2.1. SEIR model in a Bayesian state-space framework

We model the spread of infectious diseases using a stochastic SEIR framework that explicitly tracks both latent epidemic states and the time-varying transmission rate. While we present the SEIR model here as a concrete example, the methodology is applicable to a wide range of compartmental epidemic models. Let the latent state at time t be , where S_t, E_t, I_t, R_t denote the numbers of susceptible, exposed (infected but not yet infectious), infectious, and removed (recovered or deceased) individuals, respectively. The variable Z_t is not a dynamical compartment but an auxiliary quantity representing the new incidence over each reporting interval, included for convenience to directly match the observation model. The variable is incorporated into the latent state to allow inference on the time-varying transmission rate. The latent dynamics follow the system

(1)

where is the constant population size, and and denote the average incubation and infectious periods, respectively. The transmission rate is modeled as a stochastic process on the log scale, following a Brownian motion with volatility and driven by a standard Wiener process B_t. This allows to vary over time in response to behavioral changes, public health interventions, or seasonal effects [3,27]. The parameter controls the magnitude of these fluctuations: larger values permit faster changes, while smaller values constrain variation. This formulation enables the model to capture unobserved temporal variability in transmission intensity throughout an epidemic. The effective reproduction number is defined as . To simulate sample trajectories, the continuous-time system (1) is propagated forward using a discrete-time approximation. Let denote the vector of unknown model parameters governing both the latent dynamics in (1) and the observation model described below. The latent state is updated according to

(2)

The forward process updates S_t, E_t, I_t, R_t, and Z_t deterministically via a forward Euler scheme, while evolves stochastically using the Euler–Maruyama method [28]. Formally, the deterministic updates are represented as a Dirac delta distribution within the transition density , whereas the stochasticity of introduces process noise. Details on the discretization and transition density are provided in Appendix A in S1 Text.

We assume that the available data , representing daily reported cases, are noisy measurements of the latent incidence Z_t, linked through the conditional distribution . A natural starting point is to model the counts as realizations of a Poisson random variable. In practice, however, daily case counts often fluctuate more than a Poisson model would predict, a phenomenon known as overdispersion. To account for this, we also consider a Negative Binomial formulation.

Specifically, the number of reported cases y_t in a reporting interval (t − 1, t], , is modeled as either

(3a)

(3b)

The Negative Binomial is parameterized by its mean and variance , where is the overdispersion parameter; the Poisson model, with , is recovered as . This mean-variance parameterization is used directly in the ensemble-based observation variance described in Section 2.2 below, ensuring consistency between the observation model and the EnKF update step. is the reporting fraction, informed by expert knowledge. Specifying helps mitigate practical non-identifiability issues highlighted in [27], since decreases in observed cases could otherwise be attributed equally to a lower reporting rate, a reduced transmission rate, or a combination of both. Practical considerations regarding the structural identifiability of parameters in such models were also discussed in [11].

Together, Eqs (2)–(3) define our complete state-space model [4]. Let denote the latent states up to time t. Prior knowledge about and initial state x₀ is encoded in the distributions and , respectively. The joint posterior of states and parameters given the observations is then

(4)

In this sequential learning setting, the inferential objectives at each time t are twofold. First, for a given fixed value of , filtering aims to infer the current state x_t conditional on all observations up to that time, corresponding to:

(5)

Second, parameter estimation focuses on the marginal posterior of ,

(6)

where we adopt the convention that . The filtering distribution that accounts for parameter uncertainty is obtained by marginalising the conditional filtering distribution over the parameter posterior, i.e.,

(7)

For nonlinear state-space models, Eqs (5) and (6) cannot be solved analytically and are generally intractable. Consequently, practical inference relies on numerical approximations such as particle filtering or EnKF-based methods to generate samples from the filtering distribution or to estimate the marginal likelihood. In this work, motivated by the computational efficiency and scalability considerations discussed in the introduction, we focus on EnKF-based inference.

2.2. Ensemble Kalman filter

We now describe the ensemble Kalman filter (EnKF) [15], a sequential inference method for state-space models with nonlinear latent dynamics. While the EnKF was originally developed for systems with Gaussian observations, many real-world datasets, including those arising in epidemic surveillance often deviate from this assumption and may exhibit overdispersion. To accommodate such data, we follow the approach of Ebeigbe et al. [25] and extend it by constructing observation noise terms consistent with the observation models in Eq (3).

The EnKF alternates between prediction and update operations at each time step to sequentially refine the latent state estimate based on incoming observations (Fig 1). During the prediction step, the ensemble is propagated forward through the system dynamics, while in the update step, the ensemble is adjusted using the most recent observation.

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Fig 1. Schematic illustration of the sequential update–prediction cycle in the EnKF.

Each time step alternates between an update stage, where observations y_t are assimilated to refine the latent state estimate, and a prediction stage, where the ensemble is propagated forward through the system dynamics.

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Let us first suppose that the parameters and a sequence of observations y_1:T (in our case, the daily incidence) are known. At each time step t, the EnKF aims to infer the latent state x_t given the data y_1:t. Let the ensemble at time t − 1 be denoted by , representing samples from the conditional . The forecast ensemble is generated by propagating each particle forward through the transition dynamics, that is,

(8)

The forecast density is then approximated by:

(9)

where and are the sample mean and covariance of the forecast ensemble.

A key adaptation for epidemic data is the specification of the observation error variance V_t. Classical Kalman filtering assumes Gaussian observations with constant variance (or at least does not depend on the latent state), an assumption often violated for epidemiological models, where the variability of reported incidence typically scales with its magnitude [25]. A principled approach is to set V_t to match the expected conditional variance of the observation given the latent state , as implied by the observation model (3). In practice, we approximate the expectation above using the forecast ensemble. We define the ensemble estimate of the observation error variance by

(10)

where projects the model state onto the observed component, i.e., , . To prevent numerical instability when is very small (or near zero), we regularize the variance estimate by setting

(11)

where in our implementation we set , small relative to the typical magnitude of . We found that results are insensitive to this choice provided . Appendix A in S1 Text provides a detailed justification for this specification. This specification allows the EnKF update to capture key features of the observation distribution (3), though it can overestimate uncertainty when case counts are extremely low. Such low-incidence scenarios are not the primary focus of our study, and alternative sequential methods, such as the Lifebelt particle filter [29], may be preferred in those cases. A similar treatment of adaptive variance specification is discussed in (Section 3.2, [30]).

During the analysis (update) step, each forecast ensemble member is adjusted using the Kalman gain and the innovation. In the stochastic (perturbed-observation) EnKF, the update is

(12)

where the Kalman gain is given by

(13)

Using the Gaussian approximation of the forecast density in (9), the one-step predictive likelihood can be expressed as

(14)

Only the key components of the EnKF algorithm for state inference are presented above; for a comprehensive overview, see [22]. For linear–Gaussian SSMs, the likelihood approximation in (14) converges to the true likelihood as , but for finite ensembles it remains biased due to sampling variability in the forecast mean and covariance [23]. Importantly, even if the forecast ensemble is exactly Gaussian, the estimated density is not an unbiased estimate of the true likelihood. This occurs because the empirical normal density , computed from the finite-ensemble mean and covariance , is a biased estimator of the true density for any finite N_x.

Nevertheless, empirical results in the literature [21,23,24] show that the EnKF likelihood performs remarkably well for inference in both linear and nonlinear systems, with posterior estimates only weakly dependent on the ensemble size [23]. To mitigate the finite-sample bias, we adopt the unbiased Gaussian density estimator of Ghurye and Olkin [31], later employed by Price et al. [32] and Drovandi et al. [23]. This estimator yields an unbiased estimate of provided N_x > d_y + 3, where d_y is the dimension of y. It is defined as

(15)

where , and if A is positive definite, and 0 otherwise. Here, |A| denotes the determinant of the matrix A and the constants c(d, v) are defined as

(16)

We can now replace the standard EnKF likelihood term in (14) with its unbiased counterpart:

(17)

The approximation of the marginal likelihood up to the time t is then given by:

(18)

Unlike in the case of the particle filter [8], the resulting posterior approximation , still does not target the true posterior (due to the Gaussian assumption). However, when the Gaussian assumption holds, the resulting posterior no longer depends on N_x, eliminating the finite-ensemble bias and yielding a stable target for Bayesian inference [23,31]. The EnKF procedure, including unbiased likelihood estimation, is outlined in Algorithm 1. This estimator can then be incorporated into a Metropolis–Hastings algorithm to target the sequence of posterior distributions .

Algorithm 1 Ensemble Kalman filter

Operations involving index i must be performed for .

Inputs: Observation y_1:T, Number of ensemble members N_x, Parameter vector .

Output: Filtering ensembles and likelihood estimate

1: Sample initial particles:

2: for t = 1 to T do

3:  Sample forecast ensemble as:

4:  Compute the observation variance, ensemble forecast mean, and variance: , , and .

5:  Compute the incremental Likelihood:

6: Shift each ensemble member as:

7: end for

8: Compute overall likelihood:

2.3. Ensemble SMC²

The EnKF algorithm described previously assumes that the model parameters are known. In practice, these parameters are rarely known, and it is preferable to assign a distribution that reflects prior knowledge. To account for this uncertainty, inference on can be performed by embedding the EnKF within an SMC sampler over the parameter space [33]. This approach follows the general principle of SMC² [9], which traditionally employs a particle filter to estimate the likelihood. At each observation time t, the posterior distribution of the parameters given data y_1:t satisfies the recursive relation

(19)

A practical way to approximate this posterior is to represent it with a finite set of weighted parameter particles. Let’s assume that is particle approximation from the previous posterior . The updated posterior can then be expressed as:

(20)

where , for are the normalized weights given by

(21)

As the incremental likelihood is intractable in a nonlinear system, at time t, the weight of the -particle is updated using an estimator obtained from the associated EnKF. This differs from the related work of Wu et al. [34], where the EnKF is used to construct proposal kernels in the SMC sampler but not as a likelihood estimator. As in standard SMC², particle degeneracy may occur over time. To mitigate this, resampling (described in Appendix B in S1 Text) is triggered whenever the effective sample size (ESS) drops below [9,35]. This is followed by a jittering mechanism that rejuvenates the particles by proposing new candidates, which are then accepted or rejected through a few iterations of Particle Marginal Metropolis–Hastings (PMMH). Since the weighted particles already approximate (under the Gaussian assumption), applying a Metropolis--Hastings update to each one can only enhance this approximation (see Algorithm 2).

Algorithm 2 PMMH mutation

Inputs: Observations y_1:t, Current parameter , current likelihood estimate: , Number of ensemble members N_x, Number of PMMH moves R

Output: New parameter , New likelihood:

1: for r = 1 to R do

2:  Propose

3:  Run a new EnKF (Algorithm 1) with and compute

4:  Calculate acceptance ratio:

5:  Accept with probability , otherwise retain

6: end for

In the numerical experiments presented in Section 3, we use an independent proposal distribution, . The proposal is constructed directly on the parameter space using a multivariate normal distribution. Specifically, new parameter particles are drawn from

(22)

where and denote the empirical mean and covariance of the current particle population , and c is a scaling constant that controls the covariance matrix.

The eSMC² algorithm can be viewed as a variant of the standard SMC², where the particle filter–based likelihood is replaced by an EnKF approximation in (18). From a practical perspective, this modification is expected to have only a slight effect on the posterior estimates of and x_1:t. The complete procedure is detailed in Algorithm 3, with a schematic overview in Fig 2.

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Fig 2. Flowchart of the eSMC² algorithm.

Each parameter particle carries an ensemble of state particles, which are propagated using the EnKF. Weights are updated based on an EnKF-based likelihood approximation. When the ESS falls below a threshold, parameter particles are resampled and rejuvenated via a PMMH step. This procedure is repeated for every data point, allowing sequential Bayesian learning.

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To propagate parameter uncertainty in the hidden state estimation, we approximate the marginal posterior distribution of the latent states at time t, , by integrating over a subset of parameter particles, , drawn from Algorithm 3 output [36]. In this work, we set N_c = 100, which in our numerical examples (Section 3) provides satisfactory coverage of the posterior. For each sampled , an EnKF with N_x ensemble members is run to approximate the filtering distribution , and the resulting trajectories are combined across all parameter samples to form an empirical approximation of the state posterior:

(23)

resulting in a total of trajectories that approximate the filtering distribution.

Algorithm 3 Ensemble Sequential Monte Carlo Squared (eSMC²)

Operations involving index m must be performed for .

Inputs: Observations y_1:T, prior , Number of ensemble members N_x, Number of parameter particles: , Number of PMMH moves R

Output: Parameter particles:

1: Initialize , weights , for

2: for t = 1 to T do

3:  Perform iteration t of EnKF (Algorithm 1) to estimate

4:  Update weights:

5:  Normalize weights: and compute ESS

6:  if ESS/2 then

7:   Resample parameters with replacement according to

8:  Reset weights

9:   Perform PMMH move (Algorithm 2) for each

10:  end if

11: end for

It is important to note a key theoretical distinction between SMC² and eSMC². Standard SMC² is “exact”: as the number of state and parameter particles , it converges to the true posterior . In contrast, eSMC² is generally biased because the EnKF approximates the incremental likelihood via a Gaussian assumption. Nevertheless, the EnKF likelihood typically has low variance [22], and unbiased corrections are possible under mild conditions [23]. Empirical studies show that EnKF-based likelihoods yield accurate and stable inference for both linear and moderately nonlinear SSMs [23,24]. Thus, while not formally exact, eSMC² provides practically accurate posterior estimates at much lower computational cost.

3.1. Inference on a simulated epidemic

To assess the performance of the proposed eSMC² algorithm, we first conducted three simulation studies using the SEIR model from Section 2. All algorithms were implemented in Python and executed on a desktop computer equipped with an Intel Core i7-1300H processor (3.40 GHz). Unless otherwise stated, simulations assumed a population size of N = 500,000 with initial conditions I₀ = 10, S₀ = N − I₀, and E₀ = R₀ = 0. Daily incidence y_t was generated according to the Poisson observation model in Eq (3), assuming perfect case reporting ().

The three examples differ primarily in the temporal structure of the transmission rate and the number of observation points. Example 1 depicts a moderately time-varying epidemic with an initial surge followed by a mild resurgence, capturing short-term fluctuations in transmission. Example 2, by contrast, exhibits a smoother and more prolonged pattern that sustains infections for longer and decays more gradually. The latter setting was designed to emulate the dynamic observed in the 2022 U.S. mpox epidemic, where behavioral changes and intermittent control measures led to prolonged and uneven declines in case counts. Example 3 is motivated by epidemics such as COVID-19 or seasonal influenza that exhibit secondary infection waves, arising from imperfect immunity or the activation and deactivation of control policies. To capture this behavior, we employ an oscillatory transmission rate that generates recurrent epidemic waves of declining amplitude, providing a more challenging multi-wave setting in which to evaluate algorithmic performance.

Example 1. Synthetic data were generated over T = 60 time points with time-varying transmission rate . The incubation and infectious periods were set to and , respectively. Each state ensemble was initialized using truncated normal distributions: , and E₀ = R₀ = 0. The priors were specified as

Example 2. Synthetic data were generated over T = 100 time points with a time-varying transmission rate , and fixed parameters and . The priors were specified as Initial state ensemble members were identical to Example 1.

Example 3. Synthetic data were generated over T = 100 time points with an oscillatory, multi-wave transmission rate , and fixed parameters and . The priors were specified as A more informative prior was placed on relative to Examples 1 and 2, as the oscillatory structure of introduces identifiability challenges that make the recovery rate harder to pin down from incidence data alone. Initial state ensemble members were identical to Examples 1 and 2.

For each simulation study, we compared the proposed eSMC² algorithm with the standard SMC² implementation that employs a bootstrap particle filter (BPF) in its inner layer (see Appendix A in S1 Text). In all cases, we used parameter particles and N_x = 200 ensemble members (or particles). For simplicity and comparability, the value of N_x was kept fixed throughout the timeline. Parameter rejuvenation was triggered when the effective sample size of the parameter particles dropped below 50% of , followed by five iterations of the PMMH kernel.

Filtered estimates in Figs 3–5 correspond to a single representative run of each method. The results illustrate that both eSMC² (blue) and SMC² (orange) accurately capture the epidemic dynamics in Examples 1, 2, and 3. For both methods, the predicted incidence closely follows the observed data, with 95% credible intervals consistently encompassing the actual case counts, indicating reliable uncertainty quantification. Estimates of the time-varying transmission rate and the effective reproduction number R_eff(t) are broadly consistent across methods and closely align with the ground truth, demonstrating that both approaches yield comparable assessments of the evolving transmission dynamics. The wider credible intervals for R_eff(t) relative to the filtered incidence reflect two complementary sources of uncertainty: the roughness of the random walk prior on , and parameter non-identifiability, particularly during the early epidemic phase when different combinations of , , and can yield indistinguishable incidence curves. This wider uncertainty is therefore an accurate signal that incidence data alone cannot sharply identify instantaneous transmission intensity. Smoother priors on , such as integrated Brownian motion or higher-order random walks, could potentially mitigate this variability but were not considered here and remain an avenue for future work. The filtering estimates of the unobserved latent state also closely track the true latent trajectories, with their credible intervals providing appropriate coverage (see Appendix C in S1 Text).

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Fig 3. Example 1: Filtered estimates of simulated incidence, transmission rate, and effective reproduction number.

Solid orange lines (SMC²) and dashed blue lines (eSMC²) show posterior means; shaded areas indicate 95% credible intervals. Observed incidence is shown as black dots, and the black dashed line represents the ground-truth latent quantity.

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Fig 4. Example 2: Filtered estimates of simulated incidence, transmission rate, and effective reproduction number.

Solid orange lines (SMC²) and dashed blue lines (eSMC²) show posterior means; shaded areas indicate 95% credible intervals. Observed incidence is shown as black dots, and the black dashed line represents the ground-truth latent quantity.

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Fig 5. Example 3: Filtered estimates of simulated incidence, transmission rate, and effective reproduction number.

Solid orange lines (SMC²) and dashed blue lines (eSMC²) show posterior means; shaded areas indicate 95% credible intervals. Observed incidence is shown as black dots, and the black dashed line represents the ground-truth latent quantity.

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Figs 6–8 illustrate the temporal evolution of the posterior means of the inferred parameters, along with their marginal posterior densities at the final time step, each summarizing five independent runs. Across runs, both SMC² and eSMC² accurately recover the ground truth. For the latency and recovery rates ( and ), the posterior means obtained from both algorithms are closely aligned, with credible intervals narrowing steadily as more data are assimilated, indicating progressive information gain. The only notable exception is the diffusion parameter in Examples 2 and 3, where a slightly larger discrepancy is observed. This difference likely arises from the stochastic nature of the log-transmission process combined with the approximation introduced by the EnKF-based likelihood in eSMC². Importantly, these differences do not appear to introduce substantial systematic bias or distort the inferred trajectory. The joint posterior contour plots (see Appendix C in S1 Text) further reveal strong agreement in posterior geometry and dependence structure: the contours for SMC² and eSMC² largely overlap, are centered near the true values, and display similar correlation patterns among parameters. The only visible difference appears along the dimension. Additional sensitivity analyses (Appendix D in S1 Text) assessing the effect of the number of state particles (N_x) and the reporting fraction () similarly demonstrate that posterior inferences are robust to ensemble size and exhibit only moderate variation when is correctly specified. The analyses also investigate the impact of non-informative (flat) priors, demonstrating that posterior estimates remain stable under weakly informative prior specifications.

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Fig 6. Example 1: Posterior distributions of , , and for five independent runs.

Top row shows filtered means with 95% credible intervals. Bottom row shows marginal posterior distributions at T = 60, with prior distributions overlaid (green dashed lines). Black dashed lines indicate true parameter values.

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Fig 7. Example 2: Posterior distributions of , , and for five independent runs.

The top row shows filtered means with 95% credible intervals. Bottom row shows marginal posterior distributions at T = 100, with prior distributions overlaid (green dashed lines). Black dashed lines indicate true parameter values.

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Fig 8. Example 3: Posterior distributions of , , and for five independent runs.

The top row shows filtered means with 95% credible intervals. Bottom row shows marginal posterior distributions at T = 100, with prior distributions overlaid (green dashed lines). Black dashed lines indicate true parameter values.

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Results in Table 1 summarize the accuracy and computational cost of each method, averaged over five independent runs. The mean absolute error (MAE) and root mean squared error (RMSE) quantify discrepancies between the filtering means and the ground truth for both selected latent state () and static parameters (). Across both examples, eSMC² achieves error magnitudes closely matching those of SMC², indicating that the ensemble approximation preserves inferential accuracy. In terms of computational effort, eSMC² offers a substantial advantage. In Example 1, the average CPU time is approximately 467 seconds for eSMC², compared with 2277 seconds for SMC², corresponding to a reduction by a factor of about 4.87. Similarly, in Example 2, eSMC² reduces average runtime to 639 seconds versus 3929 seconds for SMC², a factor of roughly 6.1. In Example 3, eSMC² completes in approximately 543 seconds on average, compared with 3766 seconds for SMC², corresponding to a speedup factor of roughly 6.9. The variability in the CPU across runs in both methods reflects the stochastic occurrence of rejuvenation steps, which introduce random fluctuations in computing time.

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Table 1. Comparison of goodness-of-fit and parameter estimates. Each method is evaluated based on CPU time in seconds (with standard deviation) and MAE (RMSE in parentheses) for daily incidence, effective reproduction number, transmission rate and model static parameters. Results are shown for Examples 1, 2, and 3.

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As an additional benchmark, we evaluated eSMC² against the Liu and West filter [37] (using 10⁴ particles, runtime ≈9 minutes) and a long PMCMC run (10⁴ iterations, runtime ≈3 hours) as a gold-standard reference, both serving as alternative approaches for joint sequential state-parameter estimation. Results provided in Appendix E in S1 Text demonstrate that eSMC² achieves superior parameter accuracy and produces narrower, more stable credible intervals compared to the Liu and West filter, while remaining in close agreement with the PMCMC posterior distributions evaluated at the last time point.

3.2. Application to the 2022 monkeypox outbreak in the USA

We applied the proposed methodology to infer the transmission dynamics of the 2022 U.S. monkeypox (mpox) outbreak, caused by the West African clade II of the monkeypox virus. This outbreak represented an unprecedented global event, with the United States reporting among the highest case counts worldwide. The typical incubation period ranges from days, and infectiousness generally lasts weeks. The dataset consists of daily confirmed mpox case counts reported by the U.S. Centers for Disease Control and Prevention (CDC) from 6 May to 31 December 2022. Recent studies have applied Bayesian filtering to mpox data. For instance, Saldaña et al. [38] estimated the instantaneous reproduction number and growth rates using sequential Bayesian updating, while Papageorgiou and Kolias [7] proposed a particle filter incorporating penalty factors specific to mpox transmission. A related stochastic Kalman-based approach, validated on the 2022 Czech Republic mpox data, further discusses the challenges of Bayesian filtering for epidemic modeling with dynamic parameters [39]. Our framework offers a complementary perspective by enabling scalable joint inference of latent epidemic states and model parameters, allowing uncertainty quantification directly from noisy daily surveillance data.

Given the pronounced day-to-day variability in the reported case counts, we adopted a Negative Binomial observation model as the primary specification. In contrast, preliminary analyses under a Poisson observation model led to unstable posterior trajectories and parameter degeneracy, indicating a poor fit to the observed overdispersion. The Poisson assumption becomes more appropriate when incidence is aggregated over 7-day intervals, which reduces temporal variability. Supporting results for the aggregated analysis are provided in Appendix F in S1 Text.

The algorithm was implemented with N_x = 200 ensemble members and parameter particles, providing a balance between stability and computational efficiency. All primary results presented below are obtained using eSMC², which constitutes the main inference framework of this study.

Fig 9 shows the estimated daily incidence and R_eff(t) over time. The filtered trajectories closely follow the observed case counts, while the credible bands reflect uncertainty arising from both stochastic epidemic dynamics and observation noise. The estimated indicates sustained transmission above one during the early phase of the outbreak, followed by a decline after the declaration of the state of emergency, and eventual stabilization below one as the outbreak subsides. For benchmarking purposes, we additionally report results obtained using the standard SMC² algorithm under identical model and prior specifications. In Fig 9, SMC² results are shown using 95% credible interval bounds (solid orange lines) together with the posterior median trajectory (dotted orange line), enabling a direct comparison of both central estimates and uncertainty quantification between the two methods.

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Fig 9. Inference of daily incidence and effective reproduction number for the 2022 U.S. monkeypox outbreak.

Left: filtered estimates of daily incidence. The solid blue line denotes the posterior median obtained using eSMC², with reported case counts shown as red dots. Right: inferred effective reproduction number . For eSMC², shaded regions represent 50%, 75%, 90%, and 95% credible intervals. The vertical dashed line indicates the declaration of the national state of emergency. Results from SMC² are shown in orange, with solid lines indicating the 95% credible interval bounds and the dotted line representing the posterior median.

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Fig 10 presents the posterior trajectories of key epidemiological parameters, with corresponding priors and posterior summaries reported in Table 2. As more data are assimilated, these parameters become increasingly well-identified, as reflected by the progressive narrowing of the posterior intervals. The analysis indicates a mean incubation period of approximately 5.3 days (95% CrI: 4.6–6.2 days), consistent with early empirical estimates reported by Madewell et al. [40], and a mean infectious period of approximately 18.2 days (95% CrI: 15.2–22.2 days). The overdispersion parameter exhibits posterior mass distinctly separated from zero, confirming substantial extra-Poisson variability. For comparison, we also implemented the standard SMC² algorithm under identical model and prior settings. Results reported in Appendix F in S1 Text show that both methods produce a very similar evolution of the filtering distributions for the parameters, indicating that the ensemble-based approximation in eSMC² accurately captures posterior uncertainty. Notably, a single run with these settings required approximately 16 minutes of CPU time, compared with nearly 2.5 hours for the equivalent SMC² implementation, demonstrating the efficiency gains achieved by the EnKF-based approach.

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Table 2. Description of the different parameters, priors, and posterior estimates at the final time step. Upper and/or lower bounds have been imposed by the observations. denotes a truncated normal distribution .

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Fig 10. Posterior distributions of key epidemiological parameters.

The left column shows filtered means with credible intervals; the right column displays marginal posterior distributions at the last time step.

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Fig 11 presents 14-day-ahead posterior-predictive forecasts initiated at multiple time points during the outbreak, assuming the transmission rate remains fixed at its last estimated value. Forecasts are obtained by propagating the latent state forward through the state-space dynamics and sampling synthetic observations from the observation model. The purpose of this analysis is to evaluate the short-term predictive performance of the proposed eSMC² framework. We compare the results with an autoregressive (AR) model (Appendix F in S1 Text). The top panel of Fig 11 shows forecasts from eSMC² and the bottom panel shows those from the AR model. Quantitative performance is summarised in Table 3, which reports mean absolute error (MAE), weighted interval score (WIS; based on the 50%, 75%, 90%, and 95% intervals), and empirical 95% prediction interval (PI) coverage for each forecast start date. Both models struggle to anticipate the turning point in early August, and around this period the autoregressive model achieves a lower forecast error. In particular, around the epidemic peak, the AR model consistently outperforms eSMC² across all evaluation metrics. This likely reflects the intrinsic difficulty of forecasting near the peak, where rapid changes in transmission dynamics and the assumption of a fixed limit the responsiveness of the mechanistic model. In contrast, the AR model, being purely data-driven, can more readily adapt to short-term trends during this highly dynamic phase. Similar behavior has been reported in related work, where data-driven Hawkes-type processes were found to outperform stochastic SEIR models during abrupt regime shifts [41]. However, from mid-September onward, eSMC² achieves lower MAE and WIS in most forecast windows and maintains coverage close to the nominal 95% level, whereas the autoregressive model exhibits declining coverage in later periods (Table 3). These results highlight a limitation of the current modeling setup in capturing abrupt turning points, mainly due to the inertia of the mechanistic dynamics, rather than an inherent limitation of SMC-based inference. At the same time, they confirm the strength of the approach in providing well-calibrated probabilistic forecasts once the epidemic enters a more stable phase. Long-term forecasts for the final epidemic phase are presented in Fig. F4 in S1 Text. The model reproduces the overall decline and achieves full 95% predicted interval coverage.

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Table 3. Evaluation of the 2 weeks-ahead forecasts of reported mpox cases during the 2022 U.S. outbreak. Bold indicates better-performing model.

https://doi.org/10.1371/journal.pcbi.1014301.t003

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Fig 11. Posterior-predictive forecasts at different starting dates.

The top panel shows eSMC² forecasts, and the bottom panel shows AR forecasts. Median forecasts are displayed in purple (eSMC²) and green (AR), with corresponding 50%, 75%, 90%, and 95% predictive intervals. Observed case counts are shown in red. Vertical orange dashed lines indicate the forecast start dates.

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