Algorithm 2 Determining the numerical solution of piecewise SEIUR model

at time $t\in \{p_i,p_i+1,\dots ,p_{i+1}-1\}$ for $i=1\dots n$

where $p_i$ is the start point of period $i$ and $p_1=0$

Input:

The initial value of variables: $S\left[0\right]$, $E\left[0\right]$, $I\left[0\right]$, $U\left[0\right]$ and $R\left[0\right]$;

The optimal parameters for each period $i$: ${\beta }_i$, ${\gamma }_i$, $f$ and $\sigma$;

Total population of the United States: $N$;

The number of iterations or days: $n$;

Output

$S\left[t\right]$, $E\left[t\right]$, $I\left[t\right]$ and $R\left[t\right]$ at all time $t\in \{1,2,\dots ,n\}$

Procedure

For$i$ in $1$ to $n$

For$j$ in $p_i$ to $p_{i+1}-1$

$S\left[j\right]=S[j-1]-\frac{{\beta }_{i}S[j-1]\left(I[j-1]\right)}{N}$

$E\left[j\right]=E[j-1]+\frac{{\beta }_{i}S[j-1]\left(I[j-1]\right)}{N}-\sigma E[j-1]$

$I\left[j\right]=I[j-1]+\sigma fE[j-1]-{\gamma }_{j}I[j-1]$

$U\left[j\right]=U[j-1]+\sigma (1-f)E[j-1]-{\gamma }_{j}U[j-1]$

$R\left[j\right]=R[j-1]+{\gamma }_j(I[j-1]+U[j-1])$

Return$S\left[j\right]$, $E\left[j\right]$, $I\left[j\right]$, $U\left[j\right]$ and $R\left[j\right]$