Predicting Disease Outbreaks - A Mathematical Modeling Approach

Abstract

Infectious disease outbreaks pose ongoing challenges to global public health, particularly due to the recurrent and dynamic nature of many pathogens. Mathematical modeling offers a powerful tool for analyzing and predicting the spread of infectious diseases. This study focuses on the SIRS (Susceptible–Infectious–Recovered–Susceptible) model, a compartmental framework that accounts for temporary immunity and its loss over time. We present two case studies—Influenza and Pertussis—to examine how variations in transmission rate, recovery rate, and immunity loss rate influence epidemic behavior. Simulations reveal that diseases with slower immunity loss, such as influenza, tend to stabilize over time with predictable seasonal peaks, whereas faster immunity loss, as in pertussis, leads to sharper and more frequent outbreaks. These insights underline the importance of tailoring vaccination strategies and public health interventions to specific epidemiological dynamics. The results demonstrate the SIRS model’s utility in forecasting disease trends and informing control policies, thus providing a foundational approach for anticipating and managing future outbreaks.

Introduction

The SIRS model (Susceptible–Infectious–Recovered–Susceptible) is a key mathematical framework in epidemiology, especially for diseases where immunity is temporary, such as influenza, pertussis, and COVID-19 variants. Unlike the SIR model, which assumes permanent immunity, the SIRS model reflects the possibility of reinfection by allowing recovered individuals to become susceptible again.

2. Importance of SIRS Models

• Mathematical modeling helps understand disease dynamics, predict outbreaks, and evaluate intervention strategies.

• SIRS models provide a more realistic view of disease behavior when immunity wanes over time.

• Useful for policymaking, especially for booster vaccine scheduling, herd immunity estimates, and outbreak forecasting.

3. Key Advancements in SIRS Modeling

• Gradual Immunity Loss: Recent models (El Khalifi & Britton, 2022–2023) include linear/exponential waning immunity.

• Vaccination Dynamics: Models now account for vaccine efficacy decay, revaccination, and heterogeneous immunity loss (Paéz Chávez et al., 2025).

• Intrinsic Oscillations: SIRS models show natural outbreak cycles even without seasonal or behavioral changes (Marenduzzo et al., 2025).

• Stochastic Modeling: Incorporates randomness and population-level variability to capture real-world uncertainty (Alahakoon et al., 2023).

• Spatial Models: Multi-region models (Lai et al., 2024) include mobility and public response to simulate disease spread in connected communities.

4. Core SIRS Model Structure

Compartments

• S(t): Susceptible individuals

• I(t): Infectious individuals

• R(t): Recovered individuals with temporary immunity

• N = S + I + R (Total population, constant)

Differential Equations

dSdt=−βSI+ξRdIdt=βSI−γIdRdt=γI−ξR\frac{dS}{dt} = -\beta S I + \xi R \\ \frac{dI}{dt} = \beta S I - \gamma I \\ \frac{dR}{dt} = \gamma I - \xi RdtdS?=−βSI+ξRdtdI?=βSI−γIdtdR?=γI−ξR

Where:

• β: Transmission rate

• γ: Recovery rate

• ξ: Immunity loss rate

5. Key Concepts

• Basic Reproduction Number (R?):

  R0=βγR? = \frac{β}{γ}R0?=γβ?

  • If R? > 1, the disease spreads.

  • If R? < 1, the disease fades out.

• Endemic Equilibrium:
  A state where disease persists over time due to ongoing reinfection caused by waning immunity.

• Oscillatory Behavior:
  SIRS models can produce recurrent outbreaks or waves due to constant replenishment of susceptible individuals.

6. Methodology of the Study

The research aims to refine the classic SIRS model to better simulate real-world diseases by integrating:

• Nonlinear waning immunity

• Periodic vaccination strategies

• Seasonal effects

• Stochastic (random) variability

• Population heterogeneity

Objectives

1. Model partial and waning immunity, reinfection, and re-vaccination.

2. Simulate different outbreak scenarios using real/synthetic data.

3. Explore disease persistence or extinction through R? and equilibrium analysis.

4. Compare deterministic vs stochastic models to account for unpredictability.

5. Validate models using case studies (e.g., influenza, COVID-19).

6. Inform public health strategies on vaccination timing and risk of resurgence.

7. Applications & Implications

• Provides tools for predicting outbreak dynamics under changing immunity and vaccination.

• Supports data-driven decisions in public health, especially for diseases with recurrent waves.

• Helps optimize booster schedules, anticipate recurrent outbreaks, and set herd immunity thresholds.

Conclusion

In this work, we have applied the SIRS (Susceptible–Infectious–Recovered–Susceptible) mathematical model to investigate the spread and recurrence of infectious diseases such as Influenza and Pertussis. By simulating disease dynamics under different parameter settings, we demonstrated how key epidemiological factors—namely the transmission rate (?), recovery rate (?), and immunity loss rate (?)—profoundly affect the outbreak pattern and long-term behavior of infectious diseases. The simulations revealed that low immunity loss (Influenza) leads to controlled, periodic outbreaks with eventual stabilization, while high immunity loss (Pertussis) produces frequent and intense infection cycles due to the rapid return of individuals to the susceptible pool. These findings highlight the importance of tailoring public health strategies to disease-specific characteristics, particularly the duration of post-infection or post-vaccination immunity. Overall, the SIRS model provides a valuable and interpretable framework for predicting disease outbreaks, designing vaccination policies, and understanding the conditions under which a disease may become endemic or epidemic. It forms a basis for more complex models that incorporate population heterogeneity, seasonality, spatial distribution, and control interventions.

References

[1] El?Khalifi, M., & Britton, T. (2022). Extending SIRS epidemics to allow for gradual waning of immunity. arXiv Preprint. [2] El?Khalifi, M., & Britton, T. (2023). SIRS epidemics with individual heterogeneity of immunity waning. arXiv Preprint. [3] Páez Chávez, J., Gökçe, A., Götz, T., & Gürbüz, B. (2025). An SIRS model considering waning efficiency and periodic re vaccination. arXiv Preprint. [4] Marenduzzo, D., Brown, A. T., Miller, C., & Ackland, G. J. (2025). Oscillation in the SIRS model. arXiv Preprint. [5] Alahakoon, P., McCaw, J. M., & Taylor, P. G. (2023). Improving estimates of waning immunity rates in stochastic SIRS models with a hierarchical framework. Infectious Disease Modelling, 8(4), 1127–1137. [6] Huang, S., Poskitt, C. M., & Shar, L. K. (2025). How oscillations in SIRS epidemic models are affected by the distribution of immunity times. European Physical Journal B, 98, 22. [7] Lai, S., Ruktanonchai, N. W., Zhou, L., et al. (2024). Modeling precaution, immunity loss and dispersal on disease dynamics: a two patch SIRS model. Advances in Continuous and Discrete Models. [8] El?Khalifi, M., & Britton, T. (2024). SIRS epidemics with individual heterogeneity of immunity waning. PubMed/Elsevier. [9] https://arxiv.org/abs/2211.09062?utm_source=chatgpt.com \"Extending SIRS epidemics to allow for gradual waning of immunity\" [10] https://arxiv.org/abs/2501.17305?utm_source=chatgpt.com \"An SIRS-model considering waning efficiency and periodic re-vaccination\" [11] https://arxiv.org/abs/2504.00670?utm_source=chatgpt.com \"Oscillation in the SIRS model\" [12] https://pmc.ncbi.nlm.nih.gov/articles/PMC10597760/?utm_source=chatgpt.com \"Improving estimates of waning immunity rates in stochastic SIRS models with a hierarchical framework - PMC\" [13] https://link.springer.com/article/10.1140/epjb/s10051-024-00858-2?utm_source=chatgpt.com \"How oscillations in SIRS epidemic models are affected by the distribution of immunity times | The European Physical Journal B\" [14] https://pubmed.ncbi.nlm.nih.gov/38614211/?utm_source=chatgpt.com \"SIRS epidemics with individual heterogeneity of immunity waning - PubMed\" [15] https://arxiv.org/abs/2311.00592?utm_source=chatgpt.com \"SIRS epidemics with individual heterogeneity of immunity waning”

Copyright

Copyright © 2025 Dr. K. Sujatha, Dr. K. Karuppiah, N. Mohana, Dr. R. Venugopal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.