The study of the relapsing-remitting dynamics of MS and other autoimmune diseases can have practical benefits for patient management. The unpredictability of relapses in MS is one of the most disturbing aspects of the diseases reported by patients. Given the key role of relapses in the management of the disease, a broad understanding of relapsing dynamics is important to perform an accurate prognosis, improving patient's management and therapeutic decisions. In order to gain insights in the mechanistic basis of this relapsing dynamics we developed a mathematical model of the adaptive immune system. We aimed to assess the hypothesis that the characteristic relapsing-remitting dynamics of autoimmune disease emerges as a result of the intrinsic control properties of the immune system. Based on our simulations, we found that immune tolerance, defined in our model as the capacity to cope the activation of effectors T cells, is an emergent property of the Teff-Treg cross-regulation. This indicates that the Teff-Treg loop is a powerful control module that regulates the adaptive immune system when activated by stochastic environmental factors. A pathological dynamic regime of the Teff-Treg loop created a pulsing dynamics in which the expansion of the Teff population transiently escapes the control of Treg population, creating the relapses typical of autoimmune diseases such as MS. In our model, relapses mainly arise upon the failure in the Treg response and were mainly driven by stochastic process that might correspond either to thymic production of new self-reacting T cells or from random sporadic infections. Interestingly, the frequency of such stochastic events where not the main factor producing relapses, but the severity in the dysfunction of the Teff-Treg cross-regulation was the main responsible of relapse frequency and severity. This finding can explain why the relapse activity in patients with MS is quite stable during the relapsing-remitting phase, because it would mainly depend on the dysfunction of the immune system, but make relapses very difficult to predict.
Our model shows that the pathologic dynamic regime in autoimmune conditions become stationary and makes the autoimmune process chronic but relapsing instead of progressive. In this scenario, autoimmunity can be considered as a dynamic disease [34, 35], in which the pathological state arises through the emergence of stationary stochastic dynamics in the immune system that overcomes immune homeostasis. Although our study was mainly inspired in the dynamics of MS, we believe it can provide insights about the dynamic of other autoimmune diseases, since many of them also have relapses (e.g. Lupus or Rheumatoid Arthritis) but also chronic, non-relapsing autoimmune diseases such as type 1 diabetes might also display relapsing inflammatory bursts . Thus, our study supports the view of autoimmune diseases as complex disease produced not by a single molecular or cellular event or governed by environmental challenges, but rather by the combination of many factors that deregulate the control mechanisms in the immune system .
Our study highlights the critical role of the cross-regulation of T cell populations in peripheral tolerance and in the generation of autoimmune diseases. Although both the loss of the central tolerance and the impairment of the Teff-Treg loop can contribute to the generation of autoimmunity, our model suggests that the breakdown of cross-regulation and not central tolerance leads to this kind of relapsing autoimmune dynamics. This result is in agreement with the fact that the majority of autoimmune diseases are sporadic and not related with mutations in genes controlling central tolerance, such as in the IPEX or APECED syndromes , and that impairment of Teff-Treg population is also required even in monogenic autoimmune diseases . Another conclusion from our study is that the autoimmune diseases can result from the weakening of the peripheral tolerance, particularly the control exerted by Treg over effector/auto-reactive T cells. There is already experimental evidence supporting a role of such control loop on the prevention of some autoimmune disease. Particularly there is evidence of several genetics defects weakening this loop that are clearly associated with autoimmunity  as well as previous mathematical models supporting this idea . Our results show that defects on central tolerance (particularly those increasing the frequency of generation of auto-reactive T cells) might not be sufficient to induce autoimmunity. There is already experimental evidence proving this fact in animal models. Particularly, transgenic mouse model have shown that a thymus generating more than 90% of its total output of a single anti-MBP auto-reactive T cell clones without causing autoimmunity, because of the peripheral control exerted by the remaining 10% of the T cell repertoire that happens to contain Treg .
Previous models of cross-regulation between effector and regulatory cells have shown bi-stable regimes [29, 40]. In one of the stable points, corresponding to the healthy states, the Treg population controls the effector population, while in the second stable point, interpreted as the autoimmune state, effector cells outcompete or predominate over regulatory cells. These models provide an explanation for the etiology and natural history of chronic-progressive autoimmune diseases such as type I Diabetes or the progressive subtypes (non-relapsing) of Lupus or MS, as the result of a switch to the pathology stable attractor. However, relapsing-remitting dynamics of autoimmune diseases such as those found typically in MS, Rheumatoid Arthritis or Lupus still lack of a theoretical framework. The major contribution of the model introduced in this paper is a dynamical explanation for such relapsing-remitting dynamics. Even though the cross-regulation makes that both populations always stay within the basin of attraction of a unique steady stable point, there are configurations whose trajectories produce temporal relapses in presence of environmental stochastic events. One basic assumption of our model is that the oscillatory dynamics of the immune systems will depend the prey-predator model with stochasticity . However, in our model the stochastic environmental factors were modeled as a train of impulses influencing the Teff growth (prey) rate and the Treg death rate (predator). For this reason, our model differs from previous predator-prey models because the stochasticity component come from introducing either Teff or Treg randomly using a train of impulses, independently of the T cell population density used in other models .
Our results also show that the onset of the relapse is triggered by stochastic events such as the random thymic generation of auto-reactive Teff or Treg cells or sporadic common infections that activate the immune system. However, the duration of each relapse and the overall relapsing frequency are under the control of the dynamics of Teff-Treg loop. This result might help to explain why previous attempts to identify external factors triggering relapses failed to provide clear explanations [2, 11]. Moreover, predicting the appearance of new relapses will be difficult, not only because we would lack sufficient knowledge about the dynamics of the immune system in a given individual at a given time, but also due to the influence of the stochastic events.
In our study we model the reversible tissue damage in order to provide a comprehensible link to the active lesions in CNS observed in patients with MS. Therefore, we compared long-term simulations of reversible tissue damage with the CEL dataset from patients with MS. The simulations obtained closely reproduced the oscillatory behavior of the CEL dataset, obtaining moderate to high correlations and reproduced the two-phase behavior (the relapsing-remitting dynamics). In this way, this model explains dynamics of autoimmunity with a basic cyclic nature. This is important, since several human autoimmune diseases are documented to have a cyclic behavior, although this is not the only class of dynamics observed for autoimmunity (e.g. type 1 diabetes). Moreover, mathematical models of T cell vaccination have explored the dynamics of the immune system after generating anti-idiotipic Treg for switching-off the autoimmune response, explaining the cyclic dynamics of autoimmune disease [42, 43]. However, the limited predictive ability suggest that either other biological factors not include in the model or, more probably, the stochasticity of T-cell activation due to random infections might account for a significant proportion of the temporal distribution of relapses. Also, although our model is able to reproduce the relapsing dynamics of MS, other models considering other factors might also be able to provide similar explanations.
From the therapeutic point of view, our results may have several implications. First, our analysis indicates that autoimmunity is a dynamic phenomenon. Thus, perturbing the Teff and Treg populations might produce different outcomes depending on the control parameters of the immune system in a given patient and the timing of the intervention, but without modifying the underlying dynamics. Accordingly, therapeutic approaches to treat autoimmune diseases that involve either decreasing Teff populations (e.g. through chemotherapy, anti-CD52 or anti-CD20 mAb therapy) or increasing Treg populations (e.g. Treg cell therapy) will not cure the disease, since they are aimed to keep the values of both populations in the range observed in healthy state but without restoring the control of the immune system to that of the healthy state. From a systems biology perspective, therapeutic interventions should be designed to restore the dynamics of the system to the healthy state or at least to a less deleterious dynamic [44, 45]. In order to efficiently modulate the dynamics of the immune system it is necessary to know in which region of parameter space the immune system of a given patient is acting at any time, as well as to identify which control mechanism can be targeted [44, 46]. As we found in the perturbation analysis, considering the outcome of increasing the Treg population or decreasing the Teff population to treat autoimmune diseases, intervening at different times and with different perturbations, might be beneficial. The timing and dose will be specific for a given patient or subgroup of patients, implying the need for personalized medicine. Nevertheless, it will be necessary to translate the critical parameters to specific molecular and cellular markers of the immune system in order to be able to apply it to human immunotherapy [44, 45].
Our study has several limitations. As commented before, despite the emerging importance of Treg in the immune system, fundamental parameters of the biology and homeostasis of these cells, such as their lifespan, turnover, and recirculation properties remain unknown. Also, while we analyze the cross-regulation of Teff and Treg populations from a systemic point of view, many other aspects of the immune response that may also be important, were not contemplated, such as the innate immune systems activity, regulation of the effector response in the tissue, the role of the T-cell immune repertoire or the control of the immune response on T-cell activation [37, 47]. However even with this simplified model of the immune system, we were able to show that autoimmune diseases can arise as a dynamic phenomenon and we could identify the critical contribution of the Teff-Treg loop in the control of the immune response, providing a theoretical framework for the understanding of the relapsing dynamics in autoimmunity. Also, the induction of an autoimmune response in our model requires some minimal changes in the parameter values. This may represent the accumulation of the genetic background (e.g. HLA class II susceptibility alleles (DRB1*1501), IL2R or IL7R for MS) , in conjunction with the shaping of the immune repertoire during ontogeny and the presence of stochastic infectious challenges for generating individuals susceptible to develop autoimmune diseases.