SIR-Solution for Slowly Time-Dependent Ratio between Recovery and Infection Rates
The temporal evolution of pandemics described by the susceptible-infectious-recovered (SIR)-compartment model is sensitively determined by the time dependence of the infection (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><seman...
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MDPI AG
2022-05-01
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| Series: | Physics |
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| Online Access: | https://www.mdpi.com/2624-8174/4/2/34 |
| _version_ | 1797483146532356096 |
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| author | Martin Kröger Reinhard Schlickeiser |
| author_facet | Martin Kröger Reinhard Schlickeiser |
| author_sort | Martin Kröger |
| collection | DOAJ |
| description | The temporal evolution of pandemics described by the susceptible-infectious-recovered (SIR)-compartment model is sensitively determined by the time dependence of the 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><mo>)</mo></mrow></semantics></math></inline-formula> rates regulating the transitions from the susceptible to the infected and from the infected to the recovered compartment, respectively. Here, approximated SIR solutions for different time dependencies of the infection and recovery rates are derived which are based on the adiabatic approximation assuming time-dependent ratios, <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>, varying slowly in comparison with the typical time characteristics of the pandemic wave. For such slow variations, the available analytical approximations from the KSSIR-model, developed by us and valid for a stationary value of the ratio <i>k</i>, are used to insert a posteriori the adopted time-dependent ratio of the two rates. Instead of investigating endless different combinations of the time dependencies of the two rates <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 <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>, a suitably parameterized reduced time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>, dependence of the ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></semantics></math></inline-formula> is adopted. Together with the definition of the reduced time, this parameterized ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></semantics></math></inline-formula> allows us to cover a great variety of different time dependencies of the infection and recovery rates. The agreement between the solutions from the adiabatic approximation in its four different studied variants and the exact numerical solutions of the SIR-equations is tolerable providing confidence in the accuracy of the proposed adiabatic approximation. |
| first_indexed | 2024-03-09T22:43:47Z |
| format | Article |
| id | doaj.art-b4405670bbbd4aaba125a7cb963021a4 |
| institution | Directory Open Access Journal |
| issn | 2624-8174 |
| language | English |
| last_indexed | 2024-03-09T22:43:47Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Physics |
| spelling | doaj.art-b4405670bbbd4aaba125a7cb963021a42023-11-23T18:33:49ZengMDPI AGPhysics2624-81742022-05-014250452410.3390/physics4020034SIR-Solution for Slowly Time-Dependent Ratio between Recovery and Infection RatesMartin Kröger0Reinhard Schlickeiser1Polymer Physics, Department of Materials, ETH Zurich, CH-8093 Zurich, SwitzerlandTheoretische Physik, Weltraum- und Astrophysik, Lehrstuhl IV, Ruhr-Universität Bochum, D-44780 Bochum, GermanyThe temporal evolution of pandemics described by the susceptible-infectious-recovered (SIR)-compartment model is sensitively determined by the time dependence of the 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><mo>)</mo></mrow></semantics></math></inline-formula> rates regulating the transitions from the susceptible to the infected and from the infected to the recovered compartment, respectively. Here, approximated SIR solutions for different time dependencies of the infection and recovery rates are derived which are based on the adiabatic approximation assuming time-dependent ratios, <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>, varying slowly in comparison with the typical time characteristics of the pandemic wave. For such slow variations, the available analytical approximations from the KSSIR-model, developed by us and valid for a stationary value of the ratio <i>k</i>, are used to insert a posteriori the adopted time-dependent ratio of the two rates. Instead of investigating endless different combinations of the time dependencies of the two rates <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 <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>, a suitably parameterized reduced time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>, dependence of the ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></semantics></math></inline-formula> is adopted. Together with the definition of the reduced time, this parameterized ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></semantics></math></inline-formula> allows us to cover a great variety of different time dependencies of the infection and recovery rates. The agreement between the solutions from the adiabatic approximation in its four different studied variants and the exact numerical solutions of the SIR-equations is tolerable providing confidence in the accuracy of the proposed adiabatic approximation.https://www.mdpi.com/2624-8174/4/2/34epidemiologystatistical analysistime-scale separationdifferential equationsadiabatic approximation |
| spellingShingle | Martin Kröger Reinhard Schlickeiser SIR-Solution for Slowly Time-Dependent Ratio between Recovery and Infection Rates Physics epidemiology statistical analysis time-scale separation differential equations adiabatic approximation |
| title | SIR-Solution for Slowly Time-Dependent Ratio between Recovery and Infection Rates |
| title_full | SIR-Solution for Slowly Time-Dependent Ratio between Recovery and Infection Rates |
| title_fullStr | SIR-Solution for Slowly Time-Dependent Ratio between Recovery and Infection Rates |
| title_full_unstemmed | SIR-Solution for Slowly Time-Dependent Ratio between Recovery and Infection Rates |
| title_short | SIR-Solution for Slowly Time-Dependent Ratio between Recovery and Infection Rates |
| title_sort | sir solution for slowly time dependent ratio between recovery and infection rates |
| topic | epidemiology statistical analysis time-scale separation differential equations adiabatic approximation |
| url | https://www.mdpi.com/2624-8174/4/2/34 |
| work_keys_str_mv | AT martinkroger sirsolutionforslowlytimedependentratiobetweenrecoveryandinfectionrates AT reinhardschlickeiser sirsolutionforslowlytimedependentratiobetweenrecoveryandinfectionrates |