| Summary: | The susceptible–infected–recovered/removed–vaccinated (SIRV) epidemic model is an important generalization of the SIR epidemic model, as it accounts quantitatively for the effects of vaccination campaigns on the temporal evolution of epidemic outbreaks. Additional to the time-dependent infection (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>) and recovery (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo mathvariant="italic">μ</mo><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>) rates, regulating the transitions between the compartments <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>→</mo><mi>I</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>→</mo><mi>R</mi></mrow></semantics></math></inline-formula>, respectively, the time-dependent vaccination rate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> accounts for the transition between the compartments <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>→</mo><mi>V</mi></mrow></semantics></math></inline-formula> of susceptible to vaccinated fractions. An accurate analytical approximation is derived for arbitrary and different temporal dependencies of the rates, which is valid for all times after the start of the epidemics for which the cumulative fraction of new infections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>≪</mo><mn>1</mn></mrow></semantics></math></inline-formula>. As vaccination campaigns automatically reduce the rate of new infections by transferring persons from susceptible to vaccinated, the limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>≪</mo><mn>1</mn></mrow></semantics></math></inline-formula> is even better fulfilled than in the SIR-epidemic model. The comparison of the analytical approximation for the temporal dependence of the rate of new infections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>J</mi><mo>˚</mo></mover><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>S</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>I</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the corresponding cumulative fraction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, respectively, with the exact numerical solution of the SIRV-equations for different illustrative examples proves the accuracy of our approach. The considered illustrative examples include the cases of stationary ratios with a delayed start of vaccinations, and an oscillating ratio of recovery to infection rate with a delayed vaccination at constant rate. The proposed analytical approximation is self-regulating as the final analytical expression for the cumulative fraction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mo>∞</mo></msub></semantics></math></inline-formula> after infinite time allows us to check the validity of the original assumption <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><msub><mi>J</mi><mo>∞</mo></msub><mo>≪</mo><mn>1</mn></mrow></semantics></math></inline-formula>.
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