How can contemporary climate research help understand epidemic dynamics? Ensemble approach and snapshot attractors

Abstract

Standard epidemic models based on compartmental differential equations are investigated under continuous parameter change as external forcing. We show that seasonal modulation of the contact parameter superimposed upon a monotonic decay needs a different description from that of the standard chaotic dynamics. The concept of snapshot attractors and their natural distribution has been adopted from the field of the latest climate change research. This shows the importance of the finite-time chaotic effect and ensemble interpretation while investigating the spread of a disease. By defining statistical measures over the ensemble, we can interpret the internal variability of the epidemic as the onset of complex dynamics—even for those values of contact parameters where originally regular behaviour is expected. We argue that anomalous outbreaks of the infectious class cannot die out until transient chaos is presented in the system. Nevertheless, this fact becomes apparent by using an ensemble approach rather than a single trajectory representation. These findings are applicable generally in explicitly time-dependent epidemic systems regardless of parameter values and time scales.

1. Background

Although nowadays many data-driven models and methods are developed and successfully used [1,2], it turns out that the dynamical system approach with a moderate number of degrees of freedom works well in disease propagation. To estimate short- and long-term behaviour, system parameters, possible control strategies and the social impact of disease modelling the celebrated susceptible–infectious–recovered (SIR) model [3] is a reasonable choice. This representation splits the population into three disjoint parts and deals with the number of individuals in these subpopulations as time goes on.

The SIR model and its variants (SI, SIS, SEIR, SEIRS, RAS) show qualitatively similar dynamics and are in good agreement with observations. In a homogeneous environment, these models possess a globally stable fixed point attractor as a disease equilibrium [4]. Omitting the stochastic nature of real-world disease spread [5–8], the deterministic SIR-like models present diverse and rich dynamics. The nonlinearity of the model [9–12] and also the time-dependent internal forcing can be considered as a source of complexity.

Sufficiently large seasonal forcing or different mixing rates between subpopulations can cause large oscillations and also period-doubling cascades [13–19]. It has also been demonstrated that for certain values of system parameters the non-autonomous models of recurrent epidemics (measles, mumps, rubella, H5N1 avian influenza) show chaotic behaviour [7,20–22]. Traditional SIR-like epidemic models are dissipative nonlinear low-dimensional systems with constant, periodic, quasi-periodic [23] or term-time [4,24] external forcing whose dynamics is governed by (chaotic) attractors in the phase space. Furthermore, even if the long-term dynamics is regular and the final state of the system is a fixed-point attractor the route to this condition might be rather complex. Many studies have examined the role of finite-time irregularity in ecological models [25–28] as well as in epidemic dynamics [10,24,29,30], concluding the relevance of transient behaviour.

It is known from dynamical systems theory that chaotic attractors can be characterized by the natural measure corresponding to the distribution of possible states in the phase space [31–33]. More precisely, one can define the frequency of visits with which typical orbits visit a certain part of the attractor when the orbit length goes to infinity. If these frequencies are the same for different initial conditions in the basin of attraction, then we say that these frequencies define the natural measures of the attractor. The time evolution of distributions is governed by the Frobenius–Perron equation, resulting in a sequence of distributions converging to the natural—stationary, time-independent—distribution. Thus, it is independent of the initial distribution. That is, the effect of the initial inhomogeneities and/or different initial domains dies out upon reaching the attractor.

It turned out, however, that traditional numerical methods, such as monitoring a single chaotic long-term trajectory, fail in the case of arbitrary external forcing. Obviously, if one wants to model such a system this shortcoming has to be overcome. The mathematical concept of snapshot attractors introduced in the context of random dynamical systems has been known for many years in the (theoretical and experimental) dynamical systems community [34–41], and entirely fulfils our wish. Readers seeking a deeper understanding of mathematics are referred to appendix A and the references therein.

Since snapshot attractors are time-dependent objects in the phase space of non-autonomous dynamical systems their shape is changing in time while the fractal dimension might even remain constant [39]. Furthermore, obtaining one single trajectory in a system with arbitrary driving force does not provide the same result as the ensemble approach (many trajectories emerging from slightly different initial conditions) in the same system at a given time instant. This effect is the consequence of the fact that ergodicity is not satisfied as the system is driven aperiodically [42]. This conclusion generated the recent opinion that the changing climate should be scrutinized by an ensemble approach (parallel climate realizations) rather than averaging a single long-term time series [43–48].

Both climatic and epidemic evolution are sensitive to seasonal changes and to exogenous forcing: in the case of the climate, the latter is clearly affected by anthropogenic changes in the atmospheric composition caused by greenhouse gases and aerosols; in the case of epidemics, it is vaccination and voluntary or compulsory changes in the population’s defensive behaviour.

In this work a seasonally forced deterministic epidemic model with a monotonically changing contact rate (due to, for instance, vaccination or restricting the movement of the population) is presented. We propose that the statement by [43]—climate change can be seen as the evolution of snapshot attractors—also holds for disease spread dynamics with continuously changing contact parameters.

In §2, the epidemic model is defined. After that, the results about stationary (§3) and changing (§4) epidemics are presented. Section 5 is devoted to conclusions.

2. Standard epidemic model

Compartmental disease models describe the number of individuals in a population regarding their disease status: susceptible (S), infectious (I) or recovered (R). Although these models involve many simplifications (such as the progression of infection or the difference in response of individuals) they performed well in real-world epidemic situations [49]. There are two major groups of epidemic models: the SIR-like cluster that is characterized by lifelong immunity (e.g. measles, whooping cough) and the SIS-like class (containing mostly sexually transmitted diseases) that is characterized by repeated infections [49].

Here, we study the SEIR equations [50], which involve a fourth group in addition to previous ones. More concretely, we assume that an individual enters the population at birth as susceptible and leaves it by death. A susceptible person becomes exposed (E) when contacting one or more persons, called infective(s), who can transmit the disease. In an incubation period, the exposed individuals are infectious but are not yet infectious. After this term, they become infective and later become immune or recovered.

The mathematical model associated with the above description reads as follows:

$$\begin{array}{rl}& \frac{\mathrm{d}S}{\mathrm{d}t}=m-bSI-mS,\\ & \frac{\mathrm{d}E}{\mathrm{d}t}=bSI-(m+a)E,\\ & \frac{\mathrm{d}I}{\mathrm{d}t}=aE-(m+g)I,\\ & \frac{\mathrm{d}R}{\mathrm{d}t}=gI-mR,\end{array}$$ 2.1

with the following notations and assumptions [51]: (i) S, E, I and R are smooth functions of time and the size of the whole population remains unchanged, S + E + I + R = 1; (ii) there are equal birth and death rates (m); (iii) the probability that an exposed individual remains in this class for a time period τ after the first contact is exp(− aτ), where 1/a is the mean (or characteristic) latent period; (iv) similarly 1/g gives the mean infectious period, that is, the time period that an individual spends as infectious before recovery; (v) immunity is permanent and recovered individuals do not re-enter the susceptible class. The contact rate b(t), which is the average number of susceptibles contacted per single infective per unit time, is the origin of the spread of the disease. In the case of annual periodicity

$$b(t)=b_0(t)[1+b_1\cos (2\pi t)],$$ 2.2

and t is measured in units of years.

When the system parameters are kept constant (m, a, g, b₀, b₁ = 0), the solution of equation (2.1) shows weakly damped oscillation with a globally stable equilibrium (b > g). In this case, the dynamics can be further characterized by the expected number of secondary cases caused by an infectious individual in the susceptible population. This is known as the basic reproductive ratio, and can be expressed here by R₀ = ba/[(m + a)(m + g)] [14]. R₀ essentially determines whether a disease can (R₀ > 1, endemic equilibrium) or cannot (R₀ < 1, infection-free) persist in a population. Provided that the latent and infectious periods are short, i.e. a, g ≫ m, one can use the approximation R₀ = b/g.

There are observations, for example in childhood diseases or avian influenza, that do not show damped characteristics, but rather (ir)regular cycles instead. This phenomenon can be linked to the seasonal variation in the contact rate b(t) or in the recruitment rate g [20]. It has already been noted that in the case of periodic forcing, b₀(t) = b₀ = const. in equation (2.2), chaos emerges in epidemic time series [4,22,52]. Similarly to the Lorentz84 model [53], wherein solar irradiation induces a seasonal effect, the SEIR model with a variable contact rate is a non-autonomous low-order system with a well-defined chaotic attractor in the phase space at certain parameter values. The low dimension of the phase space allows us to visualize its pattern comfortably.

As a result of the fixed population size, point (i) above, one of the four equations (traditionally the equation for R) can be omitted. In addition, the infectious and exposed groups turn out to be linearly related to the first order [14] at least (see electronic supplementary material, figure S2). Consequently, the S–I phase portrait represents the state space texture accurately. Plotting one trajectory in the S–I plane of the phase space one would observe a coil design owing to the explicit time dependence of the model, b₁ ≠ 0. A conventional method to make a periodically forced system autonomous is to take ‘pictures’ about the phase space with the same frequency as the excitation acts, i.e. making a stroboscopic map [32,33]. By choosing an initial time t₀ and purely sinusoidal force in equation (2.2) (b₀(t) = const.) one can interpret the filamentary shape of the chaotic attractor by defining the stroboscopic map $t_0=0$ with period T = 1 yr. Therefore, the periodic variation in equation (2.2) yields an annually stationary epidemic with a steady attractor.

Equation (2.2) assumes a constant average contact rate b₀. We have an annual period starting with its maximum at t = 0.¹ In a physical or biological context this means the largest contact rate, possibly due to school terms, holidays, seasonal breeding patterns in a seabird colony, etc. Equation (2.1) with the following parameters, corresponding to measles [7], shows irregular dynamics: m = 0.02 yr⁻¹, a = 35.84 yr⁻¹, g = 100 yr⁻¹, b₀ = 1800 yr⁻¹, b₁ = 0.28 yr⁻¹. The above parameters lead to R₀ ≈ 18. Numerical calculations show that for lower mean values of the contact rate periodic fixed-point attractors exist. For instance, biennial cycles at b₀ ≈ 1500. For more detailed asymptotic dynamics see the bifurcation diagram in figure 1 (blue curve).

Figure 1.

Bifurcation diagram of the SEIR model with stationary b₀. The variable S is plotted on the same year of the day starting at t₀ = 0. The classical diagram (blue curve) is obtained by integrating 2000 initial conditions for t = 15 000 yr and the last 500 points stored. For low parameter values, cycles are determined by annual periodicity. There are many period-doubling segments visible (shorter blue segments); however, the first one that routes to chaos starts at b₀ ≈ 1750. The end-points of the same trajectories are plotted after a 500 yr iteration, indicating long-lived transients (green). Clearly, finite-time chaotic motion unfolds the extra structure of the bifurcation diagram. The upper x-axis evaluates the basic reproductive ratio derived from the system parameters. R₀ = 1 is also marked (red arrow) to indicate the edge of the endemic equilibrium. The long-term dynamics for R₀ < 1 (b₀ < 100) implies an infection dying out in the population. The inset displays real permanent chaos. Top: the R₀ of common diseases for comparison only. (Ebola*: based on the 2014 Ebola outbreak.)

Until now b₀ was kept constant. Following the climate change methodology monotonic variation in b(t) through b₀(t) is established. Equation (2.3) specifies the temporal decay of the mean contact rate

$$b_0(t)=\{\begin{array}{cc}1800& \hspace{0.17em}\text{if}\hspace{0.17em}t\le t_{\mathrm{st}}\\ 1710{\text{e}}^{-\alpha (t-t_{\mathrm{st}})}+90& \hspace{0.17em}\text{if}\hspace{0.17em}t>t_{\mathrm{st}}.\end{array}$$ 2.3

Here, t_st represents the time until the epidemic is stationary, i.e. b₀ = const.; this is set to be 250 years after initialization. This amount of time seems to be enough since the convergence of trajectories to the attractors is much shorter, t_c ≈ 50 years. The exponent α is the decay rate of b₀(t). In this study, α takes three values, 0.04 yr⁻¹, 0.01 yr⁻¹ and 0.004 yr⁻¹, and b₀ always starts at 1800, i.e. from the chaotic regime. Equation (2.3) implies that lim _t→∞ b₀(t) = 90. This means that the critical value of the mean contact rate b₀ = 100 (or in terms of the basic reproduction rate, R₀ ≈ 1) is reached at different times after t_st, according to α.

The main motivation of this study is based on equation (2.3). Considering, for example, the time series of childhood measles in London, UK, after the Second World War (between 1945 and 1990), the vaccination programme (starting in 1968) changed the number of cases dramatically [4,29,54]. In addition to medical treatment, if available, other artificially forced declines in contact rate (such as a gradual lockdown prescribed by the administration) can be imitated by equation (2.3), resulting in a changing epidemic. This secular variation of parameter b₀ has a clear analogy to climate change models and the framework applied in this field.

3. Stationary epidemic

3.1. Parameter dependence

The bifurcation diagram basically reveals the long-term dynamics, excluding the initial transients, in a given parameter range. Classically, one single trajectory sampled by a stroboscopic map serves the blue shape of the bifurcation diagram in figure 1. In the case of a stationary epidemic (b₀ = const.) complex dynamics arises for certain parameter values in the SEIR model, $b_0\gtrsim 1770$. We should note, and it is going to be essential in our analysis, that the bifurcation diagram can also be established in a different way. That is, a large number of initial conditions are placed in the phase space and their evolution is monitored for a sufficiently long time, but much shorter (one or two orders of magnitude) than that of the single trajectory considered above. Because of the ergodicity of the stationary chaotic dynamics the end-points of the ensemble members portray exactly the same pattern [55].

The traditional way to explore the long-term dynamics depending on seasonal forcing is to construct a bifurcation diagram similar to the one shown in figure 1. The changing parameter is, however, b₁ along with a constant transmission rate [11,17,20,29]. Exactly the opposite applies to figure 1 here. Although we focus on b₁ = 0.28 in this study, electronic supplementary material, figure S1 shows bifurcation diagrams with some other values of the seasonal amplitude, demonstrating the main qualitative differences between these scenarios. From this, we can point out that the results discussed in the present work are consistent with larger contact rates, i.e. when b₀ > 0.28. However, for smaller b₀ where no permanent chaos is present along the bifurcation curve a new analysis is required, which will be the subject of future work.

The same applies for the phase space pattern, too. Figure 2 depicts the chaotic attractor obtained after t = 2500 yr integration and 1600 initial conditions distributed uniformly in the phase space. The same picture is obtained by one single trajectory after a 200 000 year simulation. A representative S–I phase portrait is shown in figure 2. The chaotic attractor (b₀ = 1800) was established fairly early and it remains unchanged when viewed after integer multiples of T = 1 yr. Nevertheless, choosing a different day of the year, the same applies with a different pattern in S–I plane.

Figure 2.

The stationary chaotic attractor in the S–I phase plane. The ensemble is plotted at three different time instants: t = 500, 1250, 2500 years with perfect overlap. The green circle depicts the 〈S〉 and 〈I〉 values from figure 3a, while the ellipse indicates the associated standard deviations from figure 3c. Note that the green circle is a weighted average (a kind of ‘barycentre’) of the trajectories along the attractor rather than its geometrical focus.

Transient chaos, that is, complex behaviour on finite-time scales [56], manifests before a trajectory comes to an attractor, both in dissipative as well as in conservative systems. In general, the attractor can be a simple object in the phase space, for instance a fixed point or a limit cycle. Following the time evolution of the ensemble before it approaches the attractor, one can capture other aspects of the bifurcation diagram by considering the initial transients. To do this, the uniformly distributed 2000 trajectories are integrated for 500 years and the corresponding state space positions are stored. This feature becomes visible in the green shaded domain that extends along the blue curve in (figure 1). Clearly, transient chaos has a significant contribution to the dynamics for $b_0\gtrsim 125$.

3.2. Ensemble view

The initial transients are usually thrown away in long-term dynamical analysis. However, the complexity might appear just in this phase of motion as indicated by the bifurcation diagram in figure 1, which is already in the stationary case. Illustrating the role of finite-time chaotic behaviour in the SEIR epidemic model we track the evolution of 1600 different initial conditions distributed uniformly in a cube S = [0.9999; 1], E = [0; 0.00005], I = [0; 0.00005].

Figure 3 demonstrates how the individual members of the ensemble reach the attractor at different time instants. For those values of the contact parameter when the epidemic shows biannual (b₀ = 1500) cycles one can observe clearly that the chaotic transients die out sooner or later (figure 3a). However, in the case of permanent chaos (b₀ = 1800) it is not so, since the irregular property is perpetual (figure 3b).

Figure 3.

Ensemble view of stationary dynamics. (a,b) The mean of S (blue) and I (red) variables versus time. Some of the individual trajectories are also shown sampled by a stroboscopic map (only the S coordinate). The individual trajectories arrive sooner or later at the biannual fixed-point attractor (see, for example, the cyan parallel dots for t > 230 in (a)). (c) Standard deviations for the same variables as (b) for b₀ = 1800. The constant value of the average and standard deviation after the convergence time demonstrates that the ensemble has reached the attractor and then spread along it.

Introducing classical statistical measures on the ensemble one can quantitatively keep track of the transient effect. The mean $⟨A(t)⟩=1/N\sum _{i=1}^N{A}_i(t)$ (where A_i(t) denotes an observable corresponding to the ith member at time t) designates how fast the initial irregularity terminates and also indicates the average transient time. One can observe similar characteristics for both the 〈S〉 and 〈I〉 curves, fixed-point and chaotic attractors (figure 3a,b), respectively.

In the stationary epidemic model, one might expect that after some time the extent of the attractor remains constant, suggesting that all of the individual trajectories in the ensemble have reached it. The standard deviation,

$${\sigma }_A={(⟨A^2⟩-{⟨A⟩}^2)}^{1/2},$$ 3.1

refers to the size of the attractor in the direction of A. Figure 3c depicts σ_S and σ_I in the case of b₀ = 1800. The initial small values of standard deviations show that the ensemble moves together at the beginning of integration. Then, the members spread out in the phase space (≈30–50 yr) and after a while start to approach the attractor. This time is longer for parameters b₀ = 1500 (not shown) but σ_S is smaller.

A stationary epidemic with constant σ_A essentially means that after the transients the phase-space structure does not change in time. This can be visualized by using an ensemble approach. Without loss of generality we always start our simulation at t = 0 (which corresponds to that part of the year with the highest seasonal amplitude b(t)) and take the next snapshot of the ensemble at $t=0$ that coincides with the same day of the year.

The natural distribution of a chaotic attractor [32] comes from the fact that the dots visit certain parts of the filamentary structure more frequently than others. Moreover, this distribution is stationary and does not depend on the initial conditions. Instead of investigating a three-dimensional frequency diagram (see electronic supplementary material, figure S3), we plot the projection I of trajectories wandering on the chaotic attractor in figure 2. To obtain figure 4, a fine grid is defined in the S–I phase plane and the number of points in each cell is plotted against the variable I. The histogram consists of four different ensembles. Ensembles 1, 2 and 3 cover the same volume in the phase space but involve slightly different initial state vectors. Ensemble 4 lays out the initial conditions from other parts of the phase space. The distribution clearly shows the same pattern for all four ensembles.

Figure 4.

Natural distribution on the chaotic attractor (b₀ = 1800) projected onto the I variable after t = 8500 years. The distribution does not depend on the individual members of the ensemble. The frequency has been cut at 3500 for clarity.

4. Epidemic change

In a changing epidemic, the parameter(s) of the system varies (vary) as time goes on. One might think that, say, a decrease in the contact parameter b₀ means walking along the bifurcation diagram slowly from right to left, and terminating at a safe destination with R₀ < 1. In this section, we will show that this idea is fairly naive because of the internal variability and the transient effects in the epidemic. To capture the dynamics properly in this scenario, the ensemble approach and the concept of the snapshot attractor is desirable.

4.1. Snapshot attractor geometry and natural distribution

According to the common measles parameters we start the switch-off process, also called the epidemic change, from the chaotic attractor (b₀ = 1800) according to equation (2.3).

Studies [55] have revealed that, starting from the chaotic attractor well after the trajectories have reached it, the evolution of different states might be rather diverse. That is, the transient dynamics becomes important again while the parameter change appears in the system. In figure 5, four trajectories are selected from the chaotic attractor and their time dependence is followed. It can be immediately seen that different colours reach the fixed-point attractor at different times. For instance, in the case of α = 0.004 (bottom curve), first the blue, then the yellow, red and green curves arrive at the fixed-point attractor. One can pick up other trajectories that will have longer or shorter oscillations. Furthermore, the previous order of colours, i.e. transient times, might change with α, as demonstrated by the middle curve. Therefore, we can point out that it is worth investigating several trajectories simultaneously instead of following individual ones. However, after the parameter shift is switched on, the shape of the chaotic attractor starts to vary and forms the time-dependent snapshot attractor. Thus, the snapshot attractor is an object in the phase plane that contains the whole ensemble at a given time instant. We note that this mutation is not a result of the particular choice of the initial phase (day of the year) of the mapping rather than the change of parameter. The geometrical alteration of the snapshot attractor is followed by the change of distribution on it. Nevertheless, its form and the distribution are independent of the choice of the ensemble and the onset (t_st) of parameter decay.

Figure 5.

Four trajectories plotted in blue, green, red and yellow wandering on the chaotic attractor and their fates (in the S variable) after parameter change sets in at t = 250 yr (vertical line). The two scenarios α = 0.01, 0.04 are shifted by 0.2 units, respectively, for better visualization. The very peculiar time evolution of individual trajectories for different αs suggests using ensembles.

Figure 6 shows snapshot attractors drawn from N = 10⁴ initial conditions during the epidemic change. The mean contact rate decreases from (a) to (d) as specified on each panel. Thus, every single plot corresponds to a different time instant of simulation. Figure 6a, taken at t = 219 yr, coincides with the attractor in figure 2 since the parameters of the system are the same for t < t_st = 250 yr. A more interesting aspect shows up on figure 6b,c. A filamentary structure, the fingerprint of chaotic behaviour, still dominates the pattern although the parameter b₀ is well below its original value (1800) as well as 1770, where large extended chaotic attractors are formed in the bifurcation diagram (figure 1 inset). In other words, for those contact rates (b₀ = 835, 328) no chaotic behaviour is anticipated in the stationary dynamics.

Figure 6.

The evolution of the snapshot attractor with α = 0.01. (a–d) The ensemble is taken at t = 219, 333, 447, 618 yr, respectively. (a) Corresponds to stationary dynamics (before t_st = 250 yr) with the original b₀ = 1800. Filamentary patterns persist for lower b₀ values too, where no stationary chaotic attractor exists in the phase space according to the bifurcation diagram (figure 1). Still to be noted is that the physical extension of the snapshot attractor is also time dependent and its size varies significantly in both directions. Compare the scale of the axes in different panels. The green circles and the red ellipses denote the same quantities as in figure 2. The narrow panels to the right of the S–I phase planes display the natural distribution projected onto the I variables. For b₀ = 133 most of the phase points accumulate around the fixed-point attractor (I ≈ 10⁻³); therefore, no histogram is presented. The initial conditions cover the following state space volume: S = [0.9999; 1], E = [0; 0.00005], I = [0; 0.00005].

Although figure 6 indicates the transmutation of the snapshot attractor for the decay exponent α = 0.01 one can obtain a similar alteration for different switch-off rates too. A slower scenario (e.g. α = 0.004 in equation (2.3)) allows the attractor to keep its filamentary shape and maintain chaotic dynamics longer. From another perspective, the same parameter value b₀ is reached later while α is smaller. The opposite is true for a faster parameter change, say α = 0.04. It can also be shown that approximately the same pattern belongs to the same contact value regardless of the rate of change. That is, the faster the contact rate decay, the less pronounced the complexity in the phase space.

Another interesting feature is that, although the contact rate b₀ decreases monotonically, the size of the snapshot attractors might increase (figure 6b). This fact demonstrates that the domain accessible by the dynamics can be larger. Thus the possible (S, I) pairs extend to larger domains in the phase space even for smaller contact parameters. Furthermore, the average (green circles) and the standard deviation (red ellipses) also change in time. This temporal behaviour cannot be obtained from the classical view, but only by using snapshot attractors.

To understand this phenomenon we should recall the concept of the chaotic saddle. This non-attracting set with its stable and unstable manifolds in the phase space is responsible for the finite-time chaotic behaviour [56]. To construct the saddle numerically, we define two holes in the phase space (black rectangles in figure 7). Then, a large number of initial conditions (N = 3 × 10⁵) are distributed uniformly in the region S = [0.04; 0.2], E = [0; 0.00005], I = [0.00001; 0.01] and the trajectories are integrated for t = 30 yr. Only those trajectories that do not enter the holes during the integration time are retained. The initial conditions belonging to these trajectories show the stable manifold for the saddle (red dots in figure 7), while their end-points immediately prior to the escape through the holes display the unstable manifold (green dots). The saddle itself is the intersection of its manifolds (not shown here). The average lifetime of chaos (the inverse of the escape rate, κ) can be estimated by calculating the time distribution of the non-escaped trajectories.

Figure 7.

The snapshot attractor (blue) and stable/unstable (red/green) manifolds of the stationary chaotic saddle at b₀ = 835. Discontinuities along the unstable manifold reflect the escape conditions of the numerical scheme. The long-term dynamics of the stationary model is visualized by the four tiny yellow circles around (S, I) ≈ (0.125, 0.0004) as fixed-point attractors. The snapshot attractor coincides with the phase portrait in figure 6b.

It is also known from the theory of transient chaos that the saddle’s manifolds have a filamentary design just like the snapshot attractors in figure 6. Analogously to the chaotic attractor, stationary dynamics with a constant driving amplitude defines a stationary chaotic saddle related to certain parameter values of the system. Because of the continuous adjustment of the contact rate, b(t), in the changing epidemic model, the trajectories do not have time to reach the attractor belonging to a given b₀ value. Consequently, a time-dependent chaotic saddle is considered, whose unstable manifold approximates the snapshot attractor [55]. This stationary saddle persists for very low b₀ values and its unstable manifold controls the finite-time complex epidemic dynamics (figure 7).

We emphasize at this point that the natural distribution associated with the snapshot attractor also changes in time following the geometrical reorganization of the phase-space pattern (histogram visualization in figure 6).

4.2. Parallel epidemic realizations

Similarly to climate research we can define the concept of parallel epidemic realizations. In standard disease models like equation (2.1), this can be done by using a large number of initial conditions as an ensemble in the phase space and following their evolution as discussed in §3.1. This picture can be imagined as many copies of the epidemics obeying the same physical laws and being affected by the same time-dependent forcing [48].

As we have seen before, in the case of a changing epidemic, the time-dependent forcing has an impact on the natural measure of the snapshot attractors. This fact results in a temporal change in the average values as well as internal variability. To quantify the internal variability of the system (2.1) statistical measures over the ensemble should be proposed, as in the case of a stationary epidemic. The variance or the standard deviation σ_A in equation (3.1) of the ensemble characterize the fluctuations around the averages, indicating the inherent internal variability. Similar results have been found by using the framework of non-autonomous and random dynamical systems for the El Niño–Southern Oscillation (ENSO). It turns out that seasonality plays a major role in interannual climate variability, such as the annual peaking of El Niño in December [57–59]. Recently, the correlations between the ENSO temperature and Indian summer monsoon precipitation anomalies have been investigated in a changing, i.e. externally forced, climate as an ensemble-wise correlation [60]. The results based on the snapshot attractor approach indicate a strengthening teleconnection in the late twentieth century rather than a weakening, which was previously thought to be the case by climate scientists.

Figure 8a–c shows the ensemble standard deviations σ_S (blue) and σ_I for different decay rates α. The parallel epidemic realizations contain 1600 numerically integrated trajectories sampled at integer multiples of years. The convergence time t_c, the time needed for the ensemble to reach and spread along the attractor, is found to be t_c = 50 yr. Both σ_S and σ_I are constant before t_st = 250 yr since untill then the seasonal driving is constant, i.e. b₀ = 1800 yr⁻¹. Immediately after this the contact rate starts to decrease (t > 250 yr) according to equation (2.3) and both the mean state and the internal variability of the epidemic change with time.

Figure 8.

Statistical measures of SEIR ensembles. (a–c) The standard deviation of variables S and I for various decay processes. The epidemic change starts at t_st = 250 yr well after the convergence time (t_c ≈ 50 yr). Vertical dashed lines mark the time instants corresponding to the phase-space portraits in figure 6. (d–f) The linear sizes of the snapshot attractors in both variables indicate the role of transient chaos and internal variability of the dynamics depending on the decay rate α.

In changing the epidemic, first, the graph of σ_S(t) increases; after the maximum the trend follows nearly b₀(t); then it decays to zero, illustrating that the size of the attractor shrinks and asymptotically reaches the neighbourhood of a regular fixed-point attractor, (S, I) = (1, 0), as expected for b₀ ≈ 90 (R₀ = 0.9). The larger the α, the more regular, i.e less filamentary, the phase-space geometry is at the same time (figure 8d–f).

The shape of the standard deviation can be explained by transient chaos that occurs during the parameter change. The size of the green shaded band around the main feature in the bifurcation diagram (figure 1) already indicates that the size of the phase-space region filled with transient chaos increases as b₀ reduces. To demonstrate this we draw attention to the horizontal dimension of the snapshot attractors in figure 6. The smoothly decreasing profile of the standard deviation after reaching the maximum refers to the existence of the transient effect up to very small parameter values.

Different maximum values of the σ_S(t) curves indicate a non-trivial relation between the internal variability and the changing rate of b₀. The largest value corresponds to α = 0.01 (figure 8b), the other two scenarios (α = 0.04 and 0.004) show roughly the same maximum size of the snapshot attractor, albeit at different time instants. figure 8d–f presents the maximal physical extension of the snapshot attractors versus time. These plots also support our observations that the size of the attractor first increases and then shrinks to be a fixed-point attractor.

One possible explanation of this property is the relative ‘coupling’ between the time scales, that is, the decay of forcing defined by the exponent α and the average lifetime of chaos in stationary dynamics at specific parameter b₀. From a geometrical point of view, this is how close the snapshot attractor evolves to the unstable manifold of the stationary chaotic saddle at a particular contact parameter. A detailed exploration of this feature will, however, be the subject of future study.

5. Final remarks

Parallel climate realizations, an effective and new framework in climate research, have been used in an epidemic model to explore the fading of complexity due to the systematic switch-off of the driving mechanism. The mathematical concept of snapshot attractors and their natural distribution demonstrates that single time-series analysis is not capable of reflecting the complex dynamics of a changing epidemic. Instead of monitoring isolated events, the ensemble view of trajectories—parallel epidemic realizations—and its statistical description is desirable.

The temporal change of the attractor geometry and the distribution on it reveals the internal variability of the dynamics.

The extension of the bifurcation diagram indicates the importance of transient chaos in the stationary dynamics as well as during the continuous parameter shift. No matter whether we start from a stationary state of the long-term dynamics, the switch-off process activates the hidden parts of the bifurcation diagram. Thus, the underlying non-attracting object, the chaotic saddle, or more precisely its unstable manifold, organizes the system’s evolution.

The dissipative relaxation time scale, the inverse of the phase space contraction rate based on the divergence of the vector field of the system (2.1), is extremely low, approximately 2–5 days, for measles. The other characteristic times, that is, the inverse of the switch-off rate and the escape rate from the chaotic saddle, are, for comparison, α⁻¹ ≈ 25 − 250 yr, κ⁻¹ ≈ 30 − 35 yr, respectively. The latter are obtained for a few particular contact rates. This implies that the switching-off process, with parameter α used in this study, is not quasi-static [55]. In other words, there is not enough time for trajectories to reach the stationary attractor, because it is either chaotic or regular, owing to the non-stop parameter variation. Therefore, the dynamics of the switching-off process shows much more complexity than that of the stationary scenario.

The results obtained in this study reveal the dynamics of measles with harmonic external forcing. This approach is, however, less realistic than term-time forced models [29] that fit better to observed data. Furthermore, it has also been pointed out [24] that periodic forcing can be used to describe measles but at much lower values of parameter b₁ where no permanent chaotic behaviour is produced. Considering this fact, the present analysis starting from a chaotic regime of periodically excited measles with the given α and b₁ should be considered with caution. This study highlights a novel theoretical framework of disease dynamics that seems to be a general dynamical system approach.

Although the relevant seasonal forcing amplitudes in measles do not lead to permanent irregularity, the transient form of chaos is still present at these parameter values. It is worth, therefore, studying the snapshot attractor and ensemble formalism of non-autonomous epidemic dynamics starting from this regime of parameters. Furthermore, the more trusted term-time forcing model should also be incorporated in future examinations in order to have relevant outcomes for various endemic pathogens.

In the spirit of the investigated SIR-like classical epidemic model (2.1) with parameter shift² of the contact rate b(t), we emphasize the importance of the finite-time irregular dynamics. Our results reinforce the ‘hidden’ complex transients, and also the importance of time-dependent snapshot attractors not just for an epidemic with large reproductive ratio, such as measles and chickenpox, but for those with lower values ($R_0\gtrsim 1$) too.

The main conclusion of this study sheds light on the importance of the ensemble view and parallel realizations in epidemic dynamics. Figure 6 illustrates the general features one expects in the (long-term) prediction of any epidemics. Predictions based on individual simulations are not reliable since, owing to the chaotic nature of the dynamics, they can lead to many possible results, with any point of the snapshot attractor belonging to the time instant of the prediction. The ensemble approach is able to treat all possible outcomes as a whole, and predicts even the probability of the different permitted epidemic states. When convenient, statistical moments of this distribution can be determined, e.g. the average (providing the most typical epidemic outcome) or the variance (characterizing how broad the distribution of the permitted states is), but higher order moments might also be useful. In any case, one can thus follow how the statistical prediction changes in time.

Appendix A. Pullback attractors

For completeness, it should be mentioned that a rigorous mathematical interpretation of snapshot attractors is referred to as pullback attractors in mathematical and climate research [57,58]. A pullback attractor is defined as a time-dependent set $\mathcal{A}(t)$ (−∞ < t < ∞) in the phase space X. Introducing a two-parameter family [61] of operators acting on X one can, then, write

$$\underset{s\to -\mathrm{\infty }}{lim}|X(s,t;X_0)-\mathcal{A}(t)|=0,$$ A 1

for all t and initial conditions X(s) = X₀ taken at pullback time s such that s ≪ t. We obtain that the solutions, starting in the distant past, all approach the time-dependent attracting set $\mathcal{A}(t)$ together with a time-dependent, invariant measure supported on it [62,63]. A snapshot attractor, thus, can be viewed as a section of $\mathcal{A}(t)$ at a given time instant.

Data accessibility

This article has no additional data.

Competing interests

I declare I have no competing interests.

Funding

This work was supported by the NKFIH Hungarian grant no. K125171. The support of the Bolyai Research Fellowship and ÚNKP-19-4 New National Excellence Program of the Ministry for Innovation and Technology is also acknowledged.