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The lecture decides that an epidemic occurs if the time derivative of the infected compartment is positive, i.e., $\frac{dI}{dt}>0$. This means the number of infected people is growing in the early stages of the outbreak, when the susceptible population is close to the total population and the infected population is small.

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.

The lecture decides that an epidemic occurs if the time derivative of the infected compartment is positive, i.e., $\frac{dI}{dt}>0$. This means the number of infected people is growing in the early stages of the outbreak, when the susceptible population is close to the total population and the infected population is small.

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.

The lecture states $S+I+R=N$ to express the conservation of total population in the displayed SIR model. It means that individuals move among the susceptible, infected, and recovered compartments, but the total counted population remains fixed at $N$, assuming no births, deaths unrelated to removal, immigration, or emigration are included in the accounting.

Conditions: No births, deaths unrelated to removal, immigration, or emigration are included in the displayed accounting.; N is treated as the total population size.

The lecture states $S+I+R=N$ to express the conservation of total population in the displayed SIR model. It means that individuals move among the susceptible, infected, and recovered compartments, but the total counted population remains fixed at $N$, assuming no births, deaths unrelated to removal, immigration, or emigration are included in the accounting.

Conditions: No births, deaths unrelated to removal, immigration, or emigration are included in the displayed accounting.; N is treated as the total population size.

In the displayed SIR equations, $\beta$ represents the transmission rate per pair of people, one infected and one susceptible, which governs the flow of new infections. $\gamma$ represents the removal or recovery rate, which governs the flow of infected individuals leaving the infected compartment to join the recovered or removed compartment.

Conditions: $\beta$ is the transmission rate per pair.; $\gamma$ is the removal/recovery rate.; The displayed model treats the population as divided into S, I, and R compartments.

In the displayed SIR equations, $\beta$ represents the transmission rate per pair of people, one infected and one susceptible, which governs the flow of new infections. $\gamma$ represents the removal or recovery rate, which governs the flow of infected individuals leaving the infected compartment to join the recovered or removed compartment.

Conditions: $\beta$ is the transmission rate per pair.; $\gamma$ is the removal/recovery rate.; The displayed model treats the population as divided into S, I, and R compartments.