Multiple sclerosis: Exploring the limits and implications of genetic and environmental susceptibility

Objective To explore and describe the basis and implications of genetic and environmental susceptibility to multiple sclerosis (MS) using the Canadian population-based data. Background Certain parameters of MS-epidemiology are directly observable (e.g., the recurrence-risk of MS in siblings and twins, the proportion of women among MS patients, the population-prevalence of MS, and the time-dependent changes in the sex-ratio). By contrast, other parameters can only be inferred from the observed parameters (e.g., the proportion of the population that is “genetically susceptible”, the proportion of women among susceptible individuals, the probability that a susceptible individual will experience an environment “sufficient” to cause MS, and if they do, the probability that they will develop the disease). Design/methods The “genetically susceptible” subset (G) of the population (Z) is defined to include everyone with any non-zero life-time chance of developing MS under some environmental conditions. The value for each observed and non-observed epidemiological parameter is assigned a “plausible” range. Using both a Cross-sectional Model and a Longitudinal Model, together with established parameter relationships, we explore, iteratively, trillions of potential parameter combinations and determine those combinations (i.e., solutions) that fall within the acceptable range for both the observed and non-observed parameters. Results Both Models and all analyses intersect and converge to demonstrate that probability of genetic-susceptibitly, P(G), is limited to only a fraction of the population {i.e., P(G) ≤ 0.52)} and an even smaller fraction of women {i.e., P(G│F) < 0.32)}. Consequently, most individuals (particularly women) have no chance whatsoever of developing MS, regardless of their environmental exposure. However, for any susceptible individual to develop MS, requires that they also experience a “sufficient” environment. We use the Canadian data to derive, separately, the exponential response-curves for men and women that relate the increasing likelihood of developing MS to an increasing probability that a susceptible individual experiences an environment “sufficient” to cause MS. As the probability of a “sufficient” exposure increases, we define, separately, the limiting probability of developing MS in men (c) and women (d). These Canadian data strongly suggest that: (c < d ≤ 1). If so, this observation establishes both that there must be a “truly” random factor involved in MS pathogenesis and that it is this difference, rather than any difference in genetic or environmental factors, which primarily accounts for the penetrance difference between women and men. Conclusions The development of MS (in an individual) requires both that they have an appropriate genotype (which is uncommon in the population) and that they have an environmental exposure “sufficient” to cause MS given their genotype. Nevertheless, the two principal findings of this study are that: P(G) ≤ 0.52)} and: (c < d ≤ 1). Threfore, even when the necessary genetic and environmental factors, “sufficient” for MS pathogenesis, co-occur for an individual, they still may or may not develop MS. Consequently, disease pathogenesis, even in this circumstance, seems to involve an important element of chance. Moreover, the conclusion that the macroscopic process of disease development for MS includes a “truly” random element, if replicated (either for MS or for other complex diseases), provides empiric evidence that our universe is non-deterministic.


Adjusting the MZ-twin Concordance for the Shared
subset within ( ).
For susceptible probands who are members of ( ), but who are not otherwise identified, the analogous probabilities can be written: Moreover, we note that the conditioning events ( , ) and ( , , ) both represent the same underlying event for the probandi.e., the event that the i th susceptible individual (the MZ-proband) experiences an environment "sufficient" to cause MS in them. In this circumstance, therefore: Probands both from an MZ-twinship and those from a DZ-twinship experience the same ( ) environment as their co-twin who either has or will develop MS. Therefore, we assume that: In which case, the covariance ( ), using its standard definition [39], can be expressed such that: We define the term ( ) in an analogous manner to ( )see above. Moreover, because everyone who develops MS must experience the event ( ), because both MZ-and DZ-twins disproportionately share the ( ) and ( ) environments with their co-twin, and because siblings (including twins) disproportionately share their ( ) environment with their co-sibling(s), and because everyone equally shares their ( ) environment, therefore, with respect to the environmental experiences of the proband, it will be the case that: ( │ ) = ( │ ) = ( │ , ) = ( │ , ) Moreover, from the definition of ( ) and because the ( ) environment doesn't seem to contribute to the event ( , ) and because genotype is independent of ( ), then, in this circumstance: where:  Thus, the point estimate for the impact of the shared ( ) environment on the likelihood that a proband MZ-twin has or will develop MS, given the fact that their co-twin also has or will develop MS, is essentially identical for both women and men.
Proof: During any Time Period, we consider the two ratios, (A) and (B), such that: Therefore, during any Time Period, the following two relationships must hold simultaneously:
Proof: Following the same logic as that, which is used for the demonstration of Assertion 2a (above), this assertion has been demonstrated previously [3].

2c. Current Enrichment of Women Compared to Men
Assumption: 2 = ( │ , ) 2 > ( │ , ) 2 = 2 Argument: Currently, the evidence strongly favors of an arrangement (above) in which: ( > 1) or, equivalently, in which: ( 2 > 2 ). First, the (F:M) sex ratio has increased by a similar amount (i.e., at a similar rate) between every two Time Periods in the Canadian data except one [6]. Thus, the observed increase in MS prevalence over time has disproportionately impacted the prevalence in women [6,[22][23][24][25][26][27][28][29][30]. Indeed, from Equation 1b (Main Text), during the "current" Time Period, it must be that: The value of ( ) is unknown but fixed, regardless of Time Period, and the values of ( 2 ) and ( 2 ) are also unknown but fixed during the current Time Period (see Section 6a, below). Considering

Equation S1a
, an increasing (F:M) sex ratio with increasing exposure can only be explained by the penetrance in susceptible women ( ) increasing at a faster rate compared to the rate in susceptible men ( ). If the penetrance of MS in susceptible women was changing at the same rate as in susceptible men (whether increasing, decreasing, or remaining constant), the (F:M) sex ratio, would stay constant. In contrast to the fixed value of ( ), the value of ( ′ ) depends upon the Time Period and it is an observed parameter. Thus, the only possible circumstances under which: ( 2 ≥ 2 ), are also those in which the value of: ( ) is greater than (or equal to) the "current" value for ( ′ 2 ). From our parameter estimates (see Methods #2A; Main Text), the acceptable range for the parameter ( ′ 2 ) is such that: Also, because, as discussed in Section 6c (see below), the limiting values for the exponential response curves in susceptible men and women are related such that, with increasing exposure, the ratio ( ⁄ ) will always approach the limit of ( ⁄ ). Thus: reached, it will be the case that: ( = ≤ 1); so that the (F:M) sex ratio will steadily increase until it reaches the limiting value of: However, the observed "current" value of ( ′ 2 = 0.76) in Canada [6] is greater than the proportion of women among MS patients observed during every previous Time Period represented by the Canadian population data [6] and, also, greater than that for every observation made of MS since Charcot's initial clinical description in the 19 th century [3,40,[76][77][78][79]. In such a circumstance, we would have to conclude, therefore, that men, throughout the history of MS, have had a consistently greater penetrance than women but that women are considerably more likely than men to be genetically susceptible. Such a conclusion is counter intuitive. In addition, there is an inherent tension between the conclusion that the penetrance of MS is increasing faster in women than men (which suggests that: > 1) and the conclusion that ( > ) throughout the history of MS (which suggests that: < 1)see Section 6g; below.
Second, although the current relative penetrance of MS for the sets ( , ) and ( , ), is not an observed parameter, nevertheless, the penetrance values for the sets ( , , ) and: ( , , ) are directly observable and, currently [6], this penetrance is (5.7)-fold higher for women than it is for men.
This difference in penetrance between genders is highly significant such that: 2 = 10.5 ; = 0.001 As a consequence of this penetrance difference, and considering MZ-twins who either are of unknown concordance or are known to be concordanti.e., who, respectively, are members of either the ( ) subset or the ( , ) sub-subsetthis study [7] indicates that women are being continuously enriched such that: ( ) = 0.5 < ( │ ) = 0.685 < ( │ , ) = 0.92 Because any proband MZ-twin is genetically "identical" to their ( ) co-twin, the probands of such co-twins are already "enriched" for more penetrant genotypes compared to the general population (see above). Consequently, extrapolating from the fact that this "enriched" group of women has a considerably greater penetrance than similarly "enriched", strongly suggests that, currently, the relative penetrance is such that susceptible women are more penetrant than susceptible men (see Sections 2a-b; above). Nevertheless, there are two circumstances, which might, potentially, make any such extrapolation inaccurate. First, MZ-twins share their ( ) and ( ) environments, which might cause an increase in the penetrance relative to the same genotypes under non-twin conditions [4]. If the impact of these environments were markedly disproportionate for the different two genders this could change the relative penetrance. However, this concern is probably unwarranted because, observationally, the environmental impact of sharing these environments is the same for both men and women (see Section 1d; above) and, thus, this circumstance won't impact the relative penetrance. Second, as noted above, there will be an enrichment of more penetrant genotypes that occurs in the probands of ( ) co-twins compared to non-twin conditions. If this enrichment were markedly disproportionate between the genders, then, potentially, this could also impact the relative penetrance. However, for this to explain the large observed discrepancy in the MZ-twin concordance rates for men and women (see above), would require an extreme difference in the variance of individual penetrance values between the ( , ) subset and the ( , ) subset [3,4]i.e., ( 1 2 ≫ 2 2 ); or, equivalently: ( ≫ )see Section 3a; Equations S2f-i & S3a-d (below). This seems improbable. Therefore, without making any assertions regarding the circumstances that pertain during earlier Time Periods, we assume that, currently, the penetrance in women is greater than the penetrance in men so that:

Assumption #2
The penetrance of MS for a proband MZ-twin, whose co-twin is of unknown status, is assumed to be the same as if that genotype had occurred without having an MZ co-twin (i.e., the penetrance of MS for each genotype is independent of MZ-status). This assumption translates to assuming that the impact of experiencing any particular ( ) and ( ) environments together with an MZ co-twin is the same as the impact of experiencing the same ( ) and ( ) environments alone. Alternatively, it translates to the testable hypothesis that the mere fact of having an MZ co-twin does not alter the ( ) and ( ) environments in such a way that MS becomes more or less likely in both twins. Thus, we are here assuming that, for any Time Period:

Proof of Assertion A:
From Assumption #1, it follows that: Textwe conclude that, for any Time Period: Therefore, also: and similarly:

Proof of Assertion B:
From the definition of ( )see Section 1a (above); see also Methods #1D; Main Textand from Assertion A, it directly follows that, for any Time Period:

Proof of Assertion C:
From the definitions of ( ) & ( )see Section 1a (

However, from Equation S2c
, it is the case that: , it follows that: can be rearranged to yield a quadratic equation in (x) such that: This quadratic can be solved to yield: Defining the variance of penetrance values within the subsets of susceptible women and men as ( 1 2 ) and ( 2 2 ), respectively, the same line of argument also leads to the conclusions that: and:

Proof of Assertion D:
Equation S2e has real solutions only for the range of: and, also, from Equations S2f-g: Notably, the maximum variance (

Also, Equation S2d
can be re-arranged to yield: and, similarly: and:

3b. Assertion C Solutions
Equation S2e has two solutionsthe so-called Upper Solution and the Lower Solution, depending upon the value of the (±) sign. The Upper Solution represents the gradual transition from a distribution, when ( 2 = 0), in which everyone has a penetrance of ( ′ ) to a bimodal distribution, when { 2 = ( ′ 2 ⁄ ) 2 }, in which half of the ( ) subset has a penetrance of ( ′ ) and the other half has a penetrance of zero. Although, under some environmental conditions: (∀ ∈ : > 0), as noted previously (see Methods #1A, Main Text), there may be certain environmental conditions, in which, for some individuals in the ( ) subset: Therefore, the Upper Solution, during any particular Time Period, is constrained such that: The Lower Solution represents the gradual transition from the bimodal distribution described above to increasingly extreme and asymmetric distributions [3]. The Lower Solution, however, is further constrained by the requirement of Equation S2d that when: ( 2 = 0) then: ( = ′ ). Therefore, the Lower Solution is constrained such that:

3c. Quadratic Solutions for Women and Men
Moreover, regardless of whether the Upper or Lower Solution pertains, the values that ( 1 ) and ( 2 ) can take are further constrained.
Combining these two estimates {to eliminate the ( 2 ) parameter} yields: and, finally: Rearrangement, yields a quadratic equation in ( 1 ) such that: Because of the definition that: ( 1 > 2 )see Section 3a, abovethis is solved for ( 1 ) as: (above) can then be solved for ( 2 ). Alternatively, reframing the above argument to eliminate the ( 1 ) parameter instead of ( 2 ), yields: We further define ( ) to be the cumulative hazard function (for men) at an exposure-level ( = ) such that: Similarly, we define ( ) to be the cumulative hazard function (for women) at the same exposure-level of ( = ) such that: And, if the hazards are proportional, then: For men, following the usual definition of the hazard function [39] that: together with the fact that, by definition: Consequently, the cumulative survival function is exponentially related to the integral of the underlying hazard functioni.e., the cumulative hazard function. Even in the unlikely circumstance that the hazard function is discontinuous at some points, the function will still be integrable in all realistic scenarios. In this circumstance, the failure function for susceptible men becomes: In this case, the failure probability for susceptible men during the 1 st Time Period ( 1 ), can be expressed as: If the exposure level for susceptible men during the 2 nd Time Period is { ( 2 )}, then, because ( ) is currently increasing with time, the difference in exposure for men between the 1 st and 2 nd Time Periods can be represented as the difference in the environmental exposure level between these two Time Periods ( ) such that: In this circumstance, the failure probability for susceptible men during the 2 nd Time Period ( 2 ), can be expressed as: and: And dividing the 1 st of these two rearranged Equations by the 2 nd yields: Previously, we assigned the value of these arbitrary units as ( = 1) in these Equations [3], although such an assignation may be inappropriate. Thus, this unit (whatever it is) still depends upon the actual (but unknown) level of environmental change that has taken place between the two chosen Time Periods. From Equations S6d-e, this level depends upon the value of ( ), which can range over the interval of: (1 ≥ > 2 )see Methods #2D (Main Text). Because ( ) increases with increasing exposure, the ratio on the LHS of Equation S6d (above) is always greater than unity and it increases monotonically as ( ) varies throughout its range. This ratio is at its minimum when: ( = 1) and approaches infinity as: ( →  2 ).
Therefore, we can define ( ) as the "minimum" possible exposure level change for men between these two Time Periods. In this case, this minimum exposure level change will occur when: However, this minimum exposure level change ( ) may not accurately characterize the actual (but unknown) level of environmental change, which has taken place for susceptible men between the two Time Periods. Therefore, we will refer to ( ) as the "actual" exposure-level change, which may be different from this minimum exposure-level change such that: and: where { ( 1 )} represents the exposure level in women at the 1 st Time Period and ( ) represents the "actual" level of environmental change for women, which has taken place between the two Time Periods such that: In a manner directly analogous to that presented above for the development of Equation S6e, it is also the case that: Thus, as above for susceptible men, the "minimum" value ( ) for the exposure level change in susceptible women will occur under those circumstances for which: ( = 1), so that: and: ≥

4c. Relationships for the Susceptible Population as a Whole
In an analogous manner, we can define the failure, and hazard Because genotype doesn't depend upon the environment, during any Time Period, therefore: in which case the statement above can be re-expressed as: Thus, the failure rates for the entire susceptible population ( ) can be expressed as a linear combination of the failure rates in women ( ) and men ( ).

4d. Relationship of Failure to True Survival
However Also, in true survival, both the clock and the risk of death begin immediately at time-zero and continue indefinitely into the future, so that the cumulative probability of death always increases with time. By contrast, here, it may be that the prevailing environmental conditions during some Time Period ( ) are such that: ( │ , ) = 0 ; even for quite an extended period (e.g., centuries or millennia). In addition, unlike the cumulative probability of death, here, exposure can vary in any direction with time depending upon the specific environmental conditions during ( ). Therefore, although the cumulative probability of failure (i.e., developing MS) increases monotonically with increasing exposure, it can increase, decrease, or stay constant with time.

4e. Relationship of the (F:M) Sex Ratio to Exposure
Finally, regardless of ( ), and regardless of any proportionality, during any Time Period, the failure probability for susceptible women ( ) can be expressed as: ) or: = ( │ , , ) * and, similarly, the failure probability for susceptible men ( ) can be expressed as: Dividing the 2 nd of these two Equations by the 1 st , during any Time Period, yields: Consequently, any observed disparity between ( ) and ( ), during any Time Period, must be due to a difference between men and women in the likelihood of their experiencing a "sufficient" environmental exposure, to a difference between ( ) and ( ), or to a difference in both.
Therefore, by assuming that: ( = ≤ 1), we are also assuming that any difference in disease expression between susceptible women and men is due entirely to a difference between susceptible men and women in the probability of their experiencing a "sufficient" environmental exposure, despite the fact that, for every ( ), the exposure { } is both population-wide and fixed during any Time Period ( ). Because this exposure is "available" to everyone, therefore, if the level of "sufficient" exposure differs between genders, one possibility might be that this is due to a systematic difference in behavior between susceptible women and meni.e., to an increased exposure to, or avoidance of, susceptible environments by one or the other gender (perhaps consciously or unconsciously; or perhaps due to differing gender-roles, differing occupations, differing recreational activities, etc.). However, the fact that most women behave differently from men does not mean that all women do so. Notably, if the circumstance of ( ≠ 0) were explained by a systematic difference in behavior, then the observation of ( > 0) suggests that the behavior of men leads to a greater exposure than the behavior of women. However, any general conclusion regarding such a difference in behavior between susceptible women and men cannot be rationalized with the observation that, currently: ( 2 > 2 ).see Section 6g; below.
Another possible explanation for ( > 0), which does not pose this difficulty, is that the distributions of the so-called "critical exposure intensity" levels ("thresholds") differ between men and women (see Section 6g; below). In this case, although the same exposure "intensity" may be experienced equally by the two genders, this "intensity" might be "sufficient" for a disproportionate number of women or men. This possibility is considered subsequently (see Sections 6g & 8a-b; below).
Also, regardless of whether the hazards are proportional, and because proportion of women among susceptible individuals ( ) is a constant (see Table 2; Main Text), therefore, for any solution, the ratio ( ⁄ ), during any Time Period ( ), will be proportional to the observed (F:M) sex ratio during that period (see Equation S1a).

4f. Response Curves to Increasing Exposure
Notably, also, because the response curves for both men and women are exponential, any two points of observation on these curves will define the entire curve (e.g., the values of and during Time Period #1 and Time Period #2see Equations S6a & S6b and S7a & S7b; above). Moreover, if these two curves can be plotted on the same x-axis (i.e., if men and women are responding to the same environmental events), the hazards will always be proportional where the values of ( = ⁄ ) and ( ) are determined from Equations S11c-d (Section 6a, below).

Non-proportional Hazard Models 5a. General Considerations
If the hazard functions for MS in men and women are not proportional, it is always possible that the "actual' exposure level changes for men and women are each at their "minimum" valuesi.e., ( ) and ( ). However, this is a circumstance, which is true if, and only if: ( = = 1).
Also, in the circumstance of non-proportionality, the various observed and non-observed epidemiological parameter values still limit possible solutions. However, although ( ≤ 1) and ( ≤ 1) will still be constants, respectively, for men and women, no information can be learned about them or about their relationship to each other from changes in the (F:M) sex ratio and ( ) over time. The observed changes in these parameter values over time could all simply be due to the different environmental circumstances of different times and different places. In this case, also, although men and women will still each have environmental thresholds, the parameter ( )which relates these thresholds to each otheris meaningless, and there is no hazard proportionality factor ( ).
Nevertheless, even with non-proportional hazard, the ratio ( ) ⁄ , during any Time Period, must still be proportional to the observed (F:M) sex ratio during that Time Period (see Equation S1a ; above) and, if: = ≤ 1, then any observed disparity between ( ) and ( ), must be due entirely to a difference between women and men in the likelihood of their experiencing a "sufficient" environmental exposure ( ) during that Time Period (see Equation S8; above).

Proportional Hazard Models 6a. General Considerations
By contrast, if the hazards for women and men are proportional with the proportionality factor ( ), the situation is altered. First, because ( > 0), those changes, which take place for the subsets ( , ) and Rearranging Equations S6a & S10a for any Time Period yields: and: After dividing Equation S11a by S11b, we can rearrange this result, to yield: We apply Equation S11c to exposure levels ( 1 ) and ( 2 ) and subtract the 2 nd of the resulting two Equations from the 1 st . Then, using a combination of Equations S6e & S7c, together with the definitions of ( ) and ( ) from Section 4b (above), we can rearrange this result to yield: Moreover, for those circumstances in which ( = 1), Equation S11c becomes: are fixed (but unknown) constants, so that, from Equations S11a & S11b, the values of ( ) and ( ) are also fixed at any specific exposure level { ( )}.

6b. Defining an "Apparent" Proportionality Factor
We can also define a so-called "apparent" value of the hazard proportionality factor ( ) such that: = ( ( ⁄ ), which represents the value ( ) when: ( = = 1)see Section 5a; above. This value incorporates, potentially, two fundamentally different processes. First, it may capture the increased level of "sufficient" exposure experienced by one group compared to the other. Indeed, from Equation S8, this is the only interpretation possible for circumstances where: ( = ≤ 1). Second, however, if we admit the possibility that: ( < ≤ 1), then some of ( ) will be accounted for by the difference of ( ) from unity.
For example, when ( = 1), and using a proportionate hazard Model (see Section 4b; above & Section 7a; below), the "actual" exposure level change in men ( ) has the limits: From this, we can define the "actual" hazard proportionality factor ( ), at ( = 1), such that:

≥ = ⁄
In this manner, if ( > ), a portion of the "apparent" value ( ) will be accounted for by a reduction in value of ( ) from unity, if such a reduction is possible. Moreover, if such a reduction is possible for susceptible men, then, clearly, it is also possible that the value of ( ) is also reduced from unity in susceptible women. For example, in circumstances where: ( < < 1), the "actual" exposure level in women ( ) would be greater than its minimum value ( ) such that: Consequently, in each of these circumstances, the "actual" value of ( ) may be different from its "apparent" value ( ). This does not imply that these response curves are "actually" identical. Rather, the fact that the curves for those conditions under which: ( = = < 1) can be scaled to be identical to those depicted for: ( = = 1), only indicates that the relationship between the response curve for men and that of women is the same regardless of the value of ( )i.e., any changes in the ( / ) ratio with increasing exposure or, equivalently, any changes of the F:M sex ratio (see Equations S1a & S8; above), will be the same for any value of ( ). When ( Table 2; Main Textand note that, because both ( ) and the (F:M) sex ratio are both increasing with time [6,22-30], therefore:

{NB: Considering Equations
Notably, also, from Equation S1a, during any Time Period: where ( ) is independent of the environmental conditions of any Time Period. Therefore, for all solutions, the ratio ( ⁄ ) will mirror (F:M) sex ratio (i.e., the changes in both ratios will have the same directionality).

= ≥
When: ( = 0), from Equation S11e (above): Therefore, in this circumstance, the F:M sex ratio will remain constant, regardless of the exposure level.
However, from the Summary Equations presented in Section 7a (below): This relationship indicates that: / > / So that, from Equation S11e: > 0 Thus, if ( = 1), and if both the F:M sex ratio and ( ) are increasing, it must be the case that the threshold in susceptible women is greater than that in susceptible men. In turn, Equation S12b, in this circumstance, requires that:
Therefore, again, because the conditions of: ( 2 > 2 ) and an increasing F:M sex ratio only occur together after the intersection, the posited conditions are not possible. However, it is still possible that: ( < ≤ 1)e.g., Figure 2D (Main Text).

For those Conditions, in which: ( > 1):
The proportionality constant ( ) relates to how quickly the response curves for men and women go from zero to their maximums. As such, the value of ( ) is independent of ( ) but, rather, depends only upon how quickly this transition occurs. Therefore, we are free to choose, for comparison, the response curves at any value of ( ). In this case, when ( = ) & ( = 0), Equations S6a & S10a, for any Time Period, can be multiplied by the scaling factor of: (1⁄ ), and then restated such that: when ( = ), is independent of scale (see Note; Section 6b; above). Therefore, in the circumstance where: ( = ) the value of ( ) is constant for all: ( 2 < ≤ 1). Consequently: However, if conditions are such that: ( ≤ 1), then also, ( ≤ ).

6d. Strictly Proportional Hazard: ( = )
If the hazards in men and women are "strictly" proportional to each other, then it must be the case that: ( = 0). Therefore, when: (| | > 0), as it must be when ( ≥ 1), the hazards cannot be "strictly" proportional. Indeed, for those circumstances in which ( ≥ 1) and ( = 0), the observed (F:M) sex ratio, as discussed above, either decreases or remains constant with increasing exposure (see Equations S12a-c; above), regardless of the parameter values for ( ) and ( )e.g., Figures 1A & 1B (Main Text). Consequently, the only "strictly" proportional circumstances, which are possible, are those in which men have a greater hazard than womeni.e., ( < 1). Moreover, if men have a greater hazard than women, then, as noted above, the conditions of: ( = ≤ 1) & ( = 0) are also excluded.

{NB: In these and subsequent Figures, all response curves exemplifying the conditions in which
(above). Each of these possibilities is contrary to evidence where currently ( 2 > 2 ) and, thus, where: (above). Thus, the condition of: ( < 0) is only possible, in circumstances where: ( < )e.g., Figure 2D (Main Text). and, consequently, the same will be true for "i-type" groups such that: and therefore, in most circumstances, both men and women can (at least potentially) belong to the same "i-type" group. This conclusion is also supported the available genetic evidence (see Discussion Section; Main Text). Moreover, in this conceptualization, the environmental factors that comprise each set of "sufficient" exposures within the { } family (for i-type individuals) are envisioned to be the same regardless of whether the "i-type" individual happens to be a man or a womanexcept that, when: ( > 0), a "sufficient" exposure for an "i-type" woman may need to be more "intense" than it is for an "i-type" man (see Section 6g; below).

6g. Considerations of Exposure "Intensity"
In considering the notion of exposure "intensity", three conclusions seem to be well established. First, for every proportional hazard solution that we identified (see Results Section; Main Text), we found that: ( > 1). Moreover, as demonstrated on theoretical grounds in Sections 6b-c (above), and as depicted in Figure S1 (below) & Figure 4; Main Text, in these circumstances, it must be that: Second, as demonstrated in Section 6c (above), under those circumstances, in which both ( ) and ( │ ) are increasing with time, then: Third, from the Canadian data [6], it seems inescapable that, as the probability of a "sufficient" exposure for susceptible individuals has increased over the past several decades, the probability of developing MS for susceptible women has increased at a faster rate than it has for susceptible men. Consequently, if the hazards in men and women are proportional, this faster rate of increase in susceptible women implies that one of the following two conditions must hold. Thus, either: 1) ≤ 1 in which case: < or: 2) > 1 in which case: > 0 Clearly, the first of these conditions excludes the possibility that: In considering the second of these conditions, it should be noted that both of our measures of exposure i.e., ( ) and ( )relate directly back to the parameter ( │ ), which represents the probability of the event that a randomly selected susceptible individual (either a man or a women) experiences an environmental exposure "sufficient" to cause MS in them. Therefore, this second conditioni.e., that: > 0indicates that, as the probability of a "sufficient" exposure decreases, there comes a point where only susceptible men can develop MS. This implies that, at (or below) this point: ( ≈ 0). Consequently, the requirement that ( > 1) creates a paradox in that, for the second condition to be true, susceptible women must be more likely than men to experience a "sufficient" exposure when the probability { ( │ )} is high and, yet, susceptible men must be much more likely than women to experience a "sufficient" exposure when this probability is low.
There are two obvious ways to avoid this paradox. The first is to conclude that the hazards are not proportional. Nevertheless, despite this possibility, such a conclusion also presents problems of its own (see Discussion Section; Main Text). For example, because women and men of the same "i-type" necessarily have proportional hazards (see Section 6h; below), in this case, we would also have to conclude that susceptible women and men can never be in the same "i-type" group and, therefore, that each gender requires distinct sets of environmental conditions to develop MS. Thus, we would have to further conclude that MS in women must represent a disease distinct from MS in men. Alternatively, if susceptible women and men could both be members of certain "i-type" groups but not others, we would have to conclude MS represents three distinct diseases (one in women, one in men, and a third in both). Any such conclusion seems to be at substantial variance with both the genetic and the epidemiological evidence (see Discussion Section; Main Text).
The second way to avoid the paradox, is to conclude that the first of the two possible proportional hazard conditions is truei.e., that both ( ≤ 1) and: ( < ). Notably, the condition of: ( ≤ 1) is compatible with any value of ( ). However, if ( > 0), the simultaneous condition of: ( ≤ 1), offers, at least, a more consistent interpretation of the existing data. Thus, under these conditions, at every population exposure level ( ), the probability of the event that a randomly selected susceptible man will experience a "sufficient" environmental exposure to cause MS in them is as great, or greater, than the same probability for a susceptible woman (see Figure S1 & Figures 4 & 5;Main Text). Thus, although the notion of a "critical exposure intensity" (discussed below) may be necessary to rationalize any threshold difference, it is not necessary to resolve a paradox. Nevertheless, accepting this conclusion, does require also accepting the fact that some susceptible men will never develop MS, even when the correct genetic background occurs together with an environmental exposure "sufficient" to cause MS in them.

Different Meanings of Exposure "Intensity"
When considering any other circumstance, for which ( > 0), it is important to distinguish between the "intensity" (or level) of exposure as measured by the odds ( )see Section 4a; above -and the "intensity" (or level) of exposure to the individual factors or events that, together, comprise the "sufficient" sets within each { } family. Thus, as noted above, both of our measures of exposurei.e., ( ) and ( )relate directly to the probability { ( │ )}. We describe this increasing probability as an increasing "intensity" of exposure.
Also, because each "i-type" group is different, we define ( ) to be the difference in threshold between "i-type" women and "i-type" men.

Exposure "Intensity" for a Single Factor
In this context, it is helpful, initially, to consider an example using a markedly over-simplified pathogenetic Model for MS. In this simplified Model, a single factor (e.g., vitamin D deficiency) is held to be the sole environmental factor responsible, by itself, for causing MS in both susceptible women and men. In this case, we can imagine a circumstance (e.g., Figure 5A; Main Text), in which the population vitamin D levels are such that no susceptible person (or "i-type") has any chance of having a "sufficient" deficiencyi.e., where: { ( │ ) = 0}. As the population vitamin D levels drop and deficiency becomes more prevalent in the population, it finally reaches the point where the levels can be low enough (e.g., Figure 5B; Main Text) such that some "i-types" begin to have some chance of developing MSi.e., where: {0 < ( │ ) < 1}. We can define the vitamin D level, at which the deficiency becomes "sufficient" to cause MS in a particular "i-type" both as the "critical exposure intensity", or the "threshold", level for that "i-type". We can also define this level to be the "critical" exposure level for the single factor (e.g., vitamin D deficiency) for that "i-type". For this simple pathogenic Model, these two definitions are the identical whereas, for Models with more factors, these may be different (see below). As vitamin D levels in the population decline further (i.e., as the exposure become more "intense"), more and more "i-types" will reach their "critical exposure intensity" level (e.g., Figure 5C; Main Text) until, finally (e.g., Figure 5D; Main Text), the population vitamin D deficiency becomes severe enough such that everyone experiences a "sufficient" exposurei.e., where: { ( │ ) = 1}.
In this circumstance, any "i-type" with a lower "critical exposure intensity" level compared to some other "i-type", will also have a greater probability of experiencing a "sufficient" vitamin D deficiency at every level of population exposure less than the maximumsee Sections 8a-b; Figures S1-S3. Consequently, in this simple Model, the population exposure "intensity" (as measured by vitamin D levels) is directly related to the "intensity" as measured by each of our metricsi.e., ( │ ), ( ) or ( ). Moreover, in this case, because women and men are responding to the same environmental event, the value of ( ) is simply a reflection of the difference in "critical exposure intensity" level between those susceptible "i-type" women (who require the least deficiency of any woman) and those susceptible "i-type" men (who require the least deficiency of any man). However, the value of the proportionality constant ( ) cannot be predicted from the "intensity" of exposure ( ) because this constant is determined by measuring exposure as the integral of an unknown hazard function rather than as ( )see note in Section 4a (above). Nevertheless, the probability distributions for the log-transformed "critical" (or threshold) levels of exposure for susceptible individuals (women and men considered separately), will have, respectively, both means ( and ) and variances ( 2 and 2 ) for each distribution. In this case, the variance of each distribution determines how rapidly probability of a "sufficient" exposure reaches any point on the y-axis (see Figures S1-S3), which is also exactly what the parameter ( ) determines. Therefore, while, in general, it is true the value of ( ) cannot be determined exactly in these circumstances, nevertheless, when: ( 2 = 2 ); then: ( = 1)e.g., Figure S1; Section 8b & Figure 5; Main Text; when ( 2 > 2 ), then: ( < 1)see legend of Figure S1; Section 8b; and, when: ( 2 < 2 ); then: ( > 1)e.g., Figure S2 & S3 (Section 8b).

Exposure "Intensity" for Multiple Factors
However, in contrast to this simplified Model, MS pathogenesis is known to involve multiple environmental factors [3,9], in which case, the relationship between our exposure metrics -( │ ), ( ) or ( )and the "intensity" level for any individual factor (however these are determined) becomes less clear.
As discussed above, here, we are measuring exposure as the probability of the event that a randomly selected susceptible individual experiences an environment "sufficient" to cause MS in them given the environmental cri conditions of the time ( ). This measure does not depend upon any specific environmental conditions.
Rather it indicates only that those environmental conditions, which exist at some point in time, result in this probability. Nevertheless, regardless of these uncertainties, if each factor can vary in their "critical" exposure level from one "i-type" group to another, there must be some relationship between this probability and the individual "critical" exposure level for each factor. Thus, if each factor level is at its minimum "intensity", then, in this case, presumably: { ( │ ) = 0}. Conversely, if each factor level is at its maximum "intensity", then presumably: { ( │ ) = 1}.
Without knowing the actual environmental factors involved in MS pathogenesis and how they interact, it is difficult to know how this probability might relate to the "intensity" of exposure to the different factors. Nevertheless, some possible relationships between this probability and factor "intensity" levels can be envisioned. For example, suppose that the family { } includes only one "sufficient" set, consisting of ( ) environmental factors or events, each of which has some "critical" exposure leveli.e., the exposure level, at (or above) which, the exposure becomes "sufficient" for this factor, in this set of exposures, for this "i-type" group. The terms: { 1 , 2 , … }, represent the "critical" exposure levels for each environmental factor { }. By contrast, the round-bracketed terms ( 1 ), ( 2 ), … ( ) represent the events that the exposure for a randomly selected "i-type" individual has reached (or exceeded) the "critical" exposure level for a particular environmental factor and ( 1 ), ( 2 ), … ( ) represent the probabilities of these events. We also assume that every "i-type" group requires the same set of environmental events but that each differs with respect to their "critical" exposure levels for each factor. Moreover, we assume that whether an "i-type" individual experiences a "sufficient" exposure to one factor or event, is independent of whether or not they experience a "sufficient" exposure to any other factor or event. In this case, ( ) becomes the event that a randomly selected "i-type" individual, experiences a "sufficient" exposure to every necessary factor such that: so that the probability of ( ) occurring during the Time Period ( ) is: To explain ( > 0), we suppose that some susceptible "i-type" women, compared to "i-type" men, have a higher "critical" exposure level to one or more of the factors in this set (i.e., if these factors are "genderdependent"). Thus, if these "gender-dependent" factors are not already above their maximum "intensity" levels for a given "i-type", ( ) will always be smaller in women compared to men. Moreover, if these differences in ( ) between men and women varied between "i-types", one can imagine that the distribution the ( ) levels for men and women might differ either in their means, in their variances, or in both. To explain a consistent ( > 0), however, it is necessary for men in some "i-type" groups, during any Time Period ( ), to have a smaller "critical exposure intensity" level compared to any susceptible woman in the population and, thus, for all these "i-type" groups to have ( > 0). Nevertheless, other "i-type" groups could vary in the relative likelihood of experiencing their "critical exposure intensity" levels, depending upon the environmental conditions that prevail during ( ).
Another possibility is to assume the same circumstances as those described above except that, in this case, each "i-type" group is posited to consist of only a single individual (or two in the special case of MZtwins). Clearly, with only one person (or an MZ-twin pair) per "i-type" group, the concept of ( ) is meaningless. Moreover, in this circumstance, we can define a minimum "critical" exposure level for each factor, considering every "i-type", such that: and we can also define a so-called "probability of minimal exposure", { ( )}, such that: Thus, { ( │ )}, is both a constant and, also, the minimum possible "critical exposure intensity" during any Time Period ( ). Therefore, it must be the case that:
Alternatively: 1) if certain factors are important determinants for some "i-type" groups but not for others, or: 2) if different sets within an { } family involved different environmental factors, then it is difficult to rationalize any consistent value for ( ≠ 0) based on our "intensity" measure { ( )}. However, regardless of any explanation, if this notion of a "critical exposure intensity" level is appropriate and if ( > 0), it must be that some susceptible men, during any Time Period ( ), must have a lower "critical exposure intensity" level compared to every susceptible women. Notably, however, if both women and men are (or potentially could be) members of every "i-type" group, this does not imply that for every "i-type" group ( > 0), although, this must be true for those "i-type" groups with the smallest "critical exposure intensity" levels of any. Moreover, although it seems likely that any "gender-dependent" factors for one "i-type" group, would also be "gender-dependent" for another, this may not be the case and it could be that some "i-type" women have a lower "critical exposure intensity" level compared to men of the same or a different "i-type".

Rationalization of Exposure "Intensity" when
Nevertheless, regardless of how we view the notion of a "critical exposure intensity", and regardless of the details regarding any rationalization of how such a circumstance might come to be, if we accept the proposition that both: ( > 0) & ( > 1), there are three firm conclusions that must be incorporated into any plausible explanation of the circumstances created by this paradox. First, if men and women are (or potentially could be) members of the any particular "i-type" group, the hazards must be proportional within that "i-type" group (see Sections 6f & 6h). Second, if both men and women are (or potentially could be) members of every "i-type" group, the hazards must be proportional within the population (see Sections 6f & 6h). And third, as demonstrated in Sections 4a-b (above), our measure of "intensity", { ( )}, is exponentially related to the probability that a susceptible man or woman (considered separately) either has or will develop MS. Also, this same measure of "intensity" is exponentially related to the probability that a susceptible individual (considering men and women together) either has or will develop MS and that, during any Time Period, this probability is a linear combination of the probabilities for men and women, considered separately (see Section 4c, above).

Exposure "Intensity" in Susceptible Women
Any condition for which ( > 0) indicates that there must be some environmental conditions in which only susceptible men can experience a "sufficient" exposure. As noted above, this circumstance requires that, for at least some "i-types", that: ( > 0). We will define the family of exposures { } to be the subset of exposures, within the { } family, that are "sufficient" for susceptible "i-type" women such that: where, for at least one ( ), it must be the case that: In turn, as for our earlier definition of ( )see Methods #1Bwe define the event ( ) to represent the union of the ( ) disjoint events, which exhibit the pairing of susceptible "i-type" women with "sufficient" environments, where: , ) and: ( │ , ) > ( │ , ).

Consequently, if we adjust Equation S8
to account for the fact that some "i-type" men require a less "intense" exposure than women of the same "i-type", then, the ratio, ( ⁄ ), can be re-expressed as:

6h. Exposure Variability (i.e., for & ) among "i-type" Individuals
If both men and women are (or potentially could be) members of any specific i-type group, by definition, these men and women each have a non-zero probability of developing MS in response to every one of the ( ) "sufficient" sets of exposures within the { } family for this group. As discussed earlier, in these circumstances, these specific i-types, considered separately, will necessarily exhibit proportional hazards for the two genders (see Section 4f; above). We previously defined the subset ( ) = (  Consequently, if women and men can, potentially, be members of every i-type group, the hazards for women and men will always be proportional, although the hazard proportionality factor ( ) need not be the same for every i-type group. Additionally, it is possible that the difference in threshold between women and men ( ) may be different for individuals of different i-types. We consider, first, this possibility for those circumstance in which: ( > 0). In this circumstance { = min( ) > 0} -at least among those "i-types" having the smallest "threshold" of any (see Section 6g)because, by definition, some women will begin to develop MS at this level of exposure. We can also define the difference in threshold ( ) between each susceptible woman and that of susceptible men having the smallest "threshold" ( ) of any. In this case, because ( > 0) and, by definition: ( = 0)see Methods #4Atherefore:

∀ ∈ ( , ): =
In this manner, the proportionality constants for each i-type ( > 0) and each woman ( > 0) can be replaced by a "adjusted" proportionality constants ( ′ > 0) and ( ′ > 0) such that: For the circumstance where: ( < 0), the analysis is only changed in that: { = max( )} and that, in this circumstance: ( = 0). Thus, in either case, the hazards will still be proportional. By contrast, if men and women are each responding to different environmental events, the hazards will not be proportional and the response curves for women and men would need to be plotted on separate graphs, each with a different x-axis scale (see Section 6g; above). In such a circumstance, men with MS would be envisioned as having a disease distinct from MS in women. Alternatively, perhaps, it could be that, for some autosomal genotypese.g., could be members of the same "i-type" group ( ). In this case, as noted previously, MS would then be envisioned as comprising three distinct diseasesone in men only, one in women only, and a third in both.

Summary Equations for the Longitudinal Model 7a. Derivations
We define three related ratios (see Table 2; Main Text): Using these definitions, we can derive the following Summary Equations.

8a. General (Common) Considerations for the Figures
Hypothetical relationships between the "critical exposure intensity" and disease expression that might explain the circumstance of: ( > 0), in which men disproportionately (or exclusively) experience a "sufficient" exposure at low "intensity" exposures compared to women (see Section 6g; above). In these Figures, the x-axis represents the level (or "intensity") of exposure and on the y-axis is the proportion of the susceptible population ( ) who experience a "sufficient" exposure. above. Moreover, the reason for using this transformation is that ( ) is an odds and, therefore, the use of ( ) will usually normalize the variance of these distributions [39].} In Panel ( ) of each Figure, the solid black lines represent the distribution of "actual" exposure "intensity" levels experienced by the susceptible population during a Time Period ( ). The dotted lines (red for women and blue for men) represent the distributions of the "critical exposure intensity" (or "threshold") levels for susceptible men and women (of any "i-type"). These "threshold" levels are defined such that the exposure becomes "sufficient" for an "i-type" individual only once they experience an exposure "intensity" level, at (or above), their particular "threshold" (see Section 6g; above). exposure, at the levels of population exposure depicted. In these Figures, although the exposure "intensity" is the same for everyone, different "i-types" experience a "sufficient" exposure at different "intensities".
Moreover, if susceptible men and women can be plotted on the same "intensity" scale, this implies that both groups are responding to the same "environmental conditions" and that the hazards are proportional (see Section 6g (above); see also Methods #4A; Main Text). Also, in this circumstance, it seems likely that the value of the proportionality constant ( ) is related to a difference in these distributions of "critical exposure intensity" values between susceptible men and women. Therefore, it is of note that both a value of ( > 1), and an increased variance in susceptible men, reflect the same underlying circumstance (see Section 6g; above) i.e., where susceptible men, have a larger difference in exposure "intensity" between the onset of their response curve and any point on the y-axis, compared to women (e.g., Figures 3 & 4; Main Text).
Therefore, if this construct is correct, these two parametersi.e., the proportionality factor ( ) and the variance, as measured on the ( ) scalemust be related to each other. Nevertheless, because ( ) has a non-linear relationship to ( ), any exact relationship of ( ) to the variance cannot be predicted.
Nevertheless, if men and women have the same variance, then: ( = 1)see Section 6g; above; see also  In the B-D Panels of each Figure, we plot the cumulative probability of experiencing a "sufficient" environmental exposure with increasing exposure "intensity" (under the different conditions represented by each Figure). This relationship is plotted separately for susceptible men (solid blue lines) and women (solid red lines). The black lines represent the changes in the F:M sex ratio with increasing exposure and the scale for these lines is indicated in each Figure. As in the ( ) Panels (above), these cumulative probability curves are mostly plotted for conditions where: { /(1 − ) = 1}. However, this choice doesn't impact the character of any response. Thus, these response curves, following a maximum exposure (e.g., Figure 5D; Main Text), will plateau at: { /(1 − )}, whatever this is. Moreover, from Equation S1a (Section 2c), at any point where the undefined other than as they relate to the variance of these "critical exposure intensity" distributions in susceptible men and women ( 2 and 2 ), respectively. Figure S1. See the general description of the layout of Figures S1-S3 provided in Section 8a (above). This Figure (S1) assumes that the distribution of these log-transformed "critical" (or threshold) levels of exposure for susceptible men and women have the same variance ( 2 and 2 , respectively) but different means ( and , respectively)i.e., conditions such as those depicted in Panel A. Because ( > 0), and if ( = = 1), under these conditions, a greater proportion of men will experience a "sufficient" exposure at every "intensity" level compared to women (Panel B). Consequently, the F:M sex ratio can never exceed { /(1 − )}, and thus, in the case illustrated, this ratio cannot exceed 1. Thus, the only way to achieve the "current" proportion of women among MS patients (i.e., : = 3.1), is for the proportion of women among susceptible individualsi.e., ( │ )to be greater than or equal to the current estimate for ( │ )see Section 2c (above). Such a circumstance, however, requires that ( > ) throughout the entire response curve. By contrast, if conditions were such that ( < = 1), then the response curves intersect at exposures appropriate for Figure 4 (Main Text), the F:M sex ratio is steadily increasing throughout the response, this ratio can easily exceed its currently observed value (3.1), and there is no need to invoke any extreme conditions (e.g., Panels C & D). In the circumstance of these graphs, the variance is the same, which indicates that: ( = 1)see Section 6g. However, Panels C & D would be little changed if: ( ≤ 1)e.g., Figure 4; Main Text. Nevertheless, in this circumstance, the potential paradox, created by the simultaneous conditions that:

Panel C of
( > 0) and ( > 1), would be avoided (see Section 6g). Under conditions where: ( = = 1), a "sufficient" exposure will be experienced by a larger proportion of men than women until the response curves intersect at the point where the F:M sex ratio equals { (1 − )} ⁄ . In addition, those conditions (e.g., Panels B-C), in which ( = = 1), suggest that the transition from a male-predominant MS to female predominant MS takes place relatively late in the response curve for meni.e., not as early or as rapidly as might be anticipated based upon the conditions presented in Figure 3 (Main Text). Moreover, even though the F:M sex ratio can exceed the value of { /(1 − )}, this circumstance takes rather extreme conditions (Panels B-C) and, for the currently observed sex ratio of (3.1) to be achieved [6], this would, again, require { /(1 − )} to be almost as large as this (Panel C) and for ( > ) until the point of intersection. Also, unlike Figure 3 (Main Text), for only a small portion of these response curves is the F:M sex ratio declining. By contrast, if conditions were such that ( < = 1), then, as in Figure S1, the response curves intersect at exposures appropriate for Figure 4 (Main Text), the F:M sex ratio is steadily increasing throughout the response, this ratio can easily exceed its currently observed value (3.1), and there is no need to invoke any extreme conditions (e.g.,

Panel D).
In this circumstance, because the variance in men is greater than in womeni.e., ( 2 < 2 )this Figure only pertains to conditions of: ( > 1) and, thus, the paradox created by the conditions of ( > 0) and ( > 1) would persist (see Section 6g). ( 2 and 2 , respectively), and different means ( and , respectively)i.e., conditions such as those depicted in Panel A. Under conditions where: ( = = 1), a "sufficient" exposure will be experienced by a larger proportion of men than women until the response curves intersect at the point where the F:M sex ratio equals { (1 − )} ⁄ . In addition, those conditions (e.g., Panels B-C), in which ( = = 1), suggest that the transition from a male-predominant MS to female predominant MS takes place relatively late in the response curve for meni.e., not as early or as rapidly as might be anticipated based upon the conditions presented in Figure 3 (Main Text).
Moreover, only under extreme conditions (e.g., Panel C) do the response curves even approach (or surpass) the F:M sex ratio of (3.1), which occurs at (0.86) on the y-axis. Going in the opposite direction, as the difference in variance narrows, the F:M sex ratio does not intersect the line at the (3.1) mark until ( 2 = 1.3 * 2 ). Although, in this case, the transition from a male-predominant MS to female predominant MS takes place very early the response curve, the proportion of the susceptible population who experience a "sufficient" exposure at the point where the curves intersect is miniscule ( . ., < 10 −6 ). Also, as in Figure 3 (Main Text), for much of these response curves the F:M sex ratio is declining. By contrast, as in Figure S2, if conditions were such that ( < = 1), then, as in Figure   S1, the response curves intersect at exposures appropriate for Figure 4 (Main Text), the F:M sex ratio is steadily increasing throughout the response, this ratio can easily exceed its currently observed value (3.1), and there is no need to invoke any extreme conditions (Panel D). In this circumstance, because the variance in men is greater than in womeni.e., ( 2 < 2 )this Figure only pertains to conditions of: ( > 1) and, thus, the paradox created by the conditions of ( > 0) and ( > 1) would persist (see Section 6g).