# Mathematical Model of the Role of Asymptomatic Infection in Outbreaks of Some Emerging Pathogens

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## Abstract

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## 1. Introduction

## 2. Methods

- Individuals infected by symptomatic individuals immediately become symptomatic without passing through an asymptomatic stage, whereas individuals infected by asymptomatic individuals become asymptomatic. This assumption is inspired by an exponential shedding dose-response curve, as illustrated in Figure 2.
- Individuals at earlier asymptomatic stages require further infection events to progress to the next asymptomatic stage, while individuals at later asymptomatic stages can automatically progress to the symptomatic stage.
- Individuals at earlier asymptomatic stages can only move onto the next asymptomatic stage if infected by those at higher asymptomatic stages of infection, or symptomatic individuals.
- Asymptomatic individuals can revert to earlier asymptomatic stages, but symptomatic individuals cannot revert to asymptomatic infection.
- For simplicity, we assume that pathogen mutations are not included, and thus, disease properties such as transmission, aggressivity and mortality remain unchanged in time. We also do not consider the intrinsic potential of the pathogen to lay dormant within the host, and assume that the pathogen is always active and able to infect.

#### 2.1. Model Presentation

#### 2.2. Numerical Results

## 3. Discussion and Conclusion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Existence of Oscillatory Outbreak Equilibrium Points

#### Appendix A.1. Local Stability Analysis of Disease-Free Equilibrium

#### Appendix A.2. Parameters Estimation

- (a)
- The rates of transmission from asymptomatic and symptomatic individuals are related (increasing one necessarily increases the other, for example).
- (b)
- The rate of waning immunity from R is related to the rate of loss of infectiousness of ${E}_{i}$${}^{\prime}$s.
- (c)
- As individuals progress through asymptomatic classes ${E}_{i}$, their susceptibility, transmissibility, rate of loss of infectiousness, and rate of gain of immunity all increase.

#### Appendix A.2.1. Estimate for b and μ:

#### Appendix A.2.2. Estimate for β_{I} and ${\beta}_{{E}_{i}}$, i = 1,2,…,n:

#### Appendix A.2.3. Estimate for α_{S} and ${\alpha}_{{E}_{i}}$, _{i} = 1,2,…,n:

#### Appendix A.2.4. Estimate for ${\lambda}_{{E}_{i}}$, i = 1,2,…,n:

#### Appendix A.2.5. Estimate for ${\rho}_{{E}_{i}}$, i = 1,2,…,n:

#### Appendix A.2.6. Effects of Number of Asymptomatic Cases, n

**Figure A1.**Behaviors of infectious classes relative to n. The horizontal axes represent the number of asymptomatic stages and the vertical axes represent the percentage of human populations. The disease can be eradicated for $n\le 3$. There is disease persistence when $n>3$ which eventually leads to a branch of disease eradication for n large enough.

#### Appendix A.2.7. Effects of the Asymptomatic Transmission Rates ${\beta}_{{E}_{i}{}^{\prime}}$s

**Figure A2.**Oscillatory behaviors of infectious classes relative to the ${\beta}_{{E}_{i}}$${}^{\prime}$s (represented here by ${\beta}_{I}$ as in (A3)). The horizontal axes scale the baseline ${\beta}_{I}$ and the vertical axes represent percentages of human populations. There is a regime for the values of ${\beta}_{I}$ for which the disease does not persist into the population (approximately ${\beta}_{I}<0.014$). The disease persists into the population with oscillating profile when $0.014<{\beta}_{I}<0.015$; for ${\beta}_{I}>0.015$, the disease persists into the population stationarilly.

#### Appendix A.2.8. Effects of the Loss/Recovery of Asymptomatic Infection Rates ${\rho}_{{E}_{i}{}^{\prime}}$s

**Figure A3.**Behaviors of infectious classes relative to the ${\rho}_{{E}_{i}}$${}^{\prime}$s (represented here by ${\rho}_{r}$ as in (A6)). The horizontal axes scale the parameters ${\rho}_{r}$ and the vertical axes represent human populations. The profiles for the variables depict an endemic stationary branch followed by a chaotic behavior, then another stationary branch where the disease is eradicated.

#### Appendix A.2.9. Effects of the Clinical Transmission Rates ${\alpha}_{{E}_{i}{}^{\prime}}$s

**Figure A4.**Behaviors of infectious classes relative to the ${\alpha}_{{E}_{i}}$${}^{\prime}$s (represented here by ${\alpha}_{S}$ as in (A4)). The horizontal axes scale the parameters ${\alpha}_{S}$ and the vertical axes represent percentages of human populations. The disease fatality (symptomatic cases) increases slowly (${\alpha}_{S}<0.53$), then increases rapidly (${\alpha}_{S}>0.53$).

#### Appendix A.2.10. Effects of the Gain of Immunity Rates by Asymptomatic Humans, ${\lambda}_{{E}_{i}{}^{\prime}}$s

**Figure A5.**Behaviors of infectious classes relative to the $\lambda $${}^{\prime}$s (represented here by ${\lambda}_{{E}_{1}}$ as in (A5)). The horizontal axes scale the parameter ${\lambda}_{{E}_{1}}$ and the vertical axes represent the percentage of human populations. The disease persists for ${\lambda}_{{E}_{1}}$ small, then the disease is eradicated as ${\lambda}_{{E}_{1}}$ increases.

#### Appendix A.2.11. Effects of Birth Rate, b

**Figure A6.**Behaviors of infectious classes relative to b. The horizontal axes scale the parameter b and the vertical axes represent percentage of human populations. The disease fatality (asymptomatic cases) mildly increases as b increases.

## Appendix B. Ebola (DRC 1995) and COVID-19 (New York 2020)

**Figure A7.**Reverse Hill Function—Phases of outbreaks without interventions and with interventions. The horizontal axis represents time in days, and the vertical axis represents the values of the reversed Hill function in (A9). The disease progresses without interventions when time $t<{t}_{0}$ and the ${\tilde{\beta}}_{x}={\beta}_{x}$ for all x; the disease spreads with interventions when $t>{t}_{0}$ and the ${\tilde{\beta}}_{x}\simeq 0$.

#### Appendix B.1. Ebola in the Democratic Republic of Congo (DRC) 1995

**Figure A8.**The horizontal axis represents the time in days and the vertical axis represents the cumulative number of new Ebola cases from the model simulations and actual data (data are from Figure 1 in [47]). The values of the parameters are as follows: ${N}_{0}=8$ millions, ${\beta}_{I}=0.023$ per day, ${t}_{0}=164$ days, ${n}_{H}=20$, and the values for the other parameters are as in Table 2. RE $=21.04\%$ and ${\mathcal{R}}_{0}=1.49$.

#### Appendix B.2. COVID-19 in New York State (February 29–4 August 2020)

**Figure A9.**The horizontal axis represents the time in days and the vertical axis represents the cumulative number of new COVID-19 cases from the model simulations and actual data (data are from [48,49]). The values of the parameters are as follows: $\mu =3.37\times {10}^{-5}$ per day [50], $b=6.4\times {10}^{-4}$ pop. per day [51], ${N}_{0}=19.8$ millions [51], ${\beta}_{I}=0.066$ per day, ${t}_{0}=30$ days, ${n}_{H}=5$, and the values for the other parameters are as in Table 2. RE $=23.2\%$ and ${\mathcal{R}}_{0}=12.3$.

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**Figure 1.**Framework. The nodes with reddish colors (orange and red) represent the compartments of humans that are infectious, while the nodes with greenish colors (black and green) represent the compartments of humans that are not infectious. The various interactions are explained in the descriptions of Equations (1)–(6).

**Figure 2.**Profile for the parameters ${\alpha}_{i}$${}^{\prime}$s, ${\beta}_{i}$${}^{\prime}$s and ${\lambda}_{i}$${}^{\prime}$s, represented here by $p={p}_{0}{c}_{p}^{i-(n+1)}$, $\phantom{\rule{4pt}{0ex}}i\in \{1,2,\dots ,n\}$, where ${c}_{p}>0$ is a constant. More details in Technical Appendix.

**Figure 3.**Profiles for all the variables of Model (1)–(6). The horizontal axes scale the time in days and the vertical axes scale the population sizes percentages. The first infectious case arises in the population ${E}_{1}$ and does not spread to the other infectious classes. ‘Sick’ humans are those of the class I who show clinical signs of the disease. $\mathrm{ORS}=0$.

**Figure 4.**Profiles for all the variables of Model (1)–(6). The horizontal axes scale the time in days and the vertical axes scale the population sizes in percentages. The first infectious case arises in the population ${E}_{2}$ and is able to spread to other infectious classes so that the disease is maintained in the population. ‘Sick’ humans are those of the class I who show clinical signs of the disease. $\mathrm{ORS}=1:38$.

Variables | Descriptions |
---|---|

S | number of susceptible humans |

${E}_{i}$${}^{\prime}$s | numbers of asymptomatic (latent) humans, of various stages |

I | number of infected humans who show clinical signs |

R | number of recovered humans |

Parameters | Descriptions | Values | Sources |
---|---|---|---|

${\alpha}_{S}$, ${\alpha}_{i}$${}^{\prime}$s | weights of infectiousness of S and ${E}_{i}$${}^{\prime}$s by contact with I | 0.036, ${1.5}^{i}$ | Appendix A.2.3 |

b | rate of recruitment of humans | $1.1\times {10}^{-4}\times {N}_{0}$ per day | Appendix A.2.1 |

${\beta}_{I}$, ${\beta}_{{E}_{i}}$${}^{\prime}$s | rates of transmission by contact with I and ${E}_{i}$${}^{\prime}$s | 0.125, ${1.5}^{i-(n+1)}{\beta}_{I}$ per day | Appendix A.2.2 |

${\rho}_{r}$ | rate of wane of immunity of R | $8.5\times {10}^{-3}$ per day | Appendix A.2.5 |

${\rho}_{{E}_{i}}$${}^{\prime}$s | rates of loss of infectiousness of ${E}_{i}$${}^{\prime}$s | ${1.5}^{i-(n+1)}{\rho}_{r}$ per day | Appendix A.2.5 |

${\lambda}_{{E}_{i}}$${}^{\prime}$s | rates of gain of immunity of ${E}_{i}$${}^{\prime}$s | ${1.5}^{i-(n+1)}\times 0.102$ per day | Appendix A.2.4 |

$\gamma $ | rate of removal from sick class I | 0.167 per day | [33] |

$\nu $ | rate of transition from ${E}_{n}$ to I | 0.05 per day | assumed |

$\mu $ | natural death rate of humans | $3.4\times {10}^{-5}$ per day | Appendix A.2.1 |

$\theta $ | fraction of humans I who die | 0.7 | [33] |

n | number of asymptomatic stages | 6 | assumed |

$\tilde{n}$ | number of asymptomatic stages that do not transit to higher infection stage “naturally” | 1 | assumed |

${N}_{0}$ | Total initial population size | 11.5 million | assumed |

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Siewe, N.; Greening, B., Jr.; Fefferman, N.H.
Mathematical Model of the Role of Asymptomatic Infection in Outbreaks of Some Emerging Pathogens. *Trop. Med. Infect. Dis.* **2020**, *5*, 184.
https://doi.org/10.3390/tropicalmed5040184

**AMA Style**

Siewe N, Greening B Jr., Fefferman NH.
Mathematical Model of the Role of Asymptomatic Infection in Outbreaks of Some Emerging Pathogens. *Tropical Medicine and Infectious Disease*. 2020; 5(4):184.
https://doi.org/10.3390/tropicalmed5040184

**Chicago/Turabian Style**

Siewe, Nourridine, Bradford Greening, Jr., and Nina H. Fefferman.
2020. "Mathematical Model of the Role of Asymptomatic Infection in Outbreaks of Some Emerging Pathogens" *Tropical Medicine and Infectious Disease* 5, no. 4: 184.
https://doi.org/10.3390/tropicalmed5040184