Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection Rates
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:/...
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MDPI AG
2023-12-01
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| Series: | COVID |
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| Online Access: | https://www.mdpi.com/2673-8112/3/12/123 |
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| author | Reinhard Schlickeiser Martin Kröger |
| author_facet | Reinhard Schlickeiser Martin Kröger |
| author_sort | Reinhard Schlickeiser |
| collection | DOAJ |
| description | 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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| institution | Directory Open Access Journal |
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| language | English |
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| spelling | doaj.art-d5d532da4cd24f4eb1c7561c96af6de62023-12-22T14:01:40ZengMDPI AGCOVID2673-81122023-12-013121781179610.3390/covid3120123Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection RatesReinhard Schlickeiser0Martin Kröger1Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität Bochum, D-44780 Bochum, GermanyMagnetism and Interface Physics & Computational Polymer Physics, Department of Materials, ETH Zurich, CH-8093 Zurich, SwitzerlandThe 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.https://www.mdpi.com/2673-8112/3/12/123epidemicstemporal developmentcoronavirusSARS CoV-2COVID-19 |
| spellingShingle | Reinhard Schlickeiser Martin Kröger Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection Rates COVID epidemics temporal development coronavirus SARS CoV-2 COVID-19 |
| title | Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection Rates |
| title_full | Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection Rates |
| title_fullStr | Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection Rates |
| title_full_unstemmed | Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection Rates |
| title_short | Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection Rates |
| title_sort | analytical solution of the susceptible infected recovered removed model for the not too late temporal evolution of epidemics for general time dependent recovery and infection rates |
| topic | epidemics temporal development coronavirus SARS CoV-2 COVID-19 |
| url | https://www.mdpi.com/2673-8112/3/12/123 |
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