# Modeling Epidemic Spread among a Commuting Population Using Transport Schemes

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## Abstract

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## 1. Introduction

## 2. Discrete Community Metapopulation Model

#### 2.1. Commuter Traffic as a Transport Scheme: Discrete Case

#### 2.2. Network SEIR Based on Transport Schemes

#### 2.3. Network-Based SEAIR and SE(A)IR Models for COVID-19

## 3. Continuously Distributed Model

## 4. Results

#### 4.1. Construction of the Transport Scheme

#### 4.2. Network Correction: Greater Cleveland Region

#### 4.2.1. Bayesian Parameter Estimation

#### 4.2.2. Network Correction to Parameter Estimates

#### 4.3. Pittsburgh–Cleveland–Detroit Corridor

**The role of the initial condition:**Our first simulation starts with a single infected individual in the low population density Huron County, MI (population: 31,166). Figure 8 shows the results of this simulation with our metapopulation models for the 74 counties in the corridor over a period of 21 weeks. In the first few weeks the infection spreads slowly to other counties, while never becoming highly prevalent in the county where it initiated, as shown in the top row. However, once the infection reaches the three-county core of Macomb, Oakland, and Wayne constituting the Detroit metropolitan region, the high population density allows the virus to quickly spread within the region. The three major population centers in the corridor, comprising Macomb/Oakland/Wayne, Cuyahoga (Cleveland), and Allegheny (Pittsburgh) counties, are tightly connected via the contact matrix. Therefore, once the virus takes hold in any one of the three metropolitan areas, it quickly reaches the other two centers, where it spreads rapidly due to the high population density and large number of susceptible individuals. After having run its course in the high population counties, the infection then grows dramatically in the surrounding counties, facilitated by the presence of interstates and major highways, eventually reaching also rural and more remote counties. As the infection hotspots move to rural counties, regions with high population density begin to recover.

**Mitigation by restricting travel to/from counties in a state with high prevalence:**One of the state-level strategies considered to slow down the spread of COVID-19 was to reduce traffic between neighboring states and counties. To see how, according to our model, such measures affect the spread of the infection, we dynamically modified the contact matrix $\mathsf{N}$ to reflect the reduction in traffic in and out of the counties in states experiencing spikes in infections. More specifically, when the prevalence of the infected population in the counties of interest in Michigan, Ohio, or Pennsylvania reaches 10% of the total population, traffic into and out of every county in the state is decreased by 95% of its original level, simulating a ‘lockdown’ scenario. Figure 7 displays how the contact matrix changes in response to state shutdowns. Figure 11 shows the model predictions for an initial infection in Stark County, OH, with the described mitigation measures. In the first few weeks, while the prevalence is still low, the pattern is the same as in the unmitigated case, but by week 6 the spread in Ohio slows down after the high prevalence in Cuyahoga County has triggered the reduced traffic scenario statewide. Generally, if the mitigation measures are at the state level, the infection must reach the more densely populated counties of the three states before triggering traffic reductions which slow the transmission of the disease, though it does little to reduce the prevalence of the infections in the individual counties once the disease has taken hold.

**Local mitigation measures in counties with high prevalence:**In this simulation, we consider a mitigation strategy targeting selected local hotspots, in which strict social distancing measures are introduced in counties with prevalence greater than or equal to 15%. To understand how the spread pattern is affected by the triggered local mitigation measures, we run a simulation starting with a single infected individual in Stark County, dynamically reducing the internal contact frequency by 50% in a county as soon as the infected population is more than 15% of its population. The results, displayed in Figure 12, show that the infection initially spreads in a similar manner to what was observed in the unrestricted traffic mitigation. However, because the increased prevalence triggers the local mitigation measures, the peaks of infections are not as high as in the unmitigated case, as fewer people are infected at the same time. This in turn leads to a longer period of elevated incidence, as opposed to a quick spike in the number of infected. Overall this results in more counties having an elevated number of infected people, but without the sharp spikes in individual counties observed in the unmitigated cases.

**Combined triggered local mitigation and travel restriction:**The effects of local mitigation by triggered reduced mobility and contact frequency can be seen in Figure 13, illustrating how these components together change the temporal dynamics of the virus over the 74 counties in the corridor. Noticeable are the lower peaks and longer period of elevated incidence due to the reduced contact frequency, as well as the slower spread of the virus over the 74 county regions due to reduced traffic to and from high prevalence counties. This highlights how the combination of the two mitigation strategies has the dual effect of softening the increase in new infections in each individual county by reducing local contacts, while simultaneously slowing down the spread across the region by reducing commuting traffic.

#### 4.4. Simulation with Continuous, Spatially Distributed Model

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the transport scheme across six communities. In this cartoon, the dots represent entries of the matrix of the transport scheme, with the size proportional to the radius of the dot. The red histogram ${\pi}^{\left(H\right)}$, which is the marginal distribution of the contact matrix with respect to the row index, represents the distribution of the population according to the census definition of usual residence, and the blue histogram ${\pi}^{\left(W\right)}$, the marginal with respect to the row index, is the population density during the work hours. Here, ${C}_{4}$ is an example of a center attracting population from other communities, while, for instance, ${C}_{6}$ is a bedroom community with low work place density, the principal work places being in ${C}_{1}$ and ${C}_{4}$.

**Figure 2.**Susceptible resident of the community ${C}_{j}$ in the cohort ${S}_{j}$ may be exposed and infected either in the work environment (flux ${\phi}_{j}$) or off work in the home community (flux ${\psi}_{j}$). The total transmission flux out of the susceptible population is a convex combination of these two fluxes, with coefficients that depend on the number of hours spent at work. In this diagram, the flux corresponding to a positive death rate is omitted.

**Figure 3.**(

**a**) The compartment diagram of the SEAIR model in which the infected and infectious compartment has been split into the asymptomatic (${A}_{j}$) and symptomatic (${I}_{j}$) compartments. (

**b**) The SE(A)IR compartment diagram in which the exposed non-infective cohort ${E}_{j}$ and the infective, infectious and asymptomatic compartments ${A}_{j}$ are merged into a hidden compartment $E{\left(A\right)}_{j}$ of which only indirect inference is available, as long as the data consist of daily numbers of newly infected lab-confirmed cases.

**Figure 4.**The conditional densities ${\pi}^{(W\mid H)}$ county by county in the northeast Ohio network comprising the seven counties shown in the map. The plots are on a logarithmic scale, and the total population of each county is indicated in parenthesis.

**Figure 5.**Infection rates ${B}^{(\ell )}\left(t\right)$ in the seven-county network during the first 100 days after the first recorded case in Cuyahoga county. The panels show the ensemble average (black curve) and the envelopes of two standard deviations of the independent county-by-county estimates computed by the PF algorithm, and the weighted least squares estimates (red curve) corresponding to the value of $\delta =0.02223$ that minimizes the residual over the diagonal entries. Observe that the estimation process of $X\left(t\right)$ (red curve) starts as soon as each of the seven counties has had at least one COVID-19 case (Portage county was the last one) to guarantee that the vector of individual county estimates for the infection rate is defined.

**Figure 6.**The residual $r\left(\delta \right)$ defined by the Formula (55), plotted over the interval $[{10}^{-4},10]$. The minimizer is indicated by the dashed line.

**Figure 7.**Contact matrices for the 74 counties in the Cleveland–Detroit–Pittsburgh corridor, plotted in a logarithmic scale. Each county corresponds to a row/column of the matrix, with the counties grouped by state, in the order: Michigan, Ohio, Pennsylvania and West Virginia, the latter being represented by only a few counties in the northern panhandle. The intensity of $(i,j)$th pixel is proportional to the number of individuals commuting from the jth country to the ith county. The top left panel shows the commuting patterns prior to the pandemic: the counties in the states of Michigan and Pennsylvania are more tightly internally connected than those in Ohio, as indicated by the intensity of the pixels in the first and third diagonal blocks, and it is clear that there is substantial commute among the three larger metropolitan areas. The remaining panels show how the communication network changes when the mobility to/from counties in specific states is reduced by 95%.

**Figure 8.**Simulation of the spread of the infection starting in the low-density Huron County, MI, marked by a star in Week 0. The color code indicates the number of infected individuals per 100,000 inhabitants. Observe that the infection level remains low until it reaches the nearest high density metropolitan area, but afterwards propagating quickly to the other centers. The pattern of the evolution of the infection shows that metropolitan areas work as amplifiers, rendering the second wave in the surrounding areas much stronger than the initial infection. The rural areas experience the second wave when the metropolitan areas are already recovering, in agreement with what was observed in the United States in the late summer of 2020.

**Figure 9.**Simulation starting spread beginning in the medium population Stark County, OH, marked by a star in Week 0. As in the previous simulation, the initial infection remains almost unnoticed until it reaches the nearest metropolitan area, from which it quickly spreads to the other high density communities. Again, the metropolitan centers act like amplifiers, rendering the second wave in the initial point more intense than the initial infection. The timeline is similar to the previous simulation.

**Figure 10.**Simulation of the spatial pattern of the spread when starting with a single infected individual in the large population Allegheny County, PA (Pittsburgh area). The infection quickly spreads to the neighboring communities, which concurrently reaches the other two metropolitan areas. While the geographic pattern is similar to the two previous simulations, the timeline for the spread of the infection is faster.

**Figure 11.**Time course of the spread of the infection under mitigation by drastically reducing traffic to and from the counties in a state with 10% prevalence. This simulation was started with a single infected individual in Stark County, OH (marked with a star in the first panel).

**Figure 12.**Time course of the spread of the infection under mitigation by triggered contact reduction in counties as soon as the prevalence reaches 15% of the county population. This simulation was started with a single infected individual in Stark County, OH (marked with a star in the first panel).

**Figure 13.**Time course of the spread of the infection under combined mitigation by triggered contact reduction in individual counties as soon as the prevalence reaches 15% of the county population accompanied by reduction in traffic to and from the affected states. This simulation was started with a single infected individual in Stark County, OH (marked with a star in the first panel).

**Figure 14.**Propagation of the infection. The top left panel shows the population density, in arbitrary units, with the location of the first infection marked by a red dot. Snapshots of the time history of the function ${\nu}^{I}(x,t)$ are shown in lexicographical order. Observe the different color scale in the pictures. At $t=15$, the infection has reached the nearby urban area, and at $t=28$ it peaks in all densely populated centers. At $t=40$, the small center located in bottom left of the map reaches its peak, and at $t=52$, when the infection in the urban areas has already passed, the rural areas peak.

**Figure 15.**Time traces of the frequencies ${\nu}^{S}$ (left), ${\nu}^{E}$ (middle) and ${\nu}^{I}$ (right) at three selected locations: Referring to the coordinates in Figure 14 as matrix entries with row and column indices $(i,j)$, the point labeled as “Rural” is at $(i,j)=(10,10)$, the point labeled as “Urbanized” is at $(i,j)=(30,13)$, and the point “Urban Cluster” is at $(i,j)=(52,11)$. We observe that the infection peak times increase as the local population density decreases. Interestingly, in the urban cluster, the percentage of eventually infected individuals is lower than in the rural community, indicating that the dynamics depends on the connectivity and not only on the local density.

**Figure 16.**New case count over the entire area. Observe that the new case count increases exponentially, but the decay rate of new cases is lower, with an almost linear segment which is well in line with the observed case count curves, in disagreement with the classical Farr’s law stating that the rise and fall behavior is symmetric. The different behavior of the rise and the fall comes from the spatial component of the model and the fact that the population is not well-mixed.

Parameter | Symbol | Value |
---|---|---|

infection rate | $\beta $ | $0.01$ |

daily contacts | r | $20,40,60$ |

incubation rate | $\eta $ | $1/7$ |

recovery rate | $\gamma $ | $1/14$ |

death rate | $\mu $ | $0.0035$ |

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**MDPI and ACS Style**

Calvetti, D.; Hoover, A.P.; Rose, J.; Somersalo, E.
Modeling Epidemic Spread among a Commuting Population Using Transport Schemes. *Mathematics* **2021**, *9*, 1861.
https://doi.org/10.3390/math9161861

**AMA Style**

Calvetti D, Hoover AP, Rose J, Somersalo E.
Modeling Epidemic Spread among a Commuting Population Using Transport Schemes. *Mathematics*. 2021; 9(16):1861.
https://doi.org/10.3390/math9161861

**Chicago/Turabian Style**

Calvetti, Daniela, Alexander P. Hoover, Johnie Rose, and Erkki Somersalo.
2021. "Modeling Epidemic Spread among a Commuting Population Using Transport Schemes" *Mathematics* 9, no. 16: 1861.
https://doi.org/10.3390/math9161861