Question

Sometimes viruses mutate, and a person who was immune (either through natural immunity or through vaccination) loses immunity.

At the beginning of Trial C, 50% of the population had immunity. Suppose that all of them lost their immunity halfway through the experiment. As a result, the data for Trial C would have started out the same but shown
in disease spread during the second half of the simulation.

Answer 1

Do you want that rewritten as a single clear sentence, or do you want an explanation of how to model and what the result would look like? Below I give (1) a concise rewrite you can paste into a report, (2) a short description of how to implement the change in an SIR-style simulation, and (3) what you should expect to see in the plotted data.

1) Concise rewrite (report-ready)
"At the start of Trial C, 50% of the population was immune. If all of those people lost immunity exactly halfway through the experiment, the Trial C data would match the original run during the first half but would show a renewed disease spread in the second half."

2) How to implement this in a simulation (SIR framework)
- Initial conditions: S(0)=0.5, I(0)=I0, R(0)=0.5 (population normalized to 1).
- Run the usual SIR equations until t = T/2:
dS/dt = -β S I
dI/dt = β S I - γ I
dR/dt = γ I
- At t = T/2 apply the immunity-loss event instantaneously:
S(t+)=S(t−)+R(t−), R(t+)=0, I(t+)=I(t−)
(i.e., move everyone who was recovered/immune back into susceptible)
- Continue integrating the SIR equations from t = T/2 to T with the new S, I, R.

3) Expected qualitative effect on the plotted data
- First half: with 50% immune you will likely see either no outbreak or a small outbreak (depending on R0).
- Immediately after the halfway point the susceptible fraction jumps upward (by 0.5), so the effective susceptible pool increases and if R0>1 you will observe renewed exponential growth of cases in the second half.
- The second-half epidemic curve will resemble an epidemic started in a largely susceptible population, but shifted in time and possibly altered in peak size because there may already be some infected or removed individuals from the first half.
- If immunity loss is instantaneous and complete, the second-half dynamics essentially replicate a run that began with the same susceptible fraction as after the jump; if immunity loss is gradual or partial, the second-half wave will be dampened/blurred accordingly.

If you want, I can:
- produce a toy simulation (choose β and γ or give me R0 and an infectious period) and show the two-phase curves, or
- rewrite the text differently (more technical or more layman-friendly). Which would you prefer?