Is a unified theory of homeopathy and conventional medicine possible?

Introduction

In the UK, a vituperative debate has arisen over whether complementary and alternative medical (CAM) therapies should be included within the National Health Service.¹ Homeopathy is arguably the most controversial form of CAM, not least because a highly publicised Lancet meta-analysis of clinical trials comparing homeopathy to allopathy dismissed the former as having no effects beyond placebo.² This paper’s clear bias and methodological flaws,^3-7 seemingly escaped most in the conventional medical community and the media,⁸ who are equally unaware of the many published fully peer-reviewed papers showing that homeopathy can indeed have effects beyond placebo.^9-13 So, it is not surprising when twelve of the UK’s top medical scientists signed a letter to The Times calling for the cessation of CAM funding by the National Health Service (even though it is abundantly clear many conventional medical practitioners already avail themselves of adjunctive CAM therapies)¹⁴, but also accusing homeopathy of being ‘an implausible treatment’.¹⁵

The origin of this attitude is due in no small part to increasing demands that all therapeutic procedures be subjected to double-blind placebo-controlled trials (DBRCTs),¹⁶ according to the precepts of Evidence-Based Medicine (EBM).^17,18 This could be considered hypocritical given that many of the drugs and procedures currently used in orthodox medicine have yet to be clinically confirmed by EBM.¹⁹ Apart from an almost inquisitorial overemphasis on the supremacy of one particular kind of evidence,¹⁸ the problem is how the meaning of the phrase ‘placebo effect’ is taken.

If in trials homeopathy works no better than a sugar pill, it is deemed ineffective. But the placebo effect has always been part of every doctor’s kitbag, and is enshrined in the Hippocratic Oath.¹⁴ Also homeopaths say their therapy encourages the body to heal itself, suggesting at the very least a more humane appreciation of the placebo effect.²⁰

But with none of the original material substance left, sceptics want to know how a highly diluted homeopathic remedy can exert any effect, let alone a therapeutic one.²¹ Clearly, homeopathy poses serious challenges to conventional bioscience’s dominant materialistic paradigm, and will continue to be regarded as delusional, cynical ‘quackery’, or at best the ‘placebo effect’.² Yet, one of the most recent studies performed under the same Swiss-inspired Program of Evaluation of Complementary Medicine (PEK) that produced the flawed Lancet meta-analysis, found that the ‘effectiveness of homeopathy can be supported by the clinical evidence’.²²

It is this attitude that it should not work, therefore it cannot, which acts as one of the main hindrances to the integration of homeopathy into conventional medical practice. Another is many homeopaths’ distrust of anything to do with conventional science and medicine. Thus, for true integration to occur, homeopathy and conventional medicine will have to achieve some common ground.¹⁴

quantum theory, VITALISM, AND SYMPTOM EXPRESSION

The similarity in discourse

Conventional biomedicine is reductionist. This means sickness and disease are comprehended primarily in terms of pathological cellular biochemistry, as caused by external agents (viruses, bacteria, etc) or internal imbalances (e.g., genetic abnormalities). Treatment is therefore geared towards ‘fixing’ the problem by eradicating these causative disease factors, usually with powerfully toxic substances whose long-term side-effects can compromise the homeostatic immune system.

Homeopathy on the other hand, considers that ‘In the healthy condition of man, the spiritual vital force…. retains all parts of the organism in admirable harmonious, vital operation as regards both sensations and functions….‘²³ In other words, it is holistic. From conventional medicine’s reductionist view-point, the Vital Force (Vf) is treated with contempt because ‘…it doesn’t have any identifiable source… obey any kind of (physical) law, it can’t be defined; it is simply postulated ad hoc to explain whatever effects or alleged effects need explaining: it can’t be pinned down or put to the question; its function is to provide the illusion of meaning without substance….it can’t be disproved because it is too amorphous and vague a concept.‘²⁴

However, complex systems are known to self-organise, are open, and possess a wholeness that cannot be attributed solely to any particular part or sub-system.²⁵ Non-deterministically, the Vf might be an emergent property of billions of living cells, which generates an all-pervading field that by feedback so organises the totality’s elements, it reinforces itself. This field would not originate in any one cell or body part, and being the resultant of the whole organism, resists any dissipative entropic influence. ²⁶
Such a holistic view of the Vf bares phenomenological comparison with certain conclusions derived from quantum physics.^{27, 28} Thus, the Vf is thought not to be directly ascertainable: it is observed indirectly through the symptoms it produces.²³ Similarly in quantum theory, the wave function (a multi-dimensional mathematical descriptor of a quantum system’s state) may only be inferred from the effects it produces in our reality.^{28, 29} This is through no fault in observation but depends more on the multi-dimensional mathematical language used in describing wave functions.³⁰ As a result, trying to visualise a quantum state in our usual 3-D terms^{31, 32} leads to loss of information: one reason why entanglement in quantum theory seems so paradoxical. ²⁸

Entanglement in a quantum system occurs if its seemingly separate parts are so holistically matched, measurement of one part of the system instantaneously (i.e., not limited by the speed of light and therefore without classical signal transmission) provides information about all its other parts, regardless of their separation in space and time, or their size.³³ This last point addresses naïve criticisms³⁴ often levelled at those who use quantum theory to explain the therapeutic effects of homeopathy/CAMs.^35-43

In addition, more modern interpretations of quantum theory resolve such paradoxes as entanglement by taking the quantum state of a system not as an objective ‘thing-in-itself’, separate from us, but as a subjective representation of what we can know about it.^44-49 Thus, apart from predicting a quantum system’s evolution in time, the wave function is also telling us something about the irreducible relationship between the observer and the observed quantum system. Also, a generalised form of quantum theory has recently been formulated called Weak Quantum Theory (WQT),⁵⁰ which develops the idea of ‘macro-entanglement’ and uses it to explain not only phenomenological problems arising out of information theory and psychology,⁵⁰ but also the therapeutic process in homeopathy^35-37 and CAMs.⁴¹ The subsequent development of this idea⁵¹ leads to ways of considering macro-entanglement that explains significant correlations found between the brain functions of spatially separated pairs of human subjects.^{52, 53}

In this context, Walach^{54, 55} and others⁵⁶ have demonstrated clear correlations between remedy and placebo groups during double blind homeopathic provings; observations which are difficult to account for other than in terms of macro-entanglement between the two groups.^{36, 57} While purely physical entanglement between macroscopic objects seems unlikely, generalisation of the concept could be used to help explain certain observed correlations in human relationships. Although a sufficient but not a necessary condition for applying entanglement ideas to the problem of how homeopathy might work, it leads to a metaphorical use of concepts with an otherwise unique meaning in physics.^{57, 58} It should be stressed that this use of quantum mathematics has yet to be rigorously proven.

The Vital Force as a quantum entity

Based on the discourse similarity between the Vital Force and a quantum entity’s wave function, i.e., it is only inferable from the symptoms it produces,^23,28 a new metaphor imagines the Vf as a quantised precessing gyroscope.⁵⁹ This means the Vf’s spinning ‘flywheel’ slows discreetly, so that precession about an imaginary vertical axis also changes by fixed amounts. Quantised precession (i.e., a hypothetical ‘angular momentum’ vector Vf rotates in the x-iy plane: figure 1) arises by initially defining ‘Vf shift operators’, similar to the angular momentum shift operators from the physics of

Figure 1. Schematic of the Vf gyroscope: a real gyroscope in 3-D space precesses around the z-axis sweeping out gradually increasing ‘orbits’ in the x-y plane. The metaphorical Vf gyroscope precesses in fixed quantised ‘orbits’ as shown and the y and z axes are complex. Symptoms are observed along the real x axis.

electrons in atoms.⁶⁰ Vf shift operators were derived from complementary combinations³⁰ of  primary and secondary symptoms, experienced by the patient; observed by the practitioner.⁵⁹ Thus, ‘diseases’ and ‘remedies’ are simply factors that increase or decrease the Vf’s strength by quantised amounts.

A Vf  ‘wave function’, ?_Vf, then relates Vf quantised precessional motion solely to observed secondary symptoms, S₂, (i.e., ?_Vf = A(e^ik2^S2 + e^–ik2^S2)),⁵⁹ and used to illustrate some of homeopathy’s empirical ‘laws’, e.g., bi-phasal action of remedies,⁶¹ and resonance effects among remedy provers and associates.⁶² A therapeutic analogue of the time-dependent Schroedinger equation⁶⁰ was derived, and used to investigate the practitioner’s effect on the patient’s Vf during therapeutic PPR entanglement.⁶³ Manipulating the Vf wave function further allowed remedies and diseases to be understood as ‘torques’ that respectively accelerate or brake the speed of rotation of the Vf gyroscope’s ‘flywheel’.⁶⁴ Thus, if gyroscopic precession is a metaphor for Vf symptom expression, then eradication of symptoms by the remedy corresponds to an accelerating ‘remedial’ torque on the Vf ‘flywheel’, slowing precession.

Metaphorical motion in imaginary spaces

Precession of the Vf ‘gyroscope’ occurs in a mathematically complex plane. This means that only the x axis in Figures 1 and 2 is real: the y and z axes are complex.³⁰ The ‘orbits’ represent different Vf states: the larger the precessional trajectory of the vector Vf, the greater the strength of Vf symptom expression. However, as with any quantum entity (e.g., an electron in an atom), transitions between the different states represented by these trajectories may be thought to occur instantaneously not gradually.

real axis along which symptoms are expressed

Figure 2. View of the Vf gyroscope in Figure 1, looking down the imaginary iz axis.

It follows that when the Vf ‘gyroscope’ is not precessing (i.e., its axis of rotation is co-linear with the complex z-axis), it is not producing symptoms, i.e., S₂ = 0. Consequently, the larger the ‘orbit’ of Vf precession, the greater the degree of observed symptom expression along the real x axis (representing our four space-time dimensions are conflated into one) in figures 1 and 2. Thus, by precessing in a 3-D complex space defined by one real and two complex axes, the Vf may be imagined as a 6-D (i.e., four real plus two complex dimensions) rotating entity, precessing when disturbed, and producing observable effects in our reality experienced as symptoms. In future better mathematical treatments of these ideas might be realised using the multi-dimensional algebras of spinors⁶⁵ and Lie groups.⁶⁶

The effect of the similar homeopathic remedy

In previous papers,^{59, 63, 64} wave functions were derived for the vital force and the curative homeopathic remedy. These were:-

?_Vf = A(e^ik2^S2 + e^–ik2^S2) = 2Acosk₂S₂, and

?_Rx = e^–ik2^?S2,

where i = ?-1, S₂ is the totality of secondary symptoms, and ?S₂ is the change in secondary symptoms wrought by the curative remedy. The constant A relates to the intensity of the wave form, which in turn relates to the strength of symptom expression by the Vf, and so to the sizes of precessional ‘orbits’ shown in figures 1 and 2. The constant k₂ relates to the energy and ‘moment of inertia’ of the Vf ‘gyroscope’, and expresses the ‘frequency’ of the Vf wave function ?_Vf. But this constant also appears in the wave function of the curative remedy, ?_Rx and relates to the remedy’s ability to bring about this change in secondary symptom expression, by increasing the energy and ‘moment of inertia’ of the Vf.

The previous paper⁶⁴ also made the point about the ‘torque-like’ action on the Vf of remedies and diseases; the remedy as an accelerating ‘torque’, and the disease as a braking ‘torque’. Thus, disease can be represented as the remedy’s mirror image, or mathematically, its complex conjugate,³⁰ i.e.

?_Dx =  ?_Rx^* = e^ik2^?S2

Consequently, just as k₂ relates to both the potency of the homeopathic remedy and its power to cure by providing an accelerating torque, it also expresses the power of the disease over the Vf as a braking torque. Therefore, the curative remedy at the correct potency exactly cancels out the disease, i.e.,

?_Dx X?_Rx = e^ik2^?S2 X e^–ik2^?S2 =  e⁰ =  1.

In addition, this paper⁶⁴ proposed that the ‘torque-like’ therapeutic effect of the curative remedy could be understood as the rate of change of the patient’s vital force with symptoms, derived by differentiating the Vf wave function with respect to S₂, i.e.,

if ?_Vf = A(e^ik2^S2 + e^–ik2^S2) = A(2cosk₂S₂), then d?_Vf/dS₂ = T = -2Ak₂sink₂S₂

This led to some rudimentary insight into the torque-like action of the remedy was gained by graphically representing, ?_Vf (black curve),  d?_Vf/dS₂ (white curve), and their sum ?_Vf + d?_Vf/dS₂, generating a series of diagrams representing how ?_Vf and d?_Vf/dS₂ interact to produce ?_Vf + d?_Vf/dS₂ for different k₂ values, three of which are shown below.

a                                                          b                                                          c

Figure 3. Plots of the Vital Force wave function, ?_Vf = 2Acosk₂S₂ (black curve), the remedial torque,
d?_Vf/dS₂ = -2Ak₂sink₂S₂ (white curve), and the combination, ?_Vf + d?_Vf/dS₂ = 2Acosk₂S₂ – 2Ak₂sink₂S₂ (grey curve), where A = 1, and (a), k₂ = 0.5; (b), k₂ = 1; and (c), k₂ = 2. Note how as k₂ increases, so do the frequencies of the wave forms, and the grey curve peak amplitude and position with respect to the black and white curves: at low k₂ the grey curve is like the black curve, while at high k₂ the grey curve is more like the white curve (see reference 64).

This analysis considers only the initial effect of the torque on Vf: it does not take into account how long the torque is applied, and what happens after its action ceases.

It is interesting to re-examine the wave function for the curative remedy which is given as:-^{63, 64}

?_Rx =  e^–ik2^?S2                              1

Using Euler’s transformation,³⁰ equation 1 may be rewritten in polar form as:-

?_Rx =  cos k₂?S₂ – isin k₂?S₂ 2

This shows that ?_Rx consists of real and imaginary parts. So it should be possible to contemplate remedy action in more than just physical terms, with the imaginary part of ?_Rx representing the remedy’s less physically tangible effects, e.g., perhaps on mental and emotional levels. Now, as k₂ tends to 0 (i.e., at very low potency and the remedy becomes more material), the imaginary term,
– isin k₂?S₂ disappears, while the real cos k₂?S₂ term tends to a maximum. So, as k₂ approaches zero, ?_Rx » cos k₂?S₂ and the remedy may be considered to act physically. At high k₂, the real term disappears and the imaginary term increases to a maximum so that ?_Rx = –isin k₂?S₂. The remedy is now acting mainly non-physically (which does NOT mean it exerts no physical effect). This wave function depiction of the curative remedy, allows its action on the whole person to be followed.

TOWARDS UNIFYING HOMEOPATHY AND CONVENTIONAL MEDICINE

The periodicity of wave functions

Depicting the Vf, remedy, and remedy torque as wave functions, implies a certain periodicity in their properties and behaviour. Such periodicity can indeed be exhibited by the Vital Force in its expression of symptoms; their strength varying according to periodic modalities, such as the time of day, etc.
Interestingly, there is some experimental evidence that biologically active substances at different serially diluted potencies will have periodic effects on their substrates. For example, Schiff reports alternating effects of different ultra-diluted potencies of anti-immunoglobuline E (aIgE) on the decoloursation of stained basophils, in most cases when the aIgE had been diluted and agitated beyond molecular existence:⁶⁷ some potencies enhance decolourisation while others retard it. Schiff claims this as evidence for the water memory effect.

Accordingly, k₂ values determine the periodicity of the Vf and remedy wave functions. As this constant is also associated with remedy potency (inversely proportional to material dose), it is interesting to investigate the possible effects of very low remedy k₂ values on the periodicity of ?_Rx.

From the previous section, we know that at low k₂ values, the remedy wave function approximates to ?_Rx » cos k₂?S₂. Now, the cosine function can be represented algebraically as the power series:-⁶⁸

cos x = 1 – x²/2! + x⁴/4! – x⁶/6! +….(-1)ⁿx²ⁿ/(2n)!                             3

As x becomes very small, the values of the higher powers of x effectively vanish. Therefore, for small x, equation 3 approximates to :-

cos x » 1 – x²/2!                           4

Similarly, for small values of k₂, the remedy wave function, ?_Rx approximates to:-

?_Rx » 1 – [k₂?S₂]²/2!                 5

Equation 5 is no longer a periodic: it is now a parabolic function (see steep and shallow white curves in figure 4: the steepness is determined by the value of k₂).

Figure 4. Plots of the real part of equation 2 (?_Rx = cos k₂?S₂: k₂ = 1; black curve: k₂ = 0.075; shallow grey curve), and equation 5 (?_Rx » 1 – [k₂?S₂]²/2!: k₂ = 1; steep white curve: k₂ = 0.075; shallow white curve).

Thus, for small values of k₂, the shallow grey (equation 2) and white (equation 5) curves in figure 4 are virtually indistinguishable. Only as k₂ grows does the difference between the grey and white curves become apparent. This theoretical finding may help illustrate essential differences between the homeopathic and the conventional medical approaches to the action of remedies.

The effect of remedy potency

Homeopathic philosophy maintains that a remedy becomes more potent the more it is diluted and succussed.⁶⁹ This means that even at only moderate potencies (greater than 12c and 24x), homeopathic remedies possess little or no molecules of the original material substance, yet a remedy of higher potency should have a more profound effect on the Vf than one of lower potency.

The conventional medical approach to the action of remedies predicts otherwise: the less of the remedy’s material substance is actually present, the weaker its effect in curing the disease (in conventional medicine, the idea of an all-pervading Vital Force is considered moribund: its nearest equivalent is the homeostatic immune system). This could be represented in figure 4 by the grey curve, i.e., equation 5. Its parabolic shape means it crosses the x axis only once, after which it never returns, regardless of the value of k₂. However, its response to a remedy at low k₂ (or potency, in this case in a material dose) approximates to the extended white curve in figure 4. Thus, the form of the remedy function ?_Rx shown in equation 5 could be said to represent the predictions of conventional medicine concerning the effect of highly diluted remedies, i.e., that the weaker the drug dose in material terms, the less effect the drug has on the homeostatic immune system. However, the form of the remedy wave function ?_Rx in equations 1 and 2 represents the predictions of homeopathy, i.e., ultra-highly diluted remedies can still affect the Vital Force.

There is the possibility that just as equation 5 is an approximation to equation 2, so the predictions of conventional medicine are not so much in contradiction to homeopathy but might represent an approximation of much wider ranging implications concerning the effects of remedies at different dilutions, as predicted by homeopathy. If so, it might be possible to construct from basic principles a mathematical metaphor of conventional medicine’s homeostatic immune system that delivers similar predictions as the approximated form of the remedy wave function ?_Rx represented by equation 5. Again, this comes with the proviso that there are as yet no rigorous mathematical proofs or lemmas for the use of such inherently physical ideas in these essentially biomedical/therapeutic contexts.

A mathematical metaphor for the homeostatic immune system

Conventional medicine has no concept of an all-pervading Vital Force. In homeopathy, the goal of treatment is to help the Vital Force to throw off the disease (e.g., as in the Vf gyroscopic metaphor). In contrast, conventional medicine recognises a homeostatic immune system but its drug regimes are less concerned with supporting it and more concerned with alleviating symptoms. In homeopathy, this is believed to be the cause of conventional drug side-effects, as the Vital Force reacts against the drug’s initial suppression of symptoms.⁶¹

This line represents n, the totality of symptoms expressed by Is: those to the right are disease symptoms, while those to the left are remedy ‘side-effects’.

Figure 5. Force diagram metaphor for the homeostatic immune system Is disturbed by disease and remedy.

This situation in conventional medicine is represented simplistically by figure 5. Here, a disease ‘vector’ (Dx) acts on a homeostatic immune system ‘vector’ (Is) by ‘deflecting’ it through angle ?. A drug ‘vestor’ is then prescribed (Rx) to alleviate symptoms, correcting the original angle of deflection, only to produce side-effects, and a deflection in the opposite direction by angle ?. might be simply represented in terms of a mathematical metaphor based on a force diagram (figure 5). The box below gives a simplified mathematical derivation of the action of a homeostatic immune system under these circumstances.

We see a close mathematical correspondence between equations 5 and 9 if ? (in the above box) bears a similar relationship in conventional medicine to the expression of drug side-effects by the homeostatic immune system, as k₂S₂ and k₂?S₂ in homeopathy have to the expression of and changes in secondary symptoms by the Vital Force and the curative remedy respectively. Thus, the mathematical metaphors used to represent the Vf and the homeostatic immune system approximate for very low drug potency (i.e., for values of k₂ << 1 when a remedy/drug exists in material molecular form), suggesting that homeopathy and conventionalmedicine should predict similar outcomes for the effects of remedies/drugs given in material doses. Only when the remedy is becoming ultra-diluted, and according to these metaphors do the predictions of conventional medicine and homeopathy diverge.

Conclusions

The aim of this paper has been to show that a perceived similarity in discourse between homeopathy’s concept of a Vital Force and quantum theory’s notion of a wave function, could be the starting point for a much needed rapprochement between conventional medicine and homeopathy.

As others have pointed out,^{72, 73} however, such correspondences are not yet grounded in rigorous mathematical proofs or lemmas. Particularly, more detailed treatments are needed to flesh out the meaning of wave functions and hypothetical spaces used in therapeutic contexts, so that macro-entanglement of patient, practitioner and remedy can indeed be treated as if it were a generalised quantum system. Such elaboration will hopefully appear in later papers. Meanwhile, further development of a simple mathematical metaphor of the Vital Force as a quantised gyroscope, provides insights into the action of the curative homeopathic remedy over a large potency range.

This metaphor depicts the Vital Force and the curative remedy as periodic wave functions. It turns out that at low potency (i.e., the remedy is at a physical material dose), they approximate in such a way as to lose their periodicity and therefore deliver predictions concerning the effects of highly diluted remedies that are in line with those of conventional medicine, i.e., they should have little or no effect.

In order to expand on this conclusion, another simple metaphor was proposed, in which the conventional medical notion of a homeostatic immune system was treated as a simple force vector, Is, deflected by disease and remedy vectors. Thus, disturbance one way elicits symptoms of disease, while the disturbance the other way elicits symptoms of the remedy, i.e., ‘side-effects’. The mathematical treatment of this simple metaphor generates a solution whose interpretation is conventional medicine’s conclusion concerning the lack of efficacy of highly potentised substances. This, however, is the same conclusion drawn from the Vital Force gyroscopic metaphor when it is approximated.

Parallels could be drawn here with the relationship between Newtonian and Einsteinian mechanics: the latter delivers the former when it is suitably approximated to velocities much smaller than light.⁴⁴ Thus Newton does not contradict Einstein: rather Newtonian mechanics is an approximation of Einsteinian mechanics, applicable at low velocities only. Similarly, it could be that there is no contradiction between conventional medicine and homeopathy: rather conventional medicine may perhaps be better understood as an approximation of homeopathy when remedies are given at low potencies in material doses. Thus, although only a preliminary metaphor based on several admittedly unproven assumptions, more sophisticated mathematical treatments than the ones presented here might in the future provide a platform for the possible unification of conventional medicine with homeopathy. Given the increasing scepticism being directed at homeopathy, it would indeed prove ironic if conventional medicine turned out to be a subset of a much broader paradigm that included homeopathy. “Graecia capta Romam victricem captum subducit” – “Captured Greece leads conquering Rome captive.”⁷⁴ One should always dream….

ACKNOWLEDGEMENTS

The author wishes to express his deep gratitude to Ms Barbara A Drillsma-Milgrom for all her help, encouragement, patience, forbearance and inspiration in the preparation of this manuscript.

†This is part 10 in a series of papers entitled Patient-Practitioner-Remedy (PPR) entanglement.

‡Address: Department of Chemistry, Imperial College London, South Kensington, SW7 2AZ, UK; e-mail: [email protected]

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FIGURE LEGENDS

Figure 1. Schematic of the Vf gyroscope: a real gyroscope in 3-D space precesses around the z-axis sweeping out gradually increasing ‘orbits’ in the x-y plane. The metaphorical Vf gyroscope precesses in fixed quantised ‘orbits’ as shown and the y and z axes are complex. Symptoms are observed along the real x axis.

Figure 2. View of the Vf gyroscope in figure 1, looking down the imaginary iz axis.

Figure 3. Plots of the Vital Force wave function, ?_Vf = 2Acosk₂S₂ (black curve), the remedial torque, d?_Vf/dS₂ = -2Ak₂sink₂S₂ (white curve), and the combination, ?_Vf + d?_Vf/dS₂ = 2Acosk₂S₂ – 2Ak₂sink₂S₂ (grey curve), where A = 1, and (a), k₂ = 0.5; (b), k₂ = 1; and (c), k₂ = 2. Note how as k₂ increases, so do the frequencies of the wave forms, and the grey curve peak amplitude and position with respect to the black and white curves: at low k₂ the grey curve is like the black curve, while at high k₂ the grey curve is more like the white curve. (reference 64).

Figure 4. Plots of the real part of equation 2 (?_Rx = cos k₂?S₂: k₂ = 1; black curve: k₂ = 0.075; shallow grey curve), and equation 5 (?_Rx » 1 – [k₂?S₂]²/2!: k₂ = 1; steep white curve: k₂ = 0.075; shallow white curve).

Figure 5. Force diagram showing the homeostatic immune system Is disturbed by disease and remedy.

Figure 6. Plots of the real part of equation 2 for k₂ = 1 (?_Rx = cos k₂?S₂; black curve), equations 5 and 9 (k₂?₂ » ?: k₂ = 1; steep white curve: k₂ = 0.005; shallow white curve), and equation 7 (k₂?₂ » ?: k₂ = 1; steep grey curve flattening to 0: k₂ = 0.005; shallow grey curve). Note how compared to equation 2, equation 7 goes to 0