Table 1 Numerical algorithm for the time-discrete implicit SIR solution scheme (17)

Inputs:

– Population size N

– Initial values \(S_{1} > 0\), \(I_{1} > 0\), and \(R_{1} \geq 0\)

– Time-varying transmission rate sequence \(\{ \alpha _{j} \} _{j = 2}^{M}\)

– Time-varying recovery rate sequence \(\{ \beta _{j} \} _{j = 2}^{M} \)

– Strictly increasing sequence \(\{ t_{j} \} _{j = 1}^{M}\) of time points with \(t_{1} = 0\) and \(t_{M} = T\)

Step 1:

– Compute all \(\Delta _{j + 1} = t_{j + 1} - t_{j}\) for all j∈{1,…,M − 1}

Step 2:

– Compute \(I_{j + 1}\) by (22), (21) and (20) for all j∈{1,…,M − 1}

– Compute \(S_{j + 1}\) and \(R_{j + 1}\) by (17) for all j∈{1,…,M − 1}

Outputs:

– Sequences \(\{ S_{j} \} _{j = 1}^{M}\), \(\{ I_{j} \} _{j = 1}^{M}\) and \(\{ R_{j} \} _{j = 1}^{M}\)