Dynamics of beneficial epidemics

Dynamics of beneficial epidemics Santa Fe Institute Postdocs∗

arXiv:1604.02096v1 [physics.soc-ph] 7 Apr 2016

Santa Fe Institute, Santa Fe, NM, 87501 USA [email protected]

Abstract Pathogens can spread epidemically through populations. Beneficial contagions, such as viruses that enhance host survival or technological innovations that improve quality of life, also have the potential to spread epidemically. How do the dynamics of beneficial biological and social epidemics differ from those of detrimental epidemics? We investigate this question using three theoretical approaches as well as an empirical analysis of concept propagation. First, in evolutionary models, we show that a beneficial horizontally-transmissible element, such as viral DNA, spreads super-exponentially through a population, substantially more quickly than a beneficial mutation. Second, in an epidemiological social network approach, we show that infections that cause increased connectivity lead to faster-than-exponential fixation in the population. Third, in a sociological model with strategic rewiring, we find that preferences for increased global infection accelerate spread and produce super-exponential fixation rates, while preferences for local assortativity halt epidemics by disconnecting the infected from the susceptible. Finally, in an investigation of the Google Ngram corpus, we find that new words and phrases spread super-exponentially, as anticipated by our models. We conclude that the dynamics of beneficial biological and social epidemics are characterized by the remarkably rapid spread of beneficial elements, which can be facilitated in biological systems by horizontal transmission and in social systems by active spreading strategies of infected individuals.

1

Introduction

Epidemic processes describe the spread of contagious elements such as innovations, viruses/infections, socio-political movements, and linguistic variants of words. Research in biological and medical epidemiology primarily focuses on harmful epidemics, from viruses such as influenza or dengue fever in humans [55, 4] to chytrid fungus in frogs [34] or bacterial wilt in beans [33]. The serious ∗ This manuscript is a product of 72 Hours of Science (see Appendix A). Authorship is uniformly attributed to, in alphabetical order, Andrew Berdahl, Christa Brelsford, Caterina De Bacco, Marion Dumas, Vanessa Ferdinand, Joshua A. Grochow, Laurent H´ ebert-Dufresne, Yoav Kallus, Christopher P. Kempes, Artemy Kolchinsky, Daniel B. Larremore, Eric Libby, Eleanor A. Power, Caitlin A. Stern, and Brendan D. Tracey

1

consequences of such detrimental epidemics drives their study, while comparatively little is known about beneficial epidemics [43]. In the social sciences, studies of the spread of innovations, new technologies, and socio-political movements have received a great deal of attention, but their quantitative dynamics have not been explored as rigorously as the dynamics of detrimental epidemics in biology. Biological epidemics typically involve the spread viruses, plasmids, genes, or microbes. Beneficial viruses, in particular, occur in both unicellular and multicellular organisms [15] and enhanse the survival of the host [39]. In some cases, in fact, the infection can be essential to the continuation of the host [49]. Epidemic spreads of beneficial genetic elements within a population have been described for unicellular organisms (e.g., [36]). Although many such elements have been identified (see, e.g., [54, 35, 53, 18, 8, 19, 7]), their large-scale dynamics are not particularly well-studied. Some previous studies have investigated dynamics in unstructured populations (e.g., [51, 54, 21]), but the dynamics of beneficial biological epidemics are not well-studied beyond the well-mixed case. Indeed, recent publications have highlighted many open questions concerning the dynamics of beneficial epidemics [43, 18, 54]. Social epidemics, those of beneficial behaviors, ideas, and technologies, are known to occur in both human and non-human populations. In non-humans, examples include the acquisition of new feeding techniques in birds [26] and whales [1]. In humans, examples include the adoption of new agricultural technology [28], linguistic variants [13], and social movements [45] [47]. In human social systems, the importance of beneficial ideas, information, and behavior spreading through the population has been well known for some time (e.g. [38], first published in 1962). Research in political science, economics and social psychology has explored the importance of social networks in evaluating the adoption of economic policy, the participation in religious movements, political movements, or alternative lifestyles [44, 45, 46, 47] and the reasons why there is heterogeneity in the adoption of a beneficial new technology [10, 24, 3, 28]. These studies have considered temporal dynamics, spatial proximity, network topology, and the role of where the innovation first influences the population. However, there has been little research exploring the transmission of ideas and technology in a social network from a mathematical modeling perspective, and typically the underlying network does not dynamically change due to the transmission process. To encompass both the epidemics of beneficial practices and the epidemics of beneficial genetic units such as viruses within a single conceptual framework, we define the term “bene” (pronounced “BEN-ay” or, in IPA, /’bEneI/) to mean a horizontally-transmitted beneficial unit of material, for example information, behavior, practices, or genes. At the most basic level benes have two hallmark traits: 1. they are transmitted horizontally and 2. they offer some benefit to their host. The exact forms of the benefit vary depending on the particulars of the bene. In this paper, we consider different forms of benefits conferred by benes and investigate the resulting epidemic dynamics. First, we examine the case in which the beneficial effects of benes occur as fitness improvement and explore the spread using evolutionary models. Second, we study the case when the effects of benes occur in the same generation via modification of social structure. For this we use epidemiological models and consider the consequences of epidemics whose 2

effect is to preferentially increase interactions between individuals. Third, we study the case when benes incite individuals to rewire their connections based on functions of social network structure. We use a general social modeling framework that applies to technology adoption, diffusion of innovations, and social movements. Lastly, we empirically analyze the spread of novel English words and show that their dynamics are more similar to beneficial epidemic models than standard contagion models.

2

Epidemics with fitness benefits

One way in which a bene may exert its effect is through increasing the host’s fitness, i.e. increasing the number of offspring the host leaves in subsequent generations. We consider this case by adopting an evolutionary approach and modeling the spread of a beneficial sequence of genetic material through a population. Transmission of the sequence occurs within a generation. We contrast this process with the spread of a beneficial element that is only transmitted vertically, e.g., a beneficial mutation. For completeness, we study both a model with an expanding population and a model with a fixed population size.

2.1

Expanding population model

For the expanding population model we consider two types of entities: those infected by the bene I and those that are not infected but susceptible S (Eqn. (1)). The entities infected by the bene have a growth rate advantage s such that their population increases faster than the population of S. Since the benes can be transmitted horizontally, the S entities can become I entities, i.e. they can be infected. Infections occur at rate β and depend on the interactions between S and I entities. dS = S − βSI dt (1) dI = (1 + s)I + βSI dt Without horizontal transmission (β = 0), the differential equations are separable and can be solved analytically: S(t) = S(0)et and I(t) = I(0)e(1+s)t . The proportion of the population that is infected at any time t is then 1/(1+R0 e−st ), where R0 = S(0)/I(0). If horizontal transmission does occur, then the S population is continually converted to I types. This happens until all S have become I. The speed at which this occurs is much faster than without horizontal transmission. To see this, we consider a form of Eqn. (1) that is separable and can be solved analytically. We assume that when S types are infected, they are removed from the population and do not contribute to the I population (that is, we remove the βSI term from the dI dt in this version). In this case, the I(t) solution is the same when there was no horizontal transmission. The S(t) solution is then: β

S(t) = S(0)et− 1+s e

3

(1+s)t

β + 1+s

.

This makes the proportion of the population that is infected at any time t: h

β

1 + R0 e 1+s (1−e

(1+s)t

) −st

e

i−1

.

The result is a super-exponential decrease in the frequency of S entities. If β = .0001 and s = .01 and R0 = 100 then it takes t = 920 until the infected fraction makes up 99% of the population without horizontal transmission. In contrast, the infection with horizontal transmission constitutes over 99% of the population at t ≈ 11.3 in the reduced case and t ≈ 6.8 (found by numerical integration) in the full system of (Eqn. (1)).

2.2

Fixed population size model

We now consider the same effect in a model with a finite population of size N . We use a previously published model of horizontal gene transfer [31] which is easily generalized to other mechanisms of horizontal transmission of genetic elements found in multicellular organisms such as crustaceans and insects [12]. This model (Eqn. (2)) distinguishes the effects of the transmission rate β of the bene and the selective value s of the bene. The number of I entities is n and the number of S entities is N − n. The model is based on a Moran process [30] in which a birth/death process occurs in discrete time steps. The probability of having n infected types at time t, pn (t), depends on the cumulative effects of the birth (λn ) and death (µn ) rates (see Eqn. (2)). The birth rate (λn ) includes actual births (the first term, which involves s) as well as horizontal gene transfer (the second term, which involves β). There are two stationary states of this model: either i) the entire population is un-infected (S), and ii) the entire population is infected (I). We solve for the stationary distribution of pn (t) starting with pn (0) = δn,1 (i.e all realizations start with a single infected individual (n = 1)). dpn = µn+1 pn+1 − (λn + µn )pn + λn−1 pn−1 dt N −n n(N − n) (2) λn = (1 + s)n +β N +1 N n µn = (N − n) N +1 Figure 1 shows the fixation probability and time to stationarity as a function of β for two example values of s. As β increases, the probability that the bene fixes increases (left panel). Higher transmission rates also lead to faster time to stationarity when compared to those achieved by selection alone (right panel). The decrease in time to reach a stationary solution is more dramatic when selection (s = 0.1) is lower. Thus for more modest fitness values of the bene the role of horizontal gene transfer is greater.

3

Epidemics with connectivity benefits

Until now we have considered benes whose beneficial effect occurs across generations driving the evolution of a population. It is possible, however, that the effects of the bene might appear within the same generation it infected its host. 4

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2 0.0 0

s=1 s = 0.1

Relative Time to Fix

Probability to Fix

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1

2 3 4 Transmission Rate

0.0

5

0.5

1.0 1.5 2.0 2.5 Transmission Rate

3.0

Figure 1: Probability of fixation in the evolutionary model. The probability that a novel bene fixes (completely saturates the population) as a function of the infection rate, β, when s = 1 and s = 0.1 (left panel). The time to reach a stationary state (where the bene either fixes or goes extinct) as a function of β, when s = 1 and s = 0.1 (right panel). Time is plotted relative to the time it takes a match model where β = 0. N = 100 in all cases. These same-generation effects are well-known in parasites. For example, the parasite Toxoplasma gondii, which can infect many endothermic animals (e.g., [50]), has an effect like this in cats, through their interaction with mice. When a T. gondii infection from a cat spreads to a mouse, it modifies the mouse’s behavior to make it less fearful of cats, increasing the likelihood that the mouse interacts with and is captured by a cat, thus having helped the original host within a single generation. Similarly, we assume that a bene modifies its host’s behavior in such a way that the host’s interactions with other organisms change. To model this, we consider the dynamics of a bene epidemic as it spreads across a social network. The interplay of diseases and network structure has been studied at great length [17, 37, 29, 52, 2, 27, 40]. However, as diseases have a negative impact on individuals, this interplay mostly takes the form of individuals changing their behavior to avoid or limit disease transmission. This can be either a conscious decision by the sick or healthy individuals, such as increased hand-washing, or an indirect consequence due to symptoms of the infection, perhaps causing reduced energy and activity level. Here, we consider this interplay under a novel, and more positive, perspective: the infection causes a social benefit. This benefit can take many forms, both conscious and unconscious. In the cultural case, one can think of individuals excited about a new bene and actively trying to share it. In a more biological scenario, one can imagine viruses causing slight increases in energy or general happiness, or reduced inhibitions, that would increase the number of social contacts and/or the energy that is devoted to these contacts. As a result of being infected by the beneficial bene, the connectivity of the individual increases. In this section, we generalize the classic concepts from network epidemiology to include the societal effects of beneficial epidemics.

5

3.1

Model for discrete link generation

Consider a network as a set of nodes with a variable attached to them defining their epidemic state. A node can either be infected, I, or susceptible, S. We consider an SIS model where there can be transitions from any state to any other state. These transitions are regulated by the following parameters: • susceptible nodes are infected at a transmission rate β by each of their infected neighbors. • infected nodes recover at rate r to become susceptible. • In the absence of infected nodes, we assume that the average degree of a node is k0 . • In addition, we assume that an infected node will generate ∆ new links upon infection. These links can be made to any node of either state and represent the direct or indirect social benefit of the infection. • α represents the assortative bias for what type of node is linked to by the new links. Either susceptible nodes are always selected (α = 0), susceptible nodes are selected with preference (0 < α < 1), all nodes are selected uniformly (α = 1), or infected nodes are selected with preference (α > 1). • If an infected node recovers, it loses ∆ edges at random. Define S and I as the fraction of nodes currently susceptible or infected, respectively. By definition, S + I = 1. Let [SI] be the number of edges between S and I nodes normalized by the total population size, and so on for the other two combinations [SS] and [II]. We could also define similar quantities for the number of triplets [ISI] or [ISS]. However, to limit the number of variables in our system, we use a moment-closure approximation to express these triplets in terms of the previously defined pairs. This approximation, introduced in Refs. [23, 22], considers that a triplet [abc] is roughly the number of [ab] pairs times the expected number of [bc] links per b node. Following this moment-closure approximation and the methods of Refs. [17, 37], we can write the differential equations governing this process as I˙ = −S˙ = β[SI] − rI

2[SS] k0 + I∆ ˙ + r[SI] [SS] = −β [SI] S k0 + I∆ + ∆ [SS] [SI] S k0 + I∆ ˙ [SI] = β[SI] 2 − − 1 − r[SI] + β[SI]∆ + 2 r [II] S S S + Iα k0 + I∆ + ∆ ˙ = β[SI] [SI] + 1 + β [SI] ∆ Iα − 2r[II] [II] S S + Iα (3) We can reduce this system to only three variables by using the following equations for node and edge totals S+I =1 [SS] + [SI] + [II] = 12 k0 + I∆ 6

(4)

are

The average degrees of an infected node hkI i and of a susceptible node hkS i hkI i = k0 + ∆ + I∆

hkS i = k0 + I∆

3.2

(5)

Outbreak dynamics

To determine this critical value — i.e., βc where the equilibrium I = 0 becomes ˙ and [SS]. ˙ ˙ [SI] unstable— we focus on a minimal set of ODEs: I, Linearizing these equations leads to the following Jacobian matrix:   −r β 0 2r∆k 2rk0 0  − k2rk =  k0 +∆0 β(k0 − 1 + ∆) − r − k0 +∆ . (6) J| 1 0 +∆ I=0,[SI]=0,[SS]= 2 k0 rk0 0 −βk0 + k0 +∆ 0 The Jacobian sheds light on the stability of the infection-free state. I = 0 is stable if the maximal eigenvalue is negative, whereas a small infection will invade the system if it is greater than 0. This epidemic transition occurs at βc 1 = . r k0 + ∆

(7)

If β < βc , any initial infection will die out before it is able to spread. We note in this equation that as ∆ increases the critical β needed to avoid extinction decreases. This means that as more links are generated per infected individual, the transmission rate that avoids extinction decreases. Conversely, in the case without social benefit, i.e. ∆ = 0, we recover the classic SIS dynamics [17, 37].

3.3

Steady-state Convergence

The rate of infection slows after this initial period of exponential growth. The increase in infected individuals through new infection (across [SI] links) is eventually counter-balanced by the recovery of individuals (the rate of recoveries increases as I increases). The steady state of the infection is where infection events and recovery events are perfectly balanced The fraction of individuals infected at this steady-state is analyzed by setting the derivatives in Eqn. (3) to zero. In all cases, I = 0 is an equilibrium, but it is not a stable equilibrium if β > βc . Equilibrium requires 0 = β[SI] − rI

2[SS] k0 + I∆ 0 = −β [SI] + r[SI] S k0 + I∆ + ∆ [SS] [SI] S k0 + I∆ 0 = β[SI] 2 − − 1 − r[SI] + β[SI]∆ + 2 r [II] S S S + Iα k0 + I∆ + ∆ The first equation implies [SI] = rIβ and the second one implies r k0 + I∆ [SS] = (1 − I) . 2β k0 + I∆ + ∆

(8)

These values for [SI] and [SS] as a function of I, together with the value of [II] implied by Eqn. (4), can be plugged into the third equation. We are left with 7

1

𝐼(𝑡 → ∞)

0.8

0.6

0.4 𝛼 = 0.01 𝛼 = 0.1

0.2

𝛼 = 1 𝛼 = 10 𝛼 = 100

0

0

0.5

1

1.5

2

𝛽/𝑟 Figure 2: Steady state size of the infected population in the case of instantaneous link addition. The figure shows the value of the equilibrium value of I as a function of β/r when k0 = 3 and ∆ = 2. When β < βc (here βc = r/5), the only possible equilibrium is I = 0. When β > βc , the equilibrium state I = 0 becomes unstable, and a steady equilibrium with I > 0 emerges. an algebraic equation for I, which has one root between 0 and 1 for values of β above the critical value. Figure 2 shows the steady-state fraction of infected individuals as a function of β. As β increases, so does the long-term percentage of infected individuals. The figure also depicts the effect of α on the long term percentage. As α decreases, the targeting of infected individuals improves, and thus the fraction of infected individuals increases.

3.4

Fixation dynamics

We now consider the dynamics at the conclusion of the epidemic. If we consider the case with no recovery (r = 0), then the susceptible population always tends to decrease. The rate of this decrease varies with the value of the assortative bias α and the number of new links generated per infection ∆. The governing equations in Eqn. (3) become simpler in the case where S 1 with no recovery. The term [SS] becomes negligibly small as it is second order in the number of S nodes (it requires two S nodes to be connected to one another). ˙ equation is The final term in the [SI] β[SI]

S S + Iα

(9)

Eqn 9 has two regimes with qualitatively different behavior: α > 0 and α = 0.

8

3.4.1

Case: Perfect targeting, α = 0

Eqn 9 becomes 1 for S Iα. The full system thus becomes S˙ = −β[SI] ˙ = β[SI] − [SI] − 1 + β[SI]∆. [SI] S

(10)

The solution of these coupled ODEs can be seen with a variable substitution called x. [SI] x= (11) S With this definition, we get the following relation: x˙ =

˙ [SI] [SI] S˙ − S S S

(12)

˙ we get an uncoupled By substituting the evolution equations for S˙ and [SI], equation. x˙ = βx(−x − 1 + ∆) − x(−βx) = (∆ − 1)βx (13) Using Eqn 10, we get the system of equations: x˙ = (∆ − 1)βx S˙ = −βxS

(14)

In this coupled set of equations, log x changes at a rate (∆ − 1)β, and log s changes at a rate of βx. x(t) ∼ exp[(∆ − 1)βt] Z t 0 0 S(t) = exp −β x(t )dt

(15)

0

From Eqn. (15), we see that if ∆ < 1, the proportion of S nodes decays at an exponential rate that is decaying exponentially. That is, as S decreases, the rate at which S decreases gets slower and slower. Interestingly, because of the exponentially decreasing rate, even as t → ∞ there are always individuals who are not infected. On the other hand, if ∆ > 1 the rate at which S decreases grows in time. In this regime, S never fully reaches 0, but it tends to 0 more and more quickly as the epidemic spreads. At the critical point, ∆ = 1, the rate at which S shrinks is constant. 3.4.2

Case: Imperfect targeting, α > 0

S If α > 0, S+Iα approaches Sα as S approaches 0. The coupled ODE system reduces to S˙ = −β[SI] (16) [SI] S ˙ [SI] = β[SI] − − 1 + β[SI]∆ . S α

9

0 -2

-4

-4

-6

log(I)

log(S)

0 -2

-8 -10

-6 -8 -10

-12

instantaneous ∆>1 instantaneous ∆1 instantaneous ∆ 0), when S becomes small enough that S < α, the last term in Eqn. (21) will be approximately equal to ∆S/α. Using this substitution, we get the following equation for the time evolution of the variable x = [SI]/S: ∆ x˙ = −βx + . (22) α This will eventually saturate to the steady state value x∗ = ∆/(αβ), and the susceptible rate, governed by the equation S˙ = −βxS, will decay exponentially as exp(−βx∗ t) = exp(−∆ · t/α). In contrast to the instantaneous link-addition model, the rate does not decay exponentially. Thus, the fraction of S decreases faster and as t → ∞, S → 0. However, if α = 0 (perfect targeting) then the last term of Eqn. (21) is simply ∆. Unlike all other cases, the rate of susceptible individuals will become zero at a finite time. To see why this behavior is the solution to the differential equations when S approaches zero, we use the following ansatz : S(t) = S0 (t∗ − t)a

[SI](t) = [SI]0 (t∗ − t)b . 11

(23)

The first equation of Eqn. (21) gives − aS0 (t∗ − t)a−1 = −β[SI]0 (t∗ − t)b ,

(24)

implying that a = b + 1. This also implies that near t∗ , [SI]/S 1, and therefore, the second equation of Eqn. (21) gives − b[SI]0 (t∗ − t)b−1 = −β

[SI]20 ∗ (t − t)b−1 + ∆. S0

(25)

For both right-hand-side terms to be comparable, we need b = 1. Finally, we recover the prefactors S0 = 12 β∆ and [SI]0 = ∆ from Eqn. (24) and Eqn. (25). Interestingly, if α is small but nonzero, then as in the instantaneous linkadding scenario, the dynamics of the infection will cross over from a regime that behaves as if α = 0, that is where S approaches zero quickly and appears headed to vanish at some finite time, to, once S becomes comparable to α, a regime of regular exponential decay with rate constant ∆/α. Another interesting consequence is the difference in the continuous case between perfect targeting and even slightly flawed targeting (α > 0). If the targeting is perfect, then all of the individuals will be infected in finite time for any positive ∆, the number of links created per unit time. In constrast, no matter how large the value of ∆, if α is non-zero, it still takes an infinite amount of time to infect all individuals. Error-prone infected individuals cannot convert the whole population in finite time as eventually the false-positive identifications dominate the true-positive ones as the fraction of susceptible individuals becomes increasingly small. Alternately, converting the entire population in finite time requires each infected individual to create a non-zero number of new links with susceptible individuals on average per time step. Of course, the time for the number of susceptible individuals to reach a small fraction of the population (S < ) will depend greatly on the link generation rate for either α regime. 3.5.3

Summary of epidemiological behaviors

The behaviors observed in our epidemiological models differ dramatically from classic spreading dynamics. We summarize our findings in Table 1. In short, while the addition of the connectivity benefit ∆ has a straightforward impact on the epidemic threshold and its early time spread, the fixation dynamics are sensitive to both how these new links are created and to whom. Fixation dynamics refers to the spread of the epidemic in the endemic regime, where we would expect detection of benes to be most likely. Consequently, our predictions for possible behaviors in this regime provide us with potential signals for the detection of benes in empirical data. Moreover, the sensitivity of these predictions to individual preference (i.e., α) also suggest the need for frameworks incorporating the cost and benefits of social targeting.

4

Sociological model of beneficial infections with strategic rewiring

In the context of social contagions, benes can change networks because individuals employ strategies such as actively seeking contacts to spread the bene. A 12

instantaneous ∆ α=0 α>0

continuous ∆ α=0 α>0

Epidemic threshold

βc /r = 1/(k0 + ∆)

βc = 0 since r = 0

Early time

Exponential spread

Fixation

exp[− exp(t)]

exp[− exp(−t)]

Cross over between exp. regimes (t∗ − t)2 exp(−t)

Table 1: Summary of epidemiological behaviors of benes. dramatic example is that of religious missionaries, who settle in new parts of the world and effectively create connections in the global social network where previously there were none. In this case, the advocates are driven by a desire to spread their religion as widely as possible. In social contagions more generally, people may actively seek to spread or acquire new behaviors or innovations. However, the decisions to seek or spread new behaviors or innovations may come with tradeoffs, e.g. individuals may worry about being too dissimilar to their connections. Here, we develop a sociological model to analyze how these diverse motivations affect adaptive network dynamics and the spread of contagion. As in the previous section, we consider individuals to be in one of two states: susceptible S and infected I. The benefit to an individual is a function of their state Xi (Xi = I if infected and Xi = S if susceptible), the number Ini of their infected neighbors, the number Sni of their susceptible neighbors, and the number Ig of infected individuals in the whole population: B = f (Xi , Ini , Sni , Ig ). This allows us to consider three different scales at which contagions could benefit individuals: 1) the direct benefit that infection provides to the individual (indicated by Xi ), 2) benefits from the infection status of their neighborhood (indicated Sni and Ini ), and 3) benefits from the infection status of the global population Ig . Different regimes arise depending on the scales included in the individual’s benefit function. The transmission dynamics is modeled as a simple contact process whereby an edge between an infected individual and a susceptible individual leads to the infection of the susceptible individual with probability β. The dynamics of the network is modeled as a process in which individuals attempt to rewire their neighborhood in the social network in order to improve their benefit. When individuals evaluate their expected future benefit, they take into account how their state and that of their neighbors may change if the bene were transmitted over the connections between them. They also consider the future benefit expected if they removed a connection to a neighbor and replaced it with a connection to a non-neighbor chosen at random from the population. They then perform the rewiring if this increases their expectation of future benefit. In the example of the missionary, an infected (converted) individual wants the global count of infected individuals to be as high as possible. The missionary will then tend to break ties to existing converts and seek out susceptible individuals in the expectation that they will be able to convert them. If, however, the missionary also prefers to be surrounded by others with shared beliefs, they might anticipate that any susceptible individuals that they befriend might not be successfully converted. In this case, there is a possibility that the missionary’s neighborhood will gain a non-believer, which will lower the expected benefit of

13

rewiring away from a convert. We can incorporate these types of trade-offs into the benefit function B. The expression for the expected benefit is: E[B(Xi = I, Ini , Sni , Ig )] =

Sni X j=1

Binomial(j, Sni , β)B(Xi = I, Ini + j, Sni − j, Ig + j)

E[B(Xi = S, Ini , Sni , Ig )] = (1 − (1 − β)Ini )B(Xi = I, Ini , Sni , Ig + 1) + (1 − β)Ini B(Xi = S, Ini , Sni , Ig ),

(26) where Binomial(k, n, q) = nk q k (1 − q)n−k . The first expression in Eqn. (26) is the expected future benefit of an infected individual given their neighborhood ni . It takes into account the probability that a number j = 1, . . . , Sn of their susceptible neighbors become infected because of their mutual connection. The second expression in Eqn 26 is the expected future benefit of a susceptible individual given their neighborhood n. Here we include the probability that they become infected by at least one of their In infected neighbors. Note that these expressions assume that individuals make predictions about the dynamics of epidemic spreading only across their direct connections, not the entire network.

4.1

Dynamical Regimes

We simulate the spread of infection and its accompanying network changes. In each iteration of our simulation, a randomly selected individual first attempts to adaptively rewire their neighborhood. Afterward, each infected-susceptible edge causes an infection with probability β. We consider two different ways in which an individual can rewire their neighborhood. The first case is analogous to the previous section, in which infected individuals add new connections. The relevant choice is between: 1) adding a link to an infected individual, 2) adding a link to an susceptible individual, or 3) not adding a link. This formulation includes, as a special case, the perfect targeting regime of the continuous link-addition model of Sec. 3. Specifically, it corresponds to that case if the benefit function for infected individuals encourages them to add links to susceptible individuals B = Sn , while the benefit function for susceptible individuals is 0. An example of the transmission dynamics of a bene in this regime is shown in the top left panel of Figure 4. The top right panel shows an evangelical rewiring process. This process has a super-exponential fixation rate in the number of susceptibles, which is due to the fact that many infected individuals are targeting the remaining susceptibles in order to infect them (compare to Figure 3). In the second case we consider, infected and susceptible individuals rewire some of their edges with probability prewire . Individuals in either state (I or S) make the choice between 1) rewiring from an infected individual to a susceptible individual, 2) rewiring from a susceptible individual to an infected individual or 3) not rewiring. These kinds of rewiring dynamics are used to investigate three different kinds of benefit functions, described in the next section. Among other things, we will show that these dynamics (which do not add links but only rewire them) can also produce super-exponential fixation rates.

14

Portion infected

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Adding S-I links process 1.0Evangelical rewiring process

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500 1000 1500 2000 2500 3000

500 1000 1500 2000 2500 3000 Iteration

Figure 4: Transmission dynamics for link generation and rewiring. The transmission dynamics of the process in which infected nodes add nodes preferentially to non-infected nodes (left), and the process in which infected nodes rewire preferentially to non-infected nodes (right). The top row shows the number of infected individuals across generations of a typical run in both regimes. The bottom row demonstrates that both exhibit superexponential fixation, as in the epidemiological model.

15

evangelical cool kids snobs

Infected B = Ig B = Ini − Sni B = (Ini − Sni )

Susceptible B=0 B = Ini − Sni B=0

Table 2: Benefit functions

4.2

Evangelicals, Cool Kids, and Snobs

We use our framework to analyze the dynamics of beneficial epidemics using three different benefit functions that each demonstrate different trade-offs and social preferences. In the evangelical case, the benefit for infected individuals increases when the bene is spread on the global scale (i.e., in the entire population). In the cool kids case, everyone (whether infected or not) seeks to connect to as many infected individuals as possible, and as few non-infected individuals as possible. Finally, in the snob case, infected individuals seek to connect to as many other infected individuals as possible, though the benefit for non-infected individual does not depend on the state of their neighbors. The corresponding benefit functions are specified in Table 4.2. It should be noted that our framework is sufficiently general to analyze adaptive rewiring dynamics not only in benes, but also in detrimental infections – for example, in which individuals rewire to avoid infection[14, 11] – though we do not consider such cases in this work. Each of the three cases presented here are motivated by existing discussions of social processes in economics, political science, or social psychology. The evangelical case is analogous to social movements [47, 32], where explicit recruitment into a religious organization, political movement, or some other process of collective action is occurring. The cool kids case could be thought of as the transmission of an idea or characteristic through a group, where the infected population’s assortative preference results in formation of cliques and fads (e.g. the anticonformity copying modeled in [6]). Finally, the snobs case is analogous to segregation models [41, 9] where potentially asymmetric and conflicting preferences for assortativity exist. The three benefit functions were used to simulate rewiring dynamics, starting from a 1000-node Erd˝ os-Rényi random graph. Figure 5 shows the dynamical regimes and topological effects that arise. In the evangelical case, we see super-exponential fixation dynamics reminiscent of the epidemiological model where edges are added, although here edges are only rewired and not added (see right side of Figure 4). This is driven by the strong disassortative behavior from both the infected (who are interested in spreading) and the non-infected (who are interested in joining the community of infected), as shown in the cumulative rewiring plot. This disassortative behavior, in which the infected individuals are seeking remaining susceptibles to infect, speeds up the epidemic. The last individuals to remain non-infected also receive a large number of connections from infected, causing the degree distribution of the network to become heavy-tailed (hence, infection time correlates positively with final network degree). In the cool kids case, both infected and susceptible individuals want to be connected to the infected, as to a very popular club. However, the infected want to avoid the susceptible. This creates a strong asymmetry between the behavior 16

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Figure 5: Dynamical regimes in the sociological model. The dynamical regimes arising from the three different benefit functions (evangelical, cool kids, snobs). The first row shows the number of infected individuals over time. The second row from the top shows the cumulative number of rewirings, where for example, i → i to i → n indicates the rewiring by an infected individual from an infected to a non-infected neighbor . The third row shows the degree distribution of the network after the last iteration. The fourth row shows the adjacency matrix of the network after the last iteration, where the infected (red) and the non-infected (blue) have been sorted.

17

of the infected and susceptible. The susceptible are rushing to rewire to the infected, as shown by the initially predominance of n → n to n → i rewiring. The infected unwire from the non-infected, giving rise to the growing count of i → n to i → i rewirings once the number of infected individuals rises. These two behaviors are at cross-purposes, and once the infected-driven unwiring gets going, the doors to the infected community close. The result is a network composed of two blocks: one includes the susceptible as disconnected singletons, while the other includes the infected individuals which are interconnected. This also produces an incomplete epidemic spreading process that cannot proceed further: the non-infected have no links to rewire, and the infected will never rewire to connect to them. Finally, the degree distribution of the resulting cool kids network is heavy-tailed, since the first individuals to become infected attract new edges for a longer period of time (here infection time correlates negatively with final network degree). In the snobs scenario, the epidemic makes even less progress than in the cool kids scenario. This is because the infected assort with similar individuals, again cutting out susceptible individuals from their neighborhoods. Meanwhile, the non-infected rewire in two different ways: some connect to the infected (disassortative rewiring), while others seek out the non-infected (assortative rewiring). These two different strategies correspond to non-infected individuals with different neighborhoods. If a non-infected individual is mainly surrounded by infected individuals, they can rewire to an infected, increasing their chance of receiving the bene and thereby of becoming part of the infected community. However, if they are mainly surrounded by non-infected individuals, it is risky to be in contact with an infected individual, since an infection would put them at odds with their neighborhood. The result of these rewiring strategies is that the network divides into two completely disconnected communities, and this prevents the epidemic from reaching the whole population. The characteristics of these regimes vary with transmission probability β, as shown in Figure 6 (the value of β used for Figure 5 is shown as a dashed line). When β is large, the epidemic spreads in all cases, albeit at different rates depending on the benefit function. Note that the number of rewirings decreases when transmission probabilities are sufficiently large, since adaptive network change stops once the epidemic has swept throughout the population and there are no benefit differences that can can extracted by rewiring. Interestingly, in cases where rewiring can reduce epidemic spread, such as the snobs benefit function, the number of rewirings also decreases as transmission probabilities becomes sufficiently low. In these cases, transmission is so slow that rewirings disconnect the infected from the non-infected before the infection can spread, quickly driving the network into an attractor state corresponding to a partiallycompleted epidemic. In conclusion, adaptive rewiring can affect epidemic dynamics in multiple ways. When infected individuals benefit from increasing the global number of infections, it can lead to an accelerating uptake and a much faster global spreading than an epidemic without adaptive rewiring. If instead, the infected individuals tend to assort, the epidemic can be stalled as infected individuals entirely disconnect from the non-infected. This also demonstrates that adaptive rewiring dynamics can shape network topology in important ways, such as giving rise to features like community structure and fat-tailed degree distributions.

18

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5 5.1

Empirical evidence for beneficial epidemics Population genetic signatures of benes versus beneficial mutations

How could we empirically distinguish the selective sweep of a beneficial element that is transmitted strictly vertically (e.g. a beneficial mutation) versus a horizontally transmitted bene? As we found in Section 1, benes are expected to sweep through a population substantially more quickly than beneficial mutations. If the evolutionary record were perfect such that we had access to all genomes within a population across generations, the two events could be distinguished. The same beneficial mutation is unlikely to arise multiple members of a population in a single generation. Thus, we would expect to see only a small fraction of the population carrying the mutation in the generation following its first appearance. In subsequent generations, the beneficial mutation would increase in frequency owing to the fitness advantage it confers. We contrast this with a bene which can spread horizontally. After a single generation, it would be possible to see the same genetic sequence in multiple members of the population. In practice, the evolutionary record is incomplete and may be insufficient to discriminate between the spread of a beneficial mutation or a bene. This may be especially true if the bene is somehow altered with transmission. The result would be a population with a genetic sequence exhibiting considerable diversity. This could incorrectly be interpreted as an earlier arrival of a benefical mutation which then mutated over the course of several generations.

5.2

New words as linguistic benes

New words are doubly useful to speakers. First, speakers may use new words to communicate new concepts or to communicate old concepts in a new way [5, 48]. Second, speakers can use new words to assert identity through word choice [25, 16]. For instance, the use of the phrase “personal computer” could reflect that the speaker keeps up with changing technology, and may also be an intentional signal by the speaker to show an awareness of technological change. Furthermore, adoption of new words can strongly signal a desired affiliation with some particular identity or movement. For example, the use of the word “chairperson” in the early 1970s could signal a progressive approach to gender equality, and “groovy” in the early 1960s could signal a cool personality. A new useful word spreads contagiously as nearby listeners adopt the word for use. In this section, we study a sample of such words as examples of linguistic benes. We find that new word adoption follows a common three-phase trajectory: (i) usage grows super-exponentially, (ii) usage may grow exponentially, during a transition phase to (iii) usage grows sub-exponentially, concomitant with saturation or decrease in usage. We analyzed usage trajectories I(t) of beneficial new words or phrases by gathering usage data from the Google Ngram corpus for 48 words listed in Table 3. Due to the year-resolution of word usage in Google Ngrams, we approximated the growth rate I˙ using central differences and ˙ ˙ computed normalized growth rates I/I. Constant and positive I(t)/I(t) indi˙ cates exponential growth, while increasing I(t)/I(t) indicates super-exponential ˙ growth, and decreasing I(t)/I(t) indicates sub-exponential growth. Figure 7

20

shows both the time-series of adoption and the normalized growth rates for three example terms. Due to the stochastic nature of Google Ngrams data, we ˙ from a smoothed time-series. computed I/I 100

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. Figure 7: Trajectories of the terms “chairperson”, “genomics”, and “personal computer” around the times of their emergence. (Top) Time series of usage frequency for each term, according to the Google Ngram corpus, normalized by the maximum frequency achieved by that term. (Bottom) Time ˙ illustrating characteristic ordered superseries of normalized growth rates I/I exponential, exponential and sub-exponential regimes. We compare the observed phenomena in word adoption to the models developed in previous sections of this paper. Figure 8a, shows normalized growth rate ˙ for a standard SI contagion (∆ = 0) and two benes (∆ = 0.5, 1.0), curves I/I as defined in Sec. 3. Triphasic growth characterizes both new word adoption (Figure 7) and the bene epidemics (Figure 8a, ∆ > 0), but not the standard SI epidemic (Figure 8a, ∆ = 0). The Cool Kids model introduced in Sec. 4 also exhibits the same triphasic growth, shown in Figure 8b, consistent with the hypothesis that new words infect the lexicon as benes, and not as standard contagions.1 ˙ scales consisFinally, we find that the maximum normalized growth rate I/I tently with the maximum proportion of that word’s usage U within the overall population of words, as shown in Figure 9. For the 48 words studied here, we ˙ find that max(I/I) ∝ max(U )γ with γ = 0.77 ± 0.08. In the context of a beneficial epidemic, this implies that the peak rate of spreading is related to the final saturated word usage. In other words, the potential popularity of a word determines the rate at which the word will spread: very useful and popular words spread much more rapidly than words with limited use. While mechanistically unexplained, this pattern may provide clues to the process by which the beneficial epidemic of new word spread occurs.

1 We note that simple SIS contagions which are seeded at nodes of very low degree (e.g. degree one [20]) may also exhibit subtle triphasic growth, but through a different mechanism.

21

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22

aspirin chairperson chillax clunker cocaine cold fusion computer geek crunk Democratic Republic of Congo escalator Facebook factoid flapper flash mob funk music gansta

gee whiz geez genetically modified genomics get tight groovy hate crime heroin HIV HPV information superhighway kerosene laptop linoleum metrosexual microbiome

microbrewery Mozambique nylon one night stand personal computer phonograph photovoltaic Pope John Paul proteomics retrovirus snazzy swank transcriptomics xeriscape yoga zipper

Table 3: The list of 48 emergent words or phrases studied using the Google Ngram corpus.

23

6

Discussion

The dynamics of beneficial epidemics exhibit novel characteristics compared to those of previously analyzed epidemics. The key characteristic that distinguishes the dynamics of beneficial contagions, or “benes”, is the combined effect of its benefit and horizontal transmission. One striking outcome of this combined effect is the super-exponential growth rate of infections. We found other distinct dynamics of benes in discipline-specific models: an evolutionary model, an epidemiological model, a sociological model, and an empirical linguistic case study. Why do biological epidemics of harmful genetic elements, such as diseases, seem to be much more common than biological epidemics of beneficial genetic elements? Our evolutionary models suggest that a bene that can be transferred horizontally among individuals in the same generation spreads much more quickly than a beneficial genetic element that can only spread vertically between parent and offspring. This phenomenon may explain the dearth of observations of beneficial epidemics. Very shortly after such a bene appears, it spreads throughout the population, meaning there is a fleeting opportunity to observe a population in a transient state. Once the bene is fully fixed in the population, it becomes difficult to distinguish this element from previously established genetic elements. The importance of horizontal transfer in the evolutionary case led us to consider how interactions between individuals influence the spread of a bene in a social network. We used an SIS epidemiological model where we assume that the bene’s net effect is to increase the connectivity of infected individuals. Infected individuals attempt to share the beneficial infection, and susceptible individuals themselves seek the infection. This differs from other models of infectious diseases in adaptive networks, in which individuals typically attempt to limit the spread of a disease. We find that when individuals become infected and connect more often with susceptible individuals, the transmission rate necessary to ensure an outbreak is smaller than in a traditional SIS model. This result demonstrates that individual behavior is important in determining whether a beneficial epidemic occurs. Infected individuals might attempt to spread a bene by creating a fixed number of new links immediately upon infection, or by continuously creating new links at a fixed rate while infected. However, those new links may or may not be accurately targeted at susceptible individuals. The mechanism and accuracy of new link formation interact to shape the way a bene spreads through the network. When nodes add a fixed number of links immediately upon infection, but inaccurately target susceptible nodes, the bene will not spread through the entire network, always leaving a small residual susceptible population. On the other hand, if targeting is free of errors, the bene can spread to the entire population, with the residual susceptible population approaching zero in the long time limit. When infected nodes continuously add links at any fixed rate, inaccurate targeting is less consequential: the bene may sweep the population in the long time limit. Moreover, if targeting is free of errors, the bene will sweep the population in finite time. We next consider a more general framework to understand how rewiring strategies arise from social preferences. We consider benefit functions for indi24

viduals that incorporate information about the structure of the social network. We consider several sociologically realistic cases, finding two main dynamical regimes. In the first regime, the bene fixes super-exponentially. We find this in the “evangelical case”, in which infected individuals value the global distribution of the bene and therefore strategically rewire to non-infected individuals to facilitate the bene’s spread. This suggests that the enormous success of social movements such as the spread of some religions is due to the fact that they instill in individuals the desire to convert anybody, not just their social entourage. In the second regime, the epidemic stalls and results in a disconnected network. We find this regime in both the “cool kid” and“snobs” cases. In both of these, some of the individuals prefer to conform with their neighbors. This leads to assortative rewiring on the part of infected individuals. In the sociologically realistic cases considered, these spreading dynamics have important effects on the topology. Starting from random networks, the degree distribution becomes more fat-tailed, as is commonly found in real social networks. This suggests that beneficial epidemics may be drivers of common social structures. The sociological framework developed in this paper is very general. It incorporates a wide variety of preferences for spread at the individual-level, neighborhood level and global level. To our knowledge, the interaction of preferences for spread of the bene at different scales has not been modeled before even though it seems to be a pervasive feature in processes of social change. For example, most individuals have a preference for behaviors or technologies that spread globally but would prefer them not to spread in their immediate neighborhood, a phenomenon called NIMBYISM [42], e.g. windmills. Finally, we looked for evidence of beneficial epidemics in empirical data using the Google Ngram corpus. Speakers actively spread new words or phrases through a population both to keep up with the times and to signal identity— both of which are forms of social benefit. We find that new words exhibit a triphasic growth pattern of super-exponential, exponential and sub-exponential growth, prior to saturating at near-constant usage. Not only is this consistent with the dynamics found in the various beneficial epidemics models explored in this paper, but this is the first observation of such triphasic growth dynamics in neologisms. This characteristic triphasic growth is not observed in standard epidemic models. The outcome of a beneficial epidemic may reveal the mechanisms underlying its generating dynamics. For example, if a final infected population is characterized by a disconnected and completely infected population, we might conclude that a “snob” or “cool kids” strategy was involved. Crucially, the super-exponential behavior of epidemic benes is difficult to observe, because the transition from an uninfected population to the stable infected population can be extremely rapid. For example, in our analysis of the spread of new words, new usages take over very quickly. Without sufficient time series data, these events would be hard to measure. In general, our investigations reveal qualitatively different and interesting dynamics that may have important implications for the spread of technological innovations, socio-political ideas, and biological elements within the framework of beneficial epidemics.

25

Disclaimer: This paper was produced, from conception of idea, to execution, to writing, by a team in just 72 hours (see Appendix A). Although part of our team was working on literature search, we expect that, at this point, our literature review is incomplete. We welcome any additional references for future versions at [email protected].

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A

72 Hours of Science

During the 72 hours between noon on April 4 and noon on April 7, 2016, postdocs from the Santa Fe Institute (SFI) will be conducting an experiment. This is both a description of that experiment’s particulars and an open letter declaring our commitment to those particulars. It will also be included as a supplement to any eventual publication. For most projects that we work on as postdocs, science moves slowly. Research takes months or years to evolve from early-stage ideas to coherent and crisp findings, with additional months of writing before submitting work for publication. During the 72 Hours of Science, we will actively challenge the notion that this is the only timescale for science by going from idea to submitted arXiv post in only 72 hours. Our goal is also to engage nearly the entire postdoc community at SFI with specialities including anthropology, pure mathematics, urban development, cultural evolution, physics, computer science, ecology, political science, and evolutionary biology. Thus a second basic aspect of this informal experiment is to test the limits of science not only in terms of pace but also multidisciplinarity. We think it’ll be interesting, challenging, and fun. The agreed upon rules are as follows:

Rule 1: Submit to the arXiv within 72 hours The minute the clock starts, at 12:00 noon MDT on April 4, 2016, we’ll begin to choose a project to work on. Before the clock stops, at 12:00 noon MDT on April 7, 2016, we’ll submit our written manuscript to the arXiv, along with a copy of this description which registers the informal 72h(S) experiment. During the intervening three days, we’ll decide on a project, develop the project, divide tasks among the participating postdocs, and write up our findings. It should be noted that the arXiv post will not be a description or analysis of the informal experiment, but of the generated scientific project itself. This document, which you’re reading now, is intended to catalog the process as a supplement to the scientific paper generated from the process.

Rule 2: Fresh ideas only Choosing a project will be key to the success of 72h(S), so in order to select from the most diverse set of ideas, any participant can propose a project idea to the group. However, in the spirit of a good challenge, only projects or ideas that are new to everyone are allowed—they cannot be discussed beforehand, nor can they be substantially developed by the presenter. In other words, as a group, we’re starting from scratch with a fresh idea and then developing it together. Since we did not want to rule out ideas that were mentioned, or created, during an off-hand discussion or comment during a seminar we have the formal criterion that 34 of the group should not have heard the problem statement before, and we allowed each person to informally “bounce the idea off” one person not at SFI.

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Rule 3: We’re in this together It’s a collaboration, not a competition. Over the course of three days, each participant will occupy multiple roles, from reading to writing, coding to reviewing, cooking to coffee-making. So, regardless of whose fresh idea is chosen, authorship will be Santa Fe Institute Postdocs for the arXiv submission, with an alphabetical list of authors as the first reference with a statement of uniformly equal credit.

Agreed upon by SFI Postdocs on April 4, 2016, 11:55 AM MDT.

Acknowledgments We are grateful for generous financial support from the Miller Omega Program and the Santa Fe Institute. We thank the Santa Fe Institute for encouragement in developing and executing the 72 Hours of Science experiment, and for funding. L.H.-D. acknowledges Vincent Marceau for helpful discussions. We especially thank Dave Bacon, Bruce Bertram, Ronda Butler-Villa, Jennifer Dunne, Joerael Elliott, John German, Michelle Girvan, Marcus Hamilton, Michael Lachmann, Juniper Lovato, David Krakauer, Nathan Metheny, John Miller, David Reed, Sidney Redner, Kristina Serna, Carla Shedivy, and Hilary Skolnik.

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