Algorithm 3.

Updating rule in the ODE-based MCMC algorithm

1:	Input: Parameter values from the previous interation $I_0$,$R_0$,$\gamma$,${\mathit{\delta }}_{1:T},\sigma$, geneology g. Proposal density $q_1(\cdot \mid \cdot )$, $q_2(\cdot \mid \cdot )$ for updating the initial number of infected individuals and the removal rate.
2:	Output Updated parameters values
3:	Calculate ${\mathbf{X}}_{0:T},{\mathit{\theta }}_{0:T}$ based on $I_0$,$R_0$,$\gamma$,${\mathit{\delta }}_{1:T},\sigma$.
4:	Propose $I_0^{\prime }$ based on $q_1\left(\cdot \mid I_0\right)$, then ${\mathbf{X}}_{0:T}$ will be deterministically updated to ${\mathbf{X}}_{0:T}^{\prime }$ according to $I_0^{\prime }$,$R_0$,$\gamma$,${\mathit{\delta }}_{1:T},\sigma$ based on ODE integration.
5:	Accept $\left(I_0^{\prime },{\mathbf{X}}_{D:T}^{\prime }\right)$ with acceptance probability
	$$a\leftarrow \mathrm{m}\mathrm{i}\mathrm{n}\left(1,\frac{\mathrm{P}\mathrm{r}\left(\mathbf{g}\mid {\mathit{\theta }}_{0:T}^{\prime },{\mathbf{X}}_{0:T}^{\prime }\right)\mathrm{P}\mathrm{r}\left(I_0^{\prime }\right)q_1\left(I_0\mid I_0^{\prime }\right)}{\mathrm{P}\mathrm{r}\left(\mathbf{g}\mid {\mathit{\theta }}_{0:T},{\mathbf{X}}_{\mathrm{D}:T}\right)\mathrm{P}\mathrm{r}\left(I_0\right)q_1\left(I_0^{\prime }\mid I_0\right)}\right)$$
6:	Propose ${\gamma }^{\prime }$ based on $q_2\left(\cdot ∣\gamma \right)$, then ${\mathbf{X}}_{0:T},{\mathit{\theta }}_{0:T}$ will be deterministically updated to ${\mathbf{X}}_{\mathrm{D}:T}^{\prime },{\mathit{\theta }}_{0:T}^{\prime }$ according to $I_0$,$R_0$,${\gamma }^{\prime }$,${\mathit{\delta }}_{1:T},\sigma$
7:	Accept $\left({\gamma }^{\prime },{\mathbf{X}}_{0:T}^{\prime },{\mathit{\theta }}_{0:T}^{\prime }\right)$ with acceptance probability
	$$a\leftarrow \mathrm{m}\mathrm{i}\mathrm{n}\left(1,\frac{\mathrm{P}\mathrm{r}\left(\mathbf{g}\mid {\mathit{\theta }}_{0:T}^{\prime },{\mathbf{X}}_{0:T}^{\prime }\right)\mathrm{P}\mathrm{r}\left({\gamma }^{\prime }\right)q_2\left(\gamma \mid {\gamma }^{\prime }\right)}{\mathrm{P}\mathrm{r}\left(\mathbf{g}\mid {\mathit{\theta }}_{0:T},{\mathbf{X}}_{0:T}\right)\mathrm{P}\mathrm{r}\left(\gamma \right)q_2\left({\gamma }^{\prime }\mid \gamma \right)}\right).$$
8:	Let $\mathbf{U}=\left(\mathrm{l}\mathrm{o}\mathrm{g}\left(R_0\right),{\mathit{\delta }}_{1:T},\mathrm{l}\mathrm{o}\mathrm{g}\left(\sigma \right)\right)$, then $\mathbf{U}$ a priori follows a multivariate normal distribution. Use elliptical slice sampler of obtain ${\mathbf{U}}^{\prime }$ and get the updated $R_0^{\prime },{\mathit{\delta }}_{1:T}^{\prime }$ and ${\sigma }^{\prime }.$ ${\mathbf{X}}_{\mathrm{D}:T}$ will be deterministically updated to ${\mathbf{X}}_{0:T}^{\prime }$ according to $I_0$,$R_0^{\prime }$,$\gamma$,${\mathit{\delta }}_{1:T}^{\prime },{\sigma }^{\prime }$ based on ODE integration.