key: cord-0215192-rof1xxzw authors: Rowlett, Julie; Karlsson, Carl-Joar title: Decisions and disease: a mechanism for the evolution of cooperation date: 2020-04-26 journal: nan DOI: nan sha: 667a81e6c47848211bf2da58fe795584c23cca72 doc_id: 215192 cord_uid: rof1xxzw In numerous contexts, individuals may decide whether they take actions to mitigate the spread of disease, or not. Mitigating the spread of disease requires an individual to change their routine behaviours to benefit others, resulting in a 'disease dilemma' similar to the seminal prisoner's dilemma. In the classical prisoner's dilemma, evolutionary game dynamics predict that all individuals evolve to 'defect.' We have discovered that when the rate of cooperation within a population is directly linked to the rate of spread of the disease, cooperation evolves under certain conditions. For diseases which do not confer immunity to recovered individuals, if the time scale at which individuals receive information is sufficiently rapid compared to the time scale at which the disease spreads, then cooperation emerges. Moreover, in the limit as mitigation measures become increasingly effective, the disease can be controlled, and the rate of infections tends to zero. Our model is based on theoretical mathematics and therefore unconstrained to any single context. For example, the disease spreading model considered here could also be used to describe social and group dynamics. In this sense, we may have discovered a fundamental and novel mechanism for the evolution of cooperation in a broad sense. Decisions made by individuals affect the population, not the least in disease spreading. Several researchers have investigated the interplay between diseases and decisions by combining compartmental models with game theory [1] [2] [3] [4] [5] . Common considerations are dynamics on networks or lattices [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] and well-mixed populations [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] . The former's strength is that it captures the effect of population structures, while the latter's strength is that it highlights the individuals' perception of the payoff. We consider a well-mixed society in which individuals choose to what extent they will exert preventive measures to mitigate the spread of an infectious disease. There are two choices: exert mitigating measures to prevent the spread of the disease, and do-not-mitigate, making no efforts to prevent the spread of the disease. The World Health Organisation [29] and numerous other references including [30] [31] [32] argue that it is reasonable to describe this situation with the Prisoner's Dilemma (PD); this is depicted in Figure 1 . The payoffs may be interpreted in the sense that if both Alice and Bob cooperate, then they are behaving towards their mutual common good, but they are also making personal sacrifices by changing their usual behaviour to mitigate disease spread. On the other hand, if Alice cooperates while Bob defects, then Alice's mitigation efforts prevent Bob from catching the disease, thus Bob benefits from Alices's cooperation. Alice receives no such benefit from Bob's reckless actions, therefore Alice is not only restricting her personal freedoms, she is also at risk of catching the disease due to Bob's recklessness. If both Alice and Bob defect, then they are both at risk of catching the disease, but neither makes any personal sacrifices. Consequently, this situation is described by payoffs which satisfy S < P < R < T. The unique equilibrium strategy of this game is mutual defection, and when this game is used to predict behaviours according to evolutionary game dynamics, the result is always defection [33] . Nonetheless, in many contexts which fit into a PD type game, cooperation may in fact be observed [34] [35] [36] [37] [38] [39] [40] . In the particular case of the PD, there have been numerous mechanisms proposed for the evolution of cooperation [41, 42] . To our knowledge, it has been unknown -until now -whether cooperation emerges when the payoff is a trade-off between the PD and the effect on disease spreading (changes to the infection transmission rate). Infections like those from the common cold, flu, and many sexually transmitted infections do not confer any long-lasting immunity, and individuals become susceptible once they recover from infection. These diseases are described by the SIS compartmental model. Poletti et al. [43] implemented a hybrid model in which human decisions affect the rate Figure 1 : In the 'disease dilemma' two people have the choice to cooperate, mitigating the spread of the disease, or defect, making no change to their regular behaviour. This is described by the two player non-cooperative game shown here in normal form. Image source and license: openclipart.org, CC0 1.0. at which the disease spreads. They assigned two different rates of infection corresponding to individuals either changing their behaviour to mitigate the spread of the disease, or not doing so. We follow this approach by assigning the rates of infection for cooperators and defectors, β C < β D , respectively. The infection-producing contacts (via, e.g., droplets from someone who sneezes) per unit time is weighted by the proportion of cooperators, x, and defectors, 1 − x, and is therefore We consider three timescales: (i) the disease transmission timescale t, (ii) the timescale at which individuals decide whether or not to mitigate the spread of the disease α 1 t, and (iii) the timescale at which individuals receive the PD payoffs α 2 t. The replicator equations for our hybrid SIS-PD model are therefore Above, the quantities on the left side are differentiated with respect to t = time, I is the portion of the population which is infected, x is the portion of the population which chooses to cooperate, and γ is the transmission rate. If D is the duration of the infection, then γ = 1/D. We note that 1 − I is the portion of the population which is susceptible to infection, since in this model there is no immunity. For the sake of simplicity, and since no generality is lost, we shall assume the PD payoffs (1) satisfy T − R = P − S, leading to the simplified system Since β D > β C , and T > R, the terms in the equation for the evolution of cooperators have opposite signs, resulting in a competition between avoidance of disease carriers and PD reward. Moreover, the relationship between the decision-making and PD-payoff time scales is key. Similar calculations lead to the replicator equations for the SIR-PD model Above, R is the number of recovered, deceased, or immune individuals. This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance. SinceṠ +İ +Ṙ = 0, the triplet (x, I, S) describes the complete system. For the SIR-PD model, the equilibrium points consist of the set (x, I, S) An equilibrium point with x = 0 is stable if β D S * ≤ γ, and it is unstable if the reverse inequality holds. All equilibrium points with x = 1 are unstable. The equilibrium points of the SIS-PD system are the set of (x, I): Table 1 : The asymptotically stable equilibrium points of the SIS-PD model in the specified ranges of α 1 , whereα 1 is defined in (8) . The equilibrium point, (x * , I * ), is well-defined as long as x * ∈ [0, 1], and I * ∈ [0, 1), since T > R, and α 1 , α 2 > 0. We compute that We further compute Whenever it exists, the equilibrium point (x * , I * ) is always stable (and asymptotically stable). The equilibrium point (0, 0) is stable (and asymptotically stable) if Since dI/dt < 0 when this condition is satisfied, there is no epidemic. The equilibrium point (1, 0) is never stable for PD payoffs (1). The equilibrium point ( , and it is stable (and asymptotically stable) if For PD payoffs (1), this is equivalent to The equilibrium point The values of β D and γ above were suggested [44] ; however these values can be modified to any disease parameters. Since it is the relationship between α 1 and α 2 , rather than their individual values which affects the dynamics, we simply fix α 2 = 0.1. The value of α 1 ranges along the horizontal axis. The vertical axis is the frequency within the population. For sufficiently large α 1 , the population evolves to cooperation. At the same time, the more effective the mitigation measures are, the lower β C is, which pushes the portion of infected individuals to zero. More precisely, when α 1 ≥α 1 , then lim β C γ I = 0. For PD payoffs (1), this equilibrium point is stable (and asymptotically stable) wheň Our results are summarised in Table 1 . Figure 2 shows how the evolution of cooperation and the rate of infections depend on α 1 and β C when β D = 1.68 and γ = 1/5 as suggested in [44] . We note that these values were selected merely for the sake of visualisation, as our theoretical results hold for any parameter values. If both α 1 and α 2 vary, we obtain convergence to cooperation as shown in Figure 3 . Figure 4 shows that the numerical integration agrees perfectly with the analytical results. It has been suggested that mass media could be used to reduce HIV-infections [45] , which fits well with our theoretical model. If an infectious disease does not confer immunity to those who recover from it, then SIS is a suitable model. The rate of spread for those who make no mitigation efforts, β D , is strictly larger than the rate of spread for those who make mitigation efforts, β C . Our results show that the relationship between the time scale [44] . Here the value of β C corresponds to mitigation measures which are more effective than in Figure 3 but still imperfect. of decision making, α 1 t, and the timescale of PD payoffs, α 2 t is crucial. Decision-making is influenced by the speed at which individuals access or receive information upon which to base their decisions. It is reasonable to assume that the timescale of PD payoffs is similar to the timescale t for the spread of disease, or at least on the same order of magnitude. On the other hand, the speed at which individuals can access information could be much faster. This corresponds to α 1 α 2 . When α 1 >α 1 , the equilibrium point (1, 1 − γ/β C ) exists. Consequently, for sufficiently large α 1 , the unique equilibrium point of the system corresponds to total cooperation. Moreover, in this case the portion of the population which is infected tends to 1 − γ/β C . We therefore also have This shows that in the limit towards effective mitigation measures, the rate of the population which is infected tends to zero. We summarise these insights below. In the context of a communicable disease which does not confer immunity, if information is made available to all individuals quickly relative to the spread of the disease, all rational individuals acting in their best self interest will evolve to cooperate. Moreover, if their mitigation efforts are effective, they drive the rate of spread of the disease to zero. These insights suggest a general strategy for controlling both new diseases as well as diseases which are known not to confer immunity. For a new disease, it is unknown and unknowable whether contracting and recovering from the disease grants immunity [46] . Moreover, vaccines require time for development and testing [47] . It may therefore be prudent to use the SIS model for new communicable diseases. Moreover, our results show that the evolution of cooperation does not occur in the SIR-PD model. Consequently, if the desired outcome is the evolution of cooperation and control of disease, the SIS-PD model yields the best results. The value of α 1 may be associated to the frequency of public service announcements (PSAs) explaining the recommended measures. The more frequent the PSAs, the higher the value of α 1 . Our results prove that when α 1 becomes very large, cooperation emerges, and the amount of infections can be controlled. Moreover, when mitigation measures are made increasingly effective, in the limit the frequency of infections tends to zero. The perceived benefit of defecting is defined by the PD payoffs (1), so that defecting is still perceived to offer benefits if others cooperate. The key to the evolution for cooperation is the time scale for decision making. This can be much faster than the time scale at which one can actually reap the benefits of defecting. When this is the case, the population evolves towards cooperation. Our results are not constrained to any specific disease, but rather suggest a general strategy to promote the evolution of cooperation in the classical Prisoner's Dilemma when linked to the spread of disease according to the SIS model. The SIS model has further applications to describing social and group dynamics [25] . Our model may thereby provide a mechanism for the evolution of cooperation in social contexts as well. Our equations in the SIS-PD model are: Calculation and classification of equilibrium points in the SIS-PD model. We compute that the equilibrium points of the system are the set of (x, I): To determine the nature of the equilibrium points, that is whether they are (asymptotically) stable or unstable, we compute the Jacobian matrix ∂f ∂I , ∂f ∂x ∂g ∂I , ∂g ∂x whose entries are At the equilibrium point (0, 0) the Jacobian matrix is If the real parts of all eigenvalues are negative, then the equilibrium point is stable and asymptotically stable. This holds when The Jacobian matrix at the equilibrium point (1, 0) is Since α 2 > 0, and T > R, this equilibrium point is unstable. The equilibrium point (0, 1 − γ/β D ) has Jacobian matrix Since 0 ≤ I ≤ 1, this is well-defined if and only if It is stable and asymptotically stable if For PD payoffs, this is equivalent to It has Jacobian matrix It is stable and asymptotically stable if Note that if one changes the PD payoffs so that T < R, then since β D > β C , and α 1 , α 2 > 0, this equilibrium point is always stable. The condition above is equivalent to The equilibrium point, (x * , I * ), exists as long as x * ∈ [0, 1], and I * ∈ [0, 1), since T > R, and α 1 , α 2 > 0. We compute that We further compute Since 1 < β D β D −γ , this condition immediately implies I * < 1. We note that We compute the Jacobian matrix Under these conditions, we compute that it is always stable (and asymptotically stable), because we compute that the eigenvalues of the Jacobian matrix have negative real part, since the matrix is of the form − − + 0 . Hence, the interesting values of α 1 are and this equilibrium point is stable and asymptotically stable. When α 1 is greater than or equal to this value, and this equilibrium point is stable and asymptotically stable. For β D −β C , x * > 1, so this equilibrium point ceases to exist, but and this equilibrium point is stable and asymptotically stable. The equilibria with x = 0 have Jacobian matrix The equilibrium point is stable and asymptotically stable if β D S * < γ. The equilibria with x = 1 have Jacobian matrix Since α 2 > 0, and T > R, this equilibrium is always unstable. All simulations were performed with the initial conditions x(0) = 0.5 and I(0) = 0.001 if otherwise is not stated. The choice of initial conditions does not change the convergence results, only the dynamics at small times. The ODEs are integrated using the Python routine scipy.integrate.solve_ivp with default settings. These simulations were used to produce Figure 4 which shows that the numerical integration agrees perfectly with the analytical results in the main text. 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We thank Johan Runeson for the idea to use game theory to model human behaviour in the corona pandemic. Both authors are supported by Swedish Research Council Grant GAAME 2018-03873.