A qualitative mathematical model of the immune response under the effect of stress
AA qualitative mathematical modelof the immune response under the effect of stress
Maria Elena Gonzalez Herrero and Christian KuehnFebruary 11, 2021
Abstract
In the last decades, the interest to understand the connection between brainand body has grown notably. For example, in psychoneuroimmunology manystudies associate stress, arising from many different sources and situations, tochanges in the immune system from the medical or immunological point ofview as well as from the biochemical one. In this paper we identify importantbehaviours of this interplay between the immune system and stress from medicalstudies and seek to represent them qualitatively in a paradigmatic, yet simple,mathematical model. To that end we develop a differential equation modelwith two equations for infection level and immune system, which integrates theeffects of stress as an additional parameter. We are able to reproduce a stablehealthy state for little stress, an oscillatory state between healthy and infectedstates for high stress, and a “burn-out” or stable sick state for extremely highstress. The mechanism between the different dynamics is controlled by twosaddle-node in cycle (SNIC) bifurcations. Furthermore, our model is able tocapture an induced infection upon dropping from moderate to low stress, and itpredicts increasing infection periods upon increasing before eventually reachinga burn-out state.
Keywords:
Immune system, stress, psychoneuroimmunology, mathematical mod-elling, differential equations, SNIC bifurcation.
Although it is common knowledge that stress can hurt our health, scientific resultsregarding this are fairly recent. The first studies attempting to show this connectionbetween the psychological and the physical state were developed in the first half ofthe 20th century [1, 2]. In the second half of the century the corresponding fieldof psychoneuroimmunology (PNI) established the interplay of the central nervous
Maria Elena Gonzalez HerreroDepartment of Mathematics, Technical University Munich, 85748 Garching, GermanyEmail: [email protected] KuehnDepartment of Mathematics, Technical University Munich, 85748 Garching, GermanyExternal Faculty, Complexity Science Hub Vienna, Josefst¨adter Str. 39, 1080 Vienna, AustriaEmail: [email protected] a r X i v : . [ q - b i o . Q M ] F e b ystem (CNS), the endocrine system, in other words the hormonal messenger systemof the body, and the immune system [3–6]. The first goal of this field is to specify agood definition of stress to be able to work with and thus also a way to categorizeit [7]. The most common way to do this is mainly by the duration, e.g., one maycontrast very brief public speaking to extremely long-time care of a spouse [8].Furthermore, it is important to quantify stress factors, e.g. how and to what extendwe can see sports as positive or negative stressor in the same way as psychologicalones [9]. Further, PNI wants to understand the biochemistry of how behavioral andpsychological effects, in particular stress, are reflected in our bodies, i.e. how stress“gets inside the body” [8]. Although there is still much to learn, we have now a basicunderstanding of the hormonal reactions of our body to stress and how they affectthe immune system, e.g. by specific receptors on cells responsible for the immunity[6]. In addition, there are several studies regarding immunological and medicalconsequences of stress [10, 11], where a key goal is to understand and counteractnegative consequences. In this context, the difficulties of extensive clinical trialsand the lack of a commonly accepted definition, categorisation or measurements ofstressors are most evident. This is why Segerstrom and Miller performed in 2004 ameta-analytical study of numerous results going back 30 years [8]. They showed thatin general stress has a negative effect on the immune system by either decreasing thenumber of e.g. killer cells or simply disturbing the equilibrium between the differentcomponents. Nevertheless a brief stress, like public speaking, can also have a positiveeffect and even longer short-term stressors like examination periods for students canhave a beneficial effect during but cause decrease immunity afterwards [12, 13]. Having some fundamental understanding of how our immune system reacts to stress,we want to find a simple mathematical model of differential equations which is ableto represent the basic qualitative interaction between an infection and our immunesystem taking into account the effect of stress .Since we are only interested in a qualitative analysis of the system we are goingto work with normalised dynamical variables x , representing the level of activity ofthe immune system, and y , as the level of infection or sickness, with values rangingbetween 0 and 1. The stress is included by a parameter s also normalised between0 and 1 where 0 corresponds to no stress at all and 1 is the maximal level of stresslimited by the assumption that a person cannot feel infinitely strong stress.To specify the dynamics of the infection we are going to built upon principles ofmathematical biology [14, 15], where in multiple sub-disciplines such as ecology,neuroscience, or biomechanics, one aims to generic polynomial nonlinearities to rep-resent basic effects. Based upon population dynamics, we consider logistic growthand Allee effect for the immune system intrinsic dynamics. On this regard, our firstassumption is that, as long as our immune system has some activity, the infectionshould have two stable states, the “sick”-state and the “healthy”-state. Further-more, since we are constantly coming in contact with different viruses and bacteria,we are interested in having a slightly raised base line instead of the “healthy”-state2eing y = 0. Our second assumption is that, if there is no immune system or if it istoo weak then the infection should spread up to its maximal capacity. Combiningeverything we arrive at the equation y (cid:48) = − xy + r (cid:112) − y ( q ( y − y )( y − y ) + (1 − q ) y ) (2.1)where we used a square root instead of the standard multiplicative term (1 − y ) frompopulation models. Finally we choose the parameter values r = 2, y = y = 0 . q = 0 . y , our approach for find-ing an appropriate equation for the immune system is based purely on the geometrynecessary to capture the effects we aim to model. We start with the case withoutstress. It is clear that without stress we should have a unique stable equilibriumwith 0 (cid:28) x < < y (cid:28) s < s with some threshold level of stress 0 < s < < y (cid:28)
1. As s increases the x -coordinates of both fixedpoints should slowly start decreasing and approaching each other such that for some0 < s < s the stable equilibrium crosses the original position of the unstable one.If we would now drop the stress back to s = 0 it would in fact trigger the expectedinfection before converging to the healthy state.Finally, it is reasonable to assume that an extreme (prolonged) stress can preventthe body from recovering from an infection, giving a “burn-out” state, such that for s close to 1 we want to have a fixed point where 0 (cid:28) y < x (cid:48) = y − (cid:18)
15 ( kx ) −
14 (3 x + x )( kx ) + ( x + x x )( kx ) −
12 ( x + 3 x x )( kx ) + x x ( kx ) (cid:19) + 0 . x = 0 . k = 1 . − . − s ) to control the shift to the right of the equilibria and x = 0 . − . s responsible forthe disappearing of the stable healthy state. Of course, the actual numerical valuesin (2.2) are not the key aspect but the geometry of the dynamics and the qualitativedescription of the different aspects of immune reaction, infection level, and stress.3n summary, the complete system we arrive at is x (cid:48) = y − (cid:18)
15 ( kx ) −
14 (3 x + x )( kx ) + ( x + x x )( kx ) −
12 ( x + 3 x x )( kx ) + x x ( kx ) (cid:19) + 0 . y (cid:48) = α (cid:16) − xy + 2 (cid:112) − y (cid:0) q ( y − y ) + (1 − q ) y (cid:1)(cid:17) (2.3)with the parameter values x = 0 . q = 0 . y = 0 . k = 1 . − . − s ) and x = 0 . − . s integrating the effect of stress into the equations. The additional parameter α controls the scale of change of the infection y with respect to the immune system x .Figure 1 shows the corresponding phase portraits and time series for multiple valuesof s . And indeed, our model presents different dynamic regimes upon different stresslevels, which we can observe in the phase portraits and time series shown in Figure 1. To analyse in more detail the change between the different dynamical regimes, wetake a look at the corresponding bifurcation diagram [16] in Figure 2. For s = 0 wehave 3 equilibria: an unstable spiral, a saddle and a stable node. As s increases thestable node and the saddle collide and disappear in a saddle-node in cycle (SNIC)bifurcation for s ≈ .
48. At s ≈ .
95 we find a second SNIC, where another pair ofa stable node and a saddle appear. In addition to the equilibria mentioned above,in the parameter regime between the two bifurcation points we find that all orbitsexcept the unstable equilibrium have to converge to a limit cycle as t → ∞ . Thiscan be easily shown using the Poincar´e-Bendixson Theorem on the unit square.Note that SNIC bifurcations are of co-dimension one, i.e., they are generic/typicalin planar systems with one parameter presenting a mechanism to obtain oscillations.They have been observed already in many models ranging from neuroscience [17,18]to mechanics [19]. They are accompanied by a heteroclinic loop for parameter values s < .
48 (or s > .
95) since the unstable manifolds of the saddle converge to thestable node. As s approaches the bifurcation the two equilibria collide, one of theheteroclinic connections disappears and the second one becomes a homoclinic orbit.By perturbing s further this homoclinic gives rise to a family of stable limit cyclesfor 0 . < s < . In this work, we have initialized the qualitative mathematical model development forthe interaction between immune system levels, infection dynamics, and stress level.Based upon established principles of theoretical biology, we developed a simple, yetpowerful, planar dynamical system to analyze the influence of stress as an externalparameter. Using only elementary nonlinearities, out model can represent (I) a4table healthy equilibrium for low stress, (II) the effect of sudden stress level dropfrom moderate to low levels inducing a single infection, (III) periodic outbreaks ofinfections under high stress, (IV) the transition to a burn-out state at very highstress. The key transition mechanism we have identified is a saddle-node in cycle(SNIC) bifurcation. This bifurcation transition actually makes the model predictive,e.g., it predicts that if stress levels are close to a burn-out stage but still below,then the oscillatory sick periods become longer. Although this looks like a naturalprediction, it is one that we did not anticipate to be able to extract from our systemat all during the modelling process.Having established the ability of conceptual mathematical models to contribute tothe understanding of stress, we believe that further model development and connect-ing qualitative mathematical models to more detailed biophysical principles as wellas to clinical trials, could forge a effective path mitigating, and probably even moreimportantly, predicting in advance, the positive and negative effects of stressors.
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Acknowledgements:
MEGH work was partly supported by a Hans-Fischer Se-nior Fellowship of the Technical University of Munich Institute for Advanced Study(TUM-IAS). CK acknowledges partial support of the VolkswagenStiftung via aLichtenberg Professorship. 6igure 1: Phase portraits of system (2.3) and its x -nullcline (orange) and y -nullcline(yellow) for different values of s together with the corresponding time series. Onthe left the healthy state is stable and the time series shows the isolated sickness,when changing to no stress s = 0 after a moderate stress s = 0 . s = 0 . s = 1. Again,a time series is shown, where we switched from the top to the bottom dynamicalphase portrait at the dashed black vertical line.7igure 2: Bifurcation diagram (left panel) in the ( x, s )-plane for α = 2 .
5. Blackcurves mark equilibrium points, while the blue region consists of limit cycles, in-cluding the period of the limit cycles for 0 . < s < .
95 (right panel). LP mark thelimit points or saddle-node bifurcation points (in this case saddle-node in limit cycle(SNIC) bifurcation points) while H marks a neutral saddle, where the eigenvalues λ , of the Jacobian at the equilibrium satisfy λ = − λ2