Braess's Paradox in Epidemic Game: Better Condition Results in Less Payoff
Hai-Feng Zhang, Zimo Yang, Zhi-Xi Wu, Bing-Hong Wang, Tao Zhou
aa r X i v : . [ phy s i c s . s o c - ph ] M a y Braess’s Paradox in Epidemic Game: Better Condition Results in LessPayoff
Hai-Feng Zhang, Zimo Yang, Zhi-Xi Wu, Bing-Hong Wang, and Tao Zhou ∗ School of Mathematical Science, Anhui University, Hefei 230039, P. R. China Web Sciences Center, University of Electronic Scienceand Technology of China, Chengdu 611731, P. R. China Institute of Computational Physics and Complex Systems,Lanzhou University, Lanzhou 730000, P. R. China Department of Modern Physics, University of Science and Technology of China, Hefei 230026, P. R. China
Facing the threats of infectious diseases, we take various actions to protect ourselves, butfew studies considered an evolving system with competing strategies. In view of that, wepropose an evolutionary epidemic model coupled with human behaviors, where individualshave three strategies: vaccination, self-protection and laissez faire, and could adjust theirstrategies according to their neighbors’ strategies and payoffs at the beginning of each newseason of epidemic spreading. We found a counter-intuitive phenomenon analogous to thewell-known
Braess’s Paradox , namely a better condition may lead to worse performance.Specifically speaking, increasing the successful rate of self-protection does not necessarilyreduce the epidemic size or improve the system payoff. This phenomenon is insensitiveto the network topologies, and can be well explained by a mean-field approximation. Ourstudy demonstrates an important fact that a better condition for individuals may yield aworse outcome for the society. ∗ Electronic address: [email protected]
I. INTRODUCTION
Recent outbreaks of global infectious diseases, including SARS (Severe Acute RespiratorySyndrome), H1N1 (Swine Influenza) and H5H1 (Avian Influenza), have caused major publichealthy threats owing to their potential mortalities and substantial economic impacts. Accord-ing to the report of WHO, infectious diseases cause more than 10 million deaths annually andaccounting for 23% of the global disease burden [1]. Various interventions thus have been de-veloped to control infectious diseases, such as vaccination, treatment, quarantining and behaviorchange programs (e.g., social distancing and partner reduction) [2].Though preemptive vaccination is the fundamental method for preventing transmission of in-fectious diseases as well as reducing morbidity and mortality [3–5], practically, the immunizationof individuals is more than a voluntary behavior owing to the economic costs, logistical limita-tions, religious reasons, side effects, and so on [6]. Therefore, instead of vaccinating, people mayprefer to take some self-protective actions including reducing outside activities, detouring to avoidepidemic areas, wearing face masks, washing hands frequently, and so forth [7–10]. Generallyspeaking, these self-protective actions are less costly and cannot guarantee the safety against thediseases.Under such complicated environment, an individual’s strategy usually results from a tradeoffbetween cost and risk. For instance, people may be laissez-faire to the spreading of common flu,while they will take vaccination for hepatitis B since the vaccines are very effective and hepati-tis B is very difficult to be eradicated. In contrast, people prefer to take self-protection againstHIV since its consequence is terrible while the effectivity and side effects of vaccines are bothunknown. Accordingly, game-theoretic models may be suitable to characterize these decision-making processes [3, 4, 11–14]. Bauch et al. [3, 4] analyzed population behavior under voluntaryvaccination policies for childhood diseases via a game-theoretic framework, and they found thatvoluntary vaccination is unlikely to reach the group-level optimum due to the risk perception invaccines and the effect of herd immunity. Bauch [11] studied a game model in which individualsadopt strategies according to an imitation dynamics, and found that oscillations in vaccine uptakecan emerge under different conditions, for example, vaccinating behavior is very sensitive to thechanges in disease prevalence. Vardavas et al. [12] considered the effects of voluntary vaccinationon the prevalence of influenza based on a minority game, and found that severe epidemics couldnot be prevented unless vaccination programs offer incentives. Basu et al. [13] proposed an epi-demic game model for HPV vaccination based on the survey data on actual perceptions regardingcervical caner, showing that the actual vaccination level is far lower than the overall vaccinationgoals. Perisic and Bauch [14] studied the interplay between epidemic spreading dynamics andindividual vaccinating behavior on social contact networks. Compared with the homogeneouslymixing model, they found that increasing the neighborhood size of the contact network can elim-inate the disease if individuals decide whether to vaccinate by accounting for infection risks fromneighbors.As mentioned above, in most related works, individuals are usually divided into two oppositeclasses: vaccinated and laissez-faire, while less attention is paid on other alterative strategies inbetween. In this paper, we propose an evolutionary epidemic game model to study the effects ofself-protection on the system payoff and epidemic size. We find a counter-intuitive phenomenonanalogous to the well-known
Braess’s Paradox [15] in network traffic, that is, the increasing ofsuccessful rate of self-protection may, on the contrary, decrease the system payoff. We provide amean-field solution, which well reproduces such observations. This study raises an unprecedentchallenge on how to lead the masses of people when facing an epidemic, since sufficient knowl-edge about and effective protecting skills to the infectious disease, which sound very helpful forevery individual, may eventually enlarge the epidemic size and cause losses for the society.
II. MODEL DEFINITION
Considering a seasonal flu-like disease that spreads through a social contact network [16, 17].At the beginning of a season, each individual could choose one of the three strategies: vaccination,self-protection or laissez faire. If an individual gets infected during this epidemic season, she willpay a cost r . A vaccinated individual will pay a cost c that accounts for not only the monetarycost of the vaccine, but also the perceived vaccine risks, side effects, long-term healthy impacts,and so on. We assume that the vaccine could perfectly protect vaccinated individuals. A self-protective individual will pay a less cost b , while a laissez-faire individual pays nothing. Denote δ the successful rate of self-protection, that is, a self-protective individual will be equivalent to avaccinated individual with probability δ or be equivalent to a laissez-faire individual with proba-bility − δ . This will be determined right after an individual’s decision for simplicity. Obviously, r > c > b > . Without loss of generality, we set the cost of being infected as r = 1 . Table 1presents the payoffs for different strategies and outcomes. TABLE I: The payoffs for different strategies and outcomes.Healthy InfectedLaissez-faire − Self-protected − b − − b Vaccinated − c N/A
When the strategy of every individual is fixed, all individuals can be divided into two classes:susceptible ones including laissez-faire individuals and part of self-protective individuals, andirrelevant individuals (they are equivalent to be removed from the system) including vaccinatedones and all other self-protective individuals. The susceptible individuals probably get infectedwhile the irrelevant individuals will not affect or be affected by the epidemic dynamics. Amongall susceptible individuals, I individuals are randomly selected and set to be infected initially. Thespreading dynamics follows the standard susceptible-infected-removed (SIR) model [18], where ateach time step, each infected individual will infect all her susceptible neighbors with probability λ ,and then she will turn to be a removed individual with probability µ . The spreading ends when noinfected individual exists. Then, the number of removed individuals, R ∞ , is called the epidemicsize or the prevalence.After this epidemic season, every individual updates her strategy by imitating her neighbor-hood. Firstly, she will randomly select one neighbor and then decide whether to take this neigh-bor’s strategy. We apply the Fermi rule [19, 20], namely an individual i will adopt the selectedneighbor j ’s strategy with probability W ( s i ← s j ) = 11 + exp[ − κ ( P j − P i )] , (1)where s i means the strategy of i , P i is i ’s payoff in the last season, and the parameter κ > characterizes the strength of selection: smaller κ means that individuals are less responsive topayoff difference. After the moment all individuals have decided their strategies (and thus theirroles in the epidemic spreading are also decided), a new season starts. III. RESULTS
We first study the model on square lattices with von Neumann neighborhood and periodicboundary conditions. Figure 1(a) presents the effects of the successful rate of self-protection, δ ,on the decision makings of individuals and the epidemic size. Clearly, as the increasing of δ ,the condition gets better and better. A counter-intuitive phenomenon is observed when δ lies inthe middle range (from about 0.3 to about 0.4), during which a better condition leads to a largerepidemic size. One may think that though the epidemic size becomes larger, the system payoff(the sum payoff of all individuals) could still get higher since individuals pay less in choosingself-protection than vaccination. However, as shown in figure 1(b) and 1(c), the system payoffis strongly negatively correlated with the epidemic size. That is to say, a better condition (i.e., alarger δ ) could result in worse performance in view of both the larger epidemic size and the lesssystem payoff. This is very similar to the so-called Braess’s Paradox , which states that addingextra capacity to a network when the moving entities selfishly choose their route, can in somecases reduce overall performance [15, 21].Figure 2 shows the strategy distribution patterns of four representative cases. When δ is small, itis unwise to take self-protection because of its low efficiency, and people prefer to take vaccinationor laissez faire. As shown in figure 2(b), there are only two strategies, vaccination and laissez faire,and thus δ has no effect on the epidemic size. Both infected and not infected laissez-faire individ-uals form percolating clusters, and are nearly (not fully) separated by vaccinated individuals. Ofcourse, this kind of partial separation can only be possible when the number of vaccinated individ-uals is considerable. When δ gets larger, more and more individuals take self-protection and fewerand fewer individuals take vaccination or laissez faire. Since only a fraction, δ , of self-protectiveindividuals are equivalent to the vaccinated individuals, the system contains more susceptible indi-viduals. In addition, these susceptible individuals are less protected since the number of irrelevantindividuals becomes smaller. Such two factors lead to the increase of the epidemic size and thedecrease of the system payoff. As shown in figure 2(c), the light-red percolating cluster is frag-mented into pieces due to the decrease of irrelevant individuals, which is also a reason of thedecrease of the fraction of laissez-faire individuals: being laissez-faire becomes more risky now.When δ is large, the superiority of self-protection becomes more striking and no one takes vacci-nation, then the epidemic size decreases as δ increases. As shown in figure 2(d), self-protectiveand laissez-faire individuals coexist. As the increasing of δ , though the self-protection strategy ismore efficient, the laissez faire strategy is more attractive since the irrelevant individuals becomesmore and thus for susceptible individuals, the risk of being infected becomes smaller. This is thereason why the fraction of laissez-faire individuals become more and more in the right range. Infact, when δ is very large, the laissez-faire and not infected individuals again form a percolatingcluster. Please see figure 2(e) for the example case at δ = 0 . .Figures S1-S5 verify the universality of the counter-intuitive phenomenon. Figure S1 reportsthe epidemic size as a function of δ for square lattices with different sizes, suggesting that ourmain results are insensitive to the network size. To verify the insensitivity to network structures,we implement the model on disparate networks including the Erd¨os-R´enyi (ER) networks [22], theBarab´asi-Albert (BA) networks [23] and the well-mixed networks (i.e., fully connected networksor called complete networks). As shown in figure 3, in despite to the quantitative difference, thecounter-intuitive phenomenon is observed for all kinds of networks. Figures S2-S5 present system-atical simulation results about the effects of different parameters on different kinds of networks.For every kind of networks, one can observe the counter-intuitive phenomenon when the condition < b < c < is hold.Although the phenomenon is qualitatively universal for different kinds of networks, as shown infigure 3, there are quantitative differences between square lattices and other kinds of networks: (i)in ER, BA and well-mixed networks, the self-protection strategy gets promoted and could becomethe sole strategy in a certain range of δ ; (ii) the epidemic size in ER, BA and well-mixed networks issmaller than that in square lattices. In square lattices, laissez-faire individuals could form clustersthat are guarded by the surrounding irrelevant individuals. Then they paid nothing but can escapefrom the infection. On the contrary, ER, BA and well-mixed networks do not display localizedproperty and thus to choose laissez-faire strategy is of high risk. Therefore, with delocalization,the laissez-faire strategy is depressed while the self-protection strategy gets promoted and lessindividuals will get infected.To verify the above inference, we remove a number of edges in the square lattice and randomlyadd the same number of edges. During this randomizing process, the network connectivity isalways guaranteed and the self-connections and multi-connections are always not allowed. Thenumber of removed edges, A , can be used to quantify the strength of delocalization. As shownin figure 4, with the increasing of A , the self-protection strategy gets promoted and the clustersof not infected laissez-faire individuals are fragmented into small pieces. When A gets larger andlarger, the strategy distribution pattern becomes closer and closer to that of ER, BA and well-mixednetworks. The gradually changing process in figure 4 clearly demonstrates that the main reasonresulting in the quantitative differences is the structural localization effects. In a word, the ER, BAand well-mixed display essentially the same results since they do not have many localized clusters.Lastly, we present an analytical solution based on the mean-field approximation for well-mixednetworks (see Methods ), which could reproduce the counter-intuitive phenomenon. Figure 5 com-pares the analytical prediction with simulation, indicating a good accordance.
IV. DISCUSSION
Spontaneous behavioral responses to epidemic situation are recognized to have significantimpacts on epidemic spreading, and thus to incorporating human behavior into epidemiologicalmodels can enhance the models’ utility in mimicking the reality and evaluating control measures[24–30]. To this end, we proposed an evolutionary epidemic game where individuals can choosetheir strategies towards infectious diseases and adjust their strategies according to their neighbors’strategies and payoffs.Strikingly, we found a counter-intuitive phenomenon that a better condition (i.e., larger suc-cessful rate of self-protection) may unfortunately result in less system payoff. It is because whenthe successful rate of self-protection increases, people become more speculative and less interestedin vaccination. Since a vaccinated individual indeed brings benefit to the system by statisticallyreducing the infection probability of susceptible individuals, the decreasing of vaccinated individ-uals will eventually lead to the loss of system payoff. Qualitatively speaking, the counter-intuitivephenomenon is insensitive to the network topology, while quantitatively speaking, networks withdelocalized structure (e.g., ER, BA and well-mixed networks) have more self-protective individu-als and less laissez-faire individuals than networks with localized structure (e.g., square lattices),and the epidemic size is larger in the latter case. Without the diverse behavioral responses ofindividuals, epidemic in delocalized structure usually spreads more quickly and widely than inlocalized structure [31, 32]. The opposite observation reported in the current model again resultsfrom more and more speculative choices (i.e., to be laissez-faire) at a low-risky situation. There-fore, this can be considered as another kind of “less payoff in better condition” phenomenon.The observed counter-intuitive phenomenon reminds us of the well-known Braess’s Paradoxin network traffic [15, 21]. Zhang et al. [33] showed that to remove some specific edges in anetwork can largely enhance its information throughput, and Youn et al. [34] pointed out thatsome roads in Boston, New York City and London could be closed to reduce predicted traveltimes. Actually, Seoul has removed a highway to build up a park, which, beyond all expecta-tions, maintained the same traffic but reduced the travel time [35]. Very recently, Pala et al. [36]showed that Braess’s Paradox may occur in mesoscopic electron systems, that is, adding a path forelectrons in a nanoscopic network may paradoxically reduce its conductance. This work providesanother interesting example analogous to Braess’s Paradox, namely a higher successful rate ofself-protection may eventually enlarge the epidemic size and thus cause system loss. Let’s thinkof the prisoner’s dilemma, if every prisoner stays silent, they will be fine, while one more choice,to betray, makes the situation worse for them. Analogously, if the successful rate δ is small, fewpeople will choose to be self-protective, while for larger δ , people have more choices, which mayeventually reduce the number of vaccinated people and thus enlarge the epidemic size. Basically,both the original Braess’s Paradox and the current counter-intuitive phenomenon are partially dueto the additional choices to selfish individuals. This is easy to be understood in a simple modellike the prisoner’s dilemma game, but it is impressive to observe such phenomenon in a complexepidemic game.Human-activated systems are usually much more complex than our expectation, since people’schoices and actions are influenced by the environment and at the same time their choices andactions have changed the environment. This kind of interplay leads to many unexpected collectiveresponses to both emergencies and carefully designed policies, which, fortunately, can still bemodeled and analyzed to some extent. This work raises an unprecedent challenge to the publichealth agencies about how to lead the population towards an epidemic. The government shouldtake careful consideration on how to distribute their resources and money on popularizing vaccine,hospitalization, self-protection, self-treatment, and so on. V. METHODS
Given a well-mixed network with size N , the dynamical equations are dSdt = − λN SI, (2) dIdt = λN SI − µI, (3) dRdt = µI, (4)where S , I and R stand for the fraction of susceptible, infected and recovered individuals, respec-tively. Dividing Eq. (2) by Eq. (4), one has dSdR = − R S, (5) TABLE II: The payoffs for different strategies and states. P V , P S and P L stand for average payoffs forindividuals with strategy vaccination, self-protection and laissez faire, while the superscripts H (healthy)and I (infected) represent the final states.Strategy & State Fraction PayoffVaccinated & Healthy ρ V P V = − c Self-protective & Healthy ρ HS = ρ S [ δ + (1 − δ )(1 − ω )] P HS = − b Self-protective & Infected ρ IS = ρ S (1 − δ ) ω P IS = − b − Laissez-faire & Healthy ρ HL = (1 − ρ V − ρ S )(1 − ω ) P HL = 0 Laissez-faire & Infected ρ IL = (1 − ρ V − ρ S ) ω P IL = − where R = λNµ is the basic reproduction number for the standard SIR model in well-mixedpopulation [18]. Integrating Eq. (5), we get Z S ( ∞ ) S (0) dSS = Z R ( ∞ ) R (0) − R dR, (6)which leads to the solution ln S ( ∞ ) S (0) = − R [ R ( ∞ ) − R (0)] . (7)Clearly, R (0) = 0 , R ( ∞ ) + S ( ∞ ) = 1 , and in the thermodynamic limit, S (0) ≈ . Accordingly, R ( ∞ ) = 1 − exp [ − R R ( ∞ )] . (8)Let ρ V , ρ S and ρ L be the fraction of vaccinated, self-protective and laissez-faire individuals,such that ρ V + ρ S + ρ L ≡ . Since only a fraction − ρ V + δρ S of individuals are susceptible,using the similar techniques, one can easily obtain the epidemic size as R ′ ( ∞ ) = (1 − ρ V − δρ S ) { − exp [ − R ′ R ′ ( ∞ )] } , (9)where R ′ = (1 − ρ V − δρ S ) R . Then, the probability of a susceptible individual to be infectedreads ω = R ′ ( ∞ )1 − ρ V − δρ S = 1 − exp [ − R ′ R ′ ( ∞ )] . (10)The payoffs of different strategies and states are thus easily to be obtained, which are summarizedin Table 2.0The imitation dynamics governing the time evolution of the fractions of strategies in the popu-lation is similar to the replicator dynamics of evolutionary game theory [17, 37], as dρ V dt = ( ρ V ⇄ ρ HS ) + ( ρ V ⇄ ρ IS ) + ( ρ V ⇄ ρ HL ) + ( ρ V ⇄ ρ IL ) , (11) dρ S dt = ( ρ HS ⇄ ρ V ) + ( ρ HS ⇄ ρ HL ) + ( ρ HS ⇄ ρ IL ) + ( ρ IS ⇄ ρ V )+( ρ IS ⇄ ρ HL ) + ( ρ IS ⇄ ρ IL ) , (12)where ρ V ⇄ ρ HS = ( ρ HS → ρ V ) − ( ρ V → ρ HS )= ρ V ρ HS (cid:26)
11 + exp [ − κ ( P V − P HS )] −
11 + exp [ − κ ( P HS − P V )] (cid:27) = ρ V ρ HS tanh h κ P V − P HS ) i = ρ V ρ S [ δ + (1 − δ )(1 − ω )] tanh h κ − c + b ) i , (13)and the others are similar.Denote by ρ V ( τ ) the initial fraction of vaccinated individuals before the ( τ + 1) th season ofepidemic spreading. Given ρ V (0) , ρ S (0) and ρ L (0) , and for each season, we apply the initialconditions as S (0) = ( N ′ − /N ′ , I (0) = 5 /N ′ and R (0) = 0 , where N ′ = (1 − ρ V − δρ S ) N ,depending on the distribution of strategies at this season. Then, R ′ ( ∞ ) can be obtained by Eq. (9)and ω by Eq. (10). Using the evolutionary dynamics described in Eqs. (11)-(13) and the fractionspresented in Table 2, one can obtain the values of ρ V (1) , ρ S (1) and ρ L (1) , which are also the initialfractions of strategies at the beginning of the next season. Repeat the above steps until the steadystate, then we can calculate the desired variables. Acknowledgments
This work was partially supported by the National Natural Science Foundation of China underGrant Nos. 11005001, 11005051, 11222543, 11135001, 11275186, 91024026 and 10975126.H.-F.Z. acknowledges the Doctoral Research Foundation of Anhui University under Grant No.02303319. T.Z. acknowledges the Program for New Century Excellent Talents in University under1Grant No. NCET-11-0070. [1] W. H. Organization, R. Beaglehole, A. Irwin, and T. Prentice,
The world health report 2004: Changinghistory (World Health Organization, 2004).[2] E. Enns, J. Mounzer, and M. Brandeau, Mathematical Biosciences , 138 (2011).[3] C. T. Bauch, A. P. Galvani, and D. J. D. Earn, Proceedings of the National Academy of Sciences ofthe United States of America , 10564 (2003).[4] C. Bauch and D. Earn, Proceedings of the National Academy of Sciences of the United States ofAmerica , 13391 (2004).[5] A. Perisic and C. Bauch, BMC Infectious Diseases , 77 (2009).[6] P. Schimit and L. Monteiro, Ecological Modelling , 1651 (2011).[7] S. Meloni, N. Perra, A. Arenas, S. G ´omez, Y. Moreno, and A. Vespignani, Scientific Reports , 62(2011).[8] N. Perra, D. Balcan, B. Gonc¸alves, and A. Vespignani, PLoS ONE , e23084 (2011).[9] E. Fenichel, C. Castillo-Chavez, M. Ceddia, G. Chowell, P. Parra, G. Hickling, G. Holloway, R. Horan,B. Morin, C. Perrings, et al., Proceedings of the National Academy of Sciences of the United Statesof America , 6306 (2011).[10] F. Sahneh, F. Chowdhury, and C. Scoglio, Scientific Reports , 632 (2012).[11] C. Bauch, Proceedings of the Royal Society B: Biological Sciences , 1669 (2005).[12] R. Vardavas, R. Breban, and S. Blower, PLoS Computational Biology , e85 (2007).[13] S. Basu, G. Chapman, and A. Galvani, Proceedings of the National Academy of Sciences of the UnitedStates of America , 19018 (2008).[14] A. Perisic and C. Bauch, PLoS Computational Biology , e1000280 (2009).[15] D. Braess, Unternehmensforschung , 258 (1968).[16] F. Fu, D. Rosenbloom, L. Wang, and M. Nowak, Proceedings of the Royal Society B: BiologicalSciences , 42 (2011).[17] B. Wu, F. Fu, and L. Wang, PLoS ONE , e20577 (2011).[18] R. Anderson and R. May, Infectious diseases of humans: dynamics and control (Oxford Univ Press,Oxford, UK, 1992).[19] A. Traulsen, M. Nowak, and J. Pacheco, Physical Review E , 011909 (2006). [20] M. Perc and A. Szolnoki, BioSystems , 109 (2010).[21] T. Roughgarden, Selfish Routing and the Price of Anarchy (MIT Press, 2005).[22] P. Erd˝os and A. R´enyi, Publ. Math. Debrecen , 290 (1959).[23] A. Barab´asi and R. Albert, Science , 509 (1999).[24] H. Zhang, M. Small, X. Fu, G. Sun, and B. Wang, Physica D: Nonlinear Phenomena , 1512 (2012).[25] H. Zhang, J. Zhang, C. Zhou, M. Small, and B. Wang, New Journal of Physics , 023015 (2010).[26] M. Salath´e and S. Bonhoeffer, Journal of The Royal Society Interface , 1505 (2008).[27] P. Poletti, M. Ajelli, and S. Merler, PLoS ONE , e16460 (2011).[28] S. Funk, E. Gilad, C. Watkins, and V. Jansen, Proceedings of the National Academy of Sciences of theUnited States of America , 6872 (2009).[29] F. Coelho and C. Codec¸o, PLoS Computational Biology , e1000425 (2009).[30] S. Funk, M. Salath´e, and V. Jansen, Journal of The Royal Society Interface , 1247 (2010).[31] V. M. Egu´ıluz, K. Klemm, Physical Review Letters , 108701 (2002).[32] T. Zhou, Z.-Q. Fu, B.-H. Wang, Progress in Natural Science , 452 (2006).[33] G.-Q. Zhang, D. Wang, G.-J. Li, Physical Review E , 017101 (2007).[34] H. Youn, M. T. Gastner, H. Jeong, Physical Review Letters , 128701 (2008).[35] L. Baker, Scientific American pp. 20-21, Feb. 2009.[36] M. G. Pala, S. Baltazar, P. Liu, H. Sellier, B. Hackens, F. Martins, V. Bayot, X. Wallart, L. Desplanque,S. Huant, Physical Review Letters , 076802 (2012).[37] P. Poletti, B. Caprile, M. Ajelli, A. Pugliese, and S. Merler, Journal of Theoretical Biology , 31(2009). L S V R(a) (b)(c) R -1800-1500-1200-900-600 P P R FIG. 1:
Less payoff in better condition . (a) How the fractions of the three strategies and the epidemic sizechange with the successful rate of self-protection δ . (b) The epidemic size R ∞ and the system payoff P asfunctions of δ . (c) Correlation between the system payoff P and the epidemic size R ∞ , where each datapoint corresponds to a certain δ . Panel (a) is divided into three regions by two vertical dash lines: (i) In theleft region, no self-protective individual exists and δ has no effect on the epidemic size; (ii) In the middleregion, the self-protection strategy gradually replaces vaccination and laissez faire, and the epidemic sizeincreases with δ due to the decrement of vaccination fraction; (iii) In the right region, with high successfulrate of self-protection, individuals are unwilling to take vaccination and the epidemic size decreases with δ .Parameters are set to be N = 50 ×
50 = 2500 , λ = 0 . , µ = 0 . , b = 0 . , c = 0 . , κ = 10 and I = 5 . Forthis figure and all others (except snapshots), the simulation results are calculated after 1000 seasons whenthe system is in a steady state, and each data point is obtained by averaging over 100 independent runs. FIG. 2:
Strategy distribution patterns . Subgraph (a) shows the epidemic size R ∞ as a function of δ . Thewindow is divided into three parts according to the tendency of R ∞ − δ curve. Subgraphs (b), (c), (d) and(e) are snapshots in the steady state of a season at δ = 0 . , 0.35, 0.5 and 0.95. The grey, light red, dark red,light blue and dark blue points stand for vaccinated, laissez-faire and not infected, laissez-faire and infected,self-protective and not infected, and self-protective and infected individuals, respectively. Parameters arethe same as in Figure 1. L S V R(a)Lattice L S V R ER (b) L S V R BA (c) L S V RWell-mixed Network(d)
FIG. 3:
Insensitivity to the network structures . To explore the impacts of different network structures onthe epidemic size and strategy distribution, we compare the results in square lattices (a), ER networks (b),BA networks (c) and well-mixed networks (d). The parameters are set as b = 0 . , I = 5 , c = 0 . , and κ = 10 . Each data point results from an average over 100 independent runs. The average degrees of thelattices, ER networks and BA networks are all set to be 4, and the simulations presented in subgraph (a),(b) and (c) are implemented with the same transmission and recovery rate, λ = 0 . and µ = 0 . . For thewell-mixed network (d), however, the parameters are different from others as λ = 0 . and µ = 1 for itsdifferent average degree. FIG. 4:
Delocalization promotes the self-protection strategy . The subgraphs (a)-(d) show how the frac-tions of the three strategies and the epidemic size change with δ , and subgraphs (e)-(h) are the correspondingsnapshots for (a)-(d) with δ = 0 . . From (a) to (d), the number of randomized edges, A , increases. Qual-itatively, the counter-intuitive phenomenon always exists, no matter what the value of A . Quantitatively,the delocalization reduces the advantage of the laissez-faire strategy, which leads to a larger fraction ofself-protective individuals. When A is large enough, self-protection becomes the dominating strategy for acertain range of δ . Overall speaking, the epidemic size is smaller at larger A . Parameters are the same as inFigure 1. simulation L S V R(a) L S V Rsolution(b)
FIG. 5:
The analytical solution agrees well with the simulation . The analytical prediction (b) is in goodaccordance with the simulation (a). All results are implemented on a well-mixed network with N = 1000 , c = 0 . , b = 0 . , λ = 0 . , µ = 1 . , I = 5 and κ = 10= 10