The MATH of Pandemics | Intro to the SIR Model

Scientific concepts / nature phenomena presented

• Epidemic modeling using the SIR model (a compartmental model in epidemiology)

  • The population is divided into three categories:
    • Susceptible (S): people who can catch the infection
    • Infected (I): people currently carrying/transmitting the infection
    • Recovered (R): people who have cleared the infection and (under the model’s assumptions) cannot be infected again and cannot transmit it

• Core assumptions of the SIR model (as stated)

  • At the start, everyone is susceptible except a small initial number infected.
  • After infection, individuals move from S → I, then later I → R.
  • Death rate is assumed small compared to recovery, so the model mostly tracks S, I, and R.

• Conservation of population

  • [ S(t) + I(t) + R(t) = N ] where (N) is the total population size (possibly for a specific region/town).

• Differential equations / rates of change

  • The model is expressed as a system of nonlinear differential equations describing how (S), (I), and (R) change over time (t):
    • ( \frac{dS}{dt} ) decreases as susceptibles become infected (proportional to (S \times I) interactions)
    • ( \frac{dI}{dt} ) increases via new infections (from (S \times I)) and decreases via recovery (proportional to (I))
    • ( \frac{dR}{dt} ) increases as recovered individuals come from the infected via recovery
  • Parameters mentioned qualitatively:
    • Transmission / spread parameter (denoted (a))
    • Recovery parameter (denoted (B))

• Epidemic threshold via the initial growth rate

  • Evaluate ( \frac{dI}{dt} ) near (t=0) using initial susceptible count (S_0) and initial infected (I_0).
  • The sign of ( \frac{dI}{dt} ) determines whether infection:
    • grows (epidemic) or
    • dies out initially.
  • This yields a threshold condition involving (\frac{aS_0}{B}):
    • Epidemic growth if (\frac{aS_0}{B} > 1)
    • No epidemic growth (initial decline) if (\frac{aS_0}{B} < 1)
  • This ratio is identified as being analogous to (R_0) (often called the basic reproduction number).

• Public health interpretation

  • Reducing transmission (lowering (a)):
    • Examples: quarantine, washing hands
    • Goal: make the threshold ratio smaller so it drops below 1.
  • Reducing the susceptible pool (lowering (S_0)):
    • Example: vaccination (reduces the number of susceptible people)
  • The recovery parameter (B) is described as harder to change (since recovery depends more on the disease/immune response).

• Early-time exponential growth

  • Near (t=0), assuming (S(t) \approx S_0) is roughly constant, the infected curve behaves like:

    • [ I(t) \propto e^{(aS_0 - B)t} ]

  • This explains why case counts can appear exponential early.

• Log-scale visualization

  • Plotting cases on a logarithmic scale turns exponential growth into an approximately linear trend.

Methodology / modeling steps (as described)

• Define compartments

  • Susceptible (S)
  • Infected (I)
  • Recovered (R)

• Set initial conditions

  • (I(0)=I_0), small (often “one” or a small number)
  • (S(0)=S_0), large
  • (R(0)=0)

• Write a system of differential equations for:

  • ( \frac{dS}{dt} ) from infection of susceptibles via contact with infected ((S \times I) term)
  • ( \frac{dI}{dt} ) from gains (new infections) and losses (recovery)
  • ( \frac{dR}{dt} ) from recovery of infected individuals

• Analyze

  • Qualitative behavior of curves (S(t)), (I(t)), and (R(t))
  • Epidemic vs non-epidemic initial condition using the sign of (dI/dt) at (t=0)
  • Early-time growth as exponential

• Compare to real data (illustrative example)

  • Uses epidemic curves (example given: “outside of China”) and notes that early portions look exponential, later flatten as (S(t)) decreases.

Sources / researchers mentioned

• Worldometer(s): explicitly referenced as the data source for the example curve.
• Researchers around the world are mentioned generally, but no specific researcher names are provided.