An Epidemiological Approach to the Spread of Political Third Parties

Third political parties are influential in shaping American politics. In this work we study the spread of a third party ideology in a voting population where we assume that party members/activists are more influential in recruiting new third party voters than non-member third party voters. The study uses an epidemiological metaphor to develop a theoretical model with nonlinear ordinary differential equations as applied to a case study, the Green Party. Considering long-term behavior, we identify three threshold parameters in our model that describe the different possible scenarios for the political party and its spread. We also apply the model to the study of the Green Party's growth using voting and registration data in six states and the District of Columbia to identify and explain trends over the past decade. Our system produces a backward bifurcation that helps identify conditions under which a sufficiently dedicated activist core can enable a third party to thrive, under conditions which would not normally allow it to arise. Our results explain the critical role activists play in sustaining grassroots movements under adverse conditions.


Introduction
The 2000 United States presidential election was for many a testimony to the impact of third parties in a traditionally bipartisan government. Ralph Nader, the presidential candidate for the Green Party, won 2% of the popular vote, a percentage that many attribute to the defeat of Democratic candidate Al Gore (Southwell, 2004). The Green Party captured a seemingly insignificant number of votes relative to majority percentages, yet its presence in the election ultimately served to shape American politics for the years following. This incident demonstrates how third parties, often emerging as grassroots movements (i.e., movements at the local level rather than at the center of major political activity), can ultimately impact at the national level, hence prompting the need to study their emergence and spread within a voting population.
Third parties are defined as political parties operating along with two major parties in a bipartisan system over a limited period of time (where we define a limited period of time as a range of a few years). For the purposes of this paper we apply this definition to all minor parties. Traditionally, third parties have served as venues of political dissent for voting individuals dissatisfied with the major candidates in an election. They often tackle specific issues otherwise ignored by major political parties, thus relinquishing popular support nationwide. As Supreme Court Justice Earl Warren wrote in 1957, "History has amply proved the virtue of political activity by minority, dissident groups, which innumerable times have been in the vanguard of democratic thought and whose programs were ultimately accepted" (Warren, 1957). Hence, while third parties rarely capture the majority vote, their agendas, often incorporated into major party platforms, are significant nonetheless.
Given the potential relevance of third parties to national politics, we study, qualitatively and numerically, the dynamics of the emergence and spread of third parties on a local level where growth is measured in terms of the number of third party voters and members. We restrict our study to a local level because third parties usually originate in a small group and, via a "bottom-up" method of diffusion, spread within a population by acquiring local official positions and then expanding to higher levels of government (Kowalewski, 1995). Although individual personalities and circumstances dominate the initial formation of any group, the ability of even a small group of people to make itself heard within a larger community is great enough, especially in an age of information technology, that we consider the local level large enough to be described by a collective average. That is, the voting population under study is large enough that stochastic (random) effects are dominated by the deterministic average behavior of the group.
We use an epidemiological paradigm (Brauer and Castillo-Chávez, 2001) to translate third party emergence from a political phenomenon to a mathematical one where we assume that third parties grow in a similar manner as epidemics in a population. We take this approach following in the steps of previous theoretical studies that model social issues via such methods (Bettencourt et al., 2006;Castillo-Chávez and Song, 2003;Crisosto et al., 2001;González et al., 2003;Song et al., 2006). The epidemiological metaphor is suggested by the assumption that individuals' decisions are influenced by the collective peer pressure generated by others' behavior; the "contacts" between these two groups' ideas are analogous to the contact processes that drive the spread of infectious diseases. Here we assume that a certain subpopulation of the voting individuals, defined according to certain demographic factors, is more receptive (in epidemiological terms, susceptible) to third party ideology than the rest of the voting population, and that their political behavior is therefore driven by such collective peer-pressure contacts (Southwell, 2003;Timpone, 1998;Wong, 2000).
There are many components that may affect the decision of a person when it comes to voting. During a political campaign all candidates spend a great amount of effort to assure that as many people as possible get exposed to their ideas, and most importantly, their name. People are often so overwhelmed by exposure to different candidates that they end up depending more on informal ways of obtaining information such as talking to other people.
This makes networking among people very influential during election time; in fact some studies show that voting is "contagious" (Nickerson, 2008). It has been shown that people often rely on friends, relatives, coworkers, etc. to obtain information about the candidates (Robinson, 1976). Also, there is evidence that people who support a particular candidate tend to encourage others to vote for that candidate (Huckfeldt and Sprague, 1995). This makes the interaction between people a very important factor in the voting process because both the person wanting information seeks it from other people and the ones with particular preferences want to share them.
Collective behaviors such as voting have been studied for decades, notably in the seminal work of Granovetter (1978), who considered individuals in a population to have a distribution of thresholds with regard to their willingness to participate in a particular collective behavior (he used rioting as a primary example). Other subsequent studies considered the role played by a core group of especially influential individuals (Macy, 1991;Oliver and Marwell, 1988), such as party activists recruiting voters. These studies used probabilistic and stochastic models which yielded primarily numerical results; in applying the epidemiological framework described above, we focus on the collective (rather than individual) thresholds for persistence of a behavior (here voting and party membership) which our deterministic models allow us to calculate.
While our model is designed to pertain to all third parties, we consider the Green Party as a case study. Although formally united under the Association of State Green Parties in 1996 (and later the nationalized Green Party of the United States in 2001), state-based green parties have thrived in the U.S. at the local level since 1984, when the Green Committees of Correspondence (CoC) were formed with the purpose of organizing local Green groups and working toward the founding of a national Green political party (Marks, 1997). Our particular study focuses on the growth (and in some cases decline) of the Green Party in six states and the District of Columbia in the past decade, using voting and registration data.
In comparing the predictions of our model to particular data, we consider a short time frame so that we can assume that social structure within the state in question does not change drastically, a necessary condition for assuming voting population heterogeneity.
We organize our paper as follows: Section 2 describes our theoretical model, as well as a simplification in which the entire population under study is equally receptive to third party ideas. Section 3 presents the mathematical analysis of the simplified model (which employs qualitative analysis techniques from the field of nonlinear dynamical systems, as well as sensitivity analysis) and interprets the results, and Section 4 applies them to a case study, using data to estimate model parameters. Section 5 performs a similar but more limited analysis of the more complex model, and Section 6 draws conclusions about the implications of our models for the growth of third parties from grassroots movements.

Underlying Assumptions
In developing our model, we apply epidemiological terminology to describe the growth of a third party. The assumptions we make about how individuals behave, and change their behavior, define the classes in our compartmental model and the rates at which individuals move between classes.
(1) We assume that our population is a heterogeneous mix of individuals who belong to different backgrounds according to certain demographic factors.
Our model considers a population of all voters, N, divided into two classes or subpopulations whose susceptibility to third party ideology is based on demographic factors such as education, socioeconomic status, race, gender, age, political orientation and professional occupation. Inherently, certain demographic characteristics, labelled as high affinity, make an individual more likely to subscribe to a third party's ideology, which targets a more specific audience than alternative majority agendas. That is, upon entering the voting system, certain individuals are more statistically inclined to vote a certain way. For example, a progressive environmental activist is statistically more likely to agree and vote for the Green Party agenda, which stresses communal-based economics, local government, and gender and racial equity, than a conservative corporate executive whose economic philosophy directly conflicts with that of the Green Party. For this reason we consider population heterogeneity vital to this study.
We apply the following method of dividing the entering voting population into two susceptible classes: if an individual has more high affinity factors than low affinity factors then that person directly enters the high affinity class and similarly for low affinity susceptibles.
We define high affinity factors as features of the individual based on his/her demographic profile that make him/her more inclined to vote for the third party; conversely, low affinity factors make the individual less statistically likely to subscribe to the party's platform. We assume that a constant proportion p (0 < p < 1) of new voters enters the high-affinity class H (the remaining 1 − p proportion enter L).
(2) We assume that individuals' affinity factors (demographic characteristics) remain fixed for the period of the study.
(3) We assume that a third party's agenda remains consistent over time.
We assume that individuals do not move from one susceptibility class to the other. One reason for this is the relative permanence of individuals' demographic characteristics. We limit our model to tracing the expansion of the third party; hence, we refer to a shorter time period over which we assume social structure remains constant. In other words, individuals with high affinity to the third party do not become individuals with low affinity, and vice versa.
The other reason for this consistency has to do with the parties themselves. In addition to being more specific than major party agendas (i.e., more specific in their goals and less geared to moderacy), third party platforms tend to be more consistent over time. Third parties are not pressured to constantly adjust to the shifting demands of the populace since they do not seek the majority vote. Consequently, they do not target the majority voting population. Each party has its own agenda, which appeals to certain sectors of the voting population. Hence different parties target voting populations that are more inclined to subscribe to their ideology. While one party, for example, may target individuals from a certain educational background that we, in our model, label as high affinity and that other parties may overlook, all parties nonetheless recognize that education factors into an individual's likelihood to support or refute that party's platform. It is true that individuals from varied backgrounds comprise the main parties, yet, when dealing with the specific agendas of third parties that do not strive to sway the majority vote, we assume that third parties appeal to individuals of certain demographic backgrounds more than others. Therefore, we account for the aforementioned standard set of demographic factors that parties look at when spreading their ideologies. In our paper we apply our model to an individual case study of the Green Party of Pennsylvania; however, the same methodology of distinguishing susceptibles can be applied to all third parties.
(4) We define two levels of participation in third-party politics: voting for third-party candidates, and membership. A party exists only if it has members; we define members as those who pay dues, volunteer, and preside over party affairs.
As described above, all voters enter the voting system either to the low affinity, L, or high affinity, H, susceptible class. According to our epidemiological metaphor, in addition to these two susceptible classes, our model includes three infected classes: V H , V L , and M, third party voters from the high affinity class, third party voters from the low affinity class, and party members respectively. We define party members as voters of the third party who pay dues to the party; often such members officiate, volunteer and actively campaign for voter recruitment. In epidemiological terms, V H and V L correspond to voters of a lower degree of infection and individuals of the M class are voters infected to a higher degree. We distinguish between V H and V L because of their interactions with their respective "neighbors" in H and L.
(5) We assume that third parties, emerging through resource-limited grassroots efforts, spread primarily via primary (direct) contacts between third-party supporters and susceptibles.
(6) We assume that third party members have a greater effect upon voter recruitment than do third party voters, due to members' activism.
In our model, a system of nonlinear differential equations, we consider susceptible movement into voting and member compartments as well as possible regressions back from the third party voting phase into the susceptible class. Once an individual is susceptible he/she can become 'infected' (either V H or V L ) through direct contact with the V H , V L , and M classes. Due to a lack of funding and resulting lack of mass-media exposure to the general population, these primary contacts with susceptibles involve such direct interaction as personal meetings, phone conversations, and electronic communications like personally addressed e-mails and weblog (blog) comments. We assume that the rate at which these contacts occur is proportional both to the size of the susceptible group and to the proportion of third-party supporters in the population.
We do not consider a linear term weighing the influence of media coverage from the third party (i.e., secondary contact factors) in the forward transition from both susceptible classes to third party voting classes. Instead, we focus on the nonlinear terms considering the effects of voters from the V H , V L , and M classes, where voters from V H and V L bear an affinityspecific influence β H (from H to V H ) and β L (from L to V L ) in third party voter recruitment.
Through activism, members from M influence susceptibles of each type at higher rates (αβ H and αβ L , respectively) than voters, with their increased influence measured by the parameter α (α > 1).
(7) We consider both primary (direct) contacts and secondary (indirect) contacts in the regression of third party voters to the susceptible class.
(8) We assume that all other parties exert equal influence in discouraging third party voting.
(9) We assume that individuals have more influence upon others of their same affinity class (high or low) than upon members of the other affinity class, in encouraging and discouraging third party voting.
We consider the transitions back from the third party voting to the susceptible classes to involve both linear terms, ǫ H V H and ǫ L V L , and nonlinear terms, φ H (H + σL) V H N and φ L (L + σH) V L N , contributions by secondary contacts with the opposition (i.e., media from well-funded majority voters) and direct contact with the susceptible classes respectively. Compared to primary contacts, described in a previous paragraph, secondary contacts include mass emails, media, and circulating literature.
Voters from a certain susceptibility class (with its own set of demographic factors) address issues that usually appeal more to voters deriving from the same class. Therefore, susceptible individuals with high affinity bear a greater influence in recruiting voters who came from the high affinity susceptible class back into the susceptible class than susceptible individuals with low affinity. The reduction in influence by individuals from a different affinity class is denoted by σ, so that in regressing back from V H to H, σL represents the lesser influence that L individuals exert on voters from V H than do susceptibles from H, the higher affinity class (similar reasoning applies to the V L -to-L transition). Likewise, the cross-affinity influence in recruiting third party voters is reduced by a factor of σ.
(10) We assume that third party voters become active party members through the ongoing  We first introduce a two-track model, as described immediately above, to study the dynamics between a heterogeneously mixed population of susceptible voters, third party voters, and party members. We apply the following set of ordinary differential equations to model voting dynamics, as illustrated in Figure 1.

The Simplified (One-Track) Model
In order to facilitate analysis of the two-track model, we initially consider a simplified version that does away with voting population heterogeneity (assumption (1), and consequently (9)) before exploring analysis for the more complex system. This simplified model assumes a homogeneous susceptible population (p = 0 or p = 1), reducing the two-track model to one susceptible class, S, and two infected classes: third party voters, V , and party members, M, respectively. The S class comprises those individuals who vote, but do not vote for the third party. The V class comprises the third party voters, and the M class again has third party members (i.e., party officials, donors, volunteers).
In the one-track model we omit unnecessary parameters from the heterogeneous version. Figure 2 illustrates the one-track model, and Table 2 summarizes the parameters. Figure 2: The one-track model, with per capita flow rates In this case the model reduces to the following system, which is effectively two-dimensional since N can again be seen to be constant:

Analysis of the One-Track Model
We begin our analysis by calculating equilibria for our model and determining conditions for their existence and stability. We first simplify the system in two ways. Since the total voting population N is assumed constant, we can reduce our system to two dimensions by We can now analyze the system defined by (11) and (9), as S can always be found once V and M are known.
The constancy of N also allows us to proportionalize the system by defining new variables In order to analyze stability we linearize the system and compute partial first derivatives with respect to each of the variables, v and m, obtaining the Jacobian matrix J 1 for system (12)-(13):

E 1 : Party-Free Equilibrium (PFE)
The party-free equilibrium (PFE) for the reduced system occurs at (0,0), the steady state achieved when the entire population resides in the S class (i.e., the third party has neither voters nor members and, by definition of party existence, does not exist). The PFE is essentially analogous to the disease-free equilibrium in epidemiology and always exists as a possible outcome for the voting population.
Applying the above reduced Jacobian matrix to our PFE, (0,0), where we only consider the v and m terms, we determine PFE stability: The equilibrium point (0,0) will be locally asymptotically stable (LAS) if all the eigenvalues of the matrix are negative. Assuming µ > 0, the eigenvalue −µ of the Jacobian is always negative, whereas the second eigenvalue (β − φ) − (µ + ǫ) < 0 if and only if (β−φ) µ+ǫ < 1. For ease of notation and interpretation we define the threshold quantity R 1 = (β−φ) µ+ǫ , so that the PFE is LAS if and only if R 1 < 1. We discuss the relevance of this threshold value in the last part of this section.

E 2 : Member-Free Equilibrium (MFE)
The member-free equilibrium (MFE) occurs when M = 0 but V, S = 0, i.e., the voting population subdivides between susceptibles, S, and third party voters, V . While mathematically possible, this outcome is politically unrealistic given that voters cannot vote for a party that does not exist (recall our assumption (4) that party existence depends on the presence of an M class). For mathematical consistency, however, we consider the equilibrium point (Note that the asterisk superscript denotes equilibrium values, while the numerical subscript distinguishes the equilibrium point in question.) This arises from the equilibrium condition obtained by setting dm/dt = 0 in (13): which implies that either (γv * − µ) = 0 (which we will consider later) or m * = 0.
To find v * 2 we set dv/dt = 0 and m * = 0 in (12) and rearrange terms to get This implies that either v * = 0 (the party-free equilibrium) or We consider the situation where v * = 0, solve for v * and simplify the results as follows: Finally, we can also write s * We can therefore express our member-free equilibrium as . The MFE exists if and only if R 1 > 1, since ignoring this condition leads to a negative third party voting population.
The above situation makes mathematical sense but not political sense since parties, by our original assumption, do not exist without members, and in this member-free case we deal with voters who vote for a non-existent party. We might interpret this situation as having voters still willing to vote for this party, but no party candidates for whom to vote.
Regardless of the political likelihood of MFE existence, we consider its stability. Again, we apply the method of using the reduced system's Jacobian matrix in determining the stability of the member-free equilibrium: The reduced system equilibrium point (1 − 1 R 1 , 0) is locally asymptotically stable if all the eigenvalues of the above matrix are negative. We know that, since R 1 > 1 (in order for the MFE to exist), one of the eigenvalues, We define the left hand side of the inequality as we have derived two threshold parameters R 1 and R 2 that determine equilibria stability depending on relative parameter values. (Note that they are related:

E 3 and E 4 : Survival Equilibria
In the event of survival equilibria, the voting population subdivides between susceptibles, S, third party voters, V , and members, M. We regard this as a successful state of coexistence and, given certain conditions, the point at which the party thrives. We determine the equilibrium proportions by returning to the condition (14) that dm/dt = 0. Since dm dt = (γv * − µ)m * , v * = µ γ when m * = 0. Here we impose the condition µ < γ so that v * < 1, since s * + v * + m * = 1.
Next we set dv/dt = 0 and v * = µ γ in (12): This equation, which is quadratic in m * , can be rewritten (after dividing through by −αβ) Solutions to f (m * ) = 0 are given by the quadratic formula however, in general, these solutions may be real, or may fall outside the meaningful interval Analysis of the conditions involved in determining the number of survival equilibria is considerably more involved than that for E 1 and E 2 ; details are given in the Appendix. The results, which introduce an additional threshold quantity, can be summarized as follows.
To interpret the conditions R 3a > 1 and R 3b > 1, we can rewrite them as follows: whereĥ (ǫ/γ) = 2 1 αβ Both inequalities (17) and (18) relate the minimum amounts of time taken for an individual to be influenced by a party member (M) to move from S to V , 1/αβ, and from V to M, 1/γ, to the average lifetime of an individual in the voting system, 1/µ. In order for the party to survive, a weighted sum of the first two times must be less than the average lifetime in the system, or else individuals in S will, on average, leave the system before they can "replace" the members in M who recruited them. The weights in the sum involve q = β−φ αβ , the relative effectiveness of voters V to members M in recruiting new voters from S, because the influence of party voters reduces the recruiting threshold burden on party members M to some extent. 1 (Note that 0 < q < 1.) Observe that (18) implies that If ǫ << γ, we can generate a Maclaurin (Taylor) expansion in ǫ/γ for (18) which yields This expansion suggests more quantitatively how the influence ǫ of voters for other parties S in getting third-party voters V to "defect" back to the major parties complicates the third party's survival (which would otherwise only require (19)).
Having established conditions for the existence of the two survival equilibria, E 3 and E 4 , we examine their stability. The reduced Jacobian matrix for the survival equilibria follows: The stability (LAS) criterion that the eigenvalues of this matrix have negative real part is equivalent to the conditions that the trace be negative and the determinant positive. We Since m * − < − B 2 < m * + , we see that E 4 is never stable, while the stability condition for E 3 reduces to tr J 1 ( µ γ , m * + ) < 0. Thus we calculate This is true for all m * when R 2 < 2. The case when R 2 ≥ 2 requires further algebra and is relegated to the Appendix. The result is that E 3 is always stable (LAS) when it exists, while E 4 is always unstable. Table 3 summarizes the conditions for existence and local stability of the four equilibria of the one-track model. Because system (7)- (9) can be reduced to a set of two differential equations

Global Behavior
Therefore, all solutions to the system which begin within the state space must approach an equilibrium. In particular, when only one locally stable (LAS) equilibrium exists, that equilibrium must in fact be globally stable (GAS).
The conditions in Table 3  II In region II where R 1 > 1, R 2 < 1, and R 3 < 1, E 2 , the member-free state, is the only stable equilibrium.
III In region III where both R 1 > 1 and R 2 > 1, E 3 is the only stable equilibrium; the party will inevitably approach a survival state.
IV In region IV where R 1 > 1, R 2 < 1 and R 3 > 1, E 2 and E 3 coexist as stable equilibria although, if placed in a political context, E 2 is not a realistic outcome for the party.
V In region V where R 1 < 1, R 2 < 1, and R 3 > 1, E 1 and E 3 are both stable; depending on the initial conditions the solution tends to one state or the other.
The coexistence of two locally stable equilibria (one representing party survival and the other party extinction) in regions IV and V involves a phenomenon known in epidemic modeling as a backward bifurcation, in which the exchange of stability at a threshold value (here, R 2 = 1) reverses direction, creating a situation in which the unstable equilibrium E 4 serves to split the state space into two basins of attraction for the two stable equilibria. In these regions, the equilibrium approached depends upon initial conditions: in particular, a large enough core of dedicated members M can sustain the party (toward E 3 ). What is unusual here is that the "backward" part of the bifurcation curve can extend back beyond not only the bifurcation at R 2 = 1 but also the bifurcation at R 1 = 1 (where R 2 = 0), as illustrated in Figure 4. Whenever R 1 < 1, R 2 < 0, a condition that would normally lead to the death of the party given local asymptotic stability of the PFE, there are still conditions (R 3 > 1) under which two survival equilibria exist, one of which is stable (E 3 ). In other words, the party can thrive in conditions under which it would normally die out, given that we have the necessary parameters and sufficient initial number of M individuals.
The ideal conditions for the party, of course, are when R 2 > 1 (region III), in which the party survives regardless of initial conditions. (Note R 2 > 1 implies R 1 > 1.) The last part of this section interprets all these mathematical thresholds in political terms.
3.5 Threshold Parameters R 1 , R 2 , and R 3 Our system contains three local thresholds or tipping points where population outcomes, measured as S, V , and M, depend on parameter values. By tipping point we refer to the sociological term that describes the point at which a stable phenomenon turns into a crisis, which, in a political context, corresponds to the extreme states of the party: death and growth (Gladwell, 2000). In the context of our model, for example, the party cannot sustain In our political context it is more appropriate to interpret these reproductive numbers in collective terms as regards the influence of party voters on the susceptible population. We distinguish between the aforementioned thresholds, R 1 , R 2 , and R 3 , by analyzing them qualitatively in a political context.
R 1 denotes the average number of susceptibles influenced to vote for the party by a single party voter in V , if dropped into a homogeneous population of susceptibles. The expression R 1 = β−φ µ+ǫ gives the net peer pressure, β − φ, on susceptibles S by voters V , multiplied by the average time, 1 µ+ǫ , spent in the voting class V . As mentioned above, it is most appropriate to interpret R 1 as an average number per voter taking into account the collective influence of all the party voters. Note that we are assuming that the influence of party voters on susceptibles β is stronger than the reverse influence φ of susceptibles to discourage party voters, a necessary condition for party emergence, which guarantees that R 1 > 0. ing susceptibles to vote for the party and R 1 < 1, then the value of R 2 will be negative; rather than interpreting this as a negative ability of party members to recruit voters into activism, however, it should be seen as an indication that the pool of party voters available for recruitment into party membership is not there, preventing normal party growth.
R 3 measures the extent to which party members M actively recruit susceptible individuals S into the voting class V . This activism, which sidesteps the traditional hierarchical structure of a party, is a key characteristic of growing grassroots movements, which often lack a political environment favorable to their growth in the traditional way outlined above (R 2 < 1).
Because of the two conditions required mathematically for party survival when R 2 < 1, R 3 is defined as the greater of two quantities, both of which involve party members' abilities to recruit voters, r 2 , and members, r 3 , over their political lifetimes, as well as the relative efficacy of voter peer-pressure influence to members' influence through activism on the susceptible, q. The general form r 3 (1 − 1/r 2 ) present in both components of R 3 parallels the form of R 2 = r 2 (1−1/R 1 ), with members' potential r 2 for converting third-party voters into members reducing the constraint on their ability r 3 to produce those voters in the first place. Note that r 2 must be great enough in order for R 3 even to be positive: that is, only if the members can recruit other members well can their recruitment of voters sustain the party.
We now list possible outcomes involving these thresholds and their implications: 1. When R 1 < 1 (regions I and V in Figure 3 and Table 4) the net influence of party voters V upon susceptibles is weak enough that grassroots emergence of the party is not possible. The hierarchical structure of party involvement precludes normal growth of party membership M when party voters are unable to replenish their own ranks (R 1 < 1 implies R 2 < 0). However, exceptionally, if the recruiting ability of party members M is great enough (R 3 > 1, region V), a sufficiently large dedicated core can sustain the party through its own efforts, despite the relative inefficiency or lack of influence of those merely voting for the party. This outcome illustrates the key role activists play in party survival during periods of adversity.
2. When R 1 > 1 each individual in V is converting, on average, more than one person in S into V , thus allowing the voting class V to thrive. If, in addition, R 2 < 1 (regions II and IV), we have a situation in which party activists are then unable to recruit from the voting class effectively enough to maintain the party core M (perhaps because the voting class is too small). Normally this would lead to an outcome in which a group disposed to vote for the party remains (in V ) but the party core itself dwindles away, leading effectively to party extinction (with the ideologically closest main political party perhaps adjusting a platform to capture these votes). However, as with the case when R 1 < 1, a sufficiently large initial core can ensure the party's survival when the core is effective at influencing individuals to begin voting for the party (R 3 > 1, region IV).
3. The condition R 2 > 1 explains the case where the party (V and M classes) grows normally, by recruiting members from the S and V populations, respectively. Here the party voters are influential enough in garnering new voters from S (R 1 > 1) that the party's survival does not depend on party activists' ability to recruit new voters directly (i.e., R 3 does not come into play). In epidemiological terms, this corresponds to a successful invasion: conditions are so favorable for the development of the party that it will become established even with a small initial group of members.
In general, the survival of the party is determined by the interplay among the three recruitment processes involved in the model: from S to V by members of V , as measured by R 1 ; from V to M by members of M, as measured by r 2 (R 2 incorporates both of these first two processes); and from S to V by members of M, as measured by r 3 (R 3 incorporates all three processes). Party survival requires either that the first two processes be effective enough (R 1 > 1 and R 2 > 1) for a small grassroots effort to take hold, or else that an initial membership core be large enough, and the second and third processes effective enough (R 3 > 1), that direct recruitment by party activists can sustain its membership.

Methods
As an application of the simplified one-track model analyzed in Section 3, we used Green Party registration and voting records for six states (CA, ME, MD, NY, OR, PA) and the District of Columbia (DC) to study the growth of the Green Party during the past decade or so.
We began by establishing basic demographic information for the target (study) population. A report from the Pew Research Center (2007)  The data used for this case study came from official voter registration records and election results from each state (see bibliography for sources). Voter registration records showing Green Party registration totals, given in some states as often as monthly, were fit to the size of the member class M over time. Since the dates for which Green Party registration data were given varied from state to state, the initial and final times did as well, but covered approximately the decade 1999-2008. Since voting data were given less often (in general in November of even-numbered years) these data were used, where available, as initial conditions V (0) + M(0). In other cases V (0) was estimated along with other model parameters as discussed below. Table 5 gives the time periods modeled as well as the initial conditions used. The size of each class is given as a percentage of the total target population (the size of which is also given in the table). The rescaled Green Party registration data is also shown in Figure 5. As can be seen in the graph, some states saw a noticeable change in the Green Party's trajectory following the November 2004 election, and so the data for these states (DC, CA, OR, NY) was broken into two subseries (three in the case of DC, which also separately, in order to reduce the number of free parameters.) The estimation process was iterated until the estimates were not on target interval boundaries, and this optimization was carried out 10 times for each state (with different initial guesses) to obtain the best possible fit. The resulting estimates are also given in Table 5. These estimates were then used to determine the state of the system in each state by calculating reproduction numbers and predicted end states (equilibria); results are shown in Table 6.

Results
As seen in Figure 5 Table 5 suggests that the primary reason for this change was a significant jump in the attention the target population paid to the media and the activities of the two primary political parties, as evidenced by a marked (orders of magnitude) increase in the value of the parameter ǫ (despite, in most cases, a simultaneous increase in the net peer voter influence β − φ).
The other three states in this study (ME, MD, PA), on the other hand, have continued to see healthy sustained growth in the Green Party over the past decade, although the growth in Maine appears nearly meteoric compared to the slow building up in Maryland and Pennsylvania. This difference is reflected in the much higher value of R 1 for Maine (cf. Table 6). The parameter estimates for Maine are much lower than for Maryland (cf. Table 5), likely reflecting the lower overall person-to-person contact rate (Maine has no large urban areas, whereas Maryland's population is concentrated in them) as well as perhaps lower sensitivity to others' opinions, but the ratios reflected in the threshold quantities provide fertile ground for the Green Party's growth. Our one-track model predicts (Table 6) a slightly higher equilibrium level of proportional participation in MD than in ME, but it will take much longer to reach that end state (during which time political influences may change).
Pennsylvania, meanwhile, appears to be in a more unusual situation, as the parameter estimates obtained from the data suggest the presence of a backward bifurcation as discussed in Section 3: Here the tipping-point thresholds (R 1 , R 2 ) do not appear to favor the long-term survival of the Green Party, but an initially large influx of Green voters would allow the party to thrive, even enjoying majority status among the target voting population. The reason for this phenomenon (also seen in the initial periods for DC) can be observed in the parameter estimates, which show a low general sensitivity β − φ to the passive influence of community peers, but a high sensitivity αβ to grassroots activists (relative to other PA parameters), so that a sufficiently high core of party activists could sustain the party. Despite its initial similarity to Maryland in terms of the proportional data illustrated in Figure 5, the growth of the Pennsylvania Greens is indicated by the model to be part of a transient response-it is just that, in this system, transient responses can last on the order of decades, during which political influences can change significantly.

Analysis of the Two-Track Model
Having thoroughly examined and interpreted the one-track model, we now perform analysis on the original, heterogeneous two-track model. Since the added complexity of this model precludes a complete qualitative analysis, we shall make use of numerical analysis when necessary to demonstrate behavior analogous to that exhibited by the simpler model. First, we proportionalize the two-track model in the same way we did for the one-track model to get:

The Party-Free Equilibrium (PFE) and R ′ 1
The two-track model, like the simpler version, includes a party-free equilibrium. Observing dh/dt = dl/dt = 0 to find the PFE; E 1 is (p, 1 − p, 0, 0, 0). The stability of E 1 is again tied to the first threshold quantity, which we shall denote R ′ 1 (we shall use prime superscripts to denote thresholds for the two-track model) and calculate using the next-generation operator method (e.g., Castillo-Chávez, Feng and Huang, 2002), where v H , v L and m are considered the infective classes: where respectively, are the high-affinity and low-affinity analogues of R 1 (that is, the influence of v H on h, and of v L on l), and We can further interpret the expression for R ′ 1 in political terms by observing that max(r HH , r LL ) < R ′ 1 < max(r HH , r LL ) + √ r LH r HL (again see Appendix B for details). That is, R ′ 1 is at least as great as each of the withintrack conversion efficiencies, and exceeds the maximum of the two (presumably r HH ) by less than the contribution of cross-track influences. This latter contribution is the geometric mean of two terms representing a two-stage process, in which a voter in one track converts an individual of the opposite affinity class into a third-party voter, who then influences an individual in the first track to join the voting class of the original voter, thereby completing the cycle. Because this cross-affinity cycle has two stages, the appropriate measure of its efficiency is a geometric mean of the two individual stages. In the case that there is no cross-affinity influence (σ = 0), the tracks decouple completely at this stage, and (27) reduces to The PFE is locally stable when R ′ 1 < 1, and unstable when R ′ 1 > 1. In other words, each third-party voter introduced into a population that includes high-affinity and low-affinity individuals must influence, on average, more than one person to vote for the third party during his/her voting lifetime, in order for the third-party voter classes to persist, with the average defined by R ′ 1 .

The Member-Free Equilibrium (MFE) and R ′ 2
The two-track model also has a second threshold parameter R ′ 2 . Analogous to R 2 from the one-track model, we define R ′ 2 as the average number of third-party voters (V H and/or V L ) a member can convert into M if introduced into a population of them. Since R ′ 2 is primarily concerned with the transition from third-party voting to membership, we conveniently regard M as the only infectious class, in order to apply the next-generation operator method to determine this threshold. We then calculate, from (5) (or (25)): since this is scalar we seek simply the positive part divided by the term that is subtracted (departing M): where v * H2 and v * L2 are the equilibrium values at the MFE. This expression is analogous to that for the one-track model, Solving explicitly for the MFE of the two-track model is complicated, but we can show enough to suggest that, as expected, it is unique and exists only for R ′ 1 > 1. Since we have m * = 0, any MFE must be an equilibrium of the subsystem since we can now rewrite h = p − v H and l = 1 − p − v L . The two resulting equilibrium conditions can be simplified to a single equation of degree 4, which admits up to 4 solutions.
One of these is the PFE, which can be factored out to leave a cubic equation for the MFE.
It can be shown that the constant term in this cubic equation is zero precisely when R ′ 1 = 1, so that the number of positive solutions changes by one when R ′ 1 crosses 1.
In the special case that σ = 0 (no cross-affinity influence), the system (28)-(29) decouples, are meaningful only if positive, i.e., if r HH > 1 and r LL > 1, respectively. A straightforward calculation of the Jacobian matrix shows that, within this subsystem, • if R ′ 1 < 1 (r HH < 1 and r LL < 1), the only equilibrium is E 1 , which is locally asymptotically stable (LAS); • if r HH > 1 and r LL < 1, E 1 is unstable but E 2a is LAS; • if r HH < 1 and r LL > 1, E 1 is unstable but E 2b is LAS; • if r HH > 1 and r LL > 1, E 1 , E 2a and E 2b are all unstable but E 2c is LAS.
Regardless of the value of σ, it is straightforward to show that all solutions of (28)- (29) which begin within [0, p]×[0, 1−p] remain within those bounds, by observing that dv H /dt < 0 when v H = p and 0 ≤ v L ≤ 1 − p, and that dv L /dt < 0 when v L = 1 − p and 0 ≤ v H ≤ p.
One can also exclude limit cycles from solutions of (28)-(29) under the assumptions that β H > φ H and β L > φ L , via the usual application of Dulac's Criterion: We can therefore apply the Poincaré-Bendixson Theorem to conclude that the equilibria of this system with σ = 0 identified above as locally stable, are in fact globally stable.
We can also differentiate the equilibrium conditions for (28)-(29) implicitly by σ to see what happens as σ increases from zero: for instance, .
Thus when r HH > 1 and r LL > 1, the numerator is negative and the denominator is positive, so that E 2a exits the state space as σ increases from zero. A similar calculation holds for E 2b .
Since (28)-(29) is a subsystem of (21)-(25) (in which m(t) ≡ 0), stability in the subsystem does not imply stability in the full system, but instability in the subsystem does imply instability in the full system. Thus the full system (21)-(25) has at most one stable MFE when σ = 0 (and, by continuity using the above result of implicit differentiation, for σ sufficiently small), that stability depending upon the additional dimension (m) not present in (28)-(29), as measured by R ′ 2 . While R ′ 2 is not expressed explicitly, given the implicitness of v * H2 and v * L2 , the observed uniqueness of the MFE allows us to draw conclusions about the two-track model numerically.

Survival Equilibria
Although the equilibrium conditions for (21)-(25) are too complicated to solve outright (apart from the fact that v * H + v * L = µ/γ for any survival equilibrium, from (25)), we can verify numerically not only their existence as expected when R ′ 2 > 1, but also the existence, under certain conditions, of a backward bifurcation at R ′ 2 = 1 just as observed for the one-track model. Figure 6 shows a situation analogous to that depicted in Figure 4 for the one-track model, in which survival equilibria may exist even below the PFE/MFE threshold (R 2 = 0 or R ′ 2 = 0); in Figure 7 the critical (minimum) value of R ′ 2 is between 0 and 1. Both figures, however, also demonstrate the existence of multiple locally stable survival equilibria near R ′ 2 = 1 (see the close-up in Figure 6), meaning that for some parameter values there are two different levels at which the party may stabilize, in addition to the stable (for R ′ 2 < 1) extinction equilibria PFE/MFE. In these situations, the initial number of party members plays a huge role in determining whether the party surges to major growth, languishes, or dies out entirely.
Although behavior described by the model for the two affinity classes. In addition, similar curves can be obtained using a wide range of values for model parameters. Unfortunately, the complexity of the two-track model prevents an explicit calculation of a quantity analogous to R 3 which measures the ability of the member class to recruit "susceptible" voters from both affinity classes to become third-party voters. The same interpretation applies, however: it is the work of party members in recruiting voters that enables a party to persist when third-party voters' influence is too weak.

Conclusion
The models described in this study investigate in mathematical terms the consequences of our assumptions about the factors driving the growth and persistence of a third political party arising through grassroots efforts. In particular, we assume a hierarchical structure in which party members (activists) play a different, more extensive role in party survival than party voters. We also assume that the dominant influences are primary contacts among third-party voters and members and the general public, manifested in our models as nonlinear terms involving the sizes of the two groups making contact. These nonlinearities govern the behavior of the models-that is, the fate of the party under study-through threshold quantities that describe the system's tipping points. Our results should be taken as implications of the assumptions that such primary contacts, and not other factors (apart from affinity as defined for our two-track model), drive individuals' decisions whether to support third political parties at any level.
The primary result of our analysis is that our models identify, and provide a way to measure, the three factors that determine the party's survival. Each of these factors is a reproductive number that describes the ability of a given class in the party structure to recruit others. R 1 and R ′ 1 give the average number of unaffiliated voters recruited per third-party voter, for the simplified and general model, respectively. This first threshold quantity measures the voting class's ability to replace or sustain itself. R 2 and R ′ 2 , meanwhile, describes the efficiency with which third-party members convince third-party voters to become members. This average number of new members recruited per existing member presupposes the success of the first stage of recruitment: existing third-party voters recruiting new ones; that is, R 1 > 1, or else R 2 < 0, which is meaningless except to indicate the failure of a recruitment structure in which party members play no role in recruiting new voters. Finally, R 3 quantifies the ability of party members to recruit new voters directly from the unaffiliated (with this party) public. The growth of the Green Party in states like Maine and Maryland, and its recent decline in states like California and Oregon, illustrate the effect of these tipping-point thresholds (cf. Gladwell, 2000).
The model's prediction of survival states in scenarios where the primary, hierarchical recruitment structure is not strong enough (R 2 < 1, and even R 1 < 1) to allow a party to arise-that is, political trends are not favorable and the resulting influence of peer pressure to vote for and support a given party is weak-explains the persistence of established third parties during periods of adverse conditions, when political winds blow against them. In particular, when party activists (members) are sufficiently capable of finding and recruiting new voters directly (as measured by R 3 > 1), a large enough core of committed party members can ensure the party's survival. In cases where the political environment was favorable for a time (R 2 > 1), this minimum core size is typically reached quickly. This reaching across traditional hierarchical structures (rather than party members interacting primarily with those who already vote for the party) provides a robustness to the phenomenon that manifests mathematically in the backward bifurcations illustrated in earlier sections.
Backward bifurcations also underscore the importance of initial conditions (having enough initial party members) in enabling that robustness. This scenario is typified by the case study of the Green Party in Pennsylvania.
Our general model classifies the general voting population by affinity to the ideas and goals of a given third party. The form of the expression for R ′ 1 illustrates how a party's ability to take hold in even a small subset of the population (the high-affinity track) affects the party's survival in the population as a whole: since R ′ 1 is greater than either of the withintrack voting replacement numbers (analogous to R 1 ), a successful enough recruitment within the high-affinity track can maintain the party. Furthermore, the stratification into two tracks creates the potential, as illustrated in Figures 6 and 7, for multiple stable survival states, which facilitate party growth since the additional states require fewer initial members than the original one. Our models could easily be extended to a stratification with intermediate levels of affinity, each of which would make its own differential (R 1 -like) contribution to determining the overall recruitment potential of a given third party (analogous to R ′ 1 ), in addition to increasing the diversity of possible survival states. We begin with the survival equilibrium condition f (m * ) = m * 2 + BM * + C = 0, where (from (15)) We rewrite these expressions by defining the following terms: By assumption, β > φ, γ > µ, and α > 1, so that 0 < q < 1, 0 < r < 1, and y > 1. In these terms, R 1 = qx, R 2 = y 1 − 1 qx , and In order to be meaningful, solutions must fall within the interval (0, 1 − v * ) (since m * > 1 − v * will make s * = 1 − v * − m * < 0). We calculate These two inequalities imply that any roots of f lie to the left of 1 − v * . Next note that f (0) = C = − q y 2 (R 2 − 1). Then when R 2 > 1, f (0) < 0, so there is exactly one solution m * + in (0, 1 − v * ). When R 2 < 1, f (0) > 0, so the number of solutions in (0, 1 − v * ) is even. In this case, both solutions m * ± are in (0, 1 − v * ) precisely when B 2 − 4C ≥ 0 (the solutions are real) and B < 0 (the parabola's vertex m = −B/2 lies to the right of 0).
The condition B < 0 can be shown equivalent to x > r and y >ŷ B ≡ (1 + q) x x−r . The condition C > 0 (i.e., R 2 < 1) can be shown equivalent to qx < 1 or y <ŷ C ≡ qx qx−1 . The and multiplying by x 2 y 2 we get Solving the quadratic inequality in y, this means y must not be between the two positive rootŝ Thus in order to have two survival equilibria, we must have x > r,ŷ B < y <ŷ C , and either y ≤ŷ − or y ≥ŷ + .
In order to simplify these criteria, we compare the threshold values for y. We find that or simply |qx − 1| < (1 − qr) [(1 − q)(x − r) + (1 − qr)]. Squaring both sides and simplifying leads to the compound inequality The lower bound r corresponds to the vertical asymptote in x shared byŷ ± andŷ B . Since q < 1, 1 q < 1 q 2 <x + , so at the upper bound qx − 1 > 0, and it is the second inequality in (30) that is violated. That is, when x =x + ,ŷ B =ŷ + . Thus it is always true thatŷ − <ŷ B , so the criterion y >ŷ B allows us to discard the condition y ≤ŷ − in favor of y ≥ŷ + .
If we solve the inequalityŷ B <ŷ C for x (when qx > 1), we find again the condition x <x + , indicating that the graphs ofŷ B ,ŷ C , andŷ + all cross at x =x + . Further similar computation can show thatŷ + <ŷ C except at x =x + , where they are tangent. We thus require r < x <x + andŷ + ≤ y <ŷ C (the latter inequality only for qx > 1). A graph illustrating all four curves is given in Figure 8.
To put the survival equilibrium conditions back in terms of the original parameters, we see that r < x <x + becomes each of the inequalities can be solved for αβ/µ to make Note that the last expression in the inequality above is the product of R 1 and another fraction whose value exceeds 1 precisely when R 1 does. Thus if R 1 < 1, the condition is It is not so simple to rewrite y >ŷ + ; however, we can return to the original conditions B < 0, B 2 − 4C ≥ 0, which can be rewritten more simply, either as R 3a > 1, R 3b > 1 as in the statement of Proposition 1, or as: where h(ǫ/γ) = 2 1 αβ In order to rewrite the condition B 2 − 4C ≥ 0 in the form (32), we substitute B and C from (15) and get: Expansion, summing like terms, and completing the square yields Next we take the square root of both sides, Since we require µ < γ and β > φ, then so we can drop the absolute value bars. Finally, rearranging, we get In the case that ǫ = 0, this inequality can be further simplified by factoring the right-hand side as a perfect square, taking the square root of both sides, and using the fact that (31) implies 1/αβ < 1/µ, to get We can apply the same technique to obtain (32).

A.2 Stability Analysis for E 3
From the end of Section 3.3, it remains to show that tr J 1 (E 3 ) < 0 when R 2 ≥ 2. This condition on the trace is equivalent to .
In terms of q, r, x and y as defined in the previous section,

Now the stability condition is
Here we multiply by − 1 + q + yr x 2 /q, expand, and simplify to get which is true since This completes the verification that E 3 is LAS when it exists.

B R ′ 1 for the Two-Track Model
We calculate the reproductive number R ′ 1 of the two-track model using the next-generation operator method where R ′ 1 is analogous to R 1 of the one-track model. The party-free equilibrium E 1 of (21)-(25) is (p, (1 − p), 0, 0, 0). Differentiating (23)-(25) with respect to the "infective" variables v H , v L , and m, and substituting the PFE values yields the following "mini-Jacobian" matrix: where w H = µ + ǫ H + φ H σ(1 − p) and w L = µ + ǫ L + φ L σp. We next rewrite A =M −D, where the entries ofM are nonnegative andD is a diagonal matrix: The three eigenvalues are 0 and 1 2 r HH + r LL ± (r HH − r LL ) 2 + 4r HL r LH ; R ′ 1 takes the positive square root in the latter expression. By inspection we can see that for the extreme cases p = 0 and p = 1, R ′ 1 simplifies to R 1 for the one-track model.
Since the second term inside the radical is positive, we have that R ′ 1 > 1 2 r HH + r LL + (r HH − r LL ) 2 = 1 2 [r HH + r LL + |r HH − r LL |] = max(r HH , r LL ).
Since, for positive numbers a and b, √ a + b < √ a + √ b, we also have that R ′ 1 < 1 2 r HH + r LL + (r HH − r LL ) 2 + 2 √ r HL r LH = max(r HH , r LL ) + √ r HL r LH .   φ recruitment rate of V into S by direct contact with susceptibles α factor by which the recruitment rate of S into V by third party members exceeds the recruitment rate by individuals in V γ recruitment rate of V into M by third party members µ rate at which individuals enter or leave the voting system Party-Free E 1 = (1, 0, 0) always exists Survival E 3 = (1 − µ γ − m * + , µ γ , m * + ) (i) R 2 > 1, or always stable (ii) R 2 < 1, R 3 > 1 when it exists Survival E 4 = (1 − µ γ − m * − , µ γ , m * − ) R 2 < 1, R 3 > 1 always unstable   Figure 5. Initial conditions v(0) marked with asterisks *, as well as all model parameters (except µ), were estimated by data fitting as described in the main text.
All rates are given in units of 1/days; values below 0.01 are given in scientific notation.