We present a fictional scenario that, while undeniably whimsical, provides the foundation for a unique exercise in extended problem solving, physics analysis, and quantitative model development. Starting with the foundational premise of the Wild Cards shared-world superhero universe, we demonstrate how a variety of concepts appropriate to the advanced undergraduate level—ergodicity, functional analysis, Lagrangian mechanics, and the ever-important simplifying approximation—can be combined into a rich, coherent mathematical model. The goal of this case study is to develop a useful pedagogical exercise in exploring an open-ended research question that presents, at first glance, no clear path forward. Being both eclectic and lengthy, this exercise offers a unique way for students to apply their core physics and mathematics education. It is perhaps best used within a senior honors seminar or within a brief (e.g., January term) elective class.

Readers of the American Journal of Physics may be aware that superhero comic books have proven a successful tool for introducing physics concepts both outside1 and inside2 the classroom. Situations and scenarios raised by these and other popular media can be used to devise entertaining and educational physics problems for students. A more traditional, and far more ubiquitous, tool comes in the form of textbook problems with known solutions. Yet, while the immense pedagogical value of this approach cannot be overstated, it runs contrary to the researcher's raison d'être. Although the experience of a jobbing theoretician occasionally involves checking answers (one's own, and sometimes others' via peer review), a person investigating a research problem of enduring interest rarely has the solace of knowing a straightforward, concise, and exact solution exists.

This article seeks to adapt an entertaining fictional scenario into the foundation for an extended exercise in open-ended problem solving. Starting from the general premise, we combine a variety of concepts appropriate to the advanced undergraduate level to build a straightforward yet analytically rich model. We hope this case study illustrates how rewarding physics explorations may be mined from even the most whimsical scenarios.

To that end, a fictional sandbox possibly unfamiliar to many AJP readers, even those conversant with comic-book-adjacent genre media, is the Wild Cards shared-world superhero universe.3 This multi-volume series of novels presents a setting in which the release of an extraterrestrial pathogen on Earth wrought profound changes upon human physiology, capability, and society.4 Within the series canon, the so-called “wild card” virus is observed to exhibit a more-or-less fixed statistical distribution of outcomes, regardless of the size and character of the affected population. Of every 100 latent carriers who experience viral expression within their bodies (or, in the parlance of Wild Cards, when their “cards turn”) 90 experience a fatal outcome; 9 are physically mutated, often profoundly so; and 1 obtains a superhuman ability (examples include flight, trans-dimensional portal travel, and many other abilities). The mutated carriers are known as jokers, while those fortunate few who obtain powers are known, perhaps inevitably, as aces. We call this the 90:9:1 rule: 90 fatalities for every 9 jokers and 1 ace. Individual manifestations of the non-fatal outcomes are stochastic and uncontrollable (hence the name of the virus). The result is permanent. Aces, jokers, and the 90:9:1 rule together define the foundational premise of the Wild Cards universe.

Leaving aside the inherently un-answerable question of how any virus, extraterrestrial or otherwise, could imbue human beings with a panoply of physics-abusing powers, the premise prompts us to ask: What generates the fixed empirical 90:9:1 rule?

While undeniably fantastical, this scenario offers a fruitful playground for educational problem-solving. Our strategy is to adopt the mindset of a theoretician in the Wild Cards universe (really, a stand-in for students working through a guided exercise) who desires to build a coherent mathematical framework for this viral behavior. Our overarching goal is to demonstrate the wide-ranging flexibility and utility of physical concepts by converting this vague and seemingly unapproachable problem to a straightforward dynamical system—thereby putting a wealth of conceptual and mathematical tools at students' disposal.

In fact, the system can be distilled into a short Lagrangian (with appropriately calibrated parameters).5 That destination lends a focused direction and logical strategy to our investigation. Further, by placing each step in the context of that strategy, we strive to keep the overall flow of the investigation clear and meaningful to students not yet familiar with Lagrangian mechanics. We provide analogies to related problems or concepts that students might find useful, and motivate particular choices with discussions of alternate, less successful, approaches.

This exercise is both lengthy, comprising several discrete steps, and eclectic, combining various concepts, students' grasp of which may be tenuous. We see it as a unique (and, we hope, interesting) means of augmenting their physics education, not something to be grafted onto their core curriculum. It might be best suited to use within a senior honors seminar.

The rest of this article proceeds as follows. After introducing three axioms for model development in Sec. II, we propose in Sec. III a model space that accommodates the canonically established distribution of card turns. In Sec. IV, we discuss how ergodicity might relate the time-independent statistical distribution of empirical viral outcomes and the time-dependent trajectory of a viral state vector within the model space. Constructing such an ergodic trajectory is the focus of Sec. V. In Sec. VI, we derive a Lagrangian for this trajectory, effectively distilling the problem of wild card viral outcomes to a single line of mathematics. We also discuss how Lagrangian and Hamiltonian considerations can rule out candidate models. In Sec. VII, we conclude by recommending useful means of reviewing the exercise for students and identifying avenues for further engaging their analytical thinking and physical intuition.

A bit of careful thinking suggests several axioms: the foundation for building our toy model. As these axioms are rooted in peculiarities of the Wild Cards canon that students will not and should not be expected to know, they might be motivated by inviting students to consider, in a guided way, logical consequences of the scenario summarized in Sec. I, then introducing new complications (i.e., joker–aces, see below).

A challenge for epidemiologists, and one that would bedevil our hypothetical theoretician, is to establish the true lethality of a pathogen. This is complicated by the existence of asymptomatic carriers, who by definition are difficult to identify. Epidemiologists, therefore, distinguish between the case fatality risk (CFR) and infection fatality risk (IFR) of a disease.6 The CFR represents, for a given population of confirmed disease cases caused by the pathogen, the proportion of deaths resulting from that disease. This is relatively straightforward. The IFR represents the proportion of pathogen-related deaths within the total number of infections, including asymptomatic, undiagnosed, and misdiagnosed cases. This better represents the danger presented by a pathogen, yet is much more nebulous.

Our hypothetical theoretician might, therefore, wonder if card turns could go unobserved. We argue that “subtle” manifestations would be practically inevitable.

Consider two scenarios that, while objectively absurd, lie firmly within the bounds of the Wild Cards canon. In the first, the virus paints ultraviolet racing stripes on an individual's heart but induces no further physiological changes. In the second, the virus imbues a resident of Iowa with the power of line-of-sight telepathic communication with narwhals. The discovery of either change is extremely unlikely. The first individual would be unaware of their jokerism; the second would be an ace, yet never know it. We refer to such undetected card turns as “crypto-jokers” and “crypto-aces,” respectively, or “cryptos” in general.

Such scenarios are innumerable, but possibility alone does not ensure a significant crypto population. Note, however, that the rapid global dispersion of a volatile untreatable pathogen with an apparent lethality on par with pulmonary anthrax—and potentially capable of scything through most of the human population in a few months—did not cause an apocalyptic societal collapse.7 This hints at a wide gulf between the CFR and IFR.

Cryptos are more than a theoretical diversion: concrete analogs exist in the real world. We offer the medical condition situs inversus8 as one instance of real-world crypto-jokerdom. We argue, only slightly tongue in cheek, that human tetrachromats9 could qualify as (low-grade) crypto-aces. The Wild Cards canon offers a vast range of possible subtleties.

Cryptos, therefore, represent a fourth category of viral outcomes beyond the usual fatalities, jokers, and aces. The existence of this population leads to a corollary: the canonically observed 90:9:1 rule applies only to known (or presumed) card turns, rather than all cases.

The 90:9:1 rule presents jokers and aces as mutually exclusive categories with a numerical distribution amenable to the roll of a hundred-sided die, as befits the role-playing origins of Wild Cards. Yet, the canon abounds with characters who confound this categorization: “joker–aces,” who exhibit both a physical mutation and a superhuman ability. Joker–aces put the lie to an implicit premise of the 90:9:1 rule by demonstrating that jokerness and aceness are mutually inclusive qualities that can, on occasion, overlap.

When combined with the existence of cryptos, this implies each card turn is characterized by a minimum of three qualities: degree of obviousness (i.e., was the turn objectively noticeable, or did it generate a crypto?), degree of aceness, and degree of jokerness. At this point, our hypothetical theoretician might consider mining the available data (potentially on millions of viral outcomes) for evidence of correlations or patterns. Per the Wild Cards canon, there are none. Yet, if the outcome space was fixed (i.e., time-independent) and univariate (i.e., resulting from a single die roll), and the dataset large enough, these three qualities should be linked.

Given the above, it is reasonable to assert that our model for wild card viral behavior must satisfy the following axioms:

  • Crypto-jokers and crypto-aces exist, yet the size of the crypto population is unknown and unknowable.

  • The distribution of observable card turns conforms to the 90:9:1 rule.

  • Wild card viral outcomes are determined by a multivariate probability distribution.

Our first step is to establish a framework for the problem. Bearing in mind our Lagrangian goal, we seek a convenient coordinate system linked to viral outcomes. We propose a polar model defined by two apparently random variables: a transformation severity, S0, and a mixing angle, θ[0,2π). (Throughout this article, for simplicity, we choose to work modulo 2π.) Each card turn is associated with an (S,θ) pair; this is the state vector for that viral outcome.

The severity is a measure of how much the virus changes a person: the extremity of their physical mutation or the potency of their new ability. Latent viral carries whose cards have not yet turned have S=0. Importantly, the Wild Cards canon establishes no practical upper limit on ace potency, meaning S is not ruled out. Conversely, if S is sufficiently low, the transformation may go unnoticed. This suggests a “subtlety threshold,” Sc, such that cryptos can be formally defined as all individuals with 0<SSc.

The mixing angle is our answer to the joker–ace problem. Rather than treating jokers and aces as fundamental categories, we posit that every non-fatal card turn results in a joker–ace. Let one axis of the abstract model space represent acedom, and let the orthogonal axis represent jokerdom. The mixing angle quantifies the relative admixture of jokerdom and acedom embodied by any individual card turn. (Students might balk at the mixing angle, seeing it as a non-intuitive conjuration. It might be presented as a common convenient trick for making abstract relationships geometrical, such as in particle physics.) Card turns that land sufficiently close to one axis will subjectively present as aces, while otherwise they will present as jokers or joker–aces. This suggests the existence of a joker/ace boundary angle, θ0, such that aces have 0θθ0.

Sc and θ0 are evidently subjective. The definition of subtlety would vary from individual to individual, depending on everything from personal circumstances to cultural context. The same would be true of the joker/ace boundary, with different individuals making different subjective categorization decisions. Nevertheless, and as demonstrated below, numerical values for Sc and θ0 can be calculated for specific instantiations of the polar model.

Notice furthermore that introducing θ0 turns the objectively mutually inclusive ace/joker categories into subjectively mutually exclusive categories: while the objective reality, according to this picture, is that all nonfatal card turns generate joker–aces, our hypothetical theoretician looking at the world still sees a subjective distinction between jokers and aces.

This abstract coordinate system raises an obvious question: What do negative acedom or jokerdom values physically represent? Perhaps the simplest approach is to associate those situations with fatal outcomes. We propose non-fatal card turns (aces, jokers, and cryptos) are confined to Quadrant I, while Quadrants II–IV are deadly. Figure 1 depicts the polar model space.

Fig. 1.

(Color online) Coordinate system for the polar model of wild card viral expression. Card turns in the gray region are fatal. The red circle marks the starting position of all latent carriers. The blue arrow represents a viral state vector, which denotes a card turn (red star) characterized by a severity value S and mixing angle θ. The green quarter circle marks the crypto subtlety threshold, Sc; cryptos reside in the green region. The dashed brown line is the joker/ace boundary, θ0; outcomes in the brown region subjectively present as aces. Jokers and joker–aces inhabit the non-shaded region.

Fig. 1.

(Color online) Coordinate system for the polar model of wild card viral expression. Card turns in the gray region are fatal. The red circle marks the starting position of all latent carriers. The blue arrow represents a viral state vector, which denotes a card turn (red star) characterized by a severity value S and mixing angle θ. The green quarter circle marks the crypto subtlety threshold, Sc; cryptos reside in the green region. The dashed brown line is the joker/ace boundary, θ0; outcomes in the brown region subjectively present as aces. Jokers and joker–aces inhabit the non-shaded region.

Close modal

A potential peculiarity of this setup is the discontinuous change in viral outcomes at θ=0. To avoid this, one might instead place the domain of “pure” aces in a narrow band within the nonfatal region, for instance centered at θ=45° and flanked by joker regions. This approach would appear to ameliorate the discontinuous behavior by introducing an ace/joker/fatality gradient (tacitly assuming fatal outcomes are inextricably linked to jokerness, which seems plausible). However, doing so would introduce a second angle without providing new information from which an additional constraint relationship could be derived (see Sec. II A). The result would be an under-constrained model space (e.g., the choice to center the ace band at 45° is arbitrary, perhaps even aesthetic). This observation could introduce students to a running theme of the exercise: building theoretical models, even toy ones, often requires choices. Here, they might weigh the alternatives. Is it preferable to accept discontinuous behavior in a model variable, or to exacerbate an already underdetermined problem?

The random variables S and θ together imply a joint probability distribution, P(S,θ). However, as mentioned in Sec. II B, the Wild Cards canon denies any correlation between viral outcomes. As multivariate problems are often more tractable when separable, and because the canon apparently does not prohibit it, we choose to keep the explorations as simple as possible by assuming the random variables are independent. It may be useful to emphasize this choice reflects a general practice of building the simplest possible (i.e., most easily explorable) system consistent with the available information.

Let μ(0,1) denote the unknown IFR: the true lethality of the wild card virus, here expressed as the fraction of card turns that are deadly. Let PS(S) and Pθ(θ) denote unknown probability density functions with the normalization conditions,
(1)
The probabilities of fatal and nonfatal card turns can be expressed as
(2)
while the nonfatal quadrant can be further decomposed into separate sector probabilities for aces, jokers, and cryptos via
(3)
(4)
and
(5)
These are general relationships, independent of viral behavior. In order to incorporate the crucial 90:9:1 rule, we next need to establish a relationship between outcome probabilities and outcome populations. Most generally, we would expect to find distributions of possible population sizes. However, another simplifying approximation enables us to dispense with the distributions entirely. Again, we seek to build the simplest system consistent with the available information.

The key is to consider a collection of N card turns as representing N independent trials, each of which leads to an outcome in one of four (subjectively, see above) mutually exclusive categories. When sampling with replacement, this describes a multinomial distribution, which students might recognize as the generalization of a binomial (“coin flip”) distribution. (Because the ubiquitous wild card virus is constantly creating new latent carriers, we feel the assumption of replacement is valid. The alternative is to work from a hypergeometric distribution, which may be less familiar to students than a binomial distribution.)

Let us approximate the population of each category as the multinomial distribution's expected size for that category: NkPk·N. Doing so turns the 90:9:1 rule into integral constraints on the relationship between Pθ(θ) and θ0 and between PS(S) and Sc,
(6)
and
(7)
At this point, combining the sector probabilities and 90:9:1 constraints distills the nonfatal outcome probabilities to an extremely simple final form as follows:
(8)
and, therefore,
(9)
These relationships preserve the 90:9:1 rule regardless of the unknown size of the crypto population. Furthermore, if, as we assert, the crypto population is nonzero and, therefore, Pcrypto>0, it necessarily follows that μ<0.9. Contrary to what might be inferred from the 90:9:1 rule, jokers constitute less than 9% of all turned card outcomes, and aces constitute less than 1% of the outcomes.
As a further consistency check, we can estimate the case and infection fatality risks from these categorical probabilities, using the same simplifying assumptions, as follows:
(10)
and
(11)
The larger the crypto population, the larger the disparity between the CFR and IFR.

At this point, we have a coordinate system linked to the set of possible viral outcomes and integral constraints that link random variables in that space to the 90:9:1 rule. However, how can we turn this strange scenario into a concrete physics problem? Recalling our Lagrangian end goal, we consider the viral state vector as a dynamical entity. We posit that a latent carrier's state vector constantly evolves—constantly moves through the model space—until their card turns, at which moment the state vector becomes fixed and its permanent location determines their fate. (In informal settings, we have found it useful to offer a loose conceptual analogy to the Copenhagen Interpretation of quantum mechanics to motivate this leap, but only while emphasizing the model is strictly classical, much as rolling dice resolves a spectrum of possibilities into a single fixed result.)

We need to bridge the gap between this dynamical evolution and the PDFs PS and Pθ in a way that preserves the 90:9:1 rule via Eqs. (6) and (7). To do so, we utilize ergodicity.10 (Undergraduate students might be introduced to this concept by considering the motion of a cue ball on an otherwise empty, frictionless rectangular pool table.11) If the state vector exhibits an ergodic trajectory in the (S,θ) parameter space, then the static PDFs may be treated as emergent phenomena arising from the average dwell time of the dynamical variables S(t) and θ(t) in different sectors of the parameter space. Ergodicity will be guaranteed if the individual severity and mixing angle trajectories have incommensurable periods τs and τθ. If those periods are sufficiently short and the dynamical evolution of the aperiodic ergodic system sufficiently rapid, the card turn outcomes could appear stochastic even if the individual dynamical variables follow deterministic periodic orbits.12 

In this picture, the time-independent PDF of a pseudorandom real variable x and its dynamical phase-space trajectory are connected through a simple time-averaging,
(12)
where a<b and H is the Heaviside step function. In the case of a function with period τ, the limit can be dropped. Applying this relationship to the constraints of Eqs. (6) and (7) yields three independent relationships,
(13)
(14)
(15)
Equation (13) formalizes the obvious requirement that the mixing angle trajectory spends a fraction μ of each period in the fatal quadrants. When normalized by the relevant periods, Eqs. (14) and (15) give the probability of non-ace and non-crypto outcomes, respectively. They also provide quantitative (functional) definitions of θ0 and Sc. Note the non-ace probability is guaranteed to exceed 90%. This implies θ>θ0 for most of the period, suggesting θ0 might reside near the minimum value of θ(t) for a given μ value. Similarly, Eq. (15) indicates Sc must decrease with increasing μ.

We have derived the conditions [Eqs. (13)–(15)] whereby an ergodic dynamical state vector trajectory could generate both a static 90:9:1 rule and a nonzero, possibly large, crypto population. At this point, we feel it is important to crystallize the exercise through proof of concept, by constructing periodic functions θ(t) and S(t) that satisfy the model requirements. Taking the time to derive an example trajectory, then study its behavior, is crucial for driving home the key idea that students' physics training can be a powerful tool for studying a seemingly unrelated, open-ended problem.

We attempt to emphasize the thought process, including dead ends, by which these solutions are obtained. The problem is underdetermined, so our solutions will not be unique.

Our choice of modulus means 0θ(t)<2πt[0,τθ]. We can further simplify the derivations, with no loss of generality, by stipulating θ(0) and θ(τθ) must equal 0 or 2π. Next, we introduce the phase-shifted mixing angle ϑ(t)θ(t)π/2, using the shorthand notation ϑ± to denote trajectories with ϑ(0)=3π/2 and π/2, respectively (and thus θ±(0)=2π and 0). Equation (13) will be satisfied if ϑ is positive for a fraction μ of the period and negative for the remainder 1μ of the period. In this way, identifying solutions for θ becomes a problem in root-solving, making it straightforward for students.

The oscillatory nature of ϑ makes it natural to seek candidate solutions based on trigonometric functions. However, the leap to a series expansion, which is the basis of our solution, may seem unnatural or arbitrary to students. It might be motivated by first exploring a slightly more obvious or intuitive path, then identifying the shortcomings of that approach.  Appendix A presents such an example and draws an analogy to concepts advanced undergraduates may have encountered when studying Fourier analysis.

We can express ϑ(t) as a cosine series,
(16)
where M>0 is the maximum number of half-cycle oscillations in the solution. M may be thought of as the number of times ϑ changes sign in one period.
The boundary values are
(17)
In Secs. V A 1–V A 3, we briefly consider the cases M=1, M=2, and M>2.

1. M=1

The (+) solution for M=1 requires
(18)
(19)
from which it follows a0=π/2, a1=π, and ϑ+(t¯)=0 only when t¯=(2/3)τθ. Similarly, the () solution finds the only zero at t¯=(1/3)τθ. Thus, the M=1 solution is physically disallowed unless μ=2/3.

2. M=2

The (+) solution for M=2 requires
(20)
(21)
from which it follows a1=0 and a0+a2=3π/2, meaning the solution has a single degree of freedom, which is ideal for a problem with a single unknown parameter, μ. Via the cosine double-angle formula, we get
(22)
Acceptable M=2 solutions for ϑ+ must have π/2minϑ+0. (A straightforward way to see this is simply to sketch a generic (+) solution for M=2, such as the green line in Fig. 2.) As the minimum of this symmetric solution occurs at t=τθ/2, this condition is equivalent to a range of allowable a0 values: π/2a03π/4. When a0=3π/4, ϑ+(t)0t, which implies μ=1. When a0=π/2, the zeros occur at t¯=τθ/3 and t¯=2τθ/3, which implies (1μ)τθ=τθ/3 or μ=2/3. Thus, the (+) solution for M=2 is only valid for the range 2/3μ1, with the further understanding that μ0.9 is disallowed by the nonzero crypto population.
Fig. 2.

(Color online) M=2 solutions for θ±(t) μ=0.25 (solid blue), 0.50 (solid orange), 2/3 (dashed and dashed-dotted black), and 0.75 (solid green). The μ=0.75 solution requires θ+; the μ=0.25 and 0.50 solutions require θ. Both θ± solutions for μ=2/3 are plotted, showing that this special case corresponds to solutions covering the entire range of mixing angles and which in fact are just phase-shifted copies of each other. The dotted lines denote the θ0 values obtained from numerical integration of Eq. (14): 1.4°, 1.1°, 1.0°, and 44.1° for μ=0.25, 0.50, 2/3, and 0.75, respectively. As in Fig. 1, the shaded region denotes fatal card turns.

Fig. 2.

(Color online) M=2 solutions for θ±(t) μ=0.25 (solid blue), 0.50 (solid orange), 2/3 (dashed and dashed-dotted black), and 0.75 (solid green). The μ=0.75 solution requires θ+; the μ=0.25 and 0.50 solutions require θ. Both θ± solutions for μ=2/3 are plotted, showing that this special case corresponds to solutions covering the entire range of mixing angles and which in fact are just phase-shifted copies of each other. The dotted lines denote the θ0 values obtained from numerical integration of Eq. (14): 1.4°, 1.1°, 1.0°, and 44.1° for μ=0.25, 0.50, 2/3, and 0.75, respectively. As in Fig. 1, the shaded region denotes fatal card turns.

Close modal
Similar deductions find the () solution,
(23)
is valid over the range 0<μ2/3. This dovetails nicely with the (+) solution.
Because both solutions have a1=0, their zeros satisfy
(24)
The (+) solution requires t¯2t¯1=(1μ)τθ, while the () solution requires t¯2t¯1=μτθ. Students should be able to confirm the identity
(25)
which combined with a little algebra leads to the conditions
(26)
at which point the M=2 solutions are fully determined.

Given a satisfactory trajectory, we can apply numerical integration to obtain the joker/ace boundary angle, θ0, via Eq. (14). A collection of θ(t) solutions for various μ values, along with the associated θ0 boundary angles, are plotted in Fig. 2. The convergence of the time-averaged mixing angle trajectory to the required integral constraints is confirmed by Fig. 3.

Fig. 3.

(Color online) Time-averaged sector dwell times for the M=2 mixing angle solution for the ace and joker sectors (upper and lower panels, respectively), computed using the θ0 values from Fig. 2. The upper traces converge to (1μ)/10 and the lower traces to 9(1μ)/10, both as required by Eq. (6).

Fig. 3.

(Color online) Time-averaged sector dwell times for the M=2 mixing angle solution for the ace and joker sectors (upper and lower panels, respectively), computed using the θ0 values from Fig. 2. The upper traces converge to (1μ)/10 and the lower traces to 9(1μ)/10, both as required by Eq. (6).

Close modal

3. M>2

Higher-order solutions have additional degrees of freedom, meaning the solution for a given μ value will not be unique. The problem of identifying roots of the cosine series rapidly becomes complicated with additional terms. Students with a strong background in linear algebra might be shown that such problems can sometimes be tackled by constructing a matrix with a characteristic equation equivalent to the polynomial in question, thereby equating the roots with eigenvalues of the matrix, at which point a vast and robust theoretical machinery may be brought to bear on the problem. This can be particularly efficient for problems with high degrees of symmetry,13 as in this problem.

The quest for a suitable S(t) solution is more straightforward.14 Per Sec. III, we require only that S remain non-negative throughout the period and that the trajectory allow excursions to arbitrarily large S values.

Students might be encouraged to think back on their coursework experience, drawing on their incipient physical intuition to identify physical systems with analogous properties. Appealing to well-understood analogs is a useful model-building technique and one that students should feel encouraged to exercise. For instance, as a starting point, students could be reminded of amplitude growth in a resonant oscillator (either mechanical or electrical). The key is to consider timescales. While a periodic S(t) is convenient for ergodicity, as discussed above, it also sets a hard requirement on the amplitude growth rate. Under the perturbative analysis of such a system,15,16 the exponential amplitude factor cannot grow beyond order unity within a single period, meaning this approach will not produce the desired large excursions.

A nearly identical solution unencumbered by this difficulty can be obtained with a bit of creative thinking. Students familiar with the usual damped harmonic oscillator problem might be led to consider the physical consequences of changing the sign on the damping term. In keeping with the fantastical nature of the premise, then, one might posit the severity trajectory corresponds to that of an “anti-damped” harmonic oscillator,
(27)
for real α>0. (We use the term “anti-damped” because here the sign of the exponent is opposite that of the usual damped harmonic oscillator solution.) As α is unconstrained, this solution allows for excursions to arbitrarily large, albeit finite, severities.

Convergence of the time-averaged anti-damped severity trajectory to the required limits is demonstrated in Fig. 4.

Fig. 4.

(Color online) Time-averaged sector dwell times for an “anti-damped” severity trajectory with σ0=1 and α=3, for the crypto and joker–ace sectors (upper and lower panels, respectively). When computed using Sc values obtained from numerical integration of Eq. (15) (6.7, 6.5, and 5.1, respectively, in arbitrary units) the upper and lower traces converge to (910μ)/(99μ) and μ/(99μ), as required by Eq. (7).

Fig. 4.

(Color online) Time-averaged sector dwell times for an “anti-damped” severity trajectory with σ0=1 and α=3, for the crypto and joker–ace sectors (upper and lower panels, respectively). When computed using Sc values obtained from numerical integration of Eq. (15) (6.7, 6.5, and 5.1, respectively, in arbitrary units) the upper and lower traces converge to (910μ)/(99μ) and μ/(99μ), as required by Eq. (7).

Close modal

Attentive students might notice the anti-damped severity requires the injection of substantial energy over potentially short timescales of order τs. Many common ace feats (e.g., folding space) would carry titanic power requirements. Identifying this mysterious power source would be a priority for our hypothetical theoretician.

Given solutions for θ(t) and S(t), we can plot the resulting state vector trajectory for a representative set of parameter values. Figures 5 and 6 contain two examples.

Fig. 5.

(Color online) State vector trajectory over 25 severity periods, computed for μ=0.25, M=2, α=15, σ0=1, and τθ=2τs. The solid orange line marks θ0 and the dashed green circle marks Sc. The radial (severity) coordinate is plotted in arbitrary units. The color gradient represents time, from red to blue. Compare the blue line in Fig. 2.

Fig. 5.

(Color online) State vector trajectory over 25 severity periods, computed for μ=0.25, M=2, α=15, σ0=1, and τθ=2τs. The solid orange line marks θ0 and the dashed green circle marks Sc. The radial (severity) coordinate is plotted in arbitrary units. The color gradient represents time, from red to blue. Compare the blue line in Fig. 2.

Close modal
Fig. 6.

(Color online) As Fig. 5, for μ=0.75; cf. the green line in Fig. 2.

Fig. 6.

(Color online) As Fig. 5, for μ=0.75; cf. the green line in Fig. 2.

Close modal

We find these plots aesthetically pleasing, although the variation of Sc from case to case requires close attention when cross-comparing the plots and their implications. In particular, notice when μ=0.25, the highest-severity card turns barely exceed Sc, implying a scarcity of powerful aces (which conflicts with their ubiquity within the canon). In contrast, powerful aces become plentiful when the IFR is high (possibly in conflict with the arguments put forward in Sec. II): max(S)Sc when μ=0.75.

Truly asymptotic severity trajectories, should any exist (see Sec. VI), could not be captured in full by plots such as those in Figs. 5 and 6. In that case, a technique such as stereographic projection could be used to map the trajectory onto a finite Riemann sphere.17 

At this point, we have converted the nebulous question of wild card viral outcomes into a concrete expression of classical dynamics. It is our hope this demonstration aids students' appreciation not only for the flexibility of physical concepts but also for the insights that can be obtained by applying them in wide-ranging, abstract contexts.

All that remains is to realize our goal by distilling the example solution into a Lagrangian formulation. The general problem of reverse-solving for a Lagrangian that encodes a known dynamical system, or even determining whether a given system of differential equations can be associated with a Lagrangian, is notoriously difficult. While the necessary and sufficient conditions for a Lagrangian to exist have been derived, applying these conditions is challenging in practice.18 Fortunately, for the simplistic trajectories considered here, suitable Lagrangians can be reverse-engineered by arguing from analogy with simple systems that should be easily accessible, if not already familiar, to advanced undergraduates.

Our M=2 solutions for the mixing angle trajectory in Sec. V A have the generic form θ(t)=A+Bcos(2πt/τθ). This is also the solution for the motion of a mass suspended from a spring (subject to a constant gravitational force) with the initial condition θ̇(0)=0, and where A is the equilibrium position for the suspended mass. The Lagrangian for that system is simply
(28)
After applying the equilibrium condition mg=Ak, and setting m=1 and ω2=k/m=4π2/τθ2, this yields the mixing angle Lagrangian,
(29)
where A(μ)a0±(μ)+π/2 for 0<μ2/3 and 2/3μ<0.9, as appropriate. When starting from this Lagrangian, the initial conditions θ+(0)=2π and θ(0)=0 would determine B. Note this Lagrangian is an explicit function of the infection fatality risk, μ, but not an explicit function of time.
The Lagrangian for the “anti-damped” oscillator of Eq. (27) is obtained by changing the sign on the exponent in the standard Lagrangian for a damped harmonic oscillator,
(30)
Unlike Lθ, this function contains an explicit time dependence, indicating the severity trajectory is not a closed system with a time-independent Hamiltonian. This is consistent with our earlier speculation about interactions with an unknown power source.
The overall “Wild Card Lagrangian” for our hypothetical model is the sum of the anti-damped severity and mixing angle components,
(31)
LWC distills the question of wild card viral outcomes—encapsulating the distribution of all known and inferred effects of the pathogen on populations of infected carriers—into a single mathematical expression. What began as a somewhat vague problem with no obvious starting points or handholds has become a concise, concrete problem in classical dynamics.
We find it is also instructive to attempt to construct a truly asymptotic trajectory (recall from Sec. III that S is not ruled out). The cotangent function, being both periodic and asymptotic, is particularly attractive and should be familiar to all undergraduate physics students. The function S(t)=cot[ϕ(t)], where
(32)
features an asymptote at t=t̂, with 0<t̂<τs, and is periodic: S(0)=S(τs)=0.
The general functional form S(t)=cot[ϕ(t)] is a solution to the differential equation of motion
(33)
which can be associated with a simple Lagrangian
(34)
Our particular choice of ϕ(t) is linear in time, rendering ϕ̇ a constant and eliminating the ϕ̈ term, meaning this Lagrangian features no explicit time dependence and, therefore, should be associated with a time-independent Hamiltonian. However, setting S=cot[ϕ(t)] with a linear ϕ in the Hamiltonian calculation leads to the simple result
(35)
which is discontinuous unless t̂=τs/2. Yet, a deeper problem persists even after this condition is met.
The asymptotic-solution Lagrangian has the form Ls=TsVs, where Ts is the standard kinetic energy term Ṡ2/2 and, because ϕ̈=0, the potential energy term is
(36)
This resembles the centrifugal term associated with the motion of a free particle in a non-inertial reference frame rotating with constant angular velocity ϕ̇, albeit with a quartic rather than quadratic potential. This presents an insurmountable difficulty: the force on such a particle,
(37)
is non-negative throughout the severity period. Any infinitesimal perturbation S=0ϵ will be dynamically unstable, thereby undergoing an asymptotic excursion S. But this system contains no restoring force for achieving S(τs)=0. Thus, while the proposed asymptotic severity trajectory is mathematically attractive, it is physically disallowed in the Lagrangian picture.

At the outset of this exercise, students might question the relevance of their physics education to the fantastical problem of wild card viral outcomes. By the end, they should recognize the flexibility and utility of their training: the seemingly irrelevant problem can be analyzed in terms of classical dynamics, a topic firmly within their wheelhouse.

Because the exercise presented here is one of extended reverse-engineering, we feel it is crucial to review the work in reverse order. Doing so turns the problem into a straightforward exercise of using a known Lagrangian, and helps strengthen the idea that apparently unrelated subjects—here, wild card viral behavior—can be meaningfully investigated using physics concepts.

Starting with LWC, applying the Principle of Stationary Action via the Euler–Lagrange equation leads to differential equations of motion for S and θ; solving these equations gives trajectories that, under the appropriate circumstances, can be ergodic. (It is worth dwelling on this point, to reflect on why the model relies on ergodicity.) Time-averaging those trajectories turns the deterministic dynamical variables S(t) and θ(t) into apparently stochastic variables associated with probability distributions PS(S) and Pθ(θ). When tied to the appropriate coordinate system, these yield relative proportions of viral outcomes. Finally, given appropriate values of the crypto subtlety threshold, Sc, and ace/joker boundary angle, θ0, this system—fully encoded in the one-line Lagrangian—generates the 90:9:1 rule.

This approach highlights the importance of Sc and θ0: parameters calibrated to generate the canonical ratio. Students might consider how altering Sc and θ0 changes the breakdown of fatalities, jokers, aces, and cryptos, given a fixed μ (the infection fatality risk).

In reviewing the model development steps, one could re-emphasize the various choices and simplifying assumptions utilized along the way. These include:

  • Separating the joint probability distribution P(S,θ) into independent probability density functions PS(S) and Pθ(θ);

  • Simplistically relating viral outcome probabilities to population sizes; and

  • Accepting discontinuous behavior at θ=0 rather than exacerbating the underdetermined nature of the problem.

Our informal experience has been that emphasizing the possibility of alternate choices is an easy and natural way to spur conversation. To further engage students' analytical thinking skills, they might be encouraged to consider how (or whether) the model choices were justified, consequences of those choices, and implications of changing them. For instance, is the discontinuity a genuine problem? If so, how might it be avoided? Or, can they identify other physical systems that exhibit apparently discontinuous behavior? Model choices also pertain to the handling of tricky edge cases. For instance, if a card turn induces an incurable, degenerative long-term disease (e.g., a prion disease) that ultimately proves fatal years later, should that death be attributed to the Wild Cards virus?

Ambitious students might find it an interesting challenge to seek an acceptable asymptotic trajectory or, alternatively, to investigate whether such is ruled out on general principles.

As noted briefly in Sec. V C, the specific model instantiation put forward here may have a fatal flaw: namely, the tension between the need for a low infection fatality risk (as required by logic, per Sec. II) and the ubiquity of powerful aces (as required by the Wild Cards canon). This could be highlighted as empirical evidence that potentially falsifies the model. Building on that idea, students might contemplate other information that could rule out particular model choices, and then go on to ask whether it would be possible to devise (ethical) experiments to obtain that information. For instance, if the distribution of mixing angles in a large population of turned cards could somehow be measured (assuming the concept of a joker/ace mixing angle is viable), and was subsequently found to exclude certain θ values, this would rule out all odd values of M.

Similarly, might there be ethical means of measuring, or even bounding, model parameters? Or, could more satisfying models with fewer free parameters be identified? What additional information would be necessary and sufficient for deriving a unique solution? Could that information be obtained through epidemiological studies, controlled experimentation, or some other method?

We recognize this extended model-development exercise relies on synthesizing several concepts that may be new, even to advanced students. This is an eclectic problem. It is also lengthy. A measured pace would be required: motivating each new step, anchoring it in the overall context, and getting students comfortable with the key physical and mathematical concepts could easily fill a lecture or recitation period. We view this exercise as an unusual (but interesting) way to augment students' education, not something to be grafted onto their core curriculum. It might be best explored within a senior honors seminar. Or, perhaps it could be used within a several-week elective class, e.g., during a “January term” (J-term) between the fall and spring semesters, at institutions where such are offered.

The authors thank Sage Walker, M.D. and Stephen Leigh for their insightful observations, and the late John Jos. Miller for providing his valuable perspective on the early development of the Wild Cards universe.

The authors have no conflicts to disclose. I.L.T. asserts the conception, derivation, and preparation of this work used no resources or intellectual property from Los Alamos National Laboratory; Triad National Security, LLC; the National Nuclear Security Administration; or the United States Department of Energy.

Students might be inclined to seek ϑ(t) solutions built upon a single trigonometric function. One such example is
(A1)
with the conditions x(0)=π/2, x(τθ)=0, and x[(1μ)τθ]=cos1(1/4). This solution will satisfy the mixing angle requirements laid out in Sec. V A as long as t=(1μ)τθ is the only zero. Because this solution must work for arbitrary μ(0,0.90) (recall the upper limit is required for a nonzero crypto population), x(t) cannot be a linear function of time. It is a straightforward algebra problem to show the quadratic solution
(A2)
requires
(A3)
Unfortunately, this solution fails for μ values below approximately 0.0783, at which point ϑ(t) has more than a single zero. A similar solution built from a single sine function will of course have the same problem, as does the analogous solution built from combining single sine and cosine functions with a shared x(t) argument.

Fundamentally, the problem occurs because very low μ values require these overly simple solutions for ϑ to undergo a sharp transition at t=(1μ)τθ: after staying negative over nearly the entire period, they have very little time remaining to change sign and reach the necessary endpoint, ϑ(τθ)=3π/2. Satisfying this condition requires a higher-frequency oscillation than this simple approach can accommodate. Students familiar with Fourier analysis might find this conceptually reminiscent of the Gibbs phenomenon.

While the range of viable μ values could be pushed lower by moving to a cubic solution for x(t) and/or specifically seeking solutions with multiple zeros, the algebra rapidly becomes tedious. The analogy to Fourier analysis suggests a better approach might be to seek a solution that combines multiple fixed-frequency components. To keep things simple, one might seek a solution based on a pure sine or cosine series. We opt for the latter in Sec. V A.

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James
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2.
Professor Kakalios's
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” class is a popular freshman seminar at the
University of Minnesota
.
3.
A “shared world” setting is one in which a consortium of writers collectively generate a world, its rules, and its characters, communally writing stories within that setting.
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As with any problem involving systems of differential equations, the notion of treating an epidemiological model as a dynamical system is not unusual. An extremely sophisticated Lagrangian application can be found in
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Naeem
, and
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7.
When contrasted against the societal upheavals unleashed by the global COVID-19 pandemic driven by a coronavirus with a vastly lower case fatality risk, the continuity of civilization may be the most fantastical aspect of the Wild Cards universe, rather than the comic-book superpowers.
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The concept of ergodicity occasionally appears in the epidemiological literature, such as when calculating the infection fatality risks posed by real diseases. There, however, ergodicity is an assumption pertaining to a stochastic infection process that allows an IFR to be extrapolated from a small sample population. For example, see Mieskolainen et al. (op. cit.).
11.
An excellent recreational introduction to this idea can be found in David S. Richeson, “Unfolding the Mysteries of Polygonal Billiards,” Quanta Magazine, February 15, 2024 (see https://www.quantamagazine.org/the-mysterious-math-of-billiards-tables-20240215/). While this article does not refer to ergodicity, it could be introduced by first having students practice “unfolding” a rectangular pool table with integer side lengths to identify periodic orbits and then asking them to consider what happens when the side lengths are made incommensurable.
12.
The Wild Cards canon has established that although the virus may lie dormant for decades within a latent carrier, chaotic results can occur within minutes or possibly even seconds of infection. This suggests that the inherent timescale for randomization of the state vector within each infected carrier is very rapid.
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Appropriate physical units for the hypothetical transformation severity are unknown. Severity calculations in this article are, therefore, expressed in arbitrary units.
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Published open access through an agreement withLos Alamos National Laboratory