Past immunity can reduce future epidemics

Why More Transmissible Variants Don't Always Mean

The Paradox of Winning the Epidemic War

In the long arc of a pandemic, a counterintuitive pattern keeps emerging: the variants that spread fastest and widest sometimes leave behind populations that are hardest for their successors to infect. It's as if the very success of one wave plants the seeds of restraint for the next. But not always. Sometimes a dominant variant is followed by another wave of similar size. Sometimes it is followed by near silence. And sometimes the pattern becomes wild—large, small, medium, unpredictable—cycling through epidemic sizes with no clear rhythm.

This is the puzzle at the heart of a new mathematical model developed by Ryuichi Kumata, Yuma Fujimoto, Hisashi Ohtsuki, and Akira Sasaki, published in July 2026. Their work doesn't just describe these patterns—it explains why they happen and, more surprisingly, reveals that the intensity of an epidemic is not what you might expect. The pathogens that spread most easily do not, in fact, produce the largest recurring outbreaks. There is an optimum—a sweet spot of transmissibility where epidemics are at their biggest, and moving away from that sweet spot in either direction makes them smaller.

The finding has implications for how we understand influenza, SARS-CoV-2, and any pathogen that evolves to escape the immunity built up from past infections. It also challenges the intuition that "herd immunity" always grows with each successive wave.

The Immune Footprint of an Epidemic

To understand their model, it helps to start with a question that sounds simple but turns out to be surprisingly hard to answer: When a wave of infection passes through a population, what exactly does it leave behind?

Classically, epidemiologists think in terms of the SIR framework—one of the most productive simplifications in the field. "S" stands for susceptible, the people who can still catch the disease. "I" stands for infected and infectious. "R" stands for recovered, people who have cleared the infection and gained immunity. A basic SIR model predicts that an epidemic burns through the susceptible population, depleting it until there aren't enough hosts left to sustain transmission, at which point the wave crests and fades.

But this picture assumes the pathogen doesn't change. Real pathogens do. Influenza variants drift antigenically—mutating just enough that immunity from last year's strain offers partial, but incomplete, protection against this year's. SARS-CoV-2 did the same thing with its Omicron lineage and its successors. Each variant is related to its predecessors but distinct enough that the immune system doesn't recognize it perfectly.

This is where Kumata and colleagues made their contribution. They asked: what happens when you layer SIR dynamics on top of this antigenic evolution? Specifically, when one wave ends and a new, antigenically related variant arrives months or years later, how much protection does the previous wave's immunity actually provide? And how does that protection shape the next epidemic's size?

Their model—called a recurrent SIR model—tracks what happens across successive waves. Each wave begins with some fraction of the population still susceptible to the new variant. This fraction is determined by two forces working in opposite directions. First, the previous wave's infections have given some people immunity against the new variant—call this cross-immunity. Second, the new variant has mutations that let it partially escape that immunity. The interplay between these forces determines how many hosts the new wave can find.

The model captures this with a few elegant variables. The basic reproduction number, denoted $\rho$ (rho), measures how transmissible a variant is in a fully susceptible population—the higher the rho, the more contagious the pathogen. The escape parameter, $\sigma$ (sigma), measures how much protection is lost between one variant and the next. When sigma is zero, successive variants are close enough that immunity from one largely protects against the next. When sigma is one, each variant escapes completely and behaves as if it were encountering a naive population. Intermediate values of sigma represent partial escape, which is what we see in reality with influenza and SARS-CoV-2.

A third parameter, $k$, governs how cross-immunity changes with antigenic distance. When $k = 1$, cross-immunity decays multiplicatively with each step in antigenic space—the further a future variant is from a past one, the weaker the inherited protection. When $k < 1$, cross-immunity persists more strongly across larger antigenic distances. When $k > 1$, it drops off more sharply. This parameter turns out to matter enormously for what kinds of epidemic patterns emerge.

$Figure 1: Schematic summary of our recurrent epidemic model. (A) During wave ii, an SIR epidemic reduces the susceptible density to the current variant from Si(i)(0)S_{i}^{(i)}(0) to Si(i)(∞)S_{i}^{(i)}(\infty). The susceptibility retention factor is ϕi=Si(i)(∞)/Si(i)(0)\phi_{i}=S_{i}^{(i)}(\infty)/S_{i}^{(i)}(0), and the epidemic size is ψi=Si(i)(0)−Si(i)(∞)=Si(i)(0)(1−ϕi)\psi_{i}=S_{i}^{(i)}(0)-S_{i}^{(i)}(\infty)=S_{i}^{(i)}(0)(1-\phi_{i}). (B) Each epidemic wave updates the susceptible profile for the current and future variants. During wave ii, susceptibility to variant jj is reduced as Sj(i)(∞)=ϕiωj(i)Sj(i)(0)S_{j}^{(i)}(\infty)=\phi_{i}^{\omega_{j}^{(i)}}S_{j}^{(i)}(0), and this profile is carried over to the next wave. Iterating this update gives rise to the sequences of the susceptible density faced by each introduced variant. (C) Cross-immunity generated by variant jj against variant ii decreases with antigenic separation according to ωi(j)=(1−σ)|i−j|k\omega_{i}^{(j)}=(1-\sigma)^{|i-j|^{k}}.$

Figure 1: Schematic summary of our recurrent epidemic model. (A) During wave ii, an SIR epidemic reduces the susceptible density to the current variant from Si(i)(0)S_{i}^{(i)}(0) to Si(i)(∞)S_{i}^{(i)}(\infty). The susceptibility retention factor is ϕi=Si(i)(∞)/Si(i)(0)\phi_{i}=S_{i}^{(i)}(\infty)/S_{i}^{(i)}(0), and the epidemic size is ψi=Si(i)(0)−Si(i)(∞)=Si(i)(0)(1−ϕi)\psi_{i}=S_{i}^{(i)}(0)-S_{i}^{(i)}(\infty)=S_{i}^{(i)}(0)(1-\phi_{i}). (B) Each epidemic wave updates the susceptible profile for the current and future variants. During wave ii, susceptibility to variant jj is reduced as Sj(i)(∞)=ϕiωj(i)Sj(i)(0)S_{j}^{(i)}(\infty)=\phi_{i}^{\omega_{j}^{(i)}}S_{j}^{(i)}(0), and this profile is carried over to the next wave. Iterating this update gives rise to the sequences of the susceptible density faced by each introduced variant. (C) Cross-immunity generated by variant jj against variant ii decreases with antigenic separation according to ωi(j)=(1−σ)|i−j|k\omega_{i}^{(j)}=(1-\sigma)^{|i-j|^{k}}. Source: Ryuichi Kumata, Yuma Fujimoto

Building a Wave-to-Wave Map

The mathematics of the model builds on a classic result from epidemic theory: the "final size" of an SIR epidemic can be calculated from the initial susceptible density and the reproduction number. If you know how many susceptible hosts there are when a variant arrives and you know how transmissible that variant is, you can compute how many people will ultimately be infected in that wave.

Kumata and colleagues extended this logic to track how the susceptible population changes across waves. During wave $j$, some hosts get infected and move into the recovered compartment. But recovery doesn't just protect against the current variant—it also provides partial protection against future variants, weighted by the antigenic relationship between them. The model captures this through what the authors call a "susceptibility retention factor," denoted $\phi_i$. This factor represents the fraction of hosts who remain susceptible at the end of a wave. If $\phi_i = 0.7$, then 70 percent of those who were susceptible at the start of the wave are still susceptible at the end; the other 30 percent have been infected and now carry some level of immunity.

The key insight is recursive. The susceptible density at the start of wave $i$ depends on what happened in all previous waves. Each earlier wave $j$ reduces susceptibility to the current variant by a factor of $\phi_j^{\omega_i^{(j)}}$, where $\omega_i^{(j)}$ is the cross-immunity generated by wave $j$ against wave $i$. If variants $i$ and $j$ are antigenically close, $\omega_i^{(j)}$ is large and susceptibility drops sharply. If they're far apart, $\omega_i^{(j)}$ is small and the previous wave's immunity barely matters.

This gives the authors a recurrence map—a way of calculating, wave by wave, what the population's immune landscape looks like and how large each epidemic will be. The model doesn't predict when new variants will arrive; it takes the sequence of introductions as given and traces what happens as each one spreads through the immunity left by its predecessors.

When the cross-immunity parameter $k = 1$, the math simplifies into a one-dimensional map. The susceptible density at the start of each wave can be written in terms of the susceptible density and susceptibility retention factor from the previous wave. This simplification makes the dynamics analytically tractable: the authors can derive exactly when the system is stable, when it period-doubles, and why.

The Emergence of Recurring Patterns

The first major result to emerge from the model is that equal-sized, repeating epidemics—the simplest possible pattern—are remarkably common. Across a broad range of parameter values, the system settles into a fixed point where each wave infects the same fraction of the population. Epidemiologists might call this the "endemic equilibrium"—not zero infections, but a steady state where waves keep coming but keep hitting the same size.

This period-1 dynamics appears when transmission is moderate and cross-immunity is either strong or moderate. The population develops a kind of rhythm: each wave infects enough people to reduce susceptibility to the next variant, but not so many that the next wave is strangled. Variant arrives, finds its susceptible audience, infects a predictable fraction, and fades. The next variant arrives to find a population that has been shaped by the previous wave but not decimated. And the cycle repeats.

But the model also reveals that this stability is not guaranteed. When transmission is strong—when rho is high—and antigenic escape is limited—when sigma is low—the system can destabilize. The authors call this a "period-doubling bifurcation." Instead of equal-sized epidemics, you get alternating large and small waves. One wave burns through most of the susceptible population. The next variant arrives to find almost no one left to infect, so its wave is negligible. But that negligible wave also generates little new immunity, leaving the population relatively susceptible again for the wave after that. And so the pattern alternates: boom, bust, boom, bust.

Go further still, with very high transmission and weak escape, and the dynamics become more complex still—period-4 cycles, where four different wave sizes repeat, or aperiodic oscillations that look chaotic. The epidemic sizes bounce around without settling into any clear pattern.

Figure 2: Representative recurrent epidemic dynamics. Upper panels show epidemic size, ψi\psi_{i}, and lower panels show the wave-onset reproduction number, ρonset(i)=ρSi(i)(0)\rho_{\mathrm{onset}}^{(i)}=\rho S_{i}^{(i)}(0), across successive introduced variants. (A) Period-11 dynamics, with equal-sized recurrent epidemics (k=1k=1, ρ=10\rho=10). (B) Period-22 dynamics, with alternating large and negligible epidemics (k=1k=1, ρ=21\rho=21). (C) Irregular dynamics consistent with high-period or chaos-like behavior (k=2k=2, ρ=30\rho=30). The shaded region indicates ρonset(i)≤1\rho_{\mathrm{onset}}^{(i)}\leq 1, where no epidemic occurs. Other parameter: σ=0.3\sigma=0.3. Source: Ryuichi Kumata, Yuma Fujimoto

The bifurcation diagrams in the paper make this transition vivid. For $k = 1$ and $\sigma = 0.3$, epidemics are equal-sized and stable from roughly $\rho = 1$ to $\rho \approx 20$. Above that threshold, the fixed point loses stability and period-2 cycles emerge. Push higher, to around $\rho = 30$, and period-4 cycles appear. The graph shows a characteristic "splitting" of the epidemic size values—where one line of points suddenly branches into two, then four, reflecting the emergence of more complex rhythms.

What determines whether the system stays stable or tips into these more dramatic patterns? Three factors matter: the intrinsic transmissibility (rho), the strength of antigenic escape (sigma), and the shape of cross-immunity across antigenic distances (k). Strong transmission pushes toward instability. Weak escape does the same. And when cross-immunity decays rapidly with antigenic distance—when $k > 1$—the destabilization happens at lower transmission thresholds. The system becomes much more sensitive to parameter changes.

Figure 3: Bifurcation diagrams of recurrent epidemic size (A, B, C) and wave-onset reproduction number (D, E, F). Black points show long-term epidemic sizes sampled over 500 points after transients for each ρ\rho, and red curves show the mean epidemic size. Light gray region indicates no-epidemic conditions (ρSi(i)(0)<1\rho S_{i}^{(i)}(0)<1). Panels compare different values of the cross-immunity shape parameter: (A, D) k=1k=1, (B, E) k=0.5k=0.5, and (C, F) k=2k=2. Other parameters: σ=0.3\sigma=0.3 Source: Ryuichi Kumata, Yuma Fujimoto

The Map of Possible Worlds

To help visualize these regimes, Kumata and colleagues produced a periodicity map—a two-dimensional landscape where the axes are rho and sigma, and the colors indicate what kind of dynamics the model predicts. The map is striking in its structure. A large blue region of period-1 dynamics dominates much of the parameter space, especially at lower transmission rates and higher escape. This is where equal-sized recurrent epidemics are the rule. Pushing into higher transmission and lower escape, the colors shift to period-2, then to period-4 and beyond. The transition is not abrupt: there are broad bands of higher-period dynamics before the system tips into what appears to be chaos.

For $k = 0.5$, where cross-immunity is more durable across antigenic distances, the period-1 region expands to cover nearly the entire parameter space. The population's immune memory is long, and this long memory has a stabilizing effect—each wave's immunity helps damp out fluctuations before they can grow. For $k = 2$, the opposite is true. Cross-immunity fades quickly with antigenic distance, so each wave is more isolated from its predecessors. The system is more prone to complex dynamics; period-2 and period-4 regimes appear at much lower transmission rates, and chaos-like behavior is widespread.

Figure 4: Periodicity of recurrent epidemic dynamics across ρ\rho and σ\sigma. Colors indicate the dominant period estimated from simulated epidemic sequences. Black indicates no persistent epidemic. Dashed curves in panel B show analytical stability boundaries for the period-11 and period-22 solutions. Numerical methods for periodicity detection are explained in the Appendix. Source: Ryuichi Kumata, Yuma Fujimoto

The authors verified their numerical results against analytical predictions for the $k = 1$ case, where the simplified one-dimensional map allows explicit calculation of stability boundaries. The analytical curves—shown as dashed lines on the periodicity map—match the numerical classifications closely, confirming that the dynamics are indeed driven by the mathematics of the recurrence relation rather than numerical artifacts.

The Surprising Optimum

But the most counterintuitive result concerns the size of epidemics, not their pattern. One might assume that more transmissible pathogens would produce larger recurring outbreaks—that higher rho would mean more infections per wave, everything else being equal. The model shows this isn't true.

Instead, epidemic size peaks at an intermediate value of rho and then declines. For $\sigma = 0.3$, the maximum recurrent epidemic size occurs around $\rho \approx 11$ for the $k = 1$ case. Below this value, waves are small because transmission is weak—each variant struggles to find susceptible hosts even in a relatively naive population. Above this value, waves also become smaller—but for a different reason. When rho is high, each wave is so explosive that it infects a large fraction of the population, building up strong cross-immunity against future variants. This immunity then suppresses the next wave, making it smaller. The next wave after that may be larger, depending on the dynamics, but the average over many cycles goes down.

Figure 5: Intermediate basic reproduction numbers maximize recurrent epidemic size for k=1k=1. (A) recurrent epidemic size ψ^\hat{\psi} as a function of ρ\rho, (B) initial susceptible density S^\hat{S}, and (C) infection probability among susceptible hosts, 1−ϕ^1-\hat{\phi}. Dashed lines in panel A indicate ρ∗\rho^{*} and ψ^∗\hat{\psi}^{*}. Panels D and E show how (D) the maximizing basic reproduction number ρ∗\rho^{*} and (E) the maximal recurrent epidemic size ψ^∗\hat{\psi}^{*} vary with σ\sigma. (F) Wave onset reproduction number ρonset∗\rho^{*}_{\text{onset}} that is realized value of ρonset\rho_{\text{onset}} when ρ\rho takes the value ρ∗\rho^{*} that maximizes recurrent epidemic size for a given σ\sigma. The lines are analytical solutions and points are obtained from the numerical simulation of the dynamics. Parameter: σ=0.3\sigma=0.3 Source: Ryuichi Kumata, Yuma Fujimoto

This is the central trade-off the model identifies. High transmissibility boosts infections within a wave—but it also enhances cross-immunity, reducing future susceptibility. Low transmissibility does the opposite: each wave is small and generates little immunity, leaving the population largely susceptible to the next wave. The optimum sits in between, where within-wave infection and between-wave immunity are balanced to maximize the average epidemic size.

The effect holds across different values of $k$, though the location of the peak shifts. When $k = 0.5$, cross-immunity is more durable and the peak occurs at lower rho. When $k = 2$, it occurs at higher rho. But in all cases, the non-monotonic relationship is robust: the biggest recurring epidemics don't come from the most transmissible pathogens.

This finding connects to a broader theme the authors emphasize: past success does not guarantee future success. A variant that causes a massive wave because it encounters a naive population may, by that very success, build immunity that suppresses the next wave. A variant that causes a moderate wave may leave behind a population that is more susceptible to its successor. The feedback between population immunity and epidemic dynamics means that outcomes in one wave don't predict outcomes in the next.

The Epidemic Governor

Another striking finding concerns the wave-onset reproduction number—the effective reproduction number that a variant actually faces when it arrives in the population, after past waves have reshaped immunity. One might expect that a variant with rho of 20 would experience enormous epidemic growth. The model shows otherwise.

Even when the intrinsic basic reproduction number is high—say, rho equals 20 or 30—the wave-onset reproduction number stays close to the epidemic threshold. At $k = 1$, the transition from period-1 to period-2 dynamics occurs at an intrinsic rho of around 20, but the wave-onset reproduction number at that point is only about 1.5. Population immunity is doing something that looks almost like a governor, limiting how much growth is possible even for very transmissible variants.

This is because each wave reduces the susceptible population substantially, and cross-immunity further reduces susceptibility to closely related future variants. The combination can be powerful enough to keep the effective reproduction number near 1 even when the pathogen's intrinsic capacity for spread is many times higher. The epidemic cannot grow beyond what the depleted susceptible pool allows, regardless of how contagious the variant is.

This finding has implications for thinking about pandemic control. Interventions that reduce transmission during one wave—vaccination campaigns, non-pharmaceutical interventions—don't just affect that wave. They affect the immune landscape that future variants will encounter. A population that has been "spared" a large wave may be more susceptible to the next variant, not less. Conversely, a population that experiences a large wave may be temporarily protected against the next.

Connecting to the Real World

How does this model relate to actual pathogen dynamics? The authors discuss influenza and SARS-CoV-2 as prime examples of antigenically evolving pathogens where these dynamics might apply.

Influenza is perhaps the most familiar case. The virus drifts antigenically through "antigenic space," with each season's variants evolving away from their predecessors. The result is roughly annual waves that vary in size from year to year—sometimes big, sometimes small—but generally maintain themselves as ongoing seasonal epidemics. This is consistent with the model's period-1 dynamics, which covers a large region of parameter space.

SARS-CoV-2 has followed a more dramatic trajectory. After the original strain and the Alpha and Delta waves, Omicron arrived with substantial antigenic escape, infecting large portions of the population that had been protected by previous immunity. The subsequent waves—BQ.1, XBB, and their descendants—have shown varying degrees of antigenic relationship to their predecessors, sometimes causing large surges and sometimes smaller ones. The model suggests these variations might be understood as dynamics near a bifurcation boundary, where the system is sensitive to small changes in transmission or immune escape.

The authors are careful to note that their model is a simplification. It assumes a closed population with no demographic turnover, no immune waning between waves, and variants that are introduced from outside without modeling their evolution or migration. These assumptions make the mathematics tractable but mean the model is best understood as a conceptual framework for understanding immune feedback dynamics rather than a detailed forecast for any specific pathogen.

Caveats and Open Questions

Several limitations deserve acknowledgment. First, the model assumes long-lasting immunity within each variant—once recovered, a host is permanently immune to that specific strain. This holds reasonably well for many infections over the timescale of a single epidemic, but real immunity often wanes over months or years. Incorporating immune waning would likely add complexity, potentially destabilizing the dynamics further.

Second, the model treats the antigenic relationship between variants as a simple function of their ordinal distance. In reality, antigenic evolution is more complex—some mutations confer large escape gains, others small, and the topography of antigenic space is not uniform. The simple power-law kernel in the model captures the basic idea of declining cross-immunity with antigenic distance but glosses over this complexity.

Third, the model takes the sequence of variant introductions as given and does not consider how population immunity might affect which variants succeed. In reality, variants that encounter strong population immunity are less likely to spread, which could create a feedback between epidemic dynamics and evolutionary dynamics that the model does not capture.

Fourth, the transition to chaotic dynamics—where epidemic sizes vary aperiodically—is inferred from numerical simulation. While the patterns are consistent with chaos, proving rigorously that the dynamics are chaotic (rather than merely high-period) would require additional analysis, such as computing Lyapunov exponents or showing sensitive dependence on initial conditions.

Despite these limitations, the framework offers a clear conceptual lens for thinking about how immune history shapes epidemic futures. The finding that epidemic size is maximized at intermediate transmission, and that past success does not guarantee future success, applies as long as there is feedback between population immunity and epidemic dynamics—the basic condition for any antigenically evolving pathogen.

What This Opens Up

The framework developed by Kumata and colleagues points toward several directions for future research.

One avenue is to connect the model to time series data from real pathogens. Influenza surveillance data, with its annual cycles and varying wave sizes, could potentially be used to estimate the underlying parameters—the effective reproduction number, the escape rate, the cross-immunity decay function. If the model's predictions match observed patterns of year-to-year variation, that would support its relevance as a conceptual framework.

Another direction is to consider what the model implies for vaccine design. If cross-immunity generated by infection depends on antigenic distance, then vaccines that antigenically match circulating variants might generate stronger population immunity than those that don't. The model could be extended to ask whether vaccinating with updated antigens might stabilize epidemic dynamics or reduce average wave sizes.

A third direction involves the transition to chaotic dynamics. The model shows that chaotic behavior can arise under plausible conditions—high transmission, limited escape, rapidly decaying cross-immunity. If real pathogen dynamics occasionally enter chaotic regimes, this would complicate short-term forecasting. Understanding the conditions under which epidemics become unpredictable, rather than simply variable, could help public health planners set appropriate expectations for forecast uncertainty.

More broadly, the work illustrates how simple mathematical structures can generate rich dynamical behavior from the feedback between infection and immunity. The SIR model, first developed nearly a century ago, remains a powerful tool for understanding infectious disease. Layered with antigenic evolution, it reveals that epidemic futures are not determined by pathogen properties alone—they are co-produced by the pathogen, the population's immune history, and the complex interaction between them.

The intuitive expectation—that bigger epidemics build stronger immunity, which should suppress future epidemics—turns out to be half right. But the other half is that bigger epidemics also come from more transmissible pathogens, which tend to generate stronger cross-immunity, which suppresses future epidemics even more. The net effect is that the most transmissible pathogens don't always produce the largest recurring outbreaks. The relationship between transmission and epidemic size is shaped by feedback, and feedback creates optima where intuition would not predict them.

This is a reminder that epidemic dynamics are nonlinear systems, and nonlinear systems can surprise us. The same forces that make predictions difficult also make them interesting. As we continue to track antigenically evolving pathogens through their successive waves, frameworks like this one offer a way of organizing our expectations—and a reminder that the past is always shaping the future, even when it does so in ways that are not immediately obvious.