What condition on dI/dt is used to decide whether an epidemic occurs?

Conditions

• The model is the displayed SIR system.
• The relevant comparison is made in the early stages of the outbreak, where S is close to N and I is small compared with N.

Reasoning, step by step

1. Identify the infected-compartment equation: $\frac{dI}{dt}=\beta SI-\gamma I$.
2. Define epidemic onset as growth in the number of infected individuals.
3. Set the condition for growth as $\frac{dI}{dt}>0$.
4. Restrict attention to the early stages where S is close to N and I is much smaller than N.

Example

The lecturer writes on the right side of the board the criterion 'Epidemic occurs if:' followed by $\frac{dI}{dt}>0$. He explains that an epidemic corresponds to growth in the number of infected people.

Common misconceptions

• Assuming the condition applies at all times rather than specifically in the early stages.
• Thinking an epidemic occurs if $\frac{dS}{dt}$ is negative, without checking the growth of I.

Watch the explanation

Explore next

• In the discrete one-infected example, what are the initial values of S, I, and R?
• Why does the epidemic start growing when (N-1)$β/γ$ is greater than 1?