Navigating Chaos: The Dance Between Transmissibility and Immune Evasion in Viral Evolution

by Golsa Sayyar

Imagine a world where viruses constantly change, adapting to our immune defenses while trying to spread more efficiently. This biological tug of-war raises a crucial question: can we predict the evolutionary fate of viruses? Understanding this is not just about biology – it is a mathematical puzzle.

Our recent work dives deep into this intersection of biology and mathematics, exploring how viruses evolve when faced with a trade-off between transmissibility and immune evasion. The insights? Evolution isn’t a straightforward path – it can lead to stability, cycles, or even chaos.

A Trade-Off Between Two Powers

Viruses have two main strategies to thrive:
1. Transmissibility: How effectively they spread between hosts.
2. Immune Evasion: How well they dodge the host’s immune defenses.

Here’s the catch: these traits come at a cost. A virus that evades immunity might lose some transmissibility, and vice versa. Our mathematical model captures this delicate balance using a simple trade-off function, shedding light on how these traits evolve over time.

Modeling Evolution with Math

We began by representing the dynamics of a viral strain using a Susceptible-InfectedRecovered (SIR) model. To capture the trade-off, we introduced a parameter that balances transmissibility and immune evasion. The model’s iterative process, shown in Figure 1, illustrates how new viral strains emerge and replace previous ones.

Figure 1: The model outline flowchart: beginning with the SIR model with a transmission rate β, while the system is in an endemic steady state (red). New strains with immune-evasive property p, where p ∈ [0, 1], emerge. Assuming a trade-off between transmissibility and immune evasion, denoted by f(p), the transmission rates of these new strains are βv = β + f(p) (blue). As the evolution tends to favor the strain with maximal invasion fitness (maximizing the invasion reproduction number), we consider the newly emerging strain which is the fittest with a transmission rate βv = β + f(p max) (green). The system transitions to a new endemic steady state characterized by a novel transmission rate determined by the trade-off function. The process returns to the initial step and the evolutionary process is continued.

What the Math Tells Us

We modeled this trade-off using a system of equations and tracked the evolution of viral strains over generations. Here are the standout findings:

• Stable Evolution: Under certain conditions, evolution stabilizes. Successive strains
have similar transmission rates, leading to predictable outcomes.

• Cyclic Patterns: Sometimes, evolution alternates between two strategies—a strain
that is highly transmissible is followed by one that’s better at immune evasion, and
the cycle repeats.
• Chaos: In some scenarios, evolution becomes chaotic, with no clear pattern. Predicting the future of such systems is nearly impossible.

These behaviors depend on key parameters of the trade-off. Figure 2 captures the complexity of these dynamics through a bifurcation diagram. It shows how small changes in parameters can shift the system from stability to periodic cycles and even chaotic behavior.

From Equations to Real Life

Why does this matter? The dynamics we observed mirror the real-world behavior of
viruses like SARS-CoV-2. Some variants became more transmissible but less evasive
(e.g., Alpha and Delta), while others prioritized immune evasion at the cost of transmissibility (e.g., Omicron). Our findings provide a framework to anticipate such trends and help public health efforts adapt.

The Beauty of Mathematical Models

What makes this research exciting for mathematicians? It is the interplay between simple assumptions and complex outcomes. A linear trade-off function leads to fascinating dynamical behaviors—stability, periodicity, and chaos—all captured through bifurcation diagrams and difference equations.

Visual tools like cobweb plots and bifurcation diagrams reveal how small changes in parameters can push the system from predictable to chaotic. It is a reminder of how mathematical models transform biological intuition into quantifiable predictions.

Implications and Next Steps

Understanding these patterns is more than an academic exercise:
– Public health strategies, like vaccination and monitoring, can adapt if we understand evolutionary pressures.
– The approach can extend to other pathogens and trade-offs (e.g., virulence versus
transmissibility).

But there is more to explore. What happens when we add complexity—like superinfections or competition between multiple strains? And can new mathematical techniques refine our predictions?

Conclusion

Viral evolution is a dance between two opposing forces, and mathematics is our guide to understanding its steps. From stability to chaos, our model shows how simple assumptions about trade-offs can lead to profound insights into the unpredictable nature of evolution.

For mathematicians, exploring these dynamics is not just about solving equations – it is about unveiling the secrets of nature’s most intricate systems. So let’s embrace the chaos and keep decoding the dance of evolution.

Call to Action

Interested in exploring this further? Check out our full paper for detailed equations,
proofs, and visualizations of these fascinating dynamics. Let’s keep pushing the boundaries of what math can reveal about the world around us.

https://www.biorxiv.org/ content/10.1101/2024.06.29.601333v1.