Algorithm 1 Epidemic Estimation with Stochastic SIR Model

$$\begin{array}{cc}\hfill & \mathrm{Input}:k,{\mathbf{z}}_{1:k}=[{\mathbf{z}}_1^{\mathrm{T}},\dots ,{\mathbf{z}}_k^{\mathrm{T}}],N,\mathcal{D},{\overline{\beta }}_0,{\sigma }_{\beta }^2,{\overline{\gamma }}_0,{\sigma }_{\gamma }^2,\{{\overline{\mathbf{x}}}_0^{(n)},{\overline{\mathbf{C}}}_0^{(n)}{\}}_{n=1}^N\hfill \\ \hfill & \mathrm{Outpu}:\{\mathcal{P}({\mathbf{x}}_{k^{\prime }}\mid {\theta }_{k^{\prime }},{\mathbf{z}}_{1:k^{\prime }}),\mathcal{P}({\theta }_{k^{\prime }}\mid {\mathbf{z}}_{1:k^{\prime }}),{\widehat{s}}_{k^{\prime }}^{\mathrm{E}},{\widehat{ı}}_{k^{\prime }}^{\mathrm{E}},{\widehat{\beta }}_{k^{\prime }}^{\mathrm{E}},{\widehat{\gamma }}_{k^{\prime }}^{\mathrm{E}}{\}}_{k^{\prime }=1}^k\hfill \\ \hfill & \text{INITIALIZATION}\hfill \\ \hfill & 1:k^{\prime }\leftarrow 1\hfill \\ \hfill & 2:\mathbf{for}{\theta }_{k^{\prime }}=[{\beta }_{k^{\prime }},{\gamma }_{k^{\prime }}{]}^{\mathrm{T}}\in \mathcal{D}\mathbf{do}\hfill \\ \hfill & 3:\mathcal{P}({\theta }_{k^{\prime }}\mid {\mathbf{z}}_{1:k^{\prime }-1})\leftarrow \mathcal{N}({\beta }_{k^{\prime }};{\overline{\beta }}_0,{\sigma }_{\beta }^2)\mathcal{N}({\gamma }_{k^{\prime }};{\overline{\gamma }}_0,{\sigma }_{\gamma }^2)\hfill \\ \hfill & 4:\mathbf{for}n=1\mathbf{to}N\mathbf{do}\hfill \\ \hfill & 5:w_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}\leftarrow N^{-1}\hfill \\ \hfill & 6:{\widehat{\mathbf{x}}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}\leftarrow {\overline{\mathbf{x}}}_0^{(n)}\hfill \\ \hfill & 7:{\widehat{\mathbf{C}}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}\leftarrow {\overline{\mathbf{C}}}_0^{(n)}\hfill \\ \hfill & 8:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 9:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 10:\mathbf{for}{k}^{\prime }=1\mathbf{to}k\mathbf{do}\hfill \\ \hfill & \text{UPDATE}\hfill \\ \hfill & 11:\mathbf{for}{\theta }_{k^{\prime }}\in \mathcal{D}\mathbf{do}\hfill \\ \hfill & 12:\mathbf{for}n=1\mathbf{to}N\mathbf{do}\hfill \\ \hfill & 13:{\alpha }_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}\leftarrow \mathcal{N}({\mathbf{z}}_{k^{\prime }};{\mathbf{h}}_1\left({\widehat{\mathbf{x}}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}\right)\phantom{)},\hfill \\ \hfill & \phantom{(}\mathbf{R}\left({\widehat{\mathbf{x}}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}\right)+\mathbf{H}{\widehat{\mathbf{C}}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}{\mathbf{H}}^{\mathrm{T}})\hfill \\ \hfill & 14:w_{k^{\prime }\mid k^{\prime }}^{(n,{\theta }_{k^{\prime })}}\leftarrow \frac{{\alpha }_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}{w}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}}{{\sum }_{n^{\prime }=1}^N{\alpha }_{k^{\prime }\mid k^{\prime }-1}^{(n^{\prime },{\theta }_{k^{\prime }})}{w}_{k^{\prime }\mid k^{\prime }-1}^{(n^{\prime },{\theta }_{k^{\prime }})}}\hfill \\ \hfill & 15:{\widehat{\mathbf{x}}}_{k^{\prime }\mid k^{\prime }}^{(n,{\theta }_{k^{\prime })}}\leftarrow {\widehat{\mathbf{x}}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}+{\mathbf{K}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}\left[{\mathbf{z}}_{k^{\prime }}-{\mathbf{h}}_1\left({\widehat{\mathbf{x}}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}\right)\right]\hfill \\ \hfill & 16:{\widehat{\mathbf{C}}}_{k^{\prime }\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }})}\leftarrow {\widehat{\mathbf{C}}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}-{\mathbf{K}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}\mathbf{H}{\widehat{\mathbf{C}}}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}\hfill \\ \hfill & 17:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 18:\mathcal{P}({\mathbf{x}}_{k^{\prime }}\mid {\theta }_{k^{\prime }},{\mathbf{z}}_{1:k^{\prime }})\leftarrow \hfill \\ \hfill & {\sum }_{n=1}^N{w}_{k^{\prime }\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }})}\mathcal{N}\left({\mathbf{x}}_{k^{\prime }};{\widehat{\mathbf{x}}}_{k^{\prime }\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }})},{\widehat{\mathbf{C}}}_{k^{\prime }\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }})}\right)\hfill \\ \hfill & 19:\mathcal{P}({\theta }_{k^{\prime }}\mid {\mathbf{z}}_{1:k^{\prime }})\leftarrow \hfill \\ \hfill & \frac{\mathcal{P}({\theta }_{k^{\prime }}\mid {\mathbf{z}}_{1:k^{\prime }-1}){\sum }_{n=1}^N{\alpha }_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}{w}_{k^{\prime }\mid k^{\prime }-1}^{(n,{\theta }_{k^{\prime }})}}{{\sum }_{{\theta }_{k^{\prime }}^{\prime }\in \mathcal{D}}\mathcal{P}({\theta }_{k^{\prime }}^{\prime }\mid {\mathbf{z}}_{1:k^{\prime }-1}){\sum }_{n^{\prime }=1}^N{\alpha }_{k^{\prime }\mid k^{\prime }-1}^{(n^{\prime },{\theta }_{k^{\prime }}^{\prime })}{w}_{k^{\prime }\mid k^{\prime }-1}^{(n^{\prime },{\theta }_{k^{\prime }}^{\prime })}}\hfill \\ \hfill & 20:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 21:\text{Compute}{\widehat{s}}_{k^{\prime }}^{\mathrm{E}},{\widehat{ı}}_{k^{\prime }}^{\mathrm{E}},{\widehat{\beta }}_{k^{\prime }}^{\mathrm{E}},\text{and}{\widehat{\gamma }}_{k^{\prime }}^{\mathrm{E}}\text{according to (37)}-(40)\hfill \\ \hfill & \text{PREDICTION}\hfill \\ \hfill & 22:\mathbf{for}{\theta }_{k^{\prime }+1}\in \mathcal{D}\mathbf{do}\hfill \\ \hfill & 23:\mathcal{P}({\theta }_{k^{\prime }+1}\mid {\mathbf{z}}_{1:k^{\prime }})\leftarrow {\sum }_{{\theta }_{k^{\prime }}\in \mathcal{D}}\mathcal{P}({\theta }_{k^{\prime }+1}\mid {\theta }_{k^{\prime }})\mathcal{P}({\theta }_{k^{\prime }}\mid {\mathbf{z}}_{1:k^{\prime }})\hfill \\ \hfill & 24:n^{\prime }\leftarrow 1\hfill \\ \hfill & 25:\mathbf{for}{\theta }_{k^{\prime }}\in \mathcal{D}\mathbf{do}\hfill \\ \hfill & 26:\mathbf{for}n=1\mathbf{to}N\mathbf{do}\hfill \\ \hfill & 27:w_{k^{\prime }+1\mid k^{\prime }}^{(n^{\prime },{\theta }_{k^{\prime }+1})}=w_{k^{\prime }+1\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }+1},{\theta }_{k^{\prime }})}\leftarrow \hfill \\ \hfill & \frac{\mathcal{P}({\theta }_{k^{\prime }+1},{\mid \theta }_{k^{\prime }})\mathcal{P}({\theta }_{k^{\prime }}\mid {\mathbf{z}}_{1:k^{\prime }})}{\mathcal{P}({\theta }_{k^{\prime }+1}\mid {\mathbf{z}}_{1:k^{\prime }})}{w}_{k^{\prime }\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }})}\hfill \\ \hfill & 28:{\widehat{\mathbf{x}}}_{k^{\prime }+1\mid k^{\prime }}^{(n^{\prime },{\theta }_{k^{\prime }+1})}\leftarrow {\widehat{\mathbf{x}}}_{k^{\prime }+1\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }})}(\text{via moment matching})\hfill \\ \hfill & 29:{\widehat{\mathbf{C}}}_{k^{\prime }+1\mid k^{\prime }}^{(n^{\prime },{\theta }_{k^{\prime }+1})}\leftarrow {\widehat{\mathbf{C}}}_{k^{\prime }+1\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }})}(\text{via moment matching})\hfill \\ \hfill & 30:n^{\prime }\leftarrow n^{\prime }+1\hfill \\ \hfill & 31:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 32:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 33:\mathbf{end}\mathbf{for}\hfill \\ \hfill & \text{PRUNING}\hfill \\ \hfill & 34:\mathbf{for}{\theta }_{k^{\prime }+1}\in \mathcal{D}\mathbf{do}\hfill \\ \hfill & 35:\text{Sort in descending order the}N\times D\text{mixture components according}\hfill \\ \hfill & \text{to the weights}\{w_{k^{\prime }+1\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }+1})}{\}}_{n=1}^{N\times D},\text{and retain the first}N\text{elements}\hfill \\ \hfill & 36:\mathbf{for}n=1\mathbf{to}N\mathbf{do}\hfill \\ \hfill & 37:w_{k^{\prime }+1\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }+1})}\leftarrow \frac{w_{k^{\prime }+1\mid k^{\prime }}^{(n,{\theta }_{k^{\prime }+1})}}{{\sum }_{n^{\prime }=1}^N{w}_{k^{\prime }+1\mid k^{\prime }}^{(n^{\prime },{\theta }_{k^{\prime }+1})}}\hfill \\ \hfill & 38:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 39:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 40:\mathbf{end}\mathbf{for}\hfill \end{array}$$