Analytical Solution of the SIR-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 to predict and/or analyze the temporal evolution of epidemics outbreaks. Crucial input quantities are the time-dependent infection ($a(t)$) and recovery ($\mu (t)$) rates regulating the transitions between the compartments $S\to I$ and $I\to R$, respectively. Accurate analytical approximations for the temporal dependence of the rate of new infections $\dot{J}(t)=a(t)S(t)I(t)$ and the corresponding cumulative fraction of new infections $J(t)=J(t_0)+\int_{t_0}^tdx\dot{J}(x)$ are available in the literature for either stationary infection and recovery rates and for a stationary value of the ratio $k(t)=\mu (t)/a(t)$. 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 $J(t)\ll 1$ 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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Preprint| Erscheinungsjahr: |
2024 |
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| Erschienen: |
2024 |
| Enthalten in: |
Preprints.org - (2024) vom: 04. Jan. Zur Gesamtaufnahme - year:2024 |
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| Sprache: |
Englisch |
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| Beteiligte Personen: |
Schlickeiser, Reinhard [VerfasserIn] |
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| doi: |
10.20944/preprints202311.1563.v1 |
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| PPN (Katalog-ID): |
preprintsorg041656652 |
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| 100 | 1 | |a Schlickeiser, Reinhard |e verfasserin |0 (orcid)0000-0003-3171-5079 |4 aut | |
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| 520 | |a The dynamical equations of the susceptible-infected-recovered/removed (SIR) epidemics model play an important role to predict and/or analyze the temporal evolution of epidemics outbreaks. Crucial input quantities are the time-dependent infection ($a(t)$) and recovery ($\mu (t)$) rates regulating the transitions between the compartments $S\to I$ and $I\to R$, respectively. Accurate analytical approximations for the temporal dependence of the rate of new infections $\dot{J}(t)=a(t)S(t)I(t)$ and the corresponding cumulative fraction of new infections $J(t)=J(t_0)+\int_{t_0}^tdx\dot{J}(x)$ are available in the literature for either stationary infection and recovery rates and for a stationary value of the ratio $k(t)=\mu (t)/a(t)$. 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 $J(t)\ll 1$ 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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