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Utilizing Shields to Curb COVID-19’s Second Wave

By Samuel Awoniyi and Arda Vanli

The 2000 SIAM Review article by Herbert W. Hethcote [2] is a classic for those who are interested in the mathematics of infectious diseases, even though the article mainly focuses on the susceptible-infectious-recovered (SIR) model and its variants. When the COVID-19 pandemic jolted applied mathematics communities and their sponsors into a renewed interest in mathematical modeling of epidemics, many of the research efforts that promptly emerged unsurprisingly involved SIR-type modeling [3].

Although they rely on the severe assumption that compartments change continuously, SIR-type models have been fairly dependable in predicting parts of COVID-19 spread. Perhaps as a remarkable indication of the success of recent mathematical modeling, some national news networks are beginning to refer to "reproduction number \(R_0\)" and "herd immunity"—standard SIR model terms—as though they are ordinary conversational items.

There is now something of a consensus that the "second wave" of COVID-19 is upon us. Current information [4] and data suggest that COVID-19 second-wave infections are naturally patchy because a necessary condition for COVID-19 infection is the presence of super-spreaders, like South Korea’s Patient 31. COVID-19 second-wave infection therefore seems probabilistic, and SIR-type models are understandably not quite reliable in predicting it.

Many recently-published mathematical models on curbing the spread of COVID-19 are not SIR-type models. One such non-SIR-type model [1] features the concept of a shield, called the Red Corps. We generalize this idea in the present article, as the concept of shield is well-suited to curbing a second wave of COVID-19.

Informally, a shield is a group of people (a) who have been medically tested and found not to have certain necessary signs of COVID-19 infection, and (b) whose function is to prevent sufficient conditions of COVID-19 infection from holding during in-person interaction activities between two other groups in the population. One example of such a condition is when uninfected individuals and a COVID-19 super-spreader—such as South Korea’s Patient 31—occupy a poorly-ventilated space [4].

This shield concept allows practical decision-makers to devote special attention to protecting particular groups, such as schools (operating from 7:00 a.m. to 6:00 p.m.), college dormitories, special personnel at meat packaging shops, and nursing homes for senior citizens. Our forthcoming lemma formalizes this notion.

In this article, we state and prove an infection rate lemma for shields systems. Even though our proof of the lemma utilizes a Markov chain model that is instructive in itself, we aim to present a mathematically justified tool for the design of effective shields systems against a second-wave infection of COVID-19 among special groups in the U.S. whose activities are necessary to sustain the national economy at a certain minimal level. After describing the lemma, we provide several illustrative examples.

An Infection Rate Lemma

Here we state and prove a lemma about infection rate for a system that utilizes shields against COVID-19 spread, along with a corollary that removes the need to estimate certain data and thus makes the lemma quite practical. We begin with a formal definition of the concept of a shield.

Definition 1: Imagine two sets or groups of persons—say \(G1\) and \(G2\)—in a population (like the population of a community in the U.S.), along with a set of possible in-person interaction activities—say \(A=\{a_1,...,a_n\}\)—between individuals in \(G1\) and \(G2\). Another set or group of persons \((S)\) in the population is called a shield between \(G1\) and \(G2\), relative to interaction activities \(A\equiv\{a_1,...,a_n\}\), if any interaction activity  \(a_i \in A\) between a person in \(G1\) and a person in \(G2\) must be coordinated by persons in group \(S\) to ensure that COVID-19 infection sufficient conditions do not hold.

We now state and prove our lemma, with particular reference to interaction activities that may result in COVID-19 infections.

Lemma 1: Suppose that \(S\) is a shield between two groups \(G1\) and \(G2\), relative to interaction activities \(A\equiv\{a_1,...,a_n\}\), and suppose that observations of people in the population \(G1 \cup S \cup G2\) are made at regular time intervals, such that:

  • (i) Group \(G1\) consists of individuals, some of whom may be currently infected with COVID-19. Any person in \(G1\) who is certified to have recovered from COVID-19 shall promptly move (that is, by the next observation time) into group \(G2\).
  • (ii) Group \(G2\) consists of individuals who are known to be currently free of COVID-19 infection. \(G2\) is not an empty set.
  • (iii) Each person in shield \(S\) shall always wear suitable personal protective equipment (against COVID-19 infection) during all interaction activities, and any person in \(S\) who becomes infected with COVID-19 shall promptly move (that is, by the next observation time) into group \(G1\), with a suitable replacement from group \(G2\) moving into set \(S\). 
  • (iv) The interaction activities that comprise \(A\) are the only means by which a person in shield \(S\) can possibly become infected with COVID-19. The probability that some person in \(S\) will become infected by the next observation time is \(p\). 

Under conditions (i), (ii), (iii), and (iv), the mean COVID-19 infection rate is \(p/(1+p)\) for the whole population \(G1 \cup S \cup G2\).

Proof: We arrive at infection rate \(p/(1+p)\) as the reciprocal of a sojourn-time cycle (STC) of a two-state Markov chain that tracks how COVID-19, behaving like a traveling deliverer of harmful packages, goes back and forth between group \(G1\) and shield \(S\).

Figure 1. Discrete-time Markov chain (DTMC) tracking COVID-19 infections.
Figure 1 depicts this Markov chain; it is a discrete-time Markov chain (DTMC) because observations are made at regular time intervals. 

The probability value of 1 that is shown in Figure 1 is on account of condition (iii) in the lemma statement: any person in \(S\) who becomes infected with COVID-19 shall promptly move into group \(G1\). The probability value of \(p\) is a reflection of how well the shield \(S\) is protecting itself and \(G2\) from COVID-19 infection.

Requisite Markov assumption (that is, lack-of-memory assumption) makes sense naturally in this instance, as the probability values only depend on set \(S\) functioning ordinarily as a shield between \(G1\) and \(G2\).

From basic Markov chain theory, the DTMC’s mean sojourn time at state \(G1\) (see Figure 1) is \(1/p\), and the mean sojourn time at state \(S\) is 1. Therefore, the STC at each of the two states is \((1/p)+1\). For additional clarity, an STC definition and associated computations for all types of Markov chains are available online, and several sources also contain good classical materials on Markov chain modeling [1, 4].

Based on the definition of \(p\) in condition (iv) of the lemma statement, it is clear that the mean sojourn time \((1/p)+1\) has the meaning "mean time to next infection" in the population \(G1 \cup S \cup G2\), as any new infection in \(G1 \cup S \cup G2\) can only occur for a person in \(S\) through interaction activities in \(A\equiv\{a_1,...,a_n\}\). Accordingly, the desired mean infection rate for the population \(G1 \cup S \cup G2\) is the reciprocal of \((1/p)+1\) — that is, \(p/(1+p)\). 

The mean COVID-19 infection rate in a shields system therefore depends critically on how small probability \(p\) is. The following corollary pertains to the way in which the size (i.e., cardinality) of \(S\) provides a handle for making the probability \(p\) suitably small.

Corollary 2: Suppose that \(S\) is a minimal shield in the lemma, in the sense that the number of people who comprise \(S\) is the smallest possible number needed to function effectively as the required shield. The conclusion of that lemma then becomes even stronger, namely in that the mean infection rate \(p/(1+p)\) is minimal as well.

Proof: Suppose that \(S1\) is a shield with associated probability \(p_1\) and \(S2\) is an equivalent shield (they are both shields for the same \(G1\), \(G2\), and \(A\)) with associated probability \(p_2\), such that \(\left\Vert S2\right\Vert =\left\Vert S1\right\Vert +1\). If the only difference between the shields is that \(S2\) contains one additional person, we can use elementary probability reasoning to conclude that \(p_{2}\geq p_{1}\) as well.

On account of this corollary, it is thus not necessary to estimate probability \(p\) when designing a practical shield against the spread of COVID-19; one only must ensure that no redundancies exist in the personnel of \(S\) (i.e., \(S\) is minimal). 

Some Application Instances

Here we describe two examples of practical applications of the lemma. The examples vary according to how one must configure the shields to effectively cover the activities \(A\equiv\{a_1,...,a_n\}\). 

Example 1: Shielding college students in a dormitory

\(G1\) is the set of all individuals in a specific community, some of whom may currently be infected with COVID-19.

\(G2\) is the set of all students who live in a particular college dormitory that is located inside community \(G1\). We assume that everyone in \(G2\) is currently free of COVID-19.

\(A\): \(a_1 \equiv\) activities that are required for providing and maintaining local transportation for people in \(G2\); \(a_2 \equiv\) activities that are required for obtaining food, groceries, and other essential daily needs and delivering them to everyone in \(G2\); \(a_3 \equiv\) activities that are required for hospital/medical care for individuals in \(G2\); \(a_4 \equiv\) activities that are necessary for in-person visits to people in \(G2\), with the possible inclusion of physical barriers with friendly gates to regulate access. This may also include simplified medical testing to admit only healthy persons [4].

\(S\) is the set of all individuals who are needed to ensure that the in-person interactions that comprise \(A\) are completed without causing COVID-19 infection sufficient conditions to hold. Every person in \(S\) must be certified as "not infected with COVID-19"; such a certification is quicker and more accurate than standard medical tests for the detection of COVID-19 [4]. The size of \(S\) should be kept to a minimum in order to get the benefit that we state in the lemma’s corollary.

Example 2: Shielding workers of a meat processing factory

\(G1\) is the set of all people in a specific community, some of whom may currently be infected with COVID-19.

\(G2\) is the set of certain meat-processing factory workers who live in special residential buildings or hotel accommodations that are set aside for these types of workers within the community.

In this case, \(A\) and \(S\) are similar to those in Example 1. Here, the shield \(S\) should acknowledge the fact that \(G2\) is really a set of families with more dimensions than the \(G2\) in Example 1.

Closing Comments

The second wave of COVID-19 infections is ordinarily patchy, in the sense that it affects various groups differently within the same geographical region. A plausible, data-based explanation of the patchy nature of COVID-19’s second-wave infections is available in [4]. Consequently, it makes good sense to provide different types of shield protection for different groups in the same geographical region. Regardless of their specific structure, all shields are configured to prevent COVID-19 infection sufficient conditions from holding.

The various shields that might emerge could result in a new mode of professional industry in the U.S., complete with its own best practices and professional community. Regarding the second wave’s effect on the national economy, the shields might also provide temporary or transient employment for people whose regular jobs have been suspended or displaced by COVID-19’s public health restrictions.


References
[1] Awoniyi, S. (2020, June).: An alternative system for curbing COVID-19 spread in the U.S.", SIAM News, 53(5), p.1.
[2] Hethcote, H.W. (2000). The mathematics of infectious diseases. SIAM Rev., 42(4), 599-653.
[3] Meza, J.C., Feng, Z., Luo, T., & Wang, J. (2020, June). Mathematicians Quickly Respond to the COVID-19 Pandemic. SIAM News, 53(5), p.10.
[4] Tufekci, Z. (2020, September 30). This overlooked variable is the key to the pandemic. The Atlantic. Retrieved from https://www.theatlantic.com/health/archive/2020/09/k-overlooked-variable-driving-pandemic/616548/

Further Reading
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.
Kulkarni, V.G. (2010). Introduction to Modeling and Analysis of Stochastic Systems. In Springer Texts in Statistics. New York, NY: Springer.
Solberg, J.J. (2009). Modeling Random Processes for Engineers and Managers, Hoboken, NJ: John Wiley & Sons.

Samuel Awoniyi is a professor of industrial engineering at the FAMU-FSU College of Engineering. Arda Vanli is an associate professor of industrial engineering at the FAMU-FSU College of Engineering.

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