How Do SI, SIS, and SIR Models Explain Epidemic Spread?

1 Exponential Model

Define the number of infected individuals I(t). Assuming each patient makes β effective contacts per unit time, the increment of patients over a time interval Δt can be expressed, and by moving terms to the left side and taking the limit we obtain the differential equation dI/dt = β I, whose solution is I(t) = I₀ e^{β t}. This is the classic exponential growth model, identical to population growth.

However, this model counts contacts between infected individuals as new infections, leading to double‑counting. Therefore it is necessary to distinguish between infected and susceptible individuals.

2 SI Model

We divide the population into susceptible (S) and infected (I). The model assumes the total population N = S + I remains constant (no births or deaths) and each infected individual infects β susceptible individuals per unit time.

During the study period, the total number of people does not change; the proportions of infected and healthy are I/N and S/N.

The daily contact rate β applies only to contacts with susceptible individuals.

Following the same reasoning as the exponential model, the increment of infected individuals over Δt is expressed with a term β S I / N, reflecting reduced transmission as susceptibles decrease.

The SI model yields an S‑shaped (logistic) curve for the infected proportion, reaching a maximum of 1 as time goes to infinity. It shows that, regardless of how small β is, the entire population eventually becomes infected, which contradicts diseases where recovery confers immunity.

3 SIS Model

We keep the same two groups (S and I) but allow infected individuals to recover and become susceptible again, introducing a daily recovery rate γ.

Total population remains constant.

Infection rate β applies to contacts with susceptibles.

Recovery rate γ moves individuals from I back to S.

Applying the same incremental reasoning leads to the differential equations dI/dt = β S I / N – γ I and dS/dt = –β S I / N + γ I. Because an analytical solution is difficult, a discrete approximation is used, yielding a recurrence relation.

Eventually, the numbers of infected and susceptible individuals converge to a steady state that depends on β and γ, and not all individuals become infected. The SIS model reduces to the SI model when γ = 0.

4 SIR Model

For diseases that confer permanent immunity after recovery, we add a third compartment R (removed). The compartments are S, I, and R, with total N = S + I + R.

The increments are:

Infected: dI/dt = β S I / N – γ I

Susceptible: dS/dt = –β S I / N

Removed: dR/dt = γ I

Taking the limit yields the classic SIR differential equations. The model predicts an epidemic peak followed by a decline as immunity builds.

References: ThomsonRen GitHub, 司守奎 《Python数学实验与建模》.