# A Model for Reinfections and the Transition of Epidemics

^{*}

## Abstract

**:**

_{0}. The results provide insights into the evolution of contagion when reinfection and the waning of immunity are taken into consideration. A related byproduct is the finding that the conventional SIR model is singular at large times, hence the specific quantitative estimate for herd immunity it predicts will likely not materialize.

## 1. Introduction

## 2. Mathematical Formulation and Results

#### 2.1. Some Special Results

- i.
- A general expression for r(t)

- ii.
- The θ = 0 limit

- iii.
- The large ε limit

- iv.
- Equilibrium states

#### 2.2. Numerical Results

- 1.
- The case θ = 0

- 2.
- The limit of large ε

- 3.
- The general case

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## References

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**Figure 1.**(

**a**–

**d**): Numerical simulations of the evolution of susceptible, infected and recovered fractions as a function of time for the case of zero delay (θ = 0) and four different values of ε (10, 1, 0.1 and 0.01). The initial conditions for the simulations were defined in (7). The asymptotic state corresponds to either a stable node (panel (

**a**)), or a spiral (panels (

**b**–

**d**)). It is reached in a monotonic way at large ε (panel (

**a**)), and through waves of a decaying amplitude (panels (

**b**–

**d**)) depending on whether parameter Δ, defined in (31), is negative (panel (

**a**)) or positive (panels (

**b**–

**d**)), respectively.

**Figure 2.**Numerical solution of Equation (16), for the zero-delay case θ = 0, plotted as a function of the rescaled time ω

_{0}t for different values of ε and for b = 0.25 (R

_{0}= 5). Note the fast approach to the asymptotic state of a sinusoidal wave.

**Figure 3.**(

**a**,

**b**) The demarcation in the parameter space ε, b (or R

_{0}) of the two different asymptotic behaviors. Regions above or below the curve correspond to exponential decay (Δ < 0), or a damped oscillator, (Δ > 0).

**Figure 4.**(

**a**–

**d**): Solution trajectories in the parameter space (s, t) for the case θ = 0, for four different values of ε and for b = 1. Note the attraction to a stable node (Figure 4a,b) or to a spiral (Figure 4c,d). In red is the curve corresponding to the SIR model, while in light blue is the curve at the intersection of which the trajectory reverses direction.

**Figure 5.**(

**a**–

**d**) The numerical solution of Equations (19)–(22), where $\epsilon \gg 1$, for b = 1 and for four different values of θ, corresponding to $\theta <{\theta}_{m}\left(1\right)$, ${\theta}_{m}\left(1\right)<\theta <{\theta}_{c}\left(1\right)$ (panels (

**a**,

**b**)), and $\theta >{\theta}_{c}\left(1\right)$ (panels (

**c**,

**d**)), where ${\theta}_{m}\left(1\right)=0.96$ and ${\theta}_{c}\left(1\right)=7.8$. The asymptotic behavior is either monotonic (stable node) (Figure 5a), a damped oscillator (spiral) (Figure 5b), or an oscillation of constant amplitude (Figure 5c,d).

**Figure 6.**(

**a**–

**d**) The trajectories of the solution of Figure 5, in the parameter space $\left(s,r\right)$. Note the attraction to a stable node (Figure 6, corresponding to $\theta <{\theta}_{m}\left(1\right)=0.96$), a spiral (Figure 6b, corresponding to ${\theta}_{m}\left(1\right)<\theta <{\theta}_{c}\left(1\right)=7.8$) and a limit cycle (Figure 6c,d, corresponding to ${\theta}_{m}\left(1\right)<\theta <{\theta}_{c}\left(1\right)$). In red is the curve corresponding to the SIR model (Equation (11)).

**Figure 7.**(

**a**,

**b**) The dependence of the two critical delay time values θ

_{m}and θ

_{c}on b (and hence on the reproduction number R

_{0}) for the case of large ε.

**Figure 8.**(

**a**–

**d**) Numerical simulations corresponding to $b=1$ and $\epsilon =5,$ where $\Delta <0$, for four different values of $\theta $. Note the emergence of three different regimes as $\theta $ increases (${\theta}_{m}\left(1\right)=0.58,{\theta}_{c}\left(1\right)=7.8$).

**Figure 10.**(

**a**,

**b**): The dependence of the critical delay times ${\theta}_{m}$ and ${\theta}_{c}$ on $\epsilon $ for different values of $b$ (hence the reproduction number ${R}_{0}$). Note that there is no value of ${\theta}_{m}$ if $\epsilon <{\epsilon}_{c}\left(b\right)$.

**Figure 11.**(

**a**–

**d**): The trajectories of the solution of Figure 7 in the parameter space $\left(s,t\right)$. Note the attraction to a stable node (Figure 7a, where $\theta <{\theta}_{m}$), to a spiral (Figure 7b, where ${\theta}_{m}<\theta <{\theta}_{c}$) and to a limit cycle (Figure 7c,d) where ${\theta}_{m}<\theta <{\theta}_{c}$). In red is the curve corresponding to the SIR model.

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Yortsos, Y.C.; Chang, J.
A Model for Reinfections and the Transition of Epidemics. *Viruses* **2023**, *15*, 1340.
https://doi.org/10.3390/v15061340

**AMA Style**

Yortsos YC, Chang J.
A Model for Reinfections and the Transition of Epidemics. *Viruses*. 2023; 15(6):1340.
https://doi.org/10.3390/v15061340

**Chicago/Turabian Style**

Yortsos, Yannis C., and Jincai Chang.
2023. "A Model for Reinfections and the Transition of Epidemics" *Viruses* 15, no. 6: 1340.
https://doi.org/10.3390/v15061340