Understanding Exponential, SI, SIS, and SIR Models for Disease Spread

1 Exponential Model

Let the number of infected individuals be I(t). Assuming each infected person makes β effective contacts per unit time that can transmit the disease, the increment of infected individuals over a time interval Δt can be expressed. By moving terms to one side and taking the limit as Δt → 0, we obtain the differential equation dI/dt = β I, whose solution is the classic exponential growth model I(t) = I₀ e^{βt}.

This model is identical to the exponential population growth model and provides an analytical solution directly.

However, if infected individuals only contact other infected persons, the model double‑counts infections, so it is necessary to distinguish between infected (I) and susceptible (S) individuals.

2 SI Model

The population is divided into Susceptible (S) and Infected (I). The SI model assumes a closed population (no births or deaths) and a constant total N = S + I. Each infected person makes β effective contacts per unit time, and only contacts with susceptible individuals lead to new infections.

Using the same reasoning as the exponential model, the change in infected individuals over Δt is ΔI = β S I Δt / N, where the factor S/N reflects the decreasing proportion of susceptibles.

The resulting differential equations produce an S‑shaped (logistic) growth curve for I(t), reaching a maximum proportion of 1 as time goes to infinity. This shows that, unlike pure exponential growth, the infection saturates when the entire population becomes infected.

Nevertheless, the SI model predicts that eventually everyone will be infected, which is unrealistic for diseases where recovery or immunity occurs.

3 SIS Model

The SIS model also splits the population into Susceptible (S) and Infected (I), but allows infected individuals to recover and become susceptible again. The assumptions are the same as the SI model, with an additional daily recovery rate γ.

The total population remains constant.

Each infected person makes β effective contacts per unit time.

Each infected person recovers at rate γ per day.

The change in infected individuals over Δt is ΔI = β S I Δt / N - γ I Δt. Taking the limit yields the differential equation dI/dt = β S I / N - γ I.

Because this equation does not have a simple closed‑form solution, a discrete approximation is used to derive a recurrence relation for I(t). The model predicts that the numbers of infected and susceptible individuals converge to a steady state that depends on the values of β and γ, rather than all individuals becoming infected.

When γ = 0, the SIS model reduces to the SI model.

4 SIR Model

The SIR model adds a third compartment: Removed (R), representing individuals who have recovered and gained permanent immunity. The population is divided into Susceptible (S), Infected (I), and Removed (R). The model assumes a closed population and the same contact rate β and recovery rate γ as before.

The differential equations are:

dS/dt = -β S I / N

dI/dt = β S I / N - γ I

dR/dt = γ I

These equations describe how susceptibles become infected, infected individuals either recover or remain infectious, and recovered individuals move permanently to the removed class.