Algorithm 2 Epidemic Forecasting with Stochastic SIR Model

$$\begin{array}{cc}\hfill & \mathrm{Input}:k,\mathcal{P}({\mathbf{x}}_k\mid {\mathbf{z}}_{1:k}),\mathcal{P}({\theta }_k\mid {\mathbf{z}}_{1:k}),J,L,K,P,\{{\widehat{\beta }}_{k^{\prime }}^{\mathrm{E}}{\}}_{k^{\prime }=k-L}^k\hfill \\ \hfill & \mathrm{Output}:\{{\widehat{s}}_{k^{\prime }}^{\mathrm{F}},{\widehat{ı}}_{k^{\prime }}^{\mathrm{F}},{\widehat{\beta }}_{k^{\prime }}^{\mathrm{F}},{\widehat{\gamma }}_{k^{\prime }}^{\mathrm{F}}{\}}_{k^{\prime }=k+1}^{k+K}\hfill \\ \hfill & \text{INITIALIZATION}\hfill \\ \hfill & 1:\text{Draw}J\text{samples}{\tilde{\mathbf{x}}}_k^{(j)}=[{\tilde{s}}_k^{(j)},{\tilde{ı}}_k^{(j)}{]}^{\mathrm{T}}\text{from}\mathcal{P}({\mathbf{x}}_k\mid {\mathbf{z}}_{1:k})\hfill \\ \hfill & 2:\text{Draw}J\text{samples}{\tilde{\theta }}_k^{(j)}=[{\tilde{\beta }}_k^{(j)},{\tilde{\gamma }}_k^{(j)}{]}^{\mathrm{T}}\text{from}\mathcal{P}({\theta }_k\mid {\mathbf{z}}_{1:k})\hfill \\ \hfill & 3:{\widehat{\stackrel{.}{\beta }}}_{L,k}\leftarrow \frac{{\sum }_{\ell =0}^L(\ell -L∕2)({\widehat{\beta }}_{k-L+\ell }^{\mathrm{E}}-(L+1{)}^{-1}{\sum }_{{\ell }^{\prime }=0}^L{\widehat{\beta }}_{k-{\ell }^{\prime }}^{\mathrm{E}})}{{\Delta }_t{\sum }_{\ell =0}^L(\ell -L∕2{)}^2}\hfill \\ \hfill & 4:{\widehat{\sigma }}_{\omega ,L}^2\leftarrow \frac{{\sum }_{\ell =1}^L{\left({\widehat{\beta }}_{k-\ell +1}^{\mathrm{E}}-{\widehat{\beta }}_{k-\ell }^{\mathrm{E}}-{\Delta }_t{\widehat{\stackrel{.}{\beta }}}_{L,k}\right)}^2}{L-1}\hfill \\ \hfill & 5:{\stackrel{.}{\sigma }}_{L,k}^2\leftarrow \frac{{\widehat{\sigma }}_{\omega ,L}^2}{{\Delta }_t^2{\sum }_{\ell =0}^L{\left(\ell -L∕2\right)}^2}\hfill \\ \hfill & \text{FORECASTING}\hfill \\ \hfill & 6:\mathbf{for}j=1\mathbf{to}J\mathbf{do}\hfill \\ \hfill & 7:\mathbf{for}{k}^{\prime }=k+1\mathbf{to}k+K\mathbf{do}\hfill \\ \hfill & \text{STATE EVOLUTION}\hfill \\ \hfill & 8:\text{Draw}\lambda =[{\lambda }_1,{\lambda }_2{]}^{\mathrm{T}}\sim \mathcal{N}(0,{\mathbf{I}}_2{\Delta }_t)\hfill \\ \hfill & 9:{\sigma }_{1,k^{\prime }-1}^{(j)}\leftarrow \sqrt{P^{-1}{\tilde{\beta }}_{k^{\prime }-1}^{(j)}{\tilde{s}}_{k^{\prime }-1}^{(j)}{\tilde{ı}}_{k^{\prime }-1}^{(j)}}\hfill \\ \hfill & 10:{\sigma }_{2,k^{\prime }-1}^{(j)}\leftarrow \sqrt{P^{-1}{\tilde{\gamma }}_{k^{\prime }-1}^{(j)}{\tilde{ı}}_{k^{\prime }-1}^{(j)}}\hfill \\ \hfill & 11:{\tilde{s}}_{k^{\prime }}^{(j)}\leftarrow {\tilde{s}}_{k^{\prime }-1}^{(j)}-{\tilde{\beta }}_{k^{\prime }-1}^{(j)}{\tilde{s}}_{k^{\prime }-1}^{(j)}{\tilde{ı}}_{k^{\prime }-1}^{(j)}{\Delta }_t+{\sigma }_{1,k^{\prime }-1}^{(j)}{\lambda }_1\hfill \\ \hfill & 12:{\tilde{ı}}_{k^{\prime }}^{(j)}\leftarrow {\tilde{ı}}_{k^{\prime }-1}^{(j)}+{\tilde{\beta }}_{k^{\prime }-1}^{(j)}{\tilde{s}}_{k^{\prime }-1}^{(j)}{\tilde{ı}}_{k^{\prime }-1}^{(j)}{\Delta }_t\hfill \\ \hfill & -{\tilde{\gamma }}_{k^{\prime }-1}^{(j)}{\tilde{ı}}_{k^{\prime }-1}^{(j)}{\Delta }_t-{\sigma }_{1,k^{\prime }-1}^{(j)}{\lambda }_1+{\sigma }_{2,k^{\prime }-1}^{(j)}{\lambda }_2\hfill \\ \hfill & \text{PARAMETERS EVOLUTION}\hfill \\ \hfill & 13:\text{Draw}\eta \sim \mathcal{N}(0,{\stackrel{.}{\sigma }}_{L,k}^2)\hfill \\ \hfill & 14:{\tilde{\beta }}_{k^{\prime }}^{(j)}\leftarrow {\tilde{\beta }}_{k^{\prime }-1}^{(j)}+{\widehat{\stackrel{.}{\beta }}}_{L,k}{\Delta }_t+\eta \hfill \\ \hfill & 15:{\tilde{\gamma }}_{k^{\prime }}^{(j)}\leftarrow {\tilde{\gamma }}_{k^{\prime }-1}^{(j)}\hfill \\ \hfill & \text{ENSEMBLE MEAN}\hfill \\ \hfill & 16:{\widehat{s}}_{k^{\prime }}^{\mathrm{F}}\leftarrow J^{-1}{\sum }_{j=1}^J{\tilde{s}}_{k^{\prime }}^{(j)}\hfill \\ \hfill & 17:{\widehat{ı}}_{k^{\prime }}^{\mathrm{F}}\leftarrow J^{-1}{\sum }_{j=1}^J{\tilde{ı}}_{k^{\prime }}^{(j)}\hfill \\ \hfill & 18:{\widehat{\beta }}_{k^{\prime }}^{\mathrm{F}}\leftarrow J^{-1}{\sum }_{j=1}^J{\tilde{\beta }}_{k^{\prime }}^{(j)}\hfill \\ \hfill & 19:{\widehat{\gamma }}_{k^{\prime }}^{\mathrm{F}}\leftarrow J^{-1}{\sum }_{j=1}^J{\tilde{\gamma }}_{k^{\prime }}^{(j)}\hfill \\ \hfill & 20:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 21:\mathbf{end}\mathbf{for}\hfill \end{array}$$