Evaluating vaccine allocation strategies using simulation-assisted causal modelling

The paper derives the target function s(π̃) for expected severe cases by (i) writing the intervention as MD E[S | do(T1,T2,T3 ∼ P̃(·|A))], (ii) expanding via law of total probability over A, T, W, and vaccination times, (iii) using the deterministic relationships V,W as functions of (t,t1,t2,t3), and (iv) substituting the severity-mechanism factorization Pπ̃(S=1|V,A,T,W)=f0(T) g(V,A) hV(W) f1_π̃(A,T) to obtain the four-range triple-sum in Eq. (5). The supplementary derivation shows the CI steps and the deterministic elimination V,W → ranges in Eqs. (57)–(60). The candidate solution makes the same conditioning/partition argument, explicitly enumerates the ranges for (t1,t2,t3), factors out f0(t) and f1_π̃(a,t), and recovers exactly the same triple-sum expression. Assumptions match those in the paper, including the independence of f1_π̃(A,T) from V and W via the infection-dynamics correction (Eqs. (19)–(20)). Hence both are correct and follow substantially the same proof path. See the paper’s Eq. (5) in Section 4.1 and the detailed derivation in Supplement B (Eqs. (57)–(60) and the factorization (2), (19)–(20)) .