| Summary: | The dynamical equations of the susceptible-infected-recovered/removed (SIR) epidemics model play an important role in predicting and/or analyzing the temporal evolution of epidemic outbreaks. Crucial input quantities are 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><mi>μ</mi><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. Accurate analytical approximations 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> and the corresponding cumulative fraction of new infections <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><mi>J</mi><mrow><mo>(</mo><msub><mi>t</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>+</mo><msubsup><mo>∫</mo><mrow><msub><mi>t</mi><mn>0</mn></msub></mrow><mi>t</mi></msubsup><mi>d</mi><mi>x</mi><mover accent="true"><mi>J</mi><mo>˚</mo></mover><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are available in the literature for either stationary infection and recovery rates or for a stationary value of the ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>μ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>/</mo><mi>a</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Here, a new and original accurate analytical approximation is derived for general, arbitrary, and different temporal dependencies of the infection and recovery rates, which is valid for not-too-late times after the start of the infection when the 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><mo>≪</mo><mn>1</mn></mrow></semantics></math></inline-formula> is much less than unity. The comparison of the analytical approximation with the exact numerical solution of the SIR equations for different illustrative examples proves the accuracy of the analytical approach.
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