Severe Acute Respiratory Syndrome (SARS) is a highly contagious airborne and nosocomial disease caused by the coronavirus SARS-CoV [50]. SARS was a major source of concern when it first emerged in 2003 [15]. Overall, SARS accounted for over 770 deaths and 8000 infections globally [44]. SARS has a variety of symptoms that are similar to those of influenza, including headaches, body aches, rash, and fever [39]. SARS has a high mortality (about 15%) especially among the elderly (about 50%) [39]. There is no current treatment or vaccine for SARS, but there are human vaccines in development and a successful animal vaccine has been tested [34, 43]. Although the last reported case of SARS was in 2004, the threat of its possible return cannot be understated, especially given the recent emergence of SARS-CoV2.

Mathematical models are now a common tool used to understand and control the spread of infectious diseases [2, 57]. There are many models for SARS control, including [28, 27, 18, 42, 53, 45, 60, 31, 63, 46] and there are thousands of models for COVID-19 [38].

In their seminal paper [7], the authors incorporated game theory into epidemics modeling. With the addition of voluntary disease prevention, the game-theoretic models study scenarios in which self-interested individuals take actions based on the decisions of the rest of the population. In extensive reviews [61], [59] and [16], it is argued that incorporating human behavior provides more insight and better predictions than through the standard compartmental models.

There are many ways in which the human behavior can be incorporated into the underlying epidemiological models [61]. They include population level mean-field games where the population is assumed homogeneous and the human decisions are based on (i) the perceived risks of contracting the disease; (ii) the actual spread of the disease; (iii) the perceived risks of vaccine-related side effects. For example, [22] considered the effects of information-dependent vaccine uptake in a standard SIR compartmental model. On the other hand, [6] used imitation dynamics to create a behavior-explicit model of vaccinating behavior while [8] created the behavior-explicit game-theoretic models of vaccinating behavior. In this manuscript we follow the the theoretical framework started in [8] and expanded in [7]. However, we note that there are many other models of vaccination in structured and networked populations, such as [24].

To our knowledge, no game-theoretic model of SARS transmission exists in the literature. Thus, the main objective of this paper is to fill this gap and implement a game-theoretic model of voluntary precautions against SARS. We follow the compartmental model of [28] and assume that susceptible individuals can wear masks and/or limit contacts with others in order to decrease their chances of contracting SARS. We study the effects of such actions on the spread of SARS.

Our approach is similar to [62] who considered strategic choices during disease outbreak and applied their work to SARS. They considered an infinite population divided into “local population" and “potential travelers". Each of the two subpopulations were further divided into the usual susceptible, infected, and recovered compartments. The potential travelers played a game in which they had to decide whether to travel or not. By analyzing the resulting system of ordinary differential equations, the authors were able to assign payoffs to the players. They solved the game for Nash equilibria and they found that the optimal strategy for an individual can sometimes coincide with the optimal strategy for the group. However, when the optimal strategies differ, the difference is very large and highly sensitive to changes in parameter values. The authors concluded that centrally imposed, rather than voluntary, travel restrictions are needed to prevent SARS outbreaks. Also, [29] and [30] investigated a vaccination game in an explicitly finite population. They applied their model to HIV/AIDS. In their game, the individuals engaged in safe or risky behavior and, if infected, they could buy treatment (if the treatment was available; the game also included a drug company as additional player). The players received an idealized payoff based on their choices. The authors constructed an extensive game which they solved by backward induction for optimal strategies.

Similarly to [29], we consider a game in which individuals can engage in safe (wear a mask and/or socially distance) or risky (do not wear a mask and do not socially distance) behavior. Unlike [62], where only part of the population played the game, in our model, every person is a player. Unlike [29], we do not consider other players (such as mask producers) and we evaluate the payoffs to the players based on the underlying epidemiological model of SARS transmission, similarly to what was done in [62].

2.1. Compartmental ODE model

We follow the model from [28] which is an extension of the classical SIR model [20, 57, 13]. The population is divided into six classes or compartments:

• (1)

  Susceptible (S); individuals without the SARS pathogen,

• (2)

  Exposed asymptomatic (E); asymptomatic individuals who have been exposed to the

• (3)

  Infectious symptomatic (I); individuals who developed symptoms and are not in isolation,

• (4)

  Quarantined asymptomatic (Q); those are asymptomatic exposed individuals who are

• (5)

  Isolated symptomatic (J); individuals who developed clinical symptoms and are isolated (for example by hospitalization or staying at home),

• (6)

  Recovered (R); individuals who were infected in the past and possess lasting immunity against SARS.

The symbols S,E,I,Q,J,R will be used not only to denote the compartments, but also the number of individuals in the compartments. The total population size will be denoted by N=S+E+I+Q+J+R.

The individuals enter the population as susceptible at rate Π. This parameter represents a net inflow into the region at per unit time, including new births, immigration and emigration. Individuals in any class can die of natural and SARS unrelated causes at rate μ. As in [28], we ignore the net inflow of individuals in any other class.

Susceptible individuals become exposed when they come in contact with infectious symptomatic, exposed asymptomatic, quarantined asymptomatic, or isolated symptomatic individuals. Without any precautions, the transmission rates are β, εEβ, εQβ , εJβ, respectively; here β is the transmission coefficient which includes both the contact rate and the transmission probability and εX is a modification factor for the class X∈{E,J,Q}. However, the individuals may take some precautions such as wearing masks, social distancing, or avoiding gatherings. We denote such precaution efforts by epop and, similarly as in [21, 17, 14, 54] we assume that the transmission rate is reduced by the factor (1-epop). The total force of infection is thus given by

The exposed asymptomatic individuals develop symptoms at rate κ1. They can also enter quarantine at rate γ1. As discussed in [28], in reality some susceptible individuals will likely enter quarantine, but for the purpose of our model we neglect this possibility and assume that all quarantined individuals were truly exposed and, in time κ2-1, develop symptoms and become isolated symptomatic. This assumption means that our model slightly overestimates the number of susceptible individuals.

The infectious symptomatic individuals may recover at rate σ1, enter isolation at rate γ2, or die of SARS related causes at rate d1. The isolated individuals recover at rate σ2 but can also die of SARS related causes at rate d2. We assume d1>d2 and σ2>σ1 because isolated individuals are more likely to receive treatment [12, 42].

The dynamics are summarized in Figure 1. The notation is summarized in Table 1. The parameter values are adopted from [28], specifically the Hong Kong outbreak [42].

Figure 1. Scheme of the compartmental ODE model for SARS transmission. The compartments represent Susceptible (S), Exposed and asymptomatic (E), Infectious and symptomatic (I), Quarantined asymptomatic (Q), Isolated symptomatic (J), and Recovered (R). Solid arrows represent the transitions between compartments. The letters next to the arrows specify the per capita rates of the transitions. The force of infection λ is given by (1). The parameters are explained in Table 1.

Table 1 . Model parameters. The rates are per capita per day. The values were all adopted from [28]. Ranges were estimated from the values..

Symbol	Meaning	Value	Range
Π	Net inflow of susceptible individuals	221	[100,300]
β	Transmission rate	0.15	[0.1,0.25]
μ	Natural death rate	180*365	190*365,170*365
κ1	Rate at which not quarantined exposed individuals develop symptoms	0.1	[0.05,0.15]
κ2	Rate at which quarantined exposed individuals develop symptoms	0.125	[0.1,0.15]
γ1	Rate of quarantining of people who have been in contact with an infected individual	0	[0,0.15]
γ2	Rate at which symptomatic individuals seek medical attention and are put into isolation	0	[0,0.75]
σ1	Recovery rate of infectious individuals	0.0337	140,120
σ2	Recovery rate of isolated individuals	0.0386	140,120
d1	SARS-induced death rate of I	0.0079	[0.001,0.02]
d2	SARS-induced death rate of J	0.0068	[0.001,0.02]
εE	Transmission coefficient for E	0	[0,0.18]
εQ	Transmission coefficient for Q	0	[0,0.18]
εJ	Transmission coefficient for J	0	[0,0.84]
epop	Precaution level in the population	variable
C	Cost of precautions (relative to the cost of SARS)	variable

The model yields the following system of differential equations.

Above, we assume that epop is constant in time.

2.2. Equilibria of the dynamics

The disease-free equilibrium (DFE) the system (2) – (7) is given by

The effective reproduction number, R, is derived in A.1 and is given by

where

is the basic reproduction number (in a population without any precautionary efforts) and

In the endemic equilibrium (EE), as derived in A.2, the force of the infection can be expressed as

where

The endemic equilibrium is then given by

While we do not focus on stability of the equilibria, it follows directly from [58] that the DFE is locally asymptotically stable if R<1. Also, by (15), the EE exists only if R>1. Based on the numerical simulations, we believe that DFE is globally asymptotically stable if R≤1 and EE is globally asymptotically stable if R>1.

2.3. Game-theoretic framework

We incorporate the game-theoretic component into the above transmission dynamics as done in [7].

A game is played by susceptible individuals who decide how many precautions they should take. All individuals are assumed to be rational, have complete information about the SARS epidemic, and act in their own interest. They evaluate costs and benefits of their own action and the actions of others. They choose the action that maximizes their own net payoffs (benefits minus costs).

We focus on a single individual using a strategy e when the rest of the population uses a strategy epop. If the population is large enough, the decision of a single individual will not have a significant impact on the steady state of the dynamics. Thus, the force of infection will be either 0 if R<1 and the population is in the disease-free equilibrium, or it is given by (1) if R>1.

The focal individual uses e instead of epop and thus their rate of becoming infected is λ=1-e1-epopλ*. Hence, the probability of a susceptible individual becoming exposed is given by

There is a cost associated with social distancing [51]. Let C be the cost of complete prevention (e=1) relative to the cost of getting infected by SARS. Let us assume that the cost of the precautionary efforts is linear in e. The payoff to an individual using e in a population where everyone else uses epop is given by

This equation was developed in [7] and specifies that an individual using strategy e has to pay a cost Ce, and even with this level of precaution, the individual can still contract the disease (and pay a unit cost 1) with probability πS→E. The expected payoff (expressed in the units of the disease cost) is given by (24).

In accordance with the static vaccination game model in [7], it is assumed that all individuals make their choices once, simultaneously, and based on the payoffs in the equilibrium of the dynamics (2)–(7). This is a standard assumption successfully used in many scenarios, including models of influenza epidemics [25, 52]. As a result, epop can be treated as a constant. There are ways to consider a more dynamical approach and explicitly model the time evolution of the protective strategies; see for example [6, 41, 9, 19, 10, 49, 5, 1].

The focal individual tries to maximize P(e,epop) given in (24). If the basic reproduction number, R0, given by (10), is such that R0<1, then the population will reach disease free equilibrium even if nobody uses any precautions. Hence, e=0 is the Nash equilibrium of the game.

However, for SARS, R0>1. Let us denote

the minimal precautionary efforts needed to achieve R≤1. If epop>eHI, then R<1 and the optimal action for the focal individual is e=0.

Hence, we will assume epop<eHI. We have

Consequently, the function P(e,epop) has a maximum either at e=0 or e=1 and the Nash equilibrium is a solution of

i.e., the solution of

As derived in A.3, the Nash equilibrium is given by

where

For any parameter values, the Nash equilibrium eNE, i.e., the optimal precautionary level is given by (30). The dependence of eNE on C is illustrated in Figure 2(a). When C=0, eNE=eHI. As C increases, eNE decreases. When C reaches the critical value Ccrit, eNE=0.

Figure 2. (a) Values of the optimal individual precautionary levels eNE as a function of the relative cost of the precautions C. (b) Values of the effective reproduction number R when the population uses eNE. Parameter values are as specified in Table 1; the rate at which non-quarantined cases develop symptoms varies from κ1=0.05 (blue dotted), κ1=0.075 (blue dashed), κ1=0.1 (black solid), κ1=0.125 (red dashed), κ1=0.15 (red dotted).

Similarly, we can investigate the value of the reproduction number at the population that uses the Nash equilibrium level of precautions. This is shown in Figure 2(b). When C≈0, R≈1 but still larger than 1. As C increases, R increases and reaches R0 when C≥Ccrit.

We performed uncertainty and sensitivity analysis using the Latin hyper-cube sampling with partial rank correlation coefficient (LHS-PRCC) scheme [11, 55, 47, 37]; see also A.4.

Figure 3(a) shows the results of uncertainty analysis, i.e., the distribution of eNE among all the sampled parameter values. The most frequent values of eNE are between 0.5 and 0.7 and the average value is approximately 0.47.

Figure 3. Uncertainty and sensitivity analysis. (a) Uncertainty of eNE, the average is around 0.36, (b) sensitivity of eNE, (c) uncertainty of Ccrit, the average is about 0.4, and (d) sensitivity of Ccrit . The parameter ranges are as in Table 1. Only parameters with sensitivity over 0.05 are shown in figures (b) and (d).

Figure 3(b) shows the sensitivity of eNE on various parameters. There is a strong negative correlation between the optimal voluntary precaution level eNE and the cost of the precautions, C. As seen above in Figure 2, increasing C decreases eNE. Increasing recovery rate, σ2, or isolation rate, γ2, also causes eNE to decline. On the other hand, increasing the transmission rate, β, or the transmission coefficient from isolated individuals, εJ, causes eNE to increase. The influence of other parameters on eNE is relatively small.

Figure 3(c) shows the results of uncertainty analysis and the distribution of Ccrit. The most frequent values of Ccrit are between 0.5 and 0.8; the average value is approximately 0.54. Figure 3(d) shows the sensitivity of Ccrit on various parameters. The sensitivity indices are very similar to the sensitivity of eNE.

Our model has limitations connected to the assumption of infinite homogeneous population. While this assumption is common in epidemiological modeling, other approaches are possible.

For example, the use of simulations, such as in [33, 35, 36, 40, 4, 3, 32], could allow for flexibility and realism in the modeling approach, taking into account both geographical and social heterogeneity of an explicitly finite population.

In our game-theoretic setup, we assumed that the cost is linear. As demonstrated in [54], the assumption is not necessary. If the cost is assumed to be an increasing and concave down function, then the analysis and results would be qualitatively similar. It is unclear how much the result would change if no assumptions about the second derivative were made.

Finally, to evaluate the payoffs of the game, we assumed that the underlying transmission dynamics reached an equilibrium. This assumption can be relaxed by considering adaptive dynamics as done in [6]. This dynamical approach can yield more complex outcomes. For example, temporal oscillations in the use of protective strategies can emerge; this was demonstrated for example on the use of insecticide treated nets in the prevention of visceral leishmaniasis [23] and malaria [41].

In this paper, we adapted the compartmental model of SARS transmission developed by [28]. We added the game-theoretic component incorporating voluntary precautions such as masking or limiting contacts with others. The model shows that the optimal level of precautions significantly depends on the individual costs of such actions relative to the cost of SARS. The more costly those actions are (in true or perceived costs), the lower the level of precautions the individuals will adopt. As soon as the costs rise above a relatively low threshold, the individuals will not voluntarily use any precautions. Since the cost is expressed relative to the cost of SARS, a decrease of the actual or perceived SARS infection has the same effect as the increase of the cost of precautions.

This finding is in agreement with what we see happening for COVID-19. When COVID-19 first appeared and was considered very serious, people isolated and masked as much as possible. However, as the perceived negative effects of COVID-19 continued to decline, the recommended levels of precautions were also dropping.

Another result of our model is that voluntary precautions alone are not enough to reduce the reproduction number to a values less than 1, i.e., voluntary precautions alone cannot eliminate SARS nor mitigate the risks of outbreaks. That is in agreement with the general game considered in [26] as well as models for specific diseases, e.g., yellow fever [14]. In these cases, the cost of prevention is high relative to the cost of the disease. However, and this is seen from our model as well, when the relative cost is quite low, voluntary precautions may lower the reproduction number close enough to 1 so that the disease incidence or prevalence is no longer a public health concern. This phenomenon was already observed in models of many vector-borne diseases, including dengue [21], and lymphatic filariasis [54].

Brian Lee and Aurienne Maria-Claire Reyes Cruz were supported by the Young Scholars Research Experience in Mathematics (YSREM) through the MAA Tensor SUMMA Program and Regina-Mae Dominguez by the MAA National Research Experience for Undergraduates Program (NREUP) provided by the National Science Foundation (grant DMS-1652506) awarded to H.O. and L.J.C.A. H.O. and L.J.C.A were further supported by the NSF grant DMS-2327791. D.T. was supported by the National Science Foundation grants DMS-1950015 and DMS-2327790 and the National Security Agency grant H98230-23-1-0002. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

L.J.C.A - writing - original draft, methodology, supervision, conceptualization

A.C. - writing - original draft, formal analysis

R.M.D. - writing - original draft, formal analysis

B.L. - writing - original draft, formal analysis

H.O. - writing - original draft, writing - review and editing, methodology, supervision, conceptualization, formal analysis, funding acquisition

J.R. - writing - original draft, writing - review and editing, software, visualization, methodology, supervision, conceptualization, resources

D.T. - writing - original draft, writing - review and editing, methodology, supervision, conceptualization, formal analysis, visualization, funding acquisition

A. Calculations

A.1. Reproduction number

The reproduction number is derived using the next-generation matrix method [58]. We order the compartments with the infections as E, I, Q and J. The new infections only appear in E, and thus

The vector V considers the other transfers out of these infectious compartments minus all the transfers coming into the compartments other than new infections. Thus,

The Jacobian matrix of F evaluated at DFE is given by

and the Jacobian matrix of V is

Let us denote

We get

Thus, the effective reproduction number R, calculated as the spectral radius of the next-generation matrix, FV-1, is given by

We note that [28] used an alternative, direct approach that yields the same formula for R.

A.2. Endemic equilibrium

Setting the time derivative to 0 in (2)–(7) yields

where

Starting at (42) and going one equation at a time, we get

Thus, we get

Setting

(56) yields

By (58) and (50)–(54), (48) becomes

Hence,

A.3. Nash equilibrium

As derived in the main text, the Nash equilibrium has to solve

By (61) and (9),

where

Thus, (62) becomes

This is equivalent to

and solving for epop yields

Moreover, the left-hand side of (65) can be rearranged as

which is decreasing in epop. So, the maximum occurs at epop=0 and it is

When C>Ccrit, the equation (65) does not have a positive solution and thus eNE=0.

Also, all of the above calculations made sense for R0>1. When R0≤1, the dynamics tends to the disease-free equilibrium and thus it makes sense to consider Ccrit=0.

To summarize, there is a critical value Ccrit for the cost C given by

A.4. LHS-PRCC

This section contains basic information on the uncertainty and sensitivity analysis using the LHS-PRCC scheme [11, 55]. The full description of the LHS-PRCC scheme is in [47] and the MATLAB and R implementations is in [37]. A shorter description such as the one below, can be found, for example, in [56].

For the Latin Hyper-cube Sampling (LHS), the parameter ranges are divided into intervals of equal probability. Those intervals are then sampled, independently for each parameter. LHS is also called a stratified sampling without replacement. The sampling yields an unbiased estimate of the expected model output. Compared to a simple random sampling, it needs fewer samples to achieve the same accuracy [48].

PRCC, the partial rank correlation coefficient, between a model parameter P and the model output O is defined by

where RP and RO are residuals of the rank-transformed linear regression models for P and O.