Equilibria

We first analyse the equilibrium states. Approach to equilibrium is one possible outcome of the long-term behaviour of a dynamical model with fixed parameter values, and we do not observe any periodic or chaotic behaviour in our model. We find the possible equilibrium states for our model by setting the nine equations in eqn (2) to 0. This procedure yields four solutions that are analogous to considering pairs of equilibrium solutions of the standard SIR model (eqn 1). A standard SIR model has a solution at which the contagion is extinct ($\hat{I} = 0$) and a solution at which it is endemic ($\hat{I} \gt 0$). The four equilibria in our model thus correspond to: disease and sentiment both extinct ($\hat{I} = 0$, $\hat{A} = 0$), disease extinct and sentiment endemic ($\hat{I} = 0$, $\hat{A} \gt 0$), disease endemic and sentiment extinct ($\hat{I} \gt 0$, $\hat{A} = 0$) and disease and sentiment both endemic ($\hat{I} \gt 0$, $\hat{A} \gt 0$).

For almost all possible parameter values, only one equilibrium is stable, meaning that a small deviation from the equilibrium returns to the equilibrium state. The only situations in which multiple equilibria are stable occur if all stable equilibria yield the same vector of nine compartment frequencies. We determined the local linear stability of the four equilibria by studying the eigenvalues of the Jacobian matrix of eqn (2), a standard technique for studying stability in dynamical systems (Beltrami, Reference Beltrami1998). See Appendices B–E for details and Table 3 for stability conditions.

Table 3. Conditions for local stability of the four equilibria. An equilibrium is stable for a given set of parameter values (b, r, g, v, s, c, w) if and only if the conditions for R ₀ and C ₀ are both met. The columns denoted $\hat{I}$ and $\hat{A}$ indicate presence (+) or absence (−) of disease and sentiment, respectively, at equilibrium

DSFE

The first equilibrium, the disease- and sentiment-free equilibrium (DSFE), has both disease and anti-vaccine sentiment extinct ($\hat{I} = 0$, $\hat{A} = 0$). Variables with nonzero equilibrium values are

(6)$$\eqalign{ {\widehat{{SU}}} & = { \displaystyle{b \over {w + b}}} & {} \cr {\widehat{{SP}}} & = { \left({\displaystyle{b \over {v + b}}} \right)\left({\displaystyle{w \over {w + b}}} \right)} & {} \cr {\widehat{{RP}}} & = { \left({\displaystyle{v \over {v + b}}} \right)\left({\displaystyle{w \over {w + b}}} \right).} & {} \cr } $$

Only susceptible and resistant individuals for both traits are present at the DSFE. The vaccination rate v and pro-vaccine decision rate w, which together characterize the rate at which disease-susceptible, undecided SU individuals move to the pro-vaccine, vaccinated RP state, occur together frequently in our equilibrium results and are a reflection of the overall vaccination behaviour in the population. For the DSFE, the equilibrium susceptible population is

(7)$${\hat{S}} = 1-\displaystyle{{vw} \over {\lpar {v + b} \rpar \lpar {w + b} \rpar }}.$$

The value of $\hat{S}$ (eqn 7) is driven primarily by the strength of vaccination behaviour in the population. If v and w are large, then most individuals will be vaccinated and $\hat{S}$ is close to 0.

The DSFE is stable if sentiment transmission does not outpace pro-vaccine decision-making and disease transmission does not outpace vaccination (Table 3).

DFE

The second equilibrium, the disease-free equilibrium (DFE), has endemic sentiment but extinct disease ($\hat{I} = 0$, $\hat{A} \gt 0$). Variables with nonzero equilibrium values are

(8)$$\eqalign{ {\widehat{{SU}}} & = { \displaystyle{{s + b} \over c} = \displaystyle{1 \over {C_0}}} & {} \cr {\widehat{{SA}}} & = { \displaystyle{b \over {s + b}}-\displaystyle{{w + b} \over c} = \displaystyle{b \over c}\lpar {C_0-1} \rpar -\displaystyle{w \over c}} & {} \cr {\widehat{{SP}}} & = { \left({\displaystyle{b \over {v + b}}} \right)\left({\displaystyle{s \over {s + b}} + \displaystyle{{w-s} \over c}} \right) = \left({\displaystyle{b \over {v + b}}} \right)\left[{\displaystyle{s \over c}\lpar {C_0-1} \rpar + \displaystyle{w \over c}} \right]} & {} \cr {\widehat{{RP}}} & = { \left({\displaystyle{v \over {v + b}}} \right)\left({\displaystyle{s \over {s + b}} + \displaystyle{{w-s} \over c}} \right) = \left({\displaystyle{v \over {v + b}}} \right)\left[{\displaystyle{s \over c}\lpar {C_0-1} \rpar + \displaystyle{w \over c}} \right].} & {} \cr } $$

The equilibrium susceptible population for the DFE is

(9)$$\hat{S} = 1-\left({\displaystyle{v \over {v + b}}} \right)\left[{\displaystyle{s \over c}\lpar {C_0-1} \rpar + \displaystyle{w \over c}} \right].$$

The two terms subtracted from 1 in eqn (9) reflect the two possible routes towards vaccination when sentiment is present (SU → SA → SP → RP and SU → SP → RP). The conflicting presence of the sentiment transmission rate c in the numerator and denominator of the term (s/c)(C ₀ − 1) (recalling eqn 5 for C ₀) reflects the fact that sentiment transmission and sentiment switching undermine each other but that sentiment switching only matters if sentiment is transmitted. In general, if pro-vaccine decisions are fast, sentiment transmission is slow or sentiment switching is fast, then most individuals will be vaccinated. If pro-vaccine decisions are slow, sentiment transmission is fast or sentiment switching is slow, then most individuals will be susceptible.

The DFE is stable if sentiment transmission outpaces pro-vaccine decision-making but disease transmission does not outpace vaccination (Table 3).

SFE

The third equilibrium, the sentiment-free equilibrium (SFE), has endemic disease but extinct sentiment ($\hat{I} \gt 0$, $\hat{A} = 0$). For this equilibrium, it is easiest to state the values of $\widehat{{SP}}$ and $\hat{I}$ and then write other nonzero values in terms of these two quantities. We have

(10)$$\eqalign{ {\widehat{{SP}}} \hfill & = { \displaystyle{{bR_0 + v + w-x} \over {2R_0v}}} \hfill \cr {\hat{I}} \hfill & = { \displaystyle{1 \over {2r}}\lsqb {b\lpar R_0-1\rpar -\lpar b + v + w\rpar + x} \rsqb } \hfill \cr {\widehat{{SU}}} \hfill & = { \displaystyle{{2b} \over {bR_0-v + w + x}}} \hfill \cr {\widehat{{IU}}} \hfill & = { \displaystyle{{b\lsqb {b\lpar R_0-1\rpar -\lpar b + v + w\rpar + x} \rsqb } \over {\lpar g + w + b\rpar \lpar bR_0-v + w + x\rpar }}} \hfill \cr {\widehat{{IP}}} \hfill & = { \displaystyle{{r\lpar \hat{I}\rpar \lpar \widehat{{SP}}\rpar + w\lpar \widehat{{IU}}\rpar } \over {g + b}}} \hfill \cr {\widehat{{RU}}} \hfill & = { \displaystyle{{bg\lsqb {b\lpar R_0-1\rpar -\lpar b + v + w\rpar + x} \rsqb } \over {\lpar w + b\rpar \lpar g + w + b\rpar \lpar bR_0-v + w + x\rpar }}} \hfill \cr {\widehat{{RP}}} \hfill & = { \displaystyle{{w\lpar \widehat{{RU}}\rpar + v\lpar \widehat{{SP}}\rpar + g\lpar \widehat{{IP}}\rpar } \over b}\comma \;} \hfill \cr } $$

where

$$x = \sqrt {{\lpar {bR_0 + v + w} \rpar }^2-4vw}.$$

The SFE is stable if sentiment transmission does not outpace pro-vaccine decision-making but disease transmission outpaces vaccination (Table 3).

EE

The fourth equilibrium, the endemic equilibrium (EE), has both endemic disease and endemic anti-vaccine sentiment ($\hat{I} \gt 0$, $\hat{A} \gt 0$). The analytical form of this equilibrium is obtained by considering the real positive root of a quartic equation in $r\hat{I}$ (Appendix A). From $r\hat{I}$, we divide by r to obtain $\hat{I}$, and from $\hat{I}$, we obtain $\widehat{{SP}}$:

(11)$$\widehat{{SP}} = \displaystyle{b \over v}\left({1-\displaystyle{{g + b} \over r}} \right)-\left({\displaystyle{{g + b} \over v}} \right)\hat{I}.$$

From $\hat{I}$ and $\widehat{{SP}}$, we obtain all other quantities:

(12)$$\eqalign{ {\widehat{{SU}}} \hfill & = { \displaystyle{b \over {r\hat{I} + \displaystyle{{bc} \over {s + b}}}}} \hfill \cr {\widehat{{SA}}} \hfill & = { \displaystyle{{\left( \displaystyle{{bc} \over {s + b}}-w-b\right) \lpar \widehat{{SU}}\rpar } \over {r\hat{I} + s + b}}} \hfill \cr {\widehat{{IU}}} \hfill & = { \displaystyle{{r\hat{I}\lpar \widehat{{SU}}\rpar } \over {\displaystyle{{bc} \over {s + b}} + g}}} \hfill \cr {\widehat{{IA}}} \hfill & = { \displaystyle{{r\hat{I}\lpar \widehat{{SA}}\rpar + \left( \displaystyle{{bc} \over {s + b}}-w-b\right) \lpar \widehat{{IU}}\rpar } \over {g + s + b}}} \hfill \cr {\widehat{{IP}}} \hfill & = { \displaystyle{{r\hat{I}\lpar \widehat{{SP}}\rpar + w\lpar \widehat{{IU}}\rpar } \over {g + b}}} \hfill \cr {\widehat{{RU}}} \hfill & = { \displaystyle{{g\lpar s + b\rpar } \over {bc}}\lpar \widehat{{IU}}\rpar } \hfill \cr {\widehat{{RA}}} \hfill & = { \displaystyle{{\left( \displaystyle{{bc} \over {s + b}}-w-b\right) \lpar \widehat{{RU}}\rpar + g\lpar \widehat{{IA}}\rpar } \over {s + b}}} \hfill \cr {\widehat{{RP}}} \hfill & = { \displaystyle{{w\lpar \widehat{{RU}}\rpar + v\lpar \widehat{{SP}}\rpar + g\lpar \widehat{{IP}}\rpar + s\lpar \widehat{{RA}}\rpar } \over b}.} \hfill \cr } $$

The EE is stable if sentiment transmission outpaces pro-vaccine decision-making and disease transmission outpaces vaccination (Table 3). This equilibrium potentially allows diseases with low transmission rates to be maintained in the population owing to the existence of anti-vaccine sentiment.

R ₀and disease endemism

In the standard SIR model without vaccination (eqn 1, v = 0), the basic reproduction number R ₀ (eqn 4) is the expected number of secondary cases of a disease produced by a typical infected individual in a population of susceptible individuals. This quantity determines the ultimate fate of the disease: if R ₀ < 1, then the disease goes extinct, and if R ₀ > 1, then the disease is endemic at equilibrium. With vaccination included in an SIR model with no anti-vaccine sentiment (eqn 1, v > 0), the criterion for disease endemism is

(13)$$R_0 \gt 1 + \displaystyle{v \over b}.$$

With vaccination present, R ₀ can exceed 1, the threshold for disease maintenance in an unvaccinated population, but can fail to exceed 1 + v/b, so that the disease still goes extinct. By enlarging this threshold for R ₀, vaccination makes it more difficult for the disease to persist.

Our model has two conditions for disease endemism, depending on whether sentiment is extinct or endemic. If sentiment is extinct at equilibrium, then the condition for disease endemism is

(14)$$R_0 \gt \displaystyle{1 \over {1-\left({\displaystyle{v \over {v + b}}} \right)\left({\displaystyle{w \over {w + b}}} \right)}} = \left({1 + \displaystyle{v \over b}} \right)\left({\displaystyle{{w + b} \over {w + b + v}}} \right).$$

Here, we have rewritten the result from Table 3 to demonstrate that the critical value of R ₀, the right-hand side of eqn (14), is always less than or equal to the critical value of R ₀ for a system with vaccination but no sentiment dynamics: the right-hand side of eqn (13). No sentiment is present at equilibrium in this case, so the difference between eqns (14) and (13) is due to the presence of undecided, unvaccinated individuals. Their presence hinders the effects of vaccination, as it decreases the value of R ₀ that the disease must achieve to be endemic. However, from eqn (14), the critical value of R ₀ in this case is still always greater than or equal to 1. Thus, the presence of undecided, unvaccinated individuals resulting from sentiment dynamics maintains an R ₀ critical value that lies between that of the model without vaccination and that of the model with vaccination but no sentiment.

If sentiment is endemic at equilibrium, then the condition for disease endemism is

(15)$$R_0 \gt \displaystyle{1 \over {1-\left({\displaystyle{v \over {v + b}}} \right)\left({\displaystyle{s \over {s + b}} + \displaystyle{{w-s} \over c}} \right)}}.$$

The critical value of R ₀, the right-hand side of eqn (15), is less than or equal to the critical value of R ₀ in a system with vaccination but no sentiment – the right-hand side of eqn (13) – if s/(s + b) + (w − s)/c ≤ 1. This condition is equivalent to bc ≥ (s + b)(w − s), an inequality that is satisfied for the case with sentiment endemic at equilibrium: rearranging C ₀ > 1 + w/b yields bc > (s + b)(w + b) > (s + b)w > (s + b)(w − s). Thus, as in the case with extinct sentiment (eqn 14), endemic anti-vaccine sentiment hinders the effects of vaccination by decreasing the critical value of R ₀ that the disease must achieve to be endemic. Because all parameters in our model are nonnegative, and because s/(s + b) + (w − s)/c = (s/c)(C ₀ − 1) + w/c and C ₀ > 1, the denominator of the right-hand side of eqn (15) is always less than or equal to 1, so the critical value of R ₀ in this case is always greater than or equal to 1. Thus, the effects of endemic sentiment also generate an R ₀ critical value between that of the model without vaccination and that of the model with vaccination but no sentiment: the condition for disease spread in a scenario with vaccination and sentiment is stricter than in a model with no vaccination, but not as strict as with vaccination and no sentiment.

The right-hand side of eqn (15) is greater than or equal to that of eqn (14) if and only if bC ₀ ≥ w + b. This condition is already satisfied with endemic sentiment (Table 3), so it is possible for a disease to be endemic if sentiment is endemic when it would not be endemic if sentiment were extinct.

Effects of parameters on stability of equilibria

We now study the effects of individual parameters on the stability of the equilibria. The compound parameters R ₀ and C ₀ in Table 3 suffice to determine whether or not an equilibrium is stable. To understand the effects of specific parameters, we visualize a subset of the parameter space. In particular, we focus on varying the sentiment parameters c, s and w, keeping the birth rate b and the disease parameters r, g and v fixed at values that represent well-studied diseases. As in standard SIR models in which r, g, and v are rate parameters that range from 0 to infinity (Hethcote, Reference Hethcote2000), the sentiment transmission rate c, the sentiment switching rate s, and the pro-vaccine decision rate w can range from 0 to infinity. We can therefore plot regions of the first quadrant of a plane considering two of these three variables to depict stability regions for the four equilibria. Figure 2 focuses on the c–w plane. The boundary curves in Figure 2 are derived in Appendix F.

Figure 2. Stability regimes for varying sentiment transmission rate c and pro-vaccine decision rate w. (a) R ₀ ∈ [0, 1]. (b) $R_0\in \left({1\comma \;1{\rm /}\left[{1-{{sv} \over {\lpar {s + b} \rpar \lpar {v + b} \rpar }}} \right]} \right)$. (c) $R_0\in \left[{1{\rm /}\left[{1-{{sv} \over {\lpar {s + b} \rpar \lpar {v + b} \rpar }}} \right]\comma \;1 + {v \over b}} \right)$. (d) $R_0\in \left[{1 + {v \over b}\comma \;\infty } \right)$. Parameter values for (a–d) are (b, g, v, s) = (0.002, 0.14, 0.14, 0.01). These parameters lead to transitional R ₀ values of $1{\rm /}\left[{1-{{sv} \over {\lpar {s + b} \rpar \lpar {v + b} \rpar }}} \right] = 5.61$ and 1 + v/b = 72.42. For (a), r = 0.1 so that R ₀ = 0.69. For (b), r = 0.7 so that R ₀ = 4.83. For (c), r = 1.5 so that R ₀ = 10.36. For (d), r = 11 so that R ₀ = 75.94. These parameter values were chosen so that the stability regions of all possible equilibria would be visible in each panel.

Note that in our model formulation, one might expect parameters w, the pro-vaccine decision rate, and c, the sentiment transmission rate, to be inherently inversely related, as a cultural environment with a high pro-vaccine decision rate could be incompatible with high transmissibility of anti-vaccine sentiment. Concurrently high values of w and c could arise with an underlying polarization in the population on the topic of vaccination, perhaps owing to political fracturing (e.g. Jegede, Reference Jegede2007; Dubé et al., Reference Dubé, Vivion and MacDonald2015) or the presence of a non-vaccinating religious community (e.g. Woudenberg et al., Reference Woudenberg, van Binnendijk, Sanders, Wallinga, de Melker, Ruijs and Hahné2017). Concurrently low values could arise in a situation of general indifference towards vaccines (e.g. Velan, Kaplan, Ziv, Boyko, & Lerner-Geva, Reference Velan, Kaplan, Ziv, Boyko and Lerner-Geva2011; MacDougall & Monnais, Reference MacDougall and Monnais2018). Thus, the two parameters are not necessarily inversely related and it is useful to consider the full c–w plane.

In Appendix F, it is shown that all parameter sets give rise to one of four regimes for the stability region in the c–w plane, with the regime determined by the value of R ₀. The four cases correspond to four ranges for the value of R ₀. The three R ₀ values at which the regime transitions from one type to another are discussed in Appendix F.

First, if R ₀ < 1, then the disease always goes extinct, and only the DSFE and DFE are possible (Figure 2a). This result is the same as in a standard SIR model, except that increasing the sentiment transmission rate c or decreasing the pro-vaccine decision rate w results in endemic sentiment.

Next, if

$$1 \lt R_0 \lt 1{\rm /}\left[{1-\displaystyle{{sv} \over {\lpar {s + b} \rpar \lpar {v + b} \rpar }}} \right],$$

then the disease can spread in a completely susceptible population, and it is possible for the SFE and the EE to be stable in addition to the other two equilibria (Figure 2b). This case is most relevant for large values of the sentiment switching rate s, as the range of R ₀ values that produce it is narrow for small s. As the sentiment transmission rate c increases, for large s, increased sentiment transmission increases vaccination in the population: many people transition from undecided to anti-vaccine and then, because of the high sentiment switching rate s, quickly become pro-vaccine. This situation, in which increased sentiment transmission can move the disease to extinction, represents a disease whose transmissibility cannot overcome the vaccination induced by sentiment switching after transmission of anti-vaccine sentiment.

Different behaviour occurs (Figure 2c) if

$$1{\rm /}\left[{1-\displaystyle{{sv} \over {\lpar {s + b} \rpar \lpar {v + b} \rpar }}} \right]\le R_0 \lt 1 + v{\rm /}b.$$

In this range, the disease is transmissible enough to overcome the increased vaccination induced by sentiment switching. Increasing the sentiment transmission rate c now makes the disease endemic.

Finally, if 1 + v/b ≤ R ₀, then the disease is transmissible enough to overcome vaccination without assistance from anti-vaccine sentiment. In this situation, the DSFE and DFE are no longer stable anywhere, and decreasing vaccination by decreasing the pro-vaccine decision rate w or increasing the sentiment transmission rate c first leads to stability for the SFE and then for the EE.

Disease examples

To get a sense for which equilibria are stable in practical settings, we reproduce Figure 2 using disease parameter values that are commonly used to model polio (Figure 3a) and measles (Figure 3b). The parameter values appear in Table 4.

Figure 3. Stability regimes with parameter values chosen to represent two diseases. (a) Polio. (b) Measles. Parameter values are taken from Table 4. These values correspond to Figure 2c, where $R_0\in \left[{1{\rm /}\left[{1-{{sv} \over {\lpar {s + b} \rpar \lpar {v + b} \rpar }}} \right]\comma \;1 + {v \over b}} \right)$. The sentiment-free equilibrium (SFE) is stable and lies in a region of the bottom-left corner of each plot, a region which is too small to see here.

Table 4. Parameters for polio and measles. The disease transmission rate r is calculated from the basic reproduction number R ₀, disease recovery rate g and birth rate b using eqn (4). Polio R ₀, measles R ₀ and measles g are taken from Anderson and May (Reference Anderson and May1991). Polio g is from Tebbens et al. (Reference Tebbens, Pallansch, Kew, Cáceres, Sutter and Thompson2005). The value of b is from Hamilton et al. (Reference Hamilton, Martin, Osterman, Curtin and Matthews2015). The value s = 0 is chosen for simplicity, and we expect s to be low in real populations. The value of v was chosen so that it takes about 7 days on average for a newly pro-vaccine individual to become vaccinated. The parameters r, g, v, and s are reported in units of days⁻¹; b uses units of per capita births per day. R ₀ is dimensionless. The values in the table were used to produce Figure 3

Parameter values for polio reside in case 3 (Figure 2c). Most of the parameter space in Figure 3a consists of regions where either the DSFE or the EE is stable, indicating a tight coupling between disease and sentiment endemism. In a non-trivial region where the DFE is stable, sentiment is endemic, but does not reduce vaccination sufficiently to permit disease to persist. A small region of Figure 3a where the SFE is stable is not visible but exists near the origin.

Parameter values for measles also reside in case 3 (Figure 2c). For measles, with a higher R ₀ than polio, the regions in which either the DSFE or the EE is stable now occupy almost the entire parameter space in Figure 3b, indicating extremely tight coupling between disease and sentiment endemism. The high R ₀ for measles couples the persistence of the disease to the persistence of sentiment, as disease can only persist in the population owing to sentiment endemism that reduces vaccination sufficiently for the disease to spread. This effect reflects a general result we derive in Appendix F: the greater the value of R ₀, the larger the range of parameter values over which the DSFE or the EE is stable and the smaller the range over which the DFE or the SFE is stable.

Transient oscillations

Thus far, we have been examining equilibria of the model. We now turn our attention to the transient dynamics, solving eqn (2) numerically by specifying initial conditions and using a numerical differential-equation solver (see Appendix G). We begin with a population consisting primarily of susceptible individuals, so that an epidemic occurs early in the dynamics.

For parameter values that ultimately reach the endemic equilibrium, the transient dynamics are often oscillatory. We have observed two qualitatively different oscillation patterns in the transient dynamics of eqn (2). Figure 4 displays the patterns in three examples with initial compartment frequencies SU = 0.998 and IU = SA = 0.001.

Figure 4. Transient dynamics of the disease frequency for three different parameter sets. (a) Damped oscillation towards the endemic equilibrium (EE). The parameter values are (b, r, g, v, c, s, w) = (0.02, 0.8, 0.25, 0.2, 0.8, 0, 0.1). (b) Repeated epidemic spikes separated by long pauses. Parameter values are the same as in (a) except b = 0.0002. (c) Two epidemic regimes separated by a long pause during which the disease appears to go extinct. Parameter values are the same as in (a) except w = 0.25.

The first pattern (Figure 4a) is a damped oscillation around a stable EE. Comparably to a pattern sometimes seen in a standard SIR model (Wang et al., Reference Wang, Bauch, Bhattacharyya, d'Onofrio, Manfredi, Perc and Zhao2016), the amplitude of oscillation in the frequency of infected individuals becomes negligibly small after only a few cycles. In this example, the oscillatory pattern leads to two noticeable epidemics in a year, with the second less severe, before the disease reaches a relatively high endemic frequency (about 4%). The second pattern (Figure 4b) consists of large epidemics followed by periods in which the disease is nearly extinct. This example differs in that the birth rate b is smaller by a factor of 100, slowing the influx of undecided, susceptible individuals to such an extent that the disease frequency can only recover after long periods, when enough susceptible individuals have entered the population. This pattern is also observed in the standard SIR model when the endemic disease frequency is small and the system therefore has parameter values that give rise to the EE but are close to the region that produces the DFE (Wang et al., Reference Wang, Bauch, Bhattacharyya, d'Onofrio, Manfredi, Perc and Zhao2016). In this example, a series of epidemics occurs over a period of years, interrupting periods during which the disease is nearly extinct.

Figure 4c displays a situation that possesses both the damped oscillation around the EE from Figure 4a as well as the long pause between epidemics from Figure 4b. When extended over a longer period, the dynamics in Figure 4b in fact behave similarly to those in Figure 4c, with a damped oscillation to the EE, except that Figure 4b has multiple initial epidemic spikes rather than a single one, and the equilibrium disease frequency is small. Three epidemics occur over the course of a year, with the second more severe than the first and the third less dramatic than the second. During the period between the first two epidemics, the disease appears to be extinct. After the third epidemic, the disease settles into a relatively high endemic frequency (about 3%). In all cases in Figure 4, the disease goes extinct if the initial condition has no anti-vaccine sentiment.

Figure 5 further explores the dynamics of the oscillation seen in Figure 4c, showing the same dynamics but with more compartments. Figure 5a displays the disease-infected compartments IU, IA, IP and their total, I. The first epidemic at ~10 days consists of an increase in pro-vaccine IP individuals, suggesting that this epidemic is driven by pro-vaccine individuals getting infected with the disease before they have a chance to become vaccinated. This situation occurs because the initial population contains almost all disease-susceptible individuals.

Figure 5. Dynamics of compartments during long-pause oscillation behaviour. (a) Frequencies of disease-infected individuals of all sentiment classes and the total infected frequency. (b) Frequencies of disease-susceptible individuals of all sentiment classes and the total susceptible frequency. The dashed horizontal line is the 1/R ₀ threshold frequency of disease-susceptible individuals necessary for the disease to spread. The parameter values follow Figure 4c.

In contrast, the second epidemic consists mostly of IA individuals, suggesting that this epidemic is driven by anti-vaccine individuals. The sentiment trajectories in Figure 5b support this claim, as an increase in the disease-susceptible, anti-vaccine compartment SA pushes the disease-susceptible frequency S above 1/R ₀, the threshold necessary for the disease to spread.

Migration-triggered epidemics

In long pauses between oscillations such as those seen in Figure 4b and c, the disease drops to low frequencies. For instance, in the specific scenario shown in Figure 4c, the lowest disease frequency is 1.2 × 10⁻⁵. In practical settings, such low values might correspond to extinction of the disease. Thus, because disease prevalence can be effectively 0, we explored scenarios in which a single infected individual is reintroduced to the population.

Figure 6 considers introduction of a new disease case in a population of size 1000. We simulate this event by pausing the numerical solution of our model, instantaneously increasing disease frequency by 0.001, renormalizing the frequencies to sum to 1 and then completing the solution. From a baseline pattern similar to Figure 4c, the effect of this new case depends on when it is introduced. During the interval between the first two epidemics, when the disease is effectively extinct, a critical time exists at which the introduction quickly leads to an epidemic. The critical time is the time at which anti-vaccine sentiment has generated enough unvaccinated individuals that disease frequency would begin increasing in the absence of the new case (indicated by the time at which the frequency of susceptibles crosses the horizontal dashed line in Figures 6d–f).

Figure 6. Introducing a single new case of a disease into the population can affect the long-pause oscillation of the disease frequency differently, depending on when it is introduced. (a) Introduction early in the apparent extinction of the disease. (b) Introduction late in the apparent extinction of the disease. (c) Introduction after the disease has re-established and is oscillating towards an endemic equilibrium. (d–f) Disease-susceptible (S) and anti-vaccine (A) frequencies for the trajectories in (a–c). An arrow indicates the new case. In all panels, for the grey trajectory, provided as a reference, no new case is introduced. The vertical dashed line indicates the point at which disease frequency starts increasing after the initial epidemic, during its apparent extinction. The horizontal dashed line in (d–f) is the 1/R ₀ threshold frequency of disease-susceptible individuals necessary for disease spread. Parameters used in this example are the same as in Figure 4c, except w is increased to 0.5 to provide a longer pause for illustration.

The grey trajectories in Figure 6a–c show the unperturbed disease frequency over time. In Figure 6a, a new case is introduced at day 250, prior to the critical time point, as the susceptible frequency lies below the threshold value in Figure 6d. The disease does not start epidemics, although the next epidemic does start earlier than without the new case. When the new case appears at day 1750 (Figure 6c), after the new epidemics have commenced and the susceptible frequency lies below the threshold value (Figure 6f), the oscillatory approach to endemic equilibrium is mildly perturbed but recovers quickly, with no lasting effect of the new case. However, when the new case appears at day 750 (Figure 6b), after the critical time point at which anti-vaccine sentiment has generated a sizeable population of unvaccinated individuals – indicated by the concurrent rise in the frequency of anti-vaccine sentiment and disease-susceptible individuals in Figure 6e – the new case immediately triggers epidemics. Thus, even if the disease effectively goes extinct, sentiment dynamics create a situation in which reintroducing a small number of infected individuals initiates new epidemics. In this specific example, such a migration immediately induces an epidemic only after a 15-month delay after the initial epidemic.

Assortative meeting

Homophily – or the tendency of like to interact with like – is a general property of human social behaviour (McPherson, Smith-Lovin, & Cook, Reference McPherson, Smith-Lovin and Cook2001). Vaccination behaviour exhibits homophily, so that anti-vaccine sentiment is not homogeneously distributed in populations (e.g. Salathé & Khandelwal, Reference Salathé and Khandelwal2011; Barclay et al., Reference Barclay, Smieszek, He, Cao, Rainey, Gao and Salathé2014; Onnela et al., Reference Onnela, Landon, Kahn, Ahmed, Verma, O'Malley and Christakis2016; Fu, Zimet, Latkin, & Joseph, Reference Fu, Zimet, Latkin and Joseph2019). We can extend our main model in eqn (2) to permit homophily through the phenomenon of assortative meeting (Eshel & Cavalli-Sforza, Reference Eshel and Cavalli-Sforza1982), modifying interaction probabilities by a parameter that increases the probability of interactions between individuals of the same sentiment class (A × A or P × P). The goal of this modification is to understand the influence of clustering in the population of anti-vaccine sentiment on the dynamics, as assortative meeting increases contacts among A individuals and among P individuals, and it decreases contacts between A and P individuals.

We consider an assortative meeting parameter α that affects the result of two draws with replacement from the population. We assume that undecided (U) individuals are unaffected by the rate of assortative meeting: if the first individual drawn is undecided (U), then the second draw is from the whole population. If the first individual drawn is anti-vaccine (A) or pro-vaccine (P), however, then the second draw is of type A or P, respectively, with probability α and it is from the whole population with probability 1 − α. Thus, α = 0 corresponds to the well-mixed case with no assortative meeting (eqn 2), and α = 1 corresponds to complete assortative meeting.

The new system of equations with this assortative meeting assumption is:

(16)$$\eqalign{ {{\lpar SU\rpar }^{\prime}} \hfill & = { b- \left[{rI_U + c\left({1-\displaystyle{\alpha \over 2}} \right)A + w + m} \right]\lpar SU\rpar } \hfill \cr {{\lpar SA\rpar }^{\prime}} \hfill & = { c\left({1-\displaystyle{\alpha \over 2}} \right)\lpar A\rpar \lpar SU\rpar - \lpar {rI_A + s + m} \rpar \lpar SA\rpar } \hfill \cr {{\lpar SP\rpar }^{\prime}} \hfill & = { w\lpar SU\rpar + s\lpar SA\rpar - \lpar {rI_P + v + m} \rpar \lpar SP\rpar } \hfill \cr {{\lpar IU\rpar }^{\prime}} \hfill & = { rI_U\lpar SU\rpar - \left[{c\left({1-\displaystyle{\alpha \over 2}} \right)A + g + w + m} \right]\lpar IU\rpar } \hfill \cr {{\lpar IA\rpar }^{\prime}} \hfill & = { rI_A\lpar SA\rpar + c\left({1-\displaystyle{\alpha \over 2}} \right)\lpar A\rpar \lpar IU\rpar -\lpar g + s + m\rpar \lpar IA\rpar } \hfill \cr {{\lpar IP\rpar }^{\prime}} \hfill & = { rI_P\lpar SP\rpar + w\lpar IU\rpar + s\lpar IA\rpar -\lpar g + m\rpar \lpar IP\rpar } \hfill \cr {{\lpar RU\rpar }^{\prime}} \hfill & = { g\lpar IU\rpar - \left[{c\left({1-\displaystyle{\alpha \over 2}} \right)\lpar A\rpar + w + m} \right]\lpar RU\rpar } \hfill \cr {{\lpar RA\rpar }^{\prime}} \hfill & = { c\left({1-\displaystyle{\alpha \over 2}} \right)\lpar RU\rpar + g\lpar IA\rpar -\lpar s + m\rpar \lpar RA\rpar } \hfill \cr {{\lpar RP\rpar }^{\prime}} \hfill & = { w\lpar RU\rpar + v\lpar SP\rpar + g\lpar IP\rpar + s\lpar RA\rpar -m\lpar RP\rpar \comma \;} \hfill \cr } $$

where

(17)$$\eqalign{ {I_U} & = { \lpar {IU} \rpar + \left({1-\displaystyle{\alpha \over 2}} \right)\lpar {IA + IP} \rpar } & {} \cr {I_A} & = { \left({1-\displaystyle{\alpha \over 2}} \right)\lpar {IU} \rpar + \left[{1 + \alpha \left({\displaystyle{{1-A} \over A}} \right)} \right]\lpar {IA} \rpar + \lpar {1-\alpha } \rpar \lpar {IP} \rpar } & {} \cr {I_P} & = { \left({1-\displaystyle{\alpha \over 2}} \right)\lpar {IU} \rpar + \lpar {1-\alpha } \rpar \lpar {IA} \rpar + \left[{1 + \alpha \left({\displaystyle{{1-P} \over P}} \right)} \right]\lpar {IP} \rpar.} & {} \cr } $$

In eqn (17), I _i for i = U, A, P represents a modified frequency of infected individuals available for interaction with an individual of sentiment type i (Appendix H). All three values reduce to I = IU + IA + IP if α = 0.

To examine the effect of assortative meeting on our model, we numerically solved eqn (16) using the measles parameters from Table 4 with c = 0.8, w = 0.14 and s = 0 (Appendix G). These values have the same order of magnitude as disease parameters r and v (Table 4) and lead to an endemic equilibrium (Figure 3). As an initial condition, we used SU = 0.06375, SA = 0.00525, IU = 0.001, RA = 0.06975 and RP = 0.86025, with other classes set to zero. We chose these initial frequencies to be near the 1 − 1/R ₀ threshold for measles, RA + RP = 93% vaccine coverage, with overall anti-vaccine sentiment frequency SA + RA = 7.5%, the highest statewide rate of kindergarten non-medical vaccine exemptions in the US during 2017–2018 (Mellerson et al., Reference Mellerson, Maxwell, Knighton, Kriss, Seither and Black2018).

For a variety of values of α, Figure 7a shows this solution for the first 200 days, after which the initial epidemic has subsided. We see that a high value of α can trigger an epidemic that would otherwise not have occurred at α = 0. For high values of α, the epidemic is also shorter. Increased assortative meeting spreads disease within the anti-vaccine class, increasing the disease frequency; however, it also hinders both disease transmission to the undecided class and conversion of the undecided class to the anti-vaccine class, so that disease frequency drops quickly.

Figure 7. Disease frequency over time with various levels of assortative meeting. (a) Initial behaviour for a system that starts near the herd-immunity level for measles, with 7.5% of individuals having anti-vaccine sentiment. (b) Disease frequency after 50,000, 75,000 and 100,000 days for a system that starts at the endemic equilibrium. Both panels use measles parameters from Table 4 with c = 0.8 and w = 0.14. These values are chosen so that c is the same as that used in other analyses (Figures 4–6), w is on the same scale as v (Table 4) and the EE is stable (Table 3).

To study the effects of α on the equilibrium frequency of the disease, we numerically solved eqn (16) for 100,000 time steps interpreted as days, using the same parameters as in Figure 7a but starting at the endemic equilibrium for the original α = 0 model (eqn 2). The time period of 100,000 days was chosen to be long enough to rule out long-delayed epidemics.

Figure 7b plots the resulting values at 50,000, 75,000 and 100,000 time steps of the disease frequency I as a function of α. The disease frequency monotonically decreases with α, reflecting the pattern observed in Figure 7a that α speeds up the time at which disease becomes rare. In contrast to the dramatic effect the assortative meeting parameter α has on disease frequency in the short term (Figure 7a), its long-term effect is negligible (Figure 7b).