Introduction

Models of viral spread have documented consistent factors and patterns that contribute to rapidity of spread in communities. Disease outbreaks spread globally by way of contact patterns among those who are infected and can be incredibly difficult to contain once a tipping point has been reached. Currently, the COVID-19 is spreading at an exponential rate across the globe and leaving nearly no communities untouched (Dong, Du, & Gardner, 2020)

In 2014, the Global Health Security Agenda (GHSA) presented by the World Health Organization (WHO) put forth plans to prevent avoidable epidemics with the goal of reducing the impact of naturally occuring, intentional, or accidental outbreaks of disease and also improving global access to medical and nonmedical countermeasures during health emergencies. Pandemic planning efforts have called for more representative models of preparedness planning for pandemics like the one we’re experiencing with COVID-19 that account for more nuances in human behavior and interaction patterns as well as socioeconomic considerations that could provide a better picture of how infectious disease spreads across communities and also increase the effectiveness of epidemic interventions.

Kermack & McKendrick (1927) presented an early basic model for modeling epidemics that used three independent classes to represent disease spread in a fixed community. These classes included those susceptible to the disease, those infected by the disease and those infected and then removed from the disease. Kermack & McKendrick’s SIR model assumes that individuals within a fixed population have an equal probability to become infected with the disease and they also interact with an equal number of people per unit of time.

Models like the SIR model are useful for predicting epidemic size once the epidemic is well underway, but often overestimate the final size of an epidemic as a result of their limitations (Stroud et al., 2006). Unfortunately, this type of simple mathematical model is limited in its ability to account for individual differences in human contact patterns among members of a community because they assume homogeneous mixing within a fixed population.

Some models have attempted to account for a heterogenous mixing of people in the early stages of an epidemic as a way to more effectively predict the final size of an epidemic (Del Valle, Hyman, & Chitnis, 2013). Del Valle et al.’s (2013) model does not assume that individuals within communities have an equal chance of interacting with any one person in their community. Instead, they assume that any individual will interact differently with others based on factors like age, living and work environment, and behavior traits.

Grim, Singer, Read, & Fisher (2015) have also presented important considerations for how the dynamics of social subcommunities can inform the timing and mechanisms of infection. Their findings suggest the way social communities and networks are structured, and not the degree of linkage between subnetworks, has sole influence on the time to total infection within the network structure (Grim et al., 2015). Therefore, there may be communities that are simply structured in such a way that they are more at risk to experience spread at a much quicker rate than other network structures. Indeed, a review of the studies exploring the effectiveness of travel bans, or banning certain groups from traveling to and from other countries, on slowing the spread of disease showed inconsistent conclusions (Errett, Sauer & Rutko, 2020). These findings might suggest that decreasing the number or linkages between a large amount of subnetworks might have little to no effect on slowing the rate of infection across country borders.

Socioeconomic Considerations for Models of Disease Spread

Farmer (1996) was among the first to suggest the frameworks used to explain the spread of infectious disease are also the same frameworks that obscure the origins of infectious disease. Social inequality can often explain both the genesis and maintenance of disease but the former is left unaccounted for in these frameworks (Farmer, 1996). Research has since stressed the need to account for the socioeconomic factors that drive health inequities while organizations like the WHO and governmental entities are planning for global epidemics. Within communities, those who are socially disadvantaged carry a greater burden of morbidity and mortality from infectious disease because they are often the community members who are most susceptible, most exposed, and least likely to have access to care (Cordoba & Aiello, 22016; Quinn & Kumar, 2014).

Members of lower classes experience structural disadvantages like living in apartment buildings, in a metro area, or with more people in the household and also occupational disadvantages that make it difficult for them to stay home from work due to lack of sick leave or not being able to work from home. These individuals also might find it more difficult to avoid public transportation or find appropriate daycare options which also increase their chances of exposure to infectious disease. Those with low socioeconomic status are also more susceptible to infectious disease due to rates of chronic illness, like heart disease, diabetes, immunosuppression, high blood pressure, and lung disease which are often a result of factors related to class status. Finally, these individuals are also less likely to have a regular healthcare provider and either lack the insurance or money to afford preventative healthcare options (Quinn & Kumar, 2014).

Epidemic and pandemic interventions naturally operate with the goal of reaching a level of global herd immunity to minimize and slow the impact of an infectious disease. Herd immunity occurs when a sufficiently large number of people within a population become immune to the disease, either through vaccination and/or prior illness to make the spread of the disease less likely, which also protects unvaccinated members of a population given that the disease has less opportunity to spread (CDC, 2016). Socioeconomic factors may present important challenges for the feasibility of herd immunity, such that if a particular class of individuals is directly or indirectly neglected in efforts to minimize disease spread, then the disease may continue to spread to other classes, which would have otherwise become herd-immune.

Drawing on literature examining mathematical models of disease spread, information networks and structures, as well as social determinants of health, we propose a model of viral spread with the intention of accounting for the effects of a socially stratified network structure. The present study explores the mechanics of this model.

I. Agent Properties

Our model consists of two kinds of agents, “buildings” and “people.” Every agent has an ordered pair representing their location in two-dimensional cityspace.

People are stratified into three classes - lower, middle, and upper. Buildings are stratified into four classes - lower, middle, upper, and industrial. These are assigned at setup. People are also divided into three health categories - susceptible, infected, and recovered. Thus, our model represents the same three categories of health modeled in the SIR model (Kermack & McKendrick, 1927). At setup, all people are susceptible, except one, who is infected.

Buildings have two properties, neighborhood and business. They are assigned at setup. The neighborhood of the building determines its quadrant on the cityspace. The neighborhood inhabited by people of the lower class and by the upper class are on opposing quadrants, with the middle class neighborhood and the industrial neighborhood - where nobody lives - between them. All industrial status buildings are always businesses. The uninhabited buildings in the residential neighborhoods may become businesses, but often are not.

II. Setup

Before days begin to cycle, the buildings and population spawn. Our model holds the distribution and location of buildings constant, as well as the distribution of the people. People proceed to become stratified among the three classes. In our experiment, 25% of people are in the lower class, 50% are in the middle class, and 25% are in the upper class.

People form a link with one randomly-selected building whose status matches their class - they “inhabit” their “domiciles.” This leaves some buildings uninhabited, but no homeless people. It also leaves some number of buildings with multiple inhabitants.

Uninhabited buildings, with some probability “economy”, are buildings. Businesses then form “employment” links with randomly-selected people, which may cross classes. In our experiment, “economy” is held constant at .20.

Employment established as follows. Each business forms an employment link with a number of employees whose class resides in their neighborhood. This number is between one and some number “same-status-workers.” Industrial buildings do the same for people of the upper class.

Businesses then do the same for people whose class resides in the next-highest neighborhood, with the variable “lower-status-workers.” (For businesses in the lower-class neighborhood, there are no “lower status workers” to employ at this step.) Industrial buildings employ middle-class people with the same process, using the variable “internships.”

Finally, one person becomes infected. The dynamics then begin.

III. Dynamics

After setup, days begin to cycle. They consist of four “phases” that represent opportunities to transmit their infection through “interactions.” The first phase in the day, “morning,” begins with agents at their domiciles, where they interact, spreading disease.

In the second phase, “first shift,” businesses “call in workers,” selecting some random number from one to some number called “staff”, designated by the observer. The selected people then set their location equal to that of the business, and interact. In the third phase, non-industrial businesses then call in workers again for a “second shift,” which may or not be the same people. People who were not called for the second shift remain in their prior location. Another interaction occurs.

In the fourth phase, people “mingle” with others within their class and interact. Finally, they return home, interact once more, and their illnesses progress.

During interaction, people move some number called “feet-moved” in a random direction. After that, agents within a radius of some number called “proximity” become sick at some probability, determined by class of recipient. To model availability of masks, people of the middle class have a times-two multiplier on their probability of getting infected; the lower class has a times-three multiplier. The virus can only be transmitted from sick people to susceptible people; immune people cannot transmit. In our experiment, feet-moved is held constant at 3, and proximity is held constant at 3.

During progression, sick people become immune with some probability, “immunity.”

During mingling, buildings “invite” one random person to their location, with some probability, “social-density.” This person has their class match the building’s status. Industrial-status buildings do not invite people during mingling. In our experiment, social-density was held constant at .2.

The model converges when there are no infected people - only susceptible and immune.

IV. Representation

As previously mentioned, buildings are assigned neighborhoods based on their status. Buildings with business = 0 are black and have size = 1. Non-industrial businesses are blue and have size = 2. Industrial buildings are black and have size = 5. Lower-status buildings are twice as dense as middle- and upper-class statuses, and industrial-class statuses are distributed even further apart.

Lower class people are represented by triangles, middle class people by pentagons, and upper class people circles. Susceptible people are colored gray, infected people yellow, and immune people purple.

Employment links and inhabitant links may be cleared or represented by a line, as the user chooses.

V. Experiment Parameters

To explore how some parameters interact, we ran a parameter sweep in NetLogo 6.1.1. Many parameters were varied, but many were held constant as well. Population was held constant at 200; the distribution of lower, middle, and upper class were held at .25, .50, .25 respectively; economy at .2; social-density at .5, proximity at 6; feet-moved at 3; and staff at 3. The effects of class on transmission probability are as follows. Lower-class people receive transmission with p = transmissivity; middle class people with p = transmissivity * ⅔, and upper class people with p = transmissivity * ⅓. Some variables are hard-coded in and therefore cannot be easily manipulated with code attached.

The independent variables were as follows. Immunity varied between .02, .03, and .04; transmissivity varied over .01, .03, .05, 07, and .10; lower-status-workers at 0, 3, and 6; internships at 0, 2; and same-status-workers at 4, 6. We also ran control trials, where only one class of people exist, as well as only the buildings corresponding to their potential workplaces, to compare viral spread among linked subgraphs. On these trials, lower-status-workers and internships = 0. Each trial was run with twenty replicates for a total of 34,560 runs.

The dependent variable gathered was “herd,” which is the percentage of the population still susceptible to the disease. Herd was calculated for each class each day, creating viral dynamic graphs that show how the disease spreads. In addition to viral dynamics, we calculated many relevant figures.

We also calculated the number of days it takes for the model to converge across trials, and compared the herd between classes at every trial.

Results

A. Control Trials

Figure 1 is shown above, and represents 20 replicates of one condition in a control trial, where transmissivity = .01 and immunity = .04. It is a “viral dynamic chart,” illustrating how each class individually converges. The x-axis denotes days, and the y-axis denotes herd. Color represents class; Red represents lower-class herd, blue middle-class, and green upper-class. Note that the x-axis is a logarithmic scale.

These data represent a control trial, with classes on graphs disconnected from other classes or statuses. Thus, they provide no information concerning viral spread between classes. In experimental trials, classes will have links between them allowing the virus to spread. Here we are less interested in results, and more interested in setting a baseline to contrast with the experimental trials, as well as familiarizing the reader with the structure of our charts.

Connected lines represent same-status-workers = 4; dotted lines represent same-status-workers = 6. Same-status-workers can be interpreted as a degree of connectivity within a subchart. In experimental trials, this variable manipulates the number of times workers will work in their own neighborhood rather than another. In control trials, the difference is slight, but worth showing for comparative reasons.

Differences between the classes are as follows. Relative to the middle class, the lower status neighborhood has one half the population and twice the density of buildings in the same space. The upper class have the following differences relative to the middle class. Their population is also half, but their density of buildings is the same. Upper class individuals are also employed downtown, while the middle class are not in control trials. Recall that people in the lower class are twice as likely to become infected per interaction relative to the middle class, and that people in the upper class are one-half as likely. Taken together, these explain the difference between class behavior.

Figure 2.1.1 above shows the median of each point in Figure 1. Since each trial terminates after convergence, the medians are taken from the non-converged trials. This explains why some trend lines show an uptick in herd - because enough runs have terminated to shift the median to a trial with a larger herd value.

Figure 2, shown above, represents viral dynamic charts over many conditions. Figure 2.1.1 can be found in the bottom row and the leftmost column of Figure 2. This generalizes for all subcharts - their figure name can be interpreted as coordinates in the superchart space. This, Figure 2.2.2 represents immunity = .02 and transmissivity = .02. Taken together, immunity and transmissivity is a metric of “viral strength.”

Since transmissivity is lowest in the leftmost column, and immunity the bottommost row, Figure 2.1.1 represents minimal viral strength, and Figure 2.6.3 represents maximal viral strength. It demonstrates that, when viral strength is sufficiently strong, herd = 0 for most replicates. It also shows that in most runs, the middle and lower class converge at similar values. This effect becomes more pronounced as viral strength increases.

B. Experimental Trials

In Figures 1 and 2, we analyzed viral spread within individual classes, but not between classes. Herd at the endpoint was poorly represented, as noted above. Figure 3.1.1 above displays how endpoints differ between classes, as well as with a full society. Transmissivity is held constant at .01 and immunity at .04. The middle column has job-mixing = 3, and the rightmost column has job-mixing = 6.

The lower graphs indicate a control trial, indicated on the right with “D” meaning disconnected graphs and “C” meaning connected. Manipulating job mixing in control trials decreases time people spend working with others in the same neighborhood, and their chance of transmission generally. This has an observable effect - in control trials, all classes benefit from increased job-mixing.

In control trials, each class begins with one infected person and spreads throughout that class. In experimental trials, the disease must flow from the upper class to the middle and lower classes. This is represented in the chart, where one observes that the lower and middle classes in the control trials frequently end with herd = 1, and the disease failed. The effect is still visible at job-mixing = 6.

In experimental trials when job-mixing and internships = 0, there are almost no interactions between the classes, so the disease only spreads on the chance that two people walk towards each others’ neighborhood with a high degree of directedness and are significantly close to each other to interact. The data above show that, in no replicates, this occurred and resulted in transmission.

Figure 3, shown above, extends this analysis. Figure 1.1.1 consists of the bottom-left three charts and the three charts above them. Figure 3.4.3 consists of the upper-right three charts and the three below them. As one would expect, increases in transmissivity and decreases in immunity result in higher terminal herd values.

In some experimental trials, one can observe transmission between the classes even when job-mixing = 0. For example, when immunity = .02 and transmissivity = .03, .05, or .07, two replicates have terminal herd values < 1.

Within each experimental trial, job-mixing negatively correlates with herd in every class. For the middle and lower classes, the reason is obvious - there are more opportunities for interaction with the upper class, where the disease starts. The same results can be found in the upper class, because the disease spreads back from the lower-class to the upper.

Figure 4.1.1 above is a viral dynamic chart, comparing control trials to experimental trials. The bottom three graphs, which are control trials, show the same data as Figure 2.1.1 through Figure 2.3.1, where job-mixing has minimal effect. In the experimental trials, job-mixing shows the same patterns as in endpoint-analysis.. In the control trials, job-mixing did not substantially impact endpoints for the lower and middle classes, but did impact the time to convergence. It did significantly impact the endpoints for the upper class, which were all relatively high.

In the experimental graphs, , job-mixing = 0 resulted in no infections in the middle or lower class, as before. As job-mixing increased from 3 to 6, there was no effect on the lower or upper class, but the middle class did have its infection rate slow, the inverse results of the control trial.

This may result from the middle class’ increased work in the upper-class neighborhood, more interactions there and fewer happen in the between members of the middle class. Similarly, as the lower class gets called in to work in the middle-class neighborhood, more interactions happen between those classes, while the lower class has a lower rate of infection before the disease takes hold there.

Figures 5.1.1 and 6.1.1 show the herd at convergence and dynamics of graphs with and without internships, which increase the connectivity between the middle and lower class. Without internships, job-mixing = 0 leads to effectively zero opportunities for transmission from the upper class. However, with internships, job-mixing easily allows inter-class transmission. Interestingly, with internships, job-mixing = 3 is worse than both = 0 and = 6 for all classes. The mechanisms underlying this disparity are still to be explored. Also attached are Figures 4 and 5, which display the same patterns with no interesting interactions.