Lecture notes. Written from the material of MIT 8.591J 2014, licensed CC BY-NC-SA 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.
8. Virulence Evolution and the Red Queen¶
What this covers¶
This chapter answers two separate questions that the lecture pairs deliberately. First: given a simple compartmental model of infection, what determines whether a disease invades and persists, and — the sharper question — what does natural selection actually maximize when two strains of a parasite compete for the same host population? Second, and unrelated in mechanism but linked by theme (both are about the population consequences of a parasite): why does sexual reproduction persist at all, given that it costs a population a factor of two in growth rate? The chapter assumes comfort with ODE fixed points and their stability, and reuses — without re-deriving it — the geometric-distribution argument from the mRNA/protein burst-size problem earlier in the course.
Microparasites and macroparasites¶
A parasite, in the broadest sense, is layered onto essentially every organism: humans carry viruses and bacteria; bacteria are themselves preyed on by bacteriophage, viruses that have evolved specifically to exploit a bacterial cell's machinery. Before writing down any model, it is worth asking which details of this relationship are worth keeping and which are not — the choice is not forced by the biology, and different choices suit different parasites.
The distinction the lecture opens with is between microparasites — viruses and bacteria, which reproduce extremely fast inside a host and reach huge numbers there — and macroparasites — things like tapeworms, which do not reproduce prolifically inside a single host and are more often transmitted through the environment than by direct host-host contact. The consequence for modeling is this: for a microparasite, the number of pathogen particles inside a given host is not the useful state variable to track, both because it would be complicated and because, once a host is infected, it typically becomes sick and infectious quickly and recovers (or dies) on a similar time scale — there is a rough separation of time scales between "becoming infected" and "the details of what happens inside." That licenses collapsing all of that internal state into a small number of discrete classes per host: sensitive (uninfected), infected, and — later — resistant. Whether this simplification is actually justified for a given disease is an empirical question, not something the model can settle on its own; macroparasites, which don't blow up in numbers inside one host and often transmit environmentally rather than host-to-host, are a case where the same collapse would throw away exactly the information that matters, and need a different kind of model that this lecture does not build.
Aside on intra-host parasitism. The phage-on-bacteria layer has its own parasite-of-a-parasite story worth keeping in mind for later: when several phage particles infect the same bacterial cell, a phage with a shorter genome that cannot replicate on its own can still spread by taking over the replication machinery contributed by full-length, co-infecting phage — a "cheater" strategy. This is a small piece of a much larger point that resurfaces below, when the virulence-evolution discussion returns to what happens once more than one strain can share a host.
A basic model of infection¶
Martin's model (the lecture works from chapter 11) divides the host population into sensitive individuals, in number \(x\), and infected individuals, in number \(y\), and writes down the simplest possible dynamics for how a host moves between the two classes — treating contact between the classes the way a well-mixed chemical reaction treats collisions between two species:
Reading off the terms: \(k\) is a constant rate at which new sensitive individuals enter the population (birth, immigration, whatever keeps the population from vanishing — see the caveat below); \(u\) is the death rate that applies regardless of infection; \(\beta x y\) is the rate at which sensitive individuals become infected, proportional to how often the two classes "collide," exactly as in mass-action chemical kinetics; and \(v\) is the virulence — the extra death rate an infected individual suffers on top of the baseline \(u\). Every individual leaving the sensitive class through infection enters the infected class, which is why \(\beta xy\) appears with opposite signs in the two equations.
Without any infection at all, \(\dot x = k - ux = 0\) gives an equilibrium number of sensitive individuals \(x_0 = k/u\) — this is the population the model settles to before a parasite is ever introduced, and it reappears below inside \(R_0\).
A caveat the lecture is explicit about: entry at a constant rate \(k\) is a mathematical convenience, not a claim about biology — it takes something else (parents, migrants) to produce those \(k\) individuals per unit time, and treating that as a constant rather than, say, proportional to the existing population is a simplification made purely to keep the population from certain, uninteresting extinction (with no birth term, everyone dies eventually, infection or not, since both classes have positive death rates and nothing replenishes them). Classic epidemiological SIR models — the S here matches the sensitive class, but "I" and "R" stand for infected and resistant — sidestep this by having infected individuals recover into a resistant class rather than by inventing a birth term, and possibly cycle resistant individuals back to sensitive later. The lecture does not derive the SIR equations in full (see Sources), only notes that the same intuition about \(R_0\) carries over, and the underlying conclusions turn out to be robust to which of these bookkeeping choices you make.
\(R_0\): definition, distribution, and derivation¶
\(R_0\), the basic reproductive number, is defined operationally: introduce one infected individual into an otherwise fully sensitive population (at its infection-free equilibrium \(x_0\)), and ask for the expected number of new infections that one individual causes over its lifetime. It is a pure number, not a rate — a point worth being pedantic about, since "the expected number of secondary infections from one case" has no time unit attached to it. Whether \(R_0\) is greater or less than 1 is the whole story near the infection-free state: above 1, a single introduced infection grows in expectation (an exponentially branching process); below 1, it goes extinct in expectation. At exactly \(R_0=1\) the process is a critical branching process — neutrally stable in the mean, but it will nonetheless die out with probability 1 purely from the randomness of who infects whom, since a random walk with zero drift is recurrent to extinction.
\(R_0>1\) does not mean the whole population is doomed to become infected. The exponential growth of the infected class is a statement about the early, linearized dynamics near the infection-free state; as more of the population becomes infected, the pool of remaining sensitive individuals shrinks, which throttles the rate at which new infections can be generated. That feedback — not visible in \(R_0\) itself — is what can let the infected and sensitive classes settle into a nontrivial (generally damped-oscillatory) coexistence equilibrium rather than driving \(x\) to zero.
What distribution governs the number of secondary infections? Not Poisson: a Poisson count would answer "how many people do I infect in the next ten days, given that I stay alive that long," which fixes the observation window rather than the individual's lifetime. The actual question is how many times a randomly-chosen individual manages to transmit before leaving the infected class altogether (by dying, at rate \(u+v\)), when transmission itself happens at rate \(\beta x\). This is exactly the same race-between-two-rates structure that produced a geometric distribution for the number of times an mRNA is translated before it degrades, earlier in the course (production at one rate, removal at another; count how many times you go around the "produce" loop before the "remove" event fires). Here the loop is "infect one more person, at rate \(\beta x\)" versus "die, at rate \(u+v\)," and the number of secondary infections before the infected individual leaves the population is geometric with the corresponding success probability — even though every infected individual in this model is identical. Summing many such geometric draws (say, from an initial batch of 20 or 100 simultaneously-introduced infections) smooths the distribution into something close to Gaussian, by the same logic as any sum of many i.i.d. random variables — a geometric total for one seed does not make the total across many seeds geometric.
Real populations of infected individuals are not identical, and that matters: if individuals vary in how infectious they are, the distribution of secondary cases becomes broader than geometric even for a fixed mean \(R_0\). A larger fraction of introductions then fizzle out immediately, while a minority of highly infectious individuals produce disproportionately explosive outbreaks — the canonical illustration being "Typhoid Mary," an individual whose case (as a cook, in the version recalled in class) generated far more secondary infections than a typical carrier. This is the subject of the supplementary reading by Lloyd-Smith on heterogeneity in individual infectiousness (see Sources); it is not derived here.
Deriving \(R_0\) for this model. \(R_0\) factors into two pieces:
The expected lifetime as infected is \(1/(u+v)\), the reciprocal of the death rate in that class. The rate of causing new infections is \(\beta x\), evaluated at the sensitive population before any infection is introduced, i.e. at \(x = x_0 = k/u\). So
Vaccination and eradication¶
Because \(R_0\) is exactly the quantity that must exceed 1 for a disease to become endemic, it also tells you how hard it is to eliminate a disease by vaccination: vaccinating a fraction \(p\) of the population effectively removes those individuals from the pool of sensitives available to be infected, which lowers the effective reproductive number below 1 once
As \(R_0\) grows, this threshold approaches 1, meaning you have to vaccinate essentially everyone — which you can never fully achieve — and that is exactly why high-\(R_0\) diseases are hard to eradicate. The classic childhood diseases (measles, whooping cough, German measles, chickenpox, diphtheria, scarlet fever, mumps, poliomyelitis) have estimated \(R_0\)s roughly in the range 5–15, so vaccinating, e.g., 80% of the population is needed for \(R_0=5\). Smallpox, by contrast, has an estimated \(R_0\) around 3–5 — smaller than most of that list — which the lecture connects directly to why smallpox vaccination campaigns succeeded in eradicating it where campaigns against the higher-\(R_0\) diseases have not. A class discussion of Ebola arrived (from an audience member's recollection, not from anything the professor asserted as fact) at an estimated \(R_0\) around 2 for the 2014 West Africa outbreak, which would put the eradication threshold near 50%.
The important caveat is that \(R_0\) is not an intrinsic constant of a pathogen — it depends on the environment and behavior of the population it is measured in. A returning healthcare worker who knows to self-monitor and report a fever immediately has a very different (lower) local \(R_0\) than an outbreak in a setting without that surveillance; public health measures are, among other things, a way of directly pushing \(R_0\) down.
Two strains, no super-infection, and what selection maximizes¶
\(R_0\) tells you whether a single strain, on its own, can become endemic. It does not by itself tell you what a parasite is selected to do — that requires comparing two strains competing for the same hosts, which is the second model the lecture builds (still following Martin, but a simpler model than the "super-infection" framework covered later in that chapter).
The critical assumption is no super-infection: a host carries at most one strain at a time. Given that, the two strains can differ from each other in exactly two parameters: their transmissibility \(\beta_i\) and their virulence \(v_i\) (the extra mortality each strain imposes). Higher \(\beta\), all else equal, is intuitively good for a strain; what happens as \(v\) varies is the less obvious question, and it is exactly what the rest of this section is for.
Each strain, alone, has its own single-strain equilibrium \(E_i = (x_i^*, y_i^*)\) found by setting \(\dot y_i = 0\) at nonzero \(y_i\):
Suppose the population sits at \(E_1\) (strain 1 alone, endemic, so \(R_1>1\)), and a single individual infected with strain 2 is introduced. Does strain 2 invade? The natural criterion is the sign of \(\dot y_2\) evaluated at \(E_1\) as \(y_2\) is perturbed away from zero — i.e., the derivative of \(\dot y_2 = \beta_2 x y_2 - (u+v_2) y_2\) with respect to \(y_2\), evaluated at \(x=x_1^*\):
Substituting \(x_1^* = (u+v_1)/\beta_1\):
Multiplying both sides by \(k/u\) turns each side into exactly the \(R_0\) expression derived above:
So strain 2 invades an equilibrium held by strain 1 if and only if \(R_2 > R_1\) — and the same argument run the other way shows strain 1 can likewise invade strain 2's equilibrium only if \(R_1>R_2\). Together, these say that whichever strain has the larger \(R_0\) wins outright and drives the other extinct; there is no stable coexistence except in the non-generic (measure-zero) case \(R_1=R_2\), and even then, coexistence is fragile in the same way nearly-neutral mutations are — sorting between two strains with nearly equal \(R_0\) just takes correspondingly longer. This is the sense in which selection maximizes \(R_0\): two strains compete on nothing but the size of a single number, however many different combinations of \(\beta\) and \(v\) produced it. The lecture flags this as more surprising than it first sounds — the invasion condition was derived from a statement about competitive dynamics between two circulating strains, yet it collapses exactly onto \(R_0\), a quantity originally defined from a completely different scenario (one infected individual entering a population of sensitives).
What virulence evolves to depends on what it trades off against¶
Knowing that selection maximizes \(R_0 = \beta k /[u(u+v)]\) tells you nothing about where virulence ends up until you specify how \(\beta\) (transmissibility) depends on \(v\) (virulence) — because \(v\) appears in the denominator, more virulence is always bad for \(R_0\) on its own; whether it is worth it depends entirely on what it buys in transmissibility. The lecture works through three assumptions Martin considers.
1. \(\beta\) independent of \(v\). If a parasite's ability to spread does not depend at all on how much damage it does to the host, then \(R_0\) is strictly decreasing in \(v\), and selection drives \(v \to 0\). This is the formal version of the folk claim that a well-adapted parasite does not harm its host: if killing the host buys nothing, the only effect of virulence is to shorten the window in which the host can spread the infection further, so lower virulence is strictly better. The lecture is careful to flag this as a conclusion of a very particular (and probably not generally true) assumption, not an empirical law — Martin's own counterexample is malaria, which has coexisted with humans for a very long time and remains highly virulent.
2. \(\beta\) proportional to \(v\) (e.g. \(\beta = av\) — imagine viral load inside the host driving both how sick you get and how infectious you are). Then
which is monotonically increasing in \(v\) and has the Michaelis–Menten shape: linear for small \(v\), saturating toward a finite ceiling as \(v\) grows. Because it is monotonic with no interior peak, the value that maximizes \(R_0\) is \(v\to\infty\) — virulence evolves to be arbitrarily large.
3. \(\beta\) itself saturates in \(v\) (e.g. \(\beta = av/(c+v)\) — past some point, extra viral load no longer buys extra transmissibility; the lecture's example is that once you reliably infect everyone you sneeze on, having still more virus in you does not raise the infection rate further). Now \(R_0(v)\) can have an interior maximum, and virulence evolves to some finite, intermediate \(v^*\). As the saturation constant \(c \to \infty\), this case reduces to case 2 (transmission is approximately linear in \(v\) over the relevant range), and correspondingly the evolved virulence grows without bound as \(c\) grows.
The lecture's own assessment is that real data suggest infectivity does increase with virulence but sub-linearly (saturating) — case 3's shape — though how strong a claim that supports is disputed. And there is a further complication once super-infection is allowed (multiple strains sharing a host, which the two-strain model above ruled out by assumption): a strain that shares a host with a rival must out-compete that rival inside the host as well as transmit onward, and a strategy of high virulence — burn through the host fast and move on — avoids paying the cost of a rival strain exploiting a host kept alive for the long haul. In that setting, low virulence functions like a cooperative strategy (conserving host resources to maximize the transmission window), and it is exploitable by "cheater" high-virulence strains in exactly the sense that the earlier phage-cheater example illustrated: a strategy that does the patient thing is vulnerable to a rival that does not. Super-infection therefore tends to select for higher virulence than the single-strain-per-host model predicts.
The puzzle: why sex, given its cost?¶
Switching topics entirely (the shared thread is only that parasites reappear as the driving force): obligate, biparental sexual reproduction is astonishingly widespread among animals, despite what looks, on its face, like a severe cost. The clearest version of the cost is the twofold cost of males: compare a population that reproduces sexually, with separate males and females each producing two offspring per generation (holding population size constant), against a population that reproduces asexually — parthenogenetically or hermaphroditically — where every individual can produce offspring. Two asexual females, each having (say) two offspring, both of which can themselves reproduce, grow the population at twice the exponential rate of a sexual population that has to "spend" half of every generation producing males, who do not themselves bear young. Whenever a mutation arises in a sexual population that switches an individual to parthenogenetic reproduction, that lineage should, in principle, spread rapidly — and this is not purely hypothetical: sharks kept in captivity for years without access to a male have been documented to give birth via parthenogenesis ("virgin birth").
The cost is not unique to full biparental sex with distinct sexes — it is the most extreme point on a spectrum whose underlying theme is sharing DNA. Bacterial horizontal gene transfer carries a milder version of the same cost: a bacterium such as B. subtilis entering a competence state to take up external DNA has to stop dividing while it does so, and risks taking up DNA that is actively harmful. So the question is general: what selective advantage from recombination could be large enough to outweigh even the mildest version of this cost, let alone the twofold cost of males?
The Red Queen hypothesis¶
The paper the lecture discusses ("Running with the Red Queen," read for this session) argues for recombination's advantage via genetic diversity, in a specific form. In a purely asexual (clonal) population, two independently-arising beneficial mutations in different lineages cannot both reach fixation together — only the fitter of the two genotypes wins, and the population has to wait for the second mutation to arise again, this time in the winning background, before it can also spread. This is clonal interference, covered earlier in the course. Recombination breaks that constraint: a beneficial allele can spread through the population as a gene, moving across different genetic backgrounds by outcrossing, rather than being permanently tied to the one individual in which it first arose. A sexually-reproducing population can therefore adapt faster when adaptation requires combining multiple beneficial changes.
That advantage only matters if the environment changes enough, and often enough, to keep rewarding new combinations — and a genuinely fixed abiotic environment may not change fast or dramatically enough on its own to make this worth the twofold cost. The Red Queen hypothesis (the name is from the Red Queen in Lewis Carroll's Through the Looking-Glass, who has to keep running just to stay in the same place) proposes a specific, reliable source of that constant change: coevolving parasites. A parasite population is under selection to target whichever host genotype is currently common, because that is where it can spread most easily; the host population is, in effect, continually being "chased" away from whatever genotype the parasites have most recently adapted to. Both populations are then locked in a moving target dynamic — a treadmill of mutual adaptation — that keeps generating exactly the kind of environmental change that favors recombination.
One loose end raised in class: bacteria and other microbes also carry parasites (phage, plasmids) and also exchange genes horizontally, yet full obligate sex with two distinct sexes is overwhelmingly a feature of large, multicellular organisms. The professor's own tentative answer (offered as a guess, not as settled) is that body size and generation time are correlated, and generation time sets the pace at which a population can itself evolve: large organisms, with long generation times, evolve slowly relative to their much-faster-reproducing parasites, and are therefore precisely the organisms most in need of a mechanism — recombination — that speeds up their own rate of adaptation without waiting on new mutations one at a time.
The C. elegans / Serratia coevolution experiment¶
The paper's central experiment tests the Red Queen prediction directly, using the millimeter-scale nematode C. elegans and its bacterial pathogen Serratia marcescens. The worm side of the experiment used three mating regimes:
- wild type — able to self-fertilize or outcross with males, as normal;
- obligate outcrossing — forced to mate with males every generation;
- obligate selfing — forced to self-fertilize, with outcrossing prevented.
Each was challenged with three bacterial regimes:
- coevolution — the bacteria surviving each round of infection were propagated forward, so the pathogen population evolves alongside the host population;
- no evolution — worms were repeatedly challenged with the same, unevolving ancestral bacterial strain;
- control — no bacterial challenge at all.
Two results stood out. First, under coevolving bacteria, the obligate-selfing worm population was driven extinct — a purely clonal host population could not keep pace with a pathogen population that kept adapting to it. Second, tracking the outcrossing rate (the fraction of matings involving males) in the wild-type population over time gave a striking contrast between the two bacterial regimes: against coevolving bacteria, the outcrossing rate rose from roughly 20% to as high as 80% and stayed elevated; against the fixed ancestral strain, the outcrossing rate also rose at first but then fell back down once the population had adapted to that unchanging challenge. In other words, sex was favored specifically by a moving target, not by a fixed one — exactly the Red Queen prediction.
The paper's own closing sentence, which the lecture singles out as remarkable in its honesty: "sex can facilitate adaptation to novel environments, but the long-term maintenance of sex requires that the novelty does not wear off." That is, this experiment supports sex as a response to ongoing coevolutionary novelty, but does not by itself explain why that novelty would be sustained indefinitely in nature — an open question the lecture leaves standing rather than resolving.
Sources¶
- MIT 8.591J (Systems Biology), Fall 2014 — lecture recording
cn5k8r8ceii(transcript only; no slides or board images were supplied). Timestamps below refer to that transcript. - Microparasite/macroparasite distinction, ubiquity of parasites, phage-cheater aside: 00:00–10:25.
- The \(x,y\) infection model, \(R_0\) definition, the geometric-distribution argument, and the \(R_0=\beta k/[u(u+v)]\) derivation: 11:42–29:25.
- Vaccination threshold and the disease/\(R_0\) table (values beyond the "5–15" and "3–5" ranges quoted verbally were on a physical handout the professor held up in class and are not in the transcript): 29:25–1:02:56.
- Two-strain model, no-super-infection assumption, the \(R_2>R_1\) invasion derivation: 31:37–46:00.
- Trade-off cases for \(\beta(v)\) and the evolution of virulence, including the super-infection caveat: 47:06–57:34.
- Evolution of sex, twofold cost of males, Red Queen hypothesis, body-size aside: 1:02:56–1:14:17.
- The C. elegans/Serratia experiment and its results: 1:14:17–1:17:39.
- Referred to but not contained in the transcript: chapter 11 of Martin Nowak's book on evolutionary dynamics (the primary source for both the basic infection model and the two-strain virulence model — the super-infection extension of that chapter, covering more than two co-circulating strains, was mentioned but explicitly not covered in this lecture); the assigned paper "Running with the Red Queen" (source of the C. elegans/Serratia experiment and its closing sentence, quoted from memory in class); the optional supplementary paper by Jamie Lloyd-Smith on individual heterogeneity in infectiousness and superspreading; a classic paper by Lin Chao on cheater phage strategies under multiple infection; the standard SIR epidemic model (pointed at as "just Google SIR," not derived in class, and left to the course's problem set); and a card-based in-class polling exercise used to build up the \(R_0\) formula and the virulence-vote, neither of which is recoverable from audio alone.
← 7. Anticipatory Regulation and Bet Hedging · Contents · 9. Robustness in Bacterial Chemotaxis →