SIR compartments and biological transfers

This lesson focuses only on compartments and transfers. We use whole observed infection and recovery events to understand how one biological movement changes two compartment counts. Rates, differential equations and Euler’s method begin in later lessons.

Biological scenario

At the beginning of a short reporting interval, a closed population contains 990 susceptible, 10 infectious and no recovered people.

During the interval, 3 susceptible people become infectious and 1 infectious person recovers. What are the new compartment counts?

The three compartments

Symbol	Compartment	Meaning
\(S\)	Susceptible	Can become infected.
\(I\)	Infectious	Currently infectious and able to transmit under the model assumptions.
\(R\)	Recovered or removed	No longer participates in transmission in the basic SIR model.

The permitted biological transfers

SSusceptible

→infection

IInfectious

→recovery

RRecovered

The arrows show changes of biological state, not physical movement. This basic model permits only \(S\to I\) through infection and \(I\to R\) through recovery.

One event changes two compartments

One infection

\[S\to S-1,\qquad I\to I+1.\]

The same person leaves \(S\) and enters \(I\).

One recovery

\[I\to I-1,\qquad R\to R+1.\]

The same person leaves \(I\) and enters \(R\).

This coordinated change is the foundation of compartment modelling.

Calculate the interval by hand

With 3 infections and 1 recovery:

Compartment	Before	Change	After
Susceptible	990	−3	987
Infectious	10	+3−1	12
Recovered	0	+1	1
Total	1,000	0	1,000

Translate the transfers into Python

Step 1: Preserve the before-state

S_before = 990
I_before = 10
R_before = 0

The suffix _before makes the time meaning explicit.

Step 2: Store the observed transfers

new_infections = 3
new_recoveries = 1

These are whole event counts supplied for this interval. This lesson does not yet calculate them from rates.

Step 3: Calculate all after-values from the same before-state

S_after = S_before - new_infections
I_after = I_before + new_infections - new_recoveries
R_after = R_before + new_recoveries

Separate before- and after-names prevent an updated value from being accidentally reused in another calculation.

Run the transfer program

Interactive PythonCompartment transfers

# Compartment counts before the interval
S_before = 990
I_before = 10
R_before = 0

# Observed events during the interval
new_infections = 3
new_recoveries = 1

# Check whether the transfers are possible
if new_infections < 0 or new_recoveries < 0:
    raise ValueError("Event counts cannot be negative.")

if new_infections > S_before:
    raise ValueError("Infections cannot exceed susceptible people.")

if new_recoveries > I_before:
    raise ValueError("Recoveries cannot exceed infectious people.")

# Calculate every after-value from the before-state
S_after = S_before - new_infections
I_after = (
    I_before
    + new_infections
    - new_recoveries
)
R_after = R_before + new_recoveries

# Totals before and after
N_before = S_before + I_before + R_before
N_after = S_after + I_after + R_after

# Exact conservation check for integer transfers
assert N_after == N_before

# Display a before-change-after table
print("Compartment   Before   Change   After")
print("--------------------------------------")
print(
    f"Susceptible  {S_before:6d} "
    f"{-new_infections:8d} {S_after:7d}"
)
print(
    f"Infectious   {I_before:6d} "
    f"{new_infections - new_recoveries:8d} "
    f"{I_after:7d}"
)
print(
    f"Recovered    {R_before:6d} "
    f"{new_recoveries:8d} {R_after:7d}"
)
print("--------------------------------------")
print(f"Total        {N_before:6d} {0:8d} {N_after:7d}")

Output

Run the code to see the result.

Why the total is conserved

\[(-3)+(+3-1)+(+1)=0.\]

Every loss from one compartment is a matching gain to another. Therefore, internal transfers change the distribution of people but not the closed population total.

New programming conditions

Code	Meaning
`or`	The combined condition is true when at least one connected condition is true.
`>`	Tests whether the value on the left is greater than the value on the right.
`raise ValueError(...)`	Stops the program and explains why the supplied transfer is invalid.
`assert N_after == N_before`	Checks exact conservation because all values here are integer counts.

Boundary of this lesson

The values new_infections = 3 and new_recoveries = 1 are supplied observations. We have not yet asked how a deterministic model calculates flows per unit time.

Lesson 3 introduces rate formulas and differential equations. Lesson 6 introduces numerical updating through time.

Try another interval

Change the event counts to 6 infections and 4 recoveries.

Predict all three after-values.
Run the code.
Check that the total remains 1,000.
Explain why the infectious change is \(+6-4=+2\), not simply +6.