The SIS Epidemiological Model – Canada Mosquitoes Data Portal

🧠 The SIS Epidemiological Model: A Popular-Science Introduction

🩺 Abstract

The Susceptible–Infected–Susceptible (SIS) model is one of the foundational models in mathematical epidemiology. It represents diseases that do not confer long-lasting immunity — once recovered, individuals can become susceptible again. This article provides an accessible introduction to the SIS model, its assumptions, mathematical formulation, and real-world applications. We also explore its variants and extensions, including demographic, stochastic, and network-based versions.

🌍 1. Introduction

Mathematical models in epidemiology are designed to capture how infections spread, persist, or vanish within populations. Among these, the SIS model is particularly suited for diseases like the common cold, gonorrhea, or certain strains of influenza, where recovered individuals return to the susceptible class.

The SIS model is conceptually simpler than the famous SIR (Susceptible–Infected–Recovered) model but no less important: it helps us understand endemic persistence, thresholds for eradication, and the role of contact rates in disease control.

⚙️ 2. Model Description

In the SIS framework, the total population N is divided into two compartments:

S(t) – Number of susceptible individuals at time t
I(t) – Number of infected individuals at time t

Since recovered individuals lose immunity immediately, R(t) is omitted, and the total population remains constant:

N = S(t) + I(t)

The differential equations describing the SIS model are:

dS/dt = -β * S * I / N + γ * I
dI/dt = β * S * I / N - γ * I

where:

β (beta) = transmission (infection) rate, typically ranging from 0.1 to 1.0 per day depending on the disease and contact frequency.
γ (gamma) = recovery rate, typically between 0.05 and 0.5 per day.

Since individuals recover and return to the susceptible class, there is no lasting immunity. The term βSI/N represents the number of new infections per unit time, while γI represents recoveries.

📈 3. Parameter Definitions

Symbol	Meaning	Typical Range	Notes
β	Contact or infection rate	0.1 – 1.0 per day	Higher β implies more frequent contact or higher transmissibility.
γ	Recovery rate	0.05 – 0.5 per day	Depends on disease duration (γ ≈ 1 / infectious period).
N	Total population	Fixed	Assumes closed population with no births or deaths.
S(t)	Susceptible individuals	Varies	Decreases when infected individuals spread the disease.
I(t)	Infected individuals	Varies	Increases when infection rate exceeds recovery rate.

🧩 4. Assumptions and Applicability

The SIS model is based on several simplifying assumptions:

Homogeneous mixing: Each individual has an equal chance of contacting others.
Constant population size: Births and deaths are ignored in the basic model.
Immediate return to susceptibility: Recovered individuals instantly become susceptible again.
No latent (exposed) period: Infection and recovery occur without delay.
Deterministic transmission: The model uses average rates rather than random events.

This model applies best to short-immunity infections and large populations where random fluctuations are negligible.

🔬 5. Disease Persistence and Threshold

The basic reproduction number (R₀), a central concept in epidemiology, determines whether the disease will persist or die out:

R₀ = β / γ

If R₀ < 1, each infection leads to fewer than one new infection on average — the disease dies out.
If R₀ > 1, the infection can persist endemically in the population.

At equilibrium, the steady-state fraction of infected individuals is given by:

I* = N * (1 - 1/R₀)

where I* represents the endemic equilibrium prevalence.

🧠 6. Model Extensions and Variants

🧬 a) SIS with Vital Dynamics

To include births and deaths, we modify the equations as:

dS/dt = μN - β * S * I / N + γ * I - μS
dI/dt = β * S * I / N - γ * I - μI

Here μ is the per-capita birth/death rate. This version better reflects human and animal populations with continuous demographic turnover.

⚡ b) Stochastic SIS Model

In smaller populations, random effects play a major role. The stochastic SIS model introduces randomness in infection and recovery events, making it possible for a disease to go extinct even if R₀ > 1.

🌐 c) Network-Based SIS Model

In real societies, people do not mix uniformly. Network models replace the homogeneous assumption with structured interactions — each node (individual) has links representing possible transmission paths. This allows simulation of clustered or heterogeneous transmission, typical in sexually transmitted infections or localized outbreaks.

🧮 d) Time-Dependent Transmission (Seasonality)

In some infections, transmission rates vary seasonally:

β(t) = β₀ [1 + α * cos(2πt / T)]

where α is the seasonal amplitude (0 ≤ α ≤ 1), and T is the period (e.g., 365 days).
This captures seasonal fluctuations such as those seen in influenza.

🩸 7. Applications

The SIS model has been applied to:

Endemic sexually transmitted diseases (e.g., gonorrhea, chlamydia)
Bacterial infections with temporary immunity (e.g., strep throat)
Computer virus propagation and malware dynamics
Hospital-acquired infections with continuous re-exposure

These examples demonstrate the SIS model’s versatility beyond biological diseases.

🌟 8. Conclusion

The SIS model elegantly captures the dynamics of diseases with no lasting immunity. Despite its simplicity, it provides deep insight into the balance between infection and recovery, the conditions for persistence, and the impact of control measures. Its many extensions continue to make it a powerful framework for research in both public health and theoretical biology.

📚 9. References

Hethcote, H. W. (1989). Three Basic Epidemiological Models. SIAM Review. Link
Brauer, F., & Castillo-Chavez, C. (2012). Mathematical Models in Population Biology and Epidemiology. Springer.
Anderson, R. M., & May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press.
Keeling, M. J., & Rohani, P. (2008). Modeling Infectious Diseases in Humans and Animals. Princeton University Press.
Diekmann, O., Heesterbeek, J. A. P., & Roberts, M. G. (2010). The construction of next-generation matrices. Journal of the Royal Society Interface. Link
Allen, L. J. S. (2008). An Introduction to Stochastic Processes with Applications to Biology. CRC Press.
Hethcote, H. W. (2000). The Mathematics of Infectious Diseases. SIAM Review, 42(4). Link
Kiss, I. Z., Miller, J. C., & Simon, P. L. (2017). Mathematics of Epidemics on Networks. Springer.
World Bank (2021). An Introduction to Deterministic Infectious Disease Models. Link
R-project (2020). EpiDynamics: Dynamic Models in Epidemiology. Link