Appendix Table 1.

Summary of notation

Value	Description
t	Time epochs, t = 0, 1, 2, 3, 4, 5, T = 6
st	Patient health state at time t. States are: s1 = Healthy on treatment (H), s2 = DS TB on treatment (DS), s3 = DR TB on treatment (DR), s4 = healthy defaulted (dH), s5 = DS TB defaulted (dDS), s6 = DR TB defaulted (dDR), s7 = dead. States 4, 5, 6, and 7 are directly observable. Can be in the set S = 1, 2, 3, 4, 5, 6, 7
at	Action at time t. Can be one of DST (administer DST to the patient), wait and see (continue on first-line treatment with an SS test), wait and not see (continue on first-line treatment without an SS test).
δ	Discount factor
ft	7 × 7 transition matrix describing treatment dynamics. Element ft,ij provides probability of being in state j at time t + 1 if in state i at time t. The terms d, c, and ν represent probabilities of default, cure, and death, respectively.
zt	Observations at time t. Can be one of {SS+, SS−, default, death} if at = wait, and {No default or death, default, death} if at = DST.
Beliefs and Belief Updates
Sens	SS test sensitivity
Spec	SS test specificity
βt	Belief state at time t. A vector of probability values over each of the health states at time t.
${\overline{M}}_{z_t}\left(f_t{\beta }_t\right)$	Function that updates the belief given observation zt, dynamics ft, and prior belief βt: ${\beta }_t={\overline{M}}_{z_t}\left(f_t{\beta }_t\right)$
N+(x)	$\frac{1}{\left(1-\mathit{spec}\right)x\left(1\right)+\left(\mathit{sens}\right)\left(x\left(2\right)+x\left(3\right)\right)}$. Used in the belief update: $M_{zt=+}^{-}\left(x\right)=N_{+}\left(x\right)M_{+}x$
M+	$\left[\begin{array}{ccccccc}\left(1-\mathit{spec}\right)& 0& 0& 0& 0& 0& 0\\ 0& \left(\mathit{sens}\right)& 0& 0& 0& 0& 0\\ 0& 0& \left(\mathit{sens}\right)& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$. Used in the belief update: $M_{zt=+}^{-}\left(x\right)=N_{+}\left(x\right)M_{+}x$
N−(x)	$\frac{1}{\left(\mathit{spec}\right)x\left(1\right)+\left(1-\mathit{sens}\right)\left(x\left(2\right)+x\left(3\right)\right)}$. Used in the belief update: $M_{zt=-}^{-}\left(x\right)=N_{-}\left(x\right)M_{-}x$
M−	$\left[\begin{array}{ccccccc}\left(\mathit{spec}\right)& 0& 0& 0& 0& 0& 0\\ 0& \left(1-\mathit{sens}\right)& 0& 0& 0& 0& 0\\ 0& 0& \left(1-\mathit{sens}\right)& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$. Used in the belief update: $M_{zt=-}^{-}\left(x\right)=N_{-}\left(x\right)M_{-}x$
P(x)	$\frac{1}{x\left(4\right)+\left(x\left(5\right)+x\left(6\right)\right)}$. Used in the belief update: $M_{zt=\mathit{default}}^{-}\left(x\right)=P\left(x\right)Qx$
Q	$\left[\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$. Used in the belief update: $M_{zt=\mathit{default}}^{-}\left(x\right)=P\left(x\right)Qx$
Rewards
rwait,a	Stage reward.
rDST	Terminal rewards if at = DST. A 1 × 7 vector of net monetary benefits (NMB):
	[NMBDST,H NMBDST,DS NMBDST,DR NMBdH NMBdDS NMBdDS NMBdead].
r∼DST	Terminal reward if patient never undergoes DST. A 1 × 7 vector of net monetary benefits (NMB):
	[NMB~DST,H NMB~DST,DS NMB~DST,DR NMBdH NMBdDS NMBdDS NMBdead]