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.
I. INTRODUCTION
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.
II. AXIOMS FOR MODEL DEVELOPMENT
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. Crypto-jokers and crypto-aces
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.
B. The problem of joker–aces
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.
C. The ground rules
Given the above, it is reasonable to assert that our model for wild card viral behavior must satisfy the following axioms:
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Crypto-jokers and crypto-aces exist, yet the size of the crypto population is unknown and unknowable.
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The distribution of observable card turns conforms to the 90:9:1 rule.
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Wild card viral outcomes are determined by a multivariate probability distribution.
III. THE POLAR MODEL AND VIRAL STATE VECTOR
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, , and a mixing angle, . (Throughout this article, for simplicity, we choose to work modulo .) Each card turn is associated with an 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 . Importantly, the Wild Cards canon establishes no practical upper limit on ace potency, meaning is not ruled out. Conversely, if S is sufficiently low, the transformation may go unnoticed. This suggests a “subtlety threshold,” , such that cryptos can be formally defined as all individuals with .
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, , such that aces have .
and 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 and can be calculated for specific instantiations of the polar model.
Notice furthermore that introducing 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.
(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 . The green quarter circle marks the crypto subtlety threshold, ; cryptos reside in the green region. The dashed brown line is the joker/ace boundary, ; outcomes in the brown region subjectively present as aces. Jokers and joker–aces inhabit the non-shaded region.
(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 . The green quarter circle marks the crypto subtlety threshold, ; cryptos reside in the green region. The dashed brown line is the joker/ace boundary, ; outcomes in the brown region subjectively present as aces. Jokers and joker–aces inhabit the non-shaded region.
A potential peculiarity of this setup is the discontinuous change in viral outcomes at . To avoid this, one might instead place the domain of “pure” aces in a narrow band within the nonfatal region, for instance centered at 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 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?
A. Sector probabilities and constraints
The random variables S and together imply a joint probability distribution, . 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.
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.)
IV. ERGODICITY
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 and 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 parameter space, then the static PDFs may be treated as emergent phenomena arising from the average dwell time of the dynamical variables and in different sectors of the parameter space. Ergodicity will be guaranteed if the individual severity and mixing angle trajectories have incommensurable periods and . 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
V. STATE VECTOR TRAJECTORIES
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 and 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.
A. Mixing angle trajectory
Our choice of modulus means . We can further simplify the derivations, with no loss of generality, by stipulating and must equal 0 or . Next, we introduce the phase-shifted mixing angle , using the shorthand notation to denote trajectories with and , respectively (and thus and 0). Equation (13) will be satisfied if is positive for a fraction of the period and negative for the remainder of the period. In this way, identifying solutions for becomes a problem in root-solving, making it straightforward for students.
The oscillatory nature of 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.
1.
2.
(Color online) solutions for (solid blue), 0.50 (solid orange), (dashed and dashed-dotted black), and 0.75 (solid green). The solution requires ; the and 0.50 solutions require . Both solutions for 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 values obtained from numerical integration of Eq. (14): , , , and for , 0.50, , and 0.75, respectively. As in Fig. 1, the shaded region denotes fatal card turns.
(Color online) solutions for (solid blue), 0.50 (solid orange), (dashed and dashed-dotted black), and 0.75 (solid green). The solution requires ; the and 0.50 solutions require . Both solutions for 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 values obtained from numerical integration of Eq. (14): , , , and for , 0.50, , and 0.75, respectively. As in Fig. 1, the shaded region denotes fatal card turns.
Given a satisfactory trajectory, we can apply numerical integration to obtain the joker/ace boundary angle, , via Eq. (14). A collection of solutions for various values, along with the associated 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.
(Color online) Time-averaged sector dwell times for the mixing angle solution for the ace and joker sectors (upper and lower panels, respectively), computed using the values from Fig. 2. The upper traces converge to and the lower traces to , both as required by Eq. (6).
3.
Higher-order solutions have additional degrees of freedom, meaning the solution for a given 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.
B. Severity trajectory
The quest for a suitable 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 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.
Convergence of the time-averaged anti-damped severity trajectory to the required limits is demonstrated in Fig. 4.
(Color online) Time-averaged sector dwell times for an “anti-damped” severity trajectory with and , for the crypto and joker–ace sectors (upper and lower panels, respectively). When computed using 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 and , as required by Eq. (7).
(Color online) Time-averaged sector dwell times for an “anti-damped” severity trajectory with and , for the crypto and joker–ace sectors (upper and lower panels, respectively). When computed using 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 and , as required by Eq. (7).
Attentive students might notice the anti-damped severity requires the injection of substantial energy over potentially short timescales of order . 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.
C. Example trajectories
Given solutions for and , we can plot the resulting state vector trajectory for a representative set of parameter values. Figures 5 and 6 contain two examples.
(Color online) State vector trajectory over 25 severity periods, computed for , , , , and . The solid orange line marks and the dashed green circle marks . 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.
(Color online) State vector trajectory over 25 severity periods, computed for , , , , and . The solid orange line marks and the dashed green circle marks . 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.
We find these plots aesthetically pleasing, although the variation of from case to case requires close attention when cross-comparing the plots and their implications. In particular, notice when , the highest-severity card turns barely exceed , 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): when .
VI. STATE VECTOR LAGRANGIANS
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.
VII. FURTHER ENGAGEMENT
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 , applying the Principle of Stationary Action via the Euler–Lagrange equation leads to differential equations of motion for S and ; 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 and into apparently stochastic variables associated with probability distributions and . When tied to the appropriate coordinate system, these yield relative proportions of viral outcomes. Finally, given appropriate values of the crypto subtlety threshold, , and ace/joker boundary angle, , this system—fully encoded in the one-line Lagrangian—generates the 90:9:1 rule.
This approach highlights the importance of and : parameters calibrated to generate the canonical ratio. Students might consider how altering and changes the breakdown of fatalities, jokers, aces, and cryptos, given a fixed (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:
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Separating the joint probability distribution into independent probability density functions and ;
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Simplistically relating viral outcome probabilities to population sizes; and
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Accepting discontinuous behavior at 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 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.
ACKNOWLEDGMENTS
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.
AUTHOR DECLARATIONS
Conflict of Interest
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.
APPENDIX A: OVERLY SIMPLE MIXING ANGLE SOLUTIONS AND MOTIVATION FOR A SERIES
Fundamentally, the problem occurs because very low values require these overly simple solutions for to undergo a sharp transition at : after staying negative over nearly the entire period, they have very little time remaining to change sign and reach the necessary endpoint, . 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 values could be pushed lower by moving to a cubic solution for 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.