2.2. Model Description
The model is implemented via a set of ordinary differential equations, defined by Equations (1)–(6). It implements viral free-living survival via the “Waterborne Abiotic or other Indirect Transmission (WAIT)” modelling framework, coupling individuals and the pathogen within the environment [23,24].
Within the model, the βw term allows for individuals to become infected via viral pathogen deposited in the environment and terms 𝜎A and 𝜎I 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]. (1) dSdt=μN−S−βAA+βIIN+βWWS (2) dEdt=βAA+βIIN+βWWS−ε+μE (3) dAdt=εE−ω+μA (4) dIdt=1−pωA−v+μII (5) dRdt=pωA+vI−μR (6) dWdt=σAA+σIIN1−W−kW
Figure 1 depicts the compartmental diagram for the model. The direction of the arrows corresponds to the flow of the individuals and the pathogen through the system. Note that individuals can move directly from the asymptomatically infected compartment to the recovered compartment (bypassing the symptomatic compartment) via what we call a “mild track”. The dashed arrows represent WAIT coupling to the environment. The model is inspired by one developed to interrogate environmental transmission of SARS-CoV-2 [22].