A. General model specifications and definitions for genetic susceptibility to MS.

We consider a general population (Z), which is composed of (N) individuals (k = 1, 2, …, N) who are living under the prevailing environmental conditions during some specific Time Period (T)–conditions that are designated, generically, as (E_T). In Table 1, we define the different parameters used in our analysis and, in Table 2, we provide a set of parameter abbreviations, which are used for the purposes of notational simplicity. The subset (MS) is defined to include all individuals within (Z) who either have or will develop MS over the course of their lifetime. The occurrence of (MS) represents the event that an individual, randomly selected from (Z), belongs to this (MS) subset and the term P(MS│E_T) represents the probability of this event, given the prevailing environmental conditions of (E_T)—i.e., P(MS│E_T) = P(MS│Z, E_T). This probability is referred to as the “penetrance” of MS for the population (Z) during the Time Period (E_T). The occurrence of (G) is defined as the event that an individual, randomly selected from (Z), is a member of the (G) subset. The term P(G│E_T) represents the probability of this event given the prevailing environmental conditions of (E_T). In turn, the (G) subset is defined to include all individuals (genotypes) within (Z) who have any non-zero chance of developing MS under some (unspecified and not necessarily realized) environmental conditions, regardless of how small that chance might be or how rarely the appropriate environmental conditions might occur. Consequently, everyone who actually develops MS must belong to the (G) subset. Moreover, we assume that a person’s genotype is independent of the environmental conditions that prevail during (E_T). Therefore:

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Table 1. Definitions for the groups and epidemiological parameters used in the analysis.

https://doi.org/10.1371/journal.pone.0285599.t001

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Table 2. Principal parameter abbreviations^†.

https://doi.org/10.1371/journal.pone.0285599.t002

In this circumstance, each of the (m ≤ N) individuals in the (G) subset (i = 1, 2, …, m) has a unique genotype (G_i). The occurrence of (G_i) represents the event that an individual, randomly selected from (Z), belongs to the (G_i) subset–a subset consisting of only a single individual (i.e., the so-called “i^th susceptible individual” or “i^th individual”)–and the term {P(G_i) = 1/N} represents the probability of this event. Therefore, it follows from the definition of the (G) subset that, if every relevant environmental condition–see below–is possible during some Time Period (E_T), then, during this Time Period:

The conditional probabilities: {x_i = P(MS│G_i, E_T)} and: {x = P(MS│G, E_T)}, are referred to as the “penetrance” of MS, during (E_T), for i^th susceptible individual and for the (G) subset, respectively. Clearly, the penetrance of MS, both for the individual and for the group, will vary depending upon the likelihood of different environmental conditions during different Time Periods. If the environmental conditions during some Time Periods were such that certain members of the (G) subset have no possibility of ever developing MS, then, for these individuals, during these Time Periods:

Because, currently, some individuals can (and do) develop MS, it must be that, during our “current” Time Period:

However, if, at some other time, the environmental conditions were such that no member of (G) could ever develop MS then, during these Time Periods:

We also define a subset , which includes of all female members of the (G) subset {i.e., }, and we define the proportion of women in (G) as: p = P(F│G). In this circumstance, each of the (m * p) women in the subset (d = 1, 2, …, mp) has a unique genotype . The occurrence of represents the event that an individual, randomly selected from (Z), belongs to the subset–a subset consisting of only the d^th susceptible woman–and the term represents the probability of this event. Also, represents the probability of the event that an individual, randomly selected from (Z), belongs to the subset.

Individuals, who do not belong to the (G) subset, belong to the mutually exclusive (complimentary) subset (G^c), which consists of all individuals who have no chance, whatsoever, of developing MS, regardless of any environmental experiences that they either have had or could have had. The occurrence of (G^c) is defined as the event that an individual, randomly selected from the population (Z), is a member of the (G^c) subset. The term P(G^c│E_T) represents the probability of this event, given the environmental conditions of (E_T). Consequently, each of the (m^c = N − m) “non-susceptible” individuals in the (G^c) subset (j = 1, 2, …, m^c) has a unique genotype (G_j). As above, the occurrence of (G_j) represents the event that an individual, randomly selected from (Z), belongs to the (G_j) subset–a subset also consisting of a single individual. The term {P(G_j) = 1/N} represents the probability of this event. Thus, under any environmental conditions, during any Time Period: and thus,

Notably, MZ-twins, despite having nearly “identical” genotypes (IG), nevertheless, still have subtle genetic differences from each other. Thus, even if these subtle differences are irrelevant to MS susceptibility (as seems likely, and which we assume to be true), these differences still exist. Consequently, every individual–i.e., each complete genotype (G_i) and (G_j)–in the population is unique. Despite this uniqueness, however, we can also define a so-called “susceptibility-genotype” for the i^th susceptible individual such that this genotype consists of all (and only) those genetic factors, which are related to MS susceptibility. Because the specification of such a susceptibility-genotype necessarily includes many fewer genetic factors than the i^th individual’s complete genotype, it is possible that one or more other individuals in the population share the same susceptibility-genotype with the i^th individual. For example, in this conceptualization, MZ-twins would necessarily belong to the same susceptibility-genotype. We refer to the group of individuals, who belong to the i^th susceptibility-genotype, as the (G_is) subset within (Z). The occurrence of (G_is) represents the event that a person, randomly selected from (Z), belongs to the (G_is) subset, which consists of a single susceptibility genotype. The term P(G_is) represents probability of this event. Some members of (G) are MZ-twins and, thus, both twins are members of the same (G_is) subset. Therefore, the total number of these susceptibility-genotypes in the population (m_is) must be less than (m)–i.e., (m_is < m).

Also, it is possible that two or more “susceptibility genotypes” may share an identical family of “sufficient” environmental exposures {E_i} with the i^th individual (see Methods #2B; below). Therefore, we define the “i-type” group (G_it) to include all “susceptibility genotypes” who share the same {E_i} family. The probability {P(G_it)} represents the probability of the event that and individual, randomly selected from (Z), belongs to the (G_it) group. Also, from above, the total number of “i-type” groups in the population (m_it) must be less than (m)–i.e., (m_it ≤ m_is < m). In addition, we define the family {G_s} to include all of the “i-type” groups (G_it) within (Z) and define the event {G_s} as representing the union of the disjoint (G_it) events such that:

Because every susceptible person belongs to one, and only one, of these “i-type” groups, the probability of this event is expressed as:

We also define the set (X) to be the set of penetrance values for members of the (G) subset during some Time Period. Provided that the variance of (X) is not equal to zero {i.e., }, the subset (G) can be further partitioned into two mutually-exclusive subsets, (G1) and (G2), suitably defined, such that the penetrance of MS for the subset (G1) during a certain Time Period is greater than that for (G2). The terms P(G1) and P(G2), represent the probabilities of the events that an individual, randomly selected from (Z), is a member of the subsets (G1) and (G2), respectively. Although many such partitions are possible, for the purposes of the present manuscript, (G1) is generally considered interchangeable with the subset of susceptible women–i.e., –and (G2) is generally considered interchangeable with the subset of susceptible men–i.e., (G2) = (M ∩ G) = (M, G).

When considering the enrichment of more penetrant genotypes (see Section 2a, 2b in S1 File), the subsets (F, G) and (M, G) will each be further partitioned into high- and low-penetrance sub-subsets–i.e., (G1′) and (G2′), respectively–where the basis for this further partition into (G1′) and (G2′) sub-subsets is unspecified. The definitions of, and the probabilities for, these events mirrors that above for (G1) and (G2). Moreover, although the basis for this further partition must be something other than gender, it can be anything else that creates a partition, and it doesn’t need to be the same basis for both genders.

{NB: A note on terminology. When a claim refers to any partition of the (G) subset, the probabilities of developing MS (i.e., the penetrance of MS) for members of the (G1) and (G2) subsets) are designated, respectively, such that: x₁ = P(MS│G1) and: x₂ = P(MS│G2). When the partition is based specifically on gender, to provide clarity, and to avoid any confusion with our Time Period designations, the group of females/women are indicated, alternatively, either by an upper-case (F) or by a lower-case (w). Similarly, in these circumstances, males/men are indicated, alternatively, either by an upper-case (M) or by a lower-case (m)–see Table 2. In some circumstances, however, (when the meaning is clear), for purposes of notational simplicity, the designations of (x₁) and (x₂) continue to be used to designate the penetrance of MS for the subsets of susceptible women (x₁) and men (x₂). In other circumstances, however, greater clarity is provided by using the letter designations. For example, considering the partition of (G) into the subsets (F, G) and (M, G), the penetrance of MS for susceptible women and men are designated, respectively, as (Zw) and (Zm) such that: x₁ = Zw = P(MS│F, G) and: x₂ = Zm = P(MS│M, G)–see Table 2. When the listing of individual women within the (F, G) subset is important to an argument, the designations and are used.

Moreover, although this manuscript focuses on the gender partition for the disease MS, the Models developed pertain to any partition for any disease, which has data analogous to that found in Canada for the gender partition of MS [6, 7].}

B. General model specifications and definitions for environmental susceptibility to MS.

The term {E_i} represents the family of specific sets of environmental exposures, each of which, by itself, is “sufficient” to cause MS to develop in the i^th susceptible individual. Each set within the {E_i} family must be distinct (in some respect) from every other set within this family but, otherwise, there can be any degree of overlap between the factors or events that comprise these sets. Also, there can be any number of sets within the {E_i} family although, because (∀ G_i ∈ G: x_i > 0) under some environmental conditions, the family cannot be empty. Thus, at least one “sufficient” set of exposures must exist for every susceptible individual. If we assign (v_i) to the number of sets of sufficient exposures for the i^th (or an “i-type”) susceptible individual, then {E_i} represents the family of sets: ; and P({E_i}│E_T) represents the probability of the event that, at least, one of these sets of “sufficient” exposure occurs, given the prevailing environmental conditions of the time (E_T). Moreover, if more than one individual belongs to a particular “i-type” group, each group-member, by definition, will have the same {E_i} family of “sufficient” exposures as the i^th individual.

Notably, also, the probability, P({E_i}), depends entirely upon the actual environmental conditions that prevail during any Time Period–i.e., conditions that are fixed for any specific (E_T). Thus: where (A_i) represents an unknown constant. This constant (A_i) may be different for each {E_i} and, also, it may be different during different Time Periods. Consequently, during any (E_T), each {E_i} represents a population-wide exposure–i.e., an exposure that is “available” to everyone. However, whether anyone, in particular the i^th susceptible individual, experiences that exposure, is a different matter (see below).

Also, for MS to develop in the i^th susceptible individual, the events {E_i} and (G_i) must occur jointly–i.e., the individual (G_i) must experience at least one of the {E_i} environments. This joint occurrence is represented by the subset ({E_i}, G_i). The occurrence of ({E_i}, G_i) represents the event that an individual, randomly selected from (Z), is both in the (G_i) subset (described above) and that they experience an environment “sufficient” to cause MS in them. The probability of this event, given that this person is a member of (G) subset and given the environmental conditions of (E_T), is represented as P({E_i}, G_i│G, E_T). If the event (G_i) occurs without {E_i}, then whatever exposure does occur, it is insufficient, and the i^th individual cannot develop MS. However, the relationship between one individual’s family of “sufficient” exposures to that of others may be complex. For example, every “i-type” group may have a family with sets unique to that group or, alternatively, the families for any two or more individuals (not in the same “i-type” group) may overlap to any degree, even to the point where their families are almost identical. However, if every susceptible individual has an identical family of “sufficient” environmental exposures, then: ∀(i): P({E_i}, G_i│G, E_T) = P({E_i}│G, E_T); and everyone is a member of the same “i-type” group. If some individuals can develop MS under any environmental condition, then, for these individuals: P({E_i}, G_i│G, E_T) = P(G_i│G, E_T). And, finally, if there are (s_e) specific sets of environmental exposure (e = 1, 2, …, s_e) that are “sufficient” to cause MS in any susceptible individual, then, for the family {E_e} of these sets of environmental exposure:

{NB: It may be that some of these {E_e} environments, which are “sufficient” to cause MS in every susceptible individual, are so improbable (e.g., being inoculated with myelin basic protein together with complete Freund’s adjuvant), that they never occur spontaneously. Even so, any individual who can only develop MS under these extreme environmental conditions, is still able to develop MS under some environmental conditions and, thus, every such individual will be a member of the (G) subset. If anyone can develop MS under these extreme conditions, then everyone is a member of the (G) subset.}

Definition of the exposure (E). Although an individual (genotype) may experience more than one set of environmental exposures, which may be part of one, or more than one, {E_i} family, each individual’s total environmental experience is unique to them. Therefore, we will represent the exposure event of interest (E) as the union of the disjoint events, which exhibit the pairing of susceptible individuals with “sufficient” environments, such that: in which case: or:

Because genotype is assumed to be independent of the prevailing environmental conditions (E_T): so that:

Thus, the term P(E│G, E_T) represents the probability of the event that a member of the (G) subset, selected at random, will experience an environmental exposure “sufficient” to cause MS in them, given their unique genotype and given the prevailing environmental conditions of the time (E_T). Furthermore, from the definition of (E), this event can only occur when the event (G) also occurs, so that:

Notably, many environmental factors or events, which are part of a set within the {E_i} family, may (and likely do) represent a range of environmental experiences. For example, suppose that, for the i^th susceptible individual to develop MS, for one set of exposures, they need to experience a vitamin D deficiency of some minimum severity, lasting for some minimum amount of time, and occurring during some “critical” age-window. In this case, the definition for the environmental event of a “sufficient” vitamin D deficiency for this individual, for this set, presumably, would also include deficiencies of the same (or greater) severity, lasting the same (or a longer) amount of time, and occurring during the same (or more restrictive) age-window. In this circumstance, we can define a “critical exposure intensity” level as that vitamin D level, at (or above) which, the deficiency becomes “sufficient” for the i^th (or an “i-type”) individual. An expanded discussion of this notion of exposure “intensity” is presented elsewhere (see Sections 6g & 8a, 8b in S1 File).

Importantly, as noted previously, each set of “sufficient” environmental exposures is unspecified as to: 1) how many events or factors are involved; 2) when, during the life of an individual, these events or factors need to occur; 3) what these events or factors are; and 4) whether these factors need to be present or absent. Notably, this specification of a “sufficient” sets of exposures is completely agnostic with respect to whether these factors or events increase or decrease risk. For example, if behaving in some manner, or having some experience, protects the i^th person from getting MS, then one or more of the “sufficient” sets of exposure for this person will include not behaving in this manner or not having this experience. Nevertheless, regardless of any such complexities, each of these sets, of whatever they consist, simply needs to be “sufficient”, by themselves, to cause MS to develop in the i^th (or an “i-type”) susceptible individual. Thus, our definition of a “sufficient” set of exposures includes every environmental condition (known, suspected, or unknown), which is required (i.e., necessary) for such “sufficiency”.

Partitioning the environmental exposure. In addition, any set of environmental exposures, for any individual, can be partitioned conceptually into three mutually exclusive subsets, which we term: (E_pop, E_sib, and E_twn). The subset (E_pop) includes all those environmental experiences or events equally likely to be shared by the population generally (including siblings and twins). The occurrence of (E_pop) represents the event that a specific environmental event or factor, which an individual experiences, is a member of the (E_pop) subbset. The subset (E_sib) includes all those environmental experiences or events either more or less likely to be shared by siblings (including twins) compared to the general population. The occurrence of (E_sib) represents the event that a specific environmental event or factor, which an individual experiences, is a member of the (E_sib) subset. Presumably, the (E_sib) environmental experiences occur mostly (but not necessarily exclusively) during childhood. The subset (E_twn) includes all those environmental experiences or events more or less likely to be shared by MZ- and DZ-twins compared both to non-twin co-siblings and to the general population. The the occurrence of (E_twn) represents the event that a specific environmental event or factor, which an individual experiences, is a member of the (E_twn) subbset. Presumably, the (E_twn) environmental events occur mostly (but not necessarily exclusively) during the intrauterine and early post-natal periods. Importantly, creating this partition does not imply that any of these experiences are unique to twins or siblings–everyone experiences each environmental component. The difference is that twins and siblings are more or less likely to share certain experiences.

Each of the (v_i) sets of “sufficient” environmental exposures within the {E_i} family can be partitioned into these three mutually exclusive events. Thus, for (j = 1, 2, …, v_i), the event (E_ij) represents the occurrence of the j^th “suffiicient” set of exposures within the {E_i} family. The event (E_ij) can then be represented as the union of these three disjoint events such that:

In this circumstance, the probability of each event (E_ij) is the joint probability of these three independent component events such that:

and the event {E_i} is represented as:

The same applies to every {E_i} family–i.e., ∀(i): (i = 1, 2, …, m).

{NB: Most, if not all, environmental exposures are “population-wide” in the sense that the risk of these events is shared by everyone. For example, the amount of sunlight reaching the Earth’s surface in a particular region can be considered a “population-wide” exposure in the sense that the same amount of sunlight is “available” to everyone in that region. Despite this, however, there may be certain individuals or certain subgroups within the population who experience less sun-exposure than others (e.g., if they disproportionately use sun-screen, if they disproportionatley avoid the sun, or if they are otherwise disproportionatley protected from sun-exposure). Conversely, there may also be certain individuals or groups who experience more sun-exposurre than others. However, given the fact that a co-twin (or a non-twin co-sibling) experiences such an imbalance, unless their proband twin (or proband sibling) is either more or less likely to to experience a similar imbalance compared to others, then these exposures would still be part of the (E_pop) environment. Also, the (E_sib) environment may include experiences outside the childhood micro-environment if, for example, sharing the same biological mother made the intra-uterine environment more similar for siblings than that for the general population. In addition, if twins disproportionately shared certain childhood or adult experriences more so than other siblings or the general population, then these experiences woud be part of the (E_twn) environment.

Although it is unspecified as to what experiences consitiute each subset, nevertheless, these three subsets of environmental exposure (E_twn, E_sib, and E_pop) are envisioned to be mutually exclusive and that, together, they comprise any idividual’s unique environmental experience. Thus, as noted above, every individual experiences each of these components of enviornmental exposure, regardless of whether they are twins or non-twin siblings and regardless of whether they are members of the (G) subset. For example, even though the same intrauterine environment is shared by twins, everyone experiences some intrauterine environment. Similarly, although both twins and non-twin co-siblings experience a similar childhood environment, everyone experiences some childhood environment. Nevertheless, in considering these components of environmental exposure as they relate to the sufficent sets as described above–i.e. for (i = 1, 2, …, m) and for (j = 1, 2, …, v_i), we are here focused on the events (E_ij), for which, during any Time Period, it will be the case that:

Notably, during any specific Time Period (E_T), both P(E_ij) and its component parts are constants. In this conceptualization, however, each successive Time Period (E_T) are envisioned to overlap with each other. For example, suppose that all of the relevant (E_twn, E_sib, and E_pop) exposures need to take place before the age of 30 years. In this circumstance, for a person born in 1975, (E_T) will represent the Time Period from 1975 to 2005. By contrast, for a person born in 1980, (E_T) will represent the overlapping Time Period from 1980 to 2010.}

Impact of the (E_sib) environment. Despite this conceptual framework, however, the observations from Canada in adopted individuals, in siblings and half-siblings raised together or apart, in conjugal couples, and in brothers and sisters of different birth order, have indicated that MS-risk is not affected by the familial micro-environment but suggest, rather, that the important environmental risks (not considering twins) result from exposures that are experienced population-wide [10–16]. Thus, these studies, collectively, provide compelling evidence for the absence of any (E_sib) environmental impact on MS.

Relationships between, and limits relating to: (MS), (E), and (G). It is clear from the definitions of environmental and genetic susceptibility (above) that, for the event of (MS) to occur, both the event (G) and the event (E) must also occur. If either of these events does not occur, the event (MS) cannot occur. Therefore:

Also, using the definitions in Table 2, and both from Section 7b in S1 File and from Methods #1C & #1D (below), it must be the case that, if, currently, {P(E) ≠ 1}, then, also: and:

C. Circumstances relating to twins and siblings of individuals with MS.

The terms (MZ), (DZ), and (S) represent, respectively, the subsets of MZ-twins, DZ-twins, and non-twin sibships within (Z). The occurrence of (MZ), (DZ), or (S) represent the events that an individual, selected at random from (Z), belongs, respectively, to each of these subsets and the terms P(MZ), P(DZ), P(S) represent the respective probabilities of these events. For clarity, the randomly selected individual is always referred to as the “proband twin” or the “proband sibling” depending upon the subset to which they belong. The other member (or members) of the twinship or sibship are always referred to as the “co-twin(s)” or the “co-sibling(s)”.

Circumstances for twins and siblings of selected probands. Initially, we will consider two events for an MZ twin-pair. The first is the event that the proband, randomly selected from (Z), is a member of the (MS, MZ) subset and that their co-twin is a member of the (MZ) subset. The second is the event that the proband, randomly selected from (Z), is a member of the (MZ) subset and that their co-twin is a member of the (MS, MZ) subset. Clearly, the probability of these two events is the same. Therefore, to distinguish the circumstances of the proband from those of the co-twin, we will use the term (MZ_MS) to indicate, specifically, the status of the co-twin. Thus, during the Time Period (E_T): where {P(MS│MZ, E_T)} represents the probability of the event (MS) in the proband twin during (E_T) and {P(MZ_MS│E_T)} represents the same probability for the event (MS) in the co-twin during (E_T). Also, because any two MZ-twins have “identical” genotypes, therefore: and:

In which case:

Consequently, every proband who has an MZ co-twin in the (MS, MZ) = (MS, MZ, G) subset and who shares an “identical” genotype with their co-twin, must also be a member of the (G) subset. Therefore, summing over all susceptible individuals: and:

Similarly, the term P(MZ_E) represents the probability of the event that the co-twin of an MZ-twin proband, randomly selected from (Z), is a member of the (MZ, E) subsets. Thus:

In an analogous manner, for DZ-twinships, the status of the co-twin is indicated by the subsets and the events of: (DZ_MS) and (DZ_E). And for non-twin sibships, the status of the co-sibling is indicated by the subsets and events of: (S_MS) and (S_E).

Thus, the two terms, P(MS│MZ_MS, E_T) and P(MS│DZ_MS, E_T) represent the conditional life-time probability of the event that an individual (the proband), randomly selected from (Z), is a member of the either the (MS, MZ) or the (MS, DZ) subset, given the fact that their co-twin also belongs, respectively, to the (MS, MZ) or the (MS, DZ) subset, and given the prevailing environmental conditions of the time (E_T). These probabilities are estimated by the proband-wise concordance rate for either MZ- or DZ-twins [17]. This rate is calculated based on the number of concordant twin-pairs (C_TP) compared to the number of discordant twin-pairs (D_TP) and adjusted based upon the degree to which twins are “doubly ascertained”. The term “doubly ascertained”, in this context, represents the proportion of twin-pairs, for whom both twins were independently identified by the initial ascertainment scheme [17]. If all twin-pairs are “doubly ascertained” by this scheme, and if the sample from (Z), so ascertained, is random, then the formula for calculating the proband-wise concordance rate is:

However, if the probability of “double ascertainment” is less than unity, then this formula requires some modification [17]. In the Canadian data [6] the double-ascertainment rate for concordant MZ-twins was 54.2% (13/24).

In a similar manner, the term P(MS│S_MS, E_T) represents the conditional life-time probability of the event that an individual (the proband), randomly selected from (Z), is a member of the (MS, S) subset, given the fact that one or more of their non-twin co-siblings is a member of the (MS, S) subset and given prevailing environmental conditions of (E_T).

D. Adjustments for the shared environment of twins.

Lastly, the term P(MS│IG_MS, E_T), represents the proband-wise concordance rate for MZ-twins during (E_T)–i.e., P(MS│MZ_MS, E_T)–which has been adjusted for the fact that concordant MZ-twins, in addition to sharing their “identical” genotypes (IG), also disproportionately share their (E_twn) and (E_sib) environments with each other. Such an adjustment may be necessary because, if these disproportionately shared environmental experiences contribute to causing MS in the co-twin, they could also increase the likelihood of MS developing in the proband twin and such a circumstance could, potentially, alter any conclusions regarding the nature of genetic susceptibility in the population (see Section 1a, 1b in S1 File, for a discussion of why, and a development of how, this adjustment is made).

E. Characterizing genetic susceptibility to MS in a population.

From these Model specifications and definitions, we can use estimated values for observable population parameters to deduce the value of the non-observable parameter P(MS│G, E_T), which represents the probability of the event that an individual, randomly selected from (Z), will develop MS over the course of their lifetime, given that they are a member of the (G) subset and given the prevailing environmental conditions of (E_T). From the definition of (G), as noted in Section #1A (above):

Therefore, because genotype is assumed to be independent of the prevailing environmental conditions of (E_T):

Rearrangement of this equation, yields:

Consequently, the value of P(G) can be estimated using the observed data from any specific Time Period (E_T)–including ours–during which: P(MS│E_T) > 0. Thus, the parameter P(G) can be estimated regardless of whether some susceptible individuals have no chance of developing MS under the environmental conditions of (E_T). Therefore, considering only our “current” Time Period, this equation can be simplified to yield:

Moreover, once the value of P(G) is established, it can then be used to assess the nature of MS pathogenesis. For example, if: {P(G) = 1}–i.e., if the penetrance of MS for the population (Z) is the same as that for the subset (G)–then anyone can get MS under the appropriate environmental conditions. By contrast, if: {P(G) < 1}–i.e., if the penetrance of MS for the subset (G) is greater than that for the population (Z)–then only certain individuals in (Z) have any possibility of getting MS. Thus, a finding of: {P(G) < 1} would exclude any possibility that MS ever occurs in someone who lacks a genetic predisposition to getting the disease. In this sense (and in this case), MS must be considered a “genetic” disorder (i.e., unless a person has the appropriate genotype, they have no chance, whatsoever, of getting MS, regardless of their environmental exposure). Importantly, even if MS is “genetic” in this sense, this has no bearing upon whether disease pathogenesis also requires the co-occurrence of specific environmental events.

In this analysis, two basic Models are used to estimate the values of various unknown epidemiological parameters of MS. The first Model takes a cross-sectional approach, in which deductions are made from the epidemiological data obtained during a single Time Period (i.e., the “current” Time Period). This will be referred to as the Cross-sectional Model (see Methods #3; below; see also Section 3a, 3b in S1 File). The second Model takes a longitudinal approach, in which deductions are made both from the “current” epidemiological data and from the observed changes in MS epidemiology that have taken place over the past 4–5 decades [3, 6]. This will be referred to as the Longitudinal Model (see Methods #4; below; see also Section 4a–4c; in S1 File). These two Models are independent of each other although both incorporate many of the same observed and non-observed epidemiological parameters, which are important for MS pathogenesis. The Cross-sectional Model derives theoretical relationships between different epidemiologic parameters, but it also makes two assumptions regarding MZ-twin data to establish these relationships. These two assumptions are also commonly made by other studies, which analyze MZ-twin data, and each has observational data to support them [18–20]. Nevertheless, for the derivations for Eqs 2a–2d (Methods #3; below), these conditions need to be assumed (see Section 3a in S1 File). By contrast, the Longitudinal Model does not make either of these assumptions to estimate possible ranges for the non-observed parameters and several possible conditions for this Longitudinal Model are depicted in Figs 1–4.

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Fig 1. Response curves representing the likelihood of developing MS in genetically susceptible women (black lines) and men (red lines) with an increasing probability of a “sufficient” environmental exposure–see Methods #1B.

The curves depicted are “strictly” proportional, meaning that the environmental threshold is the same for both men and women–i.e., under conditions in which: (λ = 0)–see Text. The blue lines represent the change in the (F:M) sex ratio (plotted at various scales, indicated in each Figure) with increasing exposure. The thin grey vertical lines represent the portion of the response curve that covers the change in the (F:M) sex ratio from 2.2 to 3.2 (i.e., the actual change observed in Canada [6] between Time Periods #1 & #2). The grey lines are omitted under circumstances either where these observed (F:M) sex ratios are not possible or where both (Zw > Zm) and an increasing (F:M) sex ratio are not possible. Response curves A and B reflect conditions in which (R > 1); whereas curves C and D reflect conditions in which (R < 1). If (R = 1), the blue line would be flat. Response curves A and C reflect conditions in which (c = d = 1); whereas curves B and D reflect those conditions in which (c < d = 1). Under the conditions for curves A and B (R ≥ 1), there is no possibility that the (F:M) sex ratio will be observed to increase with increasing exposure. Under the conditions of curve C–i.e., (c = d = 1) and (R < 1)–at no exposure level is it possible that: Zw = P(MS, E│G, F, E_T) > P(MS, E│G, M, E_T) = Zm. Thus, the only “strictly” proportional model that could possibly account for an increasing (F:M) sex ratio, and for the fact that: (Zw₂ > Zm₂), is a Model in which (c < d ≤ 1) and (R < 1)–i.e., curve D.

https://doi.org/10.1371/journal.pone.0285599.g001

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Fig 2. Response curves for the likelihood of developing MS in genetically susceptible women (black lines) and men (red lines) with an increasing probability of a “sufficient” environmental exposure–see Methods #1B.

Like Fig 1, the curves depicted are also proportional although here the environmental threshold is greater for men than for women–i.e., under conditions in which: (λ < 0)–see Text. The blue lines represent the change in the (F:M) sex ratio (plotted at various scales, indicated in each Figure) with increasing exposure. The thin grey vertical lines represent the portion of the response curve that covers the change in the (F:M) sex ratio from 2.2 to 3.2 (i.e., the actual change observed in Canada [6] between Time Periods #1 & #2). The grey lines are omitted under circumstances where these observed (F:M) sex ratios are not possible. Response curves A reflects conditions in which (c = d = 1) & (R > 1); Response curves B reflects conditions in which (c = d = 1), (R < 1), & (p ≥ p′); curves C reflect conditions in which (c < d = 1) and (R < 1) and curve D reflects those conditions in which (c < d = 1) and (R < 0.5). To account for the observed increase in the (F:M) sex ratio, curves D (compared to curves C) requires a small enough value of (R) so that the (F:M) sex ratio curve dips below 2.2 and, also, a small enough value of (c) so that the curve rises above 3.2. For all points in curves A after the intersection, and for all points in curves B, (Zm > Zw), which is not possible. Curves C never even approach the (F:M) sex ratio of 2.2. By contrast, for curves D, both an appropriate increase in the (F:M) sex ratio and (Zw > Zm), can be observed.

https://doi.org/10.1371/journal.pone.0285599.g002

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Fig 3. Response curves for the likelihood of developing MS in genetically susceptible women (black lines) and men (red lines) with an increasing probability of a “sufficient” environmental exposure–see Methods #1B.

Like Fig 1, the curves depicted are also proportional (R = R^app), but, for these, the environmental threshold in women is greater than that it is in men–i.e., these are conditions in which: (λ > 0). Also, all these response curves represent actual solutions and reflect conditions in which (c = d = 1) and, as discussed in Methods #4C, are representative of all conditions in which c = d < 1). Moreover, with increasing values from (R^app ≥ 1.3), which is the minimum value of (R^app) for any solution–which is depicted in Fig A. The blue lines represent the change in the (F:M) sex ratio (plotted at various scales, indicated in each Figure) with increasing exposure. The thin grey vertical lines represent the portion of the response curve (for the depicted solution), which represents the actual change in the (F:M) sex ratio that occurred between Time Periods #1 & #2). To account for the observed increase in the (F:M) sex ratio, these curves require the Canadian observations [6] to have been made over a very small portion the response curve–i.e., for most of these response curve, the (F:M) sex ratio is decreasing. Also, for each of these response curves, including the maximum difference in the environmental threshold (i.e., λ ≤ 0.13) under conditions of (c = d = 1), which is depicted in Fig B, the ascending portion of the curve (which reflects and increasing F:M sex ratio) is very steep–a circumstance indicating that the portion of the response curve available for fitting the Canadian data [6] is quite narrow. Also, the intersection of the response curves does not occur as early as seems to be implied by an extension of the conditions of Panels C–B. Also, such a rapid transition from an MS that is “male-predominant” to an MS, which is “female-predominant” would seem to fit poorly with the gradual transition, which has taken place over the past two centuries [3, 6, 22–30, 40, 77, 78, 88].

https://doi.org/10.1371/journal.pone.0285599.g003

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Fig 4. Response curves for the likelihood of developing MS in genetically susceptible women (black lines) and men (red lines) with an increasing probability of a “sufficient” environmental exposure–see Methods #1B.

Like Fig 1, the curves depicted are also proportional (R ≤ 1), but, for these, the environmental threshold in women is greater than that it is in men–i.e., these are conditions in which: (λ > 0). Also, these curves represent the same solutions as those depicted in Fig 3 except that these are for conditions in which (c < d ≤ 1). The blue lines represent the change in the (F:M) sex ratio (plotted at various scales, indicated in each Figure) with increasing exposure. The thin grey vertical lines represent the portion of the response curve (for the depicted solution), which represents the actual change in the (F:M) sex ratio that occurred between Time Periods #1 & #2). Unlike the curves presented in Fig 3, however, an increase in the (F:M) sex ratio with increasing exposure is observed for any two-point interval along the entire response curves and, except for Fig A, the grey lines are clearly separated.

https://doi.org/10.1371/journal.pone.0285599.g004

For both Models, the first step is to assign acceptable ranges for the value of certain “observed” parameters (e.g., twin and sibling concordance rates, the population prevalence of MS, or the proportion of women among MS patients). These ranges are assigned such that they always include their calculated 95% CIs. However, for certain parameters, the ranges considered plausible are expanded beyond the limits set by the CIs. The second step is to assign acceptable ranges for the “non-observed” parameters (e.g., the proportion of susceptible persons in the population or the proportion of women among susceptible individuals). These ranges are assigned such that they cover the entire “plausible” range for each such parameter. In both Models, a “substitution” analysis is undertaken to determine those parameter combinations (i.e., solutions), that fit within the acceptable ranges for both the observed and non-observed parameters. These solutions are then used to assess their implications about the basis of genetic and environmental susceptibility to MS in the population. For each Model, the total number of parameter combinations interrogated in this manner was ~10¹¹.

A. Observed parameter values.

For notational simplicity, we sometimes use subscripts (1) and (2) to indicate the parameter values at Time Period #1 and Time Period #2 {e.g., P(MS)₂ = P(MS│E_T) at Time Period #2}. For the purposes of this analysis, those parameter-values observed for persons born between 1976 and 1980 (i.e., Time Period #2) are always taken to be the “current” values. When only this Time Period is being considered, the terms (E_T) and the subscript (2) are generally omitted entirely to simplify the notation.

{NB: In general, for individuals born during Time Period #2 (1976–1980), their MS status cannot be determined until 25–35 years later (i.e., 2001–2015). The estimates of other epidemiological parameters are from reports in the Time Period of (2001–2015), which is also when the Time Period #2 (F:M) sex ratio is reported [6, 7, 11–15]. For this reason, Time Period #2 is considered fixed as the “current” period. However, because the (F:M) sex ratio has increased between every previous 5-year epoch and Time Period #2 [6], the choice of any specific Time Period #1 is equivalent to any other. For our Time Period #1, we chose the 5-year epoch (1941–1945) because it was the earliest epoch with the narrowest CI [6].}

The 2010 Canadian census [8], reported that the proportion of women among the general Canadian population (Z) is 50.4%. Thus, men and women comprise essentially equal proportions of this population and, therefore, the probabilities of the events that an individual, randomly selected from (Z), is a woman or a man–P(F) and P(M), respectively–are each ~50%. Therefore, by definition: and:

The proband-wise concordance rate [7] for MS in MZ-twins, currently observed in Canada, is: with (n = 146) twin-pairs included in this estimation [7]. Therefore, the 95% CI for this parameter, calculated from an exact binomial test [21], is:

The estimates, from different studies, for the “current” proportion of women among the MS patients–i.e., P(F│MS)₂ –in North America ranges between 66% and 76% [3]. For the Cross-sectional Model, we expanded the “plausible” range beyond the 95% CI calculated from “current” Canadian data presented below [6]. Thus, for this Model, we considered the range:

The reason for this is because the “current” estimated range from the Canadian study is quite narrow and some solutions, which fall within the range of different estimates from other locations in North America [3], might be excluded. This choice permits a wider range of possibilities to be considered as solutions for our Cross-sectional Model.

By contrast, for our Longitudinal Model, because we were interested specifically in how the parameter P(F│MS) has changed for the Canadian population over time [6], we used the 95% CIs (from this single study) to estimate the ranges for this parameter value during each Time Period. For example, the proportion of women among the MS patients in Canada was 69% for patients born during Time Period #1 (1941–1945) and this proportion was significantly less (p < 10⁻⁶) than the 76% observed for patients born during the “current” Time Period #2 (1976–1980) [6]. Although the authors of this study, do not report the actual numbers of individuals in each 5-year epoch, they do report that their 5-year samples averaged 2, 400 individuals per epoch [6]. Also, the authors graphically present the 95% CIs for the (F:M) sex ratio during each of these 5-year epochs in the Figure of their manuscript [6]. Estimating that the number of individuals in both Time Periods #1 and #2 is ~2, 000, and using an exact binomial test [21], for our Longitudinal Model, we estimate that:

Both of these ranges exceed those (based on the 95% CIs) presented in the Figure of the manuscript [6].

The “current” proband-wise concordance rates for MS in female and male MZ-twins, observed in Canada, are 34% and 6.5%, with the total number of female and male twin-pairs included in these calculations being (n₁ = 100) and (n₂ = 46), respectively [7]. Using an exact binomial test [21], and using the definitions provided in Table 2, the CIs for these observations are:

The 95% CI for the difference in MZ-twin concordance between men and women is calculated as:

In which case:

This large and significant difference in the current relative penetrance values between susceptible women and men for their MZ-twin concordance rates, strongly suggests that, currently, the same relative penetrance also pertains to the (F, G) and (M, G) subsets (see Section 2c in S1 File). Therefore, we assume that:

Previously, we used three independent methods (based on observation) to estimate the value of P(MS)₂ [3]. The first method relied on measures of the population prevalence of MS in North America together with the observed age-distribution for MS-onset, the second method considered the age-specific prevalence of MS in the age-band of 45–54 years, and the third method considered a population-based multiple-cause-of-death study from British Columbia, which reported the proportion of death certificates that mention MS. The parameter-value range supported, collectively, by these different methods was: 0.0025 ≤ P(MS)₂ ≤ 0.0046 [3]. Nevertheless, for the purposes of the present analysis, we expanded the “plausible” range for this parameter to include:

B. Non-observed parameter values.

In addition to the observed parameter values (above), and using the definitions in (Table 2), we determined acceptable values for 12 additional parameters: P(G); p = P(F│G); x = P(MS│G)₂; x′ = P(MS│IG_MS)₂; x₁ = P(MS│F, G)₂; ; ; and the ratios: (s_a = P(MS│DZ_MS)₂)⁄P(MS│S_MS)₂; ; ; and: .

Most of these parameters vary depending upon the level of exposure–i.e., all except P(G) and P(F│G). Therefore, the acceptable ranges were estimated for the “current” Time Period #2. In several cases, there are constraints on the values that these non-observed parameters can take. For example, P(MS) has been observed to be increasing, especially (but not only) among women, in many parts of the world between the two Time Periods [6, 22–30]. Therefore, the parameter (C) is constrained in three ways. First, it must be that:

Second and third, on theoretical grounds (see Section 7a in S1 File), the value of (C) is also constrained such that: and:

In this case, using the limits, provided earlier, for the proportion of women among MS patients during different Time Periods, the ratio (C) is at its maximum possible value when:

Therefore, on these theoretical grounds, the value of (C) is further constrained such that: and:

Only Constraint #2 (above) satisfies all three, so that the maximum upper bound for (C) is 0.90. Nevertheless, the actual upper bound for (C) will depend upon the values that P(M│MS)₁ and P(M│MS)₂ take for in any specific solution. Moreover, if (C < 0.25) then there must have been a greater than 4-fold increase in P(MS) for Canada, which has taken place over a 35–40 year-interval. This seems to be an implausibly large increase based on the available data [6, 22–30]. Therefore, we conclude that:

Because MS develops in some individuals, the parameter P(G) cannot be equal to 0. Also, because both women and men can develop MS, the parameter P(F│G) cannot be equal to either 0 or 1. Therefore, the plausible ranges for these parameters are:

Furthermore, the penetrance of MS for the subsets (F, G) and (M, G) can be expressed (see above) such that:

Because everyone who develops MS must be a member of the (G) subset, therefore, considering only these subsets, the ratio of women to men, during any Time Period can be expressed as: or:

Consequently, using the definitions provided above and in Table 2, the parameters (x₁), (x₂), (x), (p) and (p′), during any Time Period for the gender partition, are related such that: (1a) or, equivalently: (1b) (1c) (1d)

These relationships require no assumptions and (x₁), (x₂), (x), (p) and (p′) must always satisfy Eqs 1a–1d during any Time Period, regardless of which Model is employed in the analysis [3].

Based on theoretical considerations (see Section 1b in S1 File) for the parameter (s_a)–see below–we demonstrate that: (s_a ≥ 1). Indeed, this relationship is confirmed observationally, where the recurrence risk of MS for a proband with a co-twin who is a member of the (DZ_MS) subset, is consistently reported to be greater than the recurrence risk of MS for a proband sibling with a co-sibling, who is a member of the (S_MS) subset [7, 31–37]. Therefore, from the definitions of (MZ_MS) and (IG_MS)–see Sections 1b & 7b in S1 File–it must be the case that, during our “current” Time Period:

Using the constraints (above) on P(MS│MZ_MS)₂, P(MS│F, MZ_MS)₂ and P(MS│M, MZ_MS)₂, therefore, the plausible ranges for (x), (x₁) & (x₂) during the 2^nd Time Period are: and:

As noted above (Methods #1D), the observed MZ-twin concordance rate may be increased due to the fact that the proband disproportionately shares the (E_twn) and (E_sib) environments with a co-twin who has (or will develop) MS. Notably, however, any such impact (if it exists) must represent an environmental influence. Therefore, the maximum probability of developing MS for susceptible individuals under optimal environmental conditions–i.e., P(MS│E)–must be greater than the currently observed MZ-twin concordance rates (see Section 7b in S1 File). Consequently, we can use the Table 2 notations, and the definitions of (c) and (d)–see Methods #4A; below–to demonstrate that, because, currently, (Zw₂ > Zm₂), and because both P(MS) and P(F│MS) are currently increasing, each of the following relationships must hold simultaneously: and:

Notably, these relationships include the possibility that: c = d = 1

Finally, as discussed in (Section 1c in S1 File), we estimate the impact of the disproportionately shared (E_twn) and (E_sib) environments for MZ-twins as: and:

In the Canadian data [7], the life-time probability of developing MS for the proband of a co-DZ-twin with MS (5.4%) was found to be greater than that for the proband of a non-twin co-sibling with MS (2.9%). From these observations, the point-estimate for (s_a) becomes:

This point estimate is approximately the same for both men and women (see Section 1d in S1 File). Thus, these observations from Canada suggest that sharing the (E_twn) environment with a co-twin who develops MS markedly increases the likelihood of the proband twin developing MS for both men and women [3]. Nevertheless, it is possible that impact of these disproportionately shared environments may be over- or under-estimated by the Canadian data [7]. In any event, based on theoretical considerations (see Section 1d in S1 File), if we use the point-estimate from the Canadian data [7] that: then:

Therefore, for the purpose of both Models, we considered the plausible range for (s_a) to be:

However, because, the point-estimate for (s_a) from the Canadian data [7] is generally greater than that reported in other similar studies [31–37], we also considered, separately, the more restrictive circumstances, in which: 1 ≤ s_a < 1.9

A. General considerations.

The Longitudinal Model is developed in detail in Sections 4a–c; 5a; & 6a–c in S1 File. Following standard survival analysis methods [39], we define the cumulative survival {S(u)} and failure {F(u)} functions where: F(u) = 1 − S(u). These functions are defined separately for men {S_m (u) and F_m (u)} and for women {S_w (u) and F_w (u)}. In addition, we define the hazard-rate functions for developing MS at different exposure-levels (u) in susceptible men and women {i.e., h(u) and k(u), respectively}. These hazard-rate functions for women and men may or may not be proportional to each other but, if they are proportional, then: k(u) = R * h(u), where (R > 0) represents the hazard proportionality factor. Furthermore, as defined previously (see Methods #1B), the term P(E│G, E_T) represents the probability of the event that a member of the (G) subset, selected at random, will experience an environmental exposure “sufficient” to cause MS, given their unique genotype and given the prevailing environmental conditions of the time (E_T). We define the exposure (u) as the odds that the event (E) occurs during the Time Period (E_T) such that:

We further define H(a) to be the cumulative hazard function (for men) at an exposure-level of (u = a) such that:

Similarly, we define K(a) to be the cumulative hazard function (for women) at the same exposure-level of (u = a) such that:

and, if the hazards are proportional, then:

In Section 4a in S1 File), we develop this Longitudinal Model and demonstrate there that these cumulative hazard functions are exponentially related to the cumulative survival. Thus: (3a) and: (3b) where: (3c) and: (3d)

Notably, also, because the response curves for both men and women are exponential, any two points of observation on these curves will determine the entire curve. Designating the fixed (but unknown) exposure level (a) at Time Period #1 as (a₁), and at Time Period #2 as (a₂), we can then use the values of Zw and Zm during these two Time Periods to determine, and thus to plot, the entire response curve separately for susceptible women and susceptible men–see Section 4a, Equations S6a & S6b and S7a & S7b in S1 File.

{NB: In this circumstance, we are using the cumulative hazard functions, H(a) and K(a), as measures of exposure for susceptible men and women, not as measures of either survival or failure. By contrast, failure, as defined here, is the event that a person develops MS over the course their life-time. As an example, for men, the term: Zm = P(MS, E│M, G, E_T) represents the probability that, during the Time Period (E_T), this failure event occurs for a randomly selected man from the (M, G) subset of (Z). Notably, also, the exposures of H(a) and K(a) are being used in preference to the, perhaps, more intuitive measure of exposure (u = a) provided above. Nevertheless, when {P(E│G, E_T) = 0}; both exposure measures are zero–i.e., {a = 0} and {H(a) = K(a) = 0}. Also, as the value of: {P(E│G, E_T) → 1}; both exposure measures become infinite–i.e., {a → ∞} and {H(a) & K(a) → ∞}. And, finally, all of these exposure measures increase monotonically with increasing P(E│G, E_T). Therefore, the mapping of the (u = a) measure to either the H(a) and K(a) measures is both one-to-one and onto. Consequently, all these measures of exposure are equivalent and the use of any of these exposure scales is appropriate. Although the relationship of both the H(a) and K(a) scales to P(E│G, E_T) is less obvious than it is for the (u = a) scale where: P(E│G, E_T) = a/(a + 1), the H(a) and K(a) scales, nonetheless, have the advantage that the probability of failure for each is an exponential function of exposure as measured by H(a) or K(a) and, thus, these scales are more mathematically tractable.}

Environmental exposure levels during different time periods. As developed in Section 4a–4c in S1 File, and because both P(MS) and P(F│MS) are increasing with time in both women and men [6, 22–30], we can define the change in the fixed (but unknown) exposure level that has taken place between the two Time Periods in men and women {i.e., (q_m) and (q_w), respectively} such that: and:

Previously, we assigned the value of these arbitrary units as (q_m = 1) and (q_w = 1) in these Equations [3], although such an assignation may be inappropriate. Thus, these units (whatever they are) still depend upon the actual (but unknown) level of environmental change that has taken place for men and women between the two chosen Time Periods. From Eqs 3a and 3b (above), these exposure levels depend upon the values of (c) and (d), which can range over the intervals of: (1 ≥ c > Zm₂) and (1 ≥ d > Zw₂); the exposure level for each gender being at its minimum value (i.e., and ) when (c = 1) for men and when (d = 1) for women–see Section 4b in S1 File.

However, these minimum exposure level changes, and , may not accurately characterize the actual (but unknown) level of environmental change, which has taken place for susceptible men and for susceptible women between the two Time Periods. Therefore, we will refer to (q_m) and (q_w) as the “actual” exposure-level changes, which may be different from these minimum exposure-level changes such that: and:

Relationship of failure to true survival. Unlike true survival (where everyone dies given sufficient time), the probability of developing MS, either for the subset of susceptible women {Zw = P(MS, E│F, G)} or for the subset of susceptible men {Zm = P(MS, E│M, G)}, may not approach 100% as the probability of exposure {P(E│G, E_T)} approaches unity (see Section 4a–4f in S1 File). Moreover, the limiting value for the probability of developing MS in susceptible men (c) need not be the same as that in susceptible women (d). Also, even though the values of the (c) and (d) parameters are unknown, they are, nonetheless, constants for any disease process, which requires environmental factors as an essential component of disease pathogenesis, and they are independent of whether the hazards are proportional. Finally, the threshold environmental exposure (at which MS becomes possible) must occur at: P(E│G, E_T) = 0; for one (or both) of these two subsets, provided that this exposure level is possible [3]. If the hazards are proportional, a difference in threshold (λ) can be defined as the difference between the threshold in susceptible women (λ_w) and the threshold in susceptible men (λ_m)–i.e., (λ = λ_w − λ_m). Thus, if the threshold in women is greater than the threshold in men, (λ) will be positive and (λ_m = 0); if the threshold in men is greater than the threshold in women, (λ) will be negative and (λ_w = 0).

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 (E_T) are such that: P(E│G, E_T) = 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 (E_T). 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.

Relationship of the (F:M) sex ratio to exposure. Finally, (see Section 4d in S1 File) regardless of (λ), and regardless of any proportionality, during any Time Period, the ratio of the failure probability in susceptible women to that in susceptible men (Zw⁄Zm) can be expressed as: (4)

Consequently, any observed disparity between (Zw) and (Zm), 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 (c) and (d), or to a difference in both. Therefore, by assuming that: (c = d ≤ 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 (i), the exposures {E_i} and {E_iw} are both population-wide and fixed during any Time Period (E_T). Because these exposures are “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 men–i.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 (λ ≠ 0) were explained by a systematic difference in behavior, then the observation of (λ > 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: (Zw₂ > Zm₂).

Another possible explanation for (λ > 0), which does not pose this difficulty, is that the distributions of the “critical exposure intensity” levels (thresholds) differ between men and women (see Section 6g in S1 File). 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 in detail elsewhere (see Sections 6g & 8a, 8b in S1 File).

Also, regardless of whether the hazards are proportional, and because proportion of women among susceptible individuals (p) is a constant (see Table 2), therefore, for any solution, the ratio (Zw⁄Zm), during any Time Period (E_T), will be proportional to the observed (F:M) sex ratio during that period (see Equation 1b; above).

The response curves to increasing exposure. As noted above, any two points of observation on these exponential response curves will define the entire curve (e.g., the values of Zw and Zm during Time Period #1 and Time Period #2). 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, in which case the values of (R = q_w⁄q_m) and (λ) are determined by a combination of the observed values for (Zw₂), (Zm₂), and the (F:M) sex ratio change and the fixed (but unknown) values of (c), (d), and (C)–see Sections 4b (Equations S6e & S7c), 6a (Equation S11c, S11d) & 7a in S1 File–see also Summary Equations (below), Moreover, the values of: (c) and (d) are independent of whether some susceptible individuals can only develop MS in response to extreme (and improbable) environmental conditions (see Section #1B, above). This is because these values are determined exclusively by the values of: (Zw), (Zm), P(MS) and the (F:M) sex ratio, at the two observation points. Nevertheless, if all susceptible individuals can develop MS in response to such extreme conditions, then: (c = d = 1).

B. Non-proportional hazard.

If the hazard functions for MS in men and women are not proportional (see Sections 5a & 6g, 6h in S1 File), it is always possible that the “actual’ exposure level changes for men and women are each at their “minimum” values–i.e., () and ()–in which case: (c = d = 1)–see Methods #4C (below). However, in this circumstance, the MS in women must be considered to be a separate disease from MS in men (see Section 6g, 6h in S1 File). Also, we should note that the condition of: (c = d = 1) is required, unless “true” randomness is a component of disease pathogenesis (see Discussion Section).

Moreover, in this non-proportional circumstance, the various observed and non-observed epidemiological parameter values still limit possible solutions. However, in this case, although (c) and (d) will still be constants, no information can be learned about them or about their relationship to each other from changes in the (F:M) sex ratio and P(MS) 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 (λ)–which relates these thresholds to each other–is meaningless, and there is no hazard proportionality factor (R).

Nevertheless, even with non-proportional hazard, the ratio (Zw⁄Zm), during any Time Period, must still be proportional to the observed (F:M) sex ratio during that Time Period (see Equation 1b) and, if: c = d ≤ 1, then any observed disparity between (Zw) and (Zm), must still be due entirely to a difference between women and men in the likelihood of their experiencing a “sufficient” environmental exposure (E) during that Time Period (see Eq 4; above).

C. Proportional hazard.

By contrast, if the hazards for women and men are proportional with the proportionality factor (R), the situation is altered. First, because (R > 0), those changes, which take place for the subsets P(F, MS) and P(M, MS) over time, must have the same directionality. Indeed, this circumstance is in accordance with our epidemiological observations where, over the past several decades, the prevalence of MS has been noted to be increasing for both women and men [6, 22–30]. Second, including a possible difference in threshold between the genders, the proportionate hazard Model can be represented by those circumstance for which: (5a) so that, from the Sections 4a & 6a in S1 File, for men and women, at the 1^st and 2^nd Time Periods, can be re-written as: (5b) (5c) (5d) and: (5e)

In this circumstance, as demonstrated in Section 6a in S1 File, during any Time Period, the parameters (λ) and (R) are determined such that: (6a) and: (6b)

When: (R = 1) these Equations simplify to: (6c) and: (6d)

For any specific exposure level {H(a) > λ}, the quantities (Zw) and (Zm) are unknown. However, considering any disease for which a proportionate hazard Model is appropriate, the parameters (c, d, R, & λ) are fixed (but unknown) constants, so that, from Equation S11a, S11b, Section 6a in S1 File, the values of (Zm) and (Zw) are also fixed at any specific exposure level {H(a)}.

Defining an “Apparent” proportionality factor. We can also define a so-called “apparent” value of the hazard proportionality factor (R^app) such that: . This “apparent” 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 Eq 4, this is the only interpretation possible for circumstances in which: (c = d ≤ 1). Second, however, if we admit the possibility that: (c < d ≤ 1), then as shown in the Section 6b in S1 File some of (R^app) will be accounted for by the difference of (c) from unity. For example, when (d = 1), it will be the case that:

In this manner, if , a portion of the “apparent” value (R^app) will be accounted for by a reduction in value of (c) 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 (d) is also reduced from unity in susceptible women, in which case: (c ≤ d < 1), where the “actual” exposure level in women (q_w) would be greater than its minimum value such that:

Consequently, in each of these circumstances, the “actual” value of (R) may be different from its “apparent” value (R^app). Nevertheless, from Eqs 6a and 6b (above), and from the Sections 4b & 6a, 6b in S1 File, under all circumstances, for which: (c = d ≤ 1), then:

Consequently, under those conditions, for which: (c = d = 1), this requires that both: or, equivalently:

By contrast, under those conditions, for which: (c = d < 1), this requires that both: or, equivalently:

Implications that the (R) value has for the values of (λ), (c) and (d). As demonstrated in Section 6c in S1 File, and based solely on the observations of an increasing P(MS) and an increasing (F:M) sex ratio with time [6, 22–30]–a circumstance which is true considering the “current” Time Period #2 together with any of the reported previous 5-year epochs as Time Period #1 [6]–we can conclude, based on purely theoretical grounds, that, if the hazards are proportional, then: and also:

D. Strictly proportional hazard: (λ = 0).

As demonstrated in Section 6c, 6d in S1 File, if (R ≥ 1) and (λ = 0), the observed (F:M) sex ratio either decreases or remains constant with increasing exposure (see Methods #4 Equations 11a–11c), regardless of the parameter values for (c) and (d)–e.g., Fig 1A & 1B. Consequently, the only “strictly” proportional circumstances, which are possible, are those in which men have a greater hazard than women and where (c < d ≤ 1)–e.g., Fig 1D.

{NB: In these and subsequent Figures, all response curves exemplifying the conditions in which (c = d ≤ 1), are depicted for the condition (c = d = 1). Nevertheless, for all those conditions where (c = d < 1), the response curves differ from the curves depicted in the Figures only in so far as the y-axis has a different scale. Therefore, the response curves, depicted at: (c = d = 1), are representative of all curves for which(c = d)–see note in Section 6b in S1 File.}

E. Intermediate proportional hazard: (λ < 0).

We can also consider another possible Model, which is intermediate between the “strictly” proportional and non-proportional hazard Models as described above. In this intermediate Model, the hazards are still held to be proportional although the onset of the response curves are offset from each other by an amount (λ ≠ 0). As noted earlier: ∀ (R ≥ 1): λ > 0 and, therefore, for those circumstances in which (λ < 0), the hazard in men must be greater than the hazard in women (see Section 6e in S1 File). Otherwise, the (F:M) sex ratio will decrease with increasing exposure, which is contrary to the evidence [6]–e.g., Fig 2A. Moreover, under those conditions, for which (c = d ≤ 1) & (R < 1) & (λ < 0), the (F:M) sex ratio will decrease with increasing exposure until the two response curves have intersected (e.g., Fig 2A & 2B), reaching a level below {p/(1 − p)}. Following this, the (F:M) sex ratio steadily increases to ultimately reach the level of {p/(1 − p)}. However, after the response curve in men intersects that in women (i.e., after this nadir), this circumstance requires that (Zm > Zw) throughout the entire remaining response curve until an (F:M) sex ratio of: {p/(1 − p)} is reached (e.g., Fig 2A & 2B). Thus, the only circumstance in which the Model of (c = d ≤ 1) is possible is one in which (p) is at least as large as the “current” value of (p′)–i.e., see Fig 2B; see also Section 2c in S1 File. Each of these possibilities is contrary to evidence where currently (Zw₂ > Zm₂) and, thus, where: {(F:M) sex ratio > p/(1 − p)}–see Equation 1b (above). Thus, the condition of: (λ < 0) is only possible, in circumstances where: (c < d)–e.g., Fig 2D.

F. Intermediate proportional hazard: (λ > 0).

Exposure “Intensity. In considering the notion of exposure “intensity”, three conclusions that seem well established. First, for every proportional hazard solution that we identified (see Results Section), we found that: (R^app > 1). Moreover, as demonstrated on theoretical grounds in Section 6b, 6c in S1 File), and as depicted in Figs 4 and 5 & S1 Fig in S1 File, in these circumstances, it must be that:

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Fig 5. Hypothetical relationship between exposure “intensity” and disease expression (see Sections 6g & 8a, 8b in S1 File).

Plotted on the x-axis is the level (or “intensity”) of exposure in units of the log-transformed exposure–log(a). Plotted on the y-axis is the proportion of the susceptible population (G) who experience an exposure “sufficient” to cause MS in them. The solid black lines represent the distribution of “actual” level of exposure experienced by the susceptible population. The dotted lines (red for women and blue for men) represent the distributions of these “critical exposure intensity” (or “threshold”) levels for susceptible men and women. These “threshold” levels for each individual are defined as that exposure level, at (or above) which, the exposure becomes “sufficient” to cause MS in that person. These threshold distributions have been plotted, arbitrarily, for conditions of (p = 0.5). Because (a) is the odds of exposure, the distribution of these “threshold” levels are expressed in units log(a), because this transformation will generally normalize the variance [39]–see also Section 8a, 8b in S1 File. In these Figures, the exposure level of: {log(a) = 0}, has been chosen as the point where the average odds of a “critical exposure intensity” level is equal to (1). No other units are provided because these are undefined other than as they relate to the variance of these “threshold” distributions in susceptible men and women ( and ), respectively. The circumstances depicted are those, in which men and women have the same variance but men have a lower mean compared to women (i.e., ). In any case, however, because (λ > 0), men must disproportionately (or exclusively) experience a “sufficient” exposure at low exposure “intensities”. In these examples, the blue shading represents those individuals who receive a “sufficient” exposure as the level of population exposure increases progressively–i.e., Fig 5A depicts the circumstance, in which the population exposure is such that no one experiences a “sufficient” exposure; Fig 5B and 5C depict circumstances, in which some (but not all) individuals experience a “sufficient” exposure; and Fig 5D depicts the circumstance where the population exposure has increased to the point where it exceeds the “critical exposure intensity” level for everyone.

https://doi.org/10.1371/journal.pone.0285599.g005

Second, as demonstrated in Section 6c in S1 File, under those circumstances, in which both P(MS) and P(F│MS) are increasing with time [6], 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) R ≤ 1 in which case: c < d
2. or: 2) R > 1 in which case: λ > 0

Clearly, the first of these conditions excludes the possibility that: c = d = 1

In considering the second condition, it should be noted that both of our measures of exposure–i.e., (a) and H(a)–relate directly back to the parameter P(E│G), 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 condition–i.e, that: λ > 0 –indicates that, as the probability of a “sufficient” exposure decreases, there comes a point {i.e., H(a) = λ} where only susceptible men can develop MS. This implies that, at (or below) this point: (R ≈ 0).). Consequently, the requirement that (R > 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 {P(E│G)} 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 potential 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). For example, because women and men of the same “i-type” necessarily have proportional hazards (see Section 6h in S1 File), in this case, we would also have to conclude that susceptible women and men can never be in the same “i-type” group and, thus, that each gender requires distinct sets of environmental conditions to develop MS. Also, 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).

The second way to avoid the paradox, is to conclude that the first of the two possible proportional hazard conditions is true–i.e., that both (R ≤ 1) and: (c < d). Notably, the condition of: (R ≤ 1) is compatible with any value of (λ). However, if (λ > 0), the simultaneous condition of: (R ≤ 1), offers, at least, a consistent interpretation of the existing data because, under these conditions, at every population exposure level (a), 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 S1 Fig in S1 File, Figs 4 & 5). Thus, the notion of a “critical exposure intensity” (discussed below), although it may be necessary to rationalize a 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 that individual.

Nevertheless, despite the paradox created by the possibility that: (λ > 0) & (R > 1), there are potential rationales for resolving it. For example, one way to explain it might be if some susceptible men and women had “purely genetic” MS [3]–i.e., that these individuals can develop MS under any environmental circumstance. If the proportion of “purely genetic” MS were equal in women and men, such individuals could, effectively, raise the x-axis for each of the response curves {i.e., increase the level of (0) on the y-axis} to the point where the response curves for men and women intersect, in which case, there would be no lag when considering “environmental” MS, alone [3]–see Fig 4. However, after this intersection (i.e., after the onset of these “effective” response curves), the conditions would be identical to those described for (λ = 0)–see Section 6c, 6d in S1 File–in which case, when (c = d = 1), the F:M sex ratio will decline with increasing exposure–a possibility that is counter factual [6, 22–30]. Naturally, the proportion of “purely genetic” MS might not be equal in women and men but, in such a circumstance, the paradox would remain unresolved [3]. Consequently, a “purely genetic” rationale is not possible when: (c = d = 1).

However, other potential rationales can also be envisioned (see Section 6g, in S1 File). To do this, we introduce the notion of a “critical exposure intensity” level, as the exposure level, at (or above) which, the exposure becomes “sufficient” for each susceptible individual (see Fig 5). If this notion is appropriate, it could help to rationalize the paradox of having both: (λ > 0) & (R > 1)–see Sections 6g & 8a, 8b in S1 File. However, to accommodate the condition of: (c = d = 1), the required circumstances are extreme–e.g., S2, S3 Figs in S1 File–and don’t match well with the response curves presented in Fig 3. By contrast, in all circumstances, those conditions for which (c < d) are much simpler, don’t require any extreme circumstances, fit with any value of (R), and match well with the response curves depicted in Fig 4 –e.g., S1–S3 Figs in S1 File.

Exposure “Intensity” in susceptible women. Any condition for which (λ > 0) indicates that there must be some environmental conditions in which only susceptible men can experience a “sufficient” exposure. This circumstance requires that the threshold difference for, at least, some “i-types” (λ_i), is such that: (λ_i > 0). We will define the family of exposures {E_iw} to be the subset of exposures, within the {E_i} family, that are “sufficient” for susceptible “i-type” women such that: where, for at least one (i), it must be the case that:

In turn, as for our earlier definition of (E)–see Methods #1B –we define the event (E_w) to represent the union of the (m_it) disjoint events, which exhibit the pairing of susceptible “i-type” women with “sufficient” environments, where: and:

Exposure variability (i.e., for R_i & λ_i) in “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 (v_i) “sufficient” sets of exposures within the{E_i} family for this group. In these circumstances, these specific i-types, considered separately, can be plotted on the same x-axis and, thus, will necessarily exhibit proportional hazards for the two genders–see Sections 4f & 6h in S1 File. Moreover, as demonstrated in Section 6h in S1 File, the condition of proportionality for the entire susceptible population will still be present, regardless of whether different i-types have different proportionality constants, and regardless of whether susceptible individuals of different i-types have different threshold differences between men and women.

Contrasting the possibilities that: (c = d) or (c < d). When (λ > 0), to account for an increase in the (F:M) sex ratio, as shown in Fig 3, although it is possible for: (c = d ≤ 1), this circumstance, nevertheless, seems unlikely. First, to achieve an F:M sex ratio, which reaches its current level (i.e., p ≈ 0.76), requires either that both the value of (λ) is small and the value of (R) is large (e.g. Fig 3B–3D), or that the value of (p) is large (e.g. Fig 3A). And second, the ascending portion of the response curve is very steep, which indicates that any change in the (F:M) sex ratio is quite large in response to small changes in exposure. Thus, the window of possible changes in environmental exposure necessary to explain the Canadian data is quite narrow [6]. Also, if notions of a “critical exposure intensity” (see above; see also Section 6g in S1 File) are correct, then neither of these conditions fit well with a transition from a male predominant MS to a female predominant MS, which takes place relatively late in the response curve, even under extreme conditions–see Section 8a, 8b in S1 File. Moreover, following this narrow window, and for most of these response curves, the (F:M) sex ratio is declining–a circumstance, which is contrary to evidence [6, 22–30]. Also, finally, the increase in failure rate (i.e., the increase in the penetrance of MS for the population), which was observed in Canada between the two Time Periods, was large (>32%) and especially prominent among women (see Section 7a in S1 File). Thus, although compatible with the condition of: (c = d = 1), each of the required circumstances seem to be at odds with the Canadian data, which demonstrates that the (F:M) sex ratio has been steadily, and gradually, increasing over many decades [6] and where, currently, the proportion of women among MS patients is quite high. By contrast, the circumstances of Figs 1D, 2D & 4A–4D & S1–S3 Figs in S1 File–i.e., where c < d ≤ 1)–result in a continuously increasing (F:M) sex ratio with increasing exposure over most (or all) of the response curves, they easily account for the magnitudes of the observed (F:M) sex ratios, they don’t invoke extreme circumstances, and, as in Figs 1D & 4A–4D, they could also account for the observation that, at an earlier Time Point in the history of MS [40], in both Europe and the United States, the proportion of men among individuals with MS seemed to substantially exceed that of women such that:

In conclusion, therefore, as indicated in Methods #4C, the condition of: (c < d ≤ 1) is necessarily true for all circumstances, in which (R ≤ 1) and also, as discussed above, seems likely to be true for those circumstances, in which: (R > 1).

Summary equations. For each of the two proportional hazard Models, we can use both the observed parameter values, the change in the (F:M) sex-ratio, and the change in P(MS) for Canada between any two Time Periods [6], and, thereby, construct each of these response curves in their entirety [3]. The values for: Zw₂, Zm₂, Zw₁, Zm₁, I, (d), P(E│G, F), P(E│G, M), (C), and (λ) can then be determined [3] as: and:

For the non-proportional Model, those parameters, which include (c) or (d), cannot be estimated from the observed changes in P(MS│F) and P(MS│M) over time. Notably, the values for P(F│MS)₁, P(F│MS)₂, and P(MS)₂ have been directly or indirectly observed [3, 6]. Also, the values of (Zw₁), (Zw₂), (Zm₁), (Zm₂), P(E│G, F)₂, P(E│G, M)₂, (c) and (d) are, not surprisingly, only related to the circumstances of either men and women, considered separately. Using a “substitution” analysis, we wrote a computer program, which incorporated the acceptable parameter ranges (see Methods #2; above) for the parameters {P(G); P(MS│MZ_MS)₂; p = P(F│G); P(MS)₂; P(MS│F, G)₂; P(F│MS)₁; r; s; s_a and C}, into the governing equations (above) and determined those combinations (i.e., solutions) that fit within the acceptable ranges for both the observed and non-observed parameters (see Methods #2). For this analysis, unlike for our Cross-sectional Model, we loosened the constraints on the values of (r) and (s) such that: