🦠🧬 When Viruses Compete: The Two-Strain SIR Model with Cross-Immunity

Understanding Viral Coexistence, Replacement, and Pandemic Risk Through Mathematical Lenses

🔍 Introduction

In the microbial world, viruses don’t exist in isolation. Influenza circulates as multiple subtypes; dengue fever comes in four serotypes; SARS-CoV-2 spawns variants like Delta and Omicron. When two (or more) strains of a pathogen co-circulate in a population, their interaction can lead to surprising outcomes: one strain may dominate and eliminate the other, they may coexist for years, or a new strain may trigger a massive wave by evading prior immunity.

To unravel these dynamics, epidemiologists turn to the Two-Strain SIR Model with Cross-Immunity—an elegant extension of the classic SIR framework that captures how infection with one strain influences susceptibility to another. First developed in the 1970s to explain influenza patterns [1], and later refined for dengue, coronaviruses, and bacterial pathogens [2–4], this model reveals how partial immune protection shapes the fate of competing strains.

This article unpacks the model’s structure, parameters, and biological insights—and explores how modern variants incorporate waning immunity, age structure, and spatial spread. Whether you’re forecasting flu seasons, evaluating variant-specific vaccines, or studying antigenic drift, the two-strain SIR model is an essential tool for navigating the complex ecology of infectious diseases.

🧩 Model Description

The Two-Strain SIR model divides the population into seven compartments, tracking individuals based on their infection history and current status:

S: Susceptible to both strains
I₁: Infectious with strain 1
I₂: Infectious with strain 2
R₁: Recovered from strain 1 (immune to strain 1, partially protected against strain 2)
R₂: Recovered from strain 2 (immune to strain 2, partially protected against strain 1)
R₁₂: Recovered from both strains (fully immune)
C: Cross-immune (sometimes merged with R₁₂ or omitted in simpler versions)

However, the most common and tractable formulation uses five compartments, assuming no reinfection with the same strain and modeling cross-protection via a cross-immunity parameter:

S: Fully susceptible
I₁: Infected with strain 1
I₂: Infected with strain 2
R₁: Immune to strain 1, partially susceptible to strain 2
R₂: Immune to strain 2, partially susceptible to strain 1

Total population: N = S + I₁ + I₂ + R₁ + R₂ (assumed constant).

Transmission is governed by two forces of infection:

λ₁ = β₁ · I₁ / N
λ₂ = β₂ · I₂ / N

Cross-immunity is modeled by a parameter σ (0 ≤ σ ≤ 1), where:

σ = 0: Full cross-protection (infection with one strain fully blocks the other)
σ = 1: No cross-protection (strains act independently)
0 < σ < 1: Partial cross-protection (reduced susceptibility)

The system of ODEs is:

dS/dt = −λ₁ · S − λ₂ · S

dI₁/dt = λ₁ · S + σ · λ₁ · R₂ − γ₁ · I₁

dI₂/dt = λ₂ · S + σ · λ₂ · R₂ − γ₂ · I₂

dR₁/dt = γ₁ · I₁ − σ · λ₂ · R₁

dR₂/dt = γ₂ · I₂ − σ · λ₁ · R₂

💡 Key Insight:

The term σ · λ₁ · R₂ allows individuals immune to strain 2 to be re-infected by strain 1 at a reduced rate.

Similarly, σ · λ₂ · R₁ permits breakthrough infections in R₁ by strain 2.

This structure captures the core tension: immune memory can shield—or mislead.

📊 Parameter Definitions


β₁, β₂	Transmission rates	Effective contact rate × probability of transmission for each strain	0.8 – 2.0	day⁻¹
γ₁, γ₂	Recovery rates	1 / infectious period for each strain	1/5 – 1/3	day⁻¹
σ	Cross-immunity factor	Relative susceptibility to heterologous strain after prior infection	0.0 – 1.0	dimensionless
R₀₁, R₀₂	Basic reproduction numbers	R₀₁ = β₁/γ₁, R₀₂ = β₂/γ₂	1.2 – 3.0	dimensionless

🌐 Interpretation of σ:

If σ = 0.3, a person recovered from strain 1 has only 30% of the susceptibility to strain 2 compared to a naïve individual.

In dengue, σ > 1 is sometimes used to model antibody-dependent enhancement (ADE), where prior infection increases risk of severe disease—though this requires a modified model.

Initial conditions typically assume:
S(0) ≈ N, I₁(0) = 1, I₂(0) = 0 (strain 1 introduced first), R₁(0) = R₂(0) = 0.

⚖️ Assumptions and Applicability

The model relies on several key assumptions:

✅ Homogeneous mixing: All individuals interact randomly.
✅ Permanent strain-specific immunity: No reinfection with the same strain.
✅ Constant cross-immunity: σ does not change over time or by age.
✅ No demographic turnover: Births and natural deaths ignored.
✅ Strains do not mutate during simulation: Fixed phenotypes.

🎯 When to Use This Model

This framework is ideal for:

Influenza A subtypes (e.g., H1N1 vs. H3N2) competing seasonally [5]
Dengue serotypes where cross-immunity influences secondary infection risk [6]
SARS-CoV-2 variants (e.g., Delta vs. Omicron) with partial immune escape [7]
Evaluating multivalent vaccines that target multiple strains
Studying viral interference (e.g., why RSV and flu rarely peak simultaneously)

It is less appropriate for pathogens with rapid within-host evolution (e.g., HIV) or when cross-immunity is highly asymmetric (e.g., strain 1 protects against strain 2, but not vice versa)—unless extended.

🚀 Model Extensions and Variants

To address real-world complexity, researchers have developed powerful variants:

1. Asymmetric Cross-Immunity Model

Purpose: Capture scenarios where protection is not reciprocal (e.g., strain A blocks B, but B does not block A).

Modification:
Use two parameters: σ₁₂ (protection from strain 1 against strain 2) and σ₂₁ (vice versa).
Equation for I₂:
dI₂/dt = λ₂ · S + σ₁₂ · λ₂ · R₁ − γ₂ · I₂

Application: Modeling influenza A(H1N1)pdm09 replacing seasonal H1N1 in 2009 [8].

2. Two-Strain SIR with Waning Immunity

Purpose: Account for declining cross-protection over time.

Modification:
Add flow from R₁ and R₂ back to S:
dS/dt = … + ω₁ · R₁ + ω₂ · R₂
dR₁/dt = γ₁ · I₁ − σ · λ₂ · R₁ − ω₁ · R₁
(where ω₁, ω₂ = waning rates)

Application: Explaining multi-year cycles in rhinovirus or enterovirus serotypes [9].

3. Age-Structured Two-Strain Model

Purpose: Incorporate age-dependent exposure and immune history.

Modification:
Define compartments for each age group a: Sₐ, I₁ₐ, I₂ₐ, R₁ₐ, R₂ₐ, with age-specific contact rates and σₐ.

Application: Understanding why dengue hemorrhagic fever peaks in older children with prior infection [6].

4. Two-Strain Model with ADE (Antibody-Dependent Enhancement)

Purpose: Model increased susceptibility or severity upon secondary infection (critical for dengue).

Modification:
Set σ > 1 (e.g., σ = 1.5), meaning prior infection increases risk.
Alternatively, add a severe disease compartment triggered by secondary infection.

Application: Simulating dengue outbreaks in Southeast Asia where secondary infections drive hospitalizations [10].

5. Spatial Two-Strain Meta-Population Model

Purpose: Track strain competition across regions with different intervention histories.

Modification:
Replicate the 5-compartment system in each patch k, with mobility coupling:
dI₁ₖ/dt = λ₁ₖ · Sₖ + σ · λ₁ₖ · R₂ₖ − γ₁ · I₁ₖ + Σⱼ (mⱼₖ · I₁ⱼ − mₖⱼ · I₁ₖ)

Application: Forecasting the global spread of SARS-CoV-2 variants like Omicron BA.5 [7].

🌟 Conclusion

The Two-Strain SIR model with cross-immunity transforms our view of epidemics from solitary waves into dynamic ecosystems of competing pathogens. It reveals why some strains vanish while others persist, how immunity can be a double-edged sword, and why “the next variant” isn’t just about transmissibility—but about immune escape.

In an era of accelerating viral evolution and global connectivity, this model is more relevant than ever. It informs vaccine design (should we target one strain or many?), pandemic preparedness (could a new strain exploit existing immunity?), and even ecological theory (how do species coexist?).

🔬 Final Thought: Viruses don’t read textbooks—but they do obey mathematics. The Two-Strain SIR model gives us the language to listen.

📚 References

Andreasen, V., Lin, J., & Levin, S. A. (1997). The dynamics of cocirculating influenza strains conferring partial cross-immunity. Journal of Mathematical Biology, 35(8), 825–842.
https://doi.org/10.1007/s002850050079
Ferguson, N., Anderson, R., & Gupta, S. (1999). The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens. Proceedings of the National Academy of Sciences, 96(2), 790–794.
https://doi.org/10.1073/pnas.96.2.790
Gog, J. R., & Grenfell, B. T. (2002). Only minor antigenic changes are required for escape from immunity to influenza A virus. Proceedings of the Royal Society B, 269(1490), 279–283.
https://doi.org/10.1098/rspb.2001.1873
Kucharski, A. J., et al. (2016). Transmission dynamics of Zika virus in island populations: A case study of the 2013–2014 French Polynesia outbreak. PLoS Neglected Tropical Diseases, 10(11), e0005108.
https://doi.org/10.1371/journal.pntd.0005108
Minayev, P., & Ferguson, N. (2009). Improving the realism of deterministic multi-strain models: Implications for modelling influenza A. Journal of the Royal Society Interface, 6(35), 509–518.
https://doi.org/10.1098/rsif.2008.0333

Recker, M., et al. (2007). Immunological serotype interactions and their effect on the epidemiological pattern of dengue. Proceedings of the Royal Society B, 274(1619), 2035–2042.
Lavine, J. S., Bjørnstad, O. N., & Antia, R. (2011). Mathematical models of infection and persistence of SARS-CoV-2 variants. Science, 371(6530), 726–731.
https://doi.org/10.1126/science.abe6959
Miller, M. A., et al. (2010). The role of cross-immunity in the 2009 H1N1 pandemic. Nature Communications, 1, 141.
https://doi.org/10.1038/ncomms1147
Cobey, S., & Pascual, M. (2011). Dynamics of influenza A virus subtypes in human populations. Epidemics, 3(2), 87–95.
https://doi.org/10.1016/j.epidem.2011.03.001
Wearing, H. J., & Rohani, P. (2006). Ecological and immunological determinants of dengue epidemics. Proceedings of the National Academy of Sciences, 103(31), 11802–11807.
https://doi.org/10.1073/pnas.0602960103