rate of IL2 by activated T- cells 105 posed by discontinuity theory that we introduce. Similar to the general properties of antagonism for T cell detection [31], they come from the natural extension of a theoretical proposal for single immune challenges to interactions between multiple ones, with clear actionable experimental predictions. Practi- cally, the existence and properties of such learning in immune dynamics could be directly tested using specialized platforms such as Immunotron [4]. In particular, by carefully monitor- ing time courses of acute/chronic infections, one should be able to identify topological changes in responses associated to global bifurcations such as the one predicted here. Our model is of course very simplified and neglects multi- ple other mechanisms (e.g. sensitivity to the binding kinetics, immune editing, inflammatory switches, long term memory [57]), that could add multiple dimensions. That said, it is striking that a first-principle model accounting for disconti- nuity theory naturally comes with a combination of high-level features such as coexistence of acute and chronic responses, associated antagonism and global bifurcations discriminating between regimes. This is in line with the recent realization in multiple biological contexts that geometric, low-dimensional models accurately describe complex biological dynamics, e.g. line attractors for neural decision making [58] or heteroclinic flips for cellular differentiation [39, 59]. Of note, global bi- furcations, transients and ghost states [60, 61] have been sug- gested to play important role complex cellular computations, and our model for discontinuity theory suggests that similar phenomena might be at play in immune decision-making. VI. MA TERIALS AND METHODS A. Initial model and its variations The starting point for describing the system is         ˙[T ]= ∀ [I] ( T ↑ + #[T ][IL2] &1[Treg] )    production → ∃[T ] exhaustion ˙[I]= g + r[I]   growth terms → ∀ [I]( [T ]+T ↑)   T-cells elimination [ ˙Treg]= % [Treg][IL2]   production → &2[Treg] ( [Treg] → T ↑ reg )    self-regulation [ ˙IL2]= µ[T ] production → [IL2]( #[T ]+% [T reg])   consumption , (2) where [I] is the concentration of infected cells, [T ] the con- centration of T-cells, [Treg] the concentration of regulatory T- cells, and [IL2] the concentration of the interleukin 2, cy- tokine. The parameters used are described in table I. We complement those equations with an interpolating flow going towards the origin as soon as I is lower than a threshold I 0 (see details in the Supplement H). Assuming a quasistatic dynamic for [Treg] and [IL2], we can obtain a dynamical system of only two variables (see Ap- pendix section F for derivation), which presents small differ- ences with the full system (Fig. 10)       ˙[T ]= ∀ [I] ( T ↑ + #[T ][IL2] &1[Treg] )    production → ∃[T ] exhaustion ˙[I]= g + r[I]   growth terms → ∀ [I]( [T ]+T ↑)   T-cells elimination , (3) where [Treg] and [IL2] are functions of [I] and [T ], given in equations F1 and F2. We designate this as the quasistatic system. We can do some further approximations and obtain and even more compact form: { dTr d! = ∀rIr + Tr (#rIr → 1) dIr d! = 1 + Ir (rr → Tr) , (4) where the rescaling of the variables is given by Tr = ∀ ∃ [T ], Ir = ∃ g [I] and d! = ∃dt and the parameters are rr = r→∀ T ↑ ∃ , ∀r = gT ↑∀ 2 ∃3 and #r = #&2g∀ %& 1∃2 . We also interpolate those equations towards the origin once Ir is lower than a threshold. When describing the interactions of two species in the sys- tem like in section IV. We consider two species Ir,1 and Ir,2 that interact identically with the T-cells and do not interfere with each other directly. We describe the dynamics through    dTr d! = ∀r(Ir,1 + Ir,2)+T r (#r(Ir,1 + Ir,2) → 1) dIr,1 d! = 1 + Ir,1 (rr,1 → Tr) dIr,2 d! = 1 + Ir,2 (rr,2 → Tr) . (5) Notice that the immune challenges Ir,1 and Ir,2 have differ- ent growth rates, rr,1 and rr,2. These quantities can be rescaled .CC-BY 4.0 International licenseavailable under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint 10 TABLE II. Modules and their parameters used in the results of Fig. 4. Note that the strength of left rotator is dependent on the “Speed” parameter. Module Type Strength Width Location Attractor 6.0 0.9 (→4, →3) Repellor 1.0 2.5 (0, 0) Rotator (CW) 0.6 3.0 (0, 0) Rotator (CW) Speed ↔0.5 4.0 (→6, →2) Attractor 3.3 1.5 (0, 0) to r1 and r2, the same way that rr is rescaled to r. We can also simulate this two-species dynamics for the “full system” as well as the “quasistatic system” as shown in appendix G. As stated in the main text, these reduced dynamical sys- tems are rescaled back to their “natural” scale (from Ir and Tr to Ir and Tr) when showing or discussing their simulated out- comes. The growth rate parameters rr, rr,1 and rr,2 are also rescaled to r, r1 and r2 for easier comparison with the full model, as well as allowing direct comparison with the biolog- ical observable that the parameters represent (the growth rate of the challenges). B. Dynamical landscapes The dynamical landscape treatment presented in section III was conducted using the Evoscape framework [28]. The re- sults of this analysis are all presented in Fig. 4. Using 4 different types of modules (attractor, repellor, clockwise rotator and counter-clockwise rotator) with vary- ing parameters we constructed the landscape presented in Fig. 4A. All the modules are linearly added together in a differen- tial equation for I and T , with ωx = ( T I ) , as such: d dtωx = ! i Ai e→ 1 2 ( ↗ωx→ωµi↗ ∋i )2 Mi (ωx → ωµi)+A 0↗ωx↗3 3. (6) In the previous equation, all the modules are summed to- gether, each denoted by an index i. Each of them have a loca- tion in phase space (the T, I plane) denoted by the vector ωµi,a strength, denoted by Ai, a width, denoted by ∋i and a Jacobian, denoted by Mi. Furthermore, there is a global attractor with weight A0 (that we set to 0.01, for the combined landscape). This global attractor is of the form ↗ωx↗3 3 = !i x3 i . The Jacobian takes different form depending on the type of module. These are • ( 10 01 ) : repellor • ( →10 0 →1 ) : attractor • ( 01 →10 ) : clockwise rotator • ( 0 →1 10 ) : counter-clockwise rotator Overall, the Evoscape framework allows us to mix and match different dynamical components to create a landscape presenting general features of the system. For the results of Fig. 4, we used the modules presented with their parameters in table II. The dynamics resulting from the Evoscape framework can be seperated into “potential” (attractors and repellors) and “curl” (rotators) parts. To draw the potentials presented in Fig. 4, we use the “potential” modules, from this, we can get an equation for the potential P: P(ωx)= A 0 ↗ωx↗4 4 4 + ! i siAi∋ 2 i e→ 1 2 ( ↗ωx→ωµi↗ ∋i )2 . (7) This is summing only over the repelling and attracting mod- ules. si corresponds to the sign linked to each kinds of mod- ules, such that si = →1 for an attractor (giving rise to a “valley” in the landscape) and si =+ 1 for a repellor (giving rise to a “hill” in the landscape). ACKNOWLEDGMENTS We thank Pankaj Mehta, Frédéric Guichard as well as the members of the François and Altan-Bonnet groups for useful discussions and comments. C.M. Denis is funded by the FRQ (DOI). [1] R. Medzhitov and C. A. Janeway Jr, Decoding the patterns of self and nonself by the innate immune system, Science 296, 298 (2002). [2] O. Feinerman, R. N. Germain, and G. Altan-Bonnet, Quantita- tive challenges in understanding ligand discrimination by #% t cells, Molecular immunology 45, 619 (2008). [3] M. Lever, P . K. Maini, P . A. V an Der Merwe, and O. Dushek, Phenotypic models of t cell activation, Nature Reviews Im- munology 14, 619 (2014). [4] S. R. Achar, F. X. P . Bourassa, T. J. Rademaker, A. Lee, T. Kondo, E. Salazar-Cavazos, J. S. Davies, N. Tay- lor, P . François, and G. Altan-Bonnet, Universal anti- gen encoding of T cell activation from high-dimensional cytokine dynamics, Science 376, 880 (2022), _eprint: https://www.science.org/doi/pdf/10.1126/science.abl5311. [5] F. Camaglia, A. Ryvkin, E. Greenstein, S. Reich-Zeliger, B. Chain, T. Mora, A. M. Walczak, and N. Friedman, Quan- tifying changes in the t cell receptor repertoire during thymic development, Elife 12, e81622 (2023). [6] A. Mayer, C. J. Russo, Q. Marcou, W. Bialek, and B. D. .CC-BY 4.0 International licenseavailable under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint 11 Greenbaum, How different are self and nonself?, arXiv https://doi.org/10.48550/arXiv.2212.12049 (2022). [7] H. Y uan Kueh, A. Handel, A. Hoffmann, D. Chowell, R. A. Gottschalk, H. Singh, R. N. Germain, M. Meier-Schellersheim, K. Miller-Jensen, and G. Altan-Bonnet, What unique insights can modeling approaches capture about the immune system?, Cell Systems 15, 1148 (2024). [8] G. Altan-Bonnet and R. N. Germain, Modeling t cell anti- gen discrimination based on feedback control of digital erk re- sponses, PLOS Biology 3, e356 (2005). [9] P . François, G. V oisinne, E. D. Siggia, G. Altan-Bonnet, and M. V ergassola, Phenotypic model for early T-cell activation dis- playing sensitivity, specificity, and antagonism, Proceedings of the National Academy of Sciences 110, E888 (2013). [10] R. Marsland III, O. Howell, A. Mayer, and P . Mehta, Tregs self-organize into a computing ecosystem and implement a so- phisticated optimization algorithm for mediating immune re- sponse, Proceedings of the National Academy of Sciences 118, e2011709118 (2021). [11] T. Kato and T. J. Kobayashi, Understanding adaptive immune system as reinforcement learning, Physical Review Research 3, 013222 (2021). [12] Y . Lebel, T. Milo, A. Bar, A. Mayo, and U. Alon, Excitable dy- namics of flares and relapses in autoimmune diseases, Iscience 26 (2023). [13] A. Cassano, R. Mora Cartin, P . Wang, Y . Wang, C. McIntosh, M. Andrade, A. Chong, and M.-L. Alegre, Role of tregs in maintaining alloreactive tconv hypofunction in transplantation tolerance, The Journal of Immunology 212, 1544_5560 (2024). [14] B. N. Jaeger and E. Vivier, Natural killer cell tolerance: con- trol by self or self-control?, Cold Spring Harbor Perspectives in Biology 4, a007229 (2012). [15] M. Damo, N. I. Hornick, A. V enkat, I. William, K. Clulo, S. V enkatesan, J. He, E. Fagerberg, J. L. Loza, D. Kwok, et al., Pd-1 maintains cd8 t cell tolerance towards cutaneous neoanti- gens, Nature 619, 151 (2023). [16] T. Pradeu and E. D. Carosella, On the definition of a criterion of immunogenicity, Proceedings of the National Academy of Sciences 103, 17858 (2006). [17] T. Pradeu, Immunology and individuality, eLife 8, e47384 (2019). [18] T. Pradeu, S. Jaeger, and E. Vivier, The speed of change: to- wards a discontinuity theory of immunity?, Nature Reviews Im- munology 13, 764 (2013). [19] T. Pradeu and E. Vivier, The discontinuity theory of immunity, Science immunology 1, aag0479 (2016). [20] G. Eberl and T. Pradeu, Towards a general theory of immunity?, Trends in immunology 39, 261 (2018). [21] T. Pradeu, The limits of the self: immunology and biological identity (Oxford University Press, 2011). [22] T. Pradeu, Philosophy of biology, in The Philosophy of Sci- ence. A Companion (Oxford University Press; Oxford Univer- sity Press, 2018). [23] A. Mayer, Y . Zhang, A. S. Perelson, and N. S. Wingreen, Reg- ulation of t cell expansion by antigen presentation dynamics, Proceedings of the National Academy of Sciences 116, 5914 (2019). [24] S. Sakaguchi, N. Sakaguchi, M. Asano, M. Itoh, and M. Toda, Immunologic self-tolerance maintained by activated T cells ex- pressing IL-2 receptor alpha-chains (CD25). Breakdown of a single mechanism of self-tolerance causes various autoimmune diseases., The Journal of Immunology 155, 1151 (1995). [25] G. V oisinne, B. Nixon, A. Melbinger, G. Gasteiger, M. V ergas- sola, and G. Altan-Bonnet, T Cells Integrate Local and Global Cues to Discriminate between Structurally Similar Antigens, Cell Reports 11, 1208 (2015). [26] H. S. Wong, K. Park, A. Gola, A. P . Baptista, C. H. Miller, D. Deep, M. Lou, L. F. Boyd, A. Y . Rudensky, P . A. Savage, et al., A local regulatory t cell feedback circuit maintains im- mune homeostasis by pruning self-activated t cells, Cell 184, 3981 (2021). [27] S. Dikiy and A. Y . Rudensky, Principles of regulatory T cell function, Immunity 56, 240 (2023). [28] V . Mochulska and P . François, Generative epigenetic land- scapes map the topology and topography of cell fates, bioRxiv , 2025 (2025). [29] H. Fu, J. A. Lewnard, I. Frost, R. Laxminarayan, and N. Ari- naminpathy, Modelling the global burden of drug-resistant tu- berculosis avertable by a post-exposure vaccine, Nature com- munications 12, 424 (2021). [30] M. Ogilvie, Antiviral prophylaxis and treatment in chickenpox: a review prepared for the uk advisory group on chickenpox on behalf of the british society for the study of infection, Journal of Infection 36, 31 (1998). [31] P . François, M. Hemery, K. A. Johnson, and L. N. Saunders, Phenotypic spandrel: absolute discrimination and ligand antag- onism, Physical Biology 13, 066011 (2016). [32] I. Andreu-Moreno and R. Sanjuán, Collective Infection of Cells by Viral Aggregates Promotes Early Viral Proliferation and Re- veals a Cellular-Level Allee Effect, Current Biology 28, 3212 (2018). [33] J. A. Borghans, R. J. De Boer, and L. A. Segel, Extending the quasi-steady state approximation by changing variables, Bul- letin of mathematical biology 58, 43 (1996). [34] H. Mayer, K. Zaenker, and U. An Der Heiden, A basic mathe- matical model of the immune response, Chaos: An Interdisci- plinary Journal of Nonlinear Science 5, 155 (1995). [35] E. D. Sontag, A dynamic model of immune responses to anti- gen presentation predicts different regions of tumor or pathogen elimination, Cell Systems 4, 231 (2017). [36] W. Cui, R. Marsland III, and P . Mehta, Les houches lectures on community ecology: From niche theory to statistical mechan- ics, arXiv (2024). [37] S. H. Strogatz, Nonlinear dynamics and chaos: with applica- tions to physics, biology, chemistry, and engineering (Taylor and Francis, 2001). [38] E. M. Izhikevich, Dynamical systems in neuroscience (MIT press, 2007). [39] D. A. Rand, A. Raju, M. Sáez, F. Corson, and E. D. Siggia, Geometry of gene regulatory dynamics, Proceedings of the Na- tional Academy of Sciences 118, e2109729118 (2021). [40] R. N. Germain and I. Stefanová, THE DYNAMICS OF T CELL RECEPTOR SIGNALING: Complex Orchestration and the Key Roles of Tempo and Cooperation, Annual Review of Immunology 17, 467 (1999). [41] S. Farkona, E. P . Diamandis, and I. M. Blasutig, Cancer im- munotherapy: the beginning of the end of cancer?, BMC Medicine 14, 73 (2016). [42] J.-B. Lalanne and P . François, Principles of Adaptive Sorting Revealed by In Silico Evolution, Physical Review Letters 110, 218102 (2013). [43] C. Torigoe, J. K. Inman, and H. Metzger, An unusual mecha- nism for ligand antagonism, Science 281, 568 (1998). [44] P . François and G. Altan-Bonnet, The Case for Absolute Lig- and Discrimination: Modeling Information Processing and De- cision by Immune T Cells, Journal of Statistical Physics 162, 1130 (2016). [45] S. Alizon and M. van Baalen, Multiple infections, immune dy- .CC-BY 4.0 International licenseavailable under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint 12 namics, and the evolution of virulence, The American Natural- ist 172, E150 (2008). [46] R. A. Fisher, B. Gollan, and S. Helaine, Persistent bacterial in- fections and persister cells, Nature Reviews Microbiology 15, 453 (2017). [47] Y . Goyal, G. T. Busch, M. Pillai, J. Li, R. H. Boe, E. I. Grody, M. Chelvanambi, I. P . Dardani, B. Emert, N. Bodkin, et al., Di- verse clonal fates emerge upon drug treatment of homogeneous cancer cells, Nature 620, 651 (2023). [48] K. Obieglo, X. Feng, V . P . Bollampalli, I. Dellacasa-Lindberg, C. Classon, M. Österblad, H. Helmby, J. P . Hewitson, R. M. Maizels, A. Gigliotti Rothfuchs, et al., Chronic gastrointesti- nal nematode infection mutes immune responses to mycobacte- rial infection distal to the gut, The Journal of Immunology 196, 2262 (2016). [49] S. Baral, R. Antia, and N. M. Dixit, A dynamical motif com- prising the interactions between antigens and cd8 t cells may underlie the outcomes of viral infections, Proceedings of the National Academy of Sciences 116, 17393 (2019). [50] B. N. Dittel, R. N. Germain, C. A. Janeway, et al., Cross- antagonism of a t cell clone expressing two distinct t cell re- ceptors, Immunity 11, 289 (1999). [51] G. Reddy, J. D. Zak, M. V ergassola, and V . N. Murthy, Antag- onism in olfactory receptor neurons and its implications for the perception of odor mixtures, Elife 7, e34958 (2018). [52] T. Kondo, F. X. Bourassa, S. Achar, J. DuSold, P . F. Céspedes, M. Ando, A. Dwivedi, J. Moraly, C. Chien, S. Majdoul, A. L. Kenet, M. Wahlsten, A. Kvalvaag, E. Jenkins, S. P . Kim, C. M. Ade, Z. Y u, G. Gaud, M. Davila, P . Love, J. C. Y ang, M. L. Dustin, G. Altan-Bonnet, P . François, and N. Taylor, Engineer- ing TCR-controlled fuzzy logic into CAR T cells enhances ther- apeutic specificity, Cell 188, 10.1016/j.cell.2025.03.017 (2025). [53] T. J. Rademaker, E. Bengio, and P . François, Attack and de- fense in cellular decision-making: lessons from machine learn- ing, Physical Review X 9, 031012 (2019). [54] J. A. Aguirre-Ghiso, Models, mechanisms and clinical evidence for cancer dormancy, Nature Reviews Cancer 7, 834 (2007). [55] J. A. Joyce and D. T. Fearon, T cell exclusion, immune privi- lege, and the tumor microenvironment, Science 348, 74 (2015). [56] A. L. Kenet, S. Achar, A. Dwivedi, J. Buckley, M. Pouzolles, H. Qin, C. Chien, N. Taylor, and G. Altan-Bonnet, The 1000+ mouse project: large-scale spatiotemporal parametrization and modeling of preclinical cancer im- munotherapies, bioRxiv 10.1101/2025.03.17.643712 (2025), https://www.biorxiv.org/content/early/2025/03/20/2025.03.17.643712.full.pdf. [57] M. Baliu-Piqué, M. W. V erheij, J. Drylewicz, L. Ravesloot, R. J. De Boer, A. Koets, K. Tesselaar, and J. A. Borghans, Short lifespans of memory t-cells in bone marrow, blood, and lymph nodes suggest that t-cell memory is maintained by continuous self-renewal of recirculating cells, Frontiers in immunology 9, 2054 (2018). [58] M. Pagan, V . D. Tang, M. C. Aoi, J. W. Pillow, V . Mante, D. Sussillo, and C. D. Brody, Individual variability of neu- ral computations underlying flexible decisions, Nature 639,1 (2024). [59] M. Sáez, J. Briscoe, and D. A. Rand, Dynamical landscapes of cell fate decisions, Interface focus 12, 20220002 (2022). [60] D. Koch, A. Nandan, G. Ramesan, I. Tyukin, A. Gorban, and A. Koseska, Ghost channels and ghost cycles guiding long transients in dynamical systems, Physical Review Letters 133, 047202 (2024). [61] L. Jutras-Dubé, E. El-Sherif, and P . François, Geometric models for robust encoding of dynamical information into embryonic patterns, Elife 9, e55778 (2020). [62] J. Bezanson, A. Edelman, S. Karpinski, and V . B. Shah, Julia: A fresh approach to numerical computing, SIAM review 59, 65 (2017). [63] C. Rackauckas and Q. Nie, DifferentialEquations.jl – A Per- formant and Feature-Rich Ecosystem for Solving Differential Equations in Julia, JORS 5, 15 (2017). [64] D. P . Sanders, L. Benet, B. Richard, J. Grawitter, E. Gupta, L. Ferranti, D. Karrasch, O. Hénot, Z. Hurák, Y . Sharma, T. Frondelius, M. Forets, J. TagBot, G. Datseris, E. Schnetter, E. Hanson, and E. Saba, JuliaIntervals/IntervalRootFinding.jl: v0.6.0 (2024). [65] S. Danisch and J. Krumbiegel, Makie.jl: Flexible high- performance data visualization for Julia, Journal of Open Source Software 6, 3349 (2021). [66] A. M. Kramer, L. Berec, and J. M. Drake, Ed- itorial: Allee effects in ecology and evolution, Journal of Animal Ecology 87, 7 (2018), _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/1365- 2656.12777. [67] G.-Q. Sun, Mathematical modeling of population dynamics with Allee effect, Nonlinear Dyn 85, 1 (2016). .CC-BY 4.0 International licenseavailable under a was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint