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A Mathematical Model of Swine Flu (H1N1) Transmission with Imperfect Behavioral Vaccination Dynamics: A Case Study of Japan

This study proposes a novel eight-compartment deterministic model incorporating time-dependent behavioral vaccination dynamics to analyze H1N1 transmission in Japan, demonstrating that the basic reproduction number governs disease stability and that effective mitigation relies on optimizing vaccination coverage, quarantine measures, and reducing perceived vaccination costs.

Original authors: Md. Zakir Hosen, Akhil Kumar Srivastav, Jun Tanimoto

Published 2026-07-21

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CC BY 4.0

Original authors: Md. Zakir Hosen, Akhil Kumar Srivastav, Jun Tanimoto

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world of infectious diseases as a giant, chaotic dance floor. In this dance, the virus is the music, and the people are the dancers. Sometimes, the music is so catchy that everyone rushes to the center, bumping into each other and spreading the rhythm everywhere. This is the realm of epidemiology, the science of tracking how diseases move through crowds. For decades, scientists have used simple maps to predict this dance, assuming that once a vaccine is available, people will line up to get it at a steady, boring pace, like waiting for a bus. But real life is messier. People don't just follow a schedule; they watch the dance floor, see who is getting sick, and decide in real-time whether it's worth the risk to get vaccinated. This paper dives into that messy, human side of the dance, asking: what happens when our fear of getting sick and our perception of the vaccine's difficulty or risk cost us effort or worry, and we change our minds based on what we see?

The researchers, a team from Kyushu University, built a new kind of mathematical model to study the swine flu (H1N1). Instead of treating vaccination as a fixed number, they introduced a "behavioral variable," a fancy way of saying they created a digital mirror of human decision-making. They asked: if people see a lot of sick people, will they get vaccinated? And if they think the vaccine is too risky or difficult compared to the flu, will they skip it? They also accounted for the fact that vaccines aren't perfect shields; sometimes, even vaccinated people can catch the flu, just less easily than those who aren't vaccinated.

The team found that the dance floor's chaos is heavily influenced by how people feel about the costs. When the "perceived cost" of getting the flu (sickness, missing school, feeling terrible) feels higher than the "perceived cost" of the vaccine (the effort to get it, the fear of side effects), more people jump on the vaccination bandwagon. This creates a feedback loop: more sickness leads to more vaccines, which slows down the sickness. However, if the vaccine's perceived cost feels too high, or if the vaccine isn't strong enough to stop "breakthrough" infections (where a vaccinated person still gets sick), the virus keeps dancing.

Using data from Japan, the team simulated this scenario and found their model matched real-world flu reports quite well, with a statistical score (R²) of 0.7247, meaning their digital dance floor looked a lot like the real one. They discovered that the most powerful moves to stop the flu aren't just about making vaccines; they are about making people feel like getting vaccinated is the smart, safe choice. If we can lower the perceived cost of the vaccine and make sure quarantine measures (like isolating sick dancers) are quick and effective, the virus loses its rhythm. The study suggests that while we can't control the virus's mutations, we can control the dance by understanding how people think, feel, and decide when the music starts.

Technical Summary: A Mathematical Model of Swine Flu (H1N1) Transmission with Imperfect Behavioral Vaccination Dynamics

Problem Statement
Influenza A (H1N1), or swine flu, remains a critical public health challenge due to its rapid transmission and potential for widespread outbreaks. While numerous mathematical models have addressed H1N1 dynamics, existing literature often treats vaccination as a constant intervention rate. This approach overlooks the critical feedback mechanism between infection prevalence and individual vaccination decision-making. Specifically, previous models frequently fail to account for how the perceived cost of infection relative to the perceived cost of vaccination drives dynamic changes in vaccination uptake, nor do they consistently integrate imperfect vaccination (where vaccinated individuals can still be infected, albeit at a lower rate) alongside behavioral dynamics. This study addresses these gaps by proposing a model that couples epidemiological transmission with time-dependent behavioral vaccination responses.

Methodology
The authors developed a deterministic compartmental model dividing the total human population into eight epidemiological compartments: Susceptible ( $S$ ), Vaccinated ( $V$ ), Exposed ( $E$ ), Symptomatic Infectious ( $I_s$ ), Asymptomatic Infectious ( $I_a$ ), Quarantined ( $Q$ ), Recovered ( $R$ ), and a behavioral variable ( $X$ ).

The core innovation is the inclusion of a behavioral differential equation for $X(t)$ , representing the proportion of individuals willing to receive vaccination:
$\frac{dX}{dt} = mX(1 - X)(C(I_s + I_a) - kC_v)$
Here, $m$ is the imitation rate, $C$ is the perceived cost of infection, $C_v$ is the perceived cost of vaccination, and $k$ is a weighting parameter. This equation models vaccination behavior as a cost-benefit analysis where individuals are more likely to vaccinate as the prevalence of infection (and thus the perceived cost of infection) rises.

The model incorporates:

Imperfect Vaccination: Vaccinated individuals ( $V$ ) have a lower transmission rate ( $\beta_1$ ) than susceptible individuals ( $\beta$ ), where $\beta > \beta_1$ .
Asymptomatic and Symptomatic Infections: Distinct compartments for $I_s$ and $I_a$ with different progression and quarantine rates.
Quarantine: Explicit modeling of the isolation of both symptomatic and asymptomatic cases.

Mathematical Analysis
The study rigorously analyzed the model's qualitative properties:

Basic Reproduction Number ( $R_0$ ): Derived using the next-generation matrix approach. Two distinct $R_0$ values were calculated corresponding to different behavioral equilibria: $R_0^{(1)}$ for the no-vaccination state ( $X=0$ ) and $R_0^{(2)}$ for the full-vaccination state ( $X=1$ ).
Equilibrium Points: The existence of disease-free equilibria (DFE) and endemic equilibria (EE) was established.
- The DFE is locally asymptotically stable (LAS) if $R_0 < 1$ .
- The EE exists and is LAS if $R_0 > 1$ , subject to Routh-Hurwitz stability conditions.
Sensitivity Analysis: Normalized forward sensitivity indices were computed to identify parameters most influencing $R_0$ . The transmission rates ( $\beta, \beta_1$ ) showed the highest positive sensitivity, while quarantine rates ( $\gamma_s, \gamma_a$ ) and vaccination rates ( $\nu$ ) showed significant negative sensitivity.
Numerical Simulations: The system was solved using the fourth-fifth order Runge-Kutta (RK45) method. The study employed Partial Rank Correlation Coefficient (PRCC) analysis, contour plots, heatmaps, and time-series simulations to explore parameter interactions.

Key Results

Behavioral Dynamics: The simulations demonstrate that vaccination behavior ( $X$ ) is not static; it increases as the perceived cost of infection rises (driven by higher $I_s + I_a$ ) and decreases if the perceived cost of vaccination is high.
Impact of Imperfect Vaccination: The model confirms that breakthrough infections (governed by $\beta_1$ ) and waning immunity ( $\phi$ ) significantly increase the effective reproduction number ( $R_{0,eff}$ ) and the peak infection size, even in the presence of vaccination.
Control Strategies: Sensitivity analysis and contour plots indicate that reducing transmission rates, minimizing waning immunity, and limiting breakthrough infections are essential. However, the study highlights that behavioral parameters are equally critical; lowering the perceived cost of vaccination ( $C_v$ ) or increasing the perceived cost of infection ( $C$ ) significantly enhances vaccination uptake and suppresses outbreaks.
Model Validation: The model was validated against weekly confirmed influenza cases in Japan from July 1, 2025, to May 30, 2026. The simulated data showed satisfactory agreement with observed data, yielding:
- Root Mean Square Error (RMSE): 109.4976
- Mean Absolute Error (MAE): 75.1195
- Coefficient of Determination ( $R^2$ ): 0.7247
- Residual analysis indicated no systematic bias, suggesting the model captures the general epidemic profile, including the peak and decline phases.

Significance and Claims
The paper claims that its primary contribution is the integration of time-dependent behavioral vaccination dynamics with imperfect vaccination and quarantine within a unified mathematical framework. By moving beyond constant vaccination rates, the model offers a more realistic representation of how public perception and cost-benefit analysis influence epidemic trajectories.

The authors assert that the model provides a useful framework for understanding swine flu transmission, suggesting that effective mitigation strategies must go beyond biological interventions. Specifically, they conclude that improving vaccination coverage, strengthening quarantine, and reducing transmission rates must be accompanied by strategies that lower the perceived cost of vaccination relative to the cost of infection to promote positive behavioral responses.

The study modestly acknowledges limitations, noting that the model assumes homogeneous population mixing and does not explicitly incorporate spatial heterogeneity, age structure, stochastic effects, media influence, or individual psychological differences. Future work is proposed to extend the framework to include these factors for a more comprehensive understanding of vaccination behavior.

Technical Summary: A Mathematical Model of Swine Flu (H1N1) Transmission with Imperfect Behavioral Vaccination Dynamics

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