# Will SARS-CoV-2 Become Just Another Seasonal Coronavirus?

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## Abstract

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## 1. Introduction

#### 1.1. Small SARS-CoV-2 Doses Are Associated with Mild COVID-19 Disease

#### 1.2. Children Develop Mild COVID-19 Infections

#### 1.3. Partial Immunity Results in Mild Disease

#### 1.4. Non-Pharmaceutical Interventions (NPI) Promote Mild Disease

#### 1.5. Vaccines

## 2. Methods

#### 2.1. Model with Two Infection Types

#### 2.2. Model with Age-Structure Only

#### 2.3. The Full Model

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

JASC | Just Another Seasonal Coronavirus |

## Appendix A

#### Appendix A.1. Full Steady State Solution for the Model with Two Infection Types

**Figure A1.**Accounting for infectiousness $\theta $ and susceptibility $\tilde{\nu}$ in the model with two infection phenotypes If high-shedding individuals are twice as infectious, the initial outbreak is larger, consists of fewer low-shedding cases, and the subsequent outbreak of low-shedding cases occurs later (dotted curve: $\theta =2$, solid curve: $\theta =1$, (

**top panel**)). If partially immune individuals are half as susceptible, it takes longer to reach the endemic steady state because subsequent outbreaks occur later (dotted curves: $\tilde{\nu}=1/2$, solid curves: $\tilde{\nu}=1$, (

**bottom panel**); all other parameters as in Figure 2).

**Figure A2.**The values of ${I}_{a}^{*}$ and ${I}_{k}^{*}$ from the age-structured model plotted against $\nu $, the relative susceptibility of children, who make up half the population. The curves are qualitatively similar to those in Figure 3 but the relationships are more pronounced since children make up a greater share of the population.

**Figure A3.**Supercritical Hopf Bifurcation in the Full Model. The red complex eigenpair is about to cross the imaginary axis as $\theta $ passes near $2.2$, resulting in a supercritical Hopf bifurcation (left). Two real eigenvalues near $-0.1$ are not shown. Near the bifurcation, transients decay slowly (right). These results were generated from the full model without calibrating to data, without vaccination, ${\nu}_{k}=1/2$, ${Q}_{ah}=0$, ${Q}_{al}={Q}_{kl}={Q}_{kh}=\tilde{{Q}_{ah}}=\tilde{{Q}_{al}}=\tilde{{Q}_{kl}}=\tilde{{Q}_{kh}}=1$, and contact rates ${\alpha}_{aa}={\alpha}_{ak}={\alpha}_{kk}=\alpha $ were chosen so ${R}_{0}=2.5$. For simplicity, we have set $\tau =\mu =0$. Hopf bifurcations are only present close the limiting case ${Q}_{ah}=0$, ${Q}_{al}={Q}_{kl}={Q}_{kh}=\tilde{{Q}_{ah}}=\tilde{{Q}_{al}}=\tilde{{Q}_{kl}}=\tilde{{Q}_{kh}}=1$.

**Figure A4.**Amplitude of solutions in the Full Model at 10,000 days. The supercritical Hopf bifurcation appears near $\theta =2.2$. For large enough $\theta $ the periodic solutions disappear through another supercritical Hopf bifurcation (eigenvalue analysis not shown). These results were obtained from the same parametrization as Figure A3.

#### Appendix A.2. Simplifying the Full Model

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**Figure 1.**Contour plots of ${H}^{*}$ in the model with two infection types for various ${Q}_{l}$ and ${\tilde{Q}}_{l}$ along the axes. The white line corresponds to ${H}^{*}=5000$. As ${Q}_{l}$ and ${\tilde{Q}}_{l}$ approach 1, ${H}^{*}$ decreases to 0 (top right corner of plots). If sterilizing immunity is long compared to partial immunity, ${H}^{*}$ is more sensitive to increases in ${Q}_{l}$, especially near 0. If partial immunity is long compared to sterilizing immunity, ${H}^{*}$ is more sensitive to ${\tilde{Q}}_{l}$, especially near 0. Increasing the duration of sterilizing immunity decreases ${H}^{*}$. All plots generated with ${R}_{0}=2.5$, $N={10}^{8}$, ${Q}_{h}=0,{\tilde{Q}}_{l}=0.8$, and $\gamma =1/10$ per day.

**Figure 2.**Partial immunity and a dose response can promote avirulence.The initial outbreak consists mainly of high-shedding cases, but these are replaced by low-shedding cases in subsequent outbreaks. The dashed curve corresponds to the approximate dynamics described in the text. Solid curves are model solutions under the parametrization $\u03f5=0.01$, where ${Q}_{ah}=\u03f5$ and ${Q}_{al}={Q}_{kl}={Q}_{kh}=\tilde{{Q}_{a}h}=\tilde{{Q}_{a}l}=\tilde{{Q}_{k}h}=\tilde{{Q}_{k}l}=1-\u03f5$.

**Figure 3.**The values of ${I}_{a}^{*}$ and ${I}_{k}^{*}$ from the age-structured model plotted against $\nu $, the relative susceptibility of children. Total infections are maximized when children are equally susceptible to adults ($\nu =1$). Infections are most concentrated in adults if children are less susceptible ($\nu <1$). Children reduce infections in adults if they are more susceptible ($\nu >1$). These relationships are more pronounced if children make up a greater share of the population (Figure A2). The population of the United States is ${N}_{a}+{N}_{k}$ = 328,239,523, and children are considered under 14 years of age (${N}_{k}$ = 56,406,387) [55].

**Figure 4.**Dose response, partial immunity, and protective effects of youth combine to limit the cumulative number of high-shedding infections which occur within the first 30 years (

**top panel**). “Dose” in isolation is identical to “None” for $\theta >1$. Each combination of the three mechanisms produces different dynamics, as shown for $\theta =2$ (

**bottom panel**, which corresponds to dotted black line in the

**top panel**). High-shedding cases are driven close to extinction only if all three mechanisms are in place (“All mechanisms” in (

**bottom panel**)). Cases are shown within adults only. Across all simulations, ${\nu}_{k}=1/2$, $\mathsf{\Omega}=3\times {10}^{6}$/day. In the “All” curves, ${Q}_{ab}=0$, and the other ${Q}_{xy},\tilde{{Q}_{xy}}=1$. In the “None” curves, ${Q}_{xy}=\tilde{{Q}_{xy}}=0$. Curves with the “Age” mechanism have ${Q}_{ky}=\tilde{{Q}_{ky}}=1$. Curves with the “Dose” mechanism have ${Q}_{xl}=\tilde{{Q}_{xl}}=1$, and “Dose only” additionally has ${Q}_{xh}=\tilde{{Q}_{xh}}=0$ (partial immunity and Age override this latter constraint). Curves with the “PI” mechanism have $\tilde{{Q}_{xy}}=1$.

**Figure 5.**With all three mechanisms in place and children less susceptible to infection (${\nu}_{k}=1/2$), a Hopf bifurcation occurs as $\theta $ passes through ∼2.3. For $\theta $ near 2.5, high-shedding cases return in greater numbers on a biennial cycle and eventually exceed their initial outbreak levels. The magnitude of outbreaks diminishes as $\theta $ increases past 2.5 but high-shedding infections make up a greater share of total infections in adults. Periodic solutions do not exist if children are more susceptible than adults (${\nu}_{k}>1$). Vertical axes are the number of active infections in adults. Except for $\theta $, all parameters as in Figure 4.

**Figure 6.**The Sweet Spot for JASC: the three mechanisms in combination limit the long-term average number of high-shedding cases if they generate low-shedding cases with probability close to 100% and provided that high-shedding cases are not too infectious ($\u03f5\approx 0,\theta \approx 1$, (

**left panel**)). Children help mitigate disease severity if they develop low-shedding infections and are more susceptible than adults (${\nu}_{k}>1$, (

**right panel**)). High-shedding infections can make up half of the total infection prevalence in adults if children do not acquire infection (${\nu}_{k}<1$, right panel, and also refer to the $\theta =5$ case in Figure 5). High-shedding infections in adults are rare and periodic outbreaks will not occur if children are equally susceptible (${\nu}_{k}=1$). Low-shedding infections in adults further decline if children are more susceptible (${\nu}_{k}\to \infty $). Parametrization in the left panel corresponds to ${\nu}_{k}=1/2$ and the parametrization in the right panel corresponds to $\u03f5=0$. See text for description of the other parameters.

**Figure 7.**Social distancing which disproportionately reduces contact between children and adults from day 20 to 460 could increase the number of high-shedding cases in adults if children always develop low-shedding infections. The effect is enhanced for a larger vaccination rate. The vaccination period begins on day 360 and ends on 460 (rate of 3 million doses per day). For the “Distancing” curve, ${\alpha}_{ak}=0$ from days 20–460. For the “No distancing” curve, ${\alpha}_{ak}$, ${\alpha}_{aa}$ and ${\alpha}_{kk}$ are set equal and are calibrated to the incidence data up to day 418, are set to values consistent with ${R}_{0}=1.3$ from days 418-460, and ${R}_{0}=2.5$ thereafter. Other parameters: ${Q}_{ab}=0$, ${Q}_{am}={Q}_{kb}={Q}_{km}=\tilde{{Q}_{ab}}=\tilde{{Q}_{am}}=\tilde{{Q}_{kb}}=\tilde{{Q}_{km}}=1$, and ${\nu}_{k}=1/2$.

**Figure 8.**As long as high-shedding cases are not too infectious ($\theta =2$) prolonged vaccination over the next thirty years is not necessary for SARS-CoV-2 to become JASC. If high-shedding cases are sufficiently infectious to cause periodic outbreaks ($\theta =3$) then regular vaccination is necessary to limit high-shedding cases. Vaccination rate $\mathsf{\Omega}$ is the number of doses (in the millions) administered to susceptible individuals per day during the intense vaccination program which begins on day 360 and lasts until day 460. After day 460, 60% of the population continues to get vaccinated at the corresponding Frequency (in years) along the vertical axis. Contact levels consistent with ${R}_{0}=1.3$ maintained from day 418 (16 March 2021) up to day 460, after which contact returns to levels consistent with ${R}_{0}=2.5$. In both panels ${Q}_{ab}=0$, ${Q}_{am}={Q}_{kb}={Q}_{km}=\tilde{{Q}_{ab}}=\tilde{{Q}_{am}}=\tilde{{Q}_{kb}}=\tilde{{Q}_{km}}=1$, and ${\nu}_{k}=1/2$.

Variable | Description | Initial Condition |
---|---|---|

${S}_{x}$ | Susceptible adults and children | 271,833,136 and 56,406,387 [55] |

${P}_{x}$ | Partially resistant adults, children | 0 |

${L}_{x}$ | Low-shedding infected adults, children | 0 |

${H}_{x}$ | High-shedding infected adults, children | 1, 0 |

${R}_{xy}$ | Resistant adults/children | 0 |

after low-shedding/high-shedding infection |

Parameter | Description | Value |
---|---|---|

$\tilde{\rho}$ | Loss rate of partial immunity | $\frac{1}{2}$ per year [58,59,60,61] |

$\rho $ | Loss rate of sterilizing immunity | $\frac{1}{2}$ per year [58,59,60,61] |

${\alpha}_{ak}$ | Contact rate between adults and children | Calibrated to |

daily incidence data | ||

or ${R}_{0}=1.3,2.5$ [56,62] | ||

${\alpha}_{aa}$ | Contact rate between adults | Calibrated to |

daily incidence data | ||

or ${R}_{0}=1.3,2.5$ [56,62] | ||

${\alpha}_{kk}$ | Contact rate between children | Calibrated to |

daily incidence data | ||

or ${R}_{0}=1.3,2.5$ [56,62] | ||

$\theta $ | Infectiousness of high-shedding cases | |

relative to low-shedding cases | [1, 5] | |

$\tau $ | Maturation rate | $\frac{1}{14}$ per year [55] |

$\mu $ | Death rate | $\frac{1}{80}$ per year [55] |

$\gamma $ | Recovery rate from infection | $\frac{1}{10}$ per day [63,64] |

${\nu}_{k}$ | Susceptibility, S child | [0, 5] |

$\tilde{{\nu}_{a}}$ | Susceptibility, P adult | $\frac{1}{2}$ |

$\tilde{{\nu}_{k}}$ | Susceptibility, P child | $\frac{1}{2}$${\nu}_{k}$ |

${Q}_{xy}$ | Probability an individual in ${S}_{x}$ develops | [0, 1] |

low-shedding infection ${L}_{x}$ upon contact with | ||

an infection of type y | ||

$\tilde{{Q}_{xy}}$ | Probability an individual in ${P}_{x}$ develops | [${Q}_{xy}$,1] |

low-shedding infection ${L}_{x}$ upon contact with | ||

an infection of type y | ||

$\mathsf{\Omega}$ | Daily vaccination rate | [${10}^{6}$, $3\times {10}^{6}$] |

from days 360–460 |

**Parameter values used in the full model.**Rates of immune loss are chosen so the duration of immunity to SARS-CoV-2 lies between the duration reported for SARS and the seasonal coronaviruses [58,59,60,61]. The average infectious period is chosen to coincide with estimated durations of infectiousness [63,64]. Contact rates are either (i) calibrated to match daily incidence, or (ii) held at fixed values to coincide with a given ${R}_{0}$ value, as explained at the end of the Methods section. Susceptibility parameters were chosen so partial immunity reduces the probability of infection by 50% in all of the figures to follow, but main results are not sensitive to the particular value (Figure A1). Other parameters which are not fixed across simulations are unknown and are varied over the ranges indicated by the intervals across simulations. Relationships between these parameters and model outputs are presented within the figures in Results.

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**MDPI and ACS Style**

Beams, A.B.; Bateman, R.; Adler, F.R.
Will SARS-CoV-2 Become Just Another Seasonal Coronavirus? *Viruses* **2021**, *13*, 854.
https://doi.org/10.3390/v13050854

**AMA Style**

Beams AB, Bateman R, Adler FR.
Will SARS-CoV-2 Become Just Another Seasonal Coronavirus? *Viruses*. 2021; 13(5):854.
https://doi.org/10.3390/v13050854

**Chicago/Turabian Style**

Beams, Alexander B., Rebecca Bateman, and Frederick R. Adler.
2021. "Will SARS-CoV-2 Become Just Another Seasonal Coronavirus?" *Viruses* 13, no. 5: 854.
https://doi.org/10.3390/v13050854