# The Epidemiological Signature of Pathogen Populations That Vary in the Relationship between Free-Living Parasite Survival and Virulence

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## Abstract

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_{0}, and on the size of the infectious peak in the population. These results highlight the importance of properly understanding the mechanistic relationship between virulence and parasite survival, as the evolution of increased survival across different relationships with virulence may have considerably different epidemiological signatures.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Motivation and Application

#### 2.2. Model Description

_{w}term allows for individuals to become infected via viral pathogen deposited in the environment and terms 𝜎

_{A}and 𝜎

_{I}allow asymptomatic and symptomatic individuals to deposit pathogens into the environment, respectively. Adapted from the more traditional SEIR (susceptible-exposed-infected-recovered) model, the SEAIR-W (susceptible-exposed-asymptomatic-infected-recovered-WAIT) model interrogates the consequences of the two hypotheses outlined above while representing the dynamics of a very relevant disease system (SARS-CoV-2) that includes an asymptomatic infectious population. While the importance of asymptomatic transmission was debated early in the pandemic, many studies have affirmed its role in the spread of disease [25,26,27]. Though environmental transmission of SARS-CoV-2 remains a controversial topic, it is plausible that asymptomatic individuals may spread disease through frequent contact with the environment, thus increasing the proportion of virus that is free-living [28]. We acknowledge that mathematical models of epidemics can be limited by “identifiability,” which can obfuscate the relative importance of some routes of transmission. In models that have both indirect and direct routes of transmission, it can be very difficult to conclude that one route is predominant [29,30,31].

#### 2.3. Simulations of Outbreaks

#### 2.4. Population Definitions and Parameter Values

#### 2.5. Virulence Definition

_{a}and 𝜎

_{I}). These calculations can be found in the Supplementary Materials.

_{I}), the incubation period of SARS-CoV-2 (𝜂), the rate of transfer from asymptomatic to symptomatic (1/⍵), the infected population recovery rate (ν), the percent of individuals that move from the asymptomatic to the recovered compartment without showing symptoms (the “mild” recovery track, p), the contact rate of people with people × the transmission probability of people to people by an asymptomatic individual (β

_{A}), the contact rate of people with people × the transmission probability of people to people by an asymptomatically infected person (β

_{I}), the contact rate of people with the environment × the probability of shedding by an asymptomatic individual to the environmental (𝜎

_{A}), the contact rate of people with the environment × the probability of symptomatically infected individuals shedding in the environment (𝜎

_{I}), and the average number of days before infection (1/ε).

#### 2.6. Survival Definition

_{w}). Table 4 outlines the direction (increasing or decreasing) in which these parameters are modulated when survival is decreased or increased. Within both models, a (percent) change in survival is defined as an equivalent uniform (percent) change in the survival parameters.

_{max}

^{−1}), the total infected population after 30 days, and the basic reproductive ratio (R

_{0}). Importantly, among these signatures, the basic reproductive ratio is the most frequently used in epidemiology and benefits from familiarity and mathematical formalism (see Section 2.7). The other signatures are determined through simulations of an epidemic for a given set of parameter values. Nonetheless, this study’s inclusion of multiple features of the epidemic allows us to examine how variation in virus life-history traits may influence different aspects of an epidemic in peculiar ways.

#### 2.7. Basic Reproductive Ratio

_{0}) for the model used in this study. This expression for R

_{0}can be deconstructed into two components. Equation (8) only contains parameters associated with person to person transmission (R

_{p}), while Equation (9) solely contains parameters associated with transmission from the environment (R

_{e}). In the Supplementary Materials, we provide additional information on these terms and their derivations. Applying the parameters values in Table 2, the numerical value of the basic reproductive ratio is given as R

_{0}~ 2.82.

## 3. Results

#### 3.1. Model Sensitivity Analysis

_{0}(Figure 2D) of the model is most sensitive to the parameters ⍵, β

_{A}, and ν. The sensitivity of R

_{0}to changes in ⍵ reflects the importance of the rate of conversion to the symptomatic state on model dynamics. In addition, β

_{A}has a very important influence on the model, consistent with other findings for COVID-19 that have emphasized the importance of asymptomatic transmission in disease spread [25,26,27].

#### 3.2. Illustrative Dynamics of Model System

#### 3.3. The Epidemic Consequences of Varying Virulence and Survival

_{0}, with variation in virulence-related traits having the largest effect on R

_{0}. Of note is how the range in R

_{0}values varies widely across virulence–survival values, from nearly 2.0 to 3.7 (Figure 4D).

#### 3.4. Implications of Virulence–Survival Relationships at Their Relative Extremes

#### 3.5. Positive Correlation Between Survival and Virulence

_{0}increases by approximately 94% (Figure 6 and Table 5).

#### 3.6. Negative Correlation Between Survival and Virulence

_{0}decreases by approximately 84% (Figure 6 and Table 6). Across all metrics considered, the effects of increased viral survival on outbreak dynamics is more extreme under the positive correlation than the negative correlation scenarios (Figure 6).

#### 3.7. Dynamics of Epidemics at Extreme Values for Virus Free-Living Survival

_{A}) or symptomatic (𝜎

_{I}) host. We observe how the high virulence, low survival simulation (Figure 7E) features a symptomatic peak that is larger in size and is prolonged relative to the lower virulence counterpart (Figure 7G). This relatively large symptomatic population sheds infectious virus into the environment for a longer period of time, contributing to the long tail of contaminated environments observed in Figure 5F.

## 4. Discussion

_{0}nearly doubles (94% increase; Table 5). These two traits are, of course, connected: the theoretical construction of the R

_{0}metric specifically applies to settings where a pathogen spreads in a population of susceptible hosts [34,35], an early window that is captured in the first 30 days.

_{0}difference between minimum and maximum survival is significant (approximately 84% decrease), the total number of infected individuals only changes by roughly 3% (Table 6). This large difference between R

_{0}at higher and lower survival values also does not translate to a difference in the total number of infected individuals in the first 30 days of an infection (the early outbreak window). In a scenario where survival and virulence are negatively correlated, a highly virulent and less virulent virus population can have similar signatures on a population with respect to the number of infected individuals in the first month. Thus, simply measuring the number of infected individuals in the first month of an outbreak is unlikely to reveal whether a pathogen population has undergone adaptive evolution or has evolved in a manner that meaningfully influences the natural history of disease.

#### Practical Implications for the Understanding of Outbreaks Caused by Emerging Viruses

_{0}is most impacted by changes in virulence and survival. In addition, the total number of infected individuals in the early window and the size of the infected “peak” would each be impacted most readily by changes in virulence–survival traits. The rate at which the epidemic peak was reached, on the other hand, showed relatively little change as survival increased or between the two correlation scenarios. Consequently, it would not serve as a useful proxy for virus evolution.

## Supplementary Materials

_{1}→ H

_{1}) and supplemental definition of virulence (A

_{2}→ H

_{2}).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Compartmental diagram of the SEAIR-W (susceptible-exposed-asymptomatic-infected-recovered) version of a-WAIT (Waterborne Abiotic or other Indirect Transmission)) model: this is based on a previously developed mathematical model used to interrogate environmental transmission of SARS-CoV-2 (see [22]).

**Figure 2.**Tornado plot showing the sensitivity of epidemic properties to individual parameter changes: (

**A**) the number of infected individuals (asymptomatic and symptomatic) at the epidemic peak; (

**B**) the rate at which the epidemic peak is reached, t

_{max}

^{−1}; (

**C**) the total infected population after 30 days; and (

**D**) the basic reproductive ratio (R

_{0}). Filled bars indicate the value of the epidemic feature when the associated parameter is increased by 5.0% from its nominal value. White bars indicate the value of a feature when the associated parameter is decreased by 5.0%. Blue coloring with checkered patterning indicates a parameter associated with survival, and orange coloring with lined patterning indicates a parameter associated with virulence.

**Figure 3.**Sample dynamics for the model system: (

**A**) the dynamics for all host compartments within the model and (

**B**) the fraction of environmental reservoirs in a setting that are contaminated with infectious virus.

**Figure 4.**Heatmap describing the impact of varying virus virulence and survival trait values assessed across four key epidemic metrics: these heatmaps express the change in (

**A**) the number of infected individuals (asymptomatic and symptomatic) at the epidemic peak; (

**B**) the rate at which the epidemic peak is reached, t

_{max}

^{−1}; (

**C**) the total infected population after 30 days; and (

**D**) the basic reproductive ratio (R

_{0}) when virulence and survival are modulated by ±5% within the model. Contour lines are available for clarity.

**Figure 5.**The expected effect of increasing survival on virulence for the two correlation models considered: here, we present a schematic of how the different hypotheses for the relationship between virulence and survival manifest on a map with a structure similar to the heat maps shown in Figure 4. The directions of the arrows depict how increasing survival would affect virulence under the two hypotheses: the blue arrow indicates the flow of an increasing positive correlation dynamic, while the direction of the orange arrow indicates an increasing negative correlation dynamic.

**Figure 6.**The percent change in SEAIR-W outbreak metrics as survival increases from −5% to +5% for different virulence–survival relationships (positive correlation and negative correlation). For each metric analyzed, we present the percent difference between the minimum and maximum survival values given the two hypotheses tested: (i) positive correlation between survival and virulence (comparing low virulence/low survival to high virulence/high survival) and (ii) negative correlation (high virulence/low survival to high survival/low virulence). The bars here correspond to the values (percent) in the third columns of Table 5 and Table 6, which denote the differences between the minimum and maximum values.

**Figure 7.**Virus outbreak dynamics for the extreme values of virulence and free-living survival considered for the two different relationships (positive or negative) between virulence and survival traits: these plots are similar to the illustrative dynamics in Figure 1. Here, we observe the dynamics of disease corresponding to the extreme values presented in Table 5 and Table 6. Subfigures (

**A**,

**C**,

**E**,

**G**) depict disease dynamics, and (

**B**,

**D**,

**F**,

**H**) depict the dynamics of contaminated environments. (

**A**–

**D**) correspond to the parameter values considered for the positive correlation scenario, while (

**E**–

**H**) correspond to the negative correlation scenario.

**Table 1.**Definitions and initial values for the populations represented by each compartment: the values here are the averages of the model values across all countries developed in a prior COVID-19 study [22].

Symbols | Value | Units | Definitions |
---|---|---|---|

S_{0} | 57.05 × 10^{6} | people | Susceptible individuals |

E_{0} | 66.50 | people | Exposed individuals |

A_{0} | 13.30 | people | Asymptomatic individuals |

I_{0} | 13.30 | people | Symptomatic individuals |

Rec_{0} | 0 | people | Recovered individuals |

W_{0} | 1% | unitless | % of viral pathogen in environment |

**Table 2.**Definitions for the nominal parameter values used in this study: parameter values were developed from empirical findings and country-level data, as discussed in another study [22].

Symbols | Values | Units | Definitions |
---|---|---|---|

𝜇 | 1/(80.3 × 365) | 1/day | Natural death rate (reciprocal of the upper bound of average human lifespan) |

𝜇_{I} | 0.00159 | 1/day | Infected death rate (natural death rate + disease-induced death rate) |

𝜂 = (⍵ − ε^{−}^{1}) | 5.5 | days | SARS-CoV-2 incubation period |

1/⍵ | 𝜂 − ε^{−1} | days | Expected time in the asymptomatic state |

ν | 0.0305 | 1/day | Recovery rate (average of 3 to 6 weeks) |

p | 95.6% | percent | Percent that moves along the “mild” recovery track |

k | 0.649 | 1/day | Waning virus rate in the environment (using the average of all material values, wood, steal, cardboard, and plastic) |

β_{a} | 0.550 | 1/day | Contact rate of people with people × transmission probability of people to people by A-person |

β_{I} | 0.491 | 1/day | Contact rate of people with people × transmission probability of people to people by I-person |

β_{W} | 0.031 | 1/day | Contact rate of person with environment × transmission probability of environment to people |

𝜎_{a} | 3.404 | 1/day | Contact rate of person with environment × probability of shedding by A-people to environment |

𝜎_{I} | 13.492 | 1/day | Contact rate of person with environment) × probability of shedding by I-people to environment |

1/ε | 2.478 | days | Average number of days before infectiousness |

**Table 3.**Virulence parameters: this is a list of uniformly modulated parameters and the direction in which they change when virulence is increased or decreased. When virulence changes, an up arrow (↑) indicates the parameter is increased (by an equivalent percent) and a down arrow (↓) indicates the parameter is decreased (by an equivalent percent).

Symbols | Definition | Virulence Increased | Virulence Decreased |
---|---|---|---|

𝜇_{I} | Infected death rate (natural death rate + disease induced death rate) | ↑ | ↓ |

𝜂 = (⍵ − ε^{−1}) | SARS-CoV-2 incubation period | ↓ | ↑ |

1/⍵ | Expected time in the asymptomatic state | ↑ | ↓ |

ν | Recovery rate (average of 3 to 6 weeks) | ↓ | ↑ |

p | Percent that moves along the “mild” recovery track | ↓ | ↑ |

β_{A} | Contact rate of people with people × transmission probability of people to people by A-person | ↑ | ↓ |

β_{I} | Contact rate of people with people × transmission probability of people to people by I-person | ↑ | ↓ |

𝜎_{A} | Contact rate of person with environment) × (probability of shedding by A-people to environment | ↑ | ↓ |

𝜎_{I} | Contact rate of person with environment × probability of shedding by I-people to environment | ↑ | ↓ |

1/ε | Average number of days before infectious | ↓ | ↑ |

**Table 4.**Survival parameters: this is a list of parameters that are uniformly modulated and the direction in which they change when survival is increased or decreased. An up arrow (↑) indicates the parameter is increased by some percent when the equivalent (percent) change in survival is applied. A down arrow (↓) indicates the parameter is decreased by some percent when the equivalent (percent) change in survival is applied.

Symbols | Definition | Survival Increased | Survival Decreased |
---|---|---|---|

k | Waning virus rate in environment (using the average of all material values, wood, steal, cardboard, and plastic) | ↓ | ↑ |

β_{W} | Contact rate of person with environment × transmission probability of environment to people | ↑ | ↓ |

**Table 5.**Positive correlation scenario: comparing epidemic metrics under low survival/low virulence versus high survival/high virulence scenarios (as in the positive correlation scenario). For each metric analyzed, these are the heatmap values for the bottom left (at “coordinates” (Vir, Sur) → (−5%, −5%)) and top right (at “coordinates” (Vir, Sur) → (+5%, +5%)) corners.

Epidemic Metric | min Virulence, min Survival | max Virulence, max Survival | % Difference between min Survival and max Survival |
---|---|---|---|

Peak total infected (people) | 5.68 × 10^{6} | 7.64 × 10^{6} | +34.51% |

t_{max}^{−1} (days^{−1}) | 1.99 × 10^{−2} | 2.23 × 10^{−2} | +12.06% |

Total after 30 days (people) | 8.18 × 10^{7} | 1.62 × 10^{8} | +98.04% |

Basic reproductive ratio (R_{0}) | 1.95 | 3.78 | +93.84% |

**Table 6.**Negative correlation scenario: comparing epidemic metrics under low survival/low virulence versus high survival/high virulence scenarios (as in the negative correlation scenario). For each metric analyzed, these are the global heatmap values for the top left (at “coordinates” (Vir, Sur) → (+5%, −5%)) and bottom right corners (at “coordinates” (Vir, Sur) → (−5%, +5%)).

Epidemic Metric | max Virulence, min Survival | min Virulence, max Survival | % Difference between min Survival and max Survival |
---|---|---|---|

Peak total infected (people) | 7.39 × 10^{6} | 5.99 × 10^{6} | −23.47% |

t_{max}^{−1} (days^{−1}) | 2.10 × 10^{−2} | 2.23 × 10^{−2} | −0.15% |

Total infected after 30 days (people) | 1.16 × 10^{8} | 1.13 × 10^{8} | −2.68% |

Basic reproductive ratio (R_{0}) | 3.67 | 1.99 | −84.39% |

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**MDPI and ACS Style**

Gomez, L.M.; Meszaros, V.A.; Turner, W.C.; Ogbunugafor, C.B. The Epidemiological Signature of Pathogen Populations That Vary in the Relationship between Free-Living Parasite Survival and Virulence. *Viruses* **2020**, *12*, 1055.
https://doi.org/10.3390/v12091055

**AMA Style**

Gomez LM, Meszaros VA, Turner WC, Ogbunugafor CB. The Epidemiological Signature of Pathogen Populations That Vary in the Relationship between Free-Living Parasite Survival and Virulence. *Viruses*. 2020; 12(9):1055.
https://doi.org/10.3390/v12091055

**Chicago/Turabian Style**

Gomez, Lourdes M., Victor A. Meszaros, Wendy C. Turner, and C. Brandon Ogbunugafor. 2020. "The Epidemiological Signature of Pathogen Populations That Vary in the Relationship between Free-Living Parasite Survival and Virulence" *Viruses* 12, no. 9: 1055.
https://doi.org/10.3390/v12091055