Algorithm A1 Adam based method for parameter optimization. Good default settings for the analyzed COVID-19 dataset are learning rate $\eta =0.05$, exponential decay rates $b_1=0.1$ and $b_2=0.1$, and $\gamma =1\times {10}^{-8}$. Algorithm tolerance $tol=1\times {10}^{-4}$. All operations are element-wise.

Initialization: maxit = 200 (maximum iteration steps), flag = 0 (convergence indicator), $e_1=0$ (first moment vector), $e_2=0$ (second moment vector), $ϵ=0$ (iteration-step indicator), $({\mathit{\pi }}_0,{\mathit{v}}_0)$, $a_0=0$, ${\tau }_0=0$.

Iteration process:

while $ϵ<=$ maxit and flag = 0 do

$ϵ=ϵ+1$

$g_{ϵ}={\mathsf{\nabla }}_{(\mathit{\pi },\mathit{v})}l({\mathit{\pi }}_{ϵ-1},{\mathit{v}}_{ϵ-1})$    (gradients of $l(\mathit{\pi },\mathit{v})$ shown in (A6) in Appendix A.2)

$e_{ϵ}^{\left(1\right)}=\frac{b_1{e}_{ϵ-1}^{\left(1\right)}-(1-b_1)g_{ϵ}}{1-b_1^{ϵ}}$    (bias-corrected first moment estimate)

$e_{ϵ}^{\left(2\right)}=\frac{b_2{e}_{ϵ-1}^{\left(2\right)}+(1-b_2)g_{ϵ}^2}{1-b_2^{ϵ}}$    (bias-corrected second raw moment estimate)

${\tau }_{ϵ}=({\mathit{\pi }}_{ϵ-1},{\mathit{v}}_{ϵ-1})-\eta \frac{e_{ϵ}^{\left(1\right)}}{\sqrt{e_{ϵ}^{\left(2\right)}}+\gamma }$    (temporarily updated parameters)

$({\mathit{\pi }}_{ϵ},{\mathit{v}}_{ϵ})=b_2{a}_{ϵ-1}+(1-b_2){\tau }_{ϵ}$    (updated parameters)

$a_{ϵ}=(1-\frac{1}{ϵ})a_{ϵ-1}+\frac{1}{ϵ}{\tau }_{ϵ}$    (averaged parameters for further iteration)

if $\frac{|({\mathit{\pi }}_{ϵ},{\mathit{v}}_{ϵ})-({\mathit{\pi }}_{ϵ-1},{\mathit{v}}_{ϵ-1})|}{\sqrt{5R+9}}<tol$    (convergence determination)

flag = 1

end while

return    $({\mathit{\pi }}_{ϵ},{\mathit{v}}_{ϵ})$    (optimal estimates)