# 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

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## Abstract

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## 1. Introduction

## 2. SIR Model

## 3. Approximate Analytical Solutions

#### 3.1. Solution in the Limit of Small $J\ll 1$

#### 3.2. Properties of the Approximate Solution (12)

## 4. Special Cases

#### 4.1. Constant Ratio $k\left(t\right)$

#### 4.2. Linearly Increasing Ratio $k\left(\tau \right)={k}_{0}+{k}_{1}\tau $

## 5. Illustrative Examples

#### 5.1. Monotonically Rising Ratio

#### 5.2. Oscillating Ratio

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Cumulative Fraction for General Reduced Time Dependencies k(τ)

## Appendix B. Solution at Late Times

#### Appendix B.1. Limit J≃J ∞ =0.7

#### Appendix B.2. Continuity Conditions

## Appendix C. Expansion of H(τ) at Small Times

## References

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**Figure 1.**Performance of the analytic approximation (23) for the crossover ${\tau}_{c}$ versus ${k}_{0}$ at $\eta ={10}^{-5}$. The numerical result is obtained from $J\left({\tau}_{c}\right)={J}_{\infty}/2$.

**Figure 2.**Left: Plot of the monotonically rising ratio (25) for (

**a**) $C=4$ and (

**b**) $C=0.15$, with $B=1.5$ and $\eta ={10}^{-5}$, respectively. Middle: Corresponding rate of new infections $j\left(\tau \right)$ as a function of the reduced time for these choices of the ratio $k\left(\tau \right)$. Right: Corresponding cumulative rate of new infections $J\left(\tau \right)$. Shown are the numerical (black dashed curve) solutions of the SIR set of equations in (6) in comparison with the analytical approximations (green curve) according to Equations (12) and (30). The agreement is almost perfect. The maximum relative deviations are smaller than (

**a**) 0.5% and (

**b**) 2.5%. In addition, we show (hardly visible red dot–dash curve) the approximant (39) in panel (

**a**) for $C>1$ and Equation (36) in panel (

**b**) for $C<1$. The agreement is against almost perfect. The black dotted lines were obtained for comparison using the method of steepest descent, Equation (45), which provides a lower limit to the cumulative fraction, because all other contributions far from the maximum are not adequately accounted for (Appendix A).

**Figure 3.**Left panels: Plot of the oscillating ratio of $k\left(\tau \right)$ according to Equation (46) for (

**a**) $\alpha =0.5$, $\beta =4$, and (

**b**) $\alpha =0.8$, $\beta =0.5$. Centered panels: The corresponding rates of new infections of $j\left(\tau \right)$ as a function of reduced time using $\eta ={10}^{-5}$. Right panels: Cumulative fraction of $J\left(\tau \right)$. Shown are the numerical (black dashed curve) solutions of the SIR set of equations in (6) in comparison with the analytical approximations (green curve) according to Equations (47) and (48). Because $\alpha /\beta \ll 1$ for case (

**a**), only the first term of expansion (48) had to be used, while the first three terms of Equation (48) were used in (

**b**) in accord with Figure 4. The agreement is almost perfect. The maximum relative deviations are smaller than 0.2% for all cases.

**Figure 4.**The integral ${\int}_{0}^{\tau}{e}^{zcos\theta}$ is approximated for all $\tau $ within $1\%$ precision by the integrated Equation (49), thus resulting in the expression shown inside the figure if the z-dependent order of the summation, N, is chosen as depicted. The required order grows as $N\propto \sqrt{a}$.

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**MDPI and ACS Style**

Schlickeiser, R.; Kröger, M.
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* **2023**, *3*, 1781-1796.
https://doi.org/10.3390/covid3120123

**AMA Style**

Schlickeiser R, Kröger M.
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*. 2023; 3(12):1781-1796.
https://doi.org/10.3390/covid3120123

**Chicago/Turabian Style**

Schlickeiser, Reinhard, and Martin Kröger.
2023. "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* 3, no. 12: 1781-1796.
https://doi.org/10.3390/covid3120123