key: cord-0480132-goq0cn4q
authors: Tian, Yurun; Sridhar, Anirudh; Wu, Chai Wah; Levin, Simon A.; Poor, H.Vincent; Yagan, Osman
title: The Role of Masks in Mitigating Viral Spread on Networks
date: 2021-10-08
journal: nan
DOI: nan
sha: 738e84f2e9b94da7e40024754df9bec355964e54
doc_id: 480132
cord_uid: goq0cn4q

Masks have remained an important mitigation strategy in the fight against COVID-19 due to their ability to prevent the transmission of respiratory droplets between individuals. In this work, we provide a comprehensive quantitative analysis of the impact of mask-wearing. To this end, we propose a novel agent-based model of viral spread on networks where agents may either wear no mask, or wear one of several types of masks with different properties (e.g., cloth or surgical). We derive analytical expressions for three key epidemiological quantities: the probability of emergence, the epidemic threshold, and the expected epidemic size. In particular, we show how the aforementioned quantities depend on the structure of the contact network, viral transmission dynamics, and the distribution of the different types of masks within the population. Through extensive simulations, we then investigate the impact of different allocations of masks within the population. We also investigate trade-offs between masks with high outward efficiency but low inward efficiency and masks with high inward efficiency but low outward efficiency. Interestingly, we find that the former type of mask is most useful for controlling the spread in the early stages of an epidemic, while the latter type is most useful in mitigating the impact of an already large spread. Lastly, we study whether degree-based mask allocation is more effective in reducing probability as well as epidemic size compared to random allocation. The result echoes the previous findings that spreading processes should be treated with two different stages that source-control before epidemic starts and self-protection after epidemic forms.

The COVID-19 pandemic has spread across the globe for over two years, impacting economies and, as of January 2022, had claimed over 5.7 million lives [2] . Even after vaccines have become available, mask-wearing remains an important measure in curbing the spread of the virus. While it is well-known that masks qualitatively mitigate viral spread by limiting the transmission of respiratory droplets [9] , [16] , [24] , [31] , [34] , [41] , many important questions about the quantitative impact of masks remain open [8] , [10] , [19] . For instance, how many individuals need to wear a mask to prevent future outbreaks [1] ? When there are not enough high-quality masks (e.g., N95 masks) for all individuals [15] , how should other types of massproduced masks (e.g., surgical or cloth masks) be allocated within the Chai Wah Wu is with Thomas J. Watson Research Center, IBM, Yorktown Heights, NY, 10598 USA population [38] ? What types of mask attributes are most desirable in preventing future outbreaks or controlling ongoing pandemics [21] , [25] , [35] ?

This work aims to answer the above questions from a principled, mathematical lens. We propose a novel agent-based model of viral spread on networks wherein individuals wear different types of masks. In particular, we study models incorporating multiple types of masks; prior work considers scenarios where individuals either wear a mask or do not wear a mask [?], [7] , [10] , [17] , [18] , [20] , [22] , [23] , [32] , [33] , [36] , [37] , [40] . Our contributions our twofold. First, we derive analytical predictions for three important epidemiological quantities: the probability of emergence, the epidemic threshold (also known as the reproductive number R 0 ) and the expected epidemic size. Specifically, we show how these quantities depend on the structure of the contact network, properties of the viral spread, and the distribution of masks within the population. Our results are established by leveraging the theory of multi-type branching processes [5] , [14] . Moreover, we show through extensive simulations that the analytical predictions we derive are in good agreement with empirical results.

We then explore a variety of mask-wearing scenarios relevant to the ongoing pandemic, focusing in particular on how the probability of emergence (PE) and expected epidemic size (ES) are affected. First, we quantify the tradeoffs between using superior vs. inferior masks (e.g., surgical vs. cloth) when all individuals wear one of either type of mask; naturally, we find that both the PE and ES are reduced when the fraction of superior masks increases in the population. We then consider scenarios in which individuals either wear superior masks, inferior masks, or no masks. We find that increasing the fraction of superior masks significantly decreases the fraction of infected non-mask-wearers when the fraction of non-maskwearers is small (< 10%). Interestingly, when the fraction of nonmask-wearers is larger (> 20%), increasing the fraction of superior masks does not significantly mitigate the infections in the nonmask-wearing population. This suggests that mask-wearing strategies are more effective when a larger fraction of the population wears inferior masks, as opposed to a smaller fraction of the population wearing superior masks. Next, we study trade-offs between masks that are "inward-good" (i.e., good at blocking respiratory droplets from the outside but poor at limiting transmission from the maskwearer to the outside), and those that are "outward-good" (i.e., good at limiting transmission from the mask-wearer to the outside but poor at blocking respiratory droplets from the outside). We find that both mask types are useful, but at different stages of the viral propagation. In particular, outward-good masks are most helpful in preventing the emergence of an epidemic, while inward-good masks are best for reducing the infections of an already ongoing epidemic. Lastly, we look into mask assignment depending on node degree, by assigning outward-good masks to top x% high (low) degree nodes, and inwardgood masks to the rest of nodes. We find a scenario in which high degree nodes wear inward-good masks and low degree nodes wear outward-good masks is efficient in reducing the probability of emergence and the opposite allocation scheme is more helpful in controlling the epidemic size extension after the epidemic forms. This result reconfirms that we need to treat the two stages of virus spread (i.e. before and after epidemic exists) with different mitigation strategies. It also indicates that high degree nodes and low degree nodes play different roles in the epidemic process. The most powerful factor leading to a pandemic as well as extending the pandemic is high degree nodes. However, before epidemic starts, removing the additional infecting paths from low degree initiator to susceptible high degree nodes is critical in preventing the epidemic from happening. After the epidemic forms, protecting susceptible low degree nodes from infected high degree nodes is more important in suppressing the propagation of the epidemic.

While our results are motivated by mask-wearing in a pandemic, our model can be applicable more broadly to other mitigation strategies. For example, our results on inward and outward efficiencies suggests that when we think of a mitigation strategy for a pandemic, we should consider the current stage of the spread: early on, it is most important to limit people spreading it to others (this can be achieved through social distancing, for instance), while later it becomes more important to protect individuals from getting the virus. This insight can help with strategies on prioritizing vaccines, limiting gatherings, etc. On a more technical note, our model generally captures heterogeneities in the capability of a node to be affected by or to spread a virus. While one interpretation of node-level heterogeneity is the type of mask used, it can also capture the effects of vaccinations or community-based interactions (i.e., if individuals tend to interact within their own community rather than withn others, this may result in a higher transmissibility between two individuals of the same community).

The structure of this paper is as follows. In Section II, we provide an overview of related epidemic models and introduce a formal description of a model for viral spread in the presence of masks of various types. Section III contains our theoretical analysis, where we derive expressions for the probability of emergence, the epidemic threshold and the expected size of the epidemic. Our theoretical results are verified in Section IV, where we explore the implications of the multi-type mask model through simulations. Finally, we conclude and discuss future avenues of research in Section V.

The literature on epidemic modeling generally falls into one of two categories: ordinary differential equation (ODE) models, and agentbased stochastic models. In typical ODE models, the evolution of the fraction of various types (e.g., susceptible individuals, infected individuals) is studied. The governing equations for the epidemic dynamics are derived from laws of mass action that are based on the way individuals interact and spread the virus (see, e.g., [6] ). A number of recent works have used the ODE approach to model viral spread with masks. Tracht et al. [37] first proposed an ODE model incorporating mask-wearing. Motivated by challenges caused by the COVID-19 pandemic, many authors have subsequently elaborated on the model of [37] , addressing for instance the effect of asymptomatic infections and mask allocation [10] , [17] , [18] , [22] , [38] . However, a common criticism of the ODE approach is that in modeling population-level phenomena, it fails to consider the possibly complex ways in which individuals interact with each other.

Agent-based stochastic models address this gap by studying how individual-level interactions facilitate viral spread. Such models reveal the rich interplay between interaction patterns and viral spread. Various agent-based models incorporating mask-wearing have been recently proposed [7] , [20] , [23] , [32] , [33] , [36] , [40] . The drawback of stochastic models is that they are often challenging to simulate for realistic parameters. To overcome this issue, a significant line of work on agent-based models focuses on deriving analytical predictions for key epidemiological quantities that accurately describe viral spread in large contact networks; the works [23] , [33] , [36] , [40] adopt this approach to quantify the effect of mask-wearing. This is also the focus of the present paper. We emphasize that all of the literature discussed above studies the effect of a single type of mask in the population. A key novelty of our work is that we study the effect of multiple types of masks which helps reveal the trade-offs between the inward and outward efficiencies of the masks involved and, more broadly, sheds light on the effectiveness of various mitigation strategies.

In the remainder of this section, we describe stochastic models of viral spread in more detail. In Section II-A, we review the basic stochastic model. We then discuss a new stochastic model with multitype masks in Section II-B.

In seminal work [27] , Newman considered an SIR model of viral spread over a contact network. Initially, there is a single infected individual in the population. When a susceptible individual and an infected individual interact, the infected individual transmits the virus to the susceptible individual with probability T , where T is called the transmissibility of the virus. After a fixed or random amount of time, infected individuals recover and can no longer transmit the virus. Through branching process techniques, Newman derived expressions for two key epidemiological quantities: the probability that an epidemic emerges (PE) and the expected size of the epidemic (ES). Importantly, his results revealed the crucial role that the structure of the contact network plays in the behavior of the PE and ES. Although the viral dynamics may appear simple, Newman's model can capture complex viral transmission and recovery mechanisms through the appropriate choice of T [27, Equations (2)- (6) ]. Several authors have generalized Newman's model to account for various agent-level heterogeneities [3] , [4] , [11] , [23] , [33] , [36] , [39] , [40] .

In this work, we consider a generalization of Newman's basic framework called the multi-type mask model. Motivated by maskwearing behaviors in response to the COVID-19 pandemic, we assume that there are M types of individuals, each wearing a different type of mask. For notational convenience, we shall say that an individual is of type-i if they wear a type-i mask (here, 1 ≤ i ≤ M ). We assume that the probability of transmission between individuals varies depending on the type of mask each individual wears. Specifically, the probability that a virus is eventually transmitted from a type-i infective to a type-j susceptible is T ij , where T is an M × M transmissibility matrix.

Typically, masks are characterized in terms of their inward and outward efficiencies (see, e.g., [30] ). The inward efficiency is the probability that respiratory droplets will pass from the outside layer of the mask to the inside; thus, inward efficiency quantifies the protection of the mask against receiving the virus. The outward efficiency is the probability that respiratory droplets will pass from the inside layer of the mask to the outside, quantifying the protection against transmitting the virus. The transmission probability from a type-i individual to a type-j individual is then given by

where ǫ out,i is the outward efficiency of a type-i mask, ǫ in,j is the inward efficiency of a type-j mask, and T is the baseline transmissibility of the virus, i.e., the probability of transmission in the presence of no masks. Note that T is not symmetric if the vectors ǫ out and ǫ in are not collinear.

In line with prior literature on stochastic epidemic models, we generate the contact network G by the configuration model [26] . Namely, we specify a distribution {p k } k≥0 with support on the nonnegative integers, where p k is the probability that an arbitrary vertex has degree k, i.e., it is connected to k other nodes via an undirected edge. According to the configuration model, the degrees of vertices in G are drawn independently from {p k } k≥0 . Equivalently, G is selected uniformly at random from among all graphs satisfying the degree distribution p k . Next, we assume that the M types of masks are randomly distributed amongst vertices in the G. Let {m 1 , . . . , m M } be a distribution over the set {1, . . . , M } where m i represents the fraction of individuals who wear a mask of type-i. We further assume that the type of mask is chosen independently 1 from {m i } M i=1 over all vertices in G. See Table I for a succinct description of our model parameters, and Figure 1 for an illustration of the spreading process. For each i = 1, . . . , M , our goal is to compute the following quantities of interest: the PE assuming that the initially infected node is of type-i, and the ES of the infected type-i individuals.

The above modification of Newman's model for the mask-wearing setting has garnered recent interest in the literature. In previous work, we studied the special case where M = 2: individuals either wear a mask, or do not [36] . We also derived expressions for the PE and ES in this setting. Lee and Zhu [23] simultaneously studied the 1 The independence assumption is for mathematical tractability, as it implies that the distribution of mask types within a vertex's neighborhood is given by

One can imagine realistic scenarios where this is not the case; for instance, individuals who wear a mask may be more likely to interact with other mask-wearers, rather than non-mask-wearers.

Description M Number of mask types

The probability that a given individual is type-i

The probability that an infected type-i individual infects a type-j neighbor (transmissibility) G Graph representing the contact network

Degree distribution for G generated via the configuration model. The results of both [36] and [23] follow as a special case of our more general model. We also mention the work of Allard et al. [4] , which is especially relevant. They study a bond percolation problem over multi-type networks, which can be viewed as a more generic version of the mask model we study here. A key difference between our work and theirs is that the distribution of mask types and the network formation are independent in our model, whereas Allard et al. consider a framework where the probability of connectivity among nodes also depend on the node types, resulting a joint generation of node types and network structure. This renders their results harder to interpret, albeit being more general. Our formulation yields simpler formulae that clearly illustrate how the structural aspects of the network, mask properties, and viral transmission dynamics interact to derive the PE and ES. In addition, our work also validates the theoretical analysis through extensive simulations and provides insights on the trade-offs between inward-good and outward-good masks. Finally, our framework still has the flexibility to correlate the mask types with the network structure by modifying the mask allocation strategy based on node degrees. In the Results section, we demonstrate this and show that allocating different types of masks based on node degree can help further reduce the probability and size of epidemics.

To model the underlying contact network, we utilize random graphs with random degree distribution generated by the configuration model [26] , [29] . The configuration model generates random graphs with specified degree sequence sampled from a degree distribution. Let G denotes the underlying contact network defined on the node set N = {1, ..., n}. The structure of G is defined through its degree distribution {p k , k = 0, 1, ...}, where p k is the probability that an arbitrary node in network G has degree k. We generate a degree sequence from Poison degree distribution under the assumption that all the moments of arbitrary order are finite. Note that if the second moment of the degree distribution is finite, when n approaches infinite, the expected clustering moment of the graphs approaches zero. This indicates that the graph is locally tree-like. In the context of configuration model, the degree distribution of a randomly chosen neighbor of a randomly chosen vertex denoted by {p k k = 1, 2, ...} is given byp k = kp k / k for k = 1, 2, ..., where k denotes the mean degree (i.e., k = k kp k ).

In this section, we present the derivation of the probability of emergence (PE), the epidemic threshold (R 0 ) and the expected epidemic size (ES) in the multi-type mask model. Formally, emergence is defined to be the event where the virus infects an infinite or unbounded number of vertices in the network. It is the complement of the extinction event, in which the virus dies out after infecting a finite number of individuals. In Section III-B, we compute the through an approach based on probability generating functions (PGFs); the works [3] , [11] , [27] , [36] use this method to derive the PE for related epidemic models.

In Section III-C, we study the epidemic threshold R 0 , also known as the reproductive number. When R 0 < 1, the epidemic dies out almost surely and when R 0 > 1, there is a positive probability that the epidemic emerges. Generally, R 0 is interpreted as the mean number of secondary infections caused by a given infective (see, e.g., [27] ), but our results show that the true picture is more subtle than that. Indeed, using classical results from multi-type branching process theory [5] , [14] , we show that R 0 is the spectral radius of a matrix that depends on {m} M i=1 , {T ij } 1≤i,j≤M as well as the first and second moments of the degree distribution.

Finally, in Section III-D, we study the ES. Specifically, we show how to compute the expected fraction of type-i individuals, conditioned on the emergence of the epidemic. To do so, we leverage the method of Gleeson and coauthors [12] , [13] , which were recently used to compute the ES in models of viral spread [11] , [36] .

Generating functions. The PGF of the degree distribution of an arbitrary vertex in the configuration model is given by

where p k is the probability that a given vertex has degree k (see Table  I ). It is also of interest to study the degree distribution of a node that is identified by following a randomly chosen edge, e.g., to characterize the distribution of the number of additional infections that a newly infected node might lead to. Specifically, let excess degree be defined as the "degree minus one" of a node reached by following one end of an edge selected uniformly at random. The PGF of the excess degree is given by

where k := ∞ k=0 kp k is the mean degree of a node. For details on the derivation, see [28, Chapter 13] .

Local structure of the configuration model. Graphs generated according to the configuration model are known to be locally tree-like in the following sense: the local neighborhood of a uniform random vertex converges in distribution to a random tree as the number of vertices tends to infinity [28, Chapter 12.4 ]. The number of children of the root is sampled from the degree distribution, and the number of children of a later-generation vertex is sampled from the excess degree distribution.

Suppose that a type-i infective -named v for convenience -has k j susceptible neighbors of type j, for 1 ≤ j ≤ M . Let X j be the number of neighbors of type j who are eventually infected by v, so that X j ∼ Binomial(k j , T ij ). Moreover, X 1 , . . . , X M are independent. Conditioned on the neighborhood profile k 1 , . . . , k M , the PGF of the number of infections of each type caused directly by v is

Our next step is to condition on the total number of neighbors of v rather than the number of neighbors of each type. Note that if we are given k :

Namely, the probability of observing a given instantiation (k 1 , . . . , k M ) is equal to

The PGF of the number of neighbors of all types directly caused by v conditioned on k can therefore be written as

(2)

Finally, to remove the conditioning on the number of neighbors, k, we take an expectation over the degree distribution of v. If v is typei, the PGF of the number of secondary infections of each type is given by

For notational convenience, we combine the PGFs of all types into a single vectorized PGF γ :

On the other hand, if v is a later-generation infective, the number of its children follows the excess degree distribution. Hence the PGF of the number of secondary infections of each type is given by

We also define the vectorized PGF Γ :

. With these PGFs in hand, we can compute the probability of extinction. Formally, for 1 ≤ i ≤ M and a positive integer n, let P (n) i denote the probability that the epidemic dies out by generation n (that is, distance n from the infection source), where the initial infective is of type i. Classical results from branching process theory

The probabilities of extinction can be found by taking the limit n → ∞. Formally, define Γ (n) to be the n-fold composition of Γ. Assuming there is a well-defined limit as n → ∞,

The probabilities of eventual extinction are therefore given by

Since the above describes the extinction event when the initial infective is type-i, the PE is 1 − P i . To formally justify taking the limit n → ∞, we need to check that the multi-type branching process with PGF Γ is positive regular and non-singular [5] , [14] . The process is singular if and only if each type has exactly one secondary infection. Our model is clearly non-singular, since each neighbor of an infective is independently infected. A sufficient condition for our process to be positive regular is if there is a positive probability that a type-i individual can infect a type-j individual, for any 1 ≤ i, j ≤ M . This is indeed the case if we assume that m i > 0 and T ij > 0 for all 1 ≤ i, j ≤ M . Both assumptions are expected to hold in practice as the former condition states that there is a positive fraction of the population wearing each type of mask, and the latter states that there is a positive probability of transmission between any two neighboring individuals.

In the special case where M = 1 (e.g., when no one wears a mask), it is well known that there exists a phase transition in the PE based on the basic reproductive number, R 0 , defined as the mean number of secondary infections in a naive population. Put differently, R 0 is the expected number of new infections generated by a newly infected node in a population where all individuals are susceptible. It is known that if R 0 is greater than one then the PE is positive, i.e., epidemics can take place. When R 0 ≤ 1 on the other hand, the PE is zero [27] . Beyond marking a phase transition, the metric R 0 measures, in a sense, the speed at which the epidemic grows and is often used by policy-makers for deciding on mitigating strategies. Thus, it is of significant importance to characterize R 0 for the multi-type mask model.

Firstly, notice from the computations in Section III-B (in particular, from (3)) that the PE is zero (P 1 = . . . = P M = 1) if and only if Q 1 = . . . = Q M = 1. Hence we focus on the phase transition for extinction/emergence of the multi-type branching process with PGF Γ. Let A ∈ R M×M be a non-negative matrix such that A ij is the expected number of type-j children of a type-i later-generation infective. Define R 0 := ρ(A) to be the spectral radius of A, and recall that the multi-type branching process with PGF Γ is positive regular and non-singular (see the discussion at the end of Section (III-B)). It is a well-known property of multi-type branching processes that if

g., see [5] , [14] .

We proceed by computing the entries of A to illustrate the dependence of R 0 on the graph topology, the mask parameters and the viral transmissibilities. To this end, note that A ij is equal to the expected number of type-j children of a later-generation infective times T ij . The expected number of children (computed over all types) is given by

where k and k 2 are the first and second moments of the degree distribution, respectively. Since the probability that a given child is type-j is m j , we have

This leads to the matrix representation

where the (i, j) th entry of the matrix T is T ij and m is a diagonal matrix with (i, i) th entry being m i . Putting everything together, we have

We next study a simpler version of (4) in a special case of interest. Typically, masks are characterized in terms of their inward and outward efficiencies (see, e.g., [30] ). The inward efficiency is the probability that respiratory droplets will pass from the outside layer of the mask to the inside; thus, inward efficiency quantifies the protection of the mask against receiving the virus. The outward efficiency is the probability that respiratory droplets will pass from the inside layer of the mask to the outside, quantifying the protection against transmitting the virus. The transmission probability from a type-i individual to a type-j individual is then given by

where ǫ out,i is the outward efficiency of a type-i mask, ǫ in,j is the inward efficiency of a type-j mask, and T is the baseline transmissibility of the virus, i.e., the probability of transmission in the presence of no masks. For notational convenience, let ǫ out be the M -dimensional vector where the i th entry is ǫ out,i ; similarly define the vector ǫ in . Then, using (4), we have

Since ǫ out ǫ i n ⊤ m is a rank one matrix, it has only one non-zero eigenvalue which is given by

This allows us to conclude that

The expression M i=1 m i T ii can be thought of as the average same-type transmissibility in the network. Interestingly, R 0 does not depend on transmissibilities between different types (although it does depend on all the ǫ in,i 's and ǫ out,i 's). It is worth noting that while the location of the phase transition does not explicitly depend on T ij for distinct i, j, the expressions for the PE certainly do depend on the T ij 's.

In this section, we compute the expected size of the epidemic -that is, the final fraction of infected individuals of each type, conditioned on the event that the epidemic does not die out in finite time. Our method follows [11] - [13] , [36] .

The analysis proceeds as follows. Let v be a vertex selected uniformly at random, and recall that the local structure of the graph around v is a tree with probability tending to 1 as the network size tends to infinity. The number of children of the root, v, follows the distribution {p k } k≥0 and the number of children of a later-generation vertex follows the excess degree distribution. We say that the root node is at level 0, and more generally, we say that the vertices with distance ℓ from the root are at level ℓ. The epidemic is initialized by specifying the infection status of vertices far away from v. Formally, we specify a large positive integer n as well as θ := {θ i } M i=1 , which is a collection of values between 0 and 1. For each type-i vertex in level n, we assume it is infected with probability 1 − θ i , and that it is not infected with probability θ i . Given this initial configuration, we denote for 1 ≤ i ≤ M and 0 ≤ ℓ ≤ n − 1 the quantity q (n) ℓ,i to be the probability that a type-i vertex in level ℓ is not infected, given the initial configuration θ at level n. The probability q (n) ℓ,i can be computed in a recursive manner, which we describe next.

Let u be a given vertex of type-i in level ℓ, so that q (n) ℓ,i is the probability that u is not infected by a vertex in level ℓ + 1. As in Section III-B, let k 1 , . . . , k M denote the number of neighbors of each type in level ℓ + 1 and let X 1 , . . . , X M denote the number of infected neighbors of each type. Conditioned on X 1 , . . . , X M , the probability that u is not infected is

We next take an expectation over the X j 's to compute the unconditional probability of non-infection. To this end, observe that the infection status of nodes in the same level are independent from each other since vertices in a given level do not have common infected descendants due to the tree-like structure of the network. Hence, since there are k j neighbors of type j, we have

Moreover, the X j 's are independent. The probability of u not being infected conditioned on k 1 , . . . , k M is therefore given by

Note that (6) is quite similar to the PGF derived in (2), with T ij replaced with T ji and s j replaced with q (n) ℓ+1,j . Following the same steps as in Section III-B, we therefore arrive at the following recursion for ℓ ≥ 1:

For the case ℓ = 1, we have

ℓ,i } is the vectorized collection of the level ℓ probabilities. For notational convenience, we also write F :

The recursions (7) and (8) imply that

. Conditioned on the event that the epidemic emerges, we may assume that θ i < 1 for all 1 ≤ i ≤ M ; that is, there is a positive probability that any given vertex in level n is infected. Taking the limit as n → ∞ shows that q 1 := limn→∞ q (n) 1 satisfies the fixed-point equation

The limiting probability that the root is not infected is then given by

Since q 0 specifies the asymptotic probabilities of non-infection, 1 − q 0,i is the probability that a uniform random type-i vertex is eventually infected. This is the same as the expected fraction of infected type-i vertices once the epidemic has run its course.

We make a few technical remarks about the computation of q 0 . Taking the limit n → ∞ is a well-defined operation since the multitype branching process corresponding to the PGF F is positive regular and non-singular under the assumption that m i > 0 and T ij > 0 for all 1 ≤ i, j ≤ M (for more details, see the discussion at the end of Section III-B). The convergence to the fixed point q 1 is guaranteed as long as the initial condition θ has positive entries [5, Theorem V.2]. Conditioned on the event where the epidemic emerges, this can be safely assumed.

Just as in the PE, there is a phase transition between an ES of zero (q 0 = 1) and a positive ES (q 0,i < 1 for all 1 ≤ i ≤ M ). Noting that the formulas for F and Γ are identical, except that T ij is replaced with T ji , the threshold for the ES is

When (9) is less than or equal to 1, q 1 = q 0 = 1. On the other hand, when (9) is greater than 1, q 0,i < 1 for all 1 ≤ i ≤ M .

It can be seen that (9) is exactly equal to the expression for R 0 defined in (4). We show this by proving T ⊤ m and Tm have the same spectrum, which in turn implies that ρ(T ⊤ m) = ρ(Tm). This can be seen through the stronger property that the characteristic polynomials of the two matrices are identical. Indeed, for any realvalued λ we have

Above, the first equality follows since the spectrum of any matrix and its transpose are the same, and the rest are due to standard manipulations of the determinant. Putting together the results of this section and Section III-C, we have shown that when R 0 ≤ 1, the epidemic dies out in finite time, whereas when R 0 > 1, the epidemic eventually infects a positive fraction of the population.

We next present extensive numerical simulations that validate our theoretical analysis. In all experiments, the contact network was generated via the configuration model with Poisson degree distribution and 1, 000, 000 vertices. We studied several values for the mean degree ranging between 0 and 10. To generate the simulation plots, we took an average over 5, 000 independent trials where, in each trial, a new contact network was generated. We adopt 0.05 as the threshold of epidemic emergence.

We assume there are three types of nodes in the population: surgical mask wearers (type-1), cloth mask wearers (type-2) and people who do not wear any masks (type -3 Figure 2(a) compares the PE when the initial spreader is wearing a surgical mask, a cloth mask, no mask and random (any of the three types). We observe that different types of initiators influence the PE differently. In particular, the PE is lowest when the initiator wears surgical masks which has better inward and outward efficiencies than cloth masks. On the contrary, the probability is highest when the initiator does not wear a mask. This is expected since mask wearing reduces the initial transmissibility of the virus from the initiator to the later propagation. depicts the final fraction of infected population conditioned on epidemic emergence. The total epidemic size is the summation of the three types of infection sizes (no mask, cloth mask, and surgical mask). As the mean degree increases, i.e., when the average number of contacts of people in the network increases, the ES tends to increase. This demonstrates the effectiveness of mitigation strategies such as social distancing in reducing the total size of the infected population during a pandemic. Figure 2 (c) presents the individual infection probability, i.e., the epidemic size of a type divided by the percentage of the same type in the population. The difference between epidemic size and individual infected probability is that the former indicates the fraction of people who are infected and of a certain type, and the latter shows the fraction of infected people within the a certain type.

Epidemic size provides us insights from a global perspective that how the infected population distributes over all types of masks, while individual infection probability gives us a view into each individual type of nodes. It is shown that no-mask wearers suffer from the largest probability of infection, followed by cloth mask and surgical mask wearers. No-mask wearers also has the highest increasing rate as the mean degree of the contact network increases. This trend also conforms with the trend of probability of emergence shown in Figure  2(a) . These results demonstrate the increased risks of infection for people wearing an inferior mask, or no mask at all.

In Figure 3 , we investigate the effect of the baseline transmissibility (i.e., the transmissibility between two non-mask-wearers) on the probability of emergence and expected epidemic size. This is useful in understanding the increased risk of infection based on mask wearing behavior in cases where high-transmissibility variants of the virus may emerge over time, e.g., the Delta variant for COVID-19. In Figure 3 , we use the using the same parameter setting as in Figure 2 except that the mean degree is set to 5 and T varies from 0.1 to 0.9. As T rises, the probability of emergence, epidemic size and individual probability are all seen to increase monotonically. Moreover, similar to Figure 2 , as the original transimissibility T increases, no-mask wearers experience the highest individual infection probability as well as the highest rate of increased risk with respect to increasing baseline transmissibility.

We now leverage our results to examine the effect of masks with different qualities. For example, an interesting question to ask is: what should be the mask wearing strategies in order to mitigate the epidemic in the most efficient manner?

We follow the inward and outward efficiencies of surgical masks and homemade cloth masks suggested by [10] : ǫ in = [0.2, 0.5], and ǫ out = [0.3, 0.5]. Under this setting, surgical masks have better inward and outward efficiencies than cloth masks, which provide us a straightforward separation of good and bad masks. In the later section, we will discuss the individual impact of inward and outward efficiencies in more details. Assume that there are only two types of nodes in the whole population: cloth mask wearers and surgical mask wearers. The proportion of surgical and cloth masks are given by m surgical and m cloth , respectively, where we have m surgical + m cloth = 1. Figure 4 illustrates the effect of changing the proportion of surgical masks from 0.1 to 0.9 under contact networks with four different mean degrees: 8, 10, 15, and 20. When the fraction of surgical mask wearers increases, the probability of pandemic with a random seed and the total epidemic size are decreasing monotonically in all four cases. In Figure 4 (a) (where the mean degree of the contact network is 8), we see that it suffices to have 30% of the population wearing the surgical mask to make the probability of emergence drop nearly to zero; i.e., to ensure that the spreading event is unlikely to turn into an epidemic. In Figures 4(b) -4(d), we see that the percentage of the population wearing surgical masks needs to be at least 50%, 80% and 90%, respectively, in order to make the probability of epidemics nearly zero, when the mean degree increases to 10, 15, and 20, respectively. This shows the trade-off between people having more contacts on average and the percentage of surgical mask wearers in preventing the pandemic. In particular, we conclude that when people are interacting with more contacts on average, significantly more people need to wear highquality masks in order to prevent the spreading process turning into These plots also show non-monotonic trends in the epidemic size among surgical mask wearers when the mean degree is 10, 15 and 20. The is due to the trade-off between the growth of m surgical and the drop of the epidemic size. The right-most column of figure 4 decouples this competition. As m surgical increases, individual infection probability of each mask type decreases monotonically. Probability of emergence (the leftmost column) shares a similar tendency with individual infection probability. Regardless of the virus spreading phase, i.e., either before or after the epidemic emerges, given various qualities of masks, our result demonstrates that it is recommended for the entire population to wear the best quality masks as much as possible to reach a most efficient virus mitigation.

Next, we explore the impact of a fraction of the population not wearing any mask in our results. In Figure 5 , we assume that x% of population do not wear any masks, while the rest wear either surgical or cloth masks. In other words, we set m = [m 1 , m 2 , m 3 ] where m 3 = x/100, m 1 + m 2 = 1 − x/100. We fix the mean degree of the network to be 10, and generate the theoretical prediction and simulation results for probability of emergence and epidemic size with x = 10, 20 and 40. We also plot "individual infection probability" in the right most column in Figure 5 , which is defined as the epidemic size of a given type of nodes (conditionally on the emergence of the epidemic) divided by the proportion of that node type. Put differently, individual infection probability quantifies the odds that an individual will eventually be infected based on their mask wearing behavior. When x = 10, i.e., if 10% of the population does not wear any masks, we see from Figure 5 (a) that even when 80% of the people wear surgical masks, the epidemics still occur with positive probability; contrast this with Figure 4(b) where it was sufficient for 50% of the people to wear surgical masks to have PE equals zero. This means that if there are 10% of no-mask-wearers in the population, epidemics can still occur despite the rest of the population wearing surgical masks. Additionally, when x increases from 10 to 40, the slope of the decreasing trend for probability of emergence and individual infection probability become less steep on average. This phenomenon implies that the larger the percentage of no-maskwearers in the population, the harder for good masks to alleviate the virus spreading. Both observations suggest that regardless of mask quality, mask wearing should be treated as a universal requirement to the entire population for efficient pandemic prevention.

In this section, we explore the trade-off between inward and outward efficiencies of the masks in use. As discussed before, inward efficiency refers to the probability of a mask blocking the pathogen from coming inside the mask, while the outward efficiency refers to the probability of a mask to stop the the pathogen being emitted to the outside world through the mask [11] . The materials and how tight people wearing masks could cause the divergence of the two efficiencies [30] . Vectors tm1 and tm2 represent the inward and outward efficiencies for all types of masks. When inward efficiency for a mask is better than its outward efficiency, we call them inwardgood masks. Similarly, we call masks with higher outward efficiency as outward-good masks. Inward-good masks are more effective for self-protection when the subject is immersed in the environment of virus particles than blocking the virus emitted from the infected person's respiratory system. Similarly, outward-good masks are better at source-control than protection of the wearer. One practical question to ask is: if the government is provided with both inward-good and outward-good masks, what should be the purchasing strategy? Should the government buy all inward-good masks, or outward-good masks? Or, should a more complicated strategy be adopted?

Assume that there are three types of masks among the population: inward-good mask wearers, outward-good mask wearers, and people who don't wear masks, represented as type-1, type-2 and type-3 nodes, respectively. We have the proportion vector of three types as m = [m 1 , m 2 , m 3 ] where m 1 + m 2 + m 3 = 1. To study the impact of mask assignment strategy for inward-good and outwardgood masks, each time we fix the proportion of no-mask-wearers at x%, and vary the proportion of outward-good-mask-wearers m 2 from 0.1 to 1-x/100. The specific efficiency parameters of the masks are selected as: ǫ in = [0.3, 0.7, 1], and ǫ out = [0.7, 0.3, 1]. Figure 6 shows the results of probability of emergence, epidemic size given emergence, and individual infection probability when the mean degree is 10, x =10, 20 and 40. An interesting results is that different strategies work best at different stages of the virus propagation process. Unlike all the previous figures, where both probability with random seed and total epidemic size given emergence show a decreasing trend when m surgical increases, Figure  6 displays an opposite trend between probability of emergence and epidemic size: the probability of emergence of a random seed is reducing monotonically while the total epidemic size is increasing, as the proportion of the outward-good mask wearers m 2 grows. Put differently, outward-good masks are more helpful in terminating the spreading process before the emergence of the pandemic, whereas the inward-good masks are essential to control the infection size when the pandemic already exists.

We believe that this result has implications that go beyond the impact of masks can be applied to other pandemic mitigation strategies including prioritization of vaccines, social distancing measures, and other non-pharmaceutical interventions. Generally, it is seen that at the early stages of the virus spreading, i.e., when the infection fraction has not reached a significant percentage, a source-control oriented strategy is crucial to prevent the epidemic to emerge. On the other hand, if an epidemic has already emerged and a significant fraction of the population has already been infected, it becomes most effective to implement a self-protection oriented strategy to reducing the final fraction of the infected population. It is thus of utmost importance to develop pandemic mitigation strategies with the two distinct stages in mind, with each stage potentially having a different optimum strategy.

In the previous discussion, we have allocated different types of masks to the population randomly, i.e., without any dependence on their degrees, etc. In this section, we seek to understand whether giving high degree nodes (e.g., cities) and low degree nodes (e.g., villages) different types of masks would be more effective in reducing the probability and size of the epidemics compared to randomly allocating the masks. In particular, we assume that x% of the nodes in the population are wearing outward-good masks while the rest are wearing inward-good masks. With nodes ranked according to their degrees from highest to lowest, we consider two different mask allocation strategies: i) top x% of the nodes (with the highest degree) wearing outward-good masks; and ii) bottom x% of the nodes (with the lowest degree) wearing outward-good masks. For comparison with the previous discussion, we also consider the case where the x% of the nodes wearing outward-good masks are selected uniformly at random from the entire set of nodes. Then, we obtain results for the probability of emergence and expected epidemic size as x varies from 0 to 100. Fig. 6 : Probability, Epidemic Size (given emergence) and Individual Infection Probability with increasing proportion of outward-good mask wearers (m 2 ) in the population under a fixed mean degree = 10, for different percentage of no-mask population x: (a)x=10, (b)x=20, (c)x=40. As m 2 grows, the probability of emergence with random seed (red) is decreasing whereas the total epidemic size given emergence (red) is increasing. This shows the two different stages these two metrics represent, where outward-good masks are not helpful before the emergence but after it. The simulation is done by 1, 000, 000 nodes and 5, 000 experiments.

The results are presented in Figure 7 where we see once again an intricate difference between pre-epidemic and post-epidemic stages of the spreading process. As seen in Figure 7 (a), compared to randomly allocating, it is better to assign outward-good masks to low degree nodes for the purpose of reducing the probability of epidemics. Probability reduces by around 0.06 when x = 60 for low degree selection (yellow curve, strategy 2). However, when the goal is the reduce the total fraction of infected nodes given that an epidemic took place, we see from Figure 7 (b) that it is instead better to allocate outward-good masks to high degree nodes. Epidemic size reduces by 0.06 when x = 30 for high degree selection (green curve, strategy 1). A possible explanation for this difference is as follows. If our mask allocation strategy (say, Strategy 1) gives outward-good masks to high-degree nodes (and inward-good masks to low-degree nodes), the probability of transmission from an infected node to a susceptible node will be the smallest from a high-degree node to a low-degree degree, but will be the largest from a low-degree node to a highdegree degree node. The situation will be exactly the opposite if we use a Strategy 2 that gives outward-good masks to low-degree nodes; i.e., the chances of transmission will be the smallest from a low-degree node to a high-degree degree node and largest from a high-degree node to a low-degree degree node. For reducing the probability of epidemics, we need to consider the early stages of the spreading process, particularly the very beginning of it. Since the seed node is selected uniformly at random, its degree will follow the degree distribution of the network. We can expect that when the seed node has a high degree, the chances of it infecting one or more of its neighbors and eventually leading to an epidemic is significant irrespective of the type of the mask they are wearing. If, on the other hand, the seed node has low degree, then there is hope that the spreading process will end early without leading to an epidemic. With this intuition, it can be seen that reducing the transmission probability from low-degree nodes to high-degree nodes would be most effective in reducing the probability of epidemics, which is achieved by Strategy 2 (i.e., by giving outward-good masks to low-degree nodes). Put differently, giving outward-good masks to high-degree nodes (which is the case for Strategy 1) leads to a situation where an infected low degree node has a high chance of transmitting the virus to a high-degree susceptible node, increasing the probability of a single node initiating a spreading process that leads to a epidemic. This intuition is confirmed in Figure 7 (a) where we see that Strategy 1 leads to a higher probability of epidemics than the case where masks are allocated randomly without any dependence on node degrees.

For reducing the expected size of an epidemic that has already taken place, we can explain the results seen in 7(b) in a similar fashion. In this post-epidemic stage of the spreading process, there is already a critical mass of infected individuals and we can expect to have little chance of preventing a high-degree node from being infected irrespective of the mask type they are wearing; e.g., out of the many of their neighbors, several would be expected to be infected in the post-epidemic stage and at least one would be likely to infect this node irrespective of the corresponding mask types. Thus, the only hope for reducing the epidemic size would be to protect low degree nodes from being infected by high degree virus-spreaders. As previously mentioned, the probability of transmission from a highdegree node to a low-degree node is the smallest in Strategy 1 where outward-good masks are allocated to high-degree nodes.

Summarizing, we see that at the early stages of the spreading process, it is critical to protect high degree nodes from being infected, while after the epidemic already forms, protecting other nodes from infected high degree nodes helps more in suppressing the extension of the virus. This result echoes with our previous findings that mitigation strategies for spreading processes should be treated with two different stages in mind, and by focusing on source-control before epidemic starts and self-protection after epidemic forms. 10 Fig. 7 : The probability of the emergence (a) and epidemic size given emergence (b) with degree selection of outward-good masks. Suppose only two mask types exist in the entire population: outward-good and inward-good. Three outward-good masks allocation schemes are considered: randomly allocate x percent of nodes in the population to wear outward-good masks (red); Allocate top x percent of high degree nodes with outward-good masks (green); Allocate top x percent of low degree nodes with outwardgood masks (yellow). x is the ratio of outward-good masks. Simulation results obtained by 1, 000, 000 nodes and 5, 000 experiments.

In this paper, we studied an agent-based model for viral spread on networks called the multi-type mask model in which agents wear masks of various types, leading to heterogenous probabilities of receiving and transmitting the virus. We performed a theoretical analysis of three important quantities: the probability of emergence (PE), the epidemic threshold (R 0 ) and the expected epidemic size (ES). We validated our theoretical results by comparing them against simulations and found a near-perfect match between them.

We then utilized the model to investigate the impact of maskwearing in realistic settings related to the control of viral spread. First, we studied the effect of allocating superior and inferior masks (e.g., surgical and cloth masks) within the population and found, naturally, that a greater prevalence of surgical masks significantly reduces the PE and ES. Interestingly, we also found that when there is a significant fraction of non-mask-wearers, increasing the fraction of superior masks among mask-wearers does not significantly reduce the probability of emergence or expected epidemic size. This highlights the importance of wearing some mask -even a potentially low-quality one -in mitigating the spread of a virus.

Next, we examined the trade-offs between masks which are good at preventing viral transmission from an infected neighbor (inward-good masks) and masks which are good at preventing viral transmission to susceptible neighbors (outward-good masks). Strikingly, we find that each type of mask is good for controlling different epidemiological quantities. Specifically, outward-good masks reduce the PE, making them ideal for controlling the early-stage spread of the virus. On the other hand, inward-good masks reduce the ES, hence they are most effective in mitigating the impact of an already-large pandemic. This distinction justifies the importance of source-control during the early propagation stage and self-protection when a relatively large percentage of population is infected.

We further investigated mask assignment based on node degree. In previous discussion, mask assignment is independent from node degree. Now we select masks for nodes according to their degrees. We find that assigning high degree nodes with inward-good masks and low degree nodes with outward-good masks can effectively reduce probability of emergence than the opposite or random allocation schemes. In contrast, for epidemic size, high degree nodes wearing outward-good masks and low degree nodes wearing inward-good masks is more helpful in suppressing the epidemic extension. The finding gives us insights on how to allocate inward-good and outwardgood masks to crowded areas and less populated areas in different stages of virus spread. Interestingly, the results also point out the distinct and changing roles that high degree nodes and low degree nodes are playing. Before the epidemic forms, low degree nodes source control is critical in preventing the epidemic from happening, by which we can remove the additional paths for a single seed to initiate the epidemic. After the epidemic already exists, high degree nodes is more important in extending the epidemic size. In this case giving high degree nodes outward-good masks and self-protection of low degree nodes become important.

Finally, we remark that there are several avenues for future research. There are several ways to augment the model, for instance by considering networks with community structure, multi-type networks, and the effect of multiple strains of the virus propagating. Besides, our model can be adapted to directed graphs to encode directional transmission as well. For example, considering vaccination, vaccinated individuals may still transfer the virus to others but they may not be infected themselves. We also plan to carry out our analysis using real-world networks, mask efficiencies and viral parameters in order to understand the implications of our model on the ongoing pandemic.

The impact of mask-wearing and shelter-in-place on COVID-19 outbreaks in the United States | Elsevier Enhanced Reader

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