If I understand well, you observed the result of a medical test over time for each patient. I am assuming the result of the medical test is binary (positive-negative), so that your data consist of a $n\times t$ dataset db of $\{0,1\}$ values for $n$ patients observed over $t$ times.
You assume a latent Markov process having 2 latent statuses (the "true" statuses or core statuses of the patients) and you want to estimate the latent transition probabilities. In statistics this kind of Hidden Markov model is often referred to as Latent Transition Analysis.
An R package for LTA is LMest. So if db is your dataset:
library(LMest)
res <- est_lm_basic(db, yv=rep(1,n), k=2, start = 0, mod = 1)
mod = 1 indicates a time-homogeneous Latent Markov Chain
yv=rep(1,n) is the vector of frequencies of each observed $t$-sequence (here I put 1 for each observation without counting them)
k=2 is the number of latent statuses
As a result, res$Psi are the conditional probabilites, i.e., the probabilities of False Positive FP, True Negative TN, ecc.., while res$Pi are the latent transition probabilties you want.
Note that for the label switching problem (typical of finite mixture models), your latent statuses can be switched, that is, res$Psi can be :
$$
\left( \begin{array}{cc}
TN & FN\\
FP & TP \end{array} \right)
\text{ or }
\left( \begin{array}{cc}
FN & TN\\
TP & FP \end{array} \right)
$$
but since you know the expected specificity and sensitivity of your test you should be able to distinguish them.
I tried a simulation with $n=1000$ and $t=15$ and it works perfectly well.
FYI you can add individual level covariates (as gender or age) with est_lm_cov_latent.
