Endemic Compartmental Models (SIS, SIRS, SIRV)
Endemic & Vaccination Compartmental Models (SIS, SIRS, SIRV) · Also known as: SIS Model, SIRS Model, SIRV Model, Endemic Disease Models
Endemic compartmental models extend the classical SIR framework to capture diseases that persist indefinitely in a population rather than burning out after a single epidemic wave. The SIS model allows recovered individuals to return to susceptibility immediately; SIRS introduces temporary immunity before loss; SIRV adds a vaccinated compartment. Together these models are foundational tools for studying diseases such as influenza, gonorrhea, and seasonal pathogens where reinfection or waning immunity is epidemiologically central.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
When to use it
Use endemic compartmental models when the disease of interest exhibits reinfection (no permanent immunity), waning immunity after recovery or vaccination, or when assessing long-term vaccination policy. The models assume homogeneous mixing, a closed or demographically steady population, and mass-action incidence. They are less appropriate for short-duration single-wave epidemics (use SIR instead), highly heterogeneous contact structures (use network or age-structured models), or when stochastic extinction near the threshold matters (use stochastic analogues).
Strengths & limitations
- Analytically tractable: closed-form endemic equilibria and R0 expressions allow direct policy thresholds without simulation.
- Flexible architecture: SIS, SIRS, and SIRV variants cover a wide spectrum of biological scenarios with minimal added complexity.
- Vaccination integration: SIRV provides explicit formulas for herd immunity thresholds and optimal vaccination rates.
- Well-established theoretical foundations with decades of mathematical analysis supporting stability and bifurcation results.
- Homogeneous mixing assumption ignores age structure, spatial heterogeneity, and social network effects that strongly influence real outbreaks.
- Deterministic ODEs do not capture stochastic fade-out, which is critical when case counts are small or the pathogen is near elimination.
- Parameter estimation (beta, gamma, delta, omega) requires reliable incidence data; misspecification propagates nonlinearly into equilibrium predictions.
- Models assume constant population size and exponential waiting times in each compartment, which may not match observed infectious-period distributions.
Frequently asked
What is the difference between SIS and SIRS?
In SIS, recovered individuals immediately return to the susceptible class, modelling diseases with no immunity whatsoever. In SIRS, a recovered compartment R grants temporary immunity for an average duration of 1/delta time units before individuals re-enter S. SIRS is therefore appropriate for diseases like influenza where post-infection immunity is real but not permanent.
How does vaccination change the endemic threshold?
In an SIRV model, vaccination reduces the effective susceptible fraction of the population. The effective reproduction number drops below R0, and herd immunity is achieved when the vaccinated fraction p/(p+omega) exceeds 1 - 1/R0. This formula directly determines the minimum vaccination rate required to drive the pathogen to extinction in the population.
Can endemic compartmental models generate oscillations?
Yes. SIRS and SEIRS models with appropriate parameter ranges can produce damped or sustained oscillations around the endemic equilibrium through Hopf bifurcation mechanisms. This oscillatory behaviour mimics the multi-year epidemic cycles observed for many respiratory diseases. Checking eigenvalues of the Jacobian at the endemic equilibrium is necessary to determine whether convergence is monotone or oscillatory.
Sources
- Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Review, 42(4), 599–653. DOI: 10.1137/S0036144500371907 ↗
How to cite this page
ScholarGate. (2026, June 2). Endemic & Vaccination Compartmental Models (SIS, SIRS, SIRV). ScholarGate. https://scholargate.app/en/epidemiology/endemic-compartmental-models
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Reproduction NumberEpidemiology↔ compare
- SEIR ModelEpidemiology↔ compare
- SIR ModelEpidemiology↔ compare