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Abstract:
Since Kermack and McKendrick have introduced their famous epidemiological SIR
model in 1927, mathematical epidemiology has grown as an interdisciplinary
research discipline including knowledge from biology, computer science, or
mathematics. Due to current threatening epidemics such as COVID-19, this interest is
continuously rising. As our main goal, we establish an implicit time-discrete SIR
(susceptible people–infectious people–recovered people) model. For this purpose,
we first introduce its continuous variant with time-varying transmission and recovery
rates and, as our first contribution, discuss thoroughly its properties. With respect to
these results, we develop different possible time-discrete SIR models, we derive our
implicit time-discrete SIR model in contrast to many other works which mainly
investigate explicit time-discrete schemes and, as our main contribution, show
unique solvability and further desirable properties compared to its continuous
version. We thoroughly show that many of the desired properties of the
time-continuous case are still valid in the time-discrete implicit case. Especially, we
prove an upper error bound for our time-discrete implicit numerical scheme. Finally,
we apply our proposed time-discrete SIR model to currently available data regarding
the spread of COVID-19 in Germany and Iran.
