# Theory of early warning signals of disease emergenceand leading indicators of elimination

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## Abstract

Anticipating infectious disease emergence and documenting progress in disease elimination are important applications for the theory of critical transitions. A key problem is the development of theory relating the dynamical processes of transmission to observable phenomena. In this paper, we consider compartmental susceptible–infectious–susceptible (SIS) and susceptible–infectious–recovered (SIR) models that are slowly forced through a critical transition. We derive expressions for the behavior of several candidate indicators, including the autocorrelation coefficient, variance, coefficient of variation, and power spectra of SIS and SIR epidemics during the approach to emergence or elimination. We validated these expressions using individual-based simulations. We further showed that moving-window estimates of these quantities may be used for anticipating critical transitions in infectious disease systems. Although leading indicators of elimination were highly predictive, we found the approach to emergence to be much more difficult to detect. It is hoped that these results, which show the anticipation of critical transitions in infectious disease systems to be theoretically possible, may be used to guide the construction of online algorithms for processing surveillance data.

## Keywords

Bifurcation delay Transcritical bifurcation SIS SIR van Kampen system size expansion Critical transition## Introduction

Infectious diseases are among the most visible and costly threats to individual and public health. Antibiotics, vaccines, and the molecular revolution in biology have not erased their mark. Millions of persons die every year from treatable ancient diseases, such as malaria (Gething et al. 2011; WHO 2012), tuberculosis (Dye et al. 2008), and measles (Orenstein and Hinman 2012; Simons et al. 2012). Sometimes, elimination of these diseases through vaccination, prophylaxis, and/or vector control is possible, but sustaining elimination campaigns is difficult as pathogens approach the point of elimination (Cohen et al. 2012). Conversely, emerging pathogens such as SARS and swine-origin influenza A (H1N1) cause excess mortality, disrupt business, terrorize vulnerable populations, and lead to new, sometimes ineradicable endemic diseases (e.g., HIV/AIDS). The benefits of accurately forecasting disease emergence would be tremendous: in the case of a low-incidence SARS-like pathogen, the savings could be tens of billions of $US (Rossi and Walker 2005; Smith et al. 2009); an illness resembling the 1918 influenza virus might take millions of lives and impose costs of the same order as a year’s gross domestic product (Osterholm 2005).

Elimination and emergence of infectious disease both involve a transmission system that is pushed over a critical point. In most cases, criticality occurs at the point where the basic reproduction number, \(R_{0}\), the number of secondary infected cases arising from a single infected case in an entirely susceptible population, is equal to one (Heffernan et al. 2005). Similar critical points occur in other complex systems (Strogatz 1994; Sole 2011). Particularly, in noisy (stochastic) systems, these critical points manifest as transitions between alternative modes of fluctuation (Scheffer 2009; Scheffer et al. 2009; Lenton 2011; Scheffer et al. 2012). We call such a stochastic transition a *critical transition* if there exists a bifurcation in a suitably constructed limit case of the mean field model. A central problem in the study of critical transitions is the identification of phenomena indicating the proximity to a critical transition in the absence of a detailed understanding on the system’s dynamical equations and/or the forcing variables causing the change (Scheffer et al. 2009; Scheffer et al. 2012; Boettiger and Hastings 2012). Recent studies have established that some noise-induced phenomena may signal the approach to a critical transition in a slowly forced dynamical system (Ives and Dakos 2012; Dakos et al. 2012a, b; Dakos et al. 2008; Carpenter and Brock 2006; Donangelo et al. 2012; Seekell et al. 2011; Carpenter and Brock 2011; Brock and Carpenter 2010; Dakos et al. 2010; Carpenter et al. 2008; Guttal and Jayaprakash 2008, 2009). If this property would apply to infectious diseases, it would suggest a model-independent route to forecasting infectious disease emergence in subcritical systems with \(R_{0}<1\) (i.e., crossing the critical point “from below”) and documenting the approach to elimination in endemic (supercritical) disease systems where \(R_{0}>1\) (crossing the critical point “from above”).

Many of these characteristic noise-induced phenomena involve *critical slowing down*, a decline in the resilience of the system to perturbations, which generally gives rise to an increase in the variance and autocorrelation of fluctuations as the system approaches the transition (van Nes and Scheffer 2007; Dakos et al. 2012a). But, these properties require that the transition be suitably regular (Hastings and Wysham 2010). Specifically, to observe critical slowing down requires that the potential function of the system, if it exists, be smooth. Also, for systems with multiple attracting sets (e.g., bistable systems), a sufficient level of noise gives rise to “flickering” which exhibits the characteristic increase in variance (but for a different reason) and not the increase in autocorrelation (Scheffer et al. 2009; Dakos et al. 2012b). Other obstacles to anticipating critical transitions in infectious diseases include that epidemiological systems are complicated by amplification of transients and oscillatory dynamics (Rohani et al. 2002; Bauch and Earn 2003), that infectious diseases are often seasonally forced (Altizer et al. 2006; Fraser and Grassly 2006) and propagate in demographically open systems subject to imported cases (Keeling and Rohani 2002), and that diseases that are close to elimination or close to emerging are, by definition, characterized by low prevalence and, therefore, both subject to demographic stochasticity and difficult to observe (Lloyd-Smith et al. 2009). Therefore, to determine whether diagnostic noise-induced phenomena accompany the critical transitions that occur in infectious disease dynamics, namely the transition to endemicity of emerging infectious diseases and the transition to extinction in disease elimination, it will be useful to have a quantitative theory, both to guide the selection of statistical quantities to be investigated and to predict the form that observable quantities should take as the transition is approached.

This paper is a contribution to this theory. We consider the compartmental *susceptible–infectious–susceptible* (SIS) and *susceptible–infectious–recovered* (SIR) models, which are widely used to represent the dynamics of a range of nonimmunizing and immunizing pathogens and may be viewed as approximations to an even broader class of infectious diseases. In their deterministic formulations, these models are characterized by a transcritical bifurcation at \(R_{0}=1\), where the endemic and disease-free steady states meet and exchange stability. The epidemic models we investigate are generalizations of the closed population SIS and SIR models that allow for immigration from external sources, of which the more familiar models without immigration that are characterized by a transcritical bifurcation are special cases. To appropriately model the slowly forced dynamical system, we first develop mean field theory for the nonstationary systems gradually approaching a bifurcation. We develop the mean field theory here because slowly forced dynamical systems approaching a bifurcation have rarely been investigated (but see Kuehn (2011)). We then develop the full stochastic description in terms of a master equation. Our strategy is to use the van Kampen system size expansion to separate the mean field dynamics from the fluctuations induced by demographic stochasticity. These fluctuations are described by a Fokker–Planck equation, from which the power spectrum, variance, autocorrelation, and coefficient of variation may be analytically obtained. Our main result is contained in Table 4, which provides formulas expressing these theoretical predictions in terms of the dominant eigenvalue of the mean field solutions, which goes to zero as the system approaches the critical transition. Hence, these formulas express how epidemiologically measurable quantities are expected to change as the system approaches the critical transition. A number of approximations are introduced in the derivation of these formulas, including separation of time scales, second-order description of the fluctuations, and continuum representation of the state space. We, therefore, performed simulations to study the possible detrimental effects of these approximations. Specifically, we examined the agreement between the theoretical predictions and “measured” statistics obtained in a moving window, as one would analyze real-world data, in both discrete state-space and continuous state-space simulations. We show that the analytical expressions are robust, provided the conditions of fast–slow systems are met. These results demonstrate that the critical transitions associated with disease emergence and elimination may be anticipated even in the absence of a detailed understanding of their underlying causes.

## Theory and methods for emergence and elimination

### SIS model

#### Mean field theory of SIS model

*N*is given by

*N*is constant, i.e., \(x+y=1\), the SIS model may be described by a single equation,

We note that Eq. (2) can encompass a variety of infectious disease systems, including closed population SI models. Thus, the system can be used to model diseases that confer no long-lasting immunity, e.g., sexually transmitted diseases or acute infections such as influenza or diseases that are fatal. If \(\eta = 0\), Eq. (2) has two equilibria: the disease-free equilibrium \(y_{0}=0\) and the endemic equilibrium \(y^{\ast } = (1-1/R_{0})\), where \(R_{0} = \beta / \gamma \) denotes the basic reproduction number. When \(R_{0}>1\), the infection can persist, but if \(R_{0}<1\), then the infection dies out. The equilibria \(y_{0}\) and \(y^{\ast }\) meet and exchange stability when the basic reproduction number \(R_{0}=1\), i.e., a transcritical bifurcation occurs. We will refer to the SIS model with no immigration as the limiting case. However, it is usually more realistic to assume that a low level of immigration occurs, i.e., \(\eta \) is positive. If \(\eta >0\), then Eq. (2) has a single positive equilibrium that is always locally stable, provided \(\beta , \gamma >0\), since the slope at the equilibrium \(\lambda = \beta - 2 \beta y^{\ast } - \gamma -\eta \) is negative. Moreover, whether \(\eta = 0\) or \(\eta >0\), the return time \(-1/\lambda \) decreases in the limit \(\beta \rightarrow \gamma \). At the critical point in the limiting case SIS model, \(\lambda =0\). Therefore, the transcritical bifurcation occurs in a suitably constructed limiting case (\(\eta =0\)), which we suggest gives an indication for the behavior of the fluctuations for the stochastic version of the model with immigration, provided \(\eta \) is small.

*f*describes the change in transmission rate. In this paper, we assume that transmission is a slowly changing linear function of time, i.e., \(\beta = \beta (t) =\beta (1-p(t))\). The proportion of the population that are vaccinated is modeled by the function \(p(t)=p_{s}+p_{0} t\), with \(p_{0}\) an incremental change in \(\beta \). Hence, \(\dot {\beta } = p_{0}\). If \(p_{0}>0\), then the rate of transmission is slowly declining over time, and if \(p_{0}<0\), it is slowly increasing. The limit case \(\epsilon \rightarrow 0\) of Eq. (3) is Eq. (2), and thus, the endemic equilibrium of the limit case assumes \(\beta = \beta (1-p_{s})\), which arises from the level of vaccination uptake \(p(t)\) (the bifurcation parameter). Figure 1 shows the transcritical bifurcation diagram for the SIS model. For small \(\eta \), the plot of the stable endemic equilibrium \(y^{\ast }\) as a function of vaccination uptake is similar to the bifurcation diagram for \(\eta =0\), except in the vicinity of the critical point \(p^{\ast }\) (inset figure of Fig. 1). For \(p > p^{\ast }\), the disease-free equilibrium is stable in the limiting case, but for the model with immigration, the infectious population is sustained at a low level due to importation of the disease from external sources (Fig. 1).

#### Stochastic description of SIS model

*dt*, \(\overline {\beta }\). Table 1 shows the variables and parameters of the SIS model.

Variables and parameters of the SIS model

Variable | Expression |
---|---|

Proportion of susceptible individuals | |

Proportion of infectious individuals | |

Recovery rate | |

Per capita birth rate | |

Immigration rate | |

Population size | |

Transmission rate prior to changes | |

Basic reproduction number | \(R_{0} = \beta /\gamma \) |

Rate of change in transmission | \(p_{0}\) |

Start value of vaccination uptake | \(p_{s}\) |

Transmission rate (\(\epsilon \rightarrow 0)\) | \(\beta (1-p_{s})\) |

Transmission function at time | \(\overline {\beta } = \beta (t) =\beta (1-(p_{s}+p_{0} t))\) |

Critical point | \(p^{\ast } = 1 - 1/R_{0}\) |

*dt*, the number of infectious individuals

*I*can either (a) increase by 1, (b) decrease by 1, or (c) not change in number. Infection and removal are the events in an SIS process that lead to these changes in state. We denote the number of individuals

*I*at time

*t*or the state of the system by \(\alpha = I\) at time

*t*and the alternative states by \(\tilde {\alpha }\), where either \(\tilde {\alpha } = I+1\) or \(\tilde {\alpha }=I-1\) in the SIS model. The system of individuals goes from a state with \(\alpha =I\) individuals at time

*t*to a state with \(\tilde {\alpha }\) individuals at time \(t+dt\) with transition probability per unit time \(T_{i}(\tilde {\alpha }|\alpha )\). The system can also go from a state with \(\tilde {\alpha }\) individuals at time

*t*to

*I*individuals at time \(t+dt\) with a probability flux \(\tilde {T_{i}}(\alpha |\tilde {\alpha })\). Table 2 outlines the events, changes in state, and the transition probability fluxes into and out of state

*I*that can occur in a stochastic SIS process with a slowly changing transmission rate.

Transition probability fluxes for SIS model into and out of state \(\alpha =I\)

Event | Change in state | Transition probability per unit time |
---|---|---|

Infection | ||

Into \(\alpha \) | \(I-1\rightarrow I\) | \(\tilde {T_{1}}(I|I-1)=\frac {\overline {\beta }(N-(I-1))(I-1)}{N} +\eta (N-(I-1))\) |

Out of \(\alpha \) | \(I\rightarrow I+1\) | \(T_{1}(I+1|I)=\frac {\overline {\beta }(N-I)I}{N}+\eta (N-I)\) |

Removal | ||

Into \(\alpha \) | \(I+1\rightarrow I\) | \(\tilde {T_{2}}(I|I+1)= \gamma (I+1)\) |

Out of \(\alpha \) | \(I\rightarrow I-1\) | \(T_{2}(I-1|I)= \gamma I\) |

*I*individuals at time

*t*evolves with time. The master equation represents the continuous time version of the dynamics of the Markov process. Derivations of master equations can be found in Van Kampen (1981) and Renshaw (1991). Letting \(P(I,t)=\text {Prob}\) \((I(t)=I)\) be the probability that the infectious state variable \(I(t)\) is equal to some nonnegative integer

*I*, the master equation for the evolution of the probability of the infectious population being in state \(\alpha = I\) at time

*t*is

*I*are as described in Table 2. This is a system of \(N+1\) ordinary differential equations, \(\alpha =0, 1, \dots N\), which can be solved with an initial condition of \(I_{0}\) individuals at time \(t=0\), i.e., \(P(\alpha , 0) = 1\) for \(\alpha = I_{0}\) and \(P(\alpha , 0) = 0\) for all \(\alpha \neq I_{0}\). Thus, the probability distribution at time \(t=0\) is a delta function. Table 3 summarizes the quantities in Eq. (4).

System | State | Alternative states \( \tilde{alpha} \) |

SIS | | \(I+1, I-1\) |

SIR | \((S, I)\) | \((S+1, I-1), (S-1, I), (S+1, I), (S, I-1), (S, I+1)\) |

System | Probability flux into \(\alpha \) | Probability flux out of \(\alpha \) |

SIS | \(\tilde {T_{1}}(\alpha |\tilde {\alpha })=\tilde {T_{1}}(I|I-1)\) | \(T_{1}(\tilde {\alpha }|\alpha ) = T_{1}(I+1|I)\) |

SIR | \(\tilde {T_{1}}(\alpha |\tilde {\alpha })= \tilde {T_{1}}((S,I)|(S+1, I-1))\) | \(T_{1}(\tilde {\alpha }|\alpha ) = T_{1}((S-1, I+1)|(S,I))\) |

We will apply the van Kampen system size expansion to the master equation. The system size expansion takes advantage of the fact that, for large population size *N*, the demographic fluctuations in the population are small, and so *N* is the expansion parameter. Thus, the key assumption for the expansion to be valid is that the system size *N* is sufficiently large (see Appendix A for details). Here, we pause to note that that the SIS model is equivalent to a simple logistic process, and hence, there are analytical expressions for the mean and variance (Allen 2003). The \(I=0\) state is absorbing, and the expected time to extinction is finite (Allen 2003). Nevertheless, if the population size is large, the probability distribution is approximately stationary for a very long time. It is the statistics of this quasi-stationary distribution that we are interested in.

Analytical expressions for quasi-stationary statistics about the endemic infectious quasi-steady state expressed in terms of the eigenvalues

Model | Power spectrum \(S_{I}(\omega )\) | Autocorrelation |
---|---|---|

SIS | \(\frac {2 Q}{\omega ^{2}+\lambda ^{2}}\) | \(\exp (-|\lambda | \tau )\) |

SIR (spiral) | \(\frac {2(\alpha _{I}+D_{22} \omega ^{2})}{(\omega ^{2}-((\mathrm {Re} \lambda )^{2}-(\mathrm {Im} \lambda )^{2}))^{2}+ 4(\mathrm {Re} \lambda )^{2} \omega ^{2}}\) | \(\displaystyle \frac {1}{\pi }\int _{0}^{\infty } S_{I}(\omega ) \cos (\omega \tau ) d\omega \) |

SIR (node) | \(\frac {2(\alpha _{I}+D_{22} \omega ^{2})}{(\omega ^{2}-\lambda _{1}\lambda _{2})^{2}+(\lambda _{1}+ \lambda _{2})^{2} \omega ^{2}}\) | \(\displaystyle \frac {1}{\pi }\int _{0}^{\infty } S_{I}(\omega ) \cos (\omega \tau ) d\omega \) |

Variance \(\sigma ^{2}\) | Coefficient of variation | |

SIS | \(\frac {Q}{2 |\lambda |}\) | \( \frac { Q^{\frac {1}{2}}}{\sqrt {2|\lambda |N} \varphi ^{\ast }}\) |

SIR (spiral) | \(\frac {\alpha _{I} +((\mathrm {Re} \lambda )^{2}-(\mathrm {Im} \lambda )^{2}) D_{22}}{4 ((\mathrm {Re} \lambda )^{2}-(\mathrm {Im} \lambda )^{2}) (\mathrm {Re} \lambda )}\) | \(\frac {(\alpha _{I} +((\mathrm {Re} \lambda )^{2}-(\mathrm {Im} \lambda )^{2})D_{22}) ^{\frac {1}{2}}} {\sqrt {4((\mathrm {Re} \lambda )^{2}-(\mathrm {Im} \lambda )^{2}) (\mathrm {Re} \lambda ) N} \varphi ^{\ast }}\) |

SIR (node) | \(\frac {\alpha _{I} +\lambda _{1}\lambda _{2} D_{22}}{2 \lambda _{1}\lambda _{2} (\lambda _{1}+ \lambda _{2})}\) | \(\frac {(\alpha _{I} +\lambda _{1}\lambda _{2} D_{22})^{\frac {1}{2}}}{\sqrt {2 \lambda _{1}\lambda _{2} (\lambda _{1}+ \lambda _{2}) N}\varphi ^{\ast }}\) |

#### Robustness of the van Kampen approximation

Since the system size expansion involves approximating the discrete random variable *I* with a normal random variable, its assumptions may break down when *I* is small. For small numbers of infectious individuals, e.g., if the system is subcritical and, thus, incidence of a disease is low, the assumption that the fluctuations are normally distributed about the mean is likely to be inappropriate. The van Kampen expansion is valid when the system is far from the absorbing boundary at \(I=0\). However, the approximation upon which the approach is built (Eq. 10 in Appendix A) cannot describe chance extinctions, which can occur if *I* is small. If \(\eta = 0\) and \(\overline {p} > 1- 1/R_{0}\), the disease-free equilibrium is stable, and so the approximation given by Eq. (10) is not valid. Due to the absorbing boundary, the probability distribution about the disease-free state for a given time *t* will be one-sided. On the the other hand, if \(\eta >0\), then, close to the system boundary, the normal distribution approximation about the deterministic mean may be poor because the distribution is bimodal due to extinction events. However, the models with immigration do not have a disease-free state; just a small infectious population when \(\overline {p}\) is approximately greater than \(1/R_{0}\). Therefore, predictions for the quasi-stationary statistics about this state can be obtained.

### SIR model

#### Mean field theory of SIR model

*N*is

*p*of the population are chosen at random for vaccination at birth and are recruited into the recovered class. The remaining unvaccinated proportion of the population enter the susceptible class at birth. Since the population size

*N*is constant, \(x+y+z=1\), system (5) is equivalent to the following system of ordinary differential equations

In the absence of vaccination and immigration, the basic reproduction number is given by \(R_{0} =\beta /(\gamma +\mu )\). If \(\eta = 0\), the SIR model (6) has two equilibria: the disease free state \((x_{0}, y_{0})=(1-p, 0)\) and the endemic equilibrium \((x^{\ast }, y^{\ast })=(1/R_{0}, \mu (1-p-1/R_0) /(\gamma+\mu)\), respectively. The SIR model undergoes a transcritical bifurcation at \(R_{0} =1\). The endemic equilibrium is locally stable if \(R_{0}>1\) and is not biologically feasible if \(R_{0}<1\), when the disease-free equilibrium is locally stable. As the vaccination uptake *p* increases, the basic reproduction number reduces by a factor \((1-p)\), i.e., the effective reproduction number is \(R_{0}(1-p)\). The vaccination uptake *p* for the effective reproduction number is 1 at the critical vaccination threshold, \(p^{\ast } = 1 - 1/R_{0}\) (Anderson and May 1991), and it is at this critical threshold that the transcritical bifurcation occurs. Again, the SIR model with no immigration is the relevant limiting case.

Temporary importation of pathogen often occurs in infectious disease systems (Keeling and Rohani 2008). Assuming that a low level of immigration occurs, i.e., \(\eta >0\), only a single positive equilibrium is biologically feasible. This equilibrium is a stable spiral when the square of the trace of the Jacobian matrix evaluated about the equilibrium is less than four times its determinant and is a stable node if it is greater than or equal to this quantity. Complex eigenvalues of the Jacobian matrix of Eq. (6) characterize a stable spiral, and the eigenvalues are real and negative if the equilibrium is a stable node. The stable node equilibrium can be thought of as a “disease-free” equilibrium if \(\eta \) is small, in the sense that the infectious population is sustained at a low level due to importation of the disease from external sources. Therefore, while the transcritical bifurcation occurs in a suitably constructed limiting case (\(\eta =0\)), the limit case may give an indication for the behavior of the fluctuations for the stochastic version of the model with immigration, provided that \(\eta \) is small.

*p*may induce a transition from endemicity or to extinction. Recruitment of susceptibles may slowly vary over long time scales as a result of demographic or evolutionary changes. Vaccination uptake rates are often not constant over time and may exhibit trends, e.g., percentage uptake of pertussis vaccine in the USA (Rohani and Drake 2011). Mathematically, we may express the SIR model approaching a transcritical bifurcation as a fast–slow system:

*f*describes the change in vaccination uptake

*p*. Again, we model vaccination uptake as a linear function of time \(p(t)=p_{s}+p_{0} t\), with \(p_{0}\) an incremental change in

*p*. Consequently, \(\dot {p} = p_{0}\). If \(p_{0} >0\), recruitment into the susceptible class is slowly declining over time and if \(p_{0}<0\), then recruitment is slowly increasing over time. In the limit \(\epsilon \rightarrow 0\), the vaccination uptake is fixed at a constant rate \(p_{s}\), and the system is stable. Figure 2 shows the transcritical bifurcation diagram for the SIR model. The infectious equilibrium \(y^{\ast }\) is plotted as a function of vaccination uptake

*p*. The bifurcation diagram indicates that the endemic infectious equilibrium declines linearly with

*p*in the models with and without immigration. However, the inset plot indicates that in the vicinity of the \(\eta = 0\) critical point, the infectious equilibrium of the immigration model is elevated relative to the disease-free equilibrium, since the infectious population is sustained at a low level due to immigration.

#### Stochastic description of SIR model

*p*is not constant in time but is a slowly changing function of time. Therefore, the recruitment rate into the susceptible class is slowly changing, but, for each small time increment

*dt*, we can treat the vaccination uptake as constant \(\overline {p}\). Table 5 presents the variables and parameters of the SIR model.

Variables and parameters of the SIR model

Variable | Expression |
---|---|

Proportion of susceptible individuals | |

Proportion of infectious individuals | |

Proportion of recovered individuals | |

Recovery rate | \(\gamma \) |

Per capita birth rate | \(\mu \) |

Immigration rate | \(\eta \) |

Population size | |

Transmission rate prior to changes | \(\beta \) |

Basic reproduction number | \(R_{0} = \beta /(\gamma +\mu )\) |

Vaccination uptake | |

Rate of change in vaccination uptake | \(p_{0}\) |

Start value of vaccination uptake | \(p_{s}\) |

Vaccination uptake (\(\epsilon \rightarrow 0\)) | |

Vaccination uptake at time | \(\overline {p} =p(t) = p_{s}+p_{0} t\) |

Critical point | \(p^{\ast } = 1 - 1/R_{0}\) |

*dt*moving slowly towards a transition include infection, recovery, recruitment to the susceptible class, and death due to natural causes. Table 6 outlines the events, changes in state, and the transition probabilities per unit time for each change in state. We can construct a master equation in the same manner as for the SIS model. Letting \(P(S, I,t)=\text {Prob}((S(t),I(t)) = (S, I))\) be the probability that the state vector \((S(t),I(t))\) is equal to some nonnegative integer \((S,I)\), the master equation for the evolution of the probability of the population being in state \(\alpha = (S,I)\) at time

*t*is

Transition probability fluxes for SIR model. Transitions can occur into and out of state \(\alpha = (S,I)\)

Event | Change in state | Transition probability per unit time |
---|---|---|

Infection | ||

Into \(\alpha \) | \((S+1,I-1) \rightarrow (S,I)\) | \(\tilde {T_{1}}((S,I)|(S+1,I-1))=\frac {\beta (S+1)(I-1)}{N}+\eta (S+1)\) |

Out of \(\alpha \) | \((S,I) \rightarrow (S-1, I+1)\) | \(T_{1}((S-1, I+1)|(S,I))=\frac {\beta SI}{N}+\eta S\) |

Death of susceptible individual | ||

Into \(\alpha \) | \((S+1, I) \rightarrow (S,I)\) | \(\tilde {T_{2}}((S,I)|(S+1, I))= \mu (S+1)\) |

Out of \(\alpha \) | \((S,I) \rightarrow (S-1, I)\) | \(T_{2}((S-1, I)|(S,I))= \mu S\) |

Death of infectious individual | ||

Into \(\alpha \) | \((S, I+1) \rightarrow (S,I) \) | \(\tilde {T_{3}}((S,I)|(S, I+1))= \mu (I+1)\) |

Out of \(\alpha \) | \((S,I) \rightarrow (S, I-1)\) | \(T_{3}((S, I-1)|(S, I))= \mu I\) |

Recovery of infectious individual | ||

Into \(\alpha \) | \((S, I+1) \rightarrow (S,I) \) | \(\tilde {T_{4}}((S,I)|(S, I+1))= \gamma (I+1)\) |

Out of \(\alpha \) | \((S,I) \rightarrow (S, I-1)\) | \(T_{4}((S, I-1)|(S, I))= \gamma I\) |

Recruitment of susceptible individual | ||

Into \(\alpha \) | \((S-1, I) \rightarrow (S,I) \) | \(\tilde {T_{5}}((S,I)|(S-1, I))= \mu N (1-\overline {p})\) |

Out of \(\alpha \) | \((S,I) \rightarrow (S+1, I)\) | \(T_{5}((S+1, I)|(S,I))= \mu N (1-\overline {p})\) |

The master Eq. (8) is nonlinear and, therefore, cannot be solved analytically. To make analytical progress with Eq. (8), again, we can use the van Kampen system size expansion (see Appendix B for details). The approach gives rise to the deterministic SIR fast–slow system (7) and a linear Fokker–Planck equation that describes the evolution of the fluctuations, which may be written as a system of stochastic differential equations. These equations can be analyzed using Fourier transformation. The solution of the Fokker–Planck equation is a bivariate normal distribution. Table 4 summarizes the indicator statistics in the stable system limiting case \(\epsilon \rightarrow 0\) for the SIR model, in terms of its eigenvalues.

### Simulations

Parameter values for the simulations

Parameter | SIS model | SIR model |
---|---|---|

Basic reproduction number \(R_{0}\) | 10 | 17 |

Recovery rate \(\gamma \) | 1/10 year | 365/22 year |

Per capita birth rate \(\mu \) | - | \(1/50~\text {year}^{-1}\) |

Imports per year \(\delta \) (elimination) | 1 | 100 |

Imports per year \(\delta \) (emergence) | 1 | 14 |

Immigration rate \(\eta \) (elimination) | 0.0002 | 0.00034 |

Immigration rate \(\eta \) (emergence) | 0.00002 | 0.0035 |

Population size | 50,000 | 5,000,000 |

Population size | 50,000 | 100,000 |

Change in transmission/recruitment \(p_{0}\) | \(1/500~\text {year}^{-1}\) | \(1/500~\text {year}^{-1}\) |

Starting value (elimination) \(p_{s}\) | 0 | 0 |

Starting value (emergence) \(p_{s}\) | 0.96 | 0.96 |

To test the robustness of this theory to the range of approximations that were introduced (fast–slow approximation, continuum description, van Kampen expansion), we simulated the approach to elimination and emergence in a variety of cases. To simulate the approach to elimination, we followed the “bottom-up” approach of Allen (2003) to derive stochastic differential equations that incorporated demographic stochasticity. This approach uses the rates in Tables 2 and 6 to build a system of stochastic differential equations. Stochastic differential equations formulated in this way are appropriate, provided that the population size is sufficiently large, because then changes in the state variables are assumed to be normally distributed. Simulations were compared with output from Gillespie’s direct method (Gillespie 1977). The simulations were qualitatively similar for population sizes greater than 50,000. To simulate the approach to emergence, we used Gillespie’s direct method as this is most appropriate for small population sizes.

We simulated the SIS and SIR stochastic models with immigration approaching elimination and emergence 500 times. To compare the statistics to the theoretical predictions in Table 4, the infectious time series approaching elimination were sampled at yearly intervals. The transcritical bifurcation in these scenarios was approached over a long time frame (e.g., 470 years when \(p_{0} = 1/500\) in the SIR model). However, the transcritical bifurcation to emergence was approached over a relatively short time frame for the SIR model (10 years). To obtain a better sampled time series, the data from the Gillespie simulations were aggregated over monthly intervals for the SIR model. The infectious time series were aggregated over yearly intervals for the SIS model approaching emergence because events did not always occur at a monthly frequency. Thus, time series obtained from the SIS system approaching emergence allowed us to examine the issues that arise from poorly sampled time series.

### Analysis over a moving window

*Van Kampen detrending*.

Gaussian filtering is another, more common, method used to remove the influence of a slowly varying mean of a data series. To compare the performance of Gaussian smoothing to van Kampen detrending, for each time series, we fit a Gaussian kernel smoothing function across the entire infectious case record up to the time that the transcritical bifurcation was predicted using a fixed bandwidth. Lenton et al. (2012) have shown that the results obtained from applying the Gaussian filter across the entire time series do not differ significantly from detrending within windows. To obtain the residuals, we subtracted the fit from each time series and divided by the square root of the population size to be consistent with van Kampen detrending. The choice of bandwidth was informed by the resemblance of the Gaussian residuals to the fluctuations obtained from the van Kampen approach. To study the changes in the statistics up to the critical transition, we calculated the lag-1 autocorrelation and the variance of the fluctuations obtained using the two detrending methods over a moving window half the length of the time series. We calculated the lag-1 autocorrelation coefficient of each replicate using the acf function in R. The coefficient of variation (CV) was calculated by calculating, over a moving window, the mean and standard deviation of each infectious replicate. The median and 95 % prediction intervals for each of the statistics were calculated over the 500 replicates of each model. The prediction intervals were calculated using the quantile function in R. To quantify trends in each statistic for each replicate, we used Kendall’s correlation coefficient \(\tau \). To determine the distribution of Kendall’s \(\tau \), we calculated the coefficient for the trend in the test statistic for each realization.

To assess the performance of the leading indicators, we followed the approach of Boettiger and Hastings (2012) and calculated receiver operating characteristic (ROC) curves from the distribution of Kendall’s \(\tau \) calculated from realizations of the models with and without transitions. The model without a transition is quasi-stationary, and we refer to it as the baseline or null model. The null models for elimination assume vaccination uptake \(p=0\) and are simulated beginning from the deterministic equilibrium at \(p=0\). The null models for emergence assume vaccination uptake \(p=0.96\) and are simulated beginning from the deterministic equilibrium at \(p=0.96\). The baseline models were simulated for the same length of time as it takes for the transition to be approached in the test models. The model with a transition is the test model. An ROC curve enables investigation of the sensitivity of leading indicators to detect differences between quasi-stationary systems and those approaching a critical transition. We simulated the baseline models 500 times and obtained fluctuations using the van Kampen approach and from Gaussian filtering. We then quantified the trend in each indicator using Kendall’s \(\tau \) for each baseline simulation. In our results, indicator statistics typically exhibited increasing trends or no trend, but for those that decreased, we multiplied Kendall’s \(\tau \) for each realization by -1 to calculate the ROC curve. The area under the ROC curve (AUC) was also calculated. An AUC close to one indicates near-perfect detection.

## Results

### Transitions to elimination and emergence in SIS systems

### Transitions to elimination and emergence in SIR systems

*p*. Figure 5a shows a solution of the fast–slow system (7) with \(\eta \neq 0\). The mean field \(\epsilon \rightarrow 0\) theory and the solution of Eq. (7) agree closely. The timing of the transcritical bifurcation in the \(\eta =0\) model occurs at \(t=470\), and the number of infectious individuals has reached a low level by this time. Figure 5b shows a stochastic realization of the SIR model with immigration approaching elimination (the details of the stochastic process are described in Table 6). The infectious population exhibits amplified oscillations, unlike the deterministic trajectory in Fig. 5a. Demographic stochasticity is expected to excite the transient oscillations of the SIR model (Bauch and Earn 2003), and we will see this in detail later when we examine the power spectrum of the fluctuations.

*t*until approximately \(t=55\) years. The solution grows more slowly than the stable equilibrium. The delay in the dynamics is also exhibited by stochastic realizations, e.g., Fig. 6b. Outbreaks remain small and sporadic for at least 10 years after the bifurcation has occurred. The delay before emergence arises because the mean predicted by system (7) remains small following the bifurcation. However, the bifurcation delay is not as marked as that in the SIS model (compare Fig. 4 with Fig. 6).

### Leading indicators of elimination

#### Power spectrum predictions

#### SIS leading indicator predictions

#### SIR leading indicator predictions

Leading indicators for SIR systems approaching elimination (Fig. 8d–f) are not always consistent with the standard expectations of increasing variance, increasing autocorrelation, and increasing coefficient of variation. In addition, the predictions for the leading indicators of SIR disease elimination evaluated about the endemic equilibrium of the stable system behave differently to SIS leading indicators, due to the presence of resonant frequencies. As the threshold to elimination is approached, the lag-1 autocorrelation increases with vaccination uptake, indicating an increase in system memory, but in contrast to the SIS model, the fluctuation variance declines. We observe in Fig. 7b, d that the area under the power spectrum declines as vaccination uptake increases, particularly in the model with \(\eta > 0\), where the height of the peak of the power spectrum, which reflects amplification of fluctuations, declines close to the transition. The coefficient of variation also increases, indicating that infectious time series should become noisier as the transition is approached but does so more dramatically in the model without immigration, because the power spectrum peak height increases as the transcritical bifurcation is approached. To examine how different changes in vaccination uptake \(p_{0}\) affect the statistics, we varied \(p_{0}\) by \(1/500\) and \(1/30~\text {year}^{-1}\) in the fast–slow model with immigration. Predictions generated from the \(p_{0}=1/500\) model are in close agreement with predictions arising from the \(\eta > 0\) stable model. However, when vaccination uptake is increasing an order of magnitude faster (i.e., \(p_{0}=1/30\)), there can be temporary fluctuations, including increases, in the variance (Fig. 8e). Such a pattern is not predicted when we evaluate the statistics about the endemic equilibrium.

#### SIS predictions over a moving window

Generally, the analytical predictions for leading indicators for SIS disease systems approaching elimination are robust over a moving window, and they perform well in comparison to statistics generated from the null stable models, as indicated by the ROC curves in Fig. 10. The ROC curves lie above the black line, showing that the indicators behave better than chance in distinguishing between realizations that have been generated by the null and test models and the reported AUC values are close to 1. Furthermore, the trends in the statistics agree with the theoretically predicted trends (compare Fig. 10a, c, e with Fig. 8a–c). However, the Gaussian smoother may not be appropriate when the trend declines rapidly, as occurs for the SIS system approaching elimination. Various bandwidths were used, but while smaller bandwidths removed the slowly varying trend successfully, they did not capture the fluctuations relevant for critical slowing down. Larger bandwidths captured these fluctuations but did not successfully remove the slowly varying trend from the last 10 years of the time series, as can be seen in the plots. This problem with Gaussian filtering has been noted before by Dakos et al. (2008).

#### SIR predictions over a moving window

The theoretical predictions for leading indicators for SIR disease systems approaching elimination are robust over a moving window. The trends in the statistics shown in Figs. 11a, c, e, and 12a, c, e agree with the theoretical predictions (Fig. 8d–f). Notably, the median fluctuation variance for the system with \(p_{0} = 1/30\) also exhibits more variability, as predicted theoretically. The ROC analysis for the SIR models indicates that the statistics from the test models perform well in comparison to statistics generated from the null stable models, but the fluctuation variance for the system with \(p_{0} = 1/30\) is an exception. The performance is poor because the trend in the variance exhibits wider variation (Fig. 12c). Finally, the Gaussian filtering performs well in comparison to the van Kampen detrending, because the mean of the SIR model declines linearly and does not exhibit any rapid changes.

### Early warning signals for emergence

#### SIS early warning signal predictions

#### SIR early warning signal predictions

In subcritical SIR systems approaching endemicity, the theoretical predictions for the SIR model agree with standard expectations, except for the coefficient of variation, which is predicted to decline (Figs. 9d–f). The limiting case theoretical predictions for the trends in the statistics for SIR systems on the threshold of emergence agree more closely with the predictions for the fluctuations about the solution of the corresponding SIR fast–slow system, in contrast to the SIS model, because the bifurcation delay, although present, is not as marked as that in the SIS model. Figure 9d, e shows that the monthly lag-1 autocorrelation and the variance increase as the bifurcation is approached. It is not surprising that the variance increases because the power spectrum increases in power as the critical threshold for emergence is approached (Fig. 7d). However, the coefficient of variation is predicted to decline, i.e., the signal should become less noisy with decreasing vaccination uptake. In the context of the power spectrum result, this again makes sense because when the system transitions from subcritical to endemic, the power spectrum changes from a relatively flat shape to one with a peak about a resonant frequency. To examine how differences in vaccination uptake \(p_{0}\) affect the statistics, we varied \(p_{0}\) by \(1/500\) and \(1/30~\text {year}^{-1}\) in the fast–slow model. The statistics predicted by the fast–slow theory do not change as rapidly as predicted in the \(\epsilon \rightarrow 0\) theory. Finally, we did not calculate the early warning signals for the disease systems without immigration because the disease-free state is stable.

#### Predictions over a moving window

In contrast to disease systems approaching elimination, the early warning signals for diseases on the verge of emergence do not perform well. No marked trends are present in the median statistics calculated from either of the van Kampen or Gaussian detrending approaches, with the exception that the variance of the SIS and SIR systems approaching emergence (Figs. 13c, d and 14c, d) increases slightly. The 95 % prediction intervals for each statistic further show that all the statistics are highly variable. Moreover, the ROC curves indicate that it may, in general, be difficult to distinguish between subcritical systems with \(R_{0}<1\) and endemic disease systems with \(R_{0}>1\) (Figs. 13b, d, f and 14b, d, f). The ROC curves show that the distributions of Kendall’s correlation coefficient generated from stable systems and systems approaching a critical transition overlap significantly. However, we are conservative in our approach because we take only the years up to the critical transition in the absence of immigration, and not the years after, when a bifurcation delay may occur, during which time prevalence remains low and so the mean prevalence may not increase for a long time after the transition. Therefore, it is not surprising that it is difficult to distinguish between the subcritical and supercritical systems.

## Discussion

### Summary

The transmission of infectious diseases is a paradigm example of a low-dimensional, noisy, nonlinear dynamical process. Critical transitions in infectious disease systems, which correspond to such socially important events as the emergence of novel pathogens and the elimination of disease, are of special interest. The development of early warning signals of emerging infectious diseases and leading indicators for disease elimination, particularly, would be of tremendous value to the advance of public health. The goal of our study was to develop the theory of such early warning signals and leading indicators for infectious disease transmission systems that meet the assumptions of the familiar SIS and SIR models and which are forced through a critical transition by changes in transmission. Our main result—analytical expressions for the change in observable statistics as the transmission system approaches the critical transition—are reported in Table 4.

### Bifurcation delays

Besides the practical goal of indicating patterns that function as early warning signals and leading indicators, our study also provides some basic insights into the dynamics of infectious diseases. Particularly, our simulations illustrated the ubiquity of *bifurcation delays*—changes in system state that lag behind the bifurcation of the limiting case—during disease emergence and elimination. Delays are of fundamental interest because they provide a mechanism for the frequent realization of far-from-equilibrium situations and because they are a dramatic example of tipping point phenomena in infectious diseases. Indeed, delays can cause a system to appear to undergo a catastrophic shift (e.g., the dramatic upturn of infectious cases associated with emergence) when, in fact, only a noncatastrophic transcritical bifurcation has occurred. Our results show that two kinds of delays occur in disease emergence and elimination. The first delay is a deterministic phenomenon referred to as a *canard* (Diener 1984). Canards occur in fast–slow systems where the solution follows an attracting slow manifold of the fast–slow system, passing close to a bifurcation point, and then follows a repelling slow manifold for a considerable period of time. Thus, for instance, in Fig. 3a, which depicts the emergence of an SIS infection, the \(\epsilon \rightarrow 0\) limit case branch rapidly increases close to the \(\eta = 0\) critical point, but the solution of the fast–slow system (\(\epsilon >0\)) tracks the disease-free equilibrium for a noticeable amount of time even after it has become unstable. The influence that the canard has on the stochastic dynamics can be observed in Fig. 3b, where the number of infectious individuals remains in the vicinity of the precritical level for a noticeable period of time. A related phenomenon, but not so dramatic, is observed in the elimination case (Fig. 2a). Following the bifurcation, the infectious population remains elevated, but unlike the emergence case, no dramatic shift to extinction occurs. Analogous phenomena are observed in the SIR model (Figs. 5 and 6).

The second class of delays is a stochastic phenomenon, caused by the nonzero time interval that occurs between the time at which the critical point is reached and the time of the event that initiates the transition (i.e., the index case of a major outbreak). This phenomenon is most obvious in the emergence of an immunizing pathogen (the emergence scenario for the SIR model). In this case, the number of infectious individuals in the population prior to the critical transition is most commonly zero. Assuming that there are no infectious individuals in the population at the time of the bifurcation event, then the number of infections in the population will remain zero until the time that an imported case arises. In view of the assumption of demographic stochasticity, importation is a Poisson process with nonzero rate parameter, which similarly ensures that a nonzero period of time must elapse before the first case appears. Indeed, even after the first case appears, there is no guarantee that the system will undergo transition, since even in a supercritical population, there is some chance of rapid extinction (Allen 2003). When the system is only slightly supercritical, this chance may not be small. For measurement purposes, we might define this stochastic bifurcation delay as the time elapsed between the occurrence of the bifurcation and the final occurrence of zero infectious individuals in the population. Clearly, this quantity is a random variable. The distributional properties of this stochastic bifurcation delay are an important problem for further research.

Delays are also important for the immediate practical reason that their existence implies that the state of the system, i.e., the presence or absence of disease, cannot be taken as reliable evidence concerning whether the system is subcritical or supercritical. In the case of disease emergence, a system may have developed to the supercritical state—a significant concern for public health—although no cases have occurred. In such a scenario, any introduction of infectious individuals may spark a major outbreak. In the case of disease elimination, a system may already exist in the subcritical state, with only lingering infections and short transmission chains enabling the pathogen to transiently persist. In such cases, the battle has already been won, but it is crucial that vaccination campaigns and other elimination activities be continued until the remaining lines of transmission are snuffed out. Otherwise, gains that may have come at great cost will fail to be maintained, and the pathogen may resurge in the population, sparked by its own embers still smoldering in the face of relaxed intervention. The development of statistical methods to identify such situations is an urgent problem for further research.

### Limiting case predictions can sometimes be misleading

Predictions for the statistics derived from the limiting case (\(\eta = 0\)) version of the SIR model can sometimes be misleading. As the extinction threshold is approached, the predicted SIR (\(\eta >0\)) lag-1 autocorrelation increases as expected, but this phenomenon gives way to a decrease close to the \(\eta = 0\) critical point (Fig. 8d). This finding is not so surprising when one recalls that the single biologically feasible equilibrium in the \(\eta >0\) model changes from a stable spiral to a stable node near the critical point as vaccination is increased. The magnitude of the real part of the eigenvalue, which approximates the rate of decay of the amplitude of the oscillations to the equilibrium, decreases as elimination is approached, but, near the threshold, the magnitude of the real part begins to increase once more, indicating a shorter return time to equilibrium and increased resistance of the system to perturbations, as the spiral begins to resemble a node. This subsequent decline in system memory is reflected in the decrease in the autocorrelation function near the critical point. The more rapid decay of the oscillations is also reflected in the power spectrum, in that the the spectrum peak declines as vaccination uptake increases, meaning that the resonant frequencies become less amplified. On the other hand, the recovery rate of the stable spiral in the limiting case decreases much more dramatically on the transition to elimination, and oscillations at the resonant frequencies become more amplified. Thus, while the limiting case is an important guide in understanding more general disease models, it is also necessary to thoroughly investigate these models to understand the behavior of systems approaching a critical transition.

### Limitations

This study has followed Kuehn (2011) and Boettiger and Hastings (2012) in the use of stochastic differential equations to understand noise-induced phenomena that occur during a critical transition. It is understood, of course, that for discrete systems (such as infectious disease transmission), this continuum representation is only an approximation, and, moreover, that the approximation breaks down as the system size—or even if the size of just one important compartment—becomes small. The case of disease emergence, where the number of infectious individuals in a population prior to the critical transition is typically zero and only occasionally one or more, is precisely this situation. It would not be surprising, therefore, to discover the approach taken in this paper to perform poorly in this situation. But, this is not the case. As Figs. 13 and 14 show, in both the SIS and SIR systems, the “measured” statistics obtained in simulation are qualitatively similar to the analytic expressions reported in Table 4, with the exception of the coefficient of variation in the SIR model. We note, further, that although emergence was difficult to predict (AUC values were not much greater than 0.5), the van Kampen detrending, which we think of as the theory-dependent approach, typically performed better than generic Gaussian detrending, which may be computed without any theoretical assumptions. It is, nevertheless, a concern that the approach taken here will be misleading under some scenarios: for instance, a very slow approach to the critical transition, when the time between immigration events may be large and the van Kampen approximation is likely to break down. Moreover, the stochastic features of bifurcation delay, particularly important in the emergence scenarios, depend heavily on (a) the sequence of immigration events by infectious individuals, and (b) the probability of nonextinction for a transmission chain initiated by an infectious individual after the critical point has been passed. These are both intrinsically discrete aspects of the emergence process. For this reason, we suggest that the study of (nonstationary) discrete transmission models may be a fruitful direction for further study. There is, of course, a large body of work on (stationary) discrete contagion models on which such a theory could build (Bailey 1964; Renshaw 1991; Daley and Gani 1999; Allen 2003).

## Conclusion

In conclusion, early warning systems for emerging infectious diseases and leading indicators of elimination, which would be of tremendous benefit to society, now appear to be potentially achievable. Their realization depends on two key developments: (1) better understanding of noise-induced phenomena exhibited by nonlinear contagion processes in the vicinity of their critical points, and (2) surveillance methods that acquire data of sufficiently high accuracy, resolution, and timeliness. Toward the first of these goals, our studies have indicated that anticipation of critical transitions is theoretically possible. It is hoped that these results, which include expressions for the key observable phenomena (Table 4), may help to guide achievement of the second, which is now the most important outstanding problem, i.e., empirical demonstration in experimental or surveillance data.

## Notes

### Acknowledgments

This research was funded by a grant from the James S. McDonnell Foundation.

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