Variation in microparasite free-living survival and indirect transmission can modulate the intensity of emerging outbreaks
A waterborne, abiotic, and other indirectly transmitted (WAIT) model for the dynamics of emergent viral outbreaks
Several models have been engineered to explore aspects of COVID-19 dynamics. For example, models have been used to investigate the role of social distancing20,42, social mixing43, the importance of undocumented infections44, the role of mobility in the early spread of disease in China45, and the potential for contact tracing as a solution46. Only a few notable models of SARS-CoV-2 transmission incorporate features of indirect or environmental transmission40,46 and none consider the dynamical properties of viral free-living survival in the environment. Such a model structure would provide an avenue towards exploring how variation in free-living survival influences disease outbreaks. Indirect transmission includes those routes where pathogen is spread through means other than from person to person, and includes transmission through environmental reservoirs. Environmental transmission models are aplenty in the literature and serve as a theoretical foundation for exploring similar concepts in newer, emerging viruses1,2,3,4,5,6,7,8,9,10.
Here, we parameterize and validate an SEIR-W model: Susceptible (S), Exposed (E), Infectious (I), Recovered (R), and WAIT (W) model. Here W represents the environmental component of the transmission cycle during the early stage of the SARS CoV-2 pandemic. This component introduces more opportunities for infection, and complex dynamics resulting from viral persistence in the environment. In this framework, both indirect and direct transmission occur via mass-action, “random” encounters.
This model is derived from a previously developed framework called “WAIT”—which stands for Waterborne, Abiotic, and other Indirectly Transmitted—that incorporates an environmental reservoir where a pathogen remains in the environment and “waits” for hosts to interact with it11,12. The supplementary information contains a much more rigorous discussion of the modeling details. In the main text, we provide select details.
Building the SEIR-W model framework for SARS-CoV-2
Here W represents the environmental component of the early stage of the SARS CoV-2 pandemic (Fig. S1). This environmental compartment refers to reservoirs that people may have contact with on a daily basis, such as doorknobs, appliances, and non-circulating air indoors. The W compartment of our model represents the fraction of these environmental reservoirs that house some sufficiently transmissible amount of infectious virus. We emphasize that the W compartment is meant to only represent reservoirs that are common sites for interaction with people. Thus, inclusion of the W compartment allows us to investigate the degree to which the environment is infectious at any given point, and its impact on the transmission dynamics of SARS CoV-2.
Model parameters are described in detail in Table 1. The system of equations in the proposed mathematical model corresponding to these dynamics are defined in Eqs. (1)–(6):
Table 1 Model population definitions and initial values denoted with subscript 0 for each state variable.
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$$frac{dS}{dt}=mu (N-S)-left(frac{{beta }_{A}{I}_{A}+{beta }_{S}{I}_{S}}{N}+{beta }_{W}Wright)S$$
(1)
$$frac{dE}{dt}=left(frac{{beta }_{A}{I}_{A}+{beta }_{S}{I}_{S}}{N}+{beta }_{W}Wright)S-(epsilon +mu )E$$
(2)
$$frac{d{I}_{A}}{dt}=epsilon E-(omega +mu ){I}_{A}$$
(3)
$$frac{d{I}_{S}}{dt}=(1-p)omega {I}_{A}-(nu +{mu }_{S}){I}_{S}$$
(4)
$$frac{dR}{dt}=pomega {I}_{A}+nu {I}_{S}-mu R$$
(5)
$$frac{dW}{dt}=left(frac{{sigma }_{A}{I}_{A}+{sigma }_{S}{I}_{S}}{N}right)left(1-Wright)-kW$$
(6)
Infection trajectories
In addition to including a compartment for the environment (W), our model also deviates from traditional SEIR form by splitting the infectious compartment into an IA-compartment (A for asymptomatic), and an IS-compartment (S for symptomatic). As we discuss below, including asymptomatic (or sub-clinical) transmission is both essential for understanding how environmental—as opposed to simply unobserved or hidden—transmission affects the ecological dynamics of pathogens and also for analyzing SARS-CoV-2. The former represents an initial infectious stage (following the non-infectious, exposed stage), from which individuals will either move on to recovery directly (representing those individuals who experienced mild to no symptoms) or move on to the IS-compartment (representing those with a more severe response). Finally, individuals in the IS-compartment will either move on to recovery or death due to the infection. This splitting of the traditional infectious compartment is motivated by mounting evidence of asymptomatic transmission of SARS CoV-244,47,48,49,50. Thus, we consider two trajectories for the course of the disease, similar to those employed in prior studies42: (1) E → IA → R and (2) E → IA → IS → R (or death). More precisely, once in the E state, an individual will transition to the infectious state IA, at a per-person rate of ε. A proportion p will move from IA to the recovered state R (at a rate of p ⍵). A proportion (1—p) of individuals in the IA state will develop more severe systems and transition to Is (at a rate of (1—p) ⍵). Individuals in the Is state recover at a per-person rate of ν or die at a per-person rate μS. In each state, normal mortality of the individual occurs at the per-person rate μ and newly susceptible (S) individuals enter the population at a rate μN. The important differences between these two trajectories are in how likely an individual is to move down one path or another, how infectious individuals are (both for people and for the environment), how long individuals spend in each trajectory, and how likely death is along each trajectory.
Clarification on the interactions between hosts and reservoirs
The model couples the environment and people in two ways: (1) people can deposit the infectious virus onto environmental reservoirs (e.g. physical surfaces, and in the case of aerosols, the ambient air) and (2) people can become infected by interacting with these reservoirs. While most of our study is focused on physical surfaces, we also include data and analysis of SARS-CoV-2 survival in aerosols. While aerosols likely play a more significant role in person-to-person transmission, they also facilitate an indirect means of transmitting. For example, because SARS-CoV-2 can remain suspended in the air, other individuals can become infected without ever having to be in especially close physical proximity to the aerosol emitter (only requires that they interact with the same stagnant air, containing infectious aerosol particles)51. That is, a hypothetical infectious person A may produce aerosols, leave a setting, and those aerosols may infect a susceptible individual B who was never in close proximity to person A. In the transmission event between person A and person B, aerosol transmission functions in a similar fashion to surface transmission, where aerosols may be exchanged in the same room where infected individuals were, rather than exchanging infectious particles on a surface.
In our model, indirect infection via aerosols is encoded into the terms associated with the W component, just as the different physical surfaces are. Alternatively, aerosol transmission that leads to direct infection between hosts is encoded in the terms associated with direct infection between susceptible individuals and those infected (see section entitled Infection Trajectories).
Environmental reservoirs infect people through the βW term (Eqs. 1 and 2), a proxy for a standard transmission coefficient, corresponding specifically to the probability of successful infectious transmission from the environment reservoir to a susceptible individual (the full rate term being βWW·S). Hence, the βW factor is defined as the fraction of people who interact with the environment daily, per fraction of the environment, times the probability of transmitting infection from environmental reservoir to people. The factor βWW (where W is the fraction of environmental reservoirs infected) represents the daily fraction of people that will interact with the infected portion of the environment and become infected themselves. The full term βWW·S is thus the total number of infections caused by the environment per day.
In an analogous manner, we model the spread of infection to the environment with the two terms σAIA·(1—W) / N and σSIS·(1—W) / N representing deposition of infection to the environment by asymptomatic individuals, in the former, and symptomatic individuals, in the latter. In this case, σA (and analogously for σS) gives the fraction of surfaces/reservoirs that interact with people at least once per day, times the probability that a person (depending on whether they are in the IA or the IS compartment) will deposit an infectious viral load to the reservoir. Thus, σAIA/ N and σSIS/ N (where N is the total population of people) represent the daily fraction of the environment that interacts with asymptomatic and symptomatic individuals, respectively. Lastly, the additional factor of (1—W) gives the fraction of reservoirs in the environment that have the potential for becoming infected, and so σAIA·(1—W) / N (and analogously for IS) gives the fraction of the environment that becomes infected by people each day. We use W to represent a fraction of the environment, although one could also have multiplied the W equation by a value representing the total number of reservoirs in the environment (expected to remain constant throughout the course of the epidemic, assuming no intervention strategies).
Parameter values estimation
Table 1 displays information on the population definitions and initial values in the model. Tables 2 and 3 contain the fixed and estimated values and their sources (respectively). The model’s estimated parameters are based on model fits to 17 countries with the highest cumulative COVID-19 cases (of the 181 total countries affected) as of 03/30/2020, who have endured outbreaks that had developed for at least 30 days following the first day with ≥ 10 cumulative infected cases within each country14 (See supplementary information Tables S1–S3). In addition, we compare country fits of the SEIR-W model to fits with a standard SEIR model. Lastly, we compare how various iterations of these mathematical models compare to one another with regards to the general model dynamics. For additional details, see the supplementary information.
Table 2 Fixed parameter values estimated based on available published literature.
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Table 3 Estimated parameter values, averaged across countries.
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Basic reproductive ratios (({mathcal{R}}_{0}))
We can express the ({mathcal{R}}_{0}) (Eq. 7) in a form that makes explicit the contributions from the environment and from person-to-person interactions. In this way, the full ({mathcal{R}}_{0}) is observed to comprise two ({mathcal{R}}_{0}) sub-components: one the number of secondary infections caused by a single infected person through person-to-person contact alone (Rp) and the other is the number of secondary infections caused by exchanging infection with the environment (Re).
$${R}_{0}=frac{{R}_{p} + sqrt{{R}_{p}^{2} + 4 {R}_{e}^{2}}}{2}$$
(7)
where Rp and Re are defined in Eqs. (8a) and (8b)
$${R}_{p}=frac{varepsilon ({beta }_{A} ({mu }_{S }+ nu ) + {beta }_{S}(1 – p) omega )}{(mu + varepsilon )(mu + omega )({mu }_{S} + nu )}$$
(8a)
$${R}_{e}^{2}=frac{varepsilon { beta }_{W} ({sigma }_{A} ({mu }_{S }+ nu ) + {sigma }_{S}(1 – p) omega )}{k (mu + varepsilon )(mu + omega )({mu }_{S} + nu )}$$
(8b)
Note that when Rp = 0, the ({mathcal{R}}_{0}) reduces to Re and when Re = 0, the ({mathcal{R}}_{0}) reduces to Rp. Thus, when person-to-person transmission is set to zero, the ({mathcal{R}}_{0}) consists only of terms associated with transmission from the environment, and when transmission from the environment is set to zero, the ({mathcal{R}}_{0}) consists only of infection directly between people. When both routes of transmission are turned on, the two ({mathcal{R}}_{0})-components combine in the manner in Eq. (7).
While Re represents the component of the ({mathcal{R}}_{0}) formula associated with infection from the environment, the square of this quantity Re2 represents the expected number of people who become infected in the two-step infection process: people → environment → people, representing the flow of infection from people to the environment, and then from the environment to people. Thus, while Rp gives the expected number of people infected by a single infected person when the environmental transmission is turned off, Re2 gives the expected number of people infected by a single infected person by way of the environmental route exclusively, with no direct person-to-person transmission. Also note that Re2/(Re2 + Rp) can be used to measure the extent of transmission that is mediated by the environment exclusively. This proportion can be used as a proxy for how important environmental transmission is in a given setting. Elaboration on formulas 8a–b—and associated derivation-discussions—appear in the supplementary information. More
