Evolution into chaos - Implications of the trade-off between transmissibility and immune evasion.

Predicting viral evolution presents a significant challenge and is a critical public health priority. In response to this challenge, we develop a novel model for viral evolution that considers a trade-off between immunity evasion and transmissibility. The model selects for a new strain with the highest invasion fitness, taking into account this trade-off. When the dominant strain of the pathogen is highly transmissible, evolution tends to favor immune evasion, whereas for less contagious strains the direction of evolution leads toward increasing transmissibility. Assuming a linear functional form of this trade-off, we can express the long-term evolutionary patterns following the emergence of subsequent strains by a non-linear difference equation. We provide sufficient criteria for when evolution converges, and successive strains exhibit similar transmissibility. We also identify scenarios characterized by a two-periodic pattern in upcoming strains, indicating a situation where a highly transmissible but not immune-evasive strain is replaced by a less transmissible but highly immune-evasive strain, and vice versa, creating a cyclic pattern. Finally, we show that under certain conditions, viral evolution becomes chaotic and thus future transmissibilites become unpredictable in the long run. Visualization via bifurcation diagrams elucidates our analytical findings, revealing complex dynamic behaviors that include the presence of multiple periodic solutions and extend to chaotic regimes. Our analysis provides valuable insights into the complexities of viral evolution in the light of the trade-off between immune evasion and transmissibility.