{"paper_id":"3861d3c9-21d1-41a9-99d3-e69fe4be5051","body_text":"Dynamical model and geometric insights in the discontinuity theory of immunity\nChristian Mauffette Denis\nDépartement de Physique, Université de Montréal, Montréal, QC, Canada\nVictoria Mochulska\nDepartment of Physics, McGill University, Montréal, QC, Canada\nMaya Dagher\nDepartment of Physics, McGill University, Montréal, QC, Canada\nCurrent address: Department of Collective Behavior , Max Planck Institute of Animal Behavior , Konstanz, Germany\nVincent V erbavatz\nDepartment of Physics, McGill University, Montréal, QC, Canada\nFrançois X.P . Bourassa\nDépartement de Biochimie et Médecine Moléculaire, Université de Montréal, Montréal, QC, Canada\nCurrent address : Center for the Physics of Biological Function, Princeton University, Princeton, NJ 08544, USA\nGrégoire Altan-Bonnet\nImmunodynamics Section, Laboratory of Integrative Cancer Immunology,\nCenter for Cancer Research, National Cancer Institute, Bethesda, MD, USA\nPaul François\nDépartement de Biochimie et Medecine Moléculaire,\nUniversité de Montréal, Montréal, QC, Canada\nMILA Québec, Montréal, QC, Canada\n(*paul.francois@umontreal.ca)\n(Dated: July 9, 2025)\nThe immune system’s most basic task is to decide what is “self” and “non-self”, but a precise definition of\nself versus non-self remains challenging. According to the discontinuity theory of immunity, effector responses\ndepend on how quickly an antigenic stimulus changes: rapid change triggers an immune response, whereas grad-\nual change fosters tolerance. We present a model of adaptive immune dynamics including T cells, Tregs and\ncytokines that reproduces the hallmarks of the discontinuity theory. The model allows for sharp discrimination\nbetween acute and chronic infections based on the growth rate of the immune challenge, and vaccination-like\nacute dynamics upon presentation of a bolus of immune challenge. We further show that the model behavior\nonly depends on a handful of testable assumptions that we map to geometric constraints in phase space. This\nsuggests that the model properties are generic and robust across alternative mechanistic details. We also ex-\namine the impact of multiple concurrent immune challenges in this model, and demonstrate the occurrence of\ndynamical antagonism, wherein, in some parameter regimes, slow-growing challenges hinder acute responses to\nfast-growing ones, with further counter-intuitive behaviors for sequential co-infections. Together, these results\nplace the discontinuity theory on firm mathematical footing and encourage further investigation of interferences\nof multi-agent immune challenges, from chronic viral co-infections to cancer immunoediting.\nI. INTRODUCTION\nThe conventional view of the immune system posits that\nit is focused on one goal: distinguishing and removing for-\neign molecules and cells (usually called “non-self”) from the\nhost body (usually called “self”) [1, 2]. Systems immunol-\nogy has made great progress in elucidating the parameters and\ncharacteristics of such immune discrimination [3], but, sur-\nprisingly, a quantitative definition of self vs non-self remains\nelusive. For instance, while biochemical parameters such as\nthe binding affinities of immunogenic peptides to T cell recep-\ntors allow us to define effective “antigenicities” [4], their map-\nping to peptide sequences remains unclear. Statistical analysis\nof TCR receptors’ change during thymic selection revealed\nonly small differences between positively and negatively se-\nlected sequences [5], and indeed self and non-self appear to\nhave almost indistinguishable distributions in sequence space\n[6]. This suggests that unknown parameters besides sequence\nmight play crucial roles in defining a proper immune response.\nCase in point: recent works have established that the immune\nenvironment in a broad sense plays a crucial role in modulat-\ning immune responses, e.g. due to cytokines and other co-\nstimulatory signals [7] or to antagonistic ligands that cancel\nthe response to normally immunogenic peptides [8, 9].\nThere are multiple lines of evidence that the self/non-self\ndichotomy is not absolute. Our interior body is a complex\ndynamical system, itself interacting with a constantly chang-\ning environment. It thus makes sense that over very long time\nscales, our immune system adapts to these changes [10, 11].\n.CC-BY 4.0 International licenseavailable under a \nwas not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint \n\n2\nThis explains for instance why one can slowly be desensitized\nto allergens or why – when the process fails – auto-immune\ndisorders appear, with (stochastic) flares of immune responses\n[12]. Understanding such adaptation is also of practical im-\nportance in treatments, e.g. to induce artificial tolerance to\ntransplants [13] or in cancer immunotherapy. We know that\nmultiple levels are implicated: for instance, it is well known\nthat macrophages or NK cells can become tolerant to new\nantigens [14]. T cell responses can be primed and modulated\nby earlier exposure [8]. More generally, many systems-level\nfeedbacks implicate specific tolerance genes [15] or Tregs\n[13]. Hence, what is recognized or not by the immune sys-\ntem might rather be a (slow) moving target, where immuno-\ngenic/nonimmunogenic categories do not perfectly align with\nself/non-self [16, 17]. To account for this feature, Pradeu,\nVivier, and co-workers have proposed an alternative frame-\nwork [18–22], illustrated in Fig. 1. As a first approximation,\nthey propose that slow or gradual changes of immune chal-\nlenges should be recognized as nonimmunogenic, allowing\nfor the emergence of (new) immune tolerance with time (Fig.\n1C). Conversely, any biochemical signal that changes rapidly\nin the organism is likely associated with a growing immune\nchallenge, so should be immunogenic and lead to an acute re-\nsponse (Fig. 1B). Pradeu et al. thus proposed that the speed of\nchange of molecular motifs with which immune cells interact\nis a determinant of immunogenicity, coining the “discontinu-\nity theory” of immunity [19].\nTo date, an implementation of the discontinuity theory re-\nmains murky in regard to how immune discrimination can\nbe determined by the speed of molecular change, both from\nthe immunological and theoretical standpoints. Y et, there\nhave been isolated observations in the field of quantitative im-\nmunology that suggest that biochemical derivatives, or at least\ntime-dependent responses, may be critical in determining im-\nmunological outcomes (activation versus tolerance). For ex-\nample, Mayer et al. explored how competition for a limited\namount of antigen could decide the time range and overall ex-\ntent of cell expansion [23]. Multiple groups also documented\nhow the dynamic competition between effector and regulatory\nT cells (Tregs), based on the secretion and consumption of the\ncytokine IL-2, may decide immunological outcomes [10, 24–\n27]. Reflecting this competition, a quantification of multiple\ncytokine dynamics showed how these signals universally en-\ncode an effective “immunological speed” determining long-\nterm kinetics [4]. These models were classically analyzed\nin terms of quality and quantity of T cell activation deciding\nwhether rapid/large IL-2 accumulation boosts effector func-\ntions, or, vice-versa, slow/meek IL-2 accumulation mostly\nboost the regulatory T cell compartment.\nHere we revisit these ideas by introducing a coarse-grained\nversion of the adaptive immune response – accounting for T\ncells, Tregs and cytokine response – and carrying out a thor-\nough theoretical analysis to explore the discontinuity theory of\nimmunology. We relate the discontinuity decision to the geo-\nmetric properties of the dynamics in phase space, in particular\nto the coexistence of two types of trajectories corresponding to\nacute and chronic responses. We show that such coexistence\nstrongly constrains the possible dynamics of the system, lead-\nFIG. 1. Illustration of the discontinuity theory of immunity. A) The\nimmune system reacts to an immune challenge. It mounts a response\nthat depends on its interaction with the challenge. B) The disconti-\nnuity theory of immunity hypothesizes that the immune system dis-\ntinguishes between immunogenic and non-immunogenic challenges\nthrough the speed of change of its surroundings. C) Fast-growing\nchallenges are perceived as immunogenic, triggering their elimina-\ntion, through an acute immune response. D) Slow-growing chal-\nlenges are perceived as non-immunogenic, which does not trigger\ntheir elimination, causing immune tolerance.\ning to specific bifurcations (in the dynamical systems theory\nsense) that naturally explain the extreme sensitivity to the rate\nof change of the immune challenge. To further establish this\npoint, we leverage a new “landscape” framework [28] to build\na purely geometric, “interaction-free” model of the system,\nrecapitulating the properties of more explicit models. This\napproach proves that many properties of the description we\npropose are generic, i.e. expected in a broad family of models\nrecapitulating the discontinuity theory of immunity, irrespec-\ntive of their details.\nLastly, we study antagonism between immune challenges,\nwhereby slow-growing challenges induce tolerance to fast-\ngrowing ones. Such effects are of primary importance for\ncases where immunological damages are close to self (e.g. tu-\nmors) or clinical trials of post-exposure vaccination against\nlatent infections (e.g. mycobacterium tuberculosis, varicella-\nzoster etc.) [29, 30]. This study also provides a dynamical\ngeneralization of an immune spandrel theorem [31], specify-\ning that absolute discrimination with respect to one kinetic\nparameter necessarily implies antagonism. Together, our re-\nsults confirm that a modelling approach based on the descrip-\ntion of simple dynamical rules (here, the co-existence of acute\nand chronic immune trajectories in response to growth rate)\ncan lead to the derivation of non-trivial geometric properties,\nwith clear experimental predictions for dynamical systems im-\nmunology.\n.CC-BY 4.0 International licenseavailable under a \nwas not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint \n\n3\nFIG. 2. A model recapitulating the discontinuity theory of immu-\nnity A) Scheme of the model describing the interactions between an\nimmune challenge, effector/regulatory T cells and cytokine (IL-2) in\nthe adaptive immune system. Each arrow corresponds to an interac-\ntion in the model (e.g. the immune challenge activating the T-cells),\nfull equations for the model are in the methods section, Eq. 2 with\nparameters in Table I. B) Different regimes in the parameter space\ndefining the immune challenge (initial quantity of challenge, growth\nrate ) for the full system. We show four typical trajectories. The\nsystem exhibits discontinuity in response to the growth rate, as il-\nlustrated by the vertical black line. For comparison, we add lines\ncorresponding to a simple threshold model where immune response\nwould be triggered only once a threshold of immune challenge is\nreached, see Supplement C and A for more details. Initial condi-\ntions: [T-Cells]= 0, [Tregs]= 10\n4, [IL2]= 0.\nII. RESULTS\nA. A simple, minimal model recapitulates properties of the\ndiscontinuity theory of immunity\nWe start with an explicit model characterizing the dynam-\nics of the adaptive immune system in response to an immune\nchallenge (e.g. viral infection of cells, or growth of tumors),\ngrowing exponentially at the rate r. Infected cells linearly in-\ncrease the activation and growth of effector T cells (called T ),\nwhich in turn kill the infected cells I. Effector T cells expo-\nnentially grow in response to I and secrete the cytokine IL-2.\nIL-2, in turn, enhances the proliferation of effector T cells and\nregulatory T cells that mostly act as a sink for IL-2 in our sim-\nplified model. We further assume that, below some threshold,\nthe immune challenge can not survive (see mathematical de-\ntails in Supplement H), e.g. because there can be no fewer\nthan 1 infected cell or because cooperative effects are neces-\nsary for an immune challenge to persist [32]. The model is\nillustrated in Fig. 2A, with equations given in Materials and\nMethods (Eqs 2). We chose parameters and interactions to\napproximate dynamics typically observed during an immune\nresponse (Table I), but we mostly use this model as an entry\npoint to explore dynamical aspects and fundamental geomet-\nric properties of the immune response.\nWe define an immune response as acute when infected cells\nare eliminated (typically following an exponential increase in\nT cell numbers, followed by a decrease in both immune chal-\nlenges and T cells), Fig. 1B. Conversely, following the discon-\ntinuity theory, for some other parameters, a regime might be\nreached where the immune challenge is not fully eliminated,\ncoexisting with some (low-level) immune response: this situ-\nation defines a chronic response, Fig. 1C.\nFig. 2B illustrates the dynamics of the model for several\nvalues of the parameters defining the immune challenge, i.e.\nits growth rate r and initial value I(0) (other parameters we\nuse are given in Table I). For graphical representation, we fo-\ncus on the dynamics of the immune challenge and effector T\ncells (see Supplement, Fig. 7 for the full dynamics of the other\nvariables). The model presents a broad variety of possible im-\nmune responses. For a high value of r, i.e. a fast-growing im-\nmune challenge, the immune response is always acute: both\nimmune challenge and T cells grow exponentially fast (with\nan increase in IL-2 and of activated Tregs) (Fig. 2B, top). In\nthis situation, the infected cells are quickly eliminated and the\nnumber of effector T cells slowly decreases until it becomes\nzero; meanwhile, both IL-2 and Treg number relax to their\ninitial values. For lower r and low value of the initial immune\nchallenge I(0), the dynamics are initially qualitatively simi-\nlar, with a slower exponential increase in the number of both\ninfected cells and effector T cells. The number of regulatory\nT cells and concentration of IL-2 also increase, but much less\nrapidly. Then, after some decrease and damped oscillations,\nall variables reach a non-zero, chronic equilibrium, character-\nized by a balance between infection of cells and effector T cell\nresponse (Fig. 2B, bottom). For even lower values of r, for our\ninitial parameter choices, we observe another chronic regime\nwhere both T cells and infected cells oscillate with time.\nThe phase diagram in Fig. 2B illustrates the extent of\nchronic versus acute responses. Strikingly, there is a sharp\nboundary between chronic and acute regimes, around r = 4,\nwith a vertical slope. This indicates that the nature of the re-\nsponse is based on the parameter r only. We contrast such a\nboundary with a hypothetical process where an immune re-\nsponse is triggered only when the immune challenge passes\na pre-defined threshold after a fixed time (red dotted lines in\n.CC-BY 4.0 International licenseavailable under a \nwas not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint \n\n4\nFig. 2B, see details in Supplement, section C).\nHence, our simple yet realistic model of immune response\nimplements the main tenant of discontinuity theory: the rate\nof change in the immune challenge, r, decides whether ef-\nfector T cells establish either an acute or a chronic response,\nirrespective of the precise initial immune challenge. We no-\ntice, however, that the chronic regime can only be reached for\na range of initial challenge I(0). In particular, for high enough\ninitial challenge I(0), the response is always acute, irrespec-\ntive of r (“Acute bolus” in Fig. 2B). This is not inconsistent\nwith discontinuity theory: mathematically, putting a high level\nof initial challenge at t = 0 is similar to having an extremely\nfast growing immune challenge challenge from the onset, so\nthat we would indeed expect the immune system to yield an\nacute reaction if I(0) is high enough.\nB. Discontinuity theory : coexistence and disappearances of\nimmune trajectories through bifurcations\nTo better understand the difference between chronic and\nacute regimes, we further reduce our model to two variables,\nI and T , by performing quasi-static approximations for [IL-2]\nand the number of Treg cells [33] (see details in Supplement\nF). This reduced model still displays similar properties of ab-\nsolute sensitivity to the growth rate r, corresponding to the\ndiscontinuity theory (see comparisons in Supplement B), Fig.\n3A. We derive an even simpler model, taking asymptotic lim-\nits for Tregs and IL-2 and rescaling variables and time (see\nmaterials and methods) such that we consider the following\nreduced 2D system:\n{\ndTr\nd! = I0 {∀rIr + Tr (#rIr → 1)}\ndIr\nd! = I0 {1 + Ir (rr → Tr)} , (1)\nWe explicitly introduce an operator, I0, indicating that the\n2D field is interpolated towards the origin below the rescaled\nvalue I\n0, corresponding to the minimal quantity of immune\nchallenge necessary to its survival (see Supplement H). rr is\nthe rescaled growth rate of the (rescaled) immune challenge,\nI\nr, which is depleted by (rescaled) T cells, Tr. Ir in turn acti-\nvates Tr with the (rescaled) rate ∀r, and the growth rate of Tr\nitself depends on Ir (#r term). Depletion/exhaustion of Tr oc-\ncur with a rescaled rate of 1. This minimal 3-parameter model\nstill displays exquisite sensitivity to the growth rate of the im-\nmune challenge, as illustrated in the phase diagram of Fig.\n3B, top, while only keeping the acute and chronic regimes, as\nshown in the bottom panel (we represent trajectories in log-\nscale). To interpret this model and to compare it to biological\nobservables, we rescale T\nr and Ir back to T and I appropriately\nin all relevant figures and discussion. The rr parameter is also\nrescaled to r for easier comparison.\nThis reduced 3-parameter dynamical model, Eq. 1, com-\nbines features of different existing models. Its basic dynam-\nics are close to previously proposed models of immune re-\nsponses [34, 35] but it assumes additional nonlinearities for T\ncell growth. Those nonlinearities render the model more sim-\nilar to generalized Lotka-V olterra models [36] with additional\nFIG. 3. A) Network of interactions of the reduced system. This cor-\nresponds to equation 1. B) The different regimes in the parameter\nspace of the reduced system. We show time series and phase portrait\ntrajectories for the simulated systems. Examples of chronic (slow\ngrowth rate, pale blue region) and acute (fast growth rate, gray re-\ngion) infections are presented. See supplement A for more details.\nC) Bifurcation diagram of the system as the growth rate (r ) is varied.\nSnapshots of the phase portrait at different points along the bifurca-\ntion are shown. The shaded gray area corresponds to the basin of\nattraction of the chronic stable fixed point (FP). In the bifurcation\ndiagram, we only shaded the region inside the unstable limit-cycle\n(LC). We use the same default parameters as in Fig. 2, Table I. Ini-\ntial conditions: [T-Cells]= 0, [I]= 1.0\nlinear rates ∀rIr and 1 in equation 1. If we integrate the model\ndefined in Eq. 1 without the interpolation I0 towards an at-\ntractive origin, we get damped solutions, similar to chronic\nones, irrespective of the value of rr (see Supplementary Fig-\nure 16 and Appendix section I). This points towards an im-\nportant role of the nonlinear dynamics of immune challenge\nelimination when few of them are left.\nIndeed, such 2D reduction of the dynamics allows us to per-\nform phase-plane analysis in the T, I plane Fig. 3C. First, we\nobserve that for all values of r (with scaling r\nr = r→∀ T ↑\n∃ ), there\nexists a stable fixed point (solid black point in Fig. 3B-C bot-\ntom), which corresponds to the chronic state at lowest value\nof r (light blue trajectory in Fig. 3B-C, bottom left). Con-\nversely, for very high r, the typical trajectory gets high in the\nI, T plane and circles back to the origin Fig. 3B, bottom right,\nas expected from an acute response. Crucially, this is true\neven if the initial challenge I(0) is very small (but non-zero).\nIn other words, such acute response corresponds to an almost\n.CC-BY 4.0 International licenseavailable under a \nwas not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint \n\n5\nhomoclinic trajectory [37] originating just above and circling\nback to the origin (light blue trajectory in 3B-C, bottom right).\nNotice that this acute trajectory circles around the stable fixed\n(solid black) point. We can then represent the basin of at-\ntraction of the fixed point (grey, Fig. 3C bottom middle and\nright), which is then limited by an unstable limit cycle sepa-\nrating the acute response trajectories from the (chronic) fixed\npoint (dashed purple lines in Fig. 3C middle and right).\nThe unstable limit cycle is a direct mathematical conse-\nquence of the co-existence of a stable limit fixed point circled\nby a (almost) homoclinic trajectory. A homoclinic trajectory\nis an orbit in phase space starting from one point (technically a\nsaddle point) and eventually converging to the same point after\na long excursion. Such orbits are well known in neuroscience\n(e.g. neurons [38]). In an immune context, homoclinic trajec-\ntories are natural since, from a situation with no immune chal-\nlenge, no T cells, a small injection of strong immune challenge\nshould trigger an immune response, eventually going back to\nthe initial (cleared) state. The existence of such dynamics is\nthe fundamental reason explaining the extreme sensitivity in\nresponse to r. Indeed, as another consequence of the coexis-\ntence between an unstable limit cycle and a stable origin, there\nis a saddle point, close to the origin (black cross in Fig. 3C).\nAs r decreases, the unstable limit cycle grows, until it collides\nwith this saddle point close to 0, Fig. 3C bottom middle, and\nthus disappears (through a subcritical homoclinic orbit bifur-\ncation [38], thus defining a true homoclinic orbit). Because\nof this bifurcation, acute trajectories such as the one in Fig. 3\nC right, where low amount of initial challenges circle back to\nthe origin, are no longer mathematically possible. Thus, one\nis left with the stable nonzero fixed point as the only stable\nattractor for many initial conditions I(0) in the absence of T\ncells above a threshold (gray zone represents the basin of at-\ntraction in Fig. 3C), ensuring chronicity for lower values of\nr.\nSo r acts as a control parameter for the bifurcation can-\nceling out the unstable limit cycle, explaining the sharp tran-\nsition between acute and chronic regime, and summarized in\nthe bifurcation diagram at the top of Fig. 3C. Sensitivity to the\ngrowth rate only associated to discontinuity theory arises from\nthe sudden change in the basin of attraction: the acute trajec-\ntories originating close to the origin for high r disappear, and\nare attracted to the chronic fixed point for low r for a broad\nrange of I(0), compare Fig. 3C left with Fig. 3C right.\nIII. LANDSCAPE GEOMETRIES FOR THE\nDISCONTINUITY THEORY OF IMMUNOLOGY.\nThe properties described in the previous section suggest\nthat the discrimination properties of the model are direct con-\nsequences of geometric features of the dynamical trajectories,\nin particular, the coexistence of acute and chronic trajecto-\nries in phase space. To confirm this intuition, we reverse en-\ngineer a minimal geometric model for discontinuity theory,\nusing the Evoscape approach [28]. This approach, inspired\nby geometric modelling in biology [39] and kernel-based ma-\nchine learning, relies on the combinations of simple dynami-\ncal modules to directly build “landscape-like” descriptions of\nbiological systems in 2D, here the Immune challenge/effector\nT cell plane. We build landscapes based on two conditions,\nFig. 4A :\n1. there is an acute response for immune challenges\nachieving high growth rates r, even for very low initial\nvalues. This imposes a near-homoclinic acute trajectory\nfrom and to the origin in the limit of very high r, fur-\nther associated to a stable fixed point at the origin. A\ncorresponding landscape is shown in Fig. 4A, left.\n2. there is a stable fixed point, topologically inside the\nacute homoclinic trajectory. A corresponding landscape\nis shown in Fig. 4A, middle\nEquations for each landscape are given in Supplement.\nThe first condition is very natural for a functioning immune\nresponse: the systems should trigger an immune response\neven with a low number of immune challenges, then return\nto homeostasis after resolving the infection, analogous to an\nexcitable-type trajectory [12, 28, 38]. Typically for such ex-\ncitable/acute response, we expect the trajectory to circle an\nunstable attractor as seen in the Evoscape description in Fig.\n4A, left. This first seems incompatible with the stability of\na fixed point associated with the chronic response, Fig. 4A,\nmiddle.\nY et, one can build a simple phase space encompassing those\ntwo constraints simply using a linear combination of those two\nlandscapes, Fig. 4A, right. As expected by design, we can\nget an almost homoclinic/excitable trajectory arising from the\ncombination of a repeller with flow (acute trajectory, condi-\ntion 1), itself around a stable fixed point in the center (chronic\ntrajectory, condition 2). As a consequence, an unstable limit\ncycle naturally emerges. By varying the relative strength of\nmodules (see details in Supplement), Fig 4B, the limit cycle\ncan appear or disappear through a homoclinic bifurcation, and\none can get either acute or chronic responses.\nTo fully match discontinuity theory, one needs to relate the\ncontrol parameter to growth rate, which requires some extra\ncondition. We thus chose to add an extra “growth” module\nclose to the y axis for the “acute” landscape only (light or-\nange disk in Fig. 4A left). The resulting model has all the\nhallmarks of discontinuity theory: fast-growing immune chal-\nlenges reach the bottom attractor (corresponding to elimina-\ntion), blue trajectories in Fig. 4C while slow-growing im-\nmune challenges reach the chronic attractor, red trajectories\nin Fig. 4C. One can then reconstruct a phase diagram delin-\neating chronic from acute regimes, Fig. 4D, which is very\nsimilar to the ones derived from the models in Figs. 2-3B.\nThe fact that we can reproduce properties of the more ex-\nplicit models using a reverse-engineered landscape suggest\nthat the observed geometry and sequence of bifurcations are\nvery natural, and should be shared by models with similar\nproperties irrespective of their precise mathematical formu-\nlation, as long as they satisfy conditions 1-2 above.\n.CC-BY 4.0 International licenseavailable under a \nwas not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint \n\n6\nFIG. 4. Reverse engineering a landscape model for the discontinuity theory of immunity A) Flows and potentials built using Evoscape[28].\nThe top row shows the flows, with the local Gaussian modules used indicated by circles. Green disks correspond to local attractors modules,\npurple rings to repelled, and orange modules to rotating flows. We also indicate examples of both acute and chronic trajectories. Landscapes\ncorresponding to the flows are shown on the bottom row, with isolevels of the potential projected underneath the surface. The left column\nillustrates how to combine modules to get almost homoclinic trajectories corresponding to acute responses, circling to a bottom left attractor\nthat would correspond to immune challenge elimination. The middle column is a simple attractor corresponding to the chronic response.\nAdding the two landscapes give a dynamic similar to what is observed in the explicit model, with acute/chronic trajectories going to two\ndifferent attractors (green disks) separated by an unstable limit cycle. Equations can be found in Supplement B) Bifurcation diagram and\nphase plane analysis as the control (speed) parameter is varied, recapitulating the behaviours showing in Fig. 2C. C) Possible dynamics of the\nImmune Challenges as we vary the control parameter. D) Corresponding phase diagram.\nIV . DYNAMICAL ANTAGONISM AND INVERSION\nThe transition from acute to chronic regime, with a bound-\nary controlled by a single parameter r is very reminiscent of\nspecific and sensitive discrimination by T cells, which is sen-\nsitive to the kinetics of interaction of ligands to T cell receptor,\nalmost irrespective of their concentration. Such behaviour has\nbeen well characterized experimentally [4, 8, 40, 41] and fur-\nther studied theoretically [9, 42]. In particular, it was proved\nthat such “absolute discrimination” mechanisms always dis-\nplay ligand antagonism [8, 43] as a phenotypic spandrel [44].\nIn those contexts, antagonistic properties occur at steady state.\nThis motivated us to study antagonistic properties in the con-\ntext of the discontinuity theory controlled by a dynamical bi-\nfurcation like here.\nTo do so, we expanded the simplified 3-parameter model\nof Eq. 1 to simulate a co-infection [45](we checked that the\nproperties we describe below also hold for the expanded ver-\nsions of the models). We consider two distinct immune chal-\nlenges (I\n1 and I2) with different rates (r 1 and r2) Fig. 5A-\nB. Dynamics of each of those immune challenges individu-\nally are presented in Fig. 5C-D. We assume that those chal-\nlenges activate the same effector cells Fig. 5E. This situation\nis immunologically–realistic when immune challenges mutate\nto alter their infectivity and/or growth rate, while maintain-\ning their antigenicity, which is now well established in multi-\nple contexts from persistent bacteria [46] to tumor cells [47].\nEquation 5 describes the dynamics of immune responses in\nthese coinfections for the reduced system.\nThe dynamics of those co-infections both as a function of\ntime and in phase space are shown in Fig. 5F. For this exam-\nple, we observe a counter-intuitive inversion in the behaviour\nof both immune challenges Fig. 5F left : while the slow grow-\ning r challenge I\n1 is eliminated, the fast growing challenge I2\noscillates then stabilizes at the stable chronic fixed point. In\nparticular, this means that a chronic immune challenge can an-\ntagonize an acute one, to give rise to a chronic regime for the\nfast-growing challenge (while the slow-growing challenge is\neliminated).\nFocusing on the phase plane trajectories, Fig. 5F right,\nwe see that the trajectory of I\n2 is very similar to the acute\ntrajectory, but converges to the chronic fixed point. To fur-\nther understand what happens, in Supplement (Fig. 18D), we\nshow the basin contours of the chronic fixed point for I\n2 as a\nfunction of the initial quantity of the chronically infected cells\nI1(0) : we see that addition of I1(0) modifies the unstable limit\ncycle, so that I2 crosses the boundary of the basin of attraction\nof the chronic fixed point very early during the simulated dy-\nnamics. As I\n1 increases, the basin contour retracts closer and\ncloser to the chronic fixed point.\nThose counter-intuitive effects depend on the respective\nvalues of growth rates and initial immune challenges, as\nshown in Fig. 5 G . We see four regimes, where each im-\nmune challenge can be either chronic or acute. The “inver-\n.CC-BY 4.0 International licenseavailable under a \nwas not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint \n\n7\nFIG. 5. Dynamical antagonism and inversion A) Illustration of the\ndynamics for a single chronic response considered I1(0)=10.0,\nr1 = 0.2 and B) for an acute response I2(0)= 10.0, r2 = 0.5, with\nphase plane dynamics and time courses displayed in C-D. Other pa-\nrameters are the same as in Fig. 3E) Given the same two species\npresented in A and B, we can observe different outcomes to their\nisolated outcomes by combining them into a single infection (given\nby equation 5). F) Left: time series of two species combined, I1\nand I2, undergoing, respectively, acute and chronic infections (the\nopposite of their original isolated outcomes, compare with dynamics\nin panels C-D). Right: trajectory in (T, I) space for each infection\nin the presence of the other one. G) Left: Infection outcomes for a\ncombination of two species for different initial I\n2 and r2. The val-\nues of I1(0) and r1, leading to a chronic state for I1 when it is the\nonly infectious species in the system, are fixed. The dotted red line\nrepresents the boundary between the chronic (inside) and acute (out-\nside) regions when I\n1 is the only infectious species. Right: Cross-\nsection of left plot, along the dotted black line. The growth rate for\nI2 (r2) is now fixed and the growth rate for I1 (r1) is allowed to move.\nI1(0)=100.0, T (0)=0.0\nFIG. 6. Illustration of the process of initial infection and subse-\nquent “treatment”. A) The first immune challenge initially causes\na chronic infection that reaches a steady state with I1 ↓= 0. Once this\nstate is reached, a second immune challenge (I 2) is added. Differ-\nent outcomes can occur where each species can be either completely\neliminated (acute) or remain active (chronic). Each case maps to\ncases presented in B-F. B) Infection outcomes for a system with ini-\ntially a single species with r1 = 0.2. It evolves towards a chronic\nstate. Once in this state, a second species is added, with its own\nr2, in amount I (ttreatment). Four possible outcomes are possible and\nare color-coded. Examples of these processes and their outcomes\nare presented in subplots C,D,E,F. Parameters are similar to Fig. 3.\nI1(0)= 10.0, T (0)= 0.0\nsion” regime corresponds to the green region where, when put\ntogether, a challenge I1 that would be chronic alone is becom-\ning acute in co-infection, while a challenge I2 acute alone be-\ncomes chronic in co-infection. This regime thus occurs on the\nright of the normal chronic/acute boundary for I\n2 (dotted red\nline in 5 G). We also observe a regime where both I1 and I2\nare chronic although I2 alone would be acute, which also ex-\ntends on the right of the chronic/acute boundary for I2, again\nindicating antagonism on I2 by I1.\nBecause antagonism here is associated with the dynamics\nof immune challenges, there are in fact multiple possibilities\ndepending on the timing when an acute or chronic challenge\nis added. We also consider the opposite limit, where one\n(chronic) challenge is already established, and another one is\nadded later, Fig. 6A. This corresponds to a two-tier infection,\nsequentially stimulating the same immune cells, which could\nhappen for instance if a mutation occurs for an immune chal-\nlenge stabilized in the chronic regime.\nFig. 6B-F illustrates the behavior of our 3-parameter model\nfor this staggered co-infection, depending on the quantity\nand growth rate of the added immune challenge parameters\nI\n2(0), r2 (see e.g. [48] for an example of such coinfection\n.CC-BY 4.0 International licenseavailable under a \nwas not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint \n\n8\nmixing helminth and bacteria). Compared to Fig. 5 when\nboth challenges are added initially, we see on the phase dia-\ngram Fig. 6B that both the antagonistic (light blue) and in-\nversion (green) regions largely extend towards higher r, over\nmore than half an order of magnitude in r. This means that\nthe slow-growing immune challenge I\n1 in this region would\ntypically drive a fast-growing one even more strongly towards\nthe chronic regime. It is worth pointing out two other bio-\nlogically relevant regimes. Panel E illustrates the inversion\ndynamics : while the initially chronic infection I\n1 is elimi-\nnated, the acute infection I2 is becoming chronic. Since I2 is\ngrowing about ten times faster than I1, this indicates a strong\nworsening of the chronic infection. Conversely, in panel C, if\nenough I\n2 is suddenly added, one can exit the chronic regime\nso that both challenges are eliminated. This is akin to a post-\ninfection “vaccination” against the chronically established im-\nmune challenge I\n1.\nV . DISCUSSION\nThe discontinuity theory of immunity, proposed by Pradeu\nand collaborators, posits that the immune system is sensitive\nto the (quantitative) “speed of change” of immune challenges,\nrather than more qualitative features such as molecular signa-\ntures. It was used as a holistic framework to explain phe-\nnomena such as immune tolerance or autoimmune disease.\nStarting from a simple model of an immune response, dis-\nplaying either acute/chronic activity depending on the growth\nrate r of the immune challenge, we derived a 3-parameter bi-\ndimensional model, recapitulating properties predicted by the\ndiscontinuity theory. In particular, the transition line sepa-\nrating acute and chronic regimes is purely controlled by the\ngrowth rate r over a broad range of initial sizes of immune\nchallenges.\nOur model presents common features to previously pro-\nposed models, for instance the co-existence of a chronic (“per-\nsistence”) state with a clearance state was proposed in [49],\nor an excitable model for auto-immunity proposed in [12].\nHowever, those models did not consider the discontinuity the-\nory framework. Our model presents unique, generic geom-\netry, with excitable trajectories going around the persistence\nstate in phase space. Such geometric constraints ensure that\nan unstable limit cycle separates the acute and chronic trajec-\ntories, and as a consequence the discontinuity detection occurs\nthrough a subcritical homoclinic orbit bifurcation depending\non the growth rate of immune challenge r, canceling the un-\nstable limit cycle. The properties described above would not\ndepend on the particular details of the models, as long as the\norbits can be reduced to 2D and a parameter such as r controls\nthe acute to chronic regime. In particular, immunologists have\ndocumented with very high degree of granularity, how diverse\nimmune responses take place depending on the type and size\nof infections (so-called Th1, Th2, Th17 etc. regimes for T cell\nresponses).\nThe generic aspect is only expected close to the bifurca-\ntion when the acute and chronic trajectories co-exist. We can\nnot exclude that different geometries and bifurcations are ob-\nserved for more complex dynamics, e.g. effectively living in\nmuch higher dimensions. However, we notice that, if the tran-\nsition from chronic to acute regime comes with a change of\ntopology of the orbits, then by definition a global bifurcation\nis expected, and if acute immune responses indeed correspond\nto excitable dynamics [38], homoclinic bifurcations are a very\nnatural scenario.\nAntagonism is a generic property of multiple decision-\nmaking or discrimination pathways, from multiple immune\nrecognition processes to olfaction [43, 50, 51], leading to\npractical application for cancer immunotherapy [52]. In [31],\nit was mathematically demonstrated that absolute discrimina-\ntion, defined as ligand-based cellular decision-making based\non one kinetic parameter irrespective of ligand concentrations,\nis necessarily associated with ligand antagonism, explain-\ning why it is observed for many immune decisions [8]. We\ndemonstrated here that discontinuity in the decision-making\nbetween acute and chronic response also displays antagonism,\nthus generalizing the results from [31] to a decision-making\nprocess based on a completely different mechanism (mathe-\nmatical speaking), namely a global bifurcation.\nThis leads to important biological predictions. In a disease\ncontext, it raises the possibility of dynamical “adversarial”\nstrategies [53] for a pathogen or a tumor to leverage dynam-\nical antagonism to escape immune responses. For instance,\na slow-growing tumor (like I\n1 in Fig. 6F) could first stabi-\nlize in the chronic regime of immunological response then\nlater mutate into a faster-growing one (like I\n2) while keeping\nthe immunological response at bay within a chronic regime.\nOne could even imagine a slow ramp-up of immune escape of\nsuch increasingly growing tumors so that a fast-growing tu-\nmor that would normally trigger an immune response could\nslowly evolve and remain “undetected” by the immune sys-\ntem. It is especially important to point out that in our model,\nslow growing challenges are not passive and actively antag-\nonize the detection of faster growing ones. Many observa-\ntions about the differential growth rates of primary tumors vs\nmetastasis, cancer dormancy as well differential immune re-\nsponses (“immune privilege”), are consistent with such sce-\nnario [54, 55]. Of note, it has been recently observed that\nupon CAR-T treatment, tumors with slow growth tend to bet-\nter survive [56]. This is consistent with both discontinuity\ntheory and the observation in other contexts that persistor cells\ntend to escape treatment because of their slowed metabolism\n[46]. Conversely, the counterintuitive effects due to the im-\nmune challenge cross interactions could be used to tailor bet-\nter treatments. For vaccine designs, bolus delivery is the nor-\nmal regime (top left, acute regime in Figure 2B) but it may\nbe tolerizing and inducing a chronic immune response if the\ninital challenge is too small/not antigenic enough. Such effect\ncould further be leveraged to rather induce tolerance.\nMore generally, the coexistence of acute and chronic tra-\njectories for the same immune challenge provides a simple,\ndynamical mechanism for an immune system to learn over\ntime, thus allowing for a dynamic redefinition of the self [22].\nThose predictions of 1. active antagonism of growing im-\nmune challenges and 2. mechanism of immune tolerance\nlearning, naturally come from the geometric constraints im-\n.CC-BY 4.0 International licenseavailable under a \nwas not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint \n\n9\nTABLE I. Parameters used in the full model of 2 with their biological\nsignification and value used in simulations. These are the values used\nfor all figures, unless otherwise indicated. The time unit is “day”, i.e.\nall rates are implicitly per day.\nSymbol Biological signification V alue\ng Linear growth rate 101\nr Exponential proliferation rate 0.2\n∀ Binding rate of T-cells to infected cells 10→4\nT ↑\nConcentration of naive T-cells at\nequilibrium\n102\nT ↑reg\nConcentration of regulatory T cells at\nequilibrium\n104\n# T-cell proliferation rate 104\n%\nConsumption rate of IL2 by regulatory T-\ncells\n10\n&1\nInhibition rate of T-cells by regulatory T-\ncells\n10→3\n&2 Self-regulation rate of regulatory T-cells 10→3\n∃ Exhaustion rate of activated T-cells 5.0\nµ\nProduction rate of IL2 by activated T-\ncells\n105\nposed by discontinuity theory that we introduce. Similar to\nthe general properties of antagonism for T cell detection [31],\nthey come from the natural extension of a theoretical proposal\nfor single immune challenges to interactions between multiple\nones, with clear actionable experimental predictions. Practi-\ncally, the existence and properties of such learning in immune\ndynamics could be directly tested using specialized platforms\nsuch as Immunotron [4]. In particular, by carefully monitor-\ning time courses of acute/chronic infections, one should be\nable to identify topological changes in responses associated to\nglobal bifurcations such as the one predicted here.\nOur model is of course very simplified and neglects multi-\nple other mechanisms (e.g. sensitivity to the binding kinetics,\nimmune editing, inflammatory switches, long term memory\n[57]), that could add multiple dimensions. That said, it is\nstriking that a first-principle model accounting for disconti-\nnuity theory naturally comes with a combination of high-level\nfeatures such as coexistence of acute and chronic responses,\nassociated antagonism and global bifurcations discriminating\nbetween regimes. This is in line with the recent realization in\nmultiple biological contexts that geometric, low-dimensional\nmodels accurately describe complex biological dynamics, e.g.\nline attractors for neural decision making [58] or heteroclinic\nflips for cellular differentiation [39, 59]. Of note, global bi-\nfurcations, transients and ghost states [60, 61] have been sug-\ngested to play important role complex cellular computations,\nand our model for discontinuity theory suggests that similar\nphenomena might be at play in immune decision-making.\nVI. MA TERIALS AND METHODS\nA. Initial model and its variations\nThe starting point for describing the system is\n\n\n\n\n\n\n\n\n˙[T ]= ∀ [I]\n(\nT\n↑ + #[T ][IL2]\n&1[Treg]\n)\n  \nproduction\n→ ∃[T ]\nexhaustion\n˙[I]= g + r[I]  \ngrowth terms\n→ ∀ [I]( [T ]+T ↑)  \nT-cells elimination\n[ ˙Treg]= % [Treg][IL2]  \nproduction\n→ &2[Treg]\n(\n[Treg] → T ↑\nreg\n)\n  \nself-regulation\n[ ˙IL2]= µ[T ]\nproduction\n→ [IL2]( #[T ]+% [T reg])  \nconsumption\n,\n(2)\nwhere [I] is the concentration of infected cells, [T ] the con-\ncentration of T-cells, [Treg] the concentration of regulatory T-\ncells, and [IL2] the concentration of the interleukin 2, cy-\ntokine. The parameters used are described in table I. We\ncomplement those equations with an interpolating flow going\ntowards the origin as soon as I is lower than a threshold I\n0 (see\ndetails in the Supplement H).\nAssuming a quasistatic dynamic for [Treg] and [IL2], we\ncan obtain a dynamical system of only two variables (see Ap-\npendix section F for derivation), which presents small differ-\nences with the full system (Fig. 10)\n\n\n\n\n\n\n˙[T ]= ∀ [I]\n(\nT\n↑ + #[T ][IL2]\n&1[Treg]\n)\n  \nproduction\n→ ∃[T ]\nexhaustion\n˙[I]= g + r[I]  \ngrowth terms\n→ ∀ [I]( [T ]+T ↑)  \nT-cells elimination\n,\n(3)\nwhere [Treg] and [IL2] are functions of [I] and [T ], given\nin equations F1 and F2. We designate this as the quasistatic\nsystem. We can do some further approximations and obtain\nand even more compact form:\n{\ndTr\nd! = ∀rIr + Tr (#rIr → 1)\ndIr\nd! = 1 + Ir (rr → Tr) , (4)\nwhere the rescaling of the variables is given by Tr = ∀\n∃ [T ],\nIr = ∃\ng [I] and d! = ∃dt and the parameters are rr = r→∀ T ↑\n∃ , ∀r =\ngT ↑∀ 2\n∃3 and #r = #&2g∀\n%& 1∃2 . We also interpolate those equations\ntowards the origin once Ir is lower than a threshold.\nWhen describing the interactions of two species in the sys-\ntem like in section IV. We consider two species Ir,1 and Ir,2\nthat interact identically with the T-cells and do not interfere\nwith each other directly. We describe the dynamics through\n\n\n\ndTr\nd! = ∀r(Ir,1 + Ir,2)+T r (#r(Ir,1 + Ir,2) → 1)\ndIr,1\nd! = 1 + Ir,1 (rr,1 → Tr)\ndIr,2\nd! = 1 + Ir,2 (rr,2 → Tr) .\n(5)\nNotice that the immune challenges Ir,1 and Ir,2 have differ-\nent growth rates, rr,1 and rr,2. These quantities can be rescaled\n.CC-BY 4.0 International licenseavailable under a \nwas not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint \n\n10\nTABLE II. Modules and their parameters used in the results of Fig.\n4. Note that the strength of left rotator is dependent on the “Speed”\nparameter.\nModule Type Strength Width Location\nAttractor 6.0 0.9 (→4, →3)\nRepellor 1.0 2.5 (0, 0)\nRotator (CW) 0.6 3.0 (0, 0)\nRotator (CW) Speed ↔0.5 4.0 (→6, →2)\nAttractor 3.3 1.5 (0, 0)\nto r1 and r2, the same way that rr is rescaled to r. We can\nalso simulate this two-species dynamics for the “full system”\nas well as the “quasistatic system” as shown in appendix G.\nAs stated in the main text, these reduced dynamical sys-\ntems are rescaled back to their “natural” scale (from Ir and Tr\nto Ir and Tr) when showing or discussing their simulated out-\ncomes. The growth rate parameters rr, rr,1 and rr,2 are also\nrescaled to r, r1 and r2 for easier comparison with the full\nmodel, as well as allowing direct comparison with the biolog-\nical observable that the parameters represent (the growth rate\nof the challenges).\nB. Dynamical landscapes\nThe dynamical landscape treatment presented in section III\nwas conducted using the Evoscape framework [28]. The re-\nsults of this analysis are all presented in Fig. 4.\nUsing 4 different types of modules (attractor, repellor,\nclockwise rotator and counter-clockwise rotator) with vary-\ning parameters we constructed the landscape presented in Fig.\n4A. All the modules are linearly added together in a differen-\ntial equation for I and T , with ωx =\n(\nT\nI\n)\n, as such:\nd\ndtωx = !\ni\nAi e→ 1\n2\n( ↗ωx→ωµi↗\n∋i\n)2\nMi (ωx → ωµi)+A 0↗ωx↗3\n3. (6)\nIn the previous equation, all the modules are summed to-\ngether, each denoted by an index i. Each of them have a loca-\ntion in phase space (the T, I plane) denoted by the vector ωµi,a\nstrength, denoted by Ai, a width, denoted by ∋i and a Jacobian,\ndenoted by Mi. Furthermore, there is a global attractor with\nweight A0 (that we set to 0.01, for the combined landscape).\nThis global attractor is of the form ↗ωx↗3\n3 = !i x3\ni . The Jacobian\ntakes different form depending on the type of module. These\nare\n•\n(\n10\n01\n)\n: repellor\n•\n(\n→10\n0 →1\n)\n: attractor\n•\n(\n01\n→10\n)\n: clockwise rotator\n•\n(\n0 →1\n10\n)\n: counter-clockwise rotator\nOverall, the Evoscape framework allows us to mix and\nmatch different dynamical components to create a landscape\npresenting general features of the system. For the results of\nFig. 4, we used the modules presented with their parameters\nin table II.\nThe dynamics resulting from the Evoscape framework can\nbe seperated into “potential” (attractors and repellors) and\n“curl” (rotators) parts. To draw the potentials presented in\nFig. 4, we use the “potential” modules, from this, we can get\nan equation for the potential P:\nP(ωx)= A\n0\n↗ωx↗4\n4\n4 + !\ni\nsiAi∋ 2\ni e→ 1\n2\n( ↗ωx→ωµi↗\n∋i\n)2\n. 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It is made \nThe copyright holder for this preprint (whichthis version posted July 18, 2025. ; https://doi.org/10.1101/2025.07.16.663927doi: bioRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}