By Samuel Awoniyi
Researchers are currently generating numerous mathematical models that predict both the proliferation and control of the ongoing COVID-19 pandemic. Here I employ Markov chain modeling to compare two possible systems for curbing COVID-19 spread in the U.S. The country is presently utilizing one of these systems, while the other is a proposed alternative system that features a “corps of suppliers and caregivers” to address certain imperfections within the current framework.
My group has designed the alternative system to more effectively control the rate of infection, demand on existing healthcare facilities, overall costs, and disruption to the national economy. One could initiate this method at both the state or community level with only a few weeks of planning, either as a follow-up to—or replacement for—the current system.
The Markov chain model aims to demonstrate the sound mathematical explanation of the proposed alternative system’s superiority. This is followed by a section on practical implementation that would benefit U.S. decision-makers, including state governors’ staff, county commissioners’ personnel, and school system planning committees, among other groups.
Description of the Two Potential Systems
Figure 1 depicts the two possible systems for COVID-19 mitigation in the U.S. In both systems, set B consists of all people in the country who currently test positive for COVID-19, including everyone who is presently hospitalized with COVID-19 symptoms. In the existing system, set A consists of the remaining U.S. population (or community of interest). However, set A is slightly smaller in the alternative method, as it comprises the rest of the population minus set C. And set C encompasses people who are officially commissioned as corps of suppliers and caregivers for everyone in set B.
Figure 1. Two possible systems for limiting COVID-19 spread in the U.S. The spongy look of set A is designed to reflect the malleable nature of the country’s relatively free society. Figure courtesy of Samuel Awoniyi.
The official function of individuals in set C is to coordinate and deliver the needs of those in set B, such as food and healthcare supplies, temporary housing, and hospitalization. To minimize infection probability in the proposed system, people in set C—henceforth called “the Red Corps”—will always have all necessary personal protective equipment” (PPE). Detailed guidelines for in-person contacts in this alternative framework are as follows:
- Guideline #1: Any person who tests positive for COVID-19 shall promptly move into set B as soon as the test result is known.
- Guideline #2: Individuals in set A shall not have any direct, in-person contact with those in set B.
- Guideline #3: Everyone in set B shall wear a preventive face mask during all in-person interactions.
- Guideline #4: Every member of the Red Corps (set C) shall limit their in-person interactions with people in set B to essential deliveries of needed supplies and healthcare resources.
- Guideline #5: Any person in set B shall move to set A after fully recovering from COVID-19; this might include possessing the requisite antibodies.
My team’s proposed alternative system somewhat resembles a compartmental model in the area of infectious disease modeling [3]. However, use of a Markov chain sojourn time cycle [1]—rather than mathematical analysis, as with typical compartmental models—makes our modeling efforts unique.
Two Corresponding Markov Chain Models
We utilize two discrete-time Markov chain models, based on the previous guidelines, to explain the superiority of our alternative system. Each model tracks the way in which COVID-19—behaving like a traveling deliverer of harmful packages—moves back and forth between sets B and A in the current system, and sets B and C in the proposed alternative system. This application justifies requisite Markov assumptions because COVID-19’s infection rate depends only on society’s diligence in following the five aforementioned guidelines.
Figure 2 depicts two discrete-time Markov chains (DTMCs). The meaning of states A, B, and C are the same as in Figure 1, and the probability of 1 in each DTMC is on account of guideline #1: Any person who tests positive for COVID-19 shall promptly move into set B as soon as the test result is known.
Figure 2. Two discrete-time Markov chains (DTMCs). Figure courtesy of Samuel Awoniyi.
Probability \(p_{1}\) reflects the relatively free in-person interactions between individuals in sets A and B in the current system. Similarly, probability \(p_{2}\) reflects the carefully controlled in-person interactions between the Red Corps (set C) and occupants of set B in the proposed alternative system. Accordingly, one would ordinarily have \(p_{1}\gg p_{2}\).
If we let \(STC1\) denote the sojourn time cycle for the current system’s DTMC and \(STC2\) denote the sojourn time cycle for the proposed alternative system’s DTMC (see Figure 2), we then have \(STC2\gg STC1\) by virtue of \(p_{1}\gg p_{2}\), because \(STC1=1/p_{1}+1\) and \(STC2=1/p_{2}+1\). A straightforward introduction to the computation of sojourn time cycles for general Markov chains is available in [1].
For instance, if \(p_{1}=0.5\) and \(p_{2}=0.1\), then \(STC1=3\) and \(STC2=11\). Because \(STC\) signifies “average time to next infection” in this application, it is evident that the average time to next infection for the proposed alternative system would ordinarily be much longer than for the current system.
Practical Implementation
Regarding practical implementation, our alternative system first and foremost requires that some form of reliable COVID-19 testing be available to everyone in the U.S. Testing of Red Corps members must be especially prompt; otherwise the proposed framework will not fare much better than the current system in terms of infection probability.
Red Corps members include healthcare teams, grocery and food supply personnel, and communications supply groups. Most Red Corps members will have preferably acquired some COVID-19 antibodies. Each member must wear suitable PPE during in-person interactions with people in set B, and a proper face mask in grocery stores and other shops. This requirement is meant to minimize the probability of a Red Corps worker infecting the general population.
Asymptomatic set B members should be housed in suitable hotels, separate from the members of set B who already show symptoms. These symptomatic individuals should be housed in hospitals that are specialised for COVID-19 patients. Each set B member must wear an appropriate face mask during in-person interactions with Red Corps members. As per guideline #2, people who are currently in set B must not have in-person interactions with those in set A.
Everyone in set A must wear a suitable face mask during in-person interactions involving four or more people. Members of set A who are certified to possess some COVID-19 antibodies can go to work as planned or needed. As such, COVID-19’s overall adverse impact on the national economy would likely be much less under the proposed system than the current system.
Since our proposed alternative system involves substantial reduction of human freedom in society, it should be suspended shortly after a viable COVID-19 vaccine or treatment becomes available. One may also suspend the alternative system about one month after set B becomes virtually empty. The fact that an infected person will either die or recover and develop the necessary antibodies guarantees set B’s eventual emptiness.
Ultimately, our proposed method is well-suited to prevent a second wave of COVID-19 infections because the Red Corps will keep set B below critical community size—the minimum size of a closed population within which a human-to-human, non-zoonotic pathogen can persist indefinitely [2, 3].
References
[1] Awoniyi, S., & Wheaton, I. (2019). Case for first courses on finite Markov chain modeling to include sojourn time cycle chart. SIAM Rev., 61(2), 347-360.
[2] Bartlett, M.S. (1960). The critical community size for measles in the United States. J. Roy. Stat. Soc.: Ser. A., 123(1), 37-44.
[3] Blackwood, J.C., & Childs, L.M. (2018). An introduction to compartmental modeling for the budding infectious disease modeler. Lett. Biomath., 5(1), 195-221.
Samuel Awoniyi is a professor of industrial engineering at the FAMU-FSU College of Engineering.