By Cesar Alzate
COVID-19 has a high transmission rate due to its novelty. The pathogen’s ease of transmission required quick governmental action, and news about COVID-19 was soon spreading in early 2020 as quickly as the virus itself. Shortly after the first cases appeared in the U.S., everyone was receiving public service announcements (PSAs) and adapting to government control methods like lockdowns and masks. This article will demonstrate how COVID-19 preventative methods help mitigation efforts and explore whether the disease’s growth rate is an accurate description of reality.
COVID-19 Growth Rate
COVID-19 is an infectious disease, which means that its growth rate is proportional to infected individuals. Its high transmission rate suggests that we can use the exponential growth rate function that is defined as
\[N(t)=e^{kt} \times N(0),\]
where \(N(t)\) is the number of total confirmed cases at time \(t\), \(k\) denotes the growth rate of COVID-19, and \(N(0)\) represents the initial number of confirmed cases. Because not everyone is tested, especially in the early stage of the pandemic, there is speculation that the growth rate is skewed incorrectly. In response, we can append a function \(F(t)\) that ranges from \(0\) to \(1\) and denotes testing capability. In this case, \(0\) means that no one is getting tested and \(1\) means that everyone is getting tested. Both scenarios are unrealistic, and we can expect \(F(t)\) to lie somewhere between \(0\) and \(1\). We also assume that the growth rate \(k\) is a function of time \(t\), meaning that the number of infected cases is now
\[N(t)=e^{k(t)\times t} \times N(0) \times F(t).\]
Even with the incomplete capability of \(F(t)\), the estimator of growth rate \(k(t)\) will become independent and negligible by the calculation of \(\log(N(t))\) since
\[\log(N(t))= \log(F(t))+k(t)\times t.\]
To understand the COVID-19 situation in North Carolina in 2020, we examined the state’s seven major metropolitan areas. The COVID-19 Data Repository by the Center for Systems Science and Engineering at Johns Hopkins University automatically collected COVID-19 data on daily confirmed cases for these metros.
Figure 1. Growth rates of COVID-19 within the Charlotte-Concord-Gastonia metro. Figure courtesy of the author.
The average incubation period of COVID-19 is roughly seven days [1], and we used this figure globally throughout our study. We split the timeline of COVID-19 cases into the following four events in 2020 that corresponded to governmental policies:
- Event One (no restriction): Date of the first reported COVID-19 case \(+\) incubation period (seven days) to March 17 \(+\) incubation period: Executive Order No. 118
- Event Two (mild restriction): March 18 \(+\) incubation period to April 8 \(+\) incubation period: Release of initial stay-at-home order
- Event Three (strict restriction): April 9 \(+\) incubation period to May 7 \(+\) incubation period: Executive order No. 131
- Event Four (reopen): May 8 to July 29: Businesses reopen.
Figure 1 displays the results for the Charlotte-Concord-Gastonia metro, which is the biggest metro within North Carolina. This metro’s growth rate is the following weighted sum that re-normalizes the population of each county within the metro:
\[\textrm{Weighted Growth Rate} = \sum_{\textrm{County}} k(t) \times \frac{\textrm{Population County}}{\textrm{Total Population of Metro}}.\]
After the initial COVID-19 infection, growth rates grew dramatically and then decayed over time. We see a low growth rate during the reopening phase, which might seem unexpected. However, this is a proper result if we consider the number of people who were already wearing masks and acknowledge that most of the public was informed about the pandemic and aware of strategies to reduce spread.
Removal Rates Using the Time-delayed SIR Model
A simple definition of “removal rate” is the rate at which individuals with COVID-19 are removed from the population. An accurate calculation of the removal rate allows us to measure COVID-19 preventative methods.
We consider a time-delayed susceptible-infected-recovered (SIR) model for the spread of disease to calculate the removal rate. Let \(t\) be the number of passing days since the condition first occurs. \(S(t)\) denotes the number of susceptible individuals, \(I(t)\) denotes the number of infected individuals, and \(R(t)\) denotes the number of removed individuals. The resulting time-delayed SIR model reads
\[\frac{dS(t)}{dt}=-\beta I(t-\tau_1)S(t),\]
\[\frac{dI(t)}{dt}=\beta I(t-\tau_1)S(t)-\gamma I(t-\tau_2),\]
\[\frac{dR(t)}{dt}=\gamma I(t-\tau_2),\]
Figure 2. Wilmington growth rate with the time-delayed susceptible-infected-recovered (SIR) model. Figure courtesy of the author.
where \(\tau_1\) is the average number of days of the incubation period and \(\tau_2\) is the average number of days of removal. We assume that \(\tau_1\ge \tau_2\) and that the individual will usually be self-isolated when symptoms are manifesting. The parameter \(\beta\) is the average infection rate and \(
\gamma\) is the average removal rate. We aimed to determine the removal rate for each metro region and test them against the model.
To implement our model, we employ the MATLAB discrete differential equation solver dde23 and fit the parameters based on actual data. Doing so yielded an accurate estimate of the removal rate \(
\gamma\) within each metro region. Tuning \(\gamma\) allowed the growth rate of the SIR model (pink line in Figure 2) to accurately depict the observed COVID-19 growth.
After attaining a reasonable fit for our model, we compared various metros to determine the success of government control methods. Figure 3 summarizes the removal results. To determine their accuracy, we can see how well the pink line (growth rate of the SIR model) in Figure 2 matches the actual confirmed cases. Studying the removal rates of our time-delayed SIR model allows us to evaluate the effect of government mandates on each metro and different stages of reopening, since effective government policy should raise removal rates. Without government intervention, significantly more spikes would have likely occurred near the first case, which would have ultimately resulted in over-hospitalization.
Despite its large population size and high level of confirmed cases, Charlotte-Concord-Gastonia had consistently high removal rates (see Figure 3). Charlotte has a large working-class population of white-collar workers, and we suspect that a significant portion of the metro adapted quickly to the pandemic since working from home was very feasible for many of these residents’ jobs. In addition, most metros had high removal rates upon the initial lockdown as compared to the pre-lockdown period, suggesting that this lockdown—among other actions by the state—effectively reduced the breakout spread of COVID-19.
Wilmington had the most notable results in terms of removal rates, in that that the number of post-lockdown cases soared. Wilmington was also somehow able to lower its removal rate upon reopening. Lower removal rates might seem contradictory until one considers the area’s geographical properties. Wilmington is near the ocean and tourism is vital for its economy. The data suggests that Wilmington did not do well to remove infected people due to the high levels of tourism. The lockdown helped slow the growth rate, but this metro area presumably needed to undergo further control measures — such as higher hygiene protocols and fever tests for tourists. We can also draw a similar conclusion for Jacksonville, as it has many nearby beaches.
Figure 3. Removal rates of the seven metro regions in North Carolina at each event period. Figure courtesy of the author.
Conclusion
Our SIR model was able to identify the high risk of the beach/tourist areas in North Carolina when compared to other metros. In future work, estimating the removal rate with this time-delayed SIR model could provide data on the way in which mask-wearing or even vaccinations have affected COVID-19 transmission. For example, face masks became required on June 26, 2020 in public spaces across the state. Adding another event into this study would allow us to assess this control measure’s effect on COVID-19 growth and removal rates. It is also important to note that our study stops checking cases just 10 days after the mask mandate, even though masks, PSAs, and other preventative measures were still in play.
Ultimately, applying the method in this article would allow governments to tailor their policies to specific metros rather than the entire state.
Cesar Alzate presented this research during a minisymposium presentation at the 2021 SIAM Conference on Computational Science and Engineering, which took place virtually in March.
Acknowledgments: Cesar Alzate wishes to thank his mentor, Xingjie Li of the University of North Carolina at Charlotte, for their joint efforts in this research project.
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Cesar Alzate currently studies computer engineering at the University of North Carolina at Charlotte and is in his junior year. While he was completing his associate degree at Mitchell Community College, he served as a physics tutor. Alzate hopes to work in the aerospace industry after completing graduate school. |